Research ArticleOn the Asymptotic Hyperstability of LinearTime-Invariant Continuous-Time Systems under a Class ofControllers Satisfying Discrete-Time Popovrsquos Inequality
M De la Sen
Institute of Research and Development of Processes IIDP Facultad de Ciencia y Tecnologia University of the Basque CountryLeioa (Bizkaia) PO Box 644 de Bilbao 48080-Bilbao Spain
Correspondence should be addressed to M De la Sen manueldelasenehueus
Received 8 January 2018 Accepted 26 April 2018 Published 4 June 2018
Academic Editor Jean Jacques Loiseau
Copyright copy 2018 M De la SenThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper is concerned with the property of asymptotic hyperstability of a continuous-time linear system under a classof continuous-time nonlinear and perhaps time-varying feedback controllers belonging to a certain class with two maincharacteristics namely (a) it satisfies discrete-type Popovrsquos inequality at sampling instants and (b) the control law within theintersample period is generated based on its value at sampling instants beingmodulated by twodesignweighting auxiliary functionsThe closed-loop continuous-time system is proved to be asymptotically hyperstable under some explicit conditions on suchweighting functions provided that the discrete feed-forward transfer function is strictly positive real
1 Introduction
Continuous-time and discrete-time positive systems havebeen studied in detail in recent years [1ndash10] If the stateand output possess the positivity property under nonnegativeinitial conditions and controls the positivity is said to beinternal or simply the system is positive If the outputpossesses such a property the system is said to be externallypositive Thus positive systems are intrinsically interestingto describe some problems like Markov chains queuingproblems certain distillation columns and biological andother physical compartmental problems where populationsor concentrations cannot be negative [2 3] It is wellknown that time-invariant dynamic linear systems whichare externally positive while having positive real or strictlypositive real transfer matrices are in addition hyperstable orasymptotically hyperstable ie globally Lyapunov stable forany nonlinear andor time-varying feedback device satisfyingPopovrsquos type inequality for all time [11 12] The converseassertion is not generically true in the sense that a hyper-stable linear system then characterized by a positive realtransfer function is not necessarily an externally positive
one In particular hyperstable linear systems can have apositive instantaneous input-output power and a positiveinput-output energy measure for all time Thus they haveidentical signs of the input and output for all time instantswhile they are not externally positive if those signals arenot everywhere nonnegative Thus external positivity of aSISO system is not implied by the positive realness of itstransfer function The property of asymptotic hyperstabilitygeneralizes that of absolute stability [13ndash15] which generalizesthe most basic concept of stability of dynamic systems Seefor instance [2 3 11 13ndash26] and references therein It iswell known that closed-loop hyperstability is by nature apowerful version of closed-loop stability since it refers to thestability of a hyperstable linear feed-forward plant (in thesense of the positive realness of the associated transfermatrix)under a wide class of feedback controllers applied Theabove important properties make very attractive potentialresearch issues for kind of more complex dynamic systemswith applied projection including those lying in the classof continuousdigital hybrid systems On the other handthe class of hybrid systems consisting of continuous-timeand discrete-time (or digital) systems are of an increasing
HindawiMathematical Problems in EngineeringVolume 2018 Article ID 7861396 17 pageshttpsdoiorg10115520187861396
2 Mathematical Problems in Engineering
interest since many existing industrial installations combineboth kinds of systems An elementary well-known case iswhen a discrete-time controller is used for a continuous-time plant Another case is related to teleoperation systemswhere certain variables evolve in a discrete-time or digitalfashion A background literature and related relevant resultsare given in [1 7 11 17 18 27ndash30] and some of the referencestherein
Some conditions have been given in the backgroundliterature to guarantee the positive realness of discrete trans-fer functions from related conditions on their continuous-time counterparts which are then maintained under dis-cretization See for instance [31] The results are useful toguarantee the hyperstability of discrete-time systems underthe hyperstability of the continuous-time ones when the classof feedback discrete-time controllers satisfies Popovrsquos typeinequality However it turns out that positive realness ofa continuous-time system does not guarantee the positiverealness of its discretized transfer function by a zero-orderhold for any sampling period and conversely the strongpositive realness of a discretized transfer function does notguarantee in some limited cases the strong positive realnessof its continuous-time counterpart [31ndash33] This paper isdevoted to a certain kind of inverse problem stated inthe following terms Given is an asymptotically hyperstablediscrete-time system of sampling period 119879 such that its feed-forward transfer function is strictly positive real and thenonlinear time-varying feedback controller belongs to a classΦ119889(119879) satisfying certain discrete Popovrsquos type inequalityThe elementary question that arises is how the asymptotichyperstability of the continuous-time system is guaranteedfor a certain modified class of controllers such that itsmember at sampling instants is in the class Φ119889(119879) Toanswer this question an intersample controller based onthe class Φ119889(119879) at sampling instants is designed whichcorrects the control law along the intersample period viathe concourse of two modulating continuous-time functionswhich have to satisfy certain reasonable constraints Thepaper is organized as follows Firstly a notation and ter-minology subsection is allocated in the subsequent sectionSection 3 presents the linear and time-invariant system ofthe feed-forward part of the closed-loop system togetherwith its analytic output expressions some auxiliary resultsof positivity and boundedness for all the sampling instantsof an input-output energy measure The continuous-timecontrol input is proposed to be generated within eachintersample period from the last sampling instant definingthe starting point of the current intersample period and somedesign auxiliary functions which modulate the control signalalong such an intersample period Some of the formulaspresented are derived in detail in the appendix On theother hand some auxiliary preparatory lemmas concerningthe input-output energy measure are also formulated whichare of usefulness for the main result Section 4 is devotedto obtain the main asymptotic hyperstability result whichis proved based on some given and previously provedauxiliary preparatory lemmas Basically it is proved thatthe continuous-time closed-loop system is asymptoticallyhyperstable provided that (a) the discretized feed-forward
transfer function is strictly positive real (b) the discretizedsystem at sampling instants is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsand (c) some extra additional conditions on smallness andboundedness are fulfilled by the abovementioned auxiliarymodulated functions which define the controls along theintersample periods Such conditions make any continuous-time controller belonging to the appropriate class to satisfycontinuous-type Popovrsquos inequality for a class of controllersprovided that the former one at sampling instants is fulfilledby the discretized controllers Some examples are discussedin Section 5 Finally conclusions end the paper
2 Notation
(i) R+ = R0+ cup 0 Z+ = Z0+ cup 0 R0+ = 119903 isin R 119903 ge 0 andZ0+ = 119911 isin Z 119911 ge 0
(ii) clR = [minusinfin +infin] is the closure of the real fieldthat is the set of real numbers together with the plusmn infinitypoints
(iii)The continuous and discrete-time arguments 119905 isin R0+and 119896 isin Z0+ are denoted with parenthesis and brackets thatis (119905) and [119896] respectively
(iv) If 119879 gt 0 is the sampling period then 119906(119905) = 119906[119896] +(119905) forall119905 isin [119896119879 (119896 + 1)119879) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+ isthe continuous-time input where the piecewise-continuousintersample incremental input is (119905) = 119906(119905) minus 119906[119896] with119906[119896] = 119906(119896119879) and [119896] = (119896119879) = 0 forall119905 isin [119896119879 (119896 +1)119879) forall119896 isin Z0+ The auxiliary purely discrete input is 119906119889[119896]such that 119906119889[119896] = 119906[119896] forall119896 isin Z0+ if (119905) = 0 forall119905 isinR0+
(v) 119910ℎ(119905) forall119905 isin R0+ and 119910ℎ[119896] = 119910ℎ(119896119879) forall119896 isin Z0+ arethe forced andhomogeneous (unforced) output of the systemrespectively
(vi) 119910119889ℎ[119896] = 119910119889ℎ(119896119879) and 119910119889119891[119896] = 119910119889119891(119896119879) forall119896 isin Z0+are the forced and homogeneous output of the system respec-tively when the input is piecewise-constant with constantvalue in-between each two consecutive sampling instants iewhen the input is generated from a zero- order-hold (ZOH)device
(vii) The Fourier transform of any real vector function119891(119905) is denoted by119891(i120596) if it exists forall119905 120596 isin R where i = radicminus1is the imaginary complex unit
(viii) For the purpose of formally using Fouriertransforms for the subsequent developments we definecontinuous-time and discrete-time truncated signalsfrom their untruncated counterparts on the respectivecontinuous-time and discrete-time intervals [0 119905] and[0 119896119879] as follows
V119905 (120591) =
V (120591) if 0 le 120591 le 1199050 if 120591 gt 1199050 if 120591 lt 0
forall119905 120591 isin R0+
Mathematical Problems in Engineering 3
V[119896] [119895] =
V [119895] if 0 le 119895 le 1198960 if 119895 gt 1198960 if 119895 lt 0
forall119895 119896 isin Z0+(1)
(ix) Let 119904 and 119911 = 119890minus119879119904 the Laplace and 119911-transforms(for sampling period 119879) complex arguments The set SPR isthe set of strictly positive real continuous 119892(119904) (respectivelydiscrete) transfer functions 119892(119911) ie they are (strictly) stableie with poles in Re 119904 lt 0 (respectively |119911| lt 1) whichsatisfy Re119892(119904) gt 0 for all Re 119904 ge 0 (respectively Re119892(119911) gt 0for all |119911| ge 1) These conditions imply that Re119892(i120596) gt0 respectively Re119892(119890i120596) gt 0 forall120596 isin R [13 31 34] Thestrictly positive continuous and discrete real sets are notdistinguished at the level of notation since they are easy toidentify them according to context
(x) The so-called set SSPR is the set of (continuous ordiscrete depending on context) strongly positive real transferfunctions which are those in SPR whose real part is strictlypositive also as |119904| rarr infin It can be pointed out that Szego-Kalman-Popov Lemma (also so-called Discrete Positive RealLemma) is a discrete version of the Yakubovich-Kalman-Popov Lemma (also so-called Positive Real Lemma) whichrelates positive realness of transfer functions to associatedstate-space realization properties states that discrete transferfunction in SPR are of relative degree (namely pole-zeroexcess) equal to zero so that they are also in SSPR and bothsets are equivalent This equivalence concern between thediscrete sets SPR and SSPR does not apply to continuoustransfer functions [13 31]
3 The Continuous-Time and Discrete-TimeLinear Time-Invariant Systems and SomeAuxiliary Results
Consider an 119899th-order linear and time-invariant dynamicsystem in state-space description
where 119909(119905) isin R119899 119906(119905) isin R and 119910(119905) isin R are the statepiecewise-continuous control input and output and 119860 isinR119899times119899 119887 isin R119899 119888 isin R119899 and 119889 isin R are the matrix ofdynamics control vector output vector and input-outputinterconnection gain The auxiliary purely discrete outputand the continuous-time and sampled output are calculatedfrom (2) via the impulse responses 119892(119905) and 119892119889[119896] whichbecome
119892 (119905 minus 120591) = 119888119879119890119860(119905minus120591)119887 + 119889120575 (119905 minus 120591) forall120591 119905 (ge 120591) isin R0+119892119889 [119896 minus 119895] = int119896119879
119895119879119892 (119896119879 minus 120591) 119889120591
= int(119896minus119895)1198790
119892 ((119896 minus 119895) 119879 minus 120591) 119889120591= int119896119879119895119879(119888119879119890119860(119896119879minus120591)119887 + 119889120575 (119896119879 minus 120591)) 119889120591
= int(119896minus119895)1198790
(119888119879119890119860((119896minus119895)119879minus120591)119887 + 119889120575 ((119896 minus 119895) 119879 minus 120591)) 119889120591forall119895 (le 119896) 119896 isin Z0+
(3)
and 119892(119905minus120591) = 0 forall120591 119905(lt 120591) isin R0+ where 120575(119905minus120591) is the Diracdistribution 119906(minus119905) = 0 forall119905 isin R+and 119892119889[119896 minus 119895] = 0 forall119895(gt119896) 119896 isin Z0+ with 119892119889[0] = 119889 The initial input and outputconditions are 119906[119895] = 119910[119895] = 0 for any integer 119895 lt 0 and119910(119905) = 119906(minus119905) = (minus119905) = 0 forall119905 isin R+ in causal dynamicsystems subject to initial conditions119909(0) = 1199090 Let the controlinput be
119892119889 [119896 minus 119895])119906[119896] [119895]+ intinfinminusinfin119892 (119896119879 minus 120591) 119896119879 (120591) 119889120591 + 119910ℎ [119896]
forall119896 isin Z0+
(7)
Expressions (5)ndash(7) are derived in the appendix
4 Mathematical Problems in Engineering
Remark 1 Note that the difference between the sampledcontinuous-time sequence 119910[119896] and the purely discreteoutput sequence 119910119889[119896] is that the first one is the exactvalue of the output at sampling instants including the effectsof the intersample input ripple while the second one is thediscretized output at sampling instants in the presence of azero-order hold As a result note that 119910[119896] = 119910119889[119896] for anygiven 119896 isin Z+ if and only if int119896119879
0119892(119896119879 minus 120591)(120591)119889120591 = 0 In
particular 119910[119896] = 119910119889[119896] if (119905) = 0 forall119905 isin R0+ Causalityimplies that (6)-(7) can be rewritten as
119910 (119905) = (infinsum119895=0
119892119889 [119896 minus 119895])119906[119896] [119895]+ (int119905119896119879119892 (119905 minus 120591) 119889120591) 119906 [119896]
The following result is directLemma 2 Any given piecewise-continuous control input canbe decomposed into purely discrete-time control plus incremen-tal intersample period ones as follows
through piecewise-continuous functions 120582 120590 cup119896isinZ0+[119896119879 (119896 +1)119879) rarr R defined by
120582 (119905) = (119905)119906 [119896] if 119906 [119896] = 00 if 119906 [119896] = 0
120590 (119905) = 0 if 119906 [119896] = 0 (119905) if 119906 [119896] = 0
(10)
The purely auxiliary input-output and true ones energymeasures on the discrete-time interval [0 119896119879] as well as thecontinuous-time ones on [0 119905] forall119905 isin [119896119879 (119896 + 1)119879) and forall119896 isinZ0+ are respectively from (5)ndash(7) and (9)-(10) given by
Remark 3 Since the control input is piecewise-continuousthe truncated input 119906119905(120591) for 120591 isin [0 119905] has Fourier transformsfor any 119905 isin R0+ so that the corresponding truncated output
119910119905(120591) for 120591 isin [0 119905] and also the energy measure have alsoFourier transforms for any 119905 isin R0+ This follows from the factthat these signals are calculated via truncated functionswhich
Mathematical Problems in Engineering 5
are then zero at infinity so that they are square-integrable andtheir sequences of sampled values of sampling period 119879 aresquare-summable as a result
The subsequent two simple auxiliary lemmas will be thenused in the next section to establish the main result
Lemma 4 Assume that there exist constants 1205741198890 (120574119889 gt 1205741198890) isinR+ such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some1198960 isin Z0+ Then there exist 1205740 (120574 gt 1205740) isin R+ such that 1205740 le119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin) if1205740 minus 120574119889 le 119864119889 (119896119879 119905)
= (119864 [119896] minus 119864119889 [119896]) + 119864 (119896119879 119896119879 + 120591)= 119864 (119905) minus 119864119889 [119896] = 119864119889 (119896119879 119905) le 120574 minus 1205741198890
119879 (119910 [119895] 119906 [119895] minus 119910119889 [119895] 119906119889 [119895])
(16)
forall119905 isin (119896119879 (119896 + 1)119879) forall119896 isin Z0+ so that (14) implies that 1205740 le119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin) provided that 1205741198890 le 119864119889[119896] le120574119889 forall119896(gt 1198960) isin Z+ for 1198960 isin Z0+Lemma 5 Assume that 120593(119909 119905) is subadditive for all 119909 isin Rie 120593(119909 + 119910 119905) le 120593(119909 119905) + 120593(119910 119905) forall119909 119910 isin R forall119905 isin R0+Then 120593(119910(119905) 119905) le 120593(119910[119896] 119896) + 120593(119910(119905) minus 119910[119896] 119905 119896119879) andas a result (119905) ge 120593(119910(119905) minus 119910[119896] 119905 119896119879) and 120593(119910[119896] 119896) geminus120593(minus119910[119896] 119896) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+
If 120593(119909 119905) is superadditive for all 119909 isin R ie 120593(119909 +119910 119905) ge 120593(119909 119905) + 120593(119910 119905) forall119909 119910 isin R forall119905 isin R0+ then120593(119910(119905) 119905) ge 120593(119910[119896] 119896119879) + 120593(119910(119905) minus 119910[119896] 119905 119896119879) as a result(119905) le 120593(119910(119905) minus 119910[119896] 119905 119896119879) and 120593(119910[119896]) le minus120593(minus119910[119896]) forall119905 isin[119896119879 (119896 + 1)119879) forall119896 isin Z0+Proof From the first identity in (9) if 120593(119909 119905) is subadditivethenminus120593 (119910 (119905)) = 119906 (119905) = 119906 [119896] + (119905)
ge minus120593 (119910 [119896] + (119910 (119905) minus 119910 [119896]))ge minus120593 (119910 [119896]) minus 120593 (119910 (119905) minus 119910 [119896])
= minus120593 (119910 [119896]) minus 120593 (119910 (119905) + (minus119910 [119896]))ge minus120593 (119910 [119896]) minus 120593 (119910 (119905)) minus 120593 (minus119910 [119896])
Definition 6 (Popovrsquos inequality) A continuous-time feed-back nonlinear and eventually time-varying continuous-timecontroller 119906(119905) = minus120593(119910(119905) 119905) and forall119905 isin R0+ where 120593 R0+ timesR0+ rarr R0+ is in the class Φ denoted as 120593 isin Φ is said tosatisfy Popovrsquos type integral inequality if for some 1205740 isin R+one has
Definition 7 (discrete-time Popovrsquos inequality) A discrete-time feedback nonlinear and eventually time-varyingdiscrete-time controller 119906(119905) = 119906[119896] = 119906(119896119879) =minus120593(119910(119896119879) 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) and forall119896 isin Z0+where 120593 R0+ times Z0+ rarr R0+ is in the class Φ119889(119879) ofsampling period 119879 isin R+ denoted as 120593 isin Φ119889(119879) is said tosatisfy a Popovrsquos type discrete-time inequality if for some120574119889 isin R+ one has119896sum119895=0
119910 (119896119879) 120593 (119896119879) ge minus120574119889119879 forall119896 isin Z0+ (22)
I D Landau refers to controllers satisfying a Popovrsquostype inequality (21) (or respectively (22)) as hyperstable
controllers of class Φ (or respectively of class Φ119889(119879))[13] bearing in mind that if any controller of such a class iscoupled to a linear time-invariant forward system of transferfunction being continuous-time (or respectively discrete-time) strictly positive real then the overall closed-loop systemis asymptotically hyperstable namely globally asymptoticallystable for any arbitrary controller belonging to the respectiveclass It can be pointed out that the above comments referredto discrete-time systems and in particular Definition 7 arealso applicable to digital systems in the same way as theyare applicable to discrete-time systems ie those which arefully described in the discrete domainwithout having specificlinks to a discretization process on a certain continuous-timesystem
It can be pointed out that Definition 7 can be alsoestablished for a digital system not being related to thetime discretization of a continuous-time system in terms ofPopovrsquos inequality of the following form
119896sum119895=0
119910 [119896] 120593 [119896] ge minus120574119889119886 forall119896 isin Z0+ (23)
for some constant 120574119889119886 isin R+ for any digital controller ofclass Φ119889 In this context the sampling period either can benonrelevantmdashthen nonmade notational explicit in the classΦ119889mdashor can even have nonsense since the inequality isapplied to a certain discrete sequence
The following result obtains useful expressions for the dis-cretized input-output energymeasure under some conditionsconcerning the subadditivity constraints on the controllers ofclass Φ119889(119879)The result is also a preparatory one for the nextLemma 9 which addresses the positivity and boundedness ofthe input-output energy measure at sampling instants
Lemma 8 The following properties are fulfilled(i) The subsequent formulas hold
(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119895 (le 119896) 119896 isin Z0+
(26)
(ii) Assume that 120593 isin Φ119889(119879) and that 120593(119909 119896119879) issubadditive for all 119909 isin R ie 120593(119909 + 119910 119896119879) le 120593(119909 119896119879) +120593(119910 119896119879) forall119909 119910 isin R forall119896 isin Z0+ Then the subsequentrelations are true if the sequence 120576[119896] is defined by 120576[119896] =120582[119896] + 120585[119896] forall119896 isin Z0+119910119889ℎ [119896] + intinfin
0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591
which yields (24) and leads to (25) Now define the sequence120576[119896] by 120576[119896] = 120582[119896] + 120585[119896] and 120585[119896][119895] = int(119895+1)119879119895119879(120582(119895119879 +120591)119892((119896 minus 119895)119879 minus 120591) minus 120582[119895]119892[(119896 minus 119895)119879])119889120591 forall119896 isin Z0+ and note
that
minus119910 [119896] = minus (119910119889 [119896] + (119910 [119896] minus 119910119889 [119896]))= [[infinsum119895=0
(int(119895+1)119879119895119879
(1 + 120576[119896] [119895]) 119892 [(119896 minus 119895) 119879])119889120591]]sdot 120593[119896] ([119910119889 [119895] + (119910 [119895] minus 119910119889 [119895]) 119895])minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 minus 119910ℎ [119896]
forall119896 isin Z0+
(31)
and then
minus (119910 [119896] minus 119910119889 [119896] minus 119910ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591)
The subsequent result establishes lower and upper-bounds for the discretized input-output energy measuresbased on the formulas obtained in Lemma 8 and strictpositive realness of the discrete-time transfer function ofthe feed-forward linear and time-invariant system Theresult invokes the boundedness and the smallness of theparameters which define the calculation of the intersam-ple control input from their preceding values at samplinginstants Note that the control deviation in the intersampleperiods related to the sampling time instants has to be suffi-ciently moderate enough so that the hyperstability propertyis kept from the discrete-time system to the continuous-time one In particular the positivity and boundedness ofthe energy measure at the sampling instants is guaranteedif sup119905isinR0+ |120590(119905)| is small enough and Φ119889(119879) is subaddi-tive
Mathematical Problems in Engineering 9
Lemma 9 Assume that 119892119889 isin 119878119875119877 and that 120593 isin Φ119889(119879)Then the following properties hold(i)The input-output energy systemmeasure of the auxiliary
discretized system satisfies
0 lt 1205740119889119886 le 1198792120587 min120596isinR0+
119879119910119889 [119896] 119906119889 [119896] le 120574119889119886forall119896 (gt 1198960) isin Z+
(34)
for some 1205740119889119886 120574119889119886(gt 1205740119889119886) isin R+ and for any nonidentically zerocontrols and zero initial conditions ie for any forced solution
(ii) Assume in addition that the class Φ119889(119879) consistsof subadditive functions Assume also that sup119905isinR0+ |120590(119905)| issufficiently small inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infinwhere 120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ and that furthermore
for some 120574119889119886 isin R+ from the first assumption of the theoremand 119906119889[119896] = minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ since Popovrsquosinequality (22) is fulfilled by any given discrete-time feedbackcontroller 120593 isin Φ119889(119879) generating a feedback control 119906119889[119896] =minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ On the other hand the initialconditions can be neglected since the assumption119892119889 isin SPRimplying that the discrete-time transfer function is (strictly)stable which makes their values irrelevant for purposes ofstability analysisThen the discrete frequency response being
applicable to piecewise-constant inputs in-between any twoconsecutive sampling instants is given by
since the subsequence 119892[minus119899] = 0 forall119899 isin Z+ and thetransfer frequency response of a sampling and zero-order-hold operator 119885 of period 119879 defined by 119885(119879 V)(119905) = V(119896119879) =V[119896] for any given V R0+ rarr R and all 119905 isin [119896119879 (119896 + 1)119879)119896 isin Z0+ is given by 119885(119879 i120596) = (1 minus 119890minusi119879120596)i120596 [31 35]It turns out that |119885(119879 i120596)| = |(1 minus 119890minusi119879120596)i120596| lt 1 forall120596( =0) isin R and |119885(119879 i0)| = 1 Thus the auxiliary discretizedsystem satisfies the subsequent equivalence relation in thefrequency domain to the discrete-time relation (11) It turnsout that Fourier transforms exist in the impulse responsesand truncated control and auxiliary functions of (11)ndash(13)since the truncated functions are square-integrable in clR =[minusinfin +infin] By using the discrete Parsevalrsquos theorem underzero initial conditions one has
The lower-bounding constraint of (37) follows by using(39) and the discrete Parsevalrsquos theorem for the equivalencebetween the input-output energy measures from the fre-quency domain to the discrete-time domain
119864119889 [119896] ge 1198792120587 min120596isinR0+
1199062119889 [119896]) ge 1205740119889forall119896 isin Z0+
(40)
from the assumption 119892119889 isin SPR iemin120596isinR0+ Re119892119889(119890minusi120596119879) ge 1205740119889119886 gt 0 forall1198960 119896 (119896 ge 1198960) isin Z+since
(a) strictly positive real discrete transfer functions havezero relative degree (ie an identical number of polesand zeros) from the Discrete Positive Real Lemma(Szego-Kalman-Popov Lemma) [13 31 34] so thatthe real parts of their frequency hodographs in theargument 119890minusi120596119879 are positively lower-bounded
10 Mathematical Problems in Engineering
(b) Im119892119889(119890minusi120596119879) = minus Im119892119889(119890i120596119879) forall120596 isin R0+ so that theirintegrals in the argument 120596 on frequency intervals[minus119896120587 119896120587] for any 119896 isin Z+ are null
Property (i) has been proved To prove Property (ii) notefrom (28) [Lemma 8(ii)] that
119892 [(119896 minus 119895) 119879])119889120591]]sdot (120593[ℓ] (119910119889 [119895] 119895) + 120593[ℓ] (minus119910119889 [119895] 119895))sdot (120593[ℓ] (119910119889 [ℓ] 119895) + 120593[ℓ] (minus119910119889 [ℓ] ℓ)) minus 119910119889ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 le 120574119889 = 119870120574119889119886
forall119896 isin Z+
(41)
for some 119870 isin R+ provided that sup119905isinR0+ |120590(119905)| is sufficientlysmall inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infin where120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ since 119892(119905) is bounded andconverges exponentially to zero since 119892119889 isin SPR and thenconvergent (that is strictly stable in the discrete context)119910119889ℎ[119896] rarr 0 and it is a bounded sequence
The next result which is a preparatory result to thenestablish the main asymptotic hyperstability result addressesthe relevant property that the input-output measure 119864(119905) ofthe continuous-time system is positively lower-bounded andfinitely upper-bounded for all time under the conditions ofLemmas 4 and 9
Lemma 10 If 120582 and 120590 are sufficiently small then
1205740 minus 120574119889 le 119864 (119905) minus 119864119889 [119896]le 1 + 12058222120587 intinfin
provided that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+Proof One gets the following from direct calculations byusing the continuous-time control laws (9)-(10) and (16) if120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)|
minus 12058222120587 intinfin
minusinfin119892 (i120596) 1003816100381610038161003816[119896] (i120596)10038161003816100381610038162 119889120596 minus 120590(2120582 sup
0le119895le119896
1199062[119895] + 120590)int1198961198790
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591 le 119864 (119905) minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119910 (i120596)
sdot (minusi120596) 119889120596 minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119892 (i120596) | (i120596)|2 119889120596 minus 119864119889 (119896119879 119905)
Now the use of (44) in Lemma 4 (14) implies thatLemma 9(ii) (36) holds implying the existence of 1205740 (120574 gt1205740) isin R+ such that 1205740 le 119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin)provided that 1205741198890 le 119864119889[119896] le 120574119889 for 1205741198890 (120574119889 gt 1205741198890) isin R+ since1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isin Z0+ if1205740 minus 120574119889 le minus 120582
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591le 120574 minus 1205741198890 forall119896 (gt 1198960) isin Z+ forall120591 isin (0 119879)
(45)
if 120582 and 120590 are sufficiently small such that (43) holds leadingto 1205740 gt 0 for any given positive 1205741198890 and 120574119889 gt 1205741198890
The main first result follows
Theorem 11 Consider the dynamic system (2) subject to acontrol
forall119896 isin Z0+ where 120593 [119896] isin Φ119889 (119879) (46)
Assume that(1) 120593(119910[119896] 119896119879) is in the class Φ119889(119879) (that is120593(119910[119896] 119896119879) isin Φ119889(119879)) defined by the discrete Popovrsquos
type sum inequality sum119896ℓ=0 119910119889[ℓ]119906[ℓ] ge minus120574119889119879 for some 120574119889 isin Rforall119896 isin Z0+(2) 119892119889 isin 119878119875119877 there exist constants 1205741198890 (120574119889 gt 1205741198890) isin R+
such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isinZ0+
(3) 120593(119909 119896119879) is subadditive for all 119909 isin R forall119896 isin Z0+(4) 120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)| are
119895119879(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)
minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119896 isin Z0+(47)
Then the following properties hold(i) The discrete closed-loop system is asymptotically hyper-
stable(ii) 119906(119905) rarr 0 as 119905 rarr infin and the control and output are
bounded for all time
(iii) Assume furthermore that 120590(119905) = 120593(119910(119905) 119905) minus (1 +120582(119905))120593(119910[119896] 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+ 119906(119905) =minus120593(119910(119905) 119905) and forall119905 isin R0+ and that the class Φ satisfiesPopovrsquos type integral inequality for all time and results in theclass Φ119889(119896119879) at sampling instantsThen the continuous- timeclosed-loop system is asymptotically hyperstable
Proof From Lemma 9 via assumptions (1)ndash(5) constraints(36) hold so that the input- output energymeasure associatedwith the forced solution is positively lower-bounded andfinitely upper-bounded for all sampling instants 119905 = 119896119879forall119896 isin Z+ Since the discrete positive transfer function isstrictly positive real then it is also strictly stable in the discretecontext (ie convergent [29 36]) so that the homogeneousresponse is irrelevant for stability analysis purposes Thus120593[119896] isin Φ119889 119906[119896] rarr 0 and 119906[119896] and 119910[119896] arebounded for all time for any given bounded initial conditionsIn addition one has from Lemma 10 that (42) holds andthe forced input-output energy measure is positively lower-bounded and finitely upper-bound for all 119905 isin R+ Then119906(119905) rarr 0 as 119905 rarr infin and 119906(119905) and 119910(119905) are bounded for alltime for any given bounded initial conditions Properties [(i)-(ii)] have been proved Property (iii) follows by equalizing119906(119905) = minus120593(119910(119905) 119905) = (1 + 120582(119905))119906[119896] + 120590(119905) and 120593 isin Φ fromthe fact that minusint119905
0119906(120591)119910(120591)119889120591 le 1205740
Note that the use of Fourier transforms is useful tocharacterize the error of the input-output energy measure inthe frequency domain along the intersample periods relatedto its values at the preceding sampling instants This toolis addressed in Lemma 10 which is used in the proof ofTheorem 11 It allows to keep easily the characterizationof the frequency response of the continuous-time transferfunction to evaluate the input-output energy measure in theintersample period and to calculate the minimum value ofthe real part of such a transfer function to guarantee thepositivity and boundedness of such an energy Note that thecharacterization of the input-output energy measure of thefeed-forward linear loop evaluated in the frequency domaintogether with Popovrsquos inequalities of the class of hyperstablecontrollers is relevant to the hyperstability proofs of theclosed-loop system
5 Examples
Example 12 The positive realness of a discretized transferfunction depends not only on the continuous-time param-eters but on the sampling period as well Therefore thepositive realness of the continuous-time system does notguarantee that of any discretized counterpart irrespective ofthe sampling period and conversely For instance the transferfunction 119892(119904) = 1(119904 + 119886) is strictly positive real for 119886 gt 0 butit is not strongly strictly positive real since Re119892(119894120596) rarr 0 as120596 rarr plusmninfin Its discretized version without the concourse ofan ideal that is instantaneous sampler and zero-order holdbecomes 119904 119892119889(119911) = 119911(119911 minus 119890minus119886119879) which is strongly positivereal for 119879 isin (0infin) and as the sampling period 119879 rarr infin since119892119889(119911) rarr 1 as 119879 rarr infin However 119892119889(119911) = 119911(119911 minus 1) as 119879 rarr 0
12 Mathematical Problems in Engineering
Since it has a critically stable pole in the discrete domain it ispositive real although it is not strictly positive real
Example 13 Consider the discrete transfer function 119892119889(119911) =(1199112 + 1198891119911 + 1198892)(1199112 minus 1198861119911 minus 1198862) with real numerator anddenominator coefficients Its strict positive realness requiresas necessary conditions its convergence (ie strict stabilityin the discrete sense) so that its poles have to be within thecomplex open unit circle as well as the convergence of itsinverse so that its zeros have also to lie inside the open unitcircle and furthermore its relative degree (ie its pole-zeroexcess has to be zero) Note that the last condition is notneeded for strict (nonstrong) positive realness of continuoustransfer functions where their positive real part can convergeto zero as |119904| rarr infin By using the homographic bilineartransformation 119911 = (1 + 119904)(1 minus 119904) which transforms theunit circle in the left-hand-side complex plane to get theauxiliary continuous transfer function 119892119886(119904) = 119892119889(119911 = (1 +119904)(1 minus 119904)) it is found in [13] that the denominator andnumerator polynomials of 119892119889(119911) are subject to 1198862 minus 1198861 lt 11198861 + 1198862 lt 1 and 1198862 gt minus1 Take the particular values 1198861 = 05and 1198862 = 04 leading to the values of poles 1199111 = 05 and1199112 = 04 and the coefficients of the numerator polynomialare fixed to 1198891 = 12 1198892 = 03 Then we fix a discretetransfer function 119892119889(119911) = (1199112 + 12119911 + 03)(1199112 minus 05119911 minus 04)yielding min|119911|=1 Re119892119889(119911) ge 009 so 119892119889 isin SSPR Thedecomposition of 119892119889(119911) in simple fractions leads to 119892119889(119911) =1 + 168(119911 minus 093) + 0758(119911 + 043) By using an ideal (iewith instantaneous sampling) sampler and zero-order-holddevice of period 119879 of transfer function (1 minus 119890minus119879119904)119904 on acontinuous transfer function 119892(119904) to be specified to generate119892119889(119911) one has the identity 119885((1 minus 119890minus119879119904)119892(119904)119904) = 119892119889(119911) where119885(119865(119904)) is an abbreviate usual informal notation to denote119865lowast(119904) = 119871(119891lowast(119905)) where 119871(sdot) and 119885(sdot) denote the Laplacetransform of (sdot) of argument 119904 and 119885-transform of (sdot) ofargument 119911 = 119890119879119904 and 119891lowast(119905) = suminfin119899=0 119891(119899119879)120575(119905 minus 119899119879) denotesthe pulsed function of the function119891(119905)with sampling period119879 As a result we get for a sampling period 119879 = 01 sec119892 (119904) = 119904119885minus1 ( 119911119911 minus 1 [1 + 168119911 minus 093 + 0758119911 + 043])
Note that 119892 isin SPR (and also 119892 isin SSPR sinceits relative degree is zero) is the continuous-time transferfunction which generates 119892119889(119911) under an ideal sampler andzero-order-hold device of period 119879 = 01 sec so thatmin120596isinclR119892(i120596) = min120596isinR0+119892(i120596) gt 0 Thus it is guaranteedthat the continuous-time energy measure is associated withthe forced output 119864(119905) gt 0 for all 119905 gt 0 (ie it ispositively lower-bounded for all time) for any piecewise-continuous control irrespectively of the particular controllerin operation In order to guarantee that any such a controlthat converges asymptotically to zero it is necessary to finitelyupper-bound such an energy for all timeThe above concerns
also guarantee that 119864[119896] = 119864(119896119879) ge 1205741198890 gt 0 for all integer 119896 gt0 under any piecewise-constant control 119906[119896] = 119906(119896119879) witheventual jumps at sampling instants of any period 119879 gt 0 Forsuch a control identical conclusion arises from the discrete119892119889(119911)which is also strongly strictly positive real Assume that119906[119896] = minus120593(119910[119896] 119896) and forall119896 isin Z+ with 120593 isin Φ119889 where theclass Φ119889 is defined as the set of sequences 120593(119910[119896] 119896)whichsatisfy Popovrsquos inequality 119879sum119896119895=0 119910[119896]120593(119910[119896] 119896) ge minus120574119889 forall119896 isinZ+ for some finite 120574119886 isin R+ which is identical to
= minus119879119872(1 minus ]119896+1)1 minus ] minus 119879 119896sum119895=0
120576 [119896] ge minus120574119889 forall119896 isin Z+(50)
for some real constants 119872 gt 0 ] isin (0 1) and some realsummable sequence 120576[119896] Then the class Φ119889 is definedby any real sequence 120593(119910[119896] 119896) being parameterized by anyreal constants 119872 gt 0 ] isin (0 1) and any real summablesequence 120576[119896] such that
forall119896 isin Z0+ and 120574119889 ge max(119879119872(1 minus ]) + 120576 120574) for some given120574 ge 1205741198890 where 120576 = 119879max119896isinZ+ sum119896119895=0 120576[119896]The continuous-timecontrol is generated as
where 120593[119896] isin Φ119889(119879) satisfies Popovrsquos type discrete inequal-ity which makes the feedback system globally asymptoticallystable under the theorem hypothesis Thus the discrete-timeclosed-loop system is asymptotically hyperstable In partic-ular under the additional assumptions in Theorem 11(iii)the closed-loop continuous-time system is asymptoticallyhyperstable
Mathematical Problems in Engineering 13
One of the referees has given a comment of interest on thisexample related to continuous versus discrete counterpartproperties which has motivated us to give a further intuitivediscussion on it Note the following In the continuous-timecase there are two kinds of strictly positive real transferfunctions as it is well known namely the so-called strictpositive real which can be zero (or respectively positive)at infinity frequency and which a realizable has relativedegree one (or respectively zero) In the first case the directinput-output interconnection gain 119889 is zero in a state-spacerepresentation (119860 119887 119888 119889 = 0) of the transfer function Thetransfer functions of the second case are often referred to asthe class of strong strict positive real transfer functions Thisclass is not a disjoint class of first above one of strict positivereal transfer functions since it is included in it as a potentialparticular case Note that the members of this strong classof strict positive real transfer functions have relative degreezero and a positive direct input-output interconnection gain119889 Related to this concern the Yakubovich-Kalman-Popov(refereed to the continuous-time domain) and the Szego-Kalman-Popov (referred to the discrete domain) lemmascan be invoked to compare positive realness of the transferfunctions with their state-space associated properties (in factthe properties aremutually equivalent fromboth sides)Thuswe see that in the discrete case the Szego-Kalman-Popovlemma needs for its own coherency that the input-outputinterconnection 119889-gain be nonzero since otherwise one ofthe relevant constraint in the lemma fails [13] So strictpositive realness in the discrete domain in this context istypically strong and then the relative degree of its transferfunction is zero so that there is a direct interconnectionpositive gain from the input to the output Since we assumethat the discrete system is strictly positive real it has a positiveinput-output interconnection from the input to the outputand the minimum value of the real part of its frequencyresponse is strictly positive even as the frequency tends toinfinity On the other hand note that a zero-order holdldquoconvertsrdquo any strictly proper (continuous-time) transferfunction of any relative degree into a strictly discrete oneof unity relative order while it ldquoconvertsrdquo a continuous oneof zero relative order (ie biproper) into a discrete onestill of zero-relative order and with the same input-outputinterconnection gain See (48) Therefore we deal in thisexample with a biproper strict positive real transfer functionof positive input-output interconnection gain identical to thatof its discrete-time counterpart got through a ZOH samplingand hold device
Example 14 Consider that a controller of the form 119906(119905) =minus120593(119910(119905) 119905) forall119905 isin R0+ is used for the continuous-time feed-forward transfer function (48) so that120593 isin Φ and Φ is classin Example 13 satisfying an integral type Popovrsquos inequality(21) for all 119905(ge 1199050) isin R+ and some finite 1199050 isin R+ That isthe control function belongs to a class satisfying continuous-time integral Popovrsquos inequality Then 119906(119905) rarr 0 as 119905 rarr infinand 119906(119905) and 119910(119905) are almost everywhere bounded for all timesince 119892 isin SSPR then being (strictly) stable with input-output energy measure being positively lower-bounded andalso finitely upper-bounded for all 119905 isin R+ from Popovrsquos-type
integral inequality Potential useful controllers are for somegiven controller gain 119896119906 isin R+
119906 (119905) = 0 for 119905 isin [0 1)1198961199061199052119910 (119905) for 119905 ge 1 (54)
119906 (119905) = 119896119906119890minus119905119910 (119905) for 119905 ge 0 (55)
119906 (119905) = 0 for 119905 isin [0 2)
119896119906(119905 ln2119905) 119910 (119905) for 119905 ge 2 (56)
119906 (119905) = 0 for 119905 isin [0 1)
119896119906(119890radic119905radic119905) 119910 (119905) for 119905 ge 1 (57)
119906 (119905) = 1198961199061199053119890minus119905119910 (119905) for 119905 ge 0 (58)
where Popovrsquos inequality on [0 +infin] of (49)ndash(52)is intinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus119896119906 that of (57) isintinfin
0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 and that of (58) isintinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 Since the respective integrands
are monotonically decreasing then there is some 1199050 isin R+ andsome 120574 = 120574(1199050) isin R+ such that int119905
1199050119910(119905)120593(119910(119905) 119905)119889119905 ge minus120574 for
all 119905(ge 1199050) isin R+ The obtained closed-loop systems are inall cases asymptotically hyperstable The piecewise-constantcontrols with eventual discontinuities at sampling instantsgenerated from any discrete-time controller (51) of class Φ119889of Example 13 lead also to asymptotic hyperstability of thediscretized system since 119892119889 isin SSPRExample 15 Consider the continuous-time open-loop sys-tem (2) under a real sampler of sampling period 119879 and datapicking-up nonzero duration 120576 isin (0 119879) The state and outputtrajectory solutions are
forall119896 isin Z0+ for some real sequence 120585[119896] such that 120585[119896] isin(119896119879 119896119879 + 120576) where 119892119889(119911) is the discretized transfer functionunder a sampler and zero-order-hold device subject to instan-taneous ideal sampling Now assume that the incrementalcontrol along intersample periods is generated from itspreceding sampled value as (119895119879+ 120591) = 120582119895119879+120576(119895119879+ 120591)119906[119895] forany 119895 isin Z0+ and 120591 isin [0 119879] where 120582119895119879+120576(119895119879+120591) is a truncatedfunction under the usual notation one gets
The above equations imply that if 119892119889 isin SPR then the forcedoutput 119910119891[119896] = 119910[119896] minus 119888119879119890119896119860119879119909[0] forall119896 isin Z0+ generates anassociate input-output energy measure which is strictly posi-tive for 119896 isin Z0+ since 120588 isin (0 1) for real sampling of duration120576 isin (0 119879)This is a necessary condition for asymptotic hyper-stability under an appropriate class of controllers satisfyingPopovrsquos inequality However the incremental input-outputenergy measure generated by the incremental forced output120576119888119879sum119896minus1119895=0 119890(119896minus119895)119860119879(int1198790 120582119895119879+120576(120591)119890minus119860120591119887 119889120591)119906[119895] is not guaranteedto be nonnegative in general A sufficient condition for that isthat lim sup119895rarrinfin|ess sup120591isin(0120576)(120582119895119879+120576(120591))| is sufficiently smallrelated to the amount ((1 minus 120588)120576)min120596isin[0120587] Re119892119889(119890i120596)6 Conclusions
This paper has been devoted to obtain and to prove theasymptotic hyperstability property of a linear and time-invariant continuous-time system based on some given andpreviously proved auxiliary preparatory lemmas Basicallyit is proved that the continuous-time closed-loop system isasymptotically hyperstable under the assumptions that thediscretized feed-forward transfer function is strictly positivereal and that the discretized system at sampling instants ofconstant sampling period is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsIt is assumed that the continuous-time controller belongs toa class defined in the following way (a) its values at samplinginstants are defined by a class of controllers which satisfydiscrete Popovrsquos type inequality (b) the intersample control isgenerated from its values at sampling instants with the use oftwomodulating auxiliary functionswhich are subject to someextra additional conditions on smallness and boundednessSuch conditions make the overall continuous-time controllerto satisfy continuous-type Popovrsquos inequality for all time
where 119896(120591) = max(119911 isin Z+ 119911119879 le 120591) and 119896 = 119896(119905) = max(119911 isinZ+ 119911119879 le 119905) forall119905 isin R0+ which yields (13)
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
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2 Mathematical Problems in Engineering
interest since many existing industrial installations combineboth kinds of systems An elementary well-known case iswhen a discrete-time controller is used for a continuous-time plant Another case is related to teleoperation systemswhere certain variables evolve in a discrete-time or digitalfashion A background literature and related relevant resultsare given in [1 7 11 17 18 27ndash30] and some of the referencestherein
Some conditions have been given in the backgroundliterature to guarantee the positive realness of discrete trans-fer functions from related conditions on their continuous-time counterparts which are then maintained under dis-cretization See for instance [31] The results are useful toguarantee the hyperstability of discrete-time systems underthe hyperstability of the continuous-time ones when the classof feedback discrete-time controllers satisfies Popovrsquos typeinequality However it turns out that positive realness ofa continuous-time system does not guarantee the positiverealness of its discretized transfer function by a zero-orderhold for any sampling period and conversely the strongpositive realness of a discretized transfer function does notguarantee in some limited cases the strong positive realnessof its continuous-time counterpart [31ndash33] This paper isdevoted to a certain kind of inverse problem stated inthe following terms Given is an asymptotically hyperstablediscrete-time system of sampling period 119879 such that its feed-forward transfer function is strictly positive real and thenonlinear time-varying feedback controller belongs to a classΦ119889(119879) satisfying certain discrete Popovrsquos type inequalityThe elementary question that arises is how the asymptotichyperstability of the continuous-time system is guaranteedfor a certain modified class of controllers such that itsmember at sampling instants is in the class Φ119889(119879) Toanswer this question an intersample controller based onthe class Φ119889(119879) at sampling instants is designed whichcorrects the control law along the intersample period viathe concourse of two modulating continuous-time functionswhich have to satisfy certain reasonable constraints Thepaper is organized as follows Firstly a notation and ter-minology subsection is allocated in the subsequent sectionSection 3 presents the linear and time-invariant system ofthe feed-forward part of the closed-loop system togetherwith its analytic output expressions some auxiliary resultsof positivity and boundedness for all the sampling instantsof an input-output energy measure The continuous-timecontrol input is proposed to be generated within eachintersample period from the last sampling instant definingthe starting point of the current intersample period and somedesign auxiliary functions which modulate the control signalalong such an intersample period Some of the formulaspresented are derived in detail in the appendix On theother hand some auxiliary preparatory lemmas concerningthe input-output energy measure are also formulated whichare of usefulness for the main result Section 4 is devotedto obtain the main asymptotic hyperstability result whichis proved based on some given and previously provedauxiliary preparatory lemmas Basically it is proved thatthe continuous-time closed-loop system is asymptoticallyhyperstable provided that (a) the discretized feed-forward
transfer function is strictly positive real (b) the discretizedsystem at sampling instants is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsand (c) some extra additional conditions on smallness andboundedness are fulfilled by the abovementioned auxiliarymodulated functions which define the controls along theintersample periods Such conditions make any continuous-time controller belonging to the appropriate class to satisfycontinuous-type Popovrsquos inequality for a class of controllersprovided that the former one at sampling instants is fulfilledby the discretized controllers Some examples are discussedin Section 5 Finally conclusions end the paper
2 Notation
(i) R+ = R0+ cup 0 Z+ = Z0+ cup 0 R0+ = 119903 isin R 119903 ge 0 andZ0+ = 119911 isin Z 119911 ge 0
(ii) clR = [minusinfin +infin] is the closure of the real fieldthat is the set of real numbers together with the plusmn infinitypoints
(iii)The continuous and discrete-time arguments 119905 isin R0+and 119896 isin Z0+ are denoted with parenthesis and brackets thatis (119905) and [119896] respectively
(iv) If 119879 gt 0 is the sampling period then 119906(119905) = 119906[119896] +(119905) forall119905 isin [119896119879 (119896 + 1)119879) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+ isthe continuous-time input where the piecewise-continuousintersample incremental input is (119905) = 119906(119905) minus 119906[119896] with119906[119896] = 119906(119896119879) and [119896] = (119896119879) = 0 forall119905 isin [119896119879 (119896 +1)119879) forall119896 isin Z0+ The auxiliary purely discrete input is 119906119889[119896]such that 119906119889[119896] = 119906[119896] forall119896 isin Z0+ if (119905) = 0 forall119905 isinR0+
(v) 119910ℎ(119905) forall119905 isin R0+ and 119910ℎ[119896] = 119910ℎ(119896119879) forall119896 isin Z0+ arethe forced andhomogeneous (unforced) output of the systemrespectively
(vi) 119910119889ℎ[119896] = 119910119889ℎ(119896119879) and 119910119889119891[119896] = 119910119889119891(119896119879) forall119896 isin Z0+are the forced and homogeneous output of the system respec-tively when the input is piecewise-constant with constantvalue in-between each two consecutive sampling instants iewhen the input is generated from a zero- order-hold (ZOH)device
(vii) The Fourier transform of any real vector function119891(119905) is denoted by119891(i120596) if it exists forall119905 120596 isin R where i = radicminus1is the imaginary complex unit
(viii) For the purpose of formally using Fouriertransforms for the subsequent developments we definecontinuous-time and discrete-time truncated signalsfrom their untruncated counterparts on the respectivecontinuous-time and discrete-time intervals [0 119905] and[0 119896119879] as follows
V119905 (120591) =
V (120591) if 0 le 120591 le 1199050 if 120591 gt 1199050 if 120591 lt 0
forall119905 120591 isin R0+
Mathematical Problems in Engineering 3
V[119896] [119895] =
V [119895] if 0 le 119895 le 1198960 if 119895 gt 1198960 if 119895 lt 0
forall119895 119896 isin Z0+(1)
(ix) Let 119904 and 119911 = 119890minus119879119904 the Laplace and 119911-transforms(for sampling period 119879) complex arguments The set SPR isthe set of strictly positive real continuous 119892(119904) (respectivelydiscrete) transfer functions 119892(119911) ie they are (strictly) stableie with poles in Re 119904 lt 0 (respectively |119911| lt 1) whichsatisfy Re119892(119904) gt 0 for all Re 119904 ge 0 (respectively Re119892(119911) gt 0for all |119911| ge 1) These conditions imply that Re119892(i120596) gt0 respectively Re119892(119890i120596) gt 0 forall120596 isin R [13 31 34] Thestrictly positive continuous and discrete real sets are notdistinguished at the level of notation since they are easy toidentify them according to context
(x) The so-called set SSPR is the set of (continuous ordiscrete depending on context) strongly positive real transferfunctions which are those in SPR whose real part is strictlypositive also as |119904| rarr infin It can be pointed out that Szego-Kalman-Popov Lemma (also so-called Discrete Positive RealLemma) is a discrete version of the Yakubovich-Kalman-Popov Lemma (also so-called Positive Real Lemma) whichrelates positive realness of transfer functions to associatedstate-space realization properties states that discrete transferfunction in SPR are of relative degree (namely pole-zeroexcess) equal to zero so that they are also in SSPR and bothsets are equivalent This equivalence concern between thediscrete sets SPR and SSPR does not apply to continuoustransfer functions [13 31]
3 The Continuous-Time and Discrete-TimeLinear Time-Invariant Systems and SomeAuxiliary Results
Consider an 119899th-order linear and time-invariant dynamicsystem in state-space description
where 119909(119905) isin R119899 119906(119905) isin R and 119910(119905) isin R are the statepiecewise-continuous control input and output and 119860 isinR119899times119899 119887 isin R119899 119888 isin R119899 and 119889 isin R are the matrix ofdynamics control vector output vector and input-outputinterconnection gain The auxiliary purely discrete outputand the continuous-time and sampled output are calculatedfrom (2) via the impulse responses 119892(119905) and 119892119889[119896] whichbecome
119892 (119905 minus 120591) = 119888119879119890119860(119905minus120591)119887 + 119889120575 (119905 minus 120591) forall120591 119905 (ge 120591) isin R0+119892119889 [119896 minus 119895] = int119896119879
119895119879119892 (119896119879 minus 120591) 119889120591
= int(119896minus119895)1198790
119892 ((119896 minus 119895) 119879 minus 120591) 119889120591= int119896119879119895119879(119888119879119890119860(119896119879minus120591)119887 + 119889120575 (119896119879 minus 120591)) 119889120591
= int(119896minus119895)1198790
(119888119879119890119860((119896minus119895)119879minus120591)119887 + 119889120575 ((119896 minus 119895) 119879 minus 120591)) 119889120591forall119895 (le 119896) 119896 isin Z0+
(3)
and 119892(119905minus120591) = 0 forall120591 119905(lt 120591) isin R0+ where 120575(119905minus120591) is the Diracdistribution 119906(minus119905) = 0 forall119905 isin R+and 119892119889[119896 minus 119895] = 0 forall119895(gt119896) 119896 isin Z0+ with 119892119889[0] = 119889 The initial input and outputconditions are 119906[119895] = 119910[119895] = 0 for any integer 119895 lt 0 and119910(119905) = 119906(minus119905) = (minus119905) = 0 forall119905 isin R+ in causal dynamicsystems subject to initial conditions119909(0) = 1199090 Let the controlinput be
119892119889 [119896 minus 119895])119906[119896] [119895]+ intinfinminusinfin119892 (119896119879 minus 120591) 119896119879 (120591) 119889120591 + 119910ℎ [119896]
forall119896 isin Z0+
(7)
Expressions (5)ndash(7) are derived in the appendix
4 Mathematical Problems in Engineering
Remark 1 Note that the difference between the sampledcontinuous-time sequence 119910[119896] and the purely discreteoutput sequence 119910119889[119896] is that the first one is the exactvalue of the output at sampling instants including the effectsof the intersample input ripple while the second one is thediscretized output at sampling instants in the presence of azero-order hold As a result note that 119910[119896] = 119910119889[119896] for anygiven 119896 isin Z+ if and only if int119896119879
0119892(119896119879 minus 120591)(120591)119889120591 = 0 In
particular 119910[119896] = 119910119889[119896] if (119905) = 0 forall119905 isin R0+ Causalityimplies that (6)-(7) can be rewritten as
119910 (119905) = (infinsum119895=0
119892119889 [119896 minus 119895])119906[119896] [119895]+ (int119905119896119879119892 (119905 minus 120591) 119889120591) 119906 [119896]
The following result is directLemma 2 Any given piecewise-continuous control input canbe decomposed into purely discrete-time control plus incremen-tal intersample period ones as follows
through piecewise-continuous functions 120582 120590 cup119896isinZ0+[119896119879 (119896 +1)119879) rarr R defined by
120582 (119905) = (119905)119906 [119896] if 119906 [119896] = 00 if 119906 [119896] = 0
120590 (119905) = 0 if 119906 [119896] = 0 (119905) if 119906 [119896] = 0
(10)
The purely auxiliary input-output and true ones energymeasures on the discrete-time interval [0 119896119879] as well as thecontinuous-time ones on [0 119905] forall119905 isin [119896119879 (119896 + 1)119879) and forall119896 isinZ0+ are respectively from (5)ndash(7) and (9)-(10) given by
Remark 3 Since the control input is piecewise-continuousthe truncated input 119906119905(120591) for 120591 isin [0 119905] has Fourier transformsfor any 119905 isin R0+ so that the corresponding truncated output
119910119905(120591) for 120591 isin [0 119905] and also the energy measure have alsoFourier transforms for any 119905 isin R0+ This follows from the factthat these signals are calculated via truncated functionswhich
Mathematical Problems in Engineering 5
are then zero at infinity so that they are square-integrable andtheir sequences of sampled values of sampling period 119879 aresquare-summable as a result
The subsequent two simple auxiliary lemmas will be thenused in the next section to establish the main result
Lemma 4 Assume that there exist constants 1205741198890 (120574119889 gt 1205741198890) isinR+ such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some1198960 isin Z0+ Then there exist 1205740 (120574 gt 1205740) isin R+ such that 1205740 le119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin) if1205740 minus 120574119889 le 119864119889 (119896119879 119905)
= (119864 [119896] minus 119864119889 [119896]) + 119864 (119896119879 119896119879 + 120591)= 119864 (119905) minus 119864119889 [119896] = 119864119889 (119896119879 119905) le 120574 minus 1205741198890
119879 (119910 [119895] 119906 [119895] minus 119910119889 [119895] 119906119889 [119895])
(16)
forall119905 isin (119896119879 (119896 + 1)119879) forall119896 isin Z0+ so that (14) implies that 1205740 le119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin) provided that 1205741198890 le 119864119889[119896] le120574119889 forall119896(gt 1198960) isin Z+ for 1198960 isin Z0+Lemma 5 Assume that 120593(119909 119905) is subadditive for all 119909 isin Rie 120593(119909 + 119910 119905) le 120593(119909 119905) + 120593(119910 119905) forall119909 119910 isin R forall119905 isin R0+Then 120593(119910(119905) 119905) le 120593(119910[119896] 119896) + 120593(119910(119905) minus 119910[119896] 119905 119896119879) andas a result (119905) ge 120593(119910(119905) minus 119910[119896] 119905 119896119879) and 120593(119910[119896] 119896) geminus120593(minus119910[119896] 119896) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+
If 120593(119909 119905) is superadditive for all 119909 isin R ie 120593(119909 +119910 119905) ge 120593(119909 119905) + 120593(119910 119905) forall119909 119910 isin R forall119905 isin R0+ then120593(119910(119905) 119905) ge 120593(119910[119896] 119896119879) + 120593(119910(119905) minus 119910[119896] 119905 119896119879) as a result(119905) le 120593(119910(119905) minus 119910[119896] 119905 119896119879) and 120593(119910[119896]) le minus120593(minus119910[119896]) forall119905 isin[119896119879 (119896 + 1)119879) forall119896 isin Z0+Proof From the first identity in (9) if 120593(119909 119905) is subadditivethenminus120593 (119910 (119905)) = 119906 (119905) = 119906 [119896] + (119905)
ge minus120593 (119910 [119896] + (119910 (119905) minus 119910 [119896]))ge minus120593 (119910 [119896]) minus 120593 (119910 (119905) minus 119910 [119896])
= minus120593 (119910 [119896]) minus 120593 (119910 (119905) + (minus119910 [119896]))ge minus120593 (119910 [119896]) minus 120593 (119910 (119905)) minus 120593 (minus119910 [119896])
Definition 6 (Popovrsquos inequality) A continuous-time feed-back nonlinear and eventually time-varying continuous-timecontroller 119906(119905) = minus120593(119910(119905) 119905) and forall119905 isin R0+ where 120593 R0+ timesR0+ rarr R0+ is in the class Φ denoted as 120593 isin Φ is said tosatisfy Popovrsquos type integral inequality if for some 1205740 isin R+one has
Definition 7 (discrete-time Popovrsquos inequality) A discrete-time feedback nonlinear and eventually time-varyingdiscrete-time controller 119906(119905) = 119906[119896] = 119906(119896119879) =minus120593(119910(119896119879) 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) and forall119896 isin Z0+where 120593 R0+ times Z0+ rarr R0+ is in the class Φ119889(119879) ofsampling period 119879 isin R+ denoted as 120593 isin Φ119889(119879) is said tosatisfy a Popovrsquos type discrete-time inequality if for some120574119889 isin R+ one has119896sum119895=0
119910 (119896119879) 120593 (119896119879) ge minus120574119889119879 forall119896 isin Z0+ (22)
I D Landau refers to controllers satisfying a Popovrsquostype inequality (21) (or respectively (22)) as hyperstable
controllers of class Φ (or respectively of class Φ119889(119879))[13] bearing in mind that if any controller of such a class iscoupled to a linear time-invariant forward system of transferfunction being continuous-time (or respectively discrete-time) strictly positive real then the overall closed-loop systemis asymptotically hyperstable namely globally asymptoticallystable for any arbitrary controller belonging to the respectiveclass It can be pointed out that the above comments referredto discrete-time systems and in particular Definition 7 arealso applicable to digital systems in the same way as theyare applicable to discrete-time systems ie those which arefully described in the discrete domainwithout having specificlinks to a discretization process on a certain continuous-timesystem
It can be pointed out that Definition 7 can be alsoestablished for a digital system not being related to thetime discretization of a continuous-time system in terms ofPopovrsquos inequality of the following form
119896sum119895=0
119910 [119896] 120593 [119896] ge minus120574119889119886 forall119896 isin Z0+ (23)
for some constant 120574119889119886 isin R+ for any digital controller ofclass Φ119889 In this context the sampling period either can benonrelevantmdashthen nonmade notational explicit in the classΦ119889mdashor can even have nonsense since the inequality isapplied to a certain discrete sequence
The following result obtains useful expressions for the dis-cretized input-output energymeasure under some conditionsconcerning the subadditivity constraints on the controllers ofclass Φ119889(119879)The result is also a preparatory one for the nextLemma 9 which addresses the positivity and boundedness ofthe input-output energy measure at sampling instants
Lemma 8 The following properties are fulfilled(i) The subsequent formulas hold
(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119895 (le 119896) 119896 isin Z0+
(26)
(ii) Assume that 120593 isin Φ119889(119879) and that 120593(119909 119896119879) issubadditive for all 119909 isin R ie 120593(119909 + 119910 119896119879) le 120593(119909 119896119879) +120593(119910 119896119879) forall119909 119910 isin R forall119896 isin Z0+ Then the subsequentrelations are true if the sequence 120576[119896] is defined by 120576[119896] =120582[119896] + 120585[119896] forall119896 isin Z0+119910119889ℎ [119896] + intinfin
0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591
which yields (24) and leads to (25) Now define the sequence120576[119896] by 120576[119896] = 120582[119896] + 120585[119896] and 120585[119896][119895] = int(119895+1)119879119895119879(120582(119895119879 +120591)119892((119896 minus 119895)119879 minus 120591) minus 120582[119895]119892[(119896 minus 119895)119879])119889120591 forall119896 isin Z0+ and note
that
minus119910 [119896] = minus (119910119889 [119896] + (119910 [119896] minus 119910119889 [119896]))= [[infinsum119895=0
(int(119895+1)119879119895119879
(1 + 120576[119896] [119895]) 119892 [(119896 minus 119895) 119879])119889120591]]sdot 120593[119896] ([119910119889 [119895] + (119910 [119895] minus 119910119889 [119895]) 119895])minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 minus 119910ℎ [119896]
forall119896 isin Z0+
(31)
and then
minus (119910 [119896] minus 119910119889 [119896] minus 119910ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591)
The subsequent result establishes lower and upper-bounds for the discretized input-output energy measuresbased on the formulas obtained in Lemma 8 and strictpositive realness of the discrete-time transfer function ofthe feed-forward linear and time-invariant system Theresult invokes the boundedness and the smallness of theparameters which define the calculation of the intersam-ple control input from their preceding values at samplinginstants Note that the control deviation in the intersampleperiods related to the sampling time instants has to be suffi-ciently moderate enough so that the hyperstability propertyis kept from the discrete-time system to the continuous-time one In particular the positivity and boundedness ofthe energy measure at the sampling instants is guaranteedif sup119905isinR0+ |120590(119905)| is small enough and Φ119889(119879) is subaddi-tive
Mathematical Problems in Engineering 9
Lemma 9 Assume that 119892119889 isin 119878119875119877 and that 120593 isin Φ119889(119879)Then the following properties hold(i)The input-output energy systemmeasure of the auxiliary
discretized system satisfies
0 lt 1205740119889119886 le 1198792120587 min120596isinR0+
119879119910119889 [119896] 119906119889 [119896] le 120574119889119886forall119896 (gt 1198960) isin Z+
(34)
for some 1205740119889119886 120574119889119886(gt 1205740119889119886) isin R+ and for any nonidentically zerocontrols and zero initial conditions ie for any forced solution
(ii) Assume in addition that the class Φ119889(119879) consistsof subadditive functions Assume also that sup119905isinR0+ |120590(119905)| issufficiently small inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infinwhere 120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ and that furthermore
for some 120574119889119886 isin R+ from the first assumption of the theoremand 119906119889[119896] = minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ since Popovrsquosinequality (22) is fulfilled by any given discrete-time feedbackcontroller 120593 isin Φ119889(119879) generating a feedback control 119906119889[119896] =minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ On the other hand the initialconditions can be neglected since the assumption119892119889 isin SPRimplying that the discrete-time transfer function is (strictly)stable which makes their values irrelevant for purposes ofstability analysisThen the discrete frequency response being
applicable to piecewise-constant inputs in-between any twoconsecutive sampling instants is given by
since the subsequence 119892[minus119899] = 0 forall119899 isin Z+ and thetransfer frequency response of a sampling and zero-order-hold operator 119885 of period 119879 defined by 119885(119879 V)(119905) = V(119896119879) =V[119896] for any given V R0+ rarr R and all 119905 isin [119896119879 (119896 + 1)119879)119896 isin Z0+ is given by 119885(119879 i120596) = (1 minus 119890minusi119879120596)i120596 [31 35]It turns out that |119885(119879 i120596)| = |(1 minus 119890minusi119879120596)i120596| lt 1 forall120596( =0) isin R and |119885(119879 i0)| = 1 Thus the auxiliary discretizedsystem satisfies the subsequent equivalence relation in thefrequency domain to the discrete-time relation (11) It turnsout that Fourier transforms exist in the impulse responsesand truncated control and auxiliary functions of (11)ndash(13)since the truncated functions are square-integrable in clR =[minusinfin +infin] By using the discrete Parsevalrsquos theorem underzero initial conditions one has
The lower-bounding constraint of (37) follows by using(39) and the discrete Parsevalrsquos theorem for the equivalencebetween the input-output energy measures from the fre-quency domain to the discrete-time domain
119864119889 [119896] ge 1198792120587 min120596isinR0+
1199062119889 [119896]) ge 1205740119889forall119896 isin Z0+
(40)
from the assumption 119892119889 isin SPR iemin120596isinR0+ Re119892119889(119890minusi120596119879) ge 1205740119889119886 gt 0 forall1198960 119896 (119896 ge 1198960) isin Z+since
(a) strictly positive real discrete transfer functions havezero relative degree (ie an identical number of polesand zeros) from the Discrete Positive Real Lemma(Szego-Kalman-Popov Lemma) [13 31 34] so thatthe real parts of their frequency hodographs in theargument 119890minusi120596119879 are positively lower-bounded
10 Mathematical Problems in Engineering
(b) Im119892119889(119890minusi120596119879) = minus Im119892119889(119890i120596119879) forall120596 isin R0+ so that theirintegrals in the argument 120596 on frequency intervals[minus119896120587 119896120587] for any 119896 isin Z+ are null
Property (i) has been proved To prove Property (ii) notefrom (28) [Lemma 8(ii)] that
119892 [(119896 minus 119895) 119879])119889120591]]sdot (120593[ℓ] (119910119889 [119895] 119895) + 120593[ℓ] (minus119910119889 [119895] 119895))sdot (120593[ℓ] (119910119889 [ℓ] 119895) + 120593[ℓ] (minus119910119889 [ℓ] ℓ)) minus 119910119889ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 le 120574119889 = 119870120574119889119886
forall119896 isin Z+
(41)
for some 119870 isin R+ provided that sup119905isinR0+ |120590(119905)| is sufficientlysmall inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infin where120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ since 119892(119905) is bounded andconverges exponentially to zero since 119892119889 isin SPR and thenconvergent (that is strictly stable in the discrete context)119910119889ℎ[119896] rarr 0 and it is a bounded sequence
The next result which is a preparatory result to thenestablish the main asymptotic hyperstability result addressesthe relevant property that the input-output measure 119864(119905) ofthe continuous-time system is positively lower-bounded andfinitely upper-bounded for all time under the conditions ofLemmas 4 and 9
Lemma 10 If 120582 and 120590 are sufficiently small then
1205740 minus 120574119889 le 119864 (119905) minus 119864119889 [119896]le 1 + 12058222120587 intinfin
provided that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+Proof One gets the following from direct calculations byusing the continuous-time control laws (9)-(10) and (16) if120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)|
minus 12058222120587 intinfin
minusinfin119892 (i120596) 1003816100381610038161003816[119896] (i120596)10038161003816100381610038162 119889120596 minus 120590(2120582 sup
0le119895le119896
1199062[119895] + 120590)int1198961198790
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591 le 119864 (119905) minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119910 (i120596)
sdot (minusi120596) 119889120596 minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119892 (i120596) | (i120596)|2 119889120596 minus 119864119889 (119896119879 119905)
Now the use of (44) in Lemma 4 (14) implies thatLemma 9(ii) (36) holds implying the existence of 1205740 (120574 gt1205740) isin R+ such that 1205740 le 119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin)provided that 1205741198890 le 119864119889[119896] le 120574119889 for 1205741198890 (120574119889 gt 1205741198890) isin R+ since1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isin Z0+ if1205740 minus 120574119889 le minus 120582
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591le 120574 minus 1205741198890 forall119896 (gt 1198960) isin Z+ forall120591 isin (0 119879)
(45)
if 120582 and 120590 are sufficiently small such that (43) holds leadingto 1205740 gt 0 for any given positive 1205741198890 and 120574119889 gt 1205741198890
The main first result follows
Theorem 11 Consider the dynamic system (2) subject to acontrol
forall119896 isin Z0+ where 120593 [119896] isin Φ119889 (119879) (46)
Assume that(1) 120593(119910[119896] 119896119879) is in the class Φ119889(119879) (that is120593(119910[119896] 119896119879) isin Φ119889(119879)) defined by the discrete Popovrsquos
type sum inequality sum119896ℓ=0 119910119889[ℓ]119906[ℓ] ge minus120574119889119879 for some 120574119889 isin Rforall119896 isin Z0+(2) 119892119889 isin 119878119875119877 there exist constants 1205741198890 (120574119889 gt 1205741198890) isin R+
such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isinZ0+
(3) 120593(119909 119896119879) is subadditive for all 119909 isin R forall119896 isin Z0+(4) 120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)| are
119895119879(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)
minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119896 isin Z0+(47)
Then the following properties hold(i) The discrete closed-loop system is asymptotically hyper-
stable(ii) 119906(119905) rarr 0 as 119905 rarr infin and the control and output are
bounded for all time
(iii) Assume furthermore that 120590(119905) = 120593(119910(119905) 119905) minus (1 +120582(119905))120593(119910[119896] 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+ 119906(119905) =minus120593(119910(119905) 119905) and forall119905 isin R0+ and that the class Φ satisfiesPopovrsquos type integral inequality for all time and results in theclass Φ119889(119896119879) at sampling instantsThen the continuous- timeclosed-loop system is asymptotically hyperstable
Proof From Lemma 9 via assumptions (1)ndash(5) constraints(36) hold so that the input- output energymeasure associatedwith the forced solution is positively lower-bounded andfinitely upper-bounded for all sampling instants 119905 = 119896119879forall119896 isin Z+ Since the discrete positive transfer function isstrictly positive real then it is also strictly stable in the discretecontext (ie convergent [29 36]) so that the homogeneousresponse is irrelevant for stability analysis purposes Thus120593[119896] isin Φ119889 119906[119896] rarr 0 and 119906[119896] and 119910[119896] arebounded for all time for any given bounded initial conditionsIn addition one has from Lemma 10 that (42) holds andthe forced input-output energy measure is positively lower-bounded and finitely upper-bound for all 119905 isin R+ Then119906(119905) rarr 0 as 119905 rarr infin and 119906(119905) and 119910(119905) are bounded for alltime for any given bounded initial conditions Properties [(i)-(ii)] have been proved Property (iii) follows by equalizing119906(119905) = minus120593(119910(119905) 119905) = (1 + 120582(119905))119906[119896] + 120590(119905) and 120593 isin Φ fromthe fact that minusint119905
0119906(120591)119910(120591)119889120591 le 1205740
Note that the use of Fourier transforms is useful tocharacterize the error of the input-output energy measure inthe frequency domain along the intersample periods relatedto its values at the preceding sampling instants This toolis addressed in Lemma 10 which is used in the proof ofTheorem 11 It allows to keep easily the characterizationof the frequency response of the continuous-time transferfunction to evaluate the input-output energy measure in theintersample period and to calculate the minimum value ofthe real part of such a transfer function to guarantee thepositivity and boundedness of such an energy Note that thecharacterization of the input-output energy measure of thefeed-forward linear loop evaluated in the frequency domaintogether with Popovrsquos inequalities of the class of hyperstablecontrollers is relevant to the hyperstability proofs of theclosed-loop system
5 Examples
Example 12 The positive realness of a discretized transferfunction depends not only on the continuous-time param-eters but on the sampling period as well Therefore thepositive realness of the continuous-time system does notguarantee that of any discretized counterpart irrespective ofthe sampling period and conversely For instance the transferfunction 119892(119904) = 1(119904 + 119886) is strictly positive real for 119886 gt 0 butit is not strongly strictly positive real since Re119892(119894120596) rarr 0 as120596 rarr plusmninfin Its discretized version without the concourse ofan ideal that is instantaneous sampler and zero-order holdbecomes 119904 119892119889(119911) = 119911(119911 minus 119890minus119886119879) which is strongly positivereal for 119879 isin (0infin) and as the sampling period 119879 rarr infin since119892119889(119911) rarr 1 as 119879 rarr infin However 119892119889(119911) = 119911(119911 minus 1) as 119879 rarr 0
12 Mathematical Problems in Engineering
Since it has a critically stable pole in the discrete domain it ispositive real although it is not strictly positive real
Example 13 Consider the discrete transfer function 119892119889(119911) =(1199112 + 1198891119911 + 1198892)(1199112 minus 1198861119911 minus 1198862) with real numerator anddenominator coefficients Its strict positive realness requiresas necessary conditions its convergence (ie strict stabilityin the discrete sense) so that its poles have to be within thecomplex open unit circle as well as the convergence of itsinverse so that its zeros have also to lie inside the open unitcircle and furthermore its relative degree (ie its pole-zeroexcess has to be zero) Note that the last condition is notneeded for strict (nonstrong) positive realness of continuoustransfer functions where their positive real part can convergeto zero as |119904| rarr infin By using the homographic bilineartransformation 119911 = (1 + 119904)(1 minus 119904) which transforms theunit circle in the left-hand-side complex plane to get theauxiliary continuous transfer function 119892119886(119904) = 119892119889(119911 = (1 +119904)(1 minus 119904)) it is found in [13] that the denominator andnumerator polynomials of 119892119889(119911) are subject to 1198862 minus 1198861 lt 11198861 + 1198862 lt 1 and 1198862 gt minus1 Take the particular values 1198861 = 05and 1198862 = 04 leading to the values of poles 1199111 = 05 and1199112 = 04 and the coefficients of the numerator polynomialare fixed to 1198891 = 12 1198892 = 03 Then we fix a discretetransfer function 119892119889(119911) = (1199112 + 12119911 + 03)(1199112 minus 05119911 minus 04)yielding min|119911|=1 Re119892119889(119911) ge 009 so 119892119889 isin SSPR Thedecomposition of 119892119889(119911) in simple fractions leads to 119892119889(119911) =1 + 168(119911 minus 093) + 0758(119911 + 043) By using an ideal (iewith instantaneous sampling) sampler and zero-order-holddevice of period 119879 of transfer function (1 minus 119890minus119879119904)119904 on acontinuous transfer function 119892(119904) to be specified to generate119892119889(119911) one has the identity 119885((1 minus 119890minus119879119904)119892(119904)119904) = 119892119889(119911) where119885(119865(119904)) is an abbreviate usual informal notation to denote119865lowast(119904) = 119871(119891lowast(119905)) where 119871(sdot) and 119885(sdot) denote the Laplacetransform of (sdot) of argument 119904 and 119885-transform of (sdot) ofargument 119911 = 119890119879119904 and 119891lowast(119905) = suminfin119899=0 119891(119899119879)120575(119905 minus 119899119879) denotesthe pulsed function of the function119891(119905)with sampling period119879 As a result we get for a sampling period 119879 = 01 sec119892 (119904) = 119904119885minus1 ( 119911119911 minus 1 [1 + 168119911 minus 093 + 0758119911 + 043])
Note that 119892 isin SPR (and also 119892 isin SSPR sinceits relative degree is zero) is the continuous-time transferfunction which generates 119892119889(119911) under an ideal sampler andzero-order-hold device of period 119879 = 01 sec so thatmin120596isinclR119892(i120596) = min120596isinR0+119892(i120596) gt 0 Thus it is guaranteedthat the continuous-time energy measure is associated withthe forced output 119864(119905) gt 0 for all 119905 gt 0 (ie it ispositively lower-bounded for all time) for any piecewise-continuous control irrespectively of the particular controllerin operation In order to guarantee that any such a controlthat converges asymptotically to zero it is necessary to finitelyupper-bound such an energy for all timeThe above concerns
also guarantee that 119864[119896] = 119864(119896119879) ge 1205741198890 gt 0 for all integer 119896 gt0 under any piecewise-constant control 119906[119896] = 119906(119896119879) witheventual jumps at sampling instants of any period 119879 gt 0 Forsuch a control identical conclusion arises from the discrete119892119889(119911)which is also strongly strictly positive real Assume that119906[119896] = minus120593(119910[119896] 119896) and forall119896 isin Z+ with 120593 isin Φ119889 where theclass Φ119889 is defined as the set of sequences 120593(119910[119896] 119896)whichsatisfy Popovrsquos inequality 119879sum119896119895=0 119910[119896]120593(119910[119896] 119896) ge minus120574119889 forall119896 isinZ+ for some finite 120574119886 isin R+ which is identical to
= minus119879119872(1 minus ]119896+1)1 minus ] minus 119879 119896sum119895=0
120576 [119896] ge minus120574119889 forall119896 isin Z+(50)
for some real constants 119872 gt 0 ] isin (0 1) and some realsummable sequence 120576[119896] Then the class Φ119889 is definedby any real sequence 120593(119910[119896] 119896) being parameterized by anyreal constants 119872 gt 0 ] isin (0 1) and any real summablesequence 120576[119896] such that
forall119896 isin Z0+ and 120574119889 ge max(119879119872(1 minus ]) + 120576 120574) for some given120574 ge 1205741198890 where 120576 = 119879max119896isinZ+ sum119896119895=0 120576[119896]The continuous-timecontrol is generated as
where 120593[119896] isin Φ119889(119879) satisfies Popovrsquos type discrete inequal-ity which makes the feedback system globally asymptoticallystable under the theorem hypothesis Thus the discrete-timeclosed-loop system is asymptotically hyperstable In partic-ular under the additional assumptions in Theorem 11(iii)the closed-loop continuous-time system is asymptoticallyhyperstable
Mathematical Problems in Engineering 13
One of the referees has given a comment of interest on thisexample related to continuous versus discrete counterpartproperties which has motivated us to give a further intuitivediscussion on it Note the following In the continuous-timecase there are two kinds of strictly positive real transferfunctions as it is well known namely the so-called strictpositive real which can be zero (or respectively positive)at infinity frequency and which a realizable has relativedegree one (or respectively zero) In the first case the directinput-output interconnection gain 119889 is zero in a state-spacerepresentation (119860 119887 119888 119889 = 0) of the transfer function Thetransfer functions of the second case are often referred to asthe class of strong strict positive real transfer functions Thisclass is not a disjoint class of first above one of strict positivereal transfer functions since it is included in it as a potentialparticular case Note that the members of this strong classof strict positive real transfer functions have relative degreezero and a positive direct input-output interconnection gain119889 Related to this concern the Yakubovich-Kalman-Popov(refereed to the continuous-time domain) and the Szego-Kalman-Popov (referred to the discrete domain) lemmascan be invoked to compare positive realness of the transferfunctions with their state-space associated properties (in factthe properties aremutually equivalent fromboth sides)Thuswe see that in the discrete case the Szego-Kalman-Popovlemma needs for its own coherency that the input-outputinterconnection 119889-gain be nonzero since otherwise one ofthe relevant constraint in the lemma fails [13] So strictpositive realness in the discrete domain in this context istypically strong and then the relative degree of its transferfunction is zero so that there is a direct interconnectionpositive gain from the input to the output Since we assumethat the discrete system is strictly positive real it has a positiveinput-output interconnection from the input to the outputand the minimum value of the real part of its frequencyresponse is strictly positive even as the frequency tends toinfinity On the other hand note that a zero-order holdldquoconvertsrdquo any strictly proper (continuous-time) transferfunction of any relative degree into a strictly discrete oneof unity relative order while it ldquoconvertsrdquo a continuous oneof zero relative order (ie biproper) into a discrete onestill of zero-relative order and with the same input-outputinterconnection gain See (48) Therefore we deal in thisexample with a biproper strict positive real transfer functionof positive input-output interconnection gain identical to thatof its discrete-time counterpart got through a ZOH samplingand hold device
Example 14 Consider that a controller of the form 119906(119905) =minus120593(119910(119905) 119905) forall119905 isin R0+ is used for the continuous-time feed-forward transfer function (48) so that120593 isin Φ and Φ is classin Example 13 satisfying an integral type Popovrsquos inequality(21) for all 119905(ge 1199050) isin R+ and some finite 1199050 isin R+ That isthe control function belongs to a class satisfying continuous-time integral Popovrsquos inequality Then 119906(119905) rarr 0 as 119905 rarr infinand 119906(119905) and 119910(119905) are almost everywhere bounded for all timesince 119892 isin SSPR then being (strictly) stable with input-output energy measure being positively lower-bounded andalso finitely upper-bounded for all 119905 isin R+ from Popovrsquos-type
integral inequality Potential useful controllers are for somegiven controller gain 119896119906 isin R+
119906 (119905) = 0 for 119905 isin [0 1)1198961199061199052119910 (119905) for 119905 ge 1 (54)
119906 (119905) = 119896119906119890minus119905119910 (119905) for 119905 ge 0 (55)
119906 (119905) = 0 for 119905 isin [0 2)
119896119906(119905 ln2119905) 119910 (119905) for 119905 ge 2 (56)
119906 (119905) = 0 for 119905 isin [0 1)
119896119906(119890radic119905radic119905) 119910 (119905) for 119905 ge 1 (57)
119906 (119905) = 1198961199061199053119890minus119905119910 (119905) for 119905 ge 0 (58)
where Popovrsquos inequality on [0 +infin] of (49)ndash(52)is intinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus119896119906 that of (57) isintinfin
0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 and that of (58) isintinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 Since the respective integrands
are monotonically decreasing then there is some 1199050 isin R+ andsome 120574 = 120574(1199050) isin R+ such that int119905
1199050119910(119905)120593(119910(119905) 119905)119889119905 ge minus120574 for
all 119905(ge 1199050) isin R+ The obtained closed-loop systems are inall cases asymptotically hyperstable The piecewise-constantcontrols with eventual discontinuities at sampling instantsgenerated from any discrete-time controller (51) of class Φ119889of Example 13 lead also to asymptotic hyperstability of thediscretized system since 119892119889 isin SSPRExample 15 Consider the continuous-time open-loop sys-tem (2) under a real sampler of sampling period 119879 and datapicking-up nonzero duration 120576 isin (0 119879) The state and outputtrajectory solutions are
forall119896 isin Z0+ for some real sequence 120585[119896] such that 120585[119896] isin(119896119879 119896119879 + 120576) where 119892119889(119911) is the discretized transfer functionunder a sampler and zero-order-hold device subject to instan-taneous ideal sampling Now assume that the incrementalcontrol along intersample periods is generated from itspreceding sampled value as (119895119879+ 120591) = 120582119895119879+120576(119895119879+ 120591)119906[119895] forany 119895 isin Z0+ and 120591 isin [0 119879] where 120582119895119879+120576(119895119879+120591) is a truncatedfunction under the usual notation one gets
The above equations imply that if 119892119889 isin SPR then the forcedoutput 119910119891[119896] = 119910[119896] minus 119888119879119890119896119860119879119909[0] forall119896 isin Z0+ generates anassociate input-output energy measure which is strictly posi-tive for 119896 isin Z0+ since 120588 isin (0 1) for real sampling of duration120576 isin (0 119879)This is a necessary condition for asymptotic hyper-stability under an appropriate class of controllers satisfyingPopovrsquos inequality However the incremental input-outputenergy measure generated by the incremental forced output120576119888119879sum119896minus1119895=0 119890(119896minus119895)119860119879(int1198790 120582119895119879+120576(120591)119890minus119860120591119887 119889120591)119906[119895] is not guaranteedto be nonnegative in general A sufficient condition for that isthat lim sup119895rarrinfin|ess sup120591isin(0120576)(120582119895119879+120576(120591))| is sufficiently smallrelated to the amount ((1 minus 120588)120576)min120596isin[0120587] Re119892119889(119890i120596)6 Conclusions
This paper has been devoted to obtain and to prove theasymptotic hyperstability property of a linear and time-invariant continuous-time system based on some given andpreviously proved auxiliary preparatory lemmas Basicallyit is proved that the continuous-time closed-loop system isasymptotically hyperstable under the assumptions that thediscretized feed-forward transfer function is strictly positivereal and that the discretized system at sampling instants ofconstant sampling period is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsIt is assumed that the continuous-time controller belongs toa class defined in the following way (a) its values at samplinginstants are defined by a class of controllers which satisfydiscrete Popovrsquos type inequality (b) the intersample control isgenerated from its values at sampling instants with the use oftwomodulating auxiliary functionswhich are subject to someextra additional conditions on smallness and boundednessSuch conditions make the overall continuous-time controllerto satisfy continuous-type Popovrsquos inequality for all time
where 119896(120591) = max(119911 isin Z+ 119911119879 le 120591) and 119896 = 119896(119905) = max(119911 isinZ+ 119911119879 le 119905) forall119905 isin R0+ which yields (13)
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
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Mathematical Problems in Engineering 3
V[119896] [119895] =
V [119895] if 0 le 119895 le 1198960 if 119895 gt 1198960 if 119895 lt 0
forall119895 119896 isin Z0+(1)
(ix) Let 119904 and 119911 = 119890minus119879119904 the Laplace and 119911-transforms(for sampling period 119879) complex arguments The set SPR isthe set of strictly positive real continuous 119892(119904) (respectivelydiscrete) transfer functions 119892(119911) ie they are (strictly) stableie with poles in Re 119904 lt 0 (respectively |119911| lt 1) whichsatisfy Re119892(119904) gt 0 for all Re 119904 ge 0 (respectively Re119892(119911) gt 0for all |119911| ge 1) These conditions imply that Re119892(i120596) gt0 respectively Re119892(119890i120596) gt 0 forall120596 isin R [13 31 34] Thestrictly positive continuous and discrete real sets are notdistinguished at the level of notation since they are easy toidentify them according to context
(x) The so-called set SSPR is the set of (continuous ordiscrete depending on context) strongly positive real transferfunctions which are those in SPR whose real part is strictlypositive also as |119904| rarr infin It can be pointed out that Szego-Kalman-Popov Lemma (also so-called Discrete Positive RealLemma) is a discrete version of the Yakubovich-Kalman-Popov Lemma (also so-called Positive Real Lemma) whichrelates positive realness of transfer functions to associatedstate-space realization properties states that discrete transferfunction in SPR are of relative degree (namely pole-zeroexcess) equal to zero so that they are also in SSPR and bothsets are equivalent This equivalence concern between thediscrete sets SPR and SSPR does not apply to continuoustransfer functions [13 31]
3 The Continuous-Time and Discrete-TimeLinear Time-Invariant Systems and SomeAuxiliary Results
Consider an 119899th-order linear and time-invariant dynamicsystem in state-space description
where 119909(119905) isin R119899 119906(119905) isin R and 119910(119905) isin R are the statepiecewise-continuous control input and output and 119860 isinR119899times119899 119887 isin R119899 119888 isin R119899 and 119889 isin R are the matrix ofdynamics control vector output vector and input-outputinterconnection gain The auxiliary purely discrete outputand the continuous-time and sampled output are calculatedfrom (2) via the impulse responses 119892(119905) and 119892119889[119896] whichbecome
119892 (119905 minus 120591) = 119888119879119890119860(119905minus120591)119887 + 119889120575 (119905 minus 120591) forall120591 119905 (ge 120591) isin R0+119892119889 [119896 minus 119895] = int119896119879
119895119879119892 (119896119879 minus 120591) 119889120591
= int(119896minus119895)1198790
119892 ((119896 minus 119895) 119879 minus 120591) 119889120591= int119896119879119895119879(119888119879119890119860(119896119879minus120591)119887 + 119889120575 (119896119879 minus 120591)) 119889120591
= int(119896minus119895)1198790
(119888119879119890119860((119896minus119895)119879minus120591)119887 + 119889120575 ((119896 minus 119895) 119879 minus 120591)) 119889120591forall119895 (le 119896) 119896 isin Z0+
(3)
and 119892(119905minus120591) = 0 forall120591 119905(lt 120591) isin R0+ where 120575(119905minus120591) is the Diracdistribution 119906(minus119905) = 0 forall119905 isin R+and 119892119889[119896 minus 119895] = 0 forall119895(gt119896) 119896 isin Z0+ with 119892119889[0] = 119889 The initial input and outputconditions are 119906[119895] = 119910[119895] = 0 for any integer 119895 lt 0 and119910(119905) = 119906(minus119905) = (minus119905) = 0 forall119905 isin R+ in causal dynamicsystems subject to initial conditions119909(0) = 1199090 Let the controlinput be
119892119889 [119896 minus 119895])119906[119896] [119895]+ intinfinminusinfin119892 (119896119879 minus 120591) 119896119879 (120591) 119889120591 + 119910ℎ [119896]
forall119896 isin Z0+
(7)
Expressions (5)ndash(7) are derived in the appendix
4 Mathematical Problems in Engineering
Remark 1 Note that the difference between the sampledcontinuous-time sequence 119910[119896] and the purely discreteoutput sequence 119910119889[119896] is that the first one is the exactvalue of the output at sampling instants including the effectsof the intersample input ripple while the second one is thediscretized output at sampling instants in the presence of azero-order hold As a result note that 119910[119896] = 119910119889[119896] for anygiven 119896 isin Z+ if and only if int119896119879
0119892(119896119879 minus 120591)(120591)119889120591 = 0 In
particular 119910[119896] = 119910119889[119896] if (119905) = 0 forall119905 isin R0+ Causalityimplies that (6)-(7) can be rewritten as
119910 (119905) = (infinsum119895=0
119892119889 [119896 minus 119895])119906[119896] [119895]+ (int119905119896119879119892 (119905 minus 120591) 119889120591) 119906 [119896]
The following result is directLemma 2 Any given piecewise-continuous control input canbe decomposed into purely discrete-time control plus incremen-tal intersample period ones as follows
through piecewise-continuous functions 120582 120590 cup119896isinZ0+[119896119879 (119896 +1)119879) rarr R defined by
120582 (119905) = (119905)119906 [119896] if 119906 [119896] = 00 if 119906 [119896] = 0
120590 (119905) = 0 if 119906 [119896] = 0 (119905) if 119906 [119896] = 0
(10)
The purely auxiliary input-output and true ones energymeasures on the discrete-time interval [0 119896119879] as well as thecontinuous-time ones on [0 119905] forall119905 isin [119896119879 (119896 + 1)119879) and forall119896 isinZ0+ are respectively from (5)ndash(7) and (9)-(10) given by
Remark 3 Since the control input is piecewise-continuousthe truncated input 119906119905(120591) for 120591 isin [0 119905] has Fourier transformsfor any 119905 isin R0+ so that the corresponding truncated output
119910119905(120591) for 120591 isin [0 119905] and also the energy measure have alsoFourier transforms for any 119905 isin R0+ This follows from the factthat these signals are calculated via truncated functionswhich
Mathematical Problems in Engineering 5
are then zero at infinity so that they are square-integrable andtheir sequences of sampled values of sampling period 119879 aresquare-summable as a result
The subsequent two simple auxiliary lemmas will be thenused in the next section to establish the main result
Lemma 4 Assume that there exist constants 1205741198890 (120574119889 gt 1205741198890) isinR+ such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some1198960 isin Z0+ Then there exist 1205740 (120574 gt 1205740) isin R+ such that 1205740 le119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin) if1205740 minus 120574119889 le 119864119889 (119896119879 119905)
= (119864 [119896] minus 119864119889 [119896]) + 119864 (119896119879 119896119879 + 120591)= 119864 (119905) minus 119864119889 [119896] = 119864119889 (119896119879 119905) le 120574 minus 1205741198890
119879 (119910 [119895] 119906 [119895] minus 119910119889 [119895] 119906119889 [119895])
(16)
forall119905 isin (119896119879 (119896 + 1)119879) forall119896 isin Z0+ so that (14) implies that 1205740 le119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin) provided that 1205741198890 le 119864119889[119896] le120574119889 forall119896(gt 1198960) isin Z+ for 1198960 isin Z0+Lemma 5 Assume that 120593(119909 119905) is subadditive for all 119909 isin Rie 120593(119909 + 119910 119905) le 120593(119909 119905) + 120593(119910 119905) forall119909 119910 isin R forall119905 isin R0+Then 120593(119910(119905) 119905) le 120593(119910[119896] 119896) + 120593(119910(119905) minus 119910[119896] 119905 119896119879) andas a result (119905) ge 120593(119910(119905) minus 119910[119896] 119905 119896119879) and 120593(119910[119896] 119896) geminus120593(minus119910[119896] 119896) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+
If 120593(119909 119905) is superadditive for all 119909 isin R ie 120593(119909 +119910 119905) ge 120593(119909 119905) + 120593(119910 119905) forall119909 119910 isin R forall119905 isin R0+ then120593(119910(119905) 119905) ge 120593(119910[119896] 119896119879) + 120593(119910(119905) minus 119910[119896] 119905 119896119879) as a result(119905) le 120593(119910(119905) minus 119910[119896] 119905 119896119879) and 120593(119910[119896]) le minus120593(minus119910[119896]) forall119905 isin[119896119879 (119896 + 1)119879) forall119896 isin Z0+Proof From the first identity in (9) if 120593(119909 119905) is subadditivethenminus120593 (119910 (119905)) = 119906 (119905) = 119906 [119896] + (119905)
ge minus120593 (119910 [119896] + (119910 (119905) minus 119910 [119896]))ge minus120593 (119910 [119896]) minus 120593 (119910 (119905) minus 119910 [119896])
= minus120593 (119910 [119896]) minus 120593 (119910 (119905) + (minus119910 [119896]))ge minus120593 (119910 [119896]) minus 120593 (119910 (119905)) minus 120593 (minus119910 [119896])
Definition 6 (Popovrsquos inequality) A continuous-time feed-back nonlinear and eventually time-varying continuous-timecontroller 119906(119905) = minus120593(119910(119905) 119905) and forall119905 isin R0+ where 120593 R0+ timesR0+ rarr R0+ is in the class Φ denoted as 120593 isin Φ is said tosatisfy Popovrsquos type integral inequality if for some 1205740 isin R+one has
Definition 7 (discrete-time Popovrsquos inequality) A discrete-time feedback nonlinear and eventually time-varyingdiscrete-time controller 119906(119905) = 119906[119896] = 119906(119896119879) =minus120593(119910(119896119879) 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) and forall119896 isin Z0+where 120593 R0+ times Z0+ rarr R0+ is in the class Φ119889(119879) ofsampling period 119879 isin R+ denoted as 120593 isin Φ119889(119879) is said tosatisfy a Popovrsquos type discrete-time inequality if for some120574119889 isin R+ one has119896sum119895=0
119910 (119896119879) 120593 (119896119879) ge minus120574119889119879 forall119896 isin Z0+ (22)
I D Landau refers to controllers satisfying a Popovrsquostype inequality (21) (or respectively (22)) as hyperstable
controllers of class Φ (or respectively of class Φ119889(119879))[13] bearing in mind that if any controller of such a class iscoupled to a linear time-invariant forward system of transferfunction being continuous-time (or respectively discrete-time) strictly positive real then the overall closed-loop systemis asymptotically hyperstable namely globally asymptoticallystable for any arbitrary controller belonging to the respectiveclass It can be pointed out that the above comments referredto discrete-time systems and in particular Definition 7 arealso applicable to digital systems in the same way as theyare applicable to discrete-time systems ie those which arefully described in the discrete domainwithout having specificlinks to a discretization process on a certain continuous-timesystem
It can be pointed out that Definition 7 can be alsoestablished for a digital system not being related to thetime discretization of a continuous-time system in terms ofPopovrsquos inequality of the following form
119896sum119895=0
119910 [119896] 120593 [119896] ge minus120574119889119886 forall119896 isin Z0+ (23)
for some constant 120574119889119886 isin R+ for any digital controller ofclass Φ119889 In this context the sampling period either can benonrelevantmdashthen nonmade notational explicit in the classΦ119889mdashor can even have nonsense since the inequality isapplied to a certain discrete sequence
The following result obtains useful expressions for the dis-cretized input-output energymeasure under some conditionsconcerning the subadditivity constraints on the controllers ofclass Φ119889(119879)The result is also a preparatory one for the nextLemma 9 which addresses the positivity and boundedness ofthe input-output energy measure at sampling instants
Lemma 8 The following properties are fulfilled(i) The subsequent formulas hold
(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119895 (le 119896) 119896 isin Z0+
(26)
(ii) Assume that 120593 isin Φ119889(119879) and that 120593(119909 119896119879) issubadditive for all 119909 isin R ie 120593(119909 + 119910 119896119879) le 120593(119909 119896119879) +120593(119910 119896119879) forall119909 119910 isin R forall119896 isin Z0+ Then the subsequentrelations are true if the sequence 120576[119896] is defined by 120576[119896] =120582[119896] + 120585[119896] forall119896 isin Z0+119910119889ℎ [119896] + intinfin
0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591
which yields (24) and leads to (25) Now define the sequence120576[119896] by 120576[119896] = 120582[119896] + 120585[119896] and 120585[119896][119895] = int(119895+1)119879119895119879(120582(119895119879 +120591)119892((119896 minus 119895)119879 minus 120591) minus 120582[119895]119892[(119896 minus 119895)119879])119889120591 forall119896 isin Z0+ and note
that
minus119910 [119896] = minus (119910119889 [119896] + (119910 [119896] minus 119910119889 [119896]))= [[infinsum119895=0
(int(119895+1)119879119895119879
(1 + 120576[119896] [119895]) 119892 [(119896 minus 119895) 119879])119889120591]]sdot 120593[119896] ([119910119889 [119895] + (119910 [119895] minus 119910119889 [119895]) 119895])minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 minus 119910ℎ [119896]
forall119896 isin Z0+
(31)
and then
minus (119910 [119896] minus 119910119889 [119896] minus 119910ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591)
The subsequent result establishes lower and upper-bounds for the discretized input-output energy measuresbased on the formulas obtained in Lemma 8 and strictpositive realness of the discrete-time transfer function ofthe feed-forward linear and time-invariant system Theresult invokes the boundedness and the smallness of theparameters which define the calculation of the intersam-ple control input from their preceding values at samplinginstants Note that the control deviation in the intersampleperiods related to the sampling time instants has to be suffi-ciently moderate enough so that the hyperstability propertyis kept from the discrete-time system to the continuous-time one In particular the positivity and boundedness ofthe energy measure at the sampling instants is guaranteedif sup119905isinR0+ |120590(119905)| is small enough and Φ119889(119879) is subaddi-tive
Mathematical Problems in Engineering 9
Lemma 9 Assume that 119892119889 isin 119878119875119877 and that 120593 isin Φ119889(119879)Then the following properties hold(i)The input-output energy systemmeasure of the auxiliary
discretized system satisfies
0 lt 1205740119889119886 le 1198792120587 min120596isinR0+
119879119910119889 [119896] 119906119889 [119896] le 120574119889119886forall119896 (gt 1198960) isin Z+
(34)
for some 1205740119889119886 120574119889119886(gt 1205740119889119886) isin R+ and for any nonidentically zerocontrols and zero initial conditions ie for any forced solution
(ii) Assume in addition that the class Φ119889(119879) consistsof subadditive functions Assume also that sup119905isinR0+ |120590(119905)| issufficiently small inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infinwhere 120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ and that furthermore
for some 120574119889119886 isin R+ from the first assumption of the theoremand 119906119889[119896] = minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ since Popovrsquosinequality (22) is fulfilled by any given discrete-time feedbackcontroller 120593 isin Φ119889(119879) generating a feedback control 119906119889[119896] =minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ On the other hand the initialconditions can be neglected since the assumption119892119889 isin SPRimplying that the discrete-time transfer function is (strictly)stable which makes their values irrelevant for purposes ofstability analysisThen the discrete frequency response being
applicable to piecewise-constant inputs in-between any twoconsecutive sampling instants is given by
since the subsequence 119892[minus119899] = 0 forall119899 isin Z+ and thetransfer frequency response of a sampling and zero-order-hold operator 119885 of period 119879 defined by 119885(119879 V)(119905) = V(119896119879) =V[119896] for any given V R0+ rarr R and all 119905 isin [119896119879 (119896 + 1)119879)119896 isin Z0+ is given by 119885(119879 i120596) = (1 minus 119890minusi119879120596)i120596 [31 35]It turns out that |119885(119879 i120596)| = |(1 minus 119890minusi119879120596)i120596| lt 1 forall120596( =0) isin R and |119885(119879 i0)| = 1 Thus the auxiliary discretizedsystem satisfies the subsequent equivalence relation in thefrequency domain to the discrete-time relation (11) It turnsout that Fourier transforms exist in the impulse responsesand truncated control and auxiliary functions of (11)ndash(13)since the truncated functions are square-integrable in clR =[minusinfin +infin] By using the discrete Parsevalrsquos theorem underzero initial conditions one has
The lower-bounding constraint of (37) follows by using(39) and the discrete Parsevalrsquos theorem for the equivalencebetween the input-output energy measures from the fre-quency domain to the discrete-time domain
119864119889 [119896] ge 1198792120587 min120596isinR0+
1199062119889 [119896]) ge 1205740119889forall119896 isin Z0+
(40)
from the assumption 119892119889 isin SPR iemin120596isinR0+ Re119892119889(119890minusi120596119879) ge 1205740119889119886 gt 0 forall1198960 119896 (119896 ge 1198960) isin Z+since
(a) strictly positive real discrete transfer functions havezero relative degree (ie an identical number of polesand zeros) from the Discrete Positive Real Lemma(Szego-Kalman-Popov Lemma) [13 31 34] so thatthe real parts of their frequency hodographs in theargument 119890minusi120596119879 are positively lower-bounded
10 Mathematical Problems in Engineering
(b) Im119892119889(119890minusi120596119879) = minus Im119892119889(119890i120596119879) forall120596 isin R0+ so that theirintegrals in the argument 120596 on frequency intervals[minus119896120587 119896120587] for any 119896 isin Z+ are null
Property (i) has been proved To prove Property (ii) notefrom (28) [Lemma 8(ii)] that
119892 [(119896 minus 119895) 119879])119889120591]]sdot (120593[ℓ] (119910119889 [119895] 119895) + 120593[ℓ] (minus119910119889 [119895] 119895))sdot (120593[ℓ] (119910119889 [ℓ] 119895) + 120593[ℓ] (minus119910119889 [ℓ] ℓ)) minus 119910119889ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 le 120574119889 = 119870120574119889119886
forall119896 isin Z+
(41)
for some 119870 isin R+ provided that sup119905isinR0+ |120590(119905)| is sufficientlysmall inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infin where120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ since 119892(119905) is bounded andconverges exponentially to zero since 119892119889 isin SPR and thenconvergent (that is strictly stable in the discrete context)119910119889ℎ[119896] rarr 0 and it is a bounded sequence
The next result which is a preparatory result to thenestablish the main asymptotic hyperstability result addressesthe relevant property that the input-output measure 119864(119905) ofthe continuous-time system is positively lower-bounded andfinitely upper-bounded for all time under the conditions ofLemmas 4 and 9
Lemma 10 If 120582 and 120590 are sufficiently small then
1205740 minus 120574119889 le 119864 (119905) minus 119864119889 [119896]le 1 + 12058222120587 intinfin
provided that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+Proof One gets the following from direct calculations byusing the continuous-time control laws (9)-(10) and (16) if120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)|
minus 12058222120587 intinfin
minusinfin119892 (i120596) 1003816100381610038161003816[119896] (i120596)10038161003816100381610038162 119889120596 minus 120590(2120582 sup
0le119895le119896
1199062[119895] + 120590)int1198961198790
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591 le 119864 (119905) minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119910 (i120596)
sdot (minusi120596) 119889120596 minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119892 (i120596) | (i120596)|2 119889120596 minus 119864119889 (119896119879 119905)
Now the use of (44) in Lemma 4 (14) implies thatLemma 9(ii) (36) holds implying the existence of 1205740 (120574 gt1205740) isin R+ such that 1205740 le 119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin)provided that 1205741198890 le 119864119889[119896] le 120574119889 for 1205741198890 (120574119889 gt 1205741198890) isin R+ since1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isin Z0+ if1205740 minus 120574119889 le minus 120582
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591le 120574 minus 1205741198890 forall119896 (gt 1198960) isin Z+ forall120591 isin (0 119879)
(45)
if 120582 and 120590 are sufficiently small such that (43) holds leadingto 1205740 gt 0 for any given positive 1205741198890 and 120574119889 gt 1205741198890
The main first result follows
Theorem 11 Consider the dynamic system (2) subject to acontrol
forall119896 isin Z0+ where 120593 [119896] isin Φ119889 (119879) (46)
Assume that(1) 120593(119910[119896] 119896119879) is in the class Φ119889(119879) (that is120593(119910[119896] 119896119879) isin Φ119889(119879)) defined by the discrete Popovrsquos
type sum inequality sum119896ℓ=0 119910119889[ℓ]119906[ℓ] ge minus120574119889119879 for some 120574119889 isin Rforall119896 isin Z0+(2) 119892119889 isin 119878119875119877 there exist constants 1205741198890 (120574119889 gt 1205741198890) isin R+
such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isinZ0+
(3) 120593(119909 119896119879) is subadditive for all 119909 isin R forall119896 isin Z0+(4) 120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)| are
119895119879(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)
minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119896 isin Z0+(47)
Then the following properties hold(i) The discrete closed-loop system is asymptotically hyper-
stable(ii) 119906(119905) rarr 0 as 119905 rarr infin and the control and output are
bounded for all time
(iii) Assume furthermore that 120590(119905) = 120593(119910(119905) 119905) minus (1 +120582(119905))120593(119910[119896] 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+ 119906(119905) =minus120593(119910(119905) 119905) and forall119905 isin R0+ and that the class Φ satisfiesPopovrsquos type integral inequality for all time and results in theclass Φ119889(119896119879) at sampling instantsThen the continuous- timeclosed-loop system is asymptotically hyperstable
Proof From Lemma 9 via assumptions (1)ndash(5) constraints(36) hold so that the input- output energymeasure associatedwith the forced solution is positively lower-bounded andfinitely upper-bounded for all sampling instants 119905 = 119896119879forall119896 isin Z+ Since the discrete positive transfer function isstrictly positive real then it is also strictly stable in the discretecontext (ie convergent [29 36]) so that the homogeneousresponse is irrelevant for stability analysis purposes Thus120593[119896] isin Φ119889 119906[119896] rarr 0 and 119906[119896] and 119910[119896] arebounded for all time for any given bounded initial conditionsIn addition one has from Lemma 10 that (42) holds andthe forced input-output energy measure is positively lower-bounded and finitely upper-bound for all 119905 isin R+ Then119906(119905) rarr 0 as 119905 rarr infin and 119906(119905) and 119910(119905) are bounded for alltime for any given bounded initial conditions Properties [(i)-(ii)] have been proved Property (iii) follows by equalizing119906(119905) = minus120593(119910(119905) 119905) = (1 + 120582(119905))119906[119896] + 120590(119905) and 120593 isin Φ fromthe fact that minusint119905
0119906(120591)119910(120591)119889120591 le 1205740
Note that the use of Fourier transforms is useful tocharacterize the error of the input-output energy measure inthe frequency domain along the intersample periods relatedto its values at the preceding sampling instants This toolis addressed in Lemma 10 which is used in the proof ofTheorem 11 It allows to keep easily the characterizationof the frequency response of the continuous-time transferfunction to evaluate the input-output energy measure in theintersample period and to calculate the minimum value ofthe real part of such a transfer function to guarantee thepositivity and boundedness of such an energy Note that thecharacterization of the input-output energy measure of thefeed-forward linear loop evaluated in the frequency domaintogether with Popovrsquos inequalities of the class of hyperstablecontrollers is relevant to the hyperstability proofs of theclosed-loop system
5 Examples
Example 12 The positive realness of a discretized transferfunction depends not only on the continuous-time param-eters but on the sampling period as well Therefore thepositive realness of the continuous-time system does notguarantee that of any discretized counterpart irrespective ofthe sampling period and conversely For instance the transferfunction 119892(119904) = 1(119904 + 119886) is strictly positive real for 119886 gt 0 butit is not strongly strictly positive real since Re119892(119894120596) rarr 0 as120596 rarr plusmninfin Its discretized version without the concourse ofan ideal that is instantaneous sampler and zero-order holdbecomes 119904 119892119889(119911) = 119911(119911 minus 119890minus119886119879) which is strongly positivereal for 119879 isin (0infin) and as the sampling period 119879 rarr infin since119892119889(119911) rarr 1 as 119879 rarr infin However 119892119889(119911) = 119911(119911 minus 1) as 119879 rarr 0
12 Mathematical Problems in Engineering
Since it has a critically stable pole in the discrete domain it ispositive real although it is not strictly positive real
Example 13 Consider the discrete transfer function 119892119889(119911) =(1199112 + 1198891119911 + 1198892)(1199112 minus 1198861119911 minus 1198862) with real numerator anddenominator coefficients Its strict positive realness requiresas necessary conditions its convergence (ie strict stabilityin the discrete sense) so that its poles have to be within thecomplex open unit circle as well as the convergence of itsinverse so that its zeros have also to lie inside the open unitcircle and furthermore its relative degree (ie its pole-zeroexcess has to be zero) Note that the last condition is notneeded for strict (nonstrong) positive realness of continuoustransfer functions where their positive real part can convergeto zero as |119904| rarr infin By using the homographic bilineartransformation 119911 = (1 + 119904)(1 minus 119904) which transforms theunit circle in the left-hand-side complex plane to get theauxiliary continuous transfer function 119892119886(119904) = 119892119889(119911 = (1 +119904)(1 minus 119904)) it is found in [13] that the denominator andnumerator polynomials of 119892119889(119911) are subject to 1198862 minus 1198861 lt 11198861 + 1198862 lt 1 and 1198862 gt minus1 Take the particular values 1198861 = 05and 1198862 = 04 leading to the values of poles 1199111 = 05 and1199112 = 04 and the coefficients of the numerator polynomialare fixed to 1198891 = 12 1198892 = 03 Then we fix a discretetransfer function 119892119889(119911) = (1199112 + 12119911 + 03)(1199112 minus 05119911 minus 04)yielding min|119911|=1 Re119892119889(119911) ge 009 so 119892119889 isin SSPR Thedecomposition of 119892119889(119911) in simple fractions leads to 119892119889(119911) =1 + 168(119911 minus 093) + 0758(119911 + 043) By using an ideal (iewith instantaneous sampling) sampler and zero-order-holddevice of period 119879 of transfer function (1 minus 119890minus119879119904)119904 on acontinuous transfer function 119892(119904) to be specified to generate119892119889(119911) one has the identity 119885((1 minus 119890minus119879119904)119892(119904)119904) = 119892119889(119911) where119885(119865(119904)) is an abbreviate usual informal notation to denote119865lowast(119904) = 119871(119891lowast(119905)) where 119871(sdot) and 119885(sdot) denote the Laplacetransform of (sdot) of argument 119904 and 119885-transform of (sdot) ofargument 119911 = 119890119879119904 and 119891lowast(119905) = suminfin119899=0 119891(119899119879)120575(119905 minus 119899119879) denotesthe pulsed function of the function119891(119905)with sampling period119879 As a result we get for a sampling period 119879 = 01 sec119892 (119904) = 119904119885minus1 ( 119911119911 minus 1 [1 + 168119911 minus 093 + 0758119911 + 043])
Note that 119892 isin SPR (and also 119892 isin SSPR sinceits relative degree is zero) is the continuous-time transferfunction which generates 119892119889(119911) under an ideal sampler andzero-order-hold device of period 119879 = 01 sec so thatmin120596isinclR119892(i120596) = min120596isinR0+119892(i120596) gt 0 Thus it is guaranteedthat the continuous-time energy measure is associated withthe forced output 119864(119905) gt 0 for all 119905 gt 0 (ie it ispositively lower-bounded for all time) for any piecewise-continuous control irrespectively of the particular controllerin operation In order to guarantee that any such a controlthat converges asymptotically to zero it is necessary to finitelyupper-bound such an energy for all timeThe above concerns
also guarantee that 119864[119896] = 119864(119896119879) ge 1205741198890 gt 0 for all integer 119896 gt0 under any piecewise-constant control 119906[119896] = 119906(119896119879) witheventual jumps at sampling instants of any period 119879 gt 0 Forsuch a control identical conclusion arises from the discrete119892119889(119911)which is also strongly strictly positive real Assume that119906[119896] = minus120593(119910[119896] 119896) and forall119896 isin Z+ with 120593 isin Φ119889 where theclass Φ119889 is defined as the set of sequences 120593(119910[119896] 119896)whichsatisfy Popovrsquos inequality 119879sum119896119895=0 119910[119896]120593(119910[119896] 119896) ge minus120574119889 forall119896 isinZ+ for some finite 120574119886 isin R+ which is identical to
= minus119879119872(1 minus ]119896+1)1 minus ] minus 119879 119896sum119895=0
120576 [119896] ge minus120574119889 forall119896 isin Z+(50)
for some real constants 119872 gt 0 ] isin (0 1) and some realsummable sequence 120576[119896] Then the class Φ119889 is definedby any real sequence 120593(119910[119896] 119896) being parameterized by anyreal constants 119872 gt 0 ] isin (0 1) and any real summablesequence 120576[119896] such that
forall119896 isin Z0+ and 120574119889 ge max(119879119872(1 minus ]) + 120576 120574) for some given120574 ge 1205741198890 where 120576 = 119879max119896isinZ+ sum119896119895=0 120576[119896]The continuous-timecontrol is generated as
where 120593[119896] isin Φ119889(119879) satisfies Popovrsquos type discrete inequal-ity which makes the feedback system globally asymptoticallystable under the theorem hypothesis Thus the discrete-timeclosed-loop system is asymptotically hyperstable In partic-ular under the additional assumptions in Theorem 11(iii)the closed-loop continuous-time system is asymptoticallyhyperstable
Mathematical Problems in Engineering 13
One of the referees has given a comment of interest on thisexample related to continuous versus discrete counterpartproperties which has motivated us to give a further intuitivediscussion on it Note the following In the continuous-timecase there are two kinds of strictly positive real transferfunctions as it is well known namely the so-called strictpositive real which can be zero (or respectively positive)at infinity frequency and which a realizable has relativedegree one (or respectively zero) In the first case the directinput-output interconnection gain 119889 is zero in a state-spacerepresentation (119860 119887 119888 119889 = 0) of the transfer function Thetransfer functions of the second case are often referred to asthe class of strong strict positive real transfer functions Thisclass is not a disjoint class of first above one of strict positivereal transfer functions since it is included in it as a potentialparticular case Note that the members of this strong classof strict positive real transfer functions have relative degreezero and a positive direct input-output interconnection gain119889 Related to this concern the Yakubovich-Kalman-Popov(refereed to the continuous-time domain) and the Szego-Kalman-Popov (referred to the discrete domain) lemmascan be invoked to compare positive realness of the transferfunctions with their state-space associated properties (in factthe properties aremutually equivalent fromboth sides)Thuswe see that in the discrete case the Szego-Kalman-Popovlemma needs for its own coherency that the input-outputinterconnection 119889-gain be nonzero since otherwise one ofthe relevant constraint in the lemma fails [13] So strictpositive realness in the discrete domain in this context istypically strong and then the relative degree of its transferfunction is zero so that there is a direct interconnectionpositive gain from the input to the output Since we assumethat the discrete system is strictly positive real it has a positiveinput-output interconnection from the input to the outputand the minimum value of the real part of its frequencyresponse is strictly positive even as the frequency tends toinfinity On the other hand note that a zero-order holdldquoconvertsrdquo any strictly proper (continuous-time) transferfunction of any relative degree into a strictly discrete oneof unity relative order while it ldquoconvertsrdquo a continuous oneof zero relative order (ie biproper) into a discrete onestill of zero-relative order and with the same input-outputinterconnection gain See (48) Therefore we deal in thisexample with a biproper strict positive real transfer functionof positive input-output interconnection gain identical to thatof its discrete-time counterpart got through a ZOH samplingand hold device
Example 14 Consider that a controller of the form 119906(119905) =minus120593(119910(119905) 119905) forall119905 isin R0+ is used for the continuous-time feed-forward transfer function (48) so that120593 isin Φ and Φ is classin Example 13 satisfying an integral type Popovrsquos inequality(21) for all 119905(ge 1199050) isin R+ and some finite 1199050 isin R+ That isthe control function belongs to a class satisfying continuous-time integral Popovrsquos inequality Then 119906(119905) rarr 0 as 119905 rarr infinand 119906(119905) and 119910(119905) are almost everywhere bounded for all timesince 119892 isin SSPR then being (strictly) stable with input-output energy measure being positively lower-bounded andalso finitely upper-bounded for all 119905 isin R+ from Popovrsquos-type
integral inequality Potential useful controllers are for somegiven controller gain 119896119906 isin R+
119906 (119905) = 0 for 119905 isin [0 1)1198961199061199052119910 (119905) for 119905 ge 1 (54)
119906 (119905) = 119896119906119890minus119905119910 (119905) for 119905 ge 0 (55)
119906 (119905) = 0 for 119905 isin [0 2)
119896119906(119905 ln2119905) 119910 (119905) for 119905 ge 2 (56)
119906 (119905) = 0 for 119905 isin [0 1)
119896119906(119890radic119905radic119905) 119910 (119905) for 119905 ge 1 (57)
119906 (119905) = 1198961199061199053119890minus119905119910 (119905) for 119905 ge 0 (58)
where Popovrsquos inequality on [0 +infin] of (49)ndash(52)is intinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus119896119906 that of (57) isintinfin
0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 and that of (58) isintinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 Since the respective integrands
are monotonically decreasing then there is some 1199050 isin R+ andsome 120574 = 120574(1199050) isin R+ such that int119905
1199050119910(119905)120593(119910(119905) 119905)119889119905 ge minus120574 for
all 119905(ge 1199050) isin R+ The obtained closed-loop systems are inall cases asymptotically hyperstable The piecewise-constantcontrols with eventual discontinuities at sampling instantsgenerated from any discrete-time controller (51) of class Φ119889of Example 13 lead also to asymptotic hyperstability of thediscretized system since 119892119889 isin SSPRExample 15 Consider the continuous-time open-loop sys-tem (2) under a real sampler of sampling period 119879 and datapicking-up nonzero duration 120576 isin (0 119879) The state and outputtrajectory solutions are
forall119896 isin Z0+ for some real sequence 120585[119896] such that 120585[119896] isin(119896119879 119896119879 + 120576) where 119892119889(119911) is the discretized transfer functionunder a sampler and zero-order-hold device subject to instan-taneous ideal sampling Now assume that the incrementalcontrol along intersample periods is generated from itspreceding sampled value as (119895119879+ 120591) = 120582119895119879+120576(119895119879+ 120591)119906[119895] forany 119895 isin Z0+ and 120591 isin [0 119879] where 120582119895119879+120576(119895119879+120591) is a truncatedfunction under the usual notation one gets
The above equations imply that if 119892119889 isin SPR then the forcedoutput 119910119891[119896] = 119910[119896] minus 119888119879119890119896119860119879119909[0] forall119896 isin Z0+ generates anassociate input-output energy measure which is strictly posi-tive for 119896 isin Z0+ since 120588 isin (0 1) for real sampling of duration120576 isin (0 119879)This is a necessary condition for asymptotic hyper-stability under an appropriate class of controllers satisfyingPopovrsquos inequality However the incremental input-outputenergy measure generated by the incremental forced output120576119888119879sum119896minus1119895=0 119890(119896minus119895)119860119879(int1198790 120582119895119879+120576(120591)119890minus119860120591119887 119889120591)119906[119895] is not guaranteedto be nonnegative in general A sufficient condition for that isthat lim sup119895rarrinfin|ess sup120591isin(0120576)(120582119895119879+120576(120591))| is sufficiently smallrelated to the amount ((1 minus 120588)120576)min120596isin[0120587] Re119892119889(119890i120596)6 Conclusions
This paper has been devoted to obtain and to prove theasymptotic hyperstability property of a linear and time-invariant continuous-time system based on some given andpreviously proved auxiliary preparatory lemmas Basicallyit is proved that the continuous-time closed-loop system isasymptotically hyperstable under the assumptions that thediscretized feed-forward transfer function is strictly positivereal and that the discretized system at sampling instants ofconstant sampling period is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsIt is assumed that the continuous-time controller belongs toa class defined in the following way (a) its values at samplinginstants are defined by a class of controllers which satisfydiscrete Popovrsquos type inequality (b) the intersample control isgenerated from its values at sampling instants with the use oftwomodulating auxiliary functionswhich are subject to someextra additional conditions on smallness and boundednessSuch conditions make the overall continuous-time controllerto satisfy continuous-type Popovrsquos inequality for all time
where 119896(120591) = max(119911 isin Z+ 119911119879 le 120591) and 119896 = 119896(119905) = max(119911 isinZ+ 119911119879 le 119905) forall119905 isin R0+ which yields (13)
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
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4 Mathematical Problems in Engineering
Remark 1 Note that the difference between the sampledcontinuous-time sequence 119910[119896] and the purely discreteoutput sequence 119910119889[119896] is that the first one is the exactvalue of the output at sampling instants including the effectsof the intersample input ripple while the second one is thediscretized output at sampling instants in the presence of azero-order hold As a result note that 119910[119896] = 119910119889[119896] for anygiven 119896 isin Z+ if and only if int119896119879
0119892(119896119879 minus 120591)(120591)119889120591 = 0 In
particular 119910[119896] = 119910119889[119896] if (119905) = 0 forall119905 isin R0+ Causalityimplies that (6)-(7) can be rewritten as
119910 (119905) = (infinsum119895=0
119892119889 [119896 minus 119895])119906[119896] [119895]+ (int119905119896119879119892 (119905 minus 120591) 119889120591) 119906 [119896]
The following result is directLemma 2 Any given piecewise-continuous control input canbe decomposed into purely discrete-time control plus incremen-tal intersample period ones as follows
through piecewise-continuous functions 120582 120590 cup119896isinZ0+[119896119879 (119896 +1)119879) rarr R defined by
120582 (119905) = (119905)119906 [119896] if 119906 [119896] = 00 if 119906 [119896] = 0
120590 (119905) = 0 if 119906 [119896] = 0 (119905) if 119906 [119896] = 0
(10)
The purely auxiliary input-output and true ones energymeasures on the discrete-time interval [0 119896119879] as well as thecontinuous-time ones on [0 119905] forall119905 isin [119896119879 (119896 + 1)119879) and forall119896 isinZ0+ are respectively from (5)ndash(7) and (9)-(10) given by
Remark 3 Since the control input is piecewise-continuousthe truncated input 119906119905(120591) for 120591 isin [0 119905] has Fourier transformsfor any 119905 isin R0+ so that the corresponding truncated output
119910119905(120591) for 120591 isin [0 119905] and also the energy measure have alsoFourier transforms for any 119905 isin R0+ This follows from the factthat these signals are calculated via truncated functionswhich
Mathematical Problems in Engineering 5
are then zero at infinity so that they are square-integrable andtheir sequences of sampled values of sampling period 119879 aresquare-summable as a result
The subsequent two simple auxiliary lemmas will be thenused in the next section to establish the main result
Lemma 4 Assume that there exist constants 1205741198890 (120574119889 gt 1205741198890) isinR+ such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some1198960 isin Z0+ Then there exist 1205740 (120574 gt 1205740) isin R+ such that 1205740 le119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin) if1205740 minus 120574119889 le 119864119889 (119896119879 119905)
= (119864 [119896] minus 119864119889 [119896]) + 119864 (119896119879 119896119879 + 120591)= 119864 (119905) minus 119864119889 [119896] = 119864119889 (119896119879 119905) le 120574 minus 1205741198890
119879 (119910 [119895] 119906 [119895] minus 119910119889 [119895] 119906119889 [119895])
(16)
forall119905 isin (119896119879 (119896 + 1)119879) forall119896 isin Z0+ so that (14) implies that 1205740 le119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin) provided that 1205741198890 le 119864119889[119896] le120574119889 forall119896(gt 1198960) isin Z+ for 1198960 isin Z0+Lemma 5 Assume that 120593(119909 119905) is subadditive for all 119909 isin Rie 120593(119909 + 119910 119905) le 120593(119909 119905) + 120593(119910 119905) forall119909 119910 isin R forall119905 isin R0+Then 120593(119910(119905) 119905) le 120593(119910[119896] 119896) + 120593(119910(119905) minus 119910[119896] 119905 119896119879) andas a result (119905) ge 120593(119910(119905) minus 119910[119896] 119905 119896119879) and 120593(119910[119896] 119896) geminus120593(minus119910[119896] 119896) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+
If 120593(119909 119905) is superadditive for all 119909 isin R ie 120593(119909 +119910 119905) ge 120593(119909 119905) + 120593(119910 119905) forall119909 119910 isin R forall119905 isin R0+ then120593(119910(119905) 119905) ge 120593(119910[119896] 119896119879) + 120593(119910(119905) minus 119910[119896] 119905 119896119879) as a result(119905) le 120593(119910(119905) minus 119910[119896] 119905 119896119879) and 120593(119910[119896]) le minus120593(minus119910[119896]) forall119905 isin[119896119879 (119896 + 1)119879) forall119896 isin Z0+Proof From the first identity in (9) if 120593(119909 119905) is subadditivethenminus120593 (119910 (119905)) = 119906 (119905) = 119906 [119896] + (119905)
ge minus120593 (119910 [119896] + (119910 (119905) minus 119910 [119896]))ge minus120593 (119910 [119896]) minus 120593 (119910 (119905) minus 119910 [119896])
= minus120593 (119910 [119896]) minus 120593 (119910 (119905) + (minus119910 [119896]))ge minus120593 (119910 [119896]) minus 120593 (119910 (119905)) minus 120593 (minus119910 [119896])
Definition 6 (Popovrsquos inequality) A continuous-time feed-back nonlinear and eventually time-varying continuous-timecontroller 119906(119905) = minus120593(119910(119905) 119905) and forall119905 isin R0+ where 120593 R0+ timesR0+ rarr R0+ is in the class Φ denoted as 120593 isin Φ is said tosatisfy Popovrsquos type integral inequality if for some 1205740 isin R+one has
Definition 7 (discrete-time Popovrsquos inequality) A discrete-time feedback nonlinear and eventually time-varyingdiscrete-time controller 119906(119905) = 119906[119896] = 119906(119896119879) =minus120593(119910(119896119879) 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) and forall119896 isin Z0+where 120593 R0+ times Z0+ rarr R0+ is in the class Φ119889(119879) ofsampling period 119879 isin R+ denoted as 120593 isin Φ119889(119879) is said tosatisfy a Popovrsquos type discrete-time inequality if for some120574119889 isin R+ one has119896sum119895=0
119910 (119896119879) 120593 (119896119879) ge minus120574119889119879 forall119896 isin Z0+ (22)
I D Landau refers to controllers satisfying a Popovrsquostype inequality (21) (or respectively (22)) as hyperstable
controllers of class Φ (or respectively of class Φ119889(119879))[13] bearing in mind that if any controller of such a class iscoupled to a linear time-invariant forward system of transferfunction being continuous-time (or respectively discrete-time) strictly positive real then the overall closed-loop systemis asymptotically hyperstable namely globally asymptoticallystable for any arbitrary controller belonging to the respectiveclass It can be pointed out that the above comments referredto discrete-time systems and in particular Definition 7 arealso applicable to digital systems in the same way as theyare applicable to discrete-time systems ie those which arefully described in the discrete domainwithout having specificlinks to a discretization process on a certain continuous-timesystem
It can be pointed out that Definition 7 can be alsoestablished for a digital system not being related to thetime discretization of a continuous-time system in terms ofPopovrsquos inequality of the following form
119896sum119895=0
119910 [119896] 120593 [119896] ge minus120574119889119886 forall119896 isin Z0+ (23)
for some constant 120574119889119886 isin R+ for any digital controller ofclass Φ119889 In this context the sampling period either can benonrelevantmdashthen nonmade notational explicit in the classΦ119889mdashor can even have nonsense since the inequality isapplied to a certain discrete sequence
The following result obtains useful expressions for the dis-cretized input-output energymeasure under some conditionsconcerning the subadditivity constraints on the controllers ofclass Φ119889(119879)The result is also a preparatory one for the nextLemma 9 which addresses the positivity and boundedness ofthe input-output energy measure at sampling instants
Lemma 8 The following properties are fulfilled(i) The subsequent formulas hold
(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119895 (le 119896) 119896 isin Z0+
(26)
(ii) Assume that 120593 isin Φ119889(119879) and that 120593(119909 119896119879) issubadditive for all 119909 isin R ie 120593(119909 + 119910 119896119879) le 120593(119909 119896119879) +120593(119910 119896119879) forall119909 119910 isin R forall119896 isin Z0+ Then the subsequentrelations are true if the sequence 120576[119896] is defined by 120576[119896] =120582[119896] + 120585[119896] forall119896 isin Z0+119910119889ℎ [119896] + intinfin
0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591
which yields (24) and leads to (25) Now define the sequence120576[119896] by 120576[119896] = 120582[119896] + 120585[119896] and 120585[119896][119895] = int(119895+1)119879119895119879(120582(119895119879 +120591)119892((119896 minus 119895)119879 minus 120591) minus 120582[119895]119892[(119896 minus 119895)119879])119889120591 forall119896 isin Z0+ and note
that
minus119910 [119896] = minus (119910119889 [119896] + (119910 [119896] minus 119910119889 [119896]))= [[infinsum119895=0
(int(119895+1)119879119895119879
(1 + 120576[119896] [119895]) 119892 [(119896 minus 119895) 119879])119889120591]]sdot 120593[119896] ([119910119889 [119895] + (119910 [119895] minus 119910119889 [119895]) 119895])minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 minus 119910ℎ [119896]
forall119896 isin Z0+
(31)
and then
minus (119910 [119896] minus 119910119889 [119896] minus 119910ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591)
The subsequent result establishes lower and upper-bounds for the discretized input-output energy measuresbased on the formulas obtained in Lemma 8 and strictpositive realness of the discrete-time transfer function ofthe feed-forward linear and time-invariant system Theresult invokes the boundedness and the smallness of theparameters which define the calculation of the intersam-ple control input from their preceding values at samplinginstants Note that the control deviation in the intersampleperiods related to the sampling time instants has to be suffi-ciently moderate enough so that the hyperstability propertyis kept from the discrete-time system to the continuous-time one In particular the positivity and boundedness ofthe energy measure at the sampling instants is guaranteedif sup119905isinR0+ |120590(119905)| is small enough and Φ119889(119879) is subaddi-tive
Mathematical Problems in Engineering 9
Lemma 9 Assume that 119892119889 isin 119878119875119877 and that 120593 isin Φ119889(119879)Then the following properties hold(i)The input-output energy systemmeasure of the auxiliary
discretized system satisfies
0 lt 1205740119889119886 le 1198792120587 min120596isinR0+
119879119910119889 [119896] 119906119889 [119896] le 120574119889119886forall119896 (gt 1198960) isin Z+
(34)
for some 1205740119889119886 120574119889119886(gt 1205740119889119886) isin R+ and for any nonidentically zerocontrols and zero initial conditions ie for any forced solution
(ii) Assume in addition that the class Φ119889(119879) consistsof subadditive functions Assume also that sup119905isinR0+ |120590(119905)| issufficiently small inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infinwhere 120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ and that furthermore
for some 120574119889119886 isin R+ from the first assumption of the theoremand 119906119889[119896] = minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ since Popovrsquosinequality (22) is fulfilled by any given discrete-time feedbackcontroller 120593 isin Φ119889(119879) generating a feedback control 119906119889[119896] =minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ On the other hand the initialconditions can be neglected since the assumption119892119889 isin SPRimplying that the discrete-time transfer function is (strictly)stable which makes their values irrelevant for purposes ofstability analysisThen the discrete frequency response being
applicable to piecewise-constant inputs in-between any twoconsecutive sampling instants is given by
since the subsequence 119892[minus119899] = 0 forall119899 isin Z+ and thetransfer frequency response of a sampling and zero-order-hold operator 119885 of period 119879 defined by 119885(119879 V)(119905) = V(119896119879) =V[119896] for any given V R0+ rarr R and all 119905 isin [119896119879 (119896 + 1)119879)119896 isin Z0+ is given by 119885(119879 i120596) = (1 minus 119890minusi119879120596)i120596 [31 35]It turns out that |119885(119879 i120596)| = |(1 minus 119890minusi119879120596)i120596| lt 1 forall120596( =0) isin R and |119885(119879 i0)| = 1 Thus the auxiliary discretizedsystem satisfies the subsequent equivalence relation in thefrequency domain to the discrete-time relation (11) It turnsout that Fourier transforms exist in the impulse responsesand truncated control and auxiliary functions of (11)ndash(13)since the truncated functions are square-integrable in clR =[minusinfin +infin] By using the discrete Parsevalrsquos theorem underzero initial conditions one has
The lower-bounding constraint of (37) follows by using(39) and the discrete Parsevalrsquos theorem for the equivalencebetween the input-output energy measures from the fre-quency domain to the discrete-time domain
119864119889 [119896] ge 1198792120587 min120596isinR0+
1199062119889 [119896]) ge 1205740119889forall119896 isin Z0+
(40)
from the assumption 119892119889 isin SPR iemin120596isinR0+ Re119892119889(119890minusi120596119879) ge 1205740119889119886 gt 0 forall1198960 119896 (119896 ge 1198960) isin Z+since
(a) strictly positive real discrete transfer functions havezero relative degree (ie an identical number of polesand zeros) from the Discrete Positive Real Lemma(Szego-Kalman-Popov Lemma) [13 31 34] so thatthe real parts of their frequency hodographs in theargument 119890minusi120596119879 are positively lower-bounded
10 Mathematical Problems in Engineering
(b) Im119892119889(119890minusi120596119879) = minus Im119892119889(119890i120596119879) forall120596 isin R0+ so that theirintegrals in the argument 120596 on frequency intervals[minus119896120587 119896120587] for any 119896 isin Z+ are null
Property (i) has been proved To prove Property (ii) notefrom (28) [Lemma 8(ii)] that
119892 [(119896 minus 119895) 119879])119889120591]]sdot (120593[ℓ] (119910119889 [119895] 119895) + 120593[ℓ] (minus119910119889 [119895] 119895))sdot (120593[ℓ] (119910119889 [ℓ] 119895) + 120593[ℓ] (minus119910119889 [ℓ] ℓ)) minus 119910119889ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 le 120574119889 = 119870120574119889119886
forall119896 isin Z+
(41)
for some 119870 isin R+ provided that sup119905isinR0+ |120590(119905)| is sufficientlysmall inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infin where120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ since 119892(119905) is bounded andconverges exponentially to zero since 119892119889 isin SPR and thenconvergent (that is strictly stable in the discrete context)119910119889ℎ[119896] rarr 0 and it is a bounded sequence
The next result which is a preparatory result to thenestablish the main asymptotic hyperstability result addressesthe relevant property that the input-output measure 119864(119905) ofthe continuous-time system is positively lower-bounded andfinitely upper-bounded for all time under the conditions ofLemmas 4 and 9
Lemma 10 If 120582 and 120590 are sufficiently small then
1205740 minus 120574119889 le 119864 (119905) minus 119864119889 [119896]le 1 + 12058222120587 intinfin
provided that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+Proof One gets the following from direct calculations byusing the continuous-time control laws (9)-(10) and (16) if120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)|
minus 12058222120587 intinfin
minusinfin119892 (i120596) 1003816100381610038161003816[119896] (i120596)10038161003816100381610038162 119889120596 minus 120590(2120582 sup
0le119895le119896
1199062[119895] + 120590)int1198961198790
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591 le 119864 (119905) minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119910 (i120596)
sdot (minusi120596) 119889120596 minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119892 (i120596) | (i120596)|2 119889120596 minus 119864119889 (119896119879 119905)
Now the use of (44) in Lemma 4 (14) implies thatLemma 9(ii) (36) holds implying the existence of 1205740 (120574 gt1205740) isin R+ such that 1205740 le 119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin)provided that 1205741198890 le 119864119889[119896] le 120574119889 for 1205741198890 (120574119889 gt 1205741198890) isin R+ since1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isin Z0+ if1205740 minus 120574119889 le minus 120582
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591le 120574 minus 1205741198890 forall119896 (gt 1198960) isin Z+ forall120591 isin (0 119879)
(45)
if 120582 and 120590 are sufficiently small such that (43) holds leadingto 1205740 gt 0 for any given positive 1205741198890 and 120574119889 gt 1205741198890
The main first result follows
Theorem 11 Consider the dynamic system (2) subject to acontrol
forall119896 isin Z0+ where 120593 [119896] isin Φ119889 (119879) (46)
Assume that(1) 120593(119910[119896] 119896119879) is in the class Φ119889(119879) (that is120593(119910[119896] 119896119879) isin Φ119889(119879)) defined by the discrete Popovrsquos
type sum inequality sum119896ℓ=0 119910119889[ℓ]119906[ℓ] ge minus120574119889119879 for some 120574119889 isin Rforall119896 isin Z0+(2) 119892119889 isin 119878119875119877 there exist constants 1205741198890 (120574119889 gt 1205741198890) isin R+
such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isinZ0+
(3) 120593(119909 119896119879) is subadditive for all 119909 isin R forall119896 isin Z0+(4) 120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)| are
119895119879(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)
minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119896 isin Z0+(47)
Then the following properties hold(i) The discrete closed-loop system is asymptotically hyper-
stable(ii) 119906(119905) rarr 0 as 119905 rarr infin and the control and output are
bounded for all time
(iii) Assume furthermore that 120590(119905) = 120593(119910(119905) 119905) minus (1 +120582(119905))120593(119910[119896] 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+ 119906(119905) =minus120593(119910(119905) 119905) and forall119905 isin R0+ and that the class Φ satisfiesPopovrsquos type integral inequality for all time and results in theclass Φ119889(119896119879) at sampling instantsThen the continuous- timeclosed-loop system is asymptotically hyperstable
Proof From Lemma 9 via assumptions (1)ndash(5) constraints(36) hold so that the input- output energymeasure associatedwith the forced solution is positively lower-bounded andfinitely upper-bounded for all sampling instants 119905 = 119896119879forall119896 isin Z+ Since the discrete positive transfer function isstrictly positive real then it is also strictly stable in the discretecontext (ie convergent [29 36]) so that the homogeneousresponse is irrelevant for stability analysis purposes Thus120593[119896] isin Φ119889 119906[119896] rarr 0 and 119906[119896] and 119910[119896] arebounded for all time for any given bounded initial conditionsIn addition one has from Lemma 10 that (42) holds andthe forced input-output energy measure is positively lower-bounded and finitely upper-bound for all 119905 isin R+ Then119906(119905) rarr 0 as 119905 rarr infin and 119906(119905) and 119910(119905) are bounded for alltime for any given bounded initial conditions Properties [(i)-(ii)] have been proved Property (iii) follows by equalizing119906(119905) = minus120593(119910(119905) 119905) = (1 + 120582(119905))119906[119896] + 120590(119905) and 120593 isin Φ fromthe fact that minusint119905
0119906(120591)119910(120591)119889120591 le 1205740
Note that the use of Fourier transforms is useful tocharacterize the error of the input-output energy measure inthe frequency domain along the intersample periods relatedto its values at the preceding sampling instants This toolis addressed in Lemma 10 which is used in the proof ofTheorem 11 It allows to keep easily the characterizationof the frequency response of the continuous-time transferfunction to evaluate the input-output energy measure in theintersample period and to calculate the minimum value ofthe real part of such a transfer function to guarantee thepositivity and boundedness of such an energy Note that thecharacterization of the input-output energy measure of thefeed-forward linear loop evaluated in the frequency domaintogether with Popovrsquos inequalities of the class of hyperstablecontrollers is relevant to the hyperstability proofs of theclosed-loop system
5 Examples
Example 12 The positive realness of a discretized transferfunction depends not only on the continuous-time param-eters but on the sampling period as well Therefore thepositive realness of the continuous-time system does notguarantee that of any discretized counterpart irrespective ofthe sampling period and conversely For instance the transferfunction 119892(119904) = 1(119904 + 119886) is strictly positive real for 119886 gt 0 butit is not strongly strictly positive real since Re119892(119894120596) rarr 0 as120596 rarr plusmninfin Its discretized version without the concourse ofan ideal that is instantaneous sampler and zero-order holdbecomes 119904 119892119889(119911) = 119911(119911 minus 119890minus119886119879) which is strongly positivereal for 119879 isin (0infin) and as the sampling period 119879 rarr infin since119892119889(119911) rarr 1 as 119879 rarr infin However 119892119889(119911) = 119911(119911 minus 1) as 119879 rarr 0
12 Mathematical Problems in Engineering
Since it has a critically stable pole in the discrete domain it ispositive real although it is not strictly positive real
Example 13 Consider the discrete transfer function 119892119889(119911) =(1199112 + 1198891119911 + 1198892)(1199112 minus 1198861119911 minus 1198862) with real numerator anddenominator coefficients Its strict positive realness requiresas necessary conditions its convergence (ie strict stabilityin the discrete sense) so that its poles have to be within thecomplex open unit circle as well as the convergence of itsinverse so that its zeros have also to lie inside the open unitcircle and furthermore its relative degree (ie its pole-zeroexcess has to be zero) Note that the last condition is notneeded for strict (nonstrong) positive realness of continuoustransfer functions where their positive real part can convergeto zero as |119904| rarr infin By using the homographic bilineartransformation 119911 = (1 + 119904)(1 minus 119904) which transforms theunit circle in the left-hand-side complex plane to get theauxiliary continuous transfer function 119892119886(119904) = 119892119889(119911 = (1 +119904)(1 minus 119904)) it is found in [13] that the denominator andnumerator polynomials of 119892119889(119911) are subject to 1198862 minus 1198861 lt 11198861 + 1198862 lt 1 and 1198862 gt minus1 Take the particular values 1198861 = 05and 1198862 = 04 leading to the values of poles 1199111 = 05 and1199112 = 04 and the coefficients of the numerator polynomialare fixed to 1198891 = 12 1198892 = 03 Then we fix a discretetransfer function 119892119889(119911) = (1199112 + 12119911 + 03)(1199112 minus 05119911 minus 04)yielding min|119911|=1 Re119892119889(119911) ge 009 so 119892119889 isin SSPR Thedecomposition of 119892119889(119911) in simple fractions leads to 119892119889(119911) =1 + 168(119911 minus 093) + 0758(119911 + 043) By using an ideal (iewith instantaneous sampling) sampler and zero-order-holddevice of period 119879 of transfer function (1 minus 119890minus119879119904)119904 on acontinuous transfer function 119892(119904) to be specified to generate119892119889(119911) one has the identity 119885((1 minus 119890minus119879119904)119892(119904)119904) = 119892119889(119911) where119885(119865(119904)) is an abbreviate usual informal notation to denote119865lowast(119904) = 119871(119891lowast(119905)) where 119871(sdot) and 119885(sdot) denote the Laplacetransform of (sdot) of argument 119904 and 119885-transform of (sdot) ofargument 119911 = 119890119879119904 and 119891lowast(119905) = suminfin119899=0 119891(119899119879)120575(119905 minus 119899119879) denotesthe pulsed function of the function119891(119905)with sampling period119879 As a result we get for a sampling period 119879 = 01 sec119892 (119904) = 119904119885minus1 ( 119911119911 minus 1 [1 + 168119911 minus 093 + 0758119911 + 043])
Note that 119892 isin SPR (and also 119892 isin SSPR sinceits relative degree is zero) is the continuous-time transferfunction which generates 119892119889(119911) under an ideal sampler andzero-order-hold device of period 119879 = 01 sec so thatmin120596isinclR119892(i120596) = min120596isinR0+119892(i120596) gt 0 Thus it is guaranteedthat the continuous-time energy measure is associated withthe forced output 119864(119905) gt 0 for all 119905 gt 0 (ie it ispositively lower-bounded for all time) for any piecewise-continuous control irrespectively of the particular controllerin operation In order to guarantee that any such a controlthat converges asymptotically to zero it is necessary to finitelyupper-bound such an energy for all timeThe above concerns
also guarantee that 119864[119896] = 119864(119896119879) ge 1205741198890 gt 0 for all integer 119896 gt0 under any piecewise-constant control 119906[119896] = 119906(119896119879) witheventual jumps at sampling instants of any period 119879 gt 0 Forsuch a control identical conclusion arises from the discrete119892119889(119911)which is also strongly strictly positive real Assume that119906[119896] = minus120593(119910[119896] 119896) and forall119896 isin Z+ with 120593 isin Φ119889 where theclass Φ119889 is defined as the set of sequences 120593(119910[119896] 119896)whichsatisfy Popovrsquos inequality 119879sum119896119895=0 119910[119896]120593(119910[119896] 119896) ge minus120574119889 forall119896 isinZ+ for some finite 120574119886 isin R+ which is identical to
= minus119879119872(1 minus ]119896+1)1 minus ] minus 119879 119896sum119895=0
120576 [119896] ge minus120574119889 forall119896 isin Z+(50)
for some real constants 119872 gt 0 ] isin (0 1) and some realsummable sequence 120576[119896] Then the class Φ119889 is definedby any real sequence 120593(119910[119896] 119896) being parameterized by anyreal constants 119872 gt 0 ] isin (0 1) and any real summablesequence 120576[119896] such that
forall119896 isin Z0+ and 120574119889 ge max(119879119872(1 minus ]) + 120576 120574) for some given120574 ge 1205741198890 where 120576 = 119879max119896isinZ+ sum119896119895=0 120576[119896]The continuous-timecontrol is generated as
where 120593[119896] isin Φ119889(119879) satisfies Popovrsquos type discrete inequal-ity which makes the feedback system globally asymptoticallystable under the theorem hypothesis Thus the discrete-timeclosed-loop system is asymptotically hyperstable In partic-ular under the additional assumptions in Theorem 11(iii)the closed-loop continuous-time system is asymptoticallyhyperstable
Mathematical Problems in Engineering 13
One of the referees has given a comment of interest on thisexample related to continuous versus discrete counterpartproperties which has motivated us to give a further intuitivediscussion on it Note the following In the continuous-timecase there are two kinds of strictly positive real transferfunctions as it is well known namely the so-called strictpositive real which can be zero (or respectively positive)at infinity frequency and which a realizable has relativedegree one (or respectively zero) In the first case the directinput-output interconnection gain 119889 is zero in a state-spacerepresentation (119860 119887 119888 119889 = 0) of the transfer function Thetransfer functions of the second case are often referred to asthe class of strong strict positive real transfer functions Thisclass is not a disjoint class of first above one of strict positivereal transfer functions since it is included in it as a potentialparticular case Note that the members of this strong classof strict positive real transfer functions have relative degreezero and a positive direct input-output interconnection gain119889 Related to this concern the Yakubovich-Kalman-Popov(refereed to the continuous-time domain) and the Szego-Kalman-Popov (referred to the discrete domain) lemmascan be invoked to compare positive realness of the transferfunctions with their state-space associated properties (in factthe properties aremutually equivalent fromboth sides)Thuswe see that in the discrete case the Szego-Kalman-Popovlemma needs for its own coherency that the input-outputinterconnection 119889-gain be nonzero since otherwise one ofthe relevant constraint in the lemma fails [13] So strictpositive realness in the discrete domain in this context istypically strong and then the relative degree of its transferfunction is zero so that there is a direct interconnectionpositive gain from the input to the output Since we assumethat the discrete system is strictly positive real it has a positiveinput-output interconnection from the input to the outputand the minimum value of the real part of its frequencyresponse is strictly positive even as the frequency tends toinfinity On the other hand note that a zero-order holdldquoconvertsrdquo any strictly proper (continuous-time) transferfunction of any relative degree into a strictly discrete oneof unity relative order while it ldquoconvertsrdquo a continuous oneof zero relative order (ie biproper) into a discrete onestill of zero-relative order and with the same input-outputinterconnection gain See (48) Therefore we deal in thisexample with a biproper strict positive real transfer functionof positive input-output interconnection gain identical to thatof its discrete-time counterpart got through a ZOH samplingand hold device
Example 14 Consider that a controller of the form 119906(119905) =minus120593(119910(119905) 119905) forall119905 isin R0+ is used for the continuous-time feed-forward transfer function (48) so that120593 isin Φ and Φ is classin Example 13 satisfying an integral type Popovrsquos inequality(21) for all 119905(ge 1199050) isin R+ and some finite 1199050 isin R+ That isthe control function belongs to a class satisfying continuous-time integral Popovrsquos inequality Then 119906(119905) rarr 0 as 119905 rarr infinand 119906(119905) and 119910(119905) are almost everywhere bounded for all timesince 119892 isin SSPR then being (strictly) stable with input-output energy measure being positively lower-bounded andalso finitely upper-bounded for all 119905 isin R+ from Popovrsquos-type
integral inequality Potential useful controllers are for somegiven controller gain 119896119906 isin R+
119906 (119905) = 0 for 119905 isin [0 1)1198961199061199052119910 (119905) for 119905 ge 1 (54)
119906 (119905) = 119896119906119890minus119905119910 (119905) for 119905 ge 0 (55)
119906 (119905) = 0 for 119905 isin [0 2)
119896119906(119905 ln2119905) 119910 (119905) for 119905 ge 2 (56)
119906 (119905) = 0 for 119905 isin [0 1)
119896119906(119890radic119905radic119905) 119910 (119905) for 119905 ge 1 (57)
119906 (119905) = 1198961199061199053119890minus119905119910 (119905) for 119905 ge 0 (58)
where Popovrsquos inequality on [0 +infin] of (49)ndash(52)is intinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus119896119906 that of (57) isintinfin
0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 and that of (58) isintinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 Since the respective integrands
are monotonically decreasing then there is some 1199050 isin R+ andsome 120574 = 120574(1199050) isin R+ such that int119905
1199050119910(119905)120593(119910(119905) 119905)119889119905 ge minus120574 for
all 119905(ge 1199050) isin R+ The obtained closed-loop systems are inall cases asymptotically hyperstable The piecewise-constantcontrols with eventual discontinuities at sampling instantsgenerated from any discrete-time controller (51) of class Φ119889of Example 13 lead also to asymptotic hyperstability of thediscretized system since 119892119889 isin SSPRExample 15 Consider the continuous-time open-loop sys-tem (2) under a real sampler of sampling period 119879 and datapicking-up nonzero duration 120576 isin (0 119879) The state and outputtrajectory solutions are
forall119896 isin Z0+ for some real sequence 120585[119896] such that 120585[119896] isin(119896119879 119896119879 + 120576) where 119892119889(119911) is the discretized transfer functionunder a sampler and zero-order-hold device subject to instan-taneous ideal sampling Now assume that the incrementalcontrol along intersample periods is generated from itspreceding sampled value as (119895119879+ 120591) = 120582119895119879+120576(119895119879+ 120591)119906[119895] forany 119895 isin Z0+ and 120591 isin [0 119879] where 120582119895119879+120576(119895119879+120591) is a truncatedfunction under the usual notation one gets
The above equations imply that if 119892119889 isin SPR then the forcedoutput 119910119891[119896] = 119910[119896] minus 119888119879119890119896119860119879119909[0] forall119896 isin Z0+ generates anassociate input-output energy measure which is strictly posi-tive for 119896 isin Z0+ since 120588 isin (0 1) for real sampling of duration120576 isin (0 119879)This is a necessary condition for asymptotic hyper-stability under an appropriate class of controllers satisfyingPopovrsquos inequality However the incremental input-outputenergy measure generated by the incremental forced output120576119888119879sum119896minus1119895=0 119890(119896minus119895)119860119879(int1198790 120582119895119879+120576(120591)119890minus119860120591119887 119889120591)119906[119895] is not guaranteedto be nonnegative in general A sufficient condition for that isthat lim sup119895rarrinfin|ess sup120591isin(0120576)(120582119895119879+120576(120591))| is sufficiently smallrelated to the amount ((1 minus 120588)120576)min120596isin[0120587] Re119892119889(119890i120596)6 Conclusions
This paper has been devoted to obtain and to prove theasymptotic hyperstability property of a linear and time-invariant continuous-time system based on some given andpreviously proved auxiliary preparatory lemmas Basicallyit is proved that the continuous-time closed-loop system isasymptotically hyperstable under the assumptions that thediscretized feed-forward transfer function is strictly positivereal and that the discretized system at sampling instants ofconstant sampling period is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsIt is assumed that the continuous-time controller belongs toa class defined in the following way (a) its values at samplinginstants are defined by a class of controllers which satisfydiscrete Popovrsquos type inequality (b) the intersample control isgenerated from its values at sampling instants with the use oftwomodulating auxiliary functionswhich are subject to someextra additional conditions on smallness and boundednessSuch conditions make the overall continuous-time controllerto satisfy continuous-type Popovrsquos inequality for all time
where 119896(120591) = max(119911 isin Z+ 119911119879 le 120591) and 119896 = 119896(119905) = max(119911 isinZ+ 119911119879 le 119905) forall119905 isin R0+ which yields (13)
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
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Mathematical Problems in Engineering 5
are then zero at infinity so that they are square-integrable andtheir sequences of sampled values of sampling period 119879 aresquare-summable as a result
The subsequent two simple auxiliary lemmas will be thenused in the next section to establish the main result
Lemma 4 Assume that there exist constants 1205741198890 (120574119889 gt 1205741198890) isinR+ such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some1198960 isin Z0+ Then there exist 1205740 (120574 gt 1205740) isin R+ such that 1205740 le119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin) if1205740 minus 120574119889 le 119864119889 (119896119879 119905)
= (119864 [119896] minus 119864119889 [119896]) + 119864 (119896119879 119896119879 + 120591)= 119864 (119905) minus 119864119889 [119896] = 119864119889 (119896119879 119905) le 120574 minus 1205741198890
119879 (119910 [119895] 119906 [119895] minus 119910119889 [119895] 119906119889 [119895])
(16)
forall119905 isin (119896119879 (119896 + 1)119879) forall119896 isin Z0+ so that (14) implies that 1205740 le119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin) provided that 1205741198890 le 119864119889[119896] le120574119889 forall119896(gt 1198960) isin Z+ for 1198960 isin Z0+Lemma 5 Assume that 120593(119909 119905) is subadditive for all 119909 isin Rie 120593(119909 + 119910 119905) le 120593(119909 119905) + 120593(119910 119905) forall119909 119910 isin R forall119905 isin R0+Then 120593(119910(119905) 119905) le 120593(119910[119896] 119896) + 120593(119910(119905) minus 119910[119896] 119905 119896119879) andas a result (119905) ge 120593(119910(119905) minus 119910[119896] 119905 119896119879) and 120593(119910[119896] 119896) geminus120593(minus119910[119896] 119896) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+
If 120593(119909 119905) is superadditive for all 119909 isin R ie 120593(119909 +119910 119905) ge 120593(119909 119905) + 120593(119910 119905) forall119909 119910 isin R forall119905 isin R0+ then120593(119910(119905) 119905) ge 120593(119910[119896] 119896119879) + 120593(119910(119905) minus 119910[119896] 119905 119896119879) as a result(119905) le 120593(119910(119905) minus 119910[119896] 119905 119896119879) and 120593(119910[119896]) le minus120593(minus119910[119896]) forall119905 isin[119896119879 (119896 + 1)119879) forall119896 isin Z0+Proof From the first identity in (9) if 120593(119909 119905) is subadditivethenminus120593 (119910 (119905)) = 119906 (119905) = 119906 [119896] + (119905)
ge minus120593 (119910 [119896] + (119910 (119905) minus 119910 [119896]))ge minus120593 (119910 [119896]) minus 120593 (119910 (119905) minus 119910 [119896])
= minus120593 (119910 [119896]) minus 120593 (119910 (119905) + (minus119910 [119896]))ge minus120593 (119910 [119896]) minus 120593 (119910 (119905)) minus 120593 (minus119910 [119896])
Definition 6 (Popovrsquos inequality) A continuous-time feed-back nonlinear and eventually time-varying continuous-timecontroller 119906(119905) = minus120593(119910(119905) 119905) and forall119905 isin R0+ where 120593 R0+ timesR0+ rarr R0+ is in the class Φ denoted as 120593 isin Φ is said tosatisfy Popovrsquos type integral inequality if for some 1205740 isin R+one has
Definition 7 (discrete-time Popovrsquos inequality) A discrete-time feedback nonlinear and eventually time-varyingdiscrete-time controller 119906(119905) = 119906[119896] = 119906(119896119879) =minus120593(119910(119896119879) 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) and forall119896 isin Z0+where 120593 R0+ times Z0+ rarr R0+ is in the class Φ119889(119879) ofsampling period 119879 isin R+ denoted as 120593 isin Φ119889(119879) is said tosatisfy a Popovrsquos type discrete-time inequality if for some120574119889 isin R+ one has119896sum119895=0
119910 (119896119879) 120593 (119896119879) ge minus120574119889119879 forall119896 isin Z0+ (22)
I D Landau refers to controllers satisfying a Popovrsquostype inequality (21) (or respectively (22)) as hyperstable
controllers of class Φ (or respectively of class Φ119889(119879))[13] bearing in mind that if any controller of such a class iscoupled to a linear time-invariant forward system of transferfunction being continuous-time (or respectively discrete-time) strictly positive real then the overall closed-loop systemis asymptotically hyperstable namely globally asymptoticallystable for any arbitrary controller belonging to the respectiveclass It can be pointed out that the above comments referredto discrete-time systems and in particular Definition 7 arealso applicable to digital systems in the same way as theyare applicable to discrete-time systems ie those which arefully described in the discrete domainwithout having specificlinks to a discretization process on a certain continuous-timesystem
It can be pointed out that Definition 7 can be alsoestablished for a digital system not being related to thetime discretization of a continuous-time system in terms ofPopovrsquos inequality of the following form
119896sum119895=0
119910 [119896] 120593 [119896] ge minus120574119889119886 forall119896 isin Z0+ (23)
for some constant 120574119889119886 isin R+ for any digital controller ofclass Φ119889 In this context the sampling period either can benonrelevantmdashthen nonmade notational explicit in the classΦ119889mdashor can even have nonsense since the inequality isapplied to a certain discrete sequence
The following result obtains useful expressions for the dis-cretized input-output energymeasure under some conditionsconcerning the subadditivity constraints on the controllers ofclass Φ119889(119879)The result is also a preparatory one for the nextLemma 9 which addresses the positivity and boundedness ofthe input-output energy measure at sampling instants
Lemma 8 The following properties are fulfilled(i) The subsequent formulas hold
(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119895 (le 119896) 119896 isin Z0+
(26)
(ii) Assume that 120593 isin Φ119889(119879) and that 120593(119909 119896119879) issubadditive for all 119909 isin R ie 120593(119909 + 119910 119896119879) le 120593(119909 119896119879) +120593(119910 119896119879) forall119909 119910 isin R forall119896 isin Z0+ Then the subsequentrelations are true if the sequence 120576[119896] is defined by 120576[119896] =120582[119896] + 120585[119896] forall119896 isin Z0+119910119889ℎ [119896] + intinfin
0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591
which yields (24) and leads to (25) Now define the sequence120576[119896] by 120576[119896] = 120582[119896] + 120585[119896] and 120585[119896][119895] = int(119895+1)119879119895119879(120582(119895119879 +120591)119892((119896 minus 119895)119879 minus 120591) minus 120582[119895]119892[(119896 minus 119895)119879])119889120591 forall119896 isin Z0+ and note
that
minus119910 [119896] = minus (119910119889 [119896] + (119910 [119896] minus 119910119889 [119896]))= [[infinsum119895=0
(int(119895+1)119879119895119879
(1 + 120576[119896] [119895]) 119892 [(119896 minus 119895) 119879])119889120591]]sdot 120593[119896] ([119910119889 [119895] + (119910 [119895] minus 119910119889 [119895]) 119895])minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 minus 119910ℎ [119896]
forall119896 isin Z0+
(31)
and then
minus (119910 [119896] minus 119910119889 [119896] minus 119910ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591)
The subsequent result establishes lower and upper-bounds for the discretized input-output energy measuresbased on the formulas obtained in Lemma 8 and strictpositive realness of the discrete-time transfer function ofthe feed-forward linear and time-invariant system Theresult invokes the boundedness and the smallness of theparameters which define the calculation of the intersam-ple control input from their preceding values at samplinginstants Note that the control deviation in the intersampleperiods related to the sampling time instants has to be suffi-ciently moderate enough so that the hyperstability propertyis kept from the discrete-time system to the continuous-time one In particular the positivity and boundedness ofthe energy measure at the sampling instants is guaranteedif sup119905isinR0+ |120590(119905)| is small enough and Φ119889(119879) is subaddi-tive
Mathematical Problems in Engineering 9
Lemma 9 Assume that 119892119889 isin 119878119875119877 and that 120593 isin Φ119889(119879)Then the following properties hold(i)The input-output energy systemmeasure of the auxiliary
discretized system satisfies
0 lt 1205740119889119886 le 1198792120587 min120596isinR0+
119879119910119889 [119896] 119906119889 [119896] le 120574119889119886forall119896 (gt 1198960) isin Z+
(34)
for some 1205740119889119886 120574119889119886(gt 1205740119889119886) isin R+ and for any nonidentically zerocontrols and zero initial conditions ie for any forced solution
(ii) Assume in addition that the class Φ119889(119879) consistsof subadditive functions Assume also that sup119905isinR0+ |120590(119905)| issufficiently small inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infinwhere 120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ and that furthermore
for some 120574119889119886 isin R+ from the first assumption of the theoremand 119906119889[119896] = minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ since Popovrsquosinequality (22) is fulfilled by any given discrete-time feedbackcontroller 120593 isin Φ119889(119879) generating a feedback control 119906119889[119896] =minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ On the other hand the initialconditions can be neglected since the assumption119892119889 isin SPRimplying that the discrete-time transfer function is (strictly)stable which makes their values irrelevant for purposes ofstability analysisThen the discrete frequency response being
applicable to piecewise-constant inputs in-between any twoconsecutive sampling instants is given by
since the subsequence 119892[minus119899] = 0 forall119899 isin Z+ and thetransfer frequency response of a sampling and zero-order-hold operator 119885 of period 119879 defined by 119885(119879 V)(119905) = V(119896119879) =V[119896] for any given V R0+ rarr R and all 119905 isin [119896119879 (119896 + 1)119879)119896 isin Z0+ is given by 119885(119879 i120596) = (1 minus 119890minusi119879120596)i120596 [31 35]It turns out that |119885(119879 i120596)| = |(1 minus 119890minusi119879120596)i120596| lt 1 forall120596( =0) isin R and |119885(119879 i0)| = 1 Thus the auxiliary discretizedsystem satisfies the subsequent equivalence relation in thefrequency domain to the discrete-time relation (11) It turnsout that Fourier transforms exist in the impulse responsesand truncated control and auxiliary functions of (11)ndash(13)since the truncated functions are square-integrable in clR =[minusinfin +infin] By using the discrete Parsevalrsquos theorem underzero initial conditions one has
The lower-bounding constraint of (37) follows by using(39) and the discrete Parsevalrsquos theorem for the equivalencebetween the input-output energy measures from the fre-quency domain to the discrete-time domain
119864119889 [119896] ge 1198792120587 min120596isinR0+
1199062119889 [119896]) ge 1205740119889forall119896 isin Z0+
(40)
from the assumption 119892119889 isin SPR iemin120596isinR0+ Re119892119889(119890minusi120596119879) ge 1205740119889119886 gt 0 forall1198960 119896 (119896 ge 1198960) isin Z+since
(a) strictly positive real discrete transfer functions havezero relative degree (ie an identical number of polesand zeros) from the Discrete Positive Real Lemma(Szego-Kalman-Popov Lemma) [13 31 34] so thatthe real parts of their frequency hodographs in theargument 119890minusi120596119879 are positively lower-bounded
10 Mathematical Problems in Engineering
(b) Im119892119889(119890minusi120596119879) = minus Im119892119889(119890i120596119879) forall120596 isin R0+ so that theirintegrals in the argument 120596 on frequency intervals[minus119896120587 119896120587] for any 119896 isin Z+ are null
Property (i) has been proved To prove Property (ii) notefrom (28) [Lemma 8(ii)] that
119892 [(119896 minus 119895) 119879])119889120591]]sdot (120593[ℓ] (119910119889 [119895] 119895) + 120593[ℓ] (minus119910119889 [119895] 119895))sdot (120593[ℓ] (119910119889 [ℓ] 119895) + 120593[ℓ] (minus119910119889 [ℓ] ℓ)) minus 119910119889ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 le 120574119889 = 119870120574119889119886
forall119896 isin Z+
(41)
for some 119870 isin R+ provided that sup119905isinR0+ |120590(119905)| is sufficientlysmall inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infin where120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ since 119892(119905) is bounded andconverges exponentially to zero since 119892119889 isin SPR and thenconvergent (that is strictly stable in the discrete context)119910119889ℎ[119896] rarr 0 and it is a bounded sequence
The next result which is a preparatory result to thenestablish the main asymptotic hyperstability result addressesthe relevant property that the input-output measure 119864(119905) ofthe continuous-time system is positively lower-bounded andfinitely upper-bounded for all time under the conditions ofLemmas 4 and 9
Lemma 10 If 120582 and 120590 are sufficiently small then
1205740 minus 120574119889 le 119864 (119905) minus 119864119889 [119896]le 1 + 12058222120587 intinfin
provided that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+Proof One gets the following from direct calculations byusing the continuous-time control laws (9)-(10) and (16) if120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)|
minus 12058222120587 intinfin
minusinfin119892 (i120596) 1003816100381610038161003816[119896] (i120596)10038161003816100381610038162 119889120596 minus 120590(2120582 sup
0le119895le119896
1199062[119895] + 120590)int1198961198790
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591 le 119864 (119905) minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119910 (i120596)
sdot (minusi120596) 119889120596 minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119892 (i120596) | (i120596)|2 119889120596 minus 119864119889 (119896119879 119905)
Now the use of (44) in Lemma 4 (14) implies thatLemma 9(ii) (36) holds implying the existence of 1205740 (120574 gt1205740) isin R+ such that 1205740 le 119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin)provided that 1205741198890 le 119864119889[119896] le 120574119889 for 1205741198890 (120574119889 gt 1205741198890) isin R+ since1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isin Z0+ if1205740 minus 120574119889 le minus 120582
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591le 120574 minus 1205741198890 forall119896 (gt 1198960) isin Z+ forall120591 isin (0 119879)
(45)
if 120582 and 120590 are sufficiently small such that (43) holds leadingto 1205740 gt 0 for any given positive 1205741198890 and 120574119889 gt 1205741198890
The main first result follows
Theorem 11 Consider the dynamic system (2) subject to acontrol
forall119896 isin Z0+ where 120593 [119896] isin Φ119889 (119879) (46)
Assume that(1) 120593(119910[119896] 119896119879) is in the class Φ119889(119879) (that is120593(119910[119896] 119896119879) isin Φ119889(119879)) defined by the discrete Popovrsquos
type sum inequality sum119896ℓ=0 119910119889[ℓ]119906[ℓ] ge minus120574119889119879 for some 120574119889 isin Rforall119896 isin Z0+(2) 119892119889 isin 119878119875119877 there exist constants 1205741198890 (120574119889 gt 1205741198890) isin R+
such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isinZ0+
(3) 120593(119909 119896119879) is subadditive for all 119909 isin R forall119896 isin Z0+(4) 120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)| are
119895119879(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)
minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119896 isin Z0+(47)
Then the following properties hold(i) The discrete closed-loop system is asymptotically hyper-
stable(ii) 119906(119905) rarr 0 as 119905 rarr infin and the control and output are
bounded for all time
(iii) Assume furthermore that 120590(119905) = 120593(119910(119905) 119905) minus (1 +120582(119905))120593(119910[119896] 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+ 119906(119905) =minus120593(119910(119905) 119905) and forall119905 isin R0+ and that the class Φ satisfiesPopovrsquos type integral inequality for all time and results in theclass Φ119889(119896119879) at sampling instantsThen the continuous- timeclosed-loop system is asymptotically hyperstable
Proof From Lemma 9 via assumptions (1)ndash(5) constraints(36) hold so that the input- output energymeasure associatedwith the forced solution is positively lower-bounded andfinitely upper-bounded for all sampling instants 119905 = 119896119879forall119896 isin Z+ Since the discrete positive transfer function isstrictly positive real then it is also strictly stable in the discretecontext (ie convergent [29 36]) so that the homogeneousresponse is irrelevant for stability analysis purposes Thus120593[119896] isin Φ119889 119906[119896] rarr 0 and 119906[119896] and 119910[119896] arebounded for all time for any given bounded initial conditionsIn addition one has from Lemma 10 that (42) holds andthe forced input-output energy measure is positively lower-bounded and finitely upper-bound for all 119905 isin R+ Then119906(119905) rarr 0 as 119905 rarr infin and 119906(119905) and 119910(119905) are bounded for alltime for any given bounded initial conditions Properties [(i)-(ii)] have been proved Property (iii) follows by equalizing119906(119905) = minus120593(119910(119905) 119905) = (1 + 120582(119905))119906[119896] + 120590(119905) and 120593 isin Φ fromthe fact that minusint119905
0119906(120591)119910(120591)119889120591 le 1205740
Note that the use of Fourier transforms is useful tocharacterize the error of the input-output energy measure inthe frequency domain along the intersample periods relatedto its values at the preceding sampling instants This toolis addressed in Lemma 10 which is used in the proof ofTheorem 11 It allows to keep easily the characterizationof the frequency response of the continuous-time transferfunction to evaluate the input-output energy measure in theintersample period and to calculate the minimum value ofthe real part of such a transfer function to guarantee thepositivity and boundedness of such an energy Note that thecharacterization of the input-output energy measure of thefeed-forward linear loop evaluated in the frequency domaintogether with Popovrsquos inequalities of the class of hyperstablecontrollers is relevant to the hyperstability proofs of theclosed-loop system
5 Examples
Example 12 The positive realness of a discretized transferfunction depends not only on the continuous-time param-eters but on the sampling period as well Therefore thepositive realness of the continuous-time system does notguarantee that of any discretized counterpart irrespective ofthe sampling period and conversely For instance the transferfunction 119892(119904) = 1(119904 + 119886) is strictly positive real for 119886 gt 0 butit is not strongly strictly positive real since Re119892(119894120596) rarr 0 as120596 rarr plusmninfin Its discretized version without the concourse ofan ideal that is instantaneous sampler and zero-order holdbecomes 119904 119892119889(119911) = 119911(119911 minus 119890minus119886119879) which is strongly positivereal for 119879 isin (0infin) and as the sampling period 119879 rarr infin since119892119889(119911) rarr 1 as 119879 rarr infin However 119892119889(119911) = 119911(119911 minus 1) as 119879 rarr 0
12 Mathematical Problems in Engineering
Since it has a critically stable pole in the discrete domain it ispositive real although it is not strictly positive real
Example 13 Consider the discrete transfer function 119892119889(119911) =(1199112 + 1198891119911 + 1198892)(1199112 minus 1198861119911 minus 1198862) with real numerator anddenominator coefficients Its strict positive realness requiresas necessary conditions its convergence (ie strict stabilityin the discrete sense) so that its poles have to be within thecomplex open unit circle as well as the convergence of itsinverse so that its zeros have also to lie inside the open unitcircle and furthermore its relative degree (ie its pole-zeroexcess has to be zero) Note that the last condition is notneeded for strict (nonstrong) positive realness of continuoustransfer functions where their positive real part can convergeto zero as |119904| rarr infin By using the homographic bilineartransformation 119911 = (1 + 119904)(1 minus 119904) which transforms theunit circle in the left-hand-side complex plane to get theauxiliary continuous transfer function 119892119886(119904) = 119892119889(119911 = (1 +119904)(1 minus 119904)) it is found in [13] that the denominator andnumerator polynomials of 119892119889(119911) are subject to 1198862 minus 1198861 lt 11198861 + 1198862 lt 1 and 1198862 gt minus1 Take the particular values 1198861 = 05and 1198862 = 04 leading to the values of poles 1199111 = 05 and1199112 = 04 and the coefficients of the numerator polynomialare fixed to 1198891 = 12 1198892 = 03 Then we fix a discretetransfer function 119892119889(119911) = (1199112 + 12119911 + 03)(1199112 minus 05119911 minus 04)yielding min|119911|=1 Re119892119889(119911) ge 009 so 119892119889 isin SSPR Thedecomposition of 119892119889(119911) in simple fractions leads to 119892119889(119911) =1 + 168(119911 minus 093) + 0758(119911 + 043) By using an ideal (iewith instantaneous sampling) sampler and zero-order-holddevice of period 119879 of transfer function (1 minus 119890minus119879119904)119904 on acontinuous transfer function 119892(119904) to be specified to generate119892119889(119911) one has the identity 119885((1 minus 119890minus119879119904)119892(119904)119904) = 119892119889(119911) where119885(119865(119904)) is an abbreviate usual informal notation to denote119865lowast(119904) = 119871(119891lowast(119905)) where 119871(sdot) and 119885(sdot) denote the Laplacetransform of (sdot) of argument 119904 and 119885-transform of (sdot) ofargument 119911 = 119890119879119904 and 119891lowast(119905) = suminfin119899=0 119891(119899119879)120575(119905 minus 119899119879) denotesthe pulsed function of the function119891(119905)with sampling period119879 As a result we get for a sampling period 119879 = 01 sec119892 (119904) = 119904119885minus1 ( 119911119911 minus 1 [1 + 168119911 minus 093 + 0758119911 + 043])
Note that 119892 isin SPR (and also 119892 isin SSPR sinceits relative degree is zero) is the continuous-time transferfunction which generates 119892119889(119911) under an ideal sampler andzero-order-hold device of period 119879 = 01 sec so thatmin120596isinclR119892(i120596) = min120596isinR0+119892(i120596) gt 0 Thus it is guaranteedthat the continuous-time energy measure is associated withthe forced output 119864(119905) gt 0 for all 119905 gt 0 (ie it ispositively lower-bounded for all time) for any piecewise-continuous control irrespectively of the particular controllerin operation In order to guarantee that any such a controlthat converges asymptotically to zero it is necessary to finitelyupper-bound such an energy for all timeThe above concerns
also guarantee that 119864[119896] = 119864(119896119879) ge 1205741198890 gt 0 for all integer 119896 gt0 under any piecewise-constant control 119906[119896] = 119906(119896119879) witheventual jumps at sampling instants of any period 119879 gt 0 Forsuch a control identical conclusion arises from the discrete119892119889(119911)which is also strongly strictly positive real Assume that119906[119896] = minus120593(119910[119896] 119896) and forall119896 isin Z+ with 120593 isin Φ119889 where theclass Φ119889 is defined as the set of sequences 120593(119910[119896] 119896)whichsatisfy Popovrsquos inequality 119879sum119896119895=0 119910[119896]120593(119910[119896] 119896) ge minus120574119889 forall119896 isinZ+ for some finite 120574119886 isin R+ which is identical to
= minus119879119872(1 minus ]119896+1)1 minus ] minus 119879 119896sum119895=0
120576 [119896] ge minus120574119889 forall119896 isin Z+(50)
for some real constants 119872 gt 0 ] isin (0 1) and some realsummable sequence 120576[119896] Then the class Φ119889 is definedby any real sequence 120593(119910[119896] 119896) being parameterized by anyreal constants 119872 gt 0 ] isin (0 1) and any real summablesequence 120576[119896] such that
forall119896 isin Z0+ and 120574119889 ge max(119879119872(1 minus ]) + 120576 120574) for some given120574 ge 1205741198890 where 120576 = 119879max119896isinZ+ sum119896119895=0 120576[119896]The continuous-timecontrol is generated as
where 120593[119896] isin Φ119889(119879) satisfies Popovrsquos type discrete inequal-ity which makes the feedback system globally asymptoticallystable under the theorem hypothesis Thus the discrete-timeclosed-loop system is asymptotically hyperstable In partic-ular under the additional assumptions in Theorem 11(iii)the closed-loop continuous-time system is asymptoticallyhyperstable
Mathematical Problems in Engineering 13
One of the referees has given a comment of interest on thisexample related to continuous versus discrete counterpartproperties which has motivated us to give a further intuitivediscussion on it Note the following In the continuous-timecase there are two kinds of strictly positive real transferfunctions as it is well known namely the so-called strictpositive real which can be zero (or respectively positive)at infinity frequency and which a realizable has relativedegree one (or respectively zero) In the first case the directinput-output interconnection gain 119889 is zero in a state-spacerepresentation (119860 119887 119888 119889 = 0) of the transfer function Thetransfer functions of the second case are often referred to asthe class of strong strict positive real transfer functions Thisclass is not a disjoint class of first above one of strict positivereal transfer functions since it is included in it as a potentialparticular case Note that the members of this strong classof strict positive real transfer functions have relative degreezero and a positive direct input-output interconnection gain119889 Related to this concern the Yakubovich-Kalman-Popov(refereed to the continuous-time domain) and the Szego-Kalman-Popov (referred to the discrete domain) lemmascan be invoked to compare positive realness of the transferfunctions with their state-space associated properties (in factthe properties aremutually equivalent fromboth sides)Thuswe see that in the discrete case the Szego-Kalman-Popovlemma needs for its own coherency that the input-outputinterconnection 119889-gain be nonzero since otherwise one ofthe relevant constraint in the lemma fails [13] So strictpositive realness in the discrete domain in this context istypically strong and then the relative degree of its transferfunction is zero so that there is a direct interconnectionpositive gain from the input to the output Since we assumethat the discrete system is strictly positive real it has a positiveinput-output interconnection from the input to the outputand the minimum value of the real part of its frequencyresponse is strictly positive even as the frequency tends toinfinity On the other hand note that a zero-order holdldquoconvertsrdquo any strictly proper (continuous-time) transferfunction of any relative degree into a strictly discrete oneof unity relative order while it ldquoconvertsrdquo a continuous oneof zero relative order (ie biproper) into a discrete onestill of zero-relative order and with the same input-outputinterconnection gain See (48) Therefore we deal in thisexample with a biproper strict positive real transfer functionof positive input-output interconnection gain identical to thatof its discrete-time counterpart got through a ZOH samplingand hold device
Example 14 Consider that a controller of the form 119906(119905) =minus120593(119910(119905) 119905) forall119905 isin R0+ is used for the continuous-time feed-forward transfer function (48) so that120593 isin Φ and Φ is classin Example 13 satisfying an integral type Popovrsquos inequality(21) for all 119905(ge 1199050) isin R+ and some finite 1199050 isin R+ That isthe control function belongs to a class satisfying continuous-time integral Popovrsquos inequality Then 119906(119905) rarr 0 as 119905 rarr infinand 119906(119905) and 119910(119905) are almost everywhere bounded for all timesince 119892 isin SSPR then being (strictly) stable with input-output energy measure being positively lower-bounded andalso finitely upper-bounded for all 119905 isin R+ from Popovrsquos-type
integral inequality Potential useful controllers are for somegiven controller gain 119896119906 isin R+
119906 (119905) = 0 for 119905 isin [0 1)1198961199061199052119910 (119905) for 119905 ge 1 (54)
119906 (119905) = 119896119906119890minus119905119910 (119905) for 119905 ge 0 (55)
119906 (119905) = 0 for 119905 isin [0 2)
119896119906(119905 ln2119905) 119910 (119905) for 119905 ge 2 (56)
119906 (119905) = 0 for 119905 isin [0 1)
119896119906(119890radic119905radic119905) 119910 (119905) for 119905 ge 1 (57)
119906 (119905) = 1198961199061199053119890minus119905119910 (119905) for 119905 ge 0 (58)
where Popovrsquos inequality on [0 +infin] of (49)ndash(52)is intinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus119896119906 that of (57) isintinfin
0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 and that of (58) isintinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 Since the respective integrands
are monotonically decreasing then there is some 1199050 isin R+ andsome 120574 = 120574(1199050) isin R+ such that int119905
1199050119910(119905)120593(119910(119905) 119905)119889119905 ge minus120574 for
all 119905(ge 1199050) isin R+ The obtained closed-loop systems are inall cases asymptotically hyperstable The piecewise-constantcontrols with eventual discontinuities at sampling instantsgenerated from any discrete-time controller (51) of class Φ119889of Example 13 lead also to asymptotic hyperstability of thediscretized system since 119892119889 isin SSPRExample 15 Consider the continuous-time open-loop sys-tem (2) under a real sampler of sampling period 119879 and datapicking-up nonzero duration 120576 isin (0 119879) The state and outputtrajectory solutions are
forall119896 isin Z0+ for some real sequence 120585[119896] such that 120585[119896] isin(119896119879 119896119879 + 120576) where 119892119889(119911) is the discretized transfer functionunder a sampler and zero-order-hold device subject to instan-taneous ideal sampling Now assume that the incrementalcontrol along intersample periods is generated from itspreceding sampled value as (119895119879+ 120591) = 120582119895119879+120576(119895119879+ 120591)119906[119895] forany 119895 isin Z0+ and 120591 isin [0 119879] where 120582119895119879+120576(119895119879+120591) is a truncatedfunction under the usual notation one gets
The above equations imply that if 119892119889 isin SPR then the forcedoutput 119910119891[119896] = 119910[119896] minus 119888119879119890119896119860119879119909[0] forall119896 isin Z0+ generates anassociate input-output energy measure which is strictly posi-tive for 119896 isin Z0+ since 120588 isin (0 1) for real sampling of duration120576 isin (0 119879)This is a necessary condition for asymptotic hyper-stability under an appropriate class of controllers satisfyingPopovrsquos inequality However the incremental input-outputenergy measure generated by the incremental forced output120576119888119879sum119896minus1119895=0 119890(119896minus119895)119860119879(int1198790 120582119895119879+120576(120591)119890minus119860120591119887 119889120591)119906[119895] is not guaranteedto be nonnegative in general A sufficient condition for that isthat lim sup119895rarrinfin|ess sup120591isin(0120576)(120582119895119879+120576(120591))| is sufficiently smallrelated to the amount ((1 minus 120588)120576)min120596isin[0120587] Re119892119889(119890i120596)6 Conclusions
This paper has been devoted to obtain and to prove theasymptotic hyperstability property of a linear and time-invariant continuous-time system based on some given andpreviously proved auxiliary preparatory lemmas Basicallyit is proved that the continuous-time closed-loop system isasymptotically hyperstable under the assumptions that thediscretized feed-forward transfer function is strictly positivereal and that the discretized system at sampling instants ofconstant sampling period is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsIt is assumed that the continuous-time controller belongs toa class defined in the following way (a) its values at samplinginstants are defined by a class of controllers which satisfydiscrete Popovrsquos type inequality (b) the intersample control isgenerated from its values at sampling instants with the use oftwomodulating auxiliary functionswhich are subject to someextra additional conditions on smallness and boundednessSuch conditions make the overall continuous-time controllerto satisfy continuous-type Popovrsquos inequality for all time
where 119896(120591) = max(119911 isin Z+ 119911119879 le 120591) and 119896 = 119896(119905) = max(119911 isinZ+ 119911119879 le 119905) forall119905 isin R0+ which yields (13)
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
Hindawiwwwhindawicom Volume 2018
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Mathematical Problems in Engineering
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OptimizationJournal of
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Engineering Mathematics
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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Definition 6 (Popovrsquos inequality) A continuous-time feed-back nonlinear and eventually time-varying continuous-timecontroller 119906(119905) = minus120593(119910(119905) 119905) and forall119905 isin R0+ where 120593 R0+ timesR0+ rarr R0+ is in the class Φ denoted as 120593 isin Φ is said tosatisfy Popovrsquos type integral inequality if for some 1205740 isin R+one has
Definition 7 (discrete-time Popovrsquos inequality) A discrete-time feedback nonlinear and eventually time-varyingdiscrete-time controller 119906(119905) = 119906[119896] = 119906(119896119879) =minus120593(119910(119896119879) 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) and forall119896 isin Z0+where 120593 R0+ times Z0+ rarr R0+ is in the class Φ119889(119879) ofsampling period 119879 isin R+ denoted as 120593 isin Φ119889(119879) is said tosatisfy a Popovrsquos type discrete-time inequality if for some120574119889 isin R+ one has119896sum119895=0
119910 (119896119879) 120593 (119896119879) ge minus120574119889119879 forall119896 isin Z0+ (22)
I D Landau refers to controllers satisfying a Popovrsquostype inequality (21) (or respectively (22)) as hyperstable
controllers of class Φ (or respectively of class Φ119889(119879))[13] bearing in mind that if any controller of such a class iscoupled to a linear time-invariant forward system of transferfunction being continuous-time (or respectively discrete-time) strictly positive real then the overall closed-loop systemis asymptotically hyperstable namely globally asymptoticallystable for any arbitrary controller belonging to the respectiveclass It can be pointed out that the above comments referredto discrete-time systems and in particular Definition 7 arealso applicable to digital systems in the same way as theyare applicable to discrete-time systems ie those which arefully described in the discrete domainwithout having specificlinks to a discretization process on a certain continuous-timesystem
It can be pointed out that Definition 7 can be alsoestablished for a digital system not being related to thetime discretization of a continuous-time system in terms ofPopovrsquos inequality of the following form
119896sum119895=0
119910 [119896] 120593 [119896] ge minus120574119889119886 forall119896 isin Z0+ (23)
for some constant 120574119889119886 isin R+ for any digital controller ofclass Φ119889 In this context the sampling period either can benonrelevantmdashthen nonmade notational explicit in the classΦ119889mdashor can even have nonsense since the inequality isapplied to a certain discrete sequence
The following result obtains useful expressions for the dis-cretized input-output energymeasure under some conditionsconcerning the subadditivity constraints on the controllers ofclass Φ119889(119879)The result is also a preparatory one for the nextLemma 9 which addresses the positivity and boundedness ofthe input-output energy measure at sampling instants
Lemma 8 The following properties are fulfilled(i) The subsequent formulas hold
(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119895 (le 119896) 119896 isin Z0+
(26)
(ii) Assume that 120593 isin Φ119889(119879) and that 120593(119909 119896119879) issubadditive for all 119909 isin R ie 120593(119909 + 119910 119896119879) le 120593(119909 119896119879) +120593(119910 119896119879) forall119909 119910 isin R forall119896 isin Z0+ Then the subsequentrelations are true if the sequence 120576[119896] is defined by 120576[119896] =120582[119896] + 120585[119896] forall119896 isin Z0+119910119889ℎ [119896] + intinfin
0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591
which yields (24) and leads to (25) Now define the sequence120576[119896] by 120576[119896] = 120582[119896] + 120585[119896] and 120585[119896][119895] = int(119895+1)119879119895119879(120582(119895119879 +120591)119892((119896 minus 119895)119879 minus 120591) minus 120582[119895]119892[(119896 minus 119895)119879])119889120591 forall119896 isin Z0+ and note
that
minus119910 [119896] = minus (119910119889 [119896] + (119910 [119896] minus 119910119889 [119896]))= [[infinsum119895=0
(int(119895+1)119879119895119879
(1 + 120576[119896] [119895]) 119892 [(119896 minus 119895) 119879])119889120591]]sdot 120593[119896] ([119910119889 [119895] + (119910 [119895] minus 119910119889 [119895]) 119895])minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 minus 119910ℎ [119896]
forall119896 isin Z0+
(31)
and then
minus (119910 [119896] minus 119910119889 [119896] minus 119910ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591)
The subsequent result establishes lower and upper-bounds for the discretized input-output energy measuresbased on the formulas obtained in Lemma 8 and strictpositive realness of the discrete-time transfer function ofthe feed-forward linear and time-invariant system Theresult invokes the boundedness and the smallness of theparameters which define the calculation of the intersam-ple control input from their preceding values at samplinginstants Note that the control deviation in the intersampleperiods related to the sampling time instants has to be suffi-ciently moderate enough so that the hyperstability propertyis kept from the discrete-time system to the continuous-time one In particular the positivity and boundedness ofthe energy measure at the sampling instants is guaranteedif sup119905isinR0+ |120590(119905)| is small enough and Φ119889(119879) is subaddi-tive
Mathematical Problems in Engineering 9
Lemma 9 Assume that 119892119889 isin 119878119875119877 and that 120593 isin Φ119889(119879)Then the following properties hold(i)The input-output energy systemmeasure of the auxiliary
discretized system satisfies
0 lt 1205740119889119886 le 1198792120587 min120596isinR0+
119879119910119889 [119896] 119906119889 [119896] le 120574119889119886forall119896 (gt 1198960) isin Z+
(34)
for some 1205740119889119886 120574119889119886(gt 1205740119889119886) isin R+ and for any nonidentically zerocontrols and zero initial conditions ie for any forced solution
(ii) Assume in addition that the class Φ119889(119879) consistsof subadditive functions Assume also that sup119905isinR0+ |120590(119905)| issufficiently small inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infinwhere 120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ and that furthermore
for some 120574119889119886 isin R+ from the first assumption of the theoremand 119906119889[119896] = minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ since Popovrsquosinequality (22) is fulfilled by any given discrete-time feedbackcontroller 120593 isin Φ119889(119879) generating a feedback control 119906119889[119896] =minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ On the other hand the initialconditions can be neglected since the assumption119892119889 isin SPRimplying that the discrete-time transfer function is (strictly)stable which makes their values irrelevant for purposes ofstability analysisThen the discrete frequency response being
applicable to piecewise-constant inputs in-between any twoconsecutive sampling instants is given by
since the subsequence 119892[minus119899] = 0 forall119899 isin Z+ and thetransfer frequency response of a sampling and zero-order-hold operator 119885 of period 119879 defined by 119885(119879 V)(119905) = V(119896119879) =V[119896] for any given V R0+ rarr R and all 119905 isin [119896119879 (119896 + 1)119879)119896 isin Z0+ is given by 119885(119879 i120596) = (1 minus 119890minusi119879120596)i120596 [31 35]It turns out that |119885(119879 i120596)| = |(1 minus 119890minusi119879120596)i120596| lt 1 forall120596( =0) isin R and |119885(119879 i0)| = 1 Thus the auxiliary discretizedsystem satisfies the subsequent equivalence relation in thefrequency domain to the discrete-time relation (11) It turnsout that Fourier transforms exist in the impulse responsesand truncated control and auxiliary functions of (11)ndash(13)since the truncated functions are square-integrable in clR =[minusinfin +infin] By using the discrete Parsevalrsquos theorem underzero initial conditions one has
The lower-bounding constraint of (37) follows by using(39) and the discrete Parsevalrsquos theorem for the equivalencebetween the input-output energy measures from the fre-quency domain to the discrete-time domain
119864119889 [119896] ge 1198792120587 min120596isinR0+
1199062119889 [119896]) ge 1205740119889forall119896 isin Z0+
(40)
from the assumption 119892119889 isin SPR iemin120596isinR0+ Re119892119889(119890minusi120596119879) ge 1205740119889119886 gt 0 forall1198960 119896 (119896 ge 1198960) isin Z+since
(a) strictly positive real discrete transfer functions havezero relative degree (ie an identical number of polesand zeros) from the Discrete Positive Real Lemma(Szego-Kalman-Popov Lemma) [13 31 34] so thatthe real parts of their frequency hodographs in theargument 119890minusi120596119879 are positively lower-bounded
10 Mathematical Problems in Engineering
(b) Im119892119889(119890minusi120596119879) = minus Im119892119889(119890i120596119879) forall120596 isin R0+ so that theirintegrals in the argument 120596 on frequency intervals[minus119896120587 119896120587] for any 119896 isin Z+ are null
Property (i) has been proved To prove Property (ii) notefrom (28) [Lemma 8(ii)] that
119892 [(119896 minus 119895) 119879])119889120591]]sdot (120593[ℓ] (119910119889 [119895] 119895) + 120593[ℓ] (minus119910119889 [119895] 119895))sdot (120593[ℓ] (119910119889 [ℓ] 119895) + 120593[ℓ] (minus119910119889 [ℓ] ℓ)) minus 119910119889ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 le 120574119889 = 119870120574119889119886
forall119896 isin Z+
(41)
for some 119870 isin R+ provided that sup119905isinR0+ |120590(119905)| is sufficientlysmall inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infin where120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ since 119892(119905) is bounded andconverges exponentially to zero since 119892119889 isin SPR and thenconvergent (that is strictly stable in the discrete context)119910119889ℎ[119896] rarr 0 and it is a bounded sequence
The next result which is a preparatory result to thenestablish the main asymptotic hyperstability result addressesthe relevant property that the input-output measure 119864(119905) ofthe continuous-time system is positively lower-bounded andfinitely upper-bounded for all time under the conditions ofLemmas 4 and 9
Lemma 10 If 120582 and 120590 are sufficiently small then
1205740 minus 120574119889 le 119864 (119905) minus 119864119889 [119896]le 1 + 12058222120587 intinfin
provided that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+Proof One gets the following from direct calculations byusing the continuous-time control laws (9)-(10) and (16) if120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)|
minus 12058222120587 intinfin
minusinfin119892 (i120596) 1003816100381610038161003816[119896] (i120596)10038161003816100381610038162 119889120596 minus 120590(2120582 sup
0le119895le119896
1199062[119895] + 120590)int1198961198790
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591 le 119864 (119905) minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119910 (i120596)
sdot (minusi120596) 119889120596 minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119892 (i120596) | (i120596)|2 119889120596 minus 119864119889 (119896119879 119905)
Now the use of (44) in Lemma 4 (14) implies thatLemma 9(ii) (36) holds implying the existence of 1205740 (120574 gt1205740) isin R+ such that 1205740 le 119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin)provided that 1205741198890 le 119864119889[119896] le 120574119889 for 1205741198890 (120574119889 gt 1205741198890) isin R+ since1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isin Z0+ if1205740 minus 120574119889 le minus 120582
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591le 120574 minus 1205741198890 forall119896 (gt 1198960) isin Z+ forall120591 isin (0 119879)
(45)
if 120582 and 120590 are sufficiently small such that (43) holds leadingto 1205740 gt 0 for any given positive 1205741198890 and 120574119889 gt 1205741198890
The main first result follows
Theorem 11 Consider the dynamic system (2) subject to acontrol
forall119896 isin Z0+ where 120593 [119896] isin Φ119889 (119879) (46)
Assume that(1) 120593(119910[119896] 119896119879) is in the class Φ119889(119879) (that is120593(119910[119896] 119896119879) isin Φ119889(119879)) defined by the discrete Popovrsquos
type sum inequality sum119896ℓ=0 119910119889[ℓ]119906[ℓ] ge minus120574119889119879 for some 120574119889 isin Rforall119896 isin Z0+(2) 119892119889 isin 119878119875119877 there exist constants 1205741198890 (120574119889 gt 1205741198890) isin R+
such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isinZ0+
(3) 120593(119909 119896119879) is subadditive for all 119909 isin R forall119896 isin Z0+(4) 120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)| are
119895119879(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)
minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119896 isin Z0+(47)
Then the following properties hold(i) The discrete closed-loop system is asymptotically hyper-
stable(ii) 119906(119905) rarr 0 as 119905 rarr infin and the control and output are
bounded for all time
(iii) Assume furthermore that 120590(119905) = 120593(119910(119905) 119905) minus (1 +120582(119905))120593(119910[119896] 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+ 119906(119905) =minus120593(119910(119905) 119905) and forall119905 isin R0+ and that the class Φ satisfiesPopovrsquos type integral inequality for all time and results in theclass Φ119889(119896119879) at sampling instantsThen the continuous- timeclosed-loop system is asymptotically hyperstable
Proof From Lemma 9 via assumptions (1)ndash(5) constraints(36) hold so that the input- output energymeasure associatedwith the forced solution is positively lower-bounded andfinitely upper-bounded for all sampling instants 119905 = 119896119879forall119896 isin Z+ Since the discrete positive transfer function isstrictly positive real then it is also strictly stable in the discretecontext (ie convergent [29 36]) so that the homogeneousresponse is irrelevant for stability analysis purposes Thus120593[119896] isin Φ119889 119906[119896] rarr 0 and 119906[119896] and 119910[119896] arebounded for all time for any given bounded initial conditionsIn addition one has from Lemma 10 that (42) holds andthe forced input-output energy measure is positively lower-bounded and finitely upper-bound for all 119905 isin R+ Then119906(119905) rarr 0 as 119905 rarr infin and 119906(119905) and 119910(119905) are bounded for alltime for any given bounded initial conditions Properties [(i)-(ii)] have been proved Property (iii) follows by equalizing119906(119905) = minus120593(119910(119905) 119905) = (1 + 120582(119905))119906[119896] + 120590(119905) and 120593 isin Φ fromthe fact that minusint119905
0119906(120591)119910(120591)119889120591 le 1205740
Note that the use of Fourier transforms is useful tocharacterize the error of the input-output energy measure inthe frequency domain along the intersample periods relatedto its values at the preceding sampling instants This toolis addressed in Lemma 10 which is used in the proof ofTheorem 11 It allows to keep easily the characterizationof the frequency response of the continuous-time transferfunction to evaluate the input-output energy measure in theintersample period and to calculate the minimum value ofthe real part of such a transfer function to guarantee thepositivity and boundedness of such an energy Note that thecharacterization of the input-output energy measure of thefeed-forward linear loop evaluated in the frequency domaintogether with Popovrsquos inequalities of the class of hyperstablecontrollers is relevant to the hyperstability proofs of theclosed-loop system
5 Examples
Example 12 The positive realness of a discretized transferfunction depends not only on the continuous-time param-eters but on the sampling period as well Therefore thepositive realness of the continuous-time system does notguarantee that of any discretized counterpart irrespective ofthe sampling period and conversely For instance the transferfunction 119892(119904) = 1(119904 + 119886) is strictly positive real for 119886 gt 0 butit is not strongly strictly positive real since Re119892(119894120596) rarr 0 as120596 rarr plusmninfin Its discretized version without the concourse ofan ideal that is instantaneous sampler and zero-order holdbecomes 119904 119892119889(119911) = 119911(119911 minus 119890minus119886119879) which is strongly positivereal for 119879 isin (0infin) and as the sampling period 119879 rarr infin since119892119889(119911) rarr 1 as 119879 rarr infin However 119892119889(119911) = 119911(119911 minus 1) as 119879 rarr 0
12 Mathematical Problems in Engineering
Since it has a critically stable pole in the discrete domain it ispositive real although it is not strictly positive real
Example 13 Consider the discrete transfer function 119892119889(119911) =(1199112 + 1198891119911 + 1198892)(1199112 minus 1198861119911 minus 1198862) with real numerator anddenominator coefficients Its strict positive realness requiresas necessary conditions its convergence (ie strict stabilityin the discrete sense) so that its poles have to be within thecomplex open unit circle as well as the convergence of itsinverse so that its zeros have also to lie inside the open unitcircle and furthermore its relative degree (ie its pole-zeroexcess has to be zero) Note that the last condition is notneeded for strict (nonstrong) positive realness of continuoustransfer functions where their positive real part can convergeto zero as |119904| rarr infin By using the homographic bilineartransformation 119911 = (1 + 119904)(1 minus 119904) which transforms theunit circle in the left-hand-side complex plane to get theauxiliary continuous transfer function 119892119886(119904) = 119892119889(119911 = (1 +119904)(1 minus 119904)) it is found in [13] that the denominator andnumerator polynomials of 119892119889(119911) are subject to 1198862 minus 1198861 lt 11198861 + 1198862 lt 1 and 1198862 gt minus1 Take the particular values 1198861 = 05and 1198862 = 04 leading to the values of poles 1199111 = 05 and1199112 = 04 and the coefficients of the numerator polynomialare fixed to 1198891 = 12 1198892 = 03 Then we fix a discretetransfer function 119892119889(119911) = (1199112 + 12119911 + 03)(1199112 minus 05119911 minus 04)yielding min|119911|=1 Re119892119889(119911) ge 009 so 119892119889 isin SSPR Thedecomposition of 119892119889(119911) in simple fractions leads to 119892119889(119911) =1 + 168(119911 minus 093) + 0758(119911 + 043) By using an ideal (iewith instantaneous sampling) sampler and zero-order-holddevice of period 119879 of transfer function (1 minus 119890minus119879119904)119904 on acontinuous transfer function 119892(119904) to be specified to generate119892119889(119911) one has the identity 119885((1 minus 119890minus119879119904)119892(119904)119904) = 119892119889(119911) where119885(119865(119904)) is an abbreviate usual informal notation to denote119865lowast(119904) = 119871(119891lowast(119905)) where 119871(sdot) and 119885(sdot) denote the Laplacetransform of (sdot) of argument 119904 and 119885-transform of (sdot) ofargument 119911 = 119890119879119904 and 119891lowast(119905) = suminfin119899=0 119891(119899119879)120575(119905 minus 119899119879) denotesthe pulsed function of the function119891(119905)with sampling period119879 As a result we get for a sampling period 119879 = 01 sec119892 (119904) = 119904119885minus1 ( 119911119911 minus 1 [1 + 168119911 minus 093 + 0758119911 + 043])
Note that 119892 isin SPR (and also 119892 isin SSPR sinceits relative degree is zero) is the continuous-time transferfunction which generates 119892119889(119911) under an ideal sampler andzero-order-hold device of period 119879 = 01 sec so thatmin120596isinclR119892(i120596) = min120596isinR0+119892(i120596) gt 0 Thus it is guaranteedthat the continuous-time energy measure is associated withthe forced output 119864(119905) gt 0 for all 119905 gt 0 (ie it ispositively lower-bounded for all time) for any piecewise-continuous control irrespectively of the particular controllerin operation In order to guarantee that any such a controlthat converges asymptotically to zero it is necessary to finitelyupper-bound such an energy for all timeThe above concerns
also guarantee that 119864[119896] = 119864(119896119879) ge 1205741198890 gt 0 for all integer 119896 gt0 under any piecewise-constant control 119906[119896] = 119906(119896119879) witheventual jumps at sampling instants of any period 119879 gt 0 Forsuch a control identical conclusion arises from the discrete119892119889(119911)which is also strongly strictly positive real Assume that119906[119896] = minus120593(119910[119896] 119896) and forall119896 isin Z+ with 120593 isin Φ119889 where theclass Φ119889 is defined as the set of sequences 120593(119910[119896] 119896)whichsatisfy Popovrsquos inequality 119879sum119896119895=0 119910[119896]120593(119910[119896] 119896) ge minus120574119889 forall119896 isinZ+ for some finite 120574119886 isin R+ which is identical to
= minus119879119872(1 minus ]119896+1)1 minus ] minus 119879 119896sum119895=0
120576 [119896] ge minus120574119889 forall119896 isin Z+(50)
for some real constants 119872 gt 0 ] isin (0 1) and some realsummable sequence 120576[119896] Then the class Φ119889 is definedby any real sequence 120593(119910[119896] 119896) being parameterized by anyreal constants 119872 gt 0 ] isin (0 1) and any real summablesequence 120576[119896] such that
forall119896 isin Z0+ and 120574119889 ge max(119879119872(1 minus ]) + 120576 120574) for some given120574 ge 1205741198890 where 120576 = 119879max119896isinZ+ sum119896119895=0 120576[119896]The continuous-timecontrol is generated as
where 120593[119896] isin Φ119889(119879) satisfies Popovrsquos type discrete inequal-ity which makes the feedback system globally asymptoticallystable under the theorem hypothesis Thus the discrete-timeclosed-loop system is asymptotically hyperstable In partic-ular under the additional assumptions in Theorem 11(iii)the closed-loop continuous-time system is asymptoticallyhyperstable
Mathematical Problems in Engineering 13
One of the referees has given a comment of interest on thisexample related to continuous versus discrete counterpartproperties which has motivated us to give a further intuitivediscussion on it Note the following In the continuous-timecase there are two kinds of strictly positive real transferfunctions as it is well known namely the so-called strictpositive real which can be zero (or respectively positive)at infinity frequency and which a realizable has relativedegree one (or respectively zero) In the first case the directinput-output interconnection gain 119889 is zero in a state-spacerepresentation (119860 119887 119888 119889 = 0) of the transfer function Thetransfer functions of the second case are often referred to asthe class of strong strict positive real transfer functions Thisclass is not a disjoint class of first above one of strict positivereal transfer functions since it is included in it as a potentialparticular case Note that the members of this strong classof strict positive real transfer functions have relative degreezero and a positive direct input-output interconnection gain119889 Related to this concern the Yakubovich-Kalman-Popov(refereed to the continuous-time domain) and the Szego-Kalman-Popov (referred to the discrete domain) lemmascan be invoked to compare positive realness of the transferfunctions with their state-space associated properties (in factthe properties aremutually equivalent fromboth sides)Thuswe see that in the discrete case the Szego-Kalman-Popovlemma needs for its own coherency that the input-outputinterconnection 119889-gain be nonzero since otherwise one ofthe relevant constraint in the lemma fails [13] So strictpositive realness in the discrete domain in this context istypically strong and then the relative degree of its transferfunction is zero so that there is a direct interconnectionpositive gain from the input to the output Since we assumethat the discrete system is strictly positive real it has a positiveinput-output interconnection from the input to the outputand the minimum value of the real part of its frequencyresponse is strictly positive even as the frequency tends toinfinity On the other hand note that a zero-order holdldquoconvertsrdquo any strictly proper (continuous-time) transferfunction of any relative degree into a strictly discrete oneof unity relative order while it ldquoconvertsrdquo a continuous oneof zero relative order (ie biproper) into a discrete onestill of zero-relative order and with the same input-outputinterconnection gain See (48) Therefore we deal in thisexample with a biproper strict positive real transfer functionof positive input-output interconnection gain identical to thatof its discrete-time counterpart got through a ZOH samplingand hold device
Example 14 Consider that a controller of the form 119906(119905) =minus120593(119910(119905) 119905) forall119905 isin R0+ is used for the continuous-time feed-forward transfer function (48) so that120593 isin Φ and Φ is classin Example 13 satisfying an integral type Popovrsquos inequality(21) for all 119905(ge 1199050) isin R+ and some finite 1199050 isin R+ That isthe control function belongs to a class satisfying continuous-time integral Popovrsquos inequality Then 119906(119905) rarr 0 as 119905 rarr infinand 119906(119905) and 119910(119905) are almost everywhere bounded for all timesince 119892 isin SSPR then being (strictly) stable with input-output energy measure being positively lower-bounded andalso finitely upper-bounded for all 119905 isin R+ from Popovrsquos-type
integral inequality Potential useful controllers are for somegiven controller gain 119896119906 isin R+
119906 (119905) = 0 for 119905 isin [0 1)1198961199061199052119910 (119905) for 119905 ge 1 (54)
119906 (119905) = 119896119906119890minus119905119910 (119905) for 119905 ge 0 (55)
119906 (119905) = 0 for 119905 isin [0 2)
119896119906(119905 ln2119905) 119910 (119905) for 119905 ge 2 (56)
119906 (119905) = 0 for 119905 isin [0 1)
119896119906(119890radic119905radic119905) 119910 (119905) for 119905 ge 1 (57)
119906 (119905) = 1198961199061199053119890minus119905119910 (119905) for 119905 ge 0 (58)
where Popovrsquos inequality on [0 +infin] of (49)ndash(52)is intinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus119896119906 that of (57) isintinfin
0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 and that of (58) isintinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 Since the respective integrands
are monotonically decreasing then there is some 1199050 isin R+ andsome 120574 = 120574(1199050) isin R+ such that int119905
1199050119910(119905)120593(119910(119905) 119905)119889119905 ge minus120574 for
all 119905(ge 1199050) isin R+ The obtained closed-loop systems are inall cases asymptotically hyperstable The piecewise-constantcontrols with eventual discontinuities at sampling instantsgenerated from any discrete-time controller (51) of class Φ119889of Example 13 lead also to asymptotic hyperstability of thediscretized system since 119892119889 isin SSPRExample 15 Consider the continuous-time open-loop sys-tem (2) under a real sampler of sampling period 119879 and datapicking-up nonzero duration 120576 isin (0 119879) The state and outputtrajectory solutions are
forall119896 isin Z0+ for some real sequence 120585[119896] such that 120585[119896] isin(119896119879 119896119879 + 120576) where 119892119889(119911) is the discretized transfer functionunder a sampler and zero-order-hold device subject to instan-taneous ideal sampling Now assume that the incrementalcontrol along intersample periods is generated from itspreceding sampled value as (119895119879+ 120591) = 120582119895119879+120576(119895119879+ 120591)119906[119895] forany 119895 isin Z0+ and 120591 isin [0 119879] where 120582119895119879+120576(119895119879+120591) is a truncatedfunction under the usual notation one gets
The above equations imply that if 119892119889 isin SPR then the forcedoutput 119910119891[119896] = 119910[119896] minus 119888119879119890119896119860119879119909[0] forall119896 isin Z0+ generates anassociate input-output energy measure which is strictly posi-tive for 119896 isin Z0+ since 120588 isin (0 1) for real sampling of duration120576 isin (0 119879)This is a necessary condition for asymptotic hyper-stability under an appropriate class of controllers satisfyingPopovrsquos inequality However the incremental input-outputenergy measure generated by the incremental forced output120576119888119879sum119896minus1119895=0 119890(119896minus119895)119860119879(int1198790 120582119895119879+120576(120591)119890minus119860120591119887 119889120591)119906[119895] is not guaranteedto be nonnegative in general A sufficient condition for that isthat lim sup119895rarrinfin|ess sup120591isin(0120576)(120582119895119879+120576(120591))| is sufficiently smallrelated to the amount ((1 minus 120588)120576)min120596isin[0120587] Re119892119889(119890i120596)6 Conclusions
This paper has been devoted to obtain and to prove theasymptotic hyperstability property of a linear and time-invariant continuous-time system based on some given andpreviously proved auxiliary preparatory lemmas Basicallyit is proved that the continuous-time closed-loop system isasymptotically hyperstable under the assumptions that thediscretized feed-forward transfer function is strictly positivereal and that the discretized system at sampling instants ofconstant sampling period is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsIt is assumed that the continuous-time controller belongs toa class defined in the following way (a) its values at samplinginstants are defined by a class of controllers which satisfydiscrete Popovrsquos type inequality (b) the intersample control isgenerated from its values at sampling instants with the use oftwomodulating auxiliary functionswhich are subject to someextra additional conditions on smallness and boundednessSuch conditions make the overall continuous-time controllerto satisfy continuous-type Popovrsquos inequality for all time
where 119896(120591) = max(119911 isin Z+ 119911119879 le 120591) and 119896 = 119896(119905) = max(119911 isinZ+ 119911119879 le 119905) forall119905 isin R0+ which yields (13)
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
Hindawiwwwhindawicom Volume 2018
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Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
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Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119895 (le 119896) 119896 isin Z0+
(26)
(ii) Assume that 120593 isin Φ119889(119879) and that 120593(119909 119896119879) issubadditive for all 119909 isin R ie 120593(119909 + 119910 119896119879) le 120593(119909 119896119879) +120593(119910 119896119879) forall119909 119910 isin R forall119896 isin Z0+ Then the subsequentrelations are true if the sequence 120576[119896] is defined by 120576[119896] =120582[119896] + 120585[119896] forall119896 isin Z0+119910119889ℎ [119896] + intinfin
0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591
which yields (24) and leads to (25) Now define the sequence120576[119896] by 120576[119896] = 120582[119896] + 120585[119896] and 120585[119896][119895] = int(119895+1)119879119895119879(120582(119895119879 +120591)119892((119896 minus 119895)119879 minus 120591) minus 120582[119895]119892[(119896 minus 119895)119879])119889120591 forall119896 isin Z0+ and note
that
minus119910 [119896] = minus (119910119889 [119896] + (119910 [119896] minus 119910119889 [119896]))= [[infinsum119895=0
(int(119895+1)119879119895119879
(1 + 120576[119896] [119895]) 119892 [(119896 minus 119895) 119879])119889120591]]sdot 120593[119896] ([119910119889 [119895] + (119910 [119895] minus 119910119889 [119895]) 119895])minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 minus 119910ℎ [119896]
forall119896 isin Z0+
(31)
and then
minus (119910 [119896] minus 119910119889 [119896] minus 119910ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591)
The subsequent result establishes lower and upper-bounds for the discretized input-output energy measuresbased on the formulas obtained in Lemma 8 and strictpositive realness of the discrete-time transfer function ofthe feed-forward linear and time-invariant system Theresult invokes the boundedness and the smallness of theparameters which define the calculation of the intersam-ple control input from their preceding values at samplinginstants Note that the control deviation in the intersampleperiods related to the sampling time instants has to be suffi-ciently moderate enough so that the hyperstability propertyis kept from the discrete-time system to the continuous-time one In particular the positivity and boundedness ofthe energy measure at the sampling instants is guaranteedif sup119905isinR0+ |120590(119905)| is small enough and Φ119889(119879) is subaddi-tive
Mathematical Problems in Engineering 9
Lemma 9 Assume that 119892119889 isin 119878119875119877 and that 120593 isin Φ119889(119879)Then the following properties hold(i)The input-output energy systemmeasure of the auxiliary
discretized system satisfies
0 lt 1205740119889119886 le 1198792120587 min120596isinR0+
119879119910119889 [119896] 119906119889 [119896] le 120574119889119886forall119896 (gt 1198960) isin Z+
(34)
for some 1205740119889119886 120574119889119886(gt 1205740119889119886) isin R+ and for any nonidentically zerocontrols and zero initial conditions ie for any forced solution
(ii) Assume in addition that the class Φ119889(119879) consistsof subadditive functions Assume also that sup119905isinR0+ |120590(119905)| issufficiently small inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infinwhere 120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ and that furthermore
for some 120574119889119886 isin R+ from the first assumption of the theoremand 119906119889[119896] = minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ since Popovrsquosinequality (22) is fulfilled by any given discrete-time feedbackcontroller 120593 isin Φ119889(119879) generating a feedback control 119906119889[119896] =minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ On the other hand the initialconditions can be neglected since the assumption119892119889 isin SPRimplying that the discrete-time transfer function is (strictly)stable which makes their values irrelevant for purposes ofstability analysisThen the discrete frequency response being
applicable to piecewise-constant inputs in-between any twoconsecutive sampling instants is given by
since the subsequence 119892[minus119899] = 0 forall119899 isin Z+ and thetransfer frequency response of a sampling and zero-order-hold operator 119885 of period 119879 defined by 119885(119879 V)(119905) = V(119896119879) =V[119896] for any given V R0+ rarr R and all 119905 isin [119896119879 (119896 + 1)119879)119896 isin Z0+ is given by 119885(119879 i120596) = (1 minus 119890minusi119879120596)i120596 [31 35]It turns out that |119885(119879 i120596)| = |(1 minus 119890minusi119879120596)i120596| lt 1 forall120596( =0) isin R and |119885(119879 i0)| = 1 Thus the auxiliary discretizedsystem satisfies the subsequent equivalence relation in thefrequency domain to the discrete-time relation (11) It turnsout that Fourier transforms exist in the impulse responsesand truncated control and auxiliary functions of (11)ndash(13)since the truncated functions are square-integrable in clR =[minusinfin +infin] By using the discrete Parsevalrsquos theorem underzero initial conditions one has
The lower-bounding constraint of (37) follows by using(39) and the discrete Parsevalrsquos theorem for the equivalencebetween the input-output energy measures from the fre-quency domain to the discrete-time domain
119864119889 [119896] ge 1198792120587 min120596isinR0+
1199062119889 [119896]) ge 1205740119889forall119896 isin Z0+
(40)
from the assumption 119892119889 isin SPR iemin120596isinR0+ Re119892119889(119890minusi120596119879) ge 1205740119889119886 gt 0 forall1198960 119896 (119896 ge 1198960) isin Z+since
(a) strictly positive real discrete transfer functions havezero relative degree (ie an identical number of polesand zeros) from the Discrete Positive Real Lemma(Szego-Kalman-Popov Lemma) [13 31 34] so thatthe real parts of their frequency hodographs in theargument 119890minusi120596119879 are positively lower-bounded
10 Mathematical Problems in Engineering
(b) Im119892119889(119890minusi120596119879) = minus Im119892119889(119890i120596119879) forall120596 isin R0+ so that theirintegrals in the argument 120596 on frequency intervals[minus119896120587 119896120587] for any 119896 isin Z+ are null
Property (i) has been proved To prove Property (ii) notefrom (28) [Lemma 8(ii)] that
119892 [(119896 minus 119895) 119879])119889120591]]sdot (120593[ℓ] (119910119889 [119895] 119895) + 120593[ℓ] (minus119910119889 [119895] 119895))sdot (120593[ℓ] (119910119889 [ℓ] 119895) + 120593[ℓ] (minus119910119889 [ℓ] ℓ)) minus 119910119889ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 le 120574119889 = 119870120574119889119886
forall119896 isin Z+
(41)
for some 119870 isin R+ provided that sup119905isinR0+ |120590(119905)| is sufficientlysmall inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infin where120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ since 119892(119905) is bounded andconverges exponentially to zero since 119892119889 isin SPR and thenconvergent (that is strictly stable in the discrete context)119910119889ℎ[119896] rarr 0 and it is a bounded sequence
The next result which is a preparatory result to thenestablish the main asymptotic hyperstability result addressesthe relevant property that the input-output measure 119864(119905) ofthe continuous-time system is positively lower-bounded andfinitely upper-bounded for all time under the conditions ofLemmas 4 and 9
Lemma 10 If 120582 and 120590 are sufficiently small then
1205740 minus 120574119889 le 119864 (119905) minus 119864119889 [119896]le 1 + 12058222120587 intinfin
provided that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+Proof One gets the following from direct calculations byusing the continuous-time control laws (9)-(10) and (16) if120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)|
minus 12058222120587 intinfin
minusinfin119892 (i120596) 1003816100381610038161003816[119896] (i120596)10038161003816100381610038162 119889120596 minus 120590(2120582 sup
0le119895le119896
1199062[119895] + 120590)int1198961198790
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591 le 119864 (119905) minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119910 (i120596)
sdot (minusi120596) 119889120596 minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119892 (i120596) | (i120596)|2 119889120596 minus 119864119889 (119896119879 119905)
Now the use of (44) in Lemma 4 (14) implies thatLemma 9(ii) (36) holds implying the existence of 1205740 (120574 gt1205740) isin R+ such that 1205740 le 119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin)provided that 1205741198890 le 119864119889[119896] le 120574119889 for 1205741198890 (120574119889 gt 1205741198890) isin R+ since1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isin Z0+ if1205740 minus 120574119889 le minus 120582
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591le 120574 minus 1205741198890 forall119896 (gt 1198960) isin Z+ forall120591 isin (0 119879)
(45)
if 120582 and 120590 are sufficiently small such that (43) holds leadingto 1205740 gt 0 for any given positive 1205741198890 and 120574119889 gt 1205741198890
The main first result follows
Theorem 11 Consider the dynamic system (2) subject to acontrol
forall119896 isin Z0+ where 120593 [119896] isin Φ119889 (119879) (46)
Assume that(1) 120593(119910[119896] 119896119879) is in the class Φ119889(119879) (that is120593(119910[119896] 119896119879) isin Φ119889(119879)) defined by the discrete Popovrsquos
type sum inequality sum119896ℓ=0 119910119889[ℓ]119906[ℓ] ge minus120574119889119879 for some 120574119889 isin Rforall119896 isin Z0+(2) 119892119889 isin 119878119875119877 there exist constants 1205741198890 (120574119889 gt 1205741198890) isin R+
such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isinZ0+
(3) 120593(119909 119896119879) is subadditive for all 119909 isin R forall119896 isin Z0+(4) 120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)| are
119895119879(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)
minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119896 isin Z0+(47)
Then the following properties hold(i) The discrete closed-loop system is asymptotically hyper-
stable(ii) 119906(119905) rarr 0 as 119905 rarr infin and the control and output are
bounded for all time
(iii) Assume furthermore that 120590(119905) = 120593(119910(119905) 119905) minus (1 +120582(119905))120593(119910[119896] 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+ 119906(119905) =minus120593(119910(119905) 119905) and forall119905 isin R0+ and that the class Φ satisfiesPopovrsquos type integral inequality for all time and results in theclass Φ119889(119896119879) at sampling instantsThen the continuous- timeclosed-loop system is asymptotically hyperstable
Proof From Lemma 9 via assumptions (1)ndash(5) constraints(36) hold so that the input- output energymeasure associatedwith the forced solution is positively lower-bounded andfinitely upper-bounded for all sampling instants 119905 = 119896119879forall119896 isin Z+ Since the discrete positive transfer function isstrictly positive real then it is also strictly stable in the discretecontext (ie convergent [29 36]) so that the homogeneousresponse is irrelevant for stability analysis purposes Thus120593[119896] isin Φ119889 119906[119896] rarr 0 and 119906[119896] and 119910[119896] arebounded for all time for any given bounded initial conditionsIn addition one has from Lemma 10 that (42) holds andthe forced input-output energy measure is positively lower-bounded and finitely upper-bound for all 119905 isin R+ Then119906(119905) rarr 0 as 119905 rarr infin and 119906(119905) and 119910(119905) are bounded for alltime for any given bounded initial conditions Properties [(i)-(ii)] have been proved Property (iii) follows by equalizing119906(119905) = minus120593(119910(119905) 119905) = (1 + 120582(119905))119906[119896] + 120590(119905) and 120593 isin Φ fromthe fact that minusint119905
0119906(120591)119910(120591)119889120591 le 1205740
Note that the use of Fourier transforms is useful tocharacterize the error of the input-output energy measure inthe frequency domain along the intersample periods relatedto its values at the preceding sampling instants This toolis addressed in Lemma 10 which is used in the proof ofTheorem 11 It allows to keep easily the characterizationof the frequency response of the continuous-time transferfunction to evaluate the input-output energy measure in theintersample period and to calculate the minimum value ofthe real part of such a transfer function to guarantee thepositivity and boundedness of such an energy Note that thecharacterization of the input-output energy measure of thefeed-forward linear loop evaluated in the frequency domaintogether with Popovrsquos inequalities of the class of hyperstablecontrollers is relevant to the hyperstability proofs of theclosed-loop system
5 Examples
Example 12 The positive realness of a discretized transferfunction depends not only on the continuous-time param-eters but on the sampling period as well Therefore thepositive realness of the continuous-time system does notguarantee that of any discretized counterpart irrespective ofthe sampling period and conversely For instance the transferfunction 119892(119904) = 1(119904 + 119886) is strictly positive real for 119886 gt 0 butit is not strongly strictly positive real since Re119892(119894120596) rarr 0 as120596 rarr plusmninfin Its discretized version without the concourse ofan ideal that is instantaneous sampler and zero-order holdbecomes 119904 119892119889(119911) = 119911(119911 minus 119890minus119886119879) which is strongly positivereal for 119879 isin (0infin) and as the sampling period 119879 rarr infin since119892119889(119911) rarr 1 as 119879 rarr infin However 119892119889(119911) = 119911(119911 minus 1) as 119879 rarr 0
12 Mathematical Problems in Engineering
Since it has a critically stable pole in the discrete domain it ispositive real although it is not strictly positive real
Example 13 Consider the discrete transfer function 119892119889(119911) =(1199112 + 1198891119911 + 1198892)(1199112 minus 1198861119911 minus 1198862) with real numerator anddenominator coefficients Its strict positive realness requiresas necessary conditions its convergence (ie strict stabilityin the discrete sense) so that its poles have to be within thecomplex open unit circle as well as the convergence of itsinverse so that its zeros have also to lie inside the open unitcircle and furthermore its relative degree (ie its pole-zeroexcess has to be zero) Note that the last condition is notneeded for strict (nonstrong) positive realness of continuoustransfer functions where their positive real part can convergeto zero as |119904| rarr infin By using the homographic bilineartransformation 119911 = (1 + 119904)(1 minus 119904) which transforms theunit circle in the left-hand-side complex plane to get theauxiliary continuous transfer function 119892119886(119904) = 119892119889(119911 = (1 +119904)(1 minus 119904)) it is found in [13] that the denominator andnumerator polynomials of 119892119889(119911) are subject to 1198862 minus 1198861 lt 11198861 + 1198862 lt 1 and 1198862 gt minus1 Take the particular values 1198861 = 05and 1198862 = 04 leading to the values of poles 1199111 = 05 and1199112 = 04 and the coefficients of the numerator polynomialare fixed to 1198891 = 12 1198892 = 03 Then we fix a discretetransfer function 119892119889(119911) = (1199112 + 12119911 + 03)(1199112 minus 05119911 minus 04)yielding min|119911|=1 Re119892119889(119911) ge 009 so 119892119889 isin SSPR Thedecomposition of 119892119889(119911) in simple fractions leads to 119892119889(119911) =1 + 168(119911 minus 093) + 0758(119911 + 043) By using an ideal (iewith instantaneous sampling) sampler and zero-order-holddevice of period 119879 of transfer function (1 minus 119890minus119879119904)119904 on acontinuous transfer function 119892(119904) to be specified to generate119892119889(119911) one has the identity 119885((1 minus 119890minus119879119904)119892(119904)119904) = 119892119889(119911) where119885(119865(119904)) is an abbreviate usual informal notation to denote119865lowast(119904) = 119871(119891lowast(119905)) where 119871(sdot) and 119885(sdot) denote the Laplacetransform of (sdot) of argument 119904 and 119885-transform of (sdot) ofargument 119911 = 119890119879119904 and 119891lowast(119905) = suminfin119899=0 119891(119899119879)120575(119905 minus 119899119879) denotesthe pulsed function of the function119891(119905)with sampling period119879 As a result we get for a sampling period 119879 = 01 sec119892 (119904) = 119904119885minus1 ( 119911119911 minus 1 [1 + 168119911 minus 093 + 0758119911 + 043])
Note that 119892 isin SPR (and also 119892 isin SSPR sinceits relative degree is zero) is the continuous-time transferfunction which generates 119892119889(119911) under an ideal sampler andzero-order-hold device of period 119879 = 01 sec so thatmin120596isinclR119892(i120596) = min120596isinR0+119892(i120596) gt 0 Thus it is guaranteedthat the continuous-time energy measure is associated withthe forced output 119864(119905) gt 0 for all 119905 gt 0 (ie it ispositively lower-bounded for all time) for any piecewise-continuous control irrespectively of the particular controllerin operation In order to guarantee that any such a controlthat converges asymptotically to zero it is necessary to finitelyupper-bound such an energy for all timeThe above concerns
also guarantee that 119864[119896] = 119864(119896119879) ge 1205741198890 gt 0 for all integer 119896 gt0 under any piecewise-constant control 119906[119896] = 119906(119896119879) witheventual jumps at sampling instants of any period 119879 gt 0 Forsuch a control identical conclusion arises from the discrete119892119889(119911)which is also strongly strictly positive real Assume that119906[119896] = minus120593(119910[119896] 119896) and forall119896 isin Z+ with 120593 isin Φ119889 where theclass Φ119889 is defined as the set of sequences 120593(119910[119896] 119896)whichsatisfy Popovrsquos inequality 119879sum119896119895=0 119910[119896]120593(119910[119896] 119896) ge minus120574119889 forall119896 isinZ+ for some finite 120574119886 isin R+ which is identical to
= minus119879119872(1 minus ]119896+1)1 minus ] minus 119879 119896sum119895=0
120576 [119896] ge minus120574119889 forall119896 isin Z+(50)
for some real constants 119872 gt 0 ] isin (0 1) and some realsummable sequence 120576[119896] Then the class Φ119889 is definedby any real sequence 120593(119910[119896] 119896) being parameterized by anyreal constants 119872 gt 0 ] isin (0 1) and any real summablesequence 120576[119896] such that
forall119896 isin Z0+ and 120574119889 ge max(119879119872(1 minus ]) + 120576 120574) for some given120574 ge 1205741198890 where 120576 = 119879max119896isinZ+ sum119896119895=0 120576[119896]The continuous-timecontrol is generated as
where 120593[119896] isin Φ119889(119879) satisfies Popovrsquos type discrete inequal-ity which makes the feedback system globally asymptoticallystable under the theorem hypothesis Thus the discrete-timeclosed-loop system is asymptotically hyperstable In partic-ular under the additional assumptions in Theorem 11(iii)the closed-loop continuous-time system is asymptoticallyhyperstable
Mathematical Problems in Engineering 13
One of the referees has given a comment of interest on thisexample related to continuous versus discrete counterpartproperties which has motivated us to give a further intuitivediscussion on it Note the following In the continuous-timecase there are two kinds of strictly positive real transferfunctions as it is well known namely the so-called strictpositive real which can be zero (or respectively positive)at infinity frequency and which a realizable has relativedegree one (or respectively zero) In the first case the directinput-output interconnection gain 119889 is zero in a state-spacerepresentation (119860 119887 119888 119889 = 0) of the transfer function Thetransfer functions of the second case are often referred to asthe class of strong strict positive real transfer functions Thisclass is not a disjoint class of first above one of strict positivereal transfer functions since it is included in it as a potentialparticular case Note that the members of this strong classof strict positive real transfer functions have relative degreezero and a positive direct input-output interconnection gain119889 Related to this concern the Yakubovich-Kalman-Popov(refereed to the continuous-time domain) and the Szego-Kalman-Popov (referred to the discrete domain) lemmascan be invoked to compare positive realness of the transferfunctions with their state-space associated properties (in factthe properties aremutually equivalent fromboth sides)Thuswe see that in the discrete case the Szego-Kalman-Popovlemma needs for its own coherency that the input-outputinterconnection 119889-gain be nonzero since otherwise one ofthe relevant constraint in the lemma fails [13] So strictpositive realness in the discrete domain in this context istypically strong and then the relative degree of its transferfunction is zero so that there is a direct interconnectionpositive gain from the input to the output Since we assumethat the discrete system is strictly positive real it has a positiveinput-output interconnection from the input to the outputand the minimum value of the real part of its frequencyresponse is strictly positive even as the frequency tends toinfinity On the other hand note that a zero-order holdldquoconvertsrdquo any strictly proper (continuous-time) transferfunction of any relative degree into a strictly discrete oneof unity relative order while it ldquoconvertsrdquo a continuous oneof zero relative order (ie biproper) into a discrete onestill of zero-relative order and with the same input-outputinterconnection gain See (48) Therefore we deal in thisexample with a biproper strict positive real transfer functionof positive input-output interconnection gain identical to thatof its discrete-time counterpart got through a ZOH samplingand hold device
Example 14 Consider that a controller of the form 119906(119905) =minus120593(119910(119905) 119905) forall119905 isin R0+ is used for the continuous-time feed-forward transfer function (48) so that120593 isin Φ and Φ is classin Example 13 satisfying an integral type Popovrsquos inequality(21) for all 119905(ge 1199050) isin R+ and some finite 1199050 isin R+ That isthe control function belongs to a class satisfying continuous-time integral Popovrsquos inequality Then 119906(119905) rarr 0 as 119905 rarr infinand 119906(119905) and 119910(119905) are almost everywhere bounded for all timesince 119892 isin SSPR then being (strictly) stable with input-output energy measure being positively lower-bounded andalso finitely upper-bounded for all 119905 isin R+ from Popovrsquos-type
integral inequality Potential useful controllers are for somegiven controller gain 119896119906 isin R+
119906 (119905) = 0 for 119905 isin [0 1)1198961199061199052119910 (119905) for 119905 ge 1 (54)
119906 (119905) = 119896119906119890minus119905119910 (119905) for 119905 ge 0 (55)
119906 (119905) = 0 for 119905 isin [0 2)
119896119906(119905 ln2119905) 119910 (119905) for 119905 ge 2 (56)
119906 (119905) = 0 for 119905 isin [0 1)
119896119906(119890radic119905radic119905) 119910 (119905) for 119905 ge 1 (57)
119906 (119905) = 1198961199061199053119890minus119905119910 (119905) for 119905 ge 0 (58)
where Popovrsquos inequality on [0 +infin] of (49)ndash(52)is intinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus119896119906 that of (57) isintinfin
0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 and that of (58) isintinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 Since the respective integrands
are monotonically decreasing then there is some 1199050 isin R+ andsome 120574 = 120574(1199050) isin R+ such that int119905
1199050119910(119905)120593(119910(119905) 119905)119889119905 ge minus120574 for
all 119905(ge 1199050) isin R+ The obtained closed-loop systems are inall cases asymptotically hyperstable The piecewise-constantcontrols with eventual discontinuities at sampling instantsgenerated from any discrete-time controller (51) of class Φ119889of Example 13 lead also to asymptotic hyperstability of thediscretized system since 119892119889 isin SSPRExample 15 Consider the continuous-time open-loop sys-tem (2) under a real sampler of sampling period 119879 and datapicking-up nonzero duration 120576 isin (0 119879) The state and outputtrajectory solutions are
forall119896 isin Z0+ for some real sequence 120585[119896] such that 120585[119896] isin(119896119879 119896119879 + 120576) where 119892119889(119911) is the discretized transfer functionunder a sampler and zero-order-hold device subject to instan-taneous ideal sampling Now assume that the incrementalcontrol along intersample periods is generated from itspreceding sampled value as (119895119879+ 120591) = 120582119895119879+120576(119895119879+ 120591)119906[119895] forany 119895 isin Z0+ and 120591 isin [0 119879] where 120582119895119879+120576(119895119879+120591) is a truncatedfunction under the usual notation one gets
The above equations imply that if 119892119889 isin SPR then the forcedoutput 119910119891[119896] = 119910[119896] minus 119888119879119890119896119860119879119909[0] forall119896 isin Z0+ generates anassociate input-output energy measure which is strictly posi-tive for 119896 isin Z0+ since 120588 isin (0 1) for real sampling of duration120576 isin (0 119879)This is a necessary condition for asymptotic hyper-stability under an appropriate class of controllers satisfyingPopovrsquos inequality However the incremental input-outputenergy measure generated by the incremental forced output120576119888119879sum119896minus1119895=0 119890(119896minus119895)119860119879(int1198790 120582119895119879+120576(120591)119890minus119860120591119887 119889120591)119906[119895] is not guaranteedto be nonnegative in general A sufficient condition for that isthat lim sup119895rarrinfin|ess sup120591isin(0120576)(120582119895119879+120576(120591))| is sufficiently smallrelated to the amount ((1 minus 120588)120576)min120596isin[0120587] Re119892119889(119890i120596)6 Conclusions
This paper has been devoted to obtain and to prove theasymptotic hyperstability property of a linear and time-invariant continuous-time system based on some given andpreviously proved auxiliary preparatory lemmas Basicallyit is proved that the continuous-time closed-loop system isasymptotically hyperstable under the assumptions that thediscretized feed-forward transfer function is strictly positivereal and that the discretized system at sampling instants ofconstant sampling period is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsIt is assumed that the continuous-time controller belongs toa class defined in the following way (a) its values at samplinginstants are defined by a class of controllers which satisfydiscrete Popovrsquos type inequality (b) the intersample control isgenerated from its values at sampling instants with the use oftwomodulating auxiliary functionswhich are subject to someextra additional conditions on smallness and boundednessSuch conditions make the overall continuous-time controllerto satisfy continuous-type Popovrsquos inequality for all time
where 119896(120591) = max(119911 isin Z+ 119911119879 le 120591) and 119896 = 119896(119905) = max(119911 isinZ+ 119911119879 le 119905) forall119905 isin R0+ which yields (13)
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
Hindawiwwwhindawicom Volume 2018
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Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
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Operations ResearchAdvances in
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Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
which yields (24) and leads to (25) Now define the sequence120576[119896] by 120576[119896] = 120582[119896] + 120585[119896] and 120585[119896][119895] = int(119895+1)119879119895119879(120582(119895119879 +120591)119892((119896 minus 119895)119879 minus 120591) minus 120582[119895]119892[(119896 minus 119895)119879])119889120591 forall119896 isin Z0+ and note
that
minus119910 [119896] = minus (119910119889 [119896] + (119910 [119896] minus 119910119889 [119896]))= [[infinsum119895=0
(int(119895+1)119879119895119879
(1 + 120576[119896] [119895]) 119892 [(119896 minus 119895) 119879])119889120591]]sdot 120593[119896] ([119910119889 [119895] + (119910 [119895] minus 119910119889 [119895]) 119895])minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 minus 119910ℎ [119896]
forall119896 isin Z0+
(31)
and then
minus (119910 [119896] minus 119910119889 [119896] minus 119910ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591)
The subsequent result establishes lower and upper-bounds for the discretized input-output energy measuresbased on the formulas obtained in Lemma 8 and strictpositive realness of the discrete-time transfer function ofthe feed-forward linear and time-invariant system Theresult invokes the boundedness and the smallness of theparameters which define the calculation of the intersam-ple control input from their preceding values at samplinginstants Note that the control deviation in the intersampleperiods related to the sampling time instants has to be suffi-ciently moderate enough so that the hyperstability propertyis kept from the discrete-time system to the continuous-time one In particular the positivity and boundedness ofthe energy measure at the sampling instants is guaranteedif sup119905isinR0+ |120590(119905)| is small enough and Φ119889(119879) is subaddi-tive
Mathematical Problems in Engineering 9
Lemma 9 Assume that 119892119889 isin 119878119875119877 and that 120593 isin Φ119889(119879)Then the following properties hold(i)The input-output energy systemmeasure of the auxiliary
discretized system satisfies
0 lt 1205740119889119886 le 1198792120587 min120596isinR0+
119879119910119889 [119896] 119906119889 [119896] le 120574119889119886forall119896 (gt 1198960) isin Z+
(34)
for some 1205740119889119886 120574119889119886(gt 1205740119889119886) isin R+ and for any nonidentically zerocontrols and zero initial conditions ie for any forced solution
(ii) Assume in addition that the class Φ119889(119879) consistsof subadditive functions Assume also that sup119905isinR0+ |120590(119905)| issufficiently small inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infinwhere 120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ and that furthermore
for some 120574119889119886 isin R+ from the first assumption of the theoremand 119906119889[119896] = minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ since Popovrsquosinequality (22) is fulfilled by any given discrete-time feedbackcontroller 120593 isin Φ119889(119879) generating a feedback control 119906119889[119896] =minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ On the other hand the initialconditions can be neglected since the assumption119892119889 isin SPRimplying that the discrete-time transfer function is (strictly)stable which makes their values irrelevant for purposes ofstability analysisThen the discrete frequency response being
applicable to piecewise-constant inputs in-between any twoconsecutive sampling instants is given by
since the subsequence 119892[minus119899] = 0 forall119899 isin Z+ and thetransfer frequency response of a sampling and zero-order-hold operator 119885 of period 119879 defined by 119885(119879 V)(119905) = V(119896119879) =V[119896] for any given V R0+ rarr R and all 119905 isin [119896119879 (119896 + 1)119879)119896 isin Z0+ is given by 119885(119879 i120596) = (1 minus 119890minusi119879120596)i120596 [31 35]It turns out that |119885(119879 i120596)| = |(1 minus 119890minusi119879120596)i120596| lt 1 forall120596( =0) isin R and |119885(119879 i0)| = 1 Thus the auxiliary discretizedsystem satisfies the subsequent equivalence relation in thefrequency domain to the discrete-time relation (11) It turnsout that Fourier transforms exist in the impulse responsesand truncated control and auxiliary functions of (11)ndash(13)since the truncated functions are square-integrable in clR =[minusinfin +infin] By using the discrete Parsevalrsquos theorem underzero initial conditions one has
The lower-bounding constraint of (37) follows by using(39) and the discrete Parsevalrsquos theorem for the equivalencebetween the input-output energy measures from the fre-quency domain to the discrete-time domain
119864119889 [119896] ge 1198792120587 min120596isinR0+
1199062119889 [119896]) ge 1205740119889forall119896 isin Z0+
(40)
from the assumption 119892119889 isin SPR iemin120596isinR0+ Re119892119889(119890minusi120596119879) ge 1205740119889119886 gt 0 forall1198960 119896 (119896 ge 1198960) isin Z+since
(a) strictly positive real discrete transfer functions havezero relative degree (ie an identical number of polesand zeros) from the Discrete Positive Real Lemma(Szego-Kalman-Popov Lemma) [13 31 34] so thatthe real parts of their frequency hodographs in theargument 119890minusi120596119879 are positively lower-bounded
10 Mathematical Problems in Engineering
(b) Im119892119889(119890minusi120596119879) = minus Im119892119889(119890i120596119879) forall120596 isin R0+ so that theirintegrals in the argument 120596 on frequency intervals[minus119896120587 119896120587] for any 119896 isin Z+ are null
Property (i) has been proved To prove Property (ii) notefrom (28) [Lemma 8(ii)] that
119892 [(119896 minus 119895) 119879])119889120591]]sdot (120593[ℓ] (119910119889 [119895] 119895) + 120593[ℓ] (minus119910119889 [119895] 119895))sdot (120593[ℓ] (119910119889 [ℓ] 119895) + 120593[ℓ] (minus119910119889 [ℓ] ℓ)) minus 119910119889ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 le 120574119889 = 119870120574119889119886
forall119896 isin Z+
(41)
for some 119870 isin R+ provided that sup119905isinR0+ |120590(119905)| is sufficientlysmall inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infin where120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ since 119892(119905) is bounded andconverges exponentially to zero since 119892119889 isin SPR and thenconvergent (that is strictly stable in the discrete context)119910119889ℎ[119896] rarr 0 and it is a bounded sequence
The next result which is a preparatory result to thenestablish the main asymptotic hyperstability result addressesthe relevant property that the input-output measure 119864(119905) ofthe continuous-time system is positively lower-bounded andfinitely upper-bounded for all time under the conditions ofLemmas 4 and 9
Lemma 10 If 120582 and 120590 are sufficiently small then
1205740 minus 120574119889 le 119864 (119905) minus 119864119889 [119896]le 1 + 12058222120587 intinfin
provided that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+Proof One gets the following from direct calculations byusing the continuous-time control laws (9)-(10) and (16) if120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)|
minus 12058222120587 intinfin
minusinfin119892 (i120596) 1003816100381610038161003816[119896] (i120596)10038161003816100381610038162 119889120596 minus 120590(2120582 sup
0le119895le119896
1199062[119895] + 120590)int1198961198790
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591 le 119864 (119905) minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119910 (i120596)
sdot (minusi120596) 119889120596 minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119892 (i120596) | (i120596)|2 119889120596 minus 119864119889 (119896119879 119905)
Now the use of (44) in Lemma 4 (14) implies thatLemma 9(ii) (36) holds implying the existence of 1205740 (120574 gt1205740) isin R+ such that 1205740 le 119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin)provided that 1205741198890 le 119864119889[119896] le 120574119889 for 1205741198890 (120574119889 gt 1205741198890) isin R+ since1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isin Z0+ if1205740 minus 120574119889 le minus 120582
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591le 120574 minus 1205741198890 forall119896 (gt 1198960) isin Z+ forall120591 isin (0 119879)
(45)
if 120582 and 120590 are sufficiently small such that (43) holds leadingto 1205740 gt 0 for any given positive 1205741198890 and 120574119889 gt 1205741198890
The main first result follows
Theorem 11 Consider the dynamic system (2) subject to acontrol
forall119896 isin Z0+ where 120593 [119896] isin Φ119889 (119879) (46)
Assume that(1) 120593(119910[119896] 119896119879) is in the class Φ119889(119879) (that is120593(119910[119896] 119896119879) isin Φ119889(119879)) defined by the discrete Popovrsquos
type sum inequality sum119896ℓ=0 119910119889[ℓ]119906[ℓ] ge minus120574119889119879 for some 120574119889 isin Rforall119896 isin Z0+(2) 119892119889 isin 119878119875119877 there exist constants 1205741198890 (120574119889 gt 1205741198890) isin R+
such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isinZ0+
(3) 120593(119909 119896119879) is subadditive for all 119909 isin R forall119896 isin Z0+(4) 120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)| are
119895119879(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)
minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119896 isin Z0+(47)
Then the following properties hold(i) The discrete closed-loop system is asymptotically hyper-
stable(ii) 119906(119905) rarr 0 as 119905 rarr infin and the control and output are
bounded for all time
(iii) Assume furthermore that 120590(119905) = 120593(119910(119905) 119905) minus (1 +120582(119905))120593(119910[119896] 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+ 119906(119905) =minus120593(119910(119905) 119905) and forall119905 isin R0+ and that the class Φ satisfiesPopovrsquos type integral inequality for all time and results in theclass Φ119889(119896119879) at sampling instantsThen the continuous- timeclosed-loop system is asymptotically hyperstable
Proof From Lemma 9 via assumptions (1)ndash(5) constraints(36) hold so that the input- output energymeasure associatedwith the forced solution is positively lower-bounded andfinitely upper-bounded for all sampling instants 119905 = 119896119879forall119896 isin Z+ Since the discrete positive transfer function isstrictly positive real then it is also strictly stable in the discretecontext (ie convergent [29 36]) so that the homogeneousresponse is irrelevant for stability analysis purposes Thus120593[119896] isin Φ119889 119906[119896] rarr 0 and 119906[119896] and 119910[119896] arebounded for all time for any given bounded initial conditionsIn addition one has from Lemma 10 that (42) holds andthe forced input-output energy measure is positively lower-bounded and finitely upper-bound for all 119905 isin R+ Then119906(119905) rarr 0 as 119905 rarr infin and 119906(119905) and 119910(119905) are bounded for alltime for any given bounded initial conditions Properties [(i)-(ii)] have been proved Property (iii) follows by equalizing119906(119905) = minus120593(119910(119905) 119905) = (1 + 120582(119905))119906[119896] + 120590(119905) and 120593 isin Φ fromthe fact that minusint119905
0119906(120591)119910(120591)119889120591 le 1205740
Note that the use of Fourier transforms is useful tocharacterize the error of the input-output energy measure inthe frequency domain along the intersample periods relatedto its values at the preceding sampling instants This toolis addressed in Lemma 10 which is used in the proof ofTheorem 11 It allows to keep easily the characterizationof the frequency response of the continuous-time transferfunction to evaluate the input-output energy measure in theintersample period and to calculate the minimum value ofthe real part of such a transfer function to guarantee thepositivity and boundedness of such an energy Note that thecharacterization of the input-output energy measure of thefeed-forward linear loop evaluated in the frequency domaintogether with Popovrsquos inequalities of the class of hyperstablecontrollers is relevant to the hyperstability proofs of theclosed-loop system
5 Examples
Example 12 The positive realness of a discretized transferfunction depends not only on the continuous-time param-eters but on the sampling period as well Therefore thepositive realness of the continuous-time system does notguarantee that of any discretized counterpart irrespective ofthe sampling period and conversely For instance the transferfunction 119892(119904) = 1(119904 + 119886) is strictly positive real for 119886 gt 0 butit is not strongly strictly positive real since Re119892(119894120596) rarr 0 as120596 rarr plusmninfin Its discretized version without the concourse ofan ideal that is instantaneous sampler and zero-order holdbecomes 119904 119892119889(119911) = 119911(119911 minus 119890minus119886119879) which is strongly positivereal for 119879 isin (0infin) and as the sampling period 119879 rarr infin since119892119889(119911) rarr 1 as 119879 rarr infin However 119892119889(119911) = 119911(119911 minus 1) as 119879 rarr 0
12 Mathematical Problems in Engineering
Since it has a critically stable pole in the discrete domain it ispositive real although it is not strictly positive real
Example 13 Consider the discrete transfer function 119892119889(119911) =(1199112 + 1198891119911 + 1198892)(1199112 minus 1198861119911 minus 1198862) with real numerator anddenominator coefficients Its strict positive realness requiresas necessary conditions its convergence (ie strict stabilityin the discrete sense) so that its poles have to be within thecomplex open unit circle as well as the convergence of itsinverse so that its zeros have also to lie inside the open unitcircle and furthermore its relative degree (ie its pole-zeroexcess has to be zero) Note that the last condition is notneeded for strict (nonstrong) positive realness of continuoustransfer functions where their positive real part can convergeto zero as |119904| rarr infin By using the homographic bilineartransformation 119911 = (1 + 119904)(1 minus 119904) which transforms theunit circle in the left-hand-side complex plane to get theauxiliary continuous transfer function 119892119886(119904) = 119892119889(119911 = (1 +119904)(1 minus 119904)) it is found in [13] that the denominator andnumerator polynomials of 119892119889(119911) are subject to 1198862 minus 1198861 lt 11198861 + 1198862 lt 1 and 1198862 gt minus1 Take the particular values 1198861 = 05and 1198862 = 04 leading to the values of poles 1199111 = 05 and1199112 = 04 and the coefficients of the numerator polynomialare fixed to 1198891 = 12 1198892 = 03 Then we fix a discretetransfer function 119892119889(119911) = (1199112 + 12119911 + 03)(1199112 minus 05119911 minus 04)yielding min|119911|=1 Re119892119889(119911) ge 009 so 119892119889 isin SSPR Thedecomposition of 119892119889(119911) in simple fractions leads to 119892119889(119911) =1 + 168(119911 minus 093) + 0758(119911 + 043) By using an ideal (iewith instantaneous sampling) sampler and zero-order-holddevice of period 119879 of transfer function (1 minus 119890minus119879119904)119904 on acontinuous transfer function 119892(119904) to be specified to generate119892119889(119911) one has the identity 119885((1 minus 119890minus119879119904)119892(119904)119904) = 119892119889(119911) where119885(119865(119904)) is an abbreviate usual informal notation to denote119865lowast(119904) = 119871(119891lowast(119905)) where 119871(sdot) and 119885(sdot) denote the Laplacetransform of (sdot) of argument 119904 and 119885-transform of (sdot) ofargument 119911 = 119890119879119904 and 119891lowast(119905) = suminfin119899=0 119891(119899119879)120575(119905 minus 119899119879) denotesthe pulsed function of the function119891(119905)with sampling period119879 As a result we get for a sampling period 119879 = 01 sec119892 (119904) = 119904119885minus1 ( 119911119911 minus 1 [1 + 168119911 minus 093 + 0758119911 + 043])
Note that 119892 isin SPR (and also 119892 isin SSPR sinceits relative degree is zero) is the continuous-time transferfunction which generates 119892119889(119911) under an ideal sampler andzero-order-hold device of period 119879 = 01 sec so thatmin120596isinclR119892(i120596) = min120596isinR0+119892(i120596) gt 0 Thus it is guaranteedthat the continuous-time energy measure is associated withthe forced output 119864(119905) gt 0 for all 119905 gt 0 (ie it ispositively lower-bounded for all time) for any piecewise-continuous control irrespectively of the particular controllerin operation In order to guarantee that any such a controlthat converges asymptotically to zero it is necessary to finitelyupper-bound such an energy for all timeThe above concerns
also guarantee that 119864[119896] = 119864(119896119879) ge 1205741198890 gt 0 for all integer 119896 gt0 under any piecewise-constant control 119906[119896] = 119906(119896119879) witheventual jumps at sampling instants of any period 119879 gt 0 Forsuch a control identical conclusion arises from the discrete119892119889(119911)which is also strongly strictly positive real Assume that119906[119896] = minus120593(119910[119896] 119896) and forall119896 isin Z+ with 120593 isin Φ119889 where theclass Φ119889 is defined as the set of sequences 120593(119910[119896] 119896)whichsatisfy Popovrsquos inequality 119879sum119896119895=0 119910[119896]120593(119910[119896] 119896) ge minus120574119889 forall119896 isinZ+ for some finite 120574119886 isin R+ which is identical to
= minus119879119872(1 minus ]119896+1)1 minus ] minus 119879 119896sum119895=0
120576 [119896] ge minus120574119889 forall119896 isin Z+(50)
for some real constants 119872 gt 0 ] isin (0 1) and some realsummable sequence 120576[119896] Then the class Φ119889 is definedby any real sequence 120593(119910[119896] 119896) being parameterized by anyreal constants 119872 gt 0 ] isin (0 1) and any real summablesequence 120576[119896] such that
forall119896 isin Z0+ and 120574119889 ge max(119879119872(1 minus ]) + 120576 120574) for some given120574 ge 1205741198890 where 120576 = 119879max119896isinZ+ sum119896119895=0 120576[119896]The continuous-timecontrol is generated as
where 120593[119896] isin Φ119889(119879) satisfies Popovrsquos type discrete inequal-ity which makes the feedback system globally asymptoticallystable under the theorem hypothesis Thus the discrete-timeclosed-loop system is asymptotically hyperstable In partic-ular under the additional assumptions in Theorem 11(iii)the closed-loop continuous-time system is asymptoticallyhyperstable
Mathematical Problems in Engineering 13
One of the referees has given a comment of interest on thisexample related to continuous versus discrete counterpartproperties which has motivated us to give a further intuitivediscussion on it Note the following In the continuous-timecase there are two kinds of strictly positive real transferfunctions as it is well known namely the so-called strictpositive real which can be zero (or respectively positive)at infinity frequency and which a realizable has relativedegree one (or respectively zero) In the first case the directinput-output interconnection gain 119889 is zero in a state-spacerepresentation (119860 119887 119888 119889 = 0) of the transfer function Thetransfer functions of the second case are often referred to asthe class of strong strict positive real transfer functions Thisclass is not a disjoint class of first above one of strict positivereal transfer functions since it is included in it as a potentialparticular case Note that the members of this strong classof strict positive real transfer functions have relative degreezero and a positive direct input-output interconnection gain119889 Related to this concern the Yakubovich-Kalman-Popov(refereed to the continuous-time domain) and the Szego-Kalman-Popov (referred to the discrete domain) lemmascan be invoked to compare positive realness of the transferfunctions with their state-space associated properties (in factthe properties aremutually equivalent fromboth sides)Thuswe see that in the discrete case the Szego-Kalman-Popovlemma needs for its own coherency that the input-outputinterconnection 119889-gain be nonzero since otherwise one ofthe relevant constraint in the lemma fails [13] So strictpositive realness in the discrete domain in this context istypically strong and then the relative degree of its transferfunction is zero so that there is a direct interconnectionpositive gain from the input to the output Since we assumethat the discrete system is strictly positive real it has a positiveinput-output interconnection from the input to the outputand the minimum value of the real part of its frequencyresponse is strictly positive even as the frequency tends toinfinity On the other hand note that a zero-order holdldquoconvertsrdquo any strictly proper (continuous-time) transferfunction of any relative degree into a strictly discrete oneof unity relative order while it ldquoconvertsrdquo a continuous oneof zero relative order (ie biproper) into a discrete onestill of zero-relative order and with the same input-outputinterconnection gain See (48) Therefore we deal in thisexample with a biproper strict positive real transfer functionof positive input-output interconnection gain identical to thatof its discrete-time counterpart got through a ZOH samplingand hold device
Example 14 Consider that a controller of the form 119906(119905) =minus120593(119910(119905) 119905) forall119905 isin R0+ is used for the continuous-time feed-forward transfer function (48) so that120593 isin Φ and Φ is classin Example 13 satisfying an integral type Popovrsquos inequality(21) for all 119905(ge 1199050) isin R+ and some finite 1199050 isin R+ That isthe control function belongs to a class satisfying continuous-time integral Popovrsquos inequality Then 119906(119905) rarr 0 as 119905 rarr infinand 119906(119905) and 119910(119905) are almost everywhere bounded for all timesince 119892 isin SSPR then being (strictly) stable with input-output energy measure being positively lower-bounded andalso finitely upper-bounded for all 119905 isin R+ from Popovrsquos-type
integral inequality Potential useful controllers are for somegiven controller gain 119896119906 isin R+
119906 (119905) = 0 for 119905 isin [0 1)1198961199061199052119910 (119905) for 119905 ge 1 (54)
119906 (119905) = 119896119906119890minus119905119910 (119905) for 119905 ge 0 (55)
119906 (119905) = 0 for 119905 isin [0 2)
119896119906(119905 ln2119905) 119910 (119905) for 119905 ge 2 (56)
119906 (119905) = 0 for 119905 isin [0 1)
119896119906(119890radic119905radic119905) 119910 (119905) for 119905 ge 1 (57)
119906 (119905) = 1198961199061199053119890minus119905119910 (119905) for 119905 ge 0 (58)
where Popovrsquos inequality on [0 +infin] of (49)ndash(52)is intinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus119896119906 that of (57) isintinfin
0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 and that of (58) isintinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 Since the respective integrands
are monotonically decreasing then there is some 1199050 isin R+ andsome 120574 = 120574(1199050) isin R+ such that int119905
1199050119910(119905)120593(119910(119905) 119905)119889119905 ge minus120574 for
all 119905(ge 1199050) isin R+ The obtained closed-loop systems are inall cases asymptotically hyperstable The piecewise-constantcontrols with eventual discontinuities at sampling instantsgenerated from any discrete-time controller (51) of class Φ119889of Example 13 lead also to asymptotic hyperstability of thediscretized system since 119892119889 isin SSPRExample 15 Consider the continuous-time open-loop sys-tem (2) under a real sampler of sampling period 119879 and datapicking-up nonzero duration 120576 isin (0 119879) The state and outputtrajectory solutions are
forall119896 isin Z0+ for some real sequence 120585[119896] such that 120585[119896] isin(119896119879 119896119879 + 120576) where 119892119889(119911) is the discretized transfer functionunder a sampler and zero-order-hold device subject to instan-taneous ideal sampling Now assume that the incrementalcontrol along intersample periods is generated from itspreceding sampled value as (119895119879+ 120591) = 120582119895119879+120576(119895119879+ 120591)119906[119895] forany 119895 isin Z0+ and 120591 isin [0 119879] where 120582119895119879+120576(119895119879+120591) is a truncatedfunction under the usual notation one gets
The above equations imply that if 119892119889 isin SPR then the forcedoutput 119910119891[119896] = 119910[119896] minus 119888119879119890119896119860119879119909[0] forall119896 isin Z0+ generates anassociate input-output energy measure which is strictly posi-tive for 119896 isin Z0+ since 120588 isin (0 1) for real sampling of duration120576 isin (0 119879)This is a necessary condition for asymptotic hyper-stability under an appropriate class of controllers satisfyingPopovrsquos inequality However the incremental input-outputenergy measure generated by the incremental forced output120576119888119879sum119896minus1119895=0 119890(119896minus119895)119860119879(int1198790 120582119895119879+120576(120591)119890minus119860120591119887 119889120591)119906[119895] is not guaranteedto be nonnegative in general A sufficient condition for that isthat lim sup119895rarrinfin|ess sup120591isin(0120576)(120582119895119879+120576(120591))| is sufficiently smallrelated to the amount ((1 minus 120588)120576)min120596isin[0120587] Re119892119889(119890i120596)6 Conclusions
This paper has been devoted to obtain and to prove theasymptotic hyperstability property of a linear and time-invariant continuous-time system based on some given andpreviously proved auxiliary preparatory lemmas Basicallyit is proved that the continuous-time closed-loop system isasymptotically hyperstable under the assumptions that thediscretized feed-forward transfer function is strictly positivereal and that the discretized system at sampling instants ofconstant sampling period is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsIt is assumed that the continuous-time controller belongs toa class defined in the following way (a) its values at samplinginstants are defined by a class of controllers which satisfydiscrete Popovrsquos type inequality (b) the intersample control isgenerated from its values at sampling instants with the use oftwomodulating auxiliary functionswhich are subject to someextra additional conditions on smallness and boundednessSuch conditions make the overall continuous-time controllerto satisfy continuous-type Popovrsquos inequality for all time
where 119896(120591) = max(119911 isin Z+ 119911119879 le 120591) and 119896 = 119896(119905) = max(119911 isinZ+ 119911119879 le 119905) forall119905 isin R0+ which yields (13)
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
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Mathematical Problems in Engineering 9
Lemma 9 Assume that 119892119889 isin 119878119875119877 and that 120593 isin Φ119889(119879)Then the following properties hold(i)The input-output energy systemmeasure of the auxiliary
discretized system satisfies
0 lt 1205740119889119886 le 1198792120587 min120596isinR0+
119879119910119889 [119896] 119906119889 [119896] le 120574119889119886forall119896 (gt 1198960) isin Z+
(34)
for some 1205740119889119886 120574119889119886(gt 1205740119889119886) isin R+ and for any nonidentically zerocontrols and zero initial conditions ie for any forced solution
(ii) Assume in addition that the class Φ119889(119879) consistsof subadditive functions Assume also that sup119905isinR0+ |120590(119905)| issufficiently small inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infinwhere 120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ and that furthermore
for some 120574119889119886 isin R+ from the first assumption of the theoremand 119906119889[119896] = minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ since Popovrsquosinequality (22) is fulfilled by any given discrete-time feedbackcontroller 120593 isin Φ119889(119879) generating a feedback control 119906119889[119896] =minus120593(119910119889[119896] 119896119879) forall119896 isin Z0+ On the other hand the initialconditions can be neglected since the assumption119892119889 isin SPRimplying that the discrete-time transfer function is (strictly)stable which makes their values irrelevant for purposes ofstability analysisThen the discrete frequency response being
applicable to piecewise-constant inputs in-between any twoconsecutive sampling instants is given by
since the subsequence 119892[minus119899] = 0 forall119899 isin Z+ and thetransfer frequency response of a sampling and zero-order-hold operator 119885 of period 119879 defined by 119885(119879 V)(119905) = V(119896119879) =V[119896] for any given V R0+ rarr R and all 119905 isin [119896119879 (119896 + 1)119879)119896 isin Z0+ is given by 119885(119879 i120596) = (1 minus 119890minusi119879120596)i120596 [31 35]It turns out that |119885(119879 i120596)| = |(1 minus 119890minusi119879120596)i120596| lt 1 forall120596( =0) isin R and |119885(119879 i0)| = 1 Thus the auxiliary discretizedsystem satisfies the subsequent equivalence relation in thefrequency domain to the discrete-time relation (11) It turnsout that Fourier transforms exist in the impulse responsesand truncated control and auxiliary functions of (11)ndash(13)since the truncated functions are square-integrable in clR =[minusinfin +infin] By using the discrete Parsevalrsquos theorem underzero initial conditions one has
The lower-bounding constraint of (37) follows by using(39) and the discrete Parsevalrsquos theorem for the equivalencebetween the input-output energy measures from the fre-quency domain to the discrete-time domain
119864119889 [119896] ge 1198792120587 min120596isinR0+
1199062119889 [119896]) ge 1205740119889forall119896 isin Z0+
(40)
from the assumption 119892119889 isin SPR iemin120596isinR0+ Re119892119889(119890minusi120596119879) ge 1205740119889119886 gt 0 forall1198960 119896 (119896 ge 1198960) isin Z+since
(a) strictly positive real discrete transfer functions havezero relative degree (ie an identical number of polesand zeros) from the Discrete Positive Real Lemma(Szego-Kalman-Popov Lemma) [13 31 34] so thatthe real parts of their frequency hodographs in theargument 119890minusi120596119879 are positively lower-bounded
10 Mathematical Problems in Engineering
(b) Im119892119889(119890minusi120596119879) = minus Im119892119889(119890i120596119879) forall120596 isin R0+ so that theirintegrals in the argument 120596 on frequency intervals[minus119896120587 119896120587] for any 119896 isin Z+ are null
Property (i) has been proved To prove Property (ii) notefrom (28) [Lemma 8(ii)] that
119892 [(119896 minus 119895) 119879])119889120591]]sdot (120593[ℓ] (119910119889 [119895] 119895) + 120593[ℓ] (minus119910119889 [119895] 119895))sdot (120593[ℓ] (119910119889 [ℓ] 119895) + 120593[ℓ] (minus119910119889 [ℓ] ℓ)) minus 119910119889ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 le 120574119889 = 119870120574119889119886
forall119896 isin Z+
(41)
for some 119870 isin R+ provided that sup119905isinR0+ |120590(119905)| is sufficientlysmall inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infin where120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ since 119892(119905) is bounded andconverges exponentially to zero since 119892119889 isin SPR and thenconvergent (that is strictly stable in the discrete context)119910119889ℎ[119896] rarr 0 and it is a bounded sequence
The next result which is a preparatory result to thenestablish the main asymptotic hyperstability result addressesthe relevant property that the input-output measure 119864(119905) ofthe continuous-time system is positively lower-bounded andfinitely upper-bounded for all time under the conditions ofLemmas 4 and 9
Lemma 10 If 120582 and 120590 are sufficiently small then
1205740 minus 120574119889 le 119864 (119905) minus 119864119889 [119896]le 1 + 12058222120587 intinfin
provided that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+Proof One gets the following from direct calculations byusing the continuous-time control laws (9)-(10) and (16) if120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)|
minus 12058222120587 intinfin
minusinfin119892 (i120596) 1003816100381610038161003816[119896] (i120596)10038161003816100381610038162 119889120596 minus 120590(2120582 sup
0le119895le119896
1199062[119895] + 120590)int1198961198790
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591 le 119864 (119905) minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119910 (i120596)
sdot (minusi120596) 119889120596 minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119892 (i120596) | (i120596)|2 119889120596 minus 119864119889 (119896119879 119905)
Now the use of (44) in Lemma 4 (14) implies thatLemma 9(ii) (36) holds implying the existence of 1205740 (120574 gt1205740) isin R+ such that 1205740 le 119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin)provided that 1205741198890 le 119864119889[119896] le 120574119889 for 1205741198890 (120574119889 gt 1205741198890) isin R+ since1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isin Z0+ if1205740 minus 120574119889 le minus 120582
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591le 120574 minus 1205741198890 forall119896 (gt 1198960) isin Z+ forall120591 isin (0 119879)
(45)
if 120582 and 120590 are sufficiently small such that (43) holds leadingto 1205740 gt 0 for any given positive 1205741198890 and 120574119889 gt 1205741198890
The main first result follows
Theorem 11 Consider the dynamic system (2) subject to acontrol
forall119896 isin Z0+ where 120593 [119896] isin Φ119889 (119879) (46)
Assume that(1) 120593(119910[119896] 119896119879) is in the class Φ119889(119879) (that is120593(119910[119896] 119896119879) isin Φ119889(119879)) defined by the discrete Popovrsquos
type sum inequality sum119896ℓ=0 119910119889[ℓ]119906[ℓ] ge minus120574119889119879 for some 120574119889 isin Rforall119896 isin Z0+(2) 119892119889 isin 119878119875119877 there exist constants 1205741198890 (120574119889 gt 1205741198890) isin R+
such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isinZ0+
(3) 120593(119909 119896119879) is subadditive for all 119909 isin R forall119896 isin Z0+(4) 120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)| are
119895119879(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)
minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119896 isin Z0+(47)
Then the following properties hold(i) The discrete closed-loop system is asymptotically hyper-
stable(ii) 119906(119905) rarr 0 as 119905 rarr infin and the control and output are
bounded for all time
(iii) Assume furthermore that 120590(119905) = 120593(119910(119905) 119905) minus (1 +120582(119905))120593(119910[119896] 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+ 119906(119905) =minus120593(119910(119905) 119905) and forall119905 isin R0+ and that the class Φ satisfiesPopovrsquos type integral inequality for all time and results in theclass Φ119889(119896119879) at sampling instantsThen the continuous- timeclosed-loop system is asymptotically hyperstable
Proof From Lemma 9 via assumptions (1)ndash(5) constraints(36) hold so that the input- output energymeasure associatedwith the forced solution is positively lower-bounded andfinitely upper-bounded for all sampling instants 119905 = 119896119879forall119896 isin Z+ Since the discrete positive transfer function isstrictly positive real then it is also strictly stable in the discretecontext (ie convergent [29 36]) so that the homogeneousresponse is irrelevant for stability analysis purposes Thus120593[119896] isin Φ119889 119906[119896] rarr 0 and 119906[119896] and 119910[119896] arebounded for all time for any given bounded initial conditionsIn addition one has from Lemma 10 that (42) holds andthe forced input-output energy measure is positively lower-bounded and finitely upper-bound for all 119905 isin R+ Then119906(119905) rarr 0 as 119905 rarr infin and 119906(119905) and 119910(119905) are bounded for alltime for any given bounded initial conditions Properties [(i)-(ii)] have been proved Property (iii) follows by equalizing119906(119905) = minus120593(119910(119905) 119905) = (1 + 120582(119905))119906[119896] + 120590(119905) and 120593 isin Φ fromthe fact that minusint119905
0119906(120591)119910(120591)119889120591 le 1205740
Note that the use of Fourier transforms is useful tocharacterize the error of the input-output energy measure inthe frequency domain along the intersample periods relatedto its values at the preceding sampling instants This toolis addressed in Lemma 10 which is used in the proof ofTheorem 11 It allows to keep easily the characterizationof the frequency response of the continuous-time transferfunction to evaluate the input-output energy measure in theintersample period and to calculate the minimum value ofthe real part of such a transfer function to guarantee thepositivity and boundedness of such an energy Note that thecharacterization of the input-output energy measure of thefeed-forward linear loop evaluated in the frequency domaintogether with Popovrsquos inequalities of the class of hyperstablecontrollers is relevant to the hyperstability proofs of theclosed-loop system
5 Examples
Example 12 The positive realness of a discretized transferfunction depends not only on the continuous-time param-eters but on the sampling period as well Therefore thepositive realness of the continuous-time system does notguarantee that of any discretized counterpart irrespective ofthe sampling period and conversely For instance the transferfunction 119892(119904) = 1(119904 + 119886) is strictly positive real for 119886 gt 0 butit is not strongly strictly positive real since Re119892(119894120596) rarr 0 as120596 rarr plusmninfin Its discretized version without the concourse ofan ideal that is instantaneous sampler and zero-order holdbecomes 119904 119892119889(119911) = 119911(119911 minus 119890minus119886119879) which is strongly positivereal for 119879 isin (0infin) and as the sampling period 119879 rarr infin since119892119889(119911) rarr 1 as 119879 rarr infin However 119892119889(119911) = 119911(119911 minus 1) as 119879 rarr 0
12 Mathematical Problems in Engineering
Since it has a critically stable pole in the discrete domain it ispositive real although it is not strictly positive real
Example 13 Consider the discrete transfer function 119892119889(119911) =(1199112 + 1198891119911 + 1198892)(1199112 minus 1198861119911 minus 1198862) with real numerator anddenominator coefficients Its strict positive realness requiresas necessary conditions its convergence (ie strict stabilityin the discrete sense) so that its poles have to be within thecomplex open unit circle as well as the convergence of itsinverse so that its zeros have also to lie inside the open unitcircle and furthermore its relative degree (ie its pole-zeroexcess has to be zero) Note that the last condition is notneeded for strict (nonstrong) positive realness of continuoustransfer functions where their positive real part can convergeto zero as |119904| rarr infin By using the homographic bilineartransformation 119911 = (1 + 119904)(1 minus 119904) which transforms theunit circle in the left-hand-side complex plane to get theauxiliary continuous transfer function 119892119886(119904) = 119892119889(119911 = (1 +119904)(1 minus 119904)) it is found in [13] that the denominator andnumerator polynomials of 119892119889(119911) are subject to 1198862 minus 1198861 lt 11198861 + 1198862 lt 1 and 1198862 gt minus1 Take the particular values 1198861 = 05and 1198862 = 04 leading to the values of poles 1199111 = 05 and1199112 = 04 and the coefficients of the numerator polynomialare fixed to 1198891 = 12 1198892 = 03 Then we fix a discretetransfer function 119892119889(119911) = (1199112 + 12119911 + 03)(1199112 minus 05119911 minus 04)yielding min|119911|=1 Re119892119889(119911) ge 009 so 119892119889 isin SSPR Thedecomposition of 119892119889(119911) in simple fractions leads to 119892119889(119911) =1 + 168(119911 minus 093) + 0758(119911 + 043) By using an ideal (iewith instantaneous sampling) sampler and zero-order-holddevice of period 119879 of transfer function (1 minus 119890minus119879119904)119904 on acontinuous transfer function 119892(119904) to be specified to generate119892119889(119911) one has the identity 119885((1 minus 119890minus119879119904)119892(119904)119904) = 119892119889(119911) where119885(119865(119904)) is an abbreviate usual informal notation to denote119865lowast(119904) = 119871(119891lowast(119905)) where 119871(sdot) and 119885(sdot) denote the Laplacetransform of (sdot) of argument 119904 and 119885-transform of (sdot) ofargument 119911 = 119890119879119904 and 119891lowast(119905) = suminfin119899=0 119891(119899119879)120575(119905 minus 119899119879) denotesthe pulsed function of the function119891(119905)with sampling period119879 As a result we get for a sampling period 119879 = 01 sec119892 (119904) = 119904119885minus1 ( 119911119911 minus 1 [1 + 168119911 minus 093 + 0758119911 + 043])
Note that 119892 isin SPR (and also 119892 isin SSPR sinceits relative degree is zero) is the continuous-time transferfunction which generates 119892119889(119911) under an ideal sampler andzero-order-hold device of period 119879 = 01 sec so thatmin120596isinclR119892(i120596) = min120596isinR0+119892(i120596) gt 0 Thus it is guaranteedthat the continuous-time energy measure is associated withthe forced output 119864(119905) gt 0 for all 119905 gt 0 (ie it ispositively lower-bounded for all time) for any piecewise-continuous control irrespectively of the particular controllerin operation In order to guarantee that any such a controlthat converges asymptotically to zero it is necessary to finitelyupper-bound such an energy for all timeThe above concerns
also guarantee that 119864[119896] = 119864(119896119879) ge 1205741198890 gt 0 for all integer 119896 gt0 under any piecewise-constant control 119906[119896] = 119906(119896119879) witheventual jumps at sampling instants of any period 119879 gt 0 Forsuch a control identical conclusion arises from the discrete119892119889(119911)which is also strongly strictly positive real Assume that119906[119896] = minus120593(119910[119896] 119896) and forall119896 isin Z+ with 120593 isin Φ119889 where theclass Φ119889 is defined as the set of sequences 120593(119910[119896] 119896)whichsatisfy Popovrsquos inequality 119879sum119896119895=0 119910[119896]120593(119910[119896] 119896) ge minus120574119889 forall119896 isinZ+ for some finite 120574119886 isin R+ which is identical to
= minus119879119872(1 minus ]119896+1)1 minus ] minus 119879 119896sum119895=0
120576 [119896] ge minus120574119889 forall119896 isin Z+(50)
for some real constants 119872 gt 0 ] isin (0 1) and some realsummable sequence 120576[119896] Then the class Φ119889 is definedby any real sequence 120593(119910[119896] 119896) being parameterized by anyreal constants 119872 gt 0 ] isin (0 1) and any real summablesequence 120576[119896] such that
forall119896 isin Z0+ and 120574119889 ge max(119879119872(1 minus ]) + 120576 120574) for some given120574 ge 1205741198890 where 120576 = 119879max119896isinZ+ sum119896119895=0 120576[119896]The continuous-timecontrol is generated as
where 120593[119896] isin Φ119889(119879) satisfies Popovrsquos type discrete inequal-ity which makes the feedback system globally asymptoticallystable under the theorem hypothesis Thus the discrete-timeclosed-loop system is asymptotically hyperstable In partic-ular under the additional assumptions in Theorem 11(iii)the closed-loop continuous-time system is asymptoticallyhyperstable
Mathematical Problems in Engineering 13
One of the referees has given a comment of interest on thisexample related to continuous versus discrete counterpartproperties which has motivated us to give a further intuitivediscussion on it Note the following In the continuous-timecase there are two kinds of strictly positive real transferfunctions as it is well known namely the so-called strictpositive real which can be zero (or respectively positive)at infinity frequency and which a realizable has relativedegree one (or respectively zero) In the first case the directinput-output interconnection gain 119889 is zero in a state-spacerepresentation (119860 119887 119888 119889 = 0) of the transfer function Thetransfer functions of the second case are often referred to asthe class of strong strict positive real transfer functions Thisclass is not a disjoint class of first above one of strict positivereal transfer functions since it is included in it as a potentialparticular case Note that the members of this strong classof strict positive real transfer functions have relative degreezero and a positive direct input-output interconnection gain119889 Related to this concern the Yakubovich-Kalman-Popov(refereed to the continuous-time domain) and the Szego-Kalman-Popov (referred to the discrete domain) lemmascan be invoked to compare positive realness of the transferfunctions with their state-space associated properties (in factthe properties aremutually equivalent fromboth sides)Thuswe see that in the discrete case the Szego-Kalman-Popovlemma needs for its own coherency that the input-outputinterconnection 119889-gain be nonzero since otherwise one ofthe relevant constraint in the lemma fails [13] So strictpositive realness in the discrete domain in this context istypically strong and then the relative degree of its transferfunction is zero so that there is a direct interconnectionpositive gain from the input to the output Since we assumethat the discrete system is strictly positive real it has a positiveinput-output interconnection from the input to the outputand the minimum value of the real part of its frequencyresponse is strictly positive even as the frequency tends toinfinity On the other hand note that a zero-order holdldquoconvertsrdquo any strictly proper (continuous-time) transferfunction of any relative degree into a strictly discrete oneof unity relative order while it ldquoconvertsrdquo a continuous oneof zero relative order (ie biproper) into a discrete onestill of zero-relative order and with the same input-outputinterconnection gain See (48) Therefore we deal in thisexample with a biproper strict positive real transfer functionof positive input-output interconnection gain identical to thatof its discrete-time counterpart got through a ZOH samplingand hold device
Example 14 Consider that a controller of the form 119906(119905) =minus120593(119910(119905) 119905) forall119905 isin R0+ is used for the continuous-time feed-forward transfer function (48) so that120593 isin Φ and Φ is classin Example 13 satisfying an integral type Popovrsquos inequality(21) for all 119905(ge 1199050) isin R+ and some finite 1199050 isin R+ That isthe control function belongs to a class satisfying continuous-time integral Popovrsquos inequality Then 119906(119905) rarr 0 as 119905 rarr infinand 119906(119905) and 119910(119905) are almost everywhere bounded for all timesince 119892 isin SSPR then being (strictly) stable with input-output energy measure being positively lower-bounded andalso finitely upper-bounded for all 119905 isin R+ from Popovrsquos-type
integral inequality Potential useful controllers are for somegiven controller gain 119896119906 isin R+
119906 (119905) = 0 for 119905 isin [0 1)1198961199061199052119910 (119905) for 119905 ge 1 (54)
119906 (119905) = 119896119906119890minus119905119910 (119905) for 119905 ge 0 (55)
119906 (119905) = 0 for 119905 isin [0 2)
119896119906(119905 ln2119905) 119910 (119905) for 119905 ge 2 (56)
119906 (119905) = 0 for 119905 isin [0 1)
119896119906(119890radic119905radic119905) 119910 (119905) for 119905 ge 1 (57)
119906 (119905) = 1198961199061199053119890minus119905119910 (119905) for 119905 ge 0 (58)
where Popovrsquos inequality on [0 +infin] of (49)ndash(52)is intinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus119896119906 that of (57) isintinfin
0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 and that of (58) isintinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 Since the respective integrands
are monotonically decreasing then there is some 1199050 isin R+ andsome 120574 = 120574(1199050) isin R+ such that int119905
1199050119910(119905)120593(119910(119905) 119905)119889119905 ge minus120574 for
all 119905(ge 1199050) isin R+ The obtained closed-loop systems are inall cases asymptotically hyperstable The piecewise-constantcontrols with eventual discontinuities at sampling instantsgenerated from any discrete-time controller (51) of class Φ119889of Example 13 lead also to asymptotic hyperstability of thediscretized system since 119892119889 isin SSPRExample 15 Consider the continuous-time open-loop sys-tem (2) under a real sampler of sampling period 119879 and datapicking-up nonzero duration 120576 isin (0 119879) The state and outputtrajectory solutions are
forall119896 isin Z0+ for some real sequence 120585[119896] such that 120585[119896] isin(119896119879 119896119879 + 120576) where 119892119889(119911) is the discretized transfer functionunder a sampler and zero-order-hold device subject to instan-taneous ideal sampling Now assume that the incrementalcontrol along intersample periods is generated from itspreceding sampled value as (119895119879+ 120591) = 120582119895119879+120576(119895119879+ 120591)119906[119895] forany 119895 isin Z0+ and 120591 isin [0 119879] where 120582119895119879+120576(119895119879+120591) is a truncatedfunction under the usual notation one gets
The above equations imply that if 119892119889 isin SPR then the forcedoutput 119910119891[119896] = 119910[119896] minus 119888119879119890119896119860119879119909[0] forall119896 isin Z0+ generates anassociate input-output energy measure which is strictly posi-tive for 119896 isin Z0+ since 120588 isin (0 1) for real sampling of duration120576 isin (0 119879)This is a necessary condition for asymptotic hyper-stability under an appropriate class of controllers satisfyingPopovrsquos inequality However the incremental input-outputenergy measure generated by the incremental forced output120576119888119879sum119896minus1119895=0 119890(119896minus119895)119860119879(int1198790 120582119895119879+120576(120591)119890minus119860120591119887 119889120591)119906[119895] is not guaranteedto be nonnegative in general A sufficient condition for that isthat lim sup119895rarrinfin|ess sup120591isin(0120576)(120582119895119879+120576(120591))| is sufficiently smallrelated to the amount ((1 minus 120588)120576)min120596isin[0120587] Re119892119889(119890i120596)6 Conclusions
This paper has been devoted to obtain and to prove theasymptotic hyperstability property of a linear and time-invariant continuous-time system based on some given andpreviously proved auxiliary preparatory lemmas Basicallyit is proved that the continuous-time closed-loop system isasymptotically hyperstable under the assumptions that thediscretized feed-forward transfer function is strictly positivereal and that the discretized system at sampling instants ofconstant sampling period is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsIt is assumed that the continuous-time controller belongs toa class defined in the following way (a) its values at samplinginstants are defined by a class of controllers which satisfydiscrete Popovrsquos type inequality (b) the intersample control isgenerated from its values at sampling instants with the use oftwomodulating auxiliary functionswhich are subject to someextra additional conditions on smallness and boundednessSuch conditions make the overall continuous-time controllerto satisfy continuous-type Popovrsquos inequality for all time
where 119896(120591) = max(119911 isin Z+ 119911119879 le 120591) and 119896 = 119896(119905) = max(119911 isinZ+ 119911119879 le 119905) forall119905 isin R0+ which yields (13)
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
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10 Mathematical Problems in Engineering
(b) Im119892119889(119890minusi120596119879) = minus Im119892119889(119890i120596119879) forall120596 isin R0+ so that theirintegrals in the argument 120596 on frequency intervals[minus119896120587 119896120587] for any 119896 isin Z+ are null
Property (i) has been proved To prove Property (ii) notefrom (28) [Lemma 8(ii)] that
119892 [(119896 minus 119895) 119879])119889120591]]sdot (120593[ℓ] (119910119889 [119895] 119895) + 120593[ℓ] (minus119910119889 [119895] 119895))sdot (120593[ℓ] (119910119889 [ℓ] 119895) + 120593[ℓ] (minus119910119889 [ℓ] ℓ)) minus 119910119889ℎ [119896]minus intinfin0[119892 (ℓ119879 minus 120591)] 120590(119896119879) (120591) 119889120591 le 120574119889 = 119870120574119889119886
forall119896 isin Z+
(41)
for some 119870 isin R+ provided that sup119905isinR0+ |120590(119905)| is sufficientlysmall inf119896isinZ0+120576[119896] gt minus1 and sup119896isinZ0+ |120576[119896]| lt infin where120576[119896] = 120582[119896] + 120585[119896] forall119896 isin Z0+ since 119892(119905) is bounded andconverges exponentially to zero since 119892119889 isin SPR and thenconvergent (that is strictly stable in the discrete context)119910119889ℎ[119896] rarr 0 and it is a bounded sequence
The next result which is a preparatory result to thenestablish the main asymptotic hyperstability result addressesthe relevant property that the input-output measure 119864(119905) ofthe continuous-time system is positively lower-bounded andfinitely upper-bounded for all time under the conditions ofLemmas 4 and 9
Lemma 10 If 120582 and 120590 are sufficiently small then
1205740 minus 120574119889 le 119864 (119905) minus 119864119889 [119896]le 1 + 12058222120587 intinfin
provided that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+Proof One gets the following from direct calculations byusing the continuous-time control laws (9)-(10) and (16) if120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)|
minus 12058222120587 intinfin
minusinfin119892 (i120596) 1003816100381610038161003816[119896] (i120596)10038161003816100381610038162 119889120596 minus 120590(2120582 sup
0le119895le119896
1199062[119895] + 120590)int1198961198790
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591 le 119864 (119905) minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119910 (i120596)
sdot (minusi120596) 119889120596 minus 119864119889 (119896119879 119905) = 12120587 intinfin
minusinfin119892 (i120596) | (i120596)|2 119889120596 minus 119864119889 (119896119879 119905)
Now the use of (44) in Lemma 4 (14) implies thatLemma 9(ii) (36) holds implying the existence of 1205740 (120574 gt1205740) isin R+ such that 1205740 le 119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin)provided that 1205741198890 le 119864119889[119896] le 120574119889 for 1205741198890 (120574119889 gt 1205741198890) isin R+ since1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isin Z0+ if1205740 minus 120574119889 le minus 120582
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591le 120574 minus 1205741198890 forall119896 (gt 1198960) isin Z+ forall120591 isin (0 119879)
(45)
if 120582 and 120590 are sufficiently small such that (43) holds leadingto 1205740 gt 0 for any given positive 1205741198890 and 120574119889 gt 1205741198890
The main first result follows
Theorem 11 Consider the dynamic system (2) subject to acontrol
forall119896 isin Z0+ where 120593 [119896] isin Φ119889 (119879) (46)
Assume that(1) 120593(119910[119896] 119896119879) is in the class Φ119889(119879) (that is120593(119910[119896] 119896119879) isin Φ119889(119879)) defined by the discrete Popovrsquos
type sum inequality sum119896ℓ=0 119910119889[ℓ]119906[ℓ] ge minus120574119889119879 for some 120574119889 isin Rforall119896 isin Z0+(2) 119892119889 isin 119878119875119877 there exist constants 1205741198890 (120574119889 gt 1205741198890) isin R+
such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isinZ0+
(3) 120593(119909 119896119879) is subadditive for all 119909 isin R forall119896 isin Z0+(4) 120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)| are
119895119879(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)
minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119896 isin Z0+(47)
Then the following properties hold(i) The discrete closed-loop system is asymptotically hyper-
stable(ii) 119906(119905) rarr 0 as 119905 rarr infin and the control and output are
bounded for all time
(iii) Assume furthermore that 120590(119905) = 120593(119910(119905) 119905) minus (1 +120582(119905))120593(119910[119896] 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+ 119906(119905) =minus120593(119910(119905) 119905) and forall119905 isin R0+ and that the class Φ satisfiesPopovrsquos type integral inequality for all time and results in theclass Φ119889(119896119879) at sampling instantsThen the continuous- timeclosed-loop system is asymptotically hyperstable
Proof From Lemma 9 via assumptions (1)ndash(5) constraints(36) hold so that the input- output energymeasure associatedwith the forced solution is positively lower-bounded andfinitely upper-bounded for all sampling instants 119905 = 119896119879forall119896 isin Z+ Since the discrete positive transfer function isstrictly positive real then it is also strictly stable in the discretecontext (ie convergent [29 36]) so that the homogeneousresponse is irrelevant for stability analysis purposes Thus120593[119896] isin Φ119889 119906[119896] rarr 0 and 119906[119896] and 119910[119896] arebounded for all time for any given bounded initial conditionsIn addition one has from Lemma 10 that (42) holds andthe forced input-output energy measure is positively lower-bounded and finitely upper-bound for all 119905 isin R+ Then119906(119905) rarr 0 as 119905 rarr infin and 119906(119905) and 119910(119905) are bounded for alltime for any given bounded initial conditions Properties [(i)-(ii)] have been proved Property (iii) follows by equalizing119906(119905) = minus120593(119910(119905) 119905) = (1 + 120582(119905))119906[119896] + 120590(119905) and 120593 isin Φ fromthe fact that minusint119905
0119906(120591)119910(120591)119889120591 le 1205740
Note that the use of Fourier transforms is useful tocharacterize the error of the input-output energy measure inthe frequency domain along the intersample periods relatedto its values at the preceding sampling instants This toolis addressed in Lemma 10 which is used in the proof ofTheorem 11 It allows to keep easily the characterizationof the frequency response of the continuous-time transferfunction to evaluate the input-output energy measure in theintersample period and to calculate the minimum value ofthe real part of such a transfer function to guarantee thepositivity and boundedness of such an energy Note that thecharacterization of the input-output energy measure of thefeed-forward linear loop evaluated in the frequency domaintogether with Popovrsquos inequalities of the class of hyperstablecontrollers is relevant to the hyperstability proofs of theclosed-loop system
5 Examples
Example 12 The positive realness of a discretized transferfunction depends not only on the continuous-time param-eters but on the sampling period as well Therefore thepositive realness of the continuous-time system does notguarantee that of any discretized counterpart irrespective ofthe sampling period and conversely For instance the transferfunction 119892(119904) = 1(119904 + 119886) is strictly positive real for 119886 gt 0 butit is not strongly strictly positive real since Re119892(119894120596) rarr 0 as120596 rarr plusmninfin Its discretized version without the concourse ofan ideal that is instantaneous sampler and zero-order holdbecomes 119904 119892119889(119911) = 119911(119911 minus 119890minus119886119879) which is strongly positivereal for 119879 isin (0infin) and as the sampling period 119879 rarr infin since119892119889(119911) rarr 1 as 119879 rarr infin However 119892119889(119911) = 119911(119911 minus 1) as 119879 rarr 0
12 Mathematical Problems in Engineering
Since it has a critically stable pole in the discrete domain it ispositive real although it is not strictly positive real
Example 13 Consider the discrete transfer function 119892119889(119911) =(1199112 + 1198891119911 + 1198892)(1199112 minus 1198861119911 minus 1198862) with real numerator anddenominator coefficients Its strict positive realness requiresas necessary conditions its convergence (ie strict stabilityin the discrete sense) so that its poles have to be within thecomplex open unit circle as well as the convergence of itsinverse so that its zeros have also to lie inside the open unitcircle and furthermore its relative degree (ie its pole-zeroexcess has to be zero) Note that the last condition is notneeded for strict (nonstrong) positive realness of continuoustransfer functions where their positive real part can convergeto zero as |119904| rarr infin By using the homographic bilineartransformation 119911 = (1 + 119904)(1 minus 119904) which transforms theunit circle in the left-hand-side complex plane to get theauxiliary continuous transfer function 119892119886(119904) = 119892119889(119911 = (1 +119904)(1 minus 119904)) it is found in [13] that the denominator andnumerator polynomials of 119892119889(119911) are subject to 1198862 minus 1198861 lt 11198861 + 1198862 lt 1 and 1198862 gt minus1 Take the particular values 1198861 = 05and 1198862 = 04 leading to the values of poles 1199111 = 05 and1199112 = 04 and the coefficients of the numerator polynomialare fixed to 1198891 = 12 1198892 = 03 Then we fix a discretetransfer function 119892119889(119911) = (1199112 + 12119911 + 03)(1199112 minus 05119911 minus 04)yielding min|119911|=1 Re119892119889(119911) ge 009 so 119892119889 isin SSPR Thedecomposition of 119892119889(119911) in simple fractions leads to 119892119889(119911) =1 + 168(119911 minus 093) + 0758(119911 + 043) By using an ideal (iewith instantaneous sampling) sampler and zero-order-holddevice of period 119879 of transfer function (1 minus 119890minus119879119904)119904 on acontinuous transfer function 119892(119904) to be specified to generate119892119889(119911) one has the identity 119885((1 minus 119890minus119879119904)119892(119904)119904) = 119892119889(119911) where119885(119865(119904)) is an abbreviate usual informal notation to denote119865lowast(119904) = 119871(119891lowast(119905)) where 119871(sdot) and 119885(sdot) denote the Laplacetransform of (sdot) of argument 119904 and 119885-transform of (sdot) ofargument 119911 = 119890119879119904 and 119891lowast(119905) = suminfin119899=0 119891(119899119879)120575(119905 minus 119899119879) denotesthe pulsed function of the function119891(119905)with sampling period119879 As a result we get for a sampling period 119879 = 01 sec119892 (119904) = 119904119885minus1 ( 119911119911 minus 1 [1 + 168119911 minus 093 + 0758119911 + 043])
Note that 119892 isin SPR (and also 119892 isin SSPR sinceits relative degree is zero) is the continuous-time transferfunction which generates 119892119889(119911) under an ideal sampler andzero-order-hold device of period 119879 = 01 sec so thatmin120596isinclR119892(i120596) = min120596isinR0+119892(i120596) gt 0 Thus it is guaranteedthat the continuous-time energy measure is associated withthe forced output 119864(119905) gt 0 for all 119905 gt 0 (ie it ispositively lower-bounded for all time) for any piecewise-continuous control irrespectively of the particular controllerin operation In order to guarantee that any such a controlthat converges asymptotically to zero it is necessary to finitelyupper-bound such an energy for all timeThe above concerns
also guarantee that 119864[119896] = 119864(119896119879) ge 1205741198890 gt 0 for all integer 119896 gt0 under any piecewise-constant control 119906[119896] = 119906(119896119879) witheventual jumps at sampling instants of any period 119879 gt 0 Forsuch a control identical conclusion arises from the discrete119892119889(119911)which is also strongly strictly positive real Assume that119906[119896] = minus120593(119910[119896] 119896) and forall119896 isin Z+ with 120593 isin Φ119889 where theclass Φ119889 is defined as the set of sequences 120593(119910[119896] 119896)whichsatisfy Popovrsquos inequality 119879sum119896119895=0 119910[119896]120593(119910[119896] 119896) ge minus120574119889 forall119896 isinZ+ for some finite 120574119886 isin R+ which is identical to
= minus119879119872(1 minus ]119896+1)1 minus ] minus 119879 119896sum119895=0
120576 [119896] ge minus120574119889 forall119896 isin Z+(50)
for some real constants 119872 gt 0 ] isin (0 1) and some realsummable sequence 120576[119896] Then the class Φ119889 is definedby any real sequence 120593(119910[119896] 119896) being parameterized by anyreal constants 119872 gt 0 ] isin (0 1) and any real summablesequence 120576[119896] such that
forall119896 isin Z0+ and 120574119889 ge max(119879119872(1 minus ]) + 120576 120574) for some given120574 ge 1205741198890 where 120576 = 119879max119896isinZ+ sum119896119895=0 120576[119896]The continuous-timecontrol is generated as
where 120593[119896] isin Φ119889(119879) satisfies Popovrsquos type discrete inequal-ity which makes the feedback system globally asymptoticallystable under the theorem hypothesis Thus the discrete-timeclosed-loop system is asymptotically hyperstable In partic-ular under the additional assumptions in Theorem 11(iii)the closed-loop continuous-time system is asymptoticallyhyperstable
Mathematical Problems in Engineering 13
One of the referees has given a comment of interest on thisexample related to continuous versus discrete counterpartproperties which has motivated us to give a further intuitivediscussion on it Note the following In the continuous-timecase there are two kinds of strictly positive real transferfunctions as it is well known namely the so-called strictpositive real which can be zero (or respectively positive)at infinity frequency and which a realizable has relativedegree one (or respectively zero) In the first case the directinput-output interconnection gain 119889 is zero in a state-spacerepresentation (119860 119887 119888 119889 = 0) of the transfer function Thetransfer functions of the second case are often referred to asthe class of strong strict positive real transfer functions Thisclass is not a disjoint class of first above one of strict positivereal transfer functions since it is included in it as a potentialparticular case Note that the members of this strong classof strict positive real transfer functions have relative degreezero and a positive direct input-output interconnection gain119889 Related to this concern the Yakubovich-Kalman-Popov(refereed to the continuous-time domain) and the Szego-Kalman-Popov (referred to the discrete domain) lemmascan be invoked to compare positive realness of the transferfunctions with their state-space associated properties (in factthe properties aremutually equivalent fromboth sides)Thuswe see that in the discrete case the Szego-Kalman-Popovlemma needs for its own coherency that the input-outputinterconnection 119889-gain be nonzero since otherwise one ofthe relevant constraint in the lemma fails [13] So strictpositive realness in the discrete domain in this context istypically strong and then the relative degree of its transferfunction is zero so that there is a direct interconnectionpositive gain from the input to the output Since we assumethat the discrete system is strictly positive real it has a positiveinput-output interconnection from the input to the outputand the minimum value of the real part of its frequencyresponse is strictly positive even as the frequency tends toinfinity On the other hand note that a zero-order holdldquoconvertsrdquo any strictly proper (continuous-time) transferfunction of any relative degree into a strictly discrete oneof unity relative order while it ldquoconvertsrdquo a continuous oneof zero relative order (ie biproper) into a discrete onestill of zero-relative order and with the same input-outputinterconnection gain See (48) Therefore we deal in thisexample with a biproper strict positive real transfer functionof positive input-output interconnection gain identical to thatof its discrete-time counterpart got through a ZOH samplingand hold device
Example 14 Consider that a controller of the form 119906(119905) =minus120593(119910(119905) 119905) forall119905 isin R0+ is used for the continuous-time feed-forward transfer function (48) so that120593 isin Φ and Φ is classin Example 13 satisfying an integral type Popovrsquos inequality(21) for all 119905(ge 1199050) isin R+ and some finite 1199050 isin R+ That isthe control function belongs to a class satisfying continuous-time integral Popovrsquos inequality Then 119906(119905) rarr 0 as 119905 rarr infinand 119906(119905) and 119910(119905) are almost everywhere bounded for all timesince 119892 isin SSPR then being (strictly) stable with input-output energy measure being positively lower-bounded andalso finitely upper-bounded for all 119905 isin R+ from Popovrsquos-type
integral inequality Potential useful controllers are for somegiven controller gain 119896119906 isin R+
119906 (119905) = 0 for 119905 isin [0 1)1198961199061199052119910 (119905) for 119905 ge 1 (54)
119906 (119905) = 119896119906119890minus119905119910 (119905) for 119905 ge 0 (55)
119906 (119905) = 0 for 119905 isin [0 2)
119896119906(119905 ln2119905) 119910 (119905) for 119905 ge 2 (56)
119906 (119905) = 0 for 119905 isin [0 1)
119896119906(119890radic119905radic119905) 119910 (119905) for 119905 ge 1 (57)
119906 (119905) = 1198961199061199053119890minus119905119910 (119905) for 119905 ge 0 (58)
where Popovrsquos inequality on [0 +infin] of (49)ndash(52)is intinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus119896119906 that of (57) isintinfin
0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 and that of (58) isintinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 Since the respective integrands
are monotonically decreasing then there is some 1199050 isin R+ andsome 120574 = 120574(1199050) isin R+ such that int119905
1199050119910(119905)120593(119910(119905) 119905)119889119905 ge minus120574 for
all 119905(ge 1199050) isin R+ The obtained closed-loop systems are inall cases asymptotically hyperstable The piecewise-constantcontrols with eventual discontinuities at sampling instantsgenerated from any discrete-time controller (51) of class Φ119889of Example 13 lead also to asymptotic hyperstability of thediscretized system since 119892119889 isin SSPRExample 15 Consider the continuous-time open-loop sys-tem (2) under a real sampler of sampling period 119879 and datapicking-up nonzero duration 120576 isin (0 119879) The state and outputtrajectory solutions are
forall119896 isin Z0+ for some real sequence 120585[119896] such that 120585[119896] isin(119896119879 119896119879 + 120576) where 119892119889(119911) is the discretized transfer functionunder a sampler and zero-order-hold device subject to instan-taneous ideal sampling Now assume that the incrementalcontrol along intersample periods is generated from itspreceding sampled value as (119895119879+ 120591) = 120582119895119879+120576(119895119879+ 120591)119906[119895] forany 119895 isin Z0+ and 120591 isin [0 119879] where 120582119895119879+120576(119895119879+120591) is a truncatedfunction under the usual notation one gets
The above equations imply that if 119892119889 isin SPR then the forcedoutput 119910119891[119896] = 119910[119896] minus 119888119879119890119896119860119879119909[0] forall119896 isin Z0+ generates anassociate input-output energy measure which is strictly posi-tive for 119896 isin Z0+ since 120588 isin (0 1) for real sampling of duration120576 isin (0 119879)This is a necessary condition for asymptotic hyper-stability under an appropriate class of controllers satisfyingPopovrsquos inequality However the incremental input-outputenergy measure generated by the incremental forced output120576119888119879sum119896minus1119895=0 119890(119896minus119895)119860119879(int1198790 120582119895119879+120576(120591)119890minus119860120591119887 119889120591)119906[119895] is not guaranteedto be nonnegative in general A sufficient condition for that isthat lim sup119895rarrinfin|ess sup120591isin(0120576)(120582119895119879+120576(120591))| is sufficiently smallrelated to the amount ((1 minus 120588)120576)min120596isin[0120587] Re119892119889(119890i120596)6 Conclusions
This paper has been devoted to obtain and to prove theasymptotic hyperstability property of a linear and time-invariant continuous-time system based on some given andpreviously proved auxiliary preparatory lemmas Basicallyit is proved that the continuous-time closed-loop system isasymptotically hyperstable under the assumptions that thediscretized feed-forward transfer function is strictly positivereal and that the discretized system at sampling instants ofconstant sampling period is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsIt is assumed that the continuous-time controller belongs toa class defined in the following way (a) its values at samplinginstants are defined by a class of controllers which satisfydiscrete Popovrsquos type inequality (b) the intersample control isgenerated from its values at sampling instants with the use oftwomodulating auxiliary functionswhich are subject to someextra additional conditions on smallness and boundednessSuch conditions make the overall continuous-time controllerto satisfy continuous-type Popovrsquos inequality for all time
where 119896(120591) = max(119911 isin Z+ 119911119879 le 120591) and 119896 = 119896(119905) = max(119911 isinZ+ 119911119879 le 119905) forall119905 isin R0+ which yields (13)
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
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Mathematical Problems in Engineering 11
Now the use of (44) in Lemma 4 (14) implies thatLemma 9(ii) (36) holds implying the existence of 1205740 (120574 gt1205740) isin R+ such that 1205740 le 119864(119905) le 120574 forall119905 isin [(1198960 + 1)119879infin)provided that 1205741198890 le 119864119889[119896] le 120574119889 for 1205741198890 (120574119889 gt 1205741198890) isin R+ since1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isin Z0+ if1205740 minus 120574119889 le minus 120582
1003816100381610038161003816119892 (119896119879 minus 120591)1003816100381610038161003816 119889120591le 120574 minus 1205741198890 forall119896 (gt 1198960) isin Z+ forall120591 isin (0 119879)
(45)
if 120582 and 120590 are sufficiently small such that (43) holds leadingto 1205740 gt 0 for any given positive 1205741198890 and 120574119889 gt 1205741198890
The main first result follows
Theorem 11 Consider the dynamic system (2) subject to acontrol
forall119896 isin Z0+ where 120593 [119896] isin Φ119889 (119879) (46)
Assume that(1) 120593(119910[119896] 119896119879) is in the class Φ119889(119879) (that is120593(119910[119896] 119896119879) isin Φ119889(119879)) defined by the discrete Popovrsquos
type sum inequality sum119896ℓ=0 119910119889[ℓ]119906[ℓ] ge minus120574119889119879 for some 120574119889 isin Rforall119896 isin Z0+(2) 119892119889 isin 119878119875119877 there exist constants 1205741198890 (120574119889 gt 1205741198890) isin R+
such that 1205741198890 le 119864119889[119896] le 120574119889 forall119896(gt 1198960) isin Z+ and some 1198960 isinZ0+
(3) 120593(119909 119896119879) is subadditive for all 119909 isin R forall119896 isin Z0+(4) 120582 = sup119905isinR0+ |120582(119905)| and 120590 = sup119905isinR0+ |120590(119905)| are
119895119879(120582 (119895119879 + 120591) 119892 ((119896 minus 119895) 119879 minus 120591)
minus 120582 [119895] 119892 [(119896 minus 119895) 119879]) 119889120591 forall119896 isin Z0+(47)
Then the following properties hold(i) The discrete closed-loop system is asymptotically hyper-
stable(ii) 119906(119905) rarr 0 as 119905 rarr infin and the control and output are
bounded for all time
(iii) Assume furthermore that 120590(119905) = 120593(119910(119905) 119905) minus (1 +120582(119905))120593(119910[119896] 119896119879) forall119905 isin [119896119879 (119896 + 1)119879) forall119896 isin Z0+ 119906(119905) =minus120593(119910(119905) 119905) and forall119905 isin R0+ and that the class Φ satisfiesPopovrsquos type integral inequality for all time and results in theclass Φ119889(119896119879) at sampling instantsThen the continuous- timeclosed-loop system is asymptotically hyperstable
Proof From Lemma 9 via assumptions (1)ndash(5) constraints(36) hold so that the input- output energymeasure associatedwith the forced solution is positively lower-bounded andfinitely upper-bounded for all sampling instants 119905 = 119896119879forall119896 isin Z+ Since the discrete positive transfer function isstrictly positive real then it is also strictly stable in the discretecontext (ie convergent [29 36]) so that the homogeneousresponse is irrelevant for stability analysis purposes Thus120593[119896] isin Φ119889 119906[119896] rarr 0 and 119906[119896] and 119910[119896] arebounded for all time for any given bounded initial conditionsIn addition one has from Lemma 10 that (42) holds andthe forced input-output energy measure is positively lower-bounded and finitely upper-bound for all 119905 isin R+ Then119906(119905) rarr 0 as 119905 rarr infin and 119906(119905) and 119910(119905) are bounded for alltime for any given bounded initial conditions Properties [(i)-(ii)] have been proved Property (iii) follows by equalizing119906(119905) = minus120593(119910(119905) 119905) = (1 + 120582(119905))119906[119896] + 120590(119905) and 120593 isin Φ fromthe fact that minusint119905
0119906(120591)119910(120591)119889120591 le 1205740
Note that the use of Fourier transforms is useful tocharacterize the error of the input-output energy measure inthe frequency domain along the intersample periods relatedto its values at the preceding sampling instants This toolis addressed in Lemma 10 which is used in the proof ofTheorem 11 It allows to keep easily the characterizationof the frequency response of the continuous-time transferfunction to evaluate the input-output energy measure in theintersample period and to calculate the minimum value ofthe real part of such a transfer function to guarantee thepositivity and boundedness of such an energy Note that thecharacterization of the input-output energy measure of thefeed-forward linear loop evaluated in the frequency domaintogether with Popovrsquos inequalities of the class of hyperstablecontrollers is relevant to the hyperstability proofs of theclosed-loop system
5 Examples
Example 12 The positive realness of a discretized transferfunction depends not only on the continuous-time param-eters but on the sampling period as well Therefore thepositive realness of the continuous-time system does notguarantee that of any discretized counterpart irrespective ofthe sampling period and conversely For instance the transferfunction 119892(119904) = 1(119904 + 119886) is strictly positive real for 119886 gt 0 butit is not strongly strictly positive real since Re119892(119894120596) rarr 0 as120596 rarr plusmninfin Its discretized version without the concourse ofan ideal that is instantaneous sampler and zero-order holdbecomes 119904 119892119889(119911) = 119911(119911 minus 119890minus119886119879) which is strongly positivereal for 119879 isin (0infin) and as the sampling period 119879 rarr infin since119892119889(119911) rarr 1 as 119879 rarr infin However 119892119889(119911) = 119911(119911 minus 1) as 119879 rarr 0
12 Mathematical Problems in Engineering
Since it has a critically stable pole in the discrete domain it ispositive real although it is not strictly positive real
Example 13 Consider the discrete transfer function 119892119889(119911) =(1199112 + 1198891119911 + 1198892)(1199112 minus 1198861119911 minus 1198862) with real numerator anddenominator coefficients Its strict positive realness requiresas necessary conditions its convergence (ie strict stabilityin the discrete sense) so that its poles have to be within thecomplex open unit circle as well as the convergence of itsinverse so that its zeros have also to lie inside the open unitcircle and furthermore its relative degree (ie its pole-zeroexcess has to be zero) Note that the last condition is notneeded for strict (nonstrong) positive realness of continuoustransfer functions where their positive real part can convergeto zero as |119904| rarr infin By using the homographic bilineartransformation 119911 = (1 + 119904)(1 minus 119904) which transforms theunit circle in the left-hand-side complex plane to get theauxiliary continuous transfer function 119892119886(119904) = 119892119889(119911 = (1 +119904)(1 minus 119904)) it is found in [13] that the denominator andnumerator polynomials of 119892119889(119911) are subject to 1198862 minus 1198861 lt 11198861 + 1198862 lt 1 and 1198862 gt minus1 Take the particular values 1198861 = 05and 1198862 = 04 leading to the values of poles 1199111 = 05 and1199112 = 04 and the coefficients of the numerator polynomialare fixed to 1198891 = 12 1198892 = 03 Then we fix a discretetransfer function 119892119889(119911) = (1199112 + 12119911 + 03)(1199112 minus 05119911 minus 04)yielding min|119911|=1 Re119892119889(119911) ge 009 so 119892119889 isin SSPR Thedecomposition of 119892119889(119911) in simple fractions leads to 119892119889(119911) =1 + 168(119911 minus 093) + 0758(119911 + 043) By using an ideal (iewith instantaneous sampling) sampler and zero-order-holddevice of period 119879 of transfer function (1 minus 119890minus119879119904)119904 on acontinuous transfer function 119892(119904) to be specified to generate119892119889(119911) one has the identity 119885((1 minus 119890minus119879119904)119892(119904)119904) = 119892119889(119911) where119885(119865(119904)) is an abbreviate usual informal notation to denote119865lowast(119904) = 119871(119891lowast(119905)) where 119871(sdot) and 119885(sdot) denote the Laplacetransform of (sdot) of argument 119904 and 119885-transform of (sdot) ofargument 119911 = 119890119879119904 and 119891lowast(119905) = suminfin119899=0 119891(119899119879)120575(119905 minus 119899119879) denotesthe pulsed function of the function119891(119905)with sampling period119879 As a result we get for a sampling period 119879 = 01 sec119892 (119904) = 119904119885minus1 ( 119911119911 minus 1 [1 + 168119911 minus 093 + 0758119911 + 043])
Note that 119892 isin SPR (and also 119892 isin SSPR sinceits relative degree is zero) is the continuous-time transferfunction which generates 119892119889(119911) under an ideal sampler andzero-order-hold device of period 119879 = 01 sec so thatmin120596isinclR119892(i120596) = min120596isinR0+119892(i120596) gt 0 Thus it is guaranteedthat the continuous-time energy measure is associated withthe forced output 119864(119905) gt 0 for all 119905 gt 0 (ie it ispositively lower-bounded for all time) for any piecewise-continuous control irrespectively of the particular controllerin operation In order to guarantee that any such a controlthat converges asymptotically to zero it is necessary to finitelyupper-bound such an energy for all timeThe above concerns
also guarantee that 119864[119896] = 119864(119896119879) ge 1205741198890 gt 0 for all integer 119896 gt0 under any piecewise-constant control 119906[119896] = 119906(119896119879) witheventual jumps at sampling instants of any period 119879 gt 0 Forsuch a control identical conclusion arises from the discrete119892119889(119911)which is also strongly strictly positive real Assume that119906[119896] = minus120593(119910[119896] 119896) and forall119896 isin Z+ with 120593 isin Φ119889 where theclass Φ119889 is defined as the set of sequences 120593(119910[119896] 119896)whichsatisfy Popovrsquos inequality 119879sum119896119895=0 119910[119896]120593(119910[119896] 119896) ge minus120574119889 forall119896 isinZ+ for some finite 120574119886 isin R+ which is identical to
= minus119879119872(1 minus ]119896+1)1 minus ] minus 119879 119896sum119895=0
120576 [119896] ge minus120574119889 forall119896 isin Z+(50)
for some real constants 119872 gt 0 ] isin (0 1) and some realsummable sequence 120576[119896] Then the class Φ119889 is definedby any real sequence 120593(119910[119896] 119896) being parameterized by anyreal constants 119872 gt 0 ] isin (0 1) and any real summablesequence 120576[119896] such that
forall119896 isin Z0+ and 120574119889 ge max(119879119872(1 minus ]) + 120576 120574) for some given120574 ge 1205741198890 where 120576 = 119879max119896isinZ+ sum119896119895=0 120576[119896]The continuous-timecontrol is generated as
where 120593[119896] isin Φ119889(119879) satisfies Popovrsquos type discrete inequal-ity which makes the feedback system globally asymptoticallystable under the theorem hypothesis Thus the discrete-timeclosed-loop system is asymptotically hyperstable In partic-ular under the additional assumptions in Theorem 11(iii)the closed-loop continuous-time system is asymptoticallyhyperstable
Mathematical Problems in Engineering 13
One of the referees has given a comment of interest on thisexample related to continuous versus discrete counterpartproperties which has motivated us to give a further intuitivediscussion on it Note the following In the continuous-timecase there are two kinds of strictly positive real transferfunctions as it is well known namely the so-called strictpositive real which can be zero (or respectively positive)at infinity frequency and which a realizable has relativedegree one (or respectively zero) In the first case the directinput-output interconnection gain 119889 is zero in a state-spacerepresentation (119860 119887 119888 119889 = 0) of the transfer function Thetransfer functions of the second case are often referred to asthe class of strong strict positive real transfer functions Thisclass is not a disjoint class of first above one of strict positivereal transfer functions since it is included in it as a potentialparticular case Note that the members of this strong classof strict positive real transfer functions have relative degreezero and a positive direct input-output interconnection gain119889 Related to this concern the Yakubovich-Kalman-Popov(refereed to the continuous-time domain) and the Szego-Kalman-Popov (referred to the discrete domain) lemmascan be invoked to compare positive realness of the transferfunctions with their state-space associated properties (in factthe properties aremutually equivalent fromboth sides)Thuswe see that in the discrete case the Szego-Kalman-Popovlemma needs for its own coherency that the input-outputinterconnection 119889-gain be nonzero since otherwise one ofthe relevant constraint in the lemma fails [13] So strictpositive realness in the discrete domain in this context istypically strong and then the relative degree of its transferfunction is zero so that there is a direct interconnectionpositive gain from the input to the output Since we assumethat the discrete system is strictly positive real it has a positiveinput-output interconnection from the input to the outputand the minimum value of the real part of its frequencyresponse is strictly positive even as the frequency tends toinfinity On the other hand note that a zero-order holdldquoconvertsrdquo any strictly proper (continuous-time) transferfunction of any relative degree into a strictly discrete oneof unity relative order while it ldquoconvertsrdquo a continuous oneof zero relative order (ie biproper) into a discrete onestill of zero-relative order and with the same input-outputinterconnection gain See (48) Therefore we deal in thisexample with a biproper strict positive real transfer functionof positive input-output interconnection gain identical to thatof its discrete-time counterpart got through a ZOH samplingand hold device
Example 14 Consider that a controller of the form 119906(119905) =minus120593(119910(119905) 119905) forall119905 isin R0+ is used for the continuous-time feed-forward transfer function (48) so that120593 isin Φ and Φ is classin Example 13 satisfying an integral type Popovrsquos inequality(21) for all 119905(ge 1199050) isin R+ and some finite 1199050 isin R+ That isthe control function belongs to a class satisfying continuous-time integral Popovrsquos inequality Then 119906(119905) rarr 0 as 119905 rarr infinand 119906(119905) and 119910(119905) are almost everywhere bounded for all timesince 119892 isin SSPR then being (strictly) stable with input-output energy measure being positively lower-bounded andalso finitely upper-bounded for all 119905 isin R+ from Popovrsquos-type
integral inequality Potential useful controllers are for somegiven controller gain 119896119906 isin R+
119906 (119905) = 0 for 119905 isin [0 1)1198961199061199052119910 (119905) for 119905 ge 1 (54)
119906 (119905) = 119896119906119890minus119905119910 (119905) for 119905 ge 0 (55)
119906 (119905) = 0 for 119905 isin [0 2)
119896119906(119905 ln2119905) 119910 (119905) for 119905 ge 2 (56)
119906 (119905) = 0 for 119905 isin [0 1)
119896119906(119890radic119905radic119905) 119910 (119905) for 119905 ge 1 (57)
119906 (119905) = 1198961199061199053119890minus119905119910 (119905) for 119905 ge 0 (58)
where Popovrsquos inequality on [0 +infin] of (49)ndash(52)is intinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus119896119906 that of (57) isintinfin
0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 and that of (58) isintinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 Since the respective integrands
are monotonically decreasing then there is some 1199050 isin R+ andsome 120574 = 120574(1199050) isin R+ such that int119905
1199050119910(119905)120593(119910(119905) 119905)119889119905 ge minus120574 for
all 119905(ge 1199050) isin R+ The obtained closed-loop systems are inall cases asymptotically hyperstable The piecewise-constantcontrols with eventual discontinuities at sampling instantsgenerated from any discrete-time controller (51) of class Φ119889of Example 13 lead also to asymptotic hyperstability of thediscretized system since 119892119889 isin SSPRExample 15 Consider the continuous-time open-loop sys-tem (2) under a real sampler of sampling period 119879 and datapicking-up nonzero duration 120576 isin (0 119879) The state and outputtrajectory solutions are
forall119896 isin Z0+ for some real sequence 120585[119896] such that 120585[119896] isin(119896119879 119896119879 + 120576) where 119892119889(119911) is the discretized transfer functionunder a sampler and zero-order-hold device subject to instan-taneous ideal sampling Now assume that the incrementalcontrol along intersample periods is generated from itspreceding sampled value as (119895119879+ 120591) = 120582119895119879+120576(119895119879+ 120591)119906[119895] forany 119895 isin Z0+ and 120591 isin [0 119879] where 120582119895119879+120576(119895119879+120591) is a truncatedfunction under the usual notation one gets
The above equations imply that if 119892119889 isin SPR then the forcedoutput 119910119891[119896] = 119910[119896] minus 119888119879119890119896119860119879119909[0] forall119896 isin Z0+ generates anassociate input-output energy measure which is strictly posi-tive for 119896 isin Z0+ since 120588 isin (0 1) for real sampling of duration120576 isin (0 119879)This is a necessary condition for asymptotic hyper-stability under an appropriate class of controllers satisfyingPopovrsquos inequality However the incremental input-outputenergy measure generated by the incremental forced output120576119888119879sum119896minus1119895=0 119890(119896minus119895)119860119879(int1198790 120582119895119879+120576(120591)119890minus119860120591119887 119889120591)119906[119895] is not guaranteedto be nonnegative in general A sufficient condition for that isthat lim sup119895rarrinfin|ess sup120591isin(0120576)(120582119895119879+120576(120591))| is sufficiently smallrelated to the amount ((1 minus 120588)120576)min120596isin[0120587] Re119892119889(119890i120596)6 Conclusions
This paper has been devoted to obtain and to prove theasymptotic hyperstability property of a linear and time-invariant continuous-time system based on some given andpreviously proved auxiliary preparatory lemmas Basicallyit is proved that the continuous-time closed-loop system isasymptotically hyperstable under the assumptions that thediscretized feed-forward transfer function is strictly positivereal and that the discretized system at sampling instants ofconstant sampling period is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsIt is assumed that the continuous-time controller belongs toa class defined in the following way (a) its values at samplinginstants are defined by a class of controllers which satisfydiscrete Popovrsquos type inequality (b) the intersample control isgenerated from its values at sampling instants with the use oftwomodulating auxiliary functionswhich are subject to someextra additional conditions on smallness and boundednessSuch conditions make the overall continuous-time controllerto satisfy continuous-type Popovrsquos inequality for all time
where 119896(120591) = max(119911 isin Z+ 119911119879 le 120591) and 119896 = 119896(119905) = max(119911 isinZ+ 119911119879 le 119905) forall119905 isin R0+ which yields (13)
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
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12 Mathematical Problems in Engineering
Since it has a critically stable pole in the discrete domain it ispositive real although it is not strictly positive real
Example 13 Consider the discrete transfer function 119892119889(119911) =(1199112 + 1198891119911 + 1198892)(1199112 minus 1198861119911 minus 1198862) with real numerator anddenominator coefficients Its strict positive realness requiresas necessary conditions its convergence (ie strict stabilityin the discrete sense) so that its poles have to be within thecomplex open unit circle as well as the convergence of itsinverse so that its zeros have also to lie inside the open unitcircle and furthermore its relative degree (ie its pole-zeroexcess has to be zero) Note that the last condition is notneeded for strict (nonstrong) positive realness of continuoustransfer functions where their positive real part can convergeto zero as |119904| rarr infin By using the homographic bilineartransformation 119911 = (1 + 119904)(1 minus 119904) which transforms theunit circle in the left-hand-side complex plane to get theauxiliary continuous transfer function 119892119886(119904) = 119892119889(119911 = (1 +119904)(1 minus 119904)) it is found in [13] that the denominator andnumerator polynomials of 119892119889(119911) are subject to 1198862 minus 1198861 lt 11198861 + 1198862 lt 1 and 1198862 gt minus1 Take the particular values 1198861 = 05and 1198862 = 04 leading to the values of poles 1199111 = 05 and1199112 = 04 and the coefficients of the numerator polynomialare fixed to 1198891 = 12 1198892 = 03 Then we fix a discretetransfer function 119892119889(119911) = (1199112 + 12119911 + 03)(1199112 minus 05119911 minus 04)yielding min|119911|=1 Re119892119889(119911) ge 009 so 119892119889 isin SSPR Thedecomposition of 119892119889(119911) in simple fractions leads to 119892119889(119911) =1 + 168(119911 minus 093) + 0758(119911 + 043) By using an ideal (iewith instantaneous sampling) sampler and zero-order-holddevice of period 119879 of transfer function (1 minus 119890minus119879119904)119904 on acontinuous transfer function 119892(119904) to be specified to generate119892119889(119911) one has the identity 119885((1 minus 119890minus119879119904)119892(119904)119904) = 119892119889(119911) where119885(119865(119904)) is an abbreviate usual informal notation to denote119865lowast(119904) = 119871(119891lowast(119905)) where 119871(sdot) and 119885(sdot) denote the Laplacetransform of (sdot) of argument 119904 and 119885-transform of (sdot) ofargument 119911 = 119890119879119904 and 119891lowast(119905) = suminfin119899=0 119891(119899119879)120575(119905 minus 119899119879) denotesthe pulsed function of the function119891(119905)with sampling period119879 As a result we get for a sampling period 119879 = 01 sec119892 (119904) = 119904119885minus1 ( 119911119911 minus 1 [1 + 168119911 minus 093 + 0758119911 + 043])
Note that 119892 isin SPR (and also 119892 isin SSPR sinceits relative degree is zero) is the continuous-time transferfunction which generates 119892119889(119911) under an ideal sampler andzero-order-hold device of period 119879 = 01 sec so thatmin120596isinclR119892(i120596) = min120596isinR0+119892(i120596) gt 0 Thus it is guaranteedthat the continuous-time energy measure is associated withthe forced output 119864(119905) gt 0 for all 119905 gt 0 (ie it ispositively lower-bounded for all time) for any piecewise-continuous control irrespectively of the particular controllerin operation In order to guarantee that any such a controlthat converges asymptotically to zero it is necessary to finitelyupper-bound such an energy for all timeThe above concerns
also guarantee that 119864[119896] = 119864(119896119879) ge 1205741198890 gt 0 for all integer 119896 gt0 under any piecewise-constant control 119906[119896] = 119906(119896119879) witheventual jumps at sampling instants of any period 119879 gt 0 Forsuch a control identical conclusion arises from the discrete119892119889(119911)which is also strongly strictly positive real Assume that119906[119896] = minus120593(119910[119896] 119896) and forall119896 isin Z+ with 120593 isin Φ119889 where theclass Φ119889 is defined as the set of sequences 120593(119910[119896] 119896)whichsatisfy Popovrsquos inequality 119879sum119896119895=0 119910[119896]120593(119910[119896] 119896) ge minus120574119889 forall119896 isinZ+ for some finite 120574119886 isin R+ which is identical to
= minus119879119872(1 minus ]119896+1)1 minus ] minus 119879 119896sum119895=0
120576 [119896] ge minus120574119889 forall119896 isin Z+(50)
for some real constants 119872 gt 0 ] isin (0 1) and some realsummable sequence 120576[119896] Then the class Φ119889 is definedby any real sequence 120593(119910[119896] 119896) being parameterized by anyreal constants 119872 gt 0 ] isin (0 1) and any real summablesequence 120576[119896] such that
forall119896 isin Z0+ and 120574119889 ge max(119879119872(1 minus ]) + 120576 120574) for some given120574 ge 1205741198890 where 120576 = 119879max119896isinZ+ sum119896119895=0 120576[119896]The continuous-timecontrol is generated as
where 120593[119896] isin Φ119889(119879) satisfies Popovrsquos type discrete inequal-ity which makes the feedback system globally asymptoticallystable under the theorem hypothesis Thus the discrete-timeclosed-loop system is asymptotically hyperstable In partic-ular under the additional assumptions in Theorem 11(iii)the closed-loop continuous-time system is asymptoticallyhyperstable
Mathematical Problems in Engineering 13
One of the referees has given a comment of interest on thisexample related to continuous versus discrete counterpartproperties which has motivated us to give a further intuitivediscussion on it Note the following In the continuous-timecase there are two kinds of strictly positive real transferfunctions as it is well known namely the so-called strictpositive real which can be zero (or respectively positive)at infinity frequency and which a realizable has relativedegree one (or respectively zero) In the first case the directinput-output interconnection gain 119889 is zero in a state-spacerepresentation (119860 119887 119888 119889 = 0) of the transfer function Thetransfer functions of the second case are often referred to asthe class of strong strict positive real transfer functions Thisclass is not a disjoint class of first above one of strict positivereal transfer functions since it is included in it as a potentialparticular case Note that the members of this strong classof strict positive real transfer functions have relative degreezero and a positive direct input-output interconnection gain119889 Related to this concern the Yakubovich-Kalman-Popov(refereed to the continuous-time domain) and the Szego-Kalman-Popov (referred to the discrete domain) lemmascan be invoked to compare positive realness of the transferfunctions with their state-space associated properties (in factthe properties aremutually equivalent fromboth sides)Thuswe see that in the discrete case the Szego-Kalman-Popovlemma needs for its own coherency that the input-outputinterconnection 119889-gain be nonzero since otherwise one ofthe relevant constraint in the lemma fails [13] So strictpositive realness in the discrete domain in this context istypically strong and then the relative degree of its transferfunction is zero so that there is a direct interconnectionpositive gain from the input to the output Since we assumethat the discrete system is strictly positive real it has a positiveinput-output interconnection from the input to the outputand the minimum value of the real part of its frequencyresponse is strictly positive even as the frequency tends toinfinity On the other hand note that a zero-order holdldquoconvertsrdquo any strictly proper (continuous-time) transferfunction of any relative degree into a strictly discrete oneof unity relative order while it ldquoconvertsrdquo a continuous oneof zero relative order (ie biproper) into a discrete onestill of zero-relative order and with the same input-outputinterconnection gain See (48) Therefore we deal in thisexample with a biproper strict positive real transfer functionof positive input-output interconnection gain identical to thatof its discrete-time counterpart got through a ZOH samplingand hold device
Example 14 Consider that a controller of the form 119906(119905) =minus120593(119910(119905) 119905) forall119905 isin R0+ is used for the continuous-time feed-forward transfer function (48) so that120593 isin Φ and Φ is classin Example 13 satisfying an integral type Popovrsquos inequality(21) for all 119905(ge 1199050) isin R+ and some finite 1199050 isin R+ That isthe control function belongs to a class satisfying continuous-time integral Popovrsquos inequality Then 119906(119905) rarr 0 as 119905 rarr infinand 119906(119905) and 119910(119905) are almost everywhere bounded for all timesince 119892 isin SSPR then being (strictly) stable with input-output energy measure being positively lower-bounded andalso finitely upper-bounded for all 119905 isin R+ from Popovrsquos-type
integral inequality Potential useful controllers are for somegiven controller gain 119896119906 isin R+
119906 (119905) = 0 for 119905 isin [0 1)1198961199061199052119910 (119905) for 119905 ge 1 (54)
119906 (119905) = 119896119906119890minus119905119910 (119905) for 119905 ge 0 (55)
119906 (119905) = 0 for 119905 isin [0 2)
119896119906(119905 ln2119905) 119910 (119905) for 119905 ge 2 (56)
119906 (119905) = 0 for 119905 isin [0 1)
119896119906(119890radic119905radic119905) 119910 (119905) for 119905 ge 1 (57)
119906 (119905) = 1198961199061199053119890minus119905119910 (119905) for 119905 ge 0 (58)
where Popovrsquos inequality on [0 +infin] of (49)ndash(52)is intinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus119896119906 that of (57) isintinfin
0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 and that of (58) isintinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 Since the respective integrands
are monotonically decreasing then there is some 1199050 isin R+ andsome 120574 = 120574(1199050) isin R+ such that int119905
1199050119910(119905)120593(119910(119905) 119905)119889119905 ge minus120574 for
all 119905(ge 1199050) isin R+ The obtained closed-loop systems are inall cases asymptotically hyperstable The piecewise-constantcontrols with eventual discontinuities at sampling instantsgenerated from any discrete-time controller (51) of class Φ119889of Example 13 lead also to asymptotic hyperstability of thediscretized system since 119892119889 isin SSPRExample 15 Consider the continuous-time open-loop sys-tem (2) under a real sampler of sampling period 119879 and datapicking-up nonzero duration 120576 isin (0 119879) The state and outputtrajectory solutions are
forall119896 isin Z0+ for some real sequence 120585[119896] such that 120585[119896] isin(119896119879 119896119879 + 120576) where 119892119889(119911) is the discretized transfer functionunder a sampler and zero-order-hold device subject to instan-taneous ideal sampling Now assume that the incrementalcontrol along intersample periods is generated from itspreceding sampled value as (119895119879+ 120591) = 120582119895119879+120576(119895119879+ 120591)119906[119895] forany 119895 isin Z0+ and 120591 isin [0 119879] where 120582119895119879+120576(119895119879+120591) is a truncatedfunction under the usual notation one gets
The above equations imply that if 119892119889 isin SPR then the forcedoutput 119910119891[119896] = 119910[119896] minus 119888119879119890119896119860119879119909[0] forall119896 isin Z0+ generates anassociate input-output energy measure which is strictly posi-tive for 119896 isin Z0+ since 120588 isin (0 1) for real sampling of duration120576 isin (0 119879)This is a necessary condition for asymptotic hyper-stability under an appropriate class of controllers satisfyingPopovrsquos inequality However the incremental input-outputenergy measure generated by the incremental forced output120576119888119879sum119896minus1119895=0 119890(119896minus119895)119860119879(int1198790 120582119895119879+120576(120591)119890minus119860120591119887 119889120591)119906[119895] is not guaranteedto be nonnegative in general A sufficient condition for that isthat lim sup119895rarrinfin|ess sup120591isin(0120576)(120582119895119879+120576(120591))| is sufficiently smallrelated to the amount ((1 minus 120588)120576)min120596isin[0120587] Re119892119889(119890i120596)6 Conclusions
This paper has been devoted to obtain and to prove theasymptotic hyperstability property of a linear and time-invariant continuous-time system based on some given andpreviously proved auxiliary preparatory lemmas Basicallyit is proved that the continuous-time closed-loop system isasymptotically hyperstable under the assumptions that thediscretized feed-forward transfer function is strictly positivereal and that the discretized system at sampling instants ofconstant sampling period is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsIt is assumed that the continuous-time controller belongs toa class defined in the following way (a) its values at samplinginstants are defined by a class of controllers which satisfydiscrete Popovrsquos type inequality (b) the intersample control isgenerated from its values at sampling instants with the use oftwomodulating auxiliary functionswhich are subject to someextra additional conditions on smallness and boundednessSuch conditions make the overall continuous-time controllerto satisfy continuous-type Popovrsquos inequality for all time
where 119896(120591) = max(119911 isin Z+ 119911119879 le 120591) and 119896 = 119896(119905) = max(119911 isinZ+ 119911119879 le 119905) forall119905 isin R0+ which yields (13)
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
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AnalysisInternational Journal of
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Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 13
One of the referees has given a comment of interest on thisexample related to continuous versus discrete counterpartproperties which has motivated us to give a further intuitivediscussion on it Note the following In the continuous-timecase there are two kinds of strictly positive real transferfunctions as it is well known namely the so-called strictpositive real which can be zero (or respectively positive)at infinity frequency and which a realizable has relativedegree one (or respectively zero) In the first case the directinput-output interconnection gain 119889 is zero in a state-spacerepresentation (119860 119887 119888 119889 = 0) of the transfer function Thetransfer functions of the second case are often referred to asthe class of strong strict positive real transfer functions Thisclass is not a disjoint class of first above one of strict positivereal transfer functions since it is included in it as a potentialparticular case Note that the members of this strong classof strict positive real transfer functions have relative degreezero and a positive direct input-output interconnection gain119889 Related to this concern the Yakubovich-Kalman-Popov(refereed to the continuous-time domain) and the Szego-Kalman-Popov (referred to the discrete domain) lemmascan be invoked to compare positive realness of the transferfunctions with their state-space associated properties (in factthe properties aremutually equivalent fromboth sides)Thuswe see that in the discrete case the Szego-Kalman-Popovlemma needs for its own coherency that the input-outputinterconnection 119889-gain be nonzero since otherwise one ofthe relevant constraint in the lemma fails [13] So strictpositive realness in the discrete domain in this context istypically strong and then the relative degree of its transferfunction is zero so that there is a direct interconnectionpositive gain from the input to the output Since we assumethat the discrete system is strictly positive real it has a positiveinput-output interconnection from the input to the outputand the minimum value of the real part of its frequencyresponse is strictly positive even as the frequency tends toinfinity On the other hand note that a zero-order holdldquoconvertsrdquo any strictly proper (continuous-time) transferfunction of any relative degree into a strictly discrete oneof unity relative order while it ldquoconvertsrdquo a continuous oneof zero relative order (ie biproper) into a discrete onestill of zero-relative order and with the same input-outputinterconnection gain See (48) Therefore we deal in thisexample with a biproper strict positive real transfer functionof positive input-output interconnection gain identical to thatof its discrete-time counterpart got through a ZOH samplingand hold device
Example 14 Consider that a controller of the form 119906(119905) =minus120593(119910(119905) 119905) forall119905 isin R0+ is used for the continuous-time feed-forward transfer function (48) so that120593 isin Φ and Φ is classin Example 13 satisfying an integral type Popovrsquos inequality(21) for all 119905(ge 1199050) isin R+ and some finite 1199050 isin R+ That isthe control function belongs to a class satisfying continuous-time integral Popovrsquos inequality Then 119906(119905) rarr 0 as 119905 rarr infinand 119906(119905) and 119910(119905) are almost everywhere bounded for all timesince 119892 isin SSPR then being (strictly) stable with input-output energy measure being positively lower-bounded andalso finitely upper-bounded for all 119905 isin R+ from Popovrsquos-type
integral inequality Potential useful controllers are for somegiven controller gain 119896119906 isin R+
119906 (119905) = 0 for 119905 isin [0 1)1198961199061199052119910 (119905) for 119905 ge 1 (54)
119906 (119905) = 119896119906119890minus119905119910 (119905) for 119905 ge 0 (55)
119906 (119905) = 0 for 119905 isin [0 2)
119896119906(119905 ln2119905) 119910 (119905) for 119905 ge 2 (56)
119906 (119905) = 0 for 119905 isin [0 1)
119896119906(119890radic119905radic119905) 119910 (119905) for 119905 ge 1 (57)
119906 (119905) = 1198961199061199053119890minus119905119910 (119905) for 119905 ge 0 (58)
where Popovrsquos inequality on [0 +infin] of (49)ndash(52)is intinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus119896119906 that of (57) isintinfin
0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 and that of (58) isintinfin0119910(119905)120593(119910(119905) 119905)119889119905 = minus2119896119906119890 Since the respective integrands
are monotonically decreasing then there is some 1199050 isin R+ andsome 120574 = 120574(1199050) isin R+ such that int119905
1199050119910(119905)120593(119910(119905) 119905)119889119905 ge minus120574 for
all 119905(ge 1199050) isin R+ The obtained closed-loop systems are inall cases asymptotically hyperstable The piecewise-constantcontrols with eventual discontinuities at sampling instantsgenerated from any discrete-time controller (51) of class Φ119889of Example 13 lead also to asymptotic hyperstability of thediscretized system since 119892119889 isin SSPRExample 15 Consider the continuous-time open-loop sys-tem (2) under a real sampler of sampling period 119879 and datapicking-up nonzero duration 120576 isin (0 119879) The state and outputtrajectory solutions are
forall119896 isin Z0+ for some real sequence 120585[119896] such that 120585[119896] isin(119896119879 119896119879 + 120576) where 119892119889(119911) is the discretized transfer functionunder a sampler and zero-order-hold device subject to instan-taneous ideal sampling Now assume that the incrementalcontrol along intersample periods is generated from itspreceding sampled value as (119895119879+ 120591) = 120582119895119879+120576(119895119879+ 120591)119906[119895] forany 119895 isin Z0+ and 120591 isin [0 119879] where 120582119895119879+120576(119895119879+120591) is a truncatedfunction under the usual notation one gets
The above equations imply that if 119892119889 isin SPR then the forcedoutput 119910119891[119896] = 119910[119896] minus 119888119879119890119896119860119879119909[0] forall119896 isin Z0+ generates anassociate input-output energy measure which is strictly posi-tive for 119896 isin Z0+ since 120588 isin (0 1) for real sampling of duration120576 isin (0 119879)This is a necessary condition for asymptotic hyper-stability under an appropriate class of controllers satisfyingPopovrsquos inequality However the incremental input-outputenergy measure generated by the incremental forced output120576119888119879sum119896minus1119895=0 119890(119896minus119895)119860119879(int1198790 120582119895119879+120576(120591)119890minus119860120591119887 119889120591)119906[119895] is not guaranteedto be nonnegative in general A sufficient condition for that isthat lim sup119895rarrinfin|ess sup120591isin(0120576)(120582119895119879+120576(120591))| is sufficiently smallrelated to the amount ((1 minus 120588)120576)min120596isin[0120587] Re119892119889(119890i120596)6 Conclusions
This paper has been devoted to obtain and to prove theasymptotic hyperstability property of a linear and time-invariant continuous-time system based on some given andpreviously proved auxiliary preparatory lemmas Basicallyit is proved that the continuous-time closed-loop system isasymptotically hyperstable under the assumptions that thediscretized feed-forward transfer function is strictly positivereal and that the discretized system at sampling instants ofconstant sampling period is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsIt is assumed that the continuous-time controller belongs toa class defined in the following way (a) its values at samplinginstants are defined by a class of controllers which satisfydiscrete Popovrsquos type inequality (b) the intersample control isgenerated from its values at sampling instants with the use oftwomodulating auxiliary functionswhich are subject to someextra additional conditions on smallness and boundednessSuch conditions make the overall continuous-time controllerto satisfy continuous-type Popovrsquos inequality for all time
where 119896(120591) = max(119911 isin Z+ 119911119879 le 120591) and 119896 = 119896(119905) = max(119911 isinZ+ 119911119879 le 119905) forall119905 isin R0+ which yields (13)
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
forall119896 isin Z0+ for some real sequence 120585[119896] such that 120585[119896] isin(119896119879 119896119879 + 120576) where 119892119889(119911) is the discretized transfer functionunder a sampler and zero-order-hold device subject to instan-taneous ideal sampling Now assume that the incrementalcontrol along intersample periods is generated from itspreceding sampled value as (119895119879+ 120591) = 120582119895119879+120576(119895119879+ 120591)119906[119895] forany 119895 isin Z0+ and 120591 isin [0 119879] where 120582119895119879+120576(119895119879+120591) is a truncatedfunction under the usual notation one gets
The above equations imply that if 119892119889 isin SPR then the forcedoutput 119910119891[119896] = 119910[119896] minus 119888119879119890119896119860119879119909[0] forall119896 isin Z0+ generates anassociate input-output energy measure which is strictly posi-tive for 119896 isin Z0+ since 120588 isin (0 1) for real sampling of duration120576 isin (0 119879)This is a necessary condition for asymptotic hyper-stability under an appropriate class of controllers satisfyingPopovrsquos inequality However the incremental input-outputenergy measure generated by the incremental forced output120576119888119879sum119896minus1119895=0 119890(119896minus119895)119860119879(int1198790 120582119895119879+120576(120591)119890minus119860120591119887 119889120591)119906[119895] is not guaranteedto be nonnegative in general A sufficient condition for that isthat lim sup119895rarrinfin|ess sup120591isin(0120576)(120582119895119879+120576(120591))| is sufficiently smallrelated to the amount ((1 minus 120588)120576)min120596isin[0120587] Re119892119889(119890i120596)6 Conclusions
This paper has been devoted to obtain and to prove theasymptotic hyperstability property of a linear and time-invariant continuous-time system based on some given andpreviously proved auxiliary preparatory lemmas Basicallyit is proved that the continuous-time closed-loop system isasymptotically hyperstable under the assumptions that thediscretized feed-forward transfer function is strictly positivereal and that the discretized system at sampling instants ofconstant sampling period is asymptotically hyperstable forall nonlinear and time-varying feedback controllers whichsatisfy certain Popovrsquos type inequality at sampling instantsIt is assumed that the continuous-time controller belongs toa class defined in the following way (a) its values at samplinginstants are defined by a class of controllers which satisfydiscrete Popovrsquos type inequality (b) the intersample control isgenerated from its values at sampling instants with the use oftwomodulating auxiliary functionswhich are subject to someextra additional conditions on smallness and boundednessSuch conditions make the overall continuous-time controllerto satisfy continuous-type Popovrsquos inequality for all time
where 119896(120591) = max(119911 isin Z+ 119911119879 le 120591) and 119896 = 119896(119905) = max(119911 isinZ+ 119911119879 le 119905) forall119905 isin R0+ which yields (13)
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
where 119896(120591) = max(119911 isin Z+ 119911119879 le 120591) and 119896 = 119896(119905) = max(119911 isinZ+ 119911119879 le 119905) forall119905 isin R0+ which yields (13)
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
16 Mathematical Problems in Engineering
Data Availability
Theunderlying data to support this study are included withinthe references
Conflicts of Interest
The author declares that he does not have any conflicts ofinterest
Acknowledgments
The author is very grateful to the Spanish Government andEuropean Fund of Regional Development FEDER for Grantno DPI2015-64766-R (MINECOFEDERUE) and to theUPVEHU for its support via Grant PGC 1733
References
[1] MDe la Sen ldquoOn the controllability and observability of hybridmultirate sampling systemsrdquo Computers amp Mathematics withApplications vol 31 no 1 pp 109ndash122 1996
[2] T Kaczorek Positive 1D and 2D Systems Communications andControl Engineering Edited by E D Sontag and M ThomaSpringer-Verlag Berlin Germany 2001
[3] L Farina and S Rinaldi Positive Linear Systems Theory andApplications Wiley-Interscience New York NY USA 2000
[4] R Bru C Coll and E Sanchez ldquoStructural properties ofpositive linear time-invariant difference-algebraic equationsrdquoLinear Algebra and its Applications vol 349 pp 1ndash10 2002
[5] M de la Sen ldquoOn positivity and stability of a class of time-delaysystemsrdquoNonlinear Analysis RealWorld Applications vol 8 no3 pp 749ndash768 2007
[6] M A Rami and D Napp ldquoDiscrete-time positive periodicsystems with state and control constraintsrdquo IEEE Transactionson Automatic Control vol 61 no 1 pp 1346ndash1349 2016
[7] M De la Sen ldquoAbout the positivity of a class of hybrid dynamiclinear systemsrdquoAppliedMathematics and Computation vol 189no 1 pp 852ndash868 2007
[8] A Rantzer ldquoOn the Kalman-Yakubovich-Popov lemma forpositive systemsrdquo IEEE Transactions on Automatic Control vol61 no 5 pp 1346ndash1349 2016
[9] F Najson ldquoOn the Kalman-Yakubovich-Popov lemma fordiscrete-time positive linear systems a novel simple proof andsome related resultsrdquo International Journal of Control vol 86no 10 pp 1813ndash1823 2013
[10] T Tanaka and C Langbort ldquoThe bounded real lemma forinternally positive systems andH-infinity structured static statefeedbackrdquo IEEE Transactions on Automatic Control vol 56 no9 pp 2218ndash2223 2011
[11] M De la Sen ldquoA result on the hyperstability of a class of hybriddynamic systemsrdquo IEEE Transactions on Automatic Control vol42 no 9 pp 1335ndash1339 1997
[12] MDe La Sen ldquoStability of composite systemswith an asymptot-ically hyperstable subsystemrdquo International Journal of Controlvol 44 no 6 pp 1769ndash1775 1986
[13] I D Landau Adaptive Control The Model Reference ApproachControl and Systems Theory Series Edited by J M MendelMarcel Dekker New York NY USA 1979
[14] W P Heath J Carrasco and M de la Sen ldquoSecond-ordercounterexamples to the discrete-time Kalman conjecturerdquoAutomatica vol 60 pp 140ndash144 2015
[15] V-M Popov Hyperstability of Control Systems Springer NewYork NY USA 1973
[16] M Alberdi M Amundarain A J Garrido I Garrido OCasquero and M de la Sen ldquoComplementary control ofoscillating water column-based wave energy conversion plantsto improve the instantaneous power outputrdquo IEEE Transactionson Energy Conversion vol 26 no 4 pp 1033ndash1040 2011
[17] V MMarchenko ldquoObservability of hybrid discrete-continuoussystemsrdquo Journal of Differential Equations vol 49 no 11 pp1389ndash1404 2013
[18] V M Marchenko ldquoHybrid discrete-continuous systems IStability and stabilizabilityrdquo Journal of Differential Equationsvol 48 no 12 pp 1623ndash1638 2012
[19] G Rajchakit T Rojsiraphisal and M Rajchakit ldquoRobuststability and stabilization of uncertain switched discrete-timesystemsrdquo Advances in Difference Equations vol 2012 no 1article 134 15 pages 2012
[20] B Brogliato R Lozano B Maschke and O Egeland Dissi-pative Systems Analysis and Control Theory and ApplicationsSpringer-Verlag London UK 2007
[21] M Syed Ali and J Yogambigai ldquoSynchronization of complexdynamical networks with hybrid coupling delays on timescales by handlingmultitudeKronecker product termsrdquoAppliedMathematics and Computation vol 291 pp 244ndash258 2016
[22] X Liu and P Stechlinski ldquoHybrid stabilization and synchro-nization of nonlinear systems with unbounded delaysrdquo AppliedMathematics and Computation vol 280 pp 140ndash161 2016
[23] L-G Yuan and Q-G Yang ldquoBifurcation invariant curve andhybrid control in a discrete-time predator-prey systemrdquoAppliedMathematical Modelling vol 39 no 8 pp 2345ndash2362 2015
[24] B Li ldquoPinning adaptive hybrid synchronization of two generalcomplex dynamical networks with mixed couplingrdquo AppliedMathematical Modelling vol 40 no 4 pp 2983ndash2998 2016
[25] M De la Sen ldquoSufficiency-type stability and stabilizationcriteria for linear time-invariant systems with constant pointdelaysrdquo Acta Applicandae Mathematicae vol 83 no 3 pp 235ndash256 2004
[26] MD Sen S Alonso-Quesada andA Ibeas ldquoOn the asymptotichyperstability of switched systems under integral-type feedbackregulation Popovian constraintsrdquo IMA Journal of MathematicalControl and Information vol 32 no 2 pp 359ndash386 2015
[27] P T Kabamba and S Hara ldquoWorst-case analysis and design ofsampled-data control systemsrdquo IEEE Transactions on AutomaticControl vol 38 no 9 pp 1337ndash1357 1994
[28] V M Marchenko ldquoControllability and observability of hybriddiscrete-continuous systems in the simplest function classesrdquoJournal of Differential Equations vol 51 no 11 pp 1461ndash14752015
[29] T Kailath Linear Systems Prentice-Hall Englewood Cliffs NJUSA 1980
[30] LWu and J Lam ldquoSlidingmode control of switched hybrid sys-tems with time-varying delayrdquo International Journal of AdaptiveControl and Signal Processing vol 22 no 10 pp 909ndash931 2008
[31] M de la Sen ldquoPreserving positive realness through discretiza-tionrdquo Positivity vol 6 no 1 pp 31ndash45 2002
[32] M de la Sen ldquoA method for general design of positive realfunctionsrdquo IEEE Transactions on Circuits and Systems I Fun-damental Theory and Applications vol 45 no 7 pp 764ndash7691998
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 17
[33] H J Marquez and C J Damaren ldquoOn the design of strictlypositive real transfer functionsrdquo IEEE Transactions on Circuitsand Systems I FundamentalTheory andApplications vol 42 no4 pp 214ndash218 1995
[34] J H Taylor ldquoStrictly positive-real functions and the Lefschetz-Kalman-Yakubovich (LKY) Lemmardquo IEEE Transactions onCircuits and Systems II Express Briefs vol 21 no 2 pp 310-3111974
[35] Y Sevely Systemes et Asservissements Lineaires EchantillonnesDunod Paris France 1969
[36] J M Ortega Numerical Analysis Academic Press 1972
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
![Asymptotic behavior of singularly perturbed control …€¦ · Asymptotic behavior of singularly perturbed control ... [Lions, Papanicolau, Varadhan 1986]; ... Asymptotic behavior](https://static.documents.pub/doc/80x56/5b7c19bc7f8b9a9d078b9b98/asymptotic-behavior-of-singularly-perturbed-control-asymptotic-behavior-of-singularly.jpg)
