Research ArticleDynamic Sliding Mode Control Design Based on an IntegralManifold for Nonlinear Uncertain Systems
Qudrat Khan1 Aamer Iqbal Bhatti2 and Antonella Ferrara3
1 Centre of Advanced Studies in Telecommunications (CAST) COMSATS Park Road Chak Shahzad Islamabad 44000 Pakistan2Department of Electronic Engineering MAJU Express Highway Kahuta Road Islamabad 44000 Pakistan3Department of Engineering University of Pavia Pavia Italy
Correspondence should be addressed to Qudrat Khan qudratullahqaugmailcom
Received 27 May 2013 Revised 21 October 2013 Accepted 23 October 2013 Published 2 January 2014
Academic Editor Huai-Ning Wu
Copyright copy 2014 Qudrat Khan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
An output feedback slidingmode control law design relying on an integralmanifold is proposed in this workThe considered class ofnonlinear systems is assumed to be affected by bothmatched and unmatched uncertaintiesThe use of the integral sliding manifoldallows one to subdivide the control design procedure into two steps First a linear control component is designed by pole placementand then a discontinuous control component is added so as to cope with the uncertainty presence In conventional sliding modethe control variable suffers from high frequency oscillations due to the discontinuous control component However in the presentproposal the designed control law is applied to the actual system after passing through a chain of integrators As a consequence thecontrol input actually fed into the system is continuous which is a positive feature in terms of chattering attenuation By applyingthe proposed controller the system output is regulated to zero even in the presence of the uncertainties In the paper the proposedcontrol law is theoretically analyzed and its performances are demonstrated in simulation
1 Introduction
Output feedback sliding mode control techniques provedthemselves to be the good candidate for systems where onlyoutput is measurable and its derivatives can be estimatedaccurately Linear systems or systems which could be easilylineralized are addressed in Edwards and Spurgeon [1]Nonlinear systems with measurable outputs are for instancedealt with via Dynamic Sliding Mode Control (DSMC) ([2ndash4] where the original system is replaced with a differentialinput-output form often called Fliess Controllable Canonicalform or Local generalized controllable canonical (LGCC)form by using some nonlinear transformation Asymptoticstabilization of LGCC forms by means of DSMC providedsatisfactory results Traditionally this control methodologybased on the sliding mode control (SMC) theory [5] refersto the case of uncertain systems with matched uncertainties(see [1] for a definition of this class of uncertainties) How-ever there are many systems affected by uncertainties whichdo not satisfy the matching condition To solve this prob-lem various methods have been proposed in the literature
(see eg [6ndash11]) These papers of Scararat Swaroop andFerrara relied on a backstepping based SMC design to relaxthe matching conditions
Nonlinear systems often do not remain robust againstuncertainties even of matched nature in the so-called reach-ing phase Therefore an approach capable of eliminating thisphase in the controlled systemevolutionwas proposed in [12]This approach is based on the design of an integral slidingmanifold and is called integral sliding mode control (ISMC)Levant and Alelishvili [13] synthesized higher order slidingmode technique (see eg [14ndash19]) with integral slidingmodetechnique to improve the robustness and to alleviate chat-tering Choi [20] proposed a linear matrix inequality (LMI)based sliding surface design method for integral slidingmode control of systems with unmatched norm boundeduncertainties Further Park et al [21] extendedChoirsquosmethodand proposed a dynamical output feedback variable structurecontrol law with high gain to deal with the same problemXiang et al [22] applied an iterative LMI method to avoidthe high gain related problems In this context da Silva et al
Hindawi Publishing CorporationJournal of Nonlinear DynamicsVolume 2014 Article ID 489364 10 pageshttpdxdoiorg1011552014489364
2 Journal of Nonlinear Dynamics
[23] developed an algorithm in which the existence and thereachability problems have been formulated using a polytopicdescription in order to tackle unmatched uncertainties withreduced chattering Cao and Xu [24] proposed a nonlinearintegral-type sliding surface for the system in the presenceof both matched and unmatched uncertainties The stabil-ity of the controlled system with unmatched uncertaintiesdepends on the controlled nominal system and on the natureand size of the equivalent unmatched uncertainties In theaforementioned approaches robustness is ensured but with acompromise on chattering alleviation Castanos and Fridman[25] analyzed the robust features of the integral sliding modeand used119867
infintheory to overtake the undesirable effects of the
uncertainties Rubagotti et al [26] extends thework presentedin [25] providing a control law which minimizes the effectof the uncertainties Chang [27] proposed a dynamic outputfeedback controller design according to an integral slidingmode approach for linear systems Note that in the aforemen-tioned papers it is assumed that all the states of the system areavailable since they are explicitly used to construct the controllaw
The main contribution in this work is that the uncertainnonlinear system operating under a class of states dependentmatched and unmatched uncertainties is transformed to ageneralized controllable canonical form which is analogousto that of Lu and Spurgeon [4] in some terms In additionan integral manifold based control law is developed whichestablishes sliding mode from the very beginning of theprocessThe control acting on the original system is obtainedas the output of a chain of integrators and is accordinglycontinuous thus attaining the aim of chattering attenuationThis can be a clear benefit in many applications such asthose of mechanical nature where a discontinuous controlaction could be nonappropriate Furthermore the controllerrobustness is analyzed in the presence of both matched andunmatched uncertaintiesThe claim is verified by consideringa very simple academic example Note that the outputfeedback control of nonlinear systems which can be put inLGCC form was previously faced in a preliminary versionin Khan et al [28]
The rest of the paper is organized as follows In Section 2the problem formulation is presented and in Section 3the design of the proposed control law is outlined InSection 4 the stability analysis in the presence of matchedand unmatched uncertainties is carried out A numericalexample is discussed in Section 5which is relevant to a systemwith relative degree two and is considered to be affectedby matched and unmatched uncertainties Some concludingremarks are reported in Section 6
2 Problem Formulation
Consider a nonlinear SISO dynamic system represented bythe state equation analogous to that considered in [24]
where 119909 isin 119877119899 is the state vector 119906 isin 119877 is scalar control input
119891(119909 119905) and 119892(119909 119905) are smooth vector fields 120575119898and Δ119892
119898(119909 119905)
are matched uncertainties 119891119906(119909 119905) is the unmatched uncer-
tainty vector and119910 = ℎ(119909) isin 119877 is a sufficiently smooth outputfunction The following assumption is introduced
Assumption 1 120575119898 Δ119892119898(119909 119905) and 119891
119906(119909 119905) are continuous and
bounded with continuous bounded time derivatives for all(119909 119905) isin 119877
119899
times 119877+ that is |Δ119892
119898(119909 119905)| le 120588
119898 |120575119898| le 1 minus 120576
119898
where 120576119898is some positive constant and 119891
119906(119909 119905) le 120588
119906
The problem we want to solve (Problem 1) is that ofsteering the output 119910 to zero asymptotically that is an outputregulation problem is considered in the presence of matchedand unmatched uncertainties
In order to design the proposed controller system (1) issuitably transformed To this end we denote with
The relative degree ldquo119903rdquo of the systemwith respect to the outputis the 119903th derivative of the output function in which the input119906 appears explicitly [29] One has
is the matched uncertainty term and 119865119906(119910 119905) is the
unmatched uncertainty term
Note 1 To transform the system (6) into a suitable form itis first assumed that there are no uncertainties in the systemTherefore by defining the transformation 119910
(119894minus1)
= 120585119894[3] 120585 =
(1205851 1205852 120585
119899) for the nominal system the uncertainties can
be represented in the transformed variables like 120577119894(120585 119905) =
119871119894
119891119906
ℎ(119909) for 119894 = 1 2 119899 and 119910 = 120585 Finally system (6) canbe written as follows
The representation in (8) is analogous to the so-called Localgeneralized controllable canonical (LGCC) form [2] in thesense that it differs from the basic LGCC form since it isalso affected by uncertainties With reference to system (8)the following assumption (which is an alternative form ofAssumption 1) is introduced
Assumption 2 Assume that |120593(120585 )| le 119862 |120574(120585)| le 119870119872
|Δ119866119898(119910 119905)| le 120573
1 |119865119906(120585 119905)| le 120573
2 and |120577
119894(120585 119905)| le
120583119894 119894 = 1 2 119899 minus 1 where 120573
1 1205732and 120583
119894are positive
constants Furthermore consider that 1205771(120585 119905) + 120577
2(120585 119905) +
sdot sdot sdot + 119865119906(120585 119905) cong ΔΦ(120585 119905) and |ΔΦ(120585 119905)| le 120591
Now note that the nominal system corresponding tosystem (8) is given by
Definition 3 The differential input output (I-O) form (orLGCC form) is termed as proper if [4]
(1) it is single input single output(2) 120595(120585 119906(119896)) isin 119862
1(3) the following regularity condition is satisfied
det[
[
120597120595 (120585 119906(119896)
)
120597119906(119896)]
]
= 0 (10)
Definition 4 The zero dynamics of the system in (9) aredefined as [4]
120595 (0 119906(119896)
) = 0 (11)
The system in (9) is called minimum phase if the zerodynamics defined in (11) are uniformly asymptotically stableNote that the zero dynamics in the I-O form are the dynamicsof the control and is the generalization of the definitionin Fliess [30] They are different from the zero dynamicsmentioned in Isidori [29] which are the dynamics of theuncontrollable states
Assumption 5 System (9) is proper and minimum phaseaccording to Definitions 3 and 4 respectively
Now the original control problem (Problem 1) can bereformulated with reference to system (8) under Assump-tion 2 and to the nominal system in (9) subject to Assump-tion 5 The new problem (Problem 2) is that of steeringthe state vector 120585 = [120585
1 1205852 120585
119899]119879 of system (8) to
zero asymptotically inspite of the presence of matched andunmatched uncertainties that is a state regulation problemis now considered Clearly the solution to Problem 2 impliesthe solution to Problem 1 since 120585
1= 119910 = ℎ(119909)
3 The Proposed Control Law Design
In analogy with Khan et al [31] where only the presence ofmatched uncertainties was considered we propose a controllaw of dynamic nature which can be expressed as
119906(119896)
= 119906(119896)
0+ 119906(119896)
1 (12)
4 Journal of Nonlinear Dynamics
The first part 119906(119896)0
isin 119877 is continuous and stabilizes the systemat the equilibrium point while the second part 119906(119896)
1isin 119877 is
discontinuous in nature and can be classified as an integralSMC Its role is to reject uncertainties In the next subsectionsthe design of 119906(119896)
0isin 119877 and 119906
(119896)
1isin 119877 will be discussed Starting
from the nominal case and then moving to the case in whichthe presence of matched and unmatched uncertainties arealso considered
31 The Nominal Case
311 Design of 119906(119896)0 The nominal system in (9) in alternative
In the design of 119906(119896)0 system (13) is first considered to be
independent of nonlinearities that is 120594(120585 119906(119896)) = 0 and itis also supposed to operate under 119906(119896)
0only Then system (13)
becomes1205851= 1205852
1205852= 1205853
120585119899= 119906(119896)
0
(15)
This is a linear system so it can be written as
120585 = 119860120585 + 119861119906(119896)
0 (16)
where
119860 = [0(119899minus1)times1
119868(119899minus1)(119899minus1)
01times1
01times(119899minus1)
]
119861 = [0(119899minus1)times1
1]
(17)
For the sake of simplicity the input 119906(119896)0
is designed via poleplacement that is
119906(119896)
0= minus119870119879
0120585 (18)
312 Design of 119906(119896)1 Now in order to achieve the desired
performance robust compensation of the uncertainties isneeded To this end we select the following sliding manifoldof integral type [12]
120590 (120585) = 1205900(120585) + 119911 (19)
where 1205900(120585) is a conventional sliding surface which is math-
ematically defined by 1205900(120585) = sum
119899
119894=1119888119894120585119894 with 119888
119899= 1 and 119911 is
the integral term The time derivative of (19) along (9) yields
(120585) =
119899minus1
sum
119894=1
119888119894120585119894+1
+ 120594 (120585 119906(119896)
) + 119906(119896)
0+ 119906(119896)
1+ (20)
Now choose z with the following expression
= minus(
119899minus1
sum
119894=1
119888119894120585119894+1
+ 119906(119896)
0)
119911 (0) = minus1205900(120585 (0))
(21)
Then (20) becomes
(120585) = 120594 (120585 119906(119896)
) + 119906(119896)
1
= 120593 (120585 ) + (120574 (120585) minus 1) 119906(119896)
0+ 120574 (120585) 119906
119896
1
(22)
This initial condition 119911(0) is adjusted in such a way to meetthe requirement 120590(0) = 0
Taking into account the reachability condition defined asfollows [5]
= minus119870 sign120590 (23)
and comparing (22) with (23) the expression of the discon-tinuous control component 119906(119896)
1becomes
119906(119896)
1=
1
120574 (120585)
(minus120593 (120585 ) minus (120574 (120585) minus 1) 119906(119896)
0minus 119870 sign120590) (24)
This control law enforces sliding mode along the slidingmanifold defined in (19) The constant 119870 can be selectedaccording to the subsequent stability analysis
Note that this control law can be implemented by integratingthe derivative of the control 119906(119896) ldquo119896rdquo times so that the controlinput actually applied to the system is continuous This canbe a benefit for various class of systems such as those ofmechanical type for which a discontinuous control actioncould be disruptive
Remark 6 Thecoefficients of the conventional sliding surfaceare chosen by tacking into the dynamic response of thesystem However in real applications these constants can alsobe optimized using LMIs methods
Remark 7 The proposed methodology needs the availabilityof the system output and of its derivatives for the controllerimplementation In case the output derivatives are not avail-able for measurements one can use for instance a finite timesliding mode differentiator like the one proposed in [16] toreconstruct them
Journal of Nonlinear Dynamics 5
4 Stability Analysis
In this section the proposed control law when applied tothe uncertain nonlinear system in question is theoreticallyanalyzed First the case in which only matched uncertaintiesare present will be discussed and then the more general caseof matched and unmatched uncertainties will be considered
41 The System Operating under Matched Uncertainties Nowwe assume that the system operates only under matcheduncertainties Thus system (8) with matched uncertaintiesbecomes
To show that this system is stabilized in finite time in thepresence ofmatched uncertainties the following theorem canbe stated
Theorem 8 Consider that Assumptions 2 and 5 are satisfiedThe sliding surface is chosen as 120590(120585) = 0 where 120590(120585) is definedin (19) and the control law 119906
(119896) is selected according to (25) Ifthe gain 119870 is chosen according to the following condition
10038161003816100381610038161003816+ (1 minus 120576
119898) 119862 + 119870
1198721205731+ 1205781]
(33)
as in (27) Note that (32) can also be written as
+ radic21205781
radic119881 lt 0 (34)
This implies that 120590(120585) converges to zero in a finite time 119905119904[1]
such that
119905119904le radic2120578
minus1
1
radic119881 (120590 (0)) (35)
which completes the proof
Corollary 9 Thedynamics of the system (26) with control law(25) and sliding manifold 120590(120585) = 0 with 120590(120585) defined in (19)in sliding mode are governed by the linear control law (18)
Proof The nonlinear system (26) can be written in thefollowing alternate form
1205851= 1205852
1205852= 1205853
120585119899= 120593 (120585 ) + (120574 (120585) minus 1) 119906
(119896)
+ 119906(119896)
0
+ 119906(119896)
1+ 120574 (120585) 119906
(119896)
120575119898+ Δ119866119898(120585 119905)
(36)
The time derivative of (19) along (36) yields
(120585) =
119899minus1
sum
119894=1
119888119894120585119894+1
+ 120593 (120585 ) + (120574 (120585) minus 1) 119906(119896)
+ 119906(119896)
0+ 119906(119896)
1+ 120574 (120585) 119906
(119896)
120575119898+ Δ119866119898(120585 119905) +
(37)
Substituting (21) into (37) posing (120585) = 0 and solving withrespect to the control variable 119906(119896) one obtains the so-calledequivalent control [5] as
where 120585119904is the state of system (26) while in sliding mode
Thus it is proved that the system in sliding mode operatesunder the continuous control law and the eigenvalues of thecontrolled transformed system in sliding mode are those of119860 minus 119861119870
119879
0
42 The System Operating under Both Matched andUnmatched Uncertainties In this subsection it is nowassumed that the considered system operates under bothmatched and unmatched uncertainties and the controlobjective is to regulate the output of the system in thepresence of these uncertainties To prove that the proposedcontrol law is capable of compensating for these uncertainterms the following theorem can be stated
Theorem 10 Consider that Assumptions 2 and 5 are satisfiedThe sliding surface is chosen as 120590(120585) = 0 where 120590(120585) is definedin (19) and the control law 119906
(119896) is selected according to (25) Ifthe gain 119870 is chosen according to the following condition
10038161003816100381610038161003816+ (1 minus 120576
119898) 119862 + 119870
1198721205731+1205782+120591]
(46)
The expression in (45) can be placed in the same formatlike that of (34) Note that the finite time 119905
119904in this case is
given by the formula in (35) with 1205782instead of 120578
1 Thus it
is confirmed that when the gain of the control law (25) isselected according to (40) the finite time enforcement of thesliding mode is guaranteed in the presence of matched andunmatched uncertainties which proves the theorem
Corollary 11 The dynamics of system (8) with control law(25) and an integral manifold 120590(120585) = 0 with 120590(120585) defined in(19) in slidingmode are governed by the linear control law (18)
Proof The proof can be performed by following the sameprocedure as in the proof of Corollary 9 with the onlydifference that in this case the equivalent control is equal to
119906(119896)
eq = minus1
120574 (120585) (1 + 120575119898)
times [120593 (120585 ) minus 119906(119896)
0+120574 (120585) Δ119866
119898(120585 119905)+ΔΦ (120585 119905)]
(47)
5 Illustrative Example (System with RelativeDegree 2)
Consider the following uncertain nonlinear system [4]
1= 1199092+ 1198911(119909 119905)
2= 1199092
1+ (1199092
2+ 1) 119906 (1 + 120575
119898) + Δ119892
119898(119909 119905) + 119909
3+ 1198912(119909 119905)
3= minus1199093+ 11990921199092
3+ 1198913(119909 119905)
(48)
where 1199091 1199092 and 119909
3are the states of the nonlinear system
The terms 120575119898 Δ119892119898(119909 119905) are matched uncertainties and
119891119894(119909 119905) are components of themismatched uncertainty which
satisfy Assumptions 1 and 2 and these terms contributeto the system uncertainty with the following mathematicalexpressions
1198911(119909 119905) = minus 119909
3+ 11990921199092
3+ (minus119909
3+ 11990921199092
3)2
+ 025 sin (119905) cos (31199092) + 026
1198912(119909 119905) = 025 sin (119905) cos (3119909
2) + 01
Journal of Nonlinear Dynamics 7
minus02
minus5
minus100 5 10 15 20 25 30
0
5
10
Time (s) Time (s)
Time (s)0 5 10 15 20 25 30
0
02
04
06
08O
utpu
tx1
regu
latio
n
x
u
1
Con
trol i
nput
u
0 5 10 15 20 25 30
0
5
10
0 5 10 15 20 25 30
0
5
10
15
minus5
minus5
minus10
minus15
Time (s)
120585 i
12058531205852
1205851
i=1
23
regu
latio
nSl
idin
g va
riabl
e120590
120590
conv
erge
nce
Figure 1 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched uncertainty
via the proposed control law
1198913(119909 119905) = minus 119909
3+ 11990921199092
3+ 3(minus119909
3+ 11990921199092
3)2
+ 025 sin (119905) cos (31199092) + 01
Δ119892119898(119909 119905) = 3 (minus119909
3+ 11990921199092
3)
120575119898= 03 cos (120587119905119909
2)
(49)
The output of interest is the state variable 1199091 The relative
degree of which is 2 Consequently the system in (48) canbe expressed in LGCC form as follows
120593 (120585 ) = 212058511205852+ 212058521205853119906 minus (120585
3minus 1205852
1minus (1205852
2+ 1) 119906)
+ 1205852(1205853minus 1205852
1minus (1205852
2+ 1) 119906)
(52)
The regularity condition mentioned in Definition 3 holdsand the zero dynamics of this system express according toDefinition 4 becomes
+ 119906 = 0 (53)
This confirms that the nominal system is minimum phaseThe corresponding linear system becomes
120585 = 119860120585 + 1198610 (54)
where
119860 = [
[
0 1 0
0 0 1
0 0 0
]
]
119861 = [
[
0
0
1
]
]
(55)
8 Journal of Nonlinear Dynamics
0 5 10 15 20 25 30
0
02
04
06
08
0
02
04
06
08
0 5 10 15 20 25 30
Time (s)
0 5 10 15 20 25 30
0
05
1
Time (s)
x1 120590
1205852
1205851u
Out
putx
1re
gula
tion
120585 ii
=1
2 re
gula
tion
Slid
ing
varia
ble120590
conv
erge
nce
minus02minus02
minus05
minus1
0
5
10
minus5
minus10
Con
trol i
nput
u
0 5 10 15 20 25 30
Figure 2 Output regulation control effort sliding variable convergence and [1205851 1205852]119879 regulation in the presence of matched uncertainty via
and the final expression of the control law takes the form
= 0+
1
120574 (120585)
[minus120593 (120585 ) minus (120574 (120585) minus 1) 0minus 119870 sign (120590)]
(59)
In this study we compare the results of the proposed controllaw with that of quasicontinuous high order sliding mode
controller proposed by Levant in [18] To apply such anapproach we denote
119904 = 1199091
119904 = 1199092
(60)
So that the expression of the quasicontinuous sliding modecontroller in case of relative degree (2-QCSMC) takes thefollowing form
119906 = minus
120572 ( 119904 + |119904|12 sign (119904))
| 119904| + |119904|12
(61)
where120572 is the controller gainwhich can be selected accordingto Bartolini et al [32] As proved in Levant [18] the controllaw (61) provides a finite time slidingmode of the systemwitha control law which is continuous everywhere except on thesecond order sliding manifold 119904 = 119904 = 0
Note 2 It is not necessary that every system whose output isavailable can be put in the form appearing in (8) and (9)
Case 1 (system operated with matched uncertainty) In thisstudy the system with matched uncertainties (ie with
Journal of Nonlinear Dynamics 9
x1 120590
u12058521205851
1205853
120585 1i
=1
23
regu
latio
n
0
0
5 10 15 20 25 30
02
04
06
08
0 5 10 15 20 25 30
0
2
4
6
8
0 5 10 15 20 25 30
05
101520
Time (s)0
0
5 10 15 20 25 30
10
20
Time (s)
Slid
ing
varia
ble120590
conv
erge
nce
minus2
minus4
minus20
minus10
minus15
minus10
minus5Con
trol i
nput
uO
utpu
tx1
regu
latio
n
minus20
Figure 3 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched and
unmatched uncertainty via the proposed control law
Table 1 The parameters values of the control law used in the simulations
Constants 1198961
1198962
1198963
1198881
1198882
1198883
119870 or 1205722-QCSMC mdash mdash mdash mdash mdash mdash 4Proposed control law with 119903 = 2 4902 1807 59 6 5 mdash 230
119891119894(119909 119905) = 0 for 119894 = 1 2 3) is simulated to confirm the afore-
mentioned claim of the compensation of uncertain termsThis test with matched uncertainty is also performed with2-QCSMC previously mentioned The results are reportedin Figures 1 and 2 In these Figures it can be seen that theoutput system with state vector [120585
1 1205852 1205853]119879 is regulated in the
presence of uncertainties It is noticeable that the proposedmethodology provides a satisfactory regulation of the systemoutput via a continuous control law The 2-QCSMC alsoprovides excellent performance yet with a control law whichbecomes discontinuous when the output regulation objectiveis attained Apart from that both the controllers need to use adifferentiator [16 18] to construct the derivatives of the outputvariable necessary in the control laws
Case 2 (system operated under matched and unmatcheduncertainties) In this section the test with both matchedand unmatched uncertainty is performed The results withthe proposed control law are depicted in Figure 3These sim-ulation results confirm the robust and chattering free natureof the proposed controller as well as its capability of efficientlysolving the regulation problem even in this particularly
critical case In view of the nature of the uncertainty now con-sidered we cannot compare our results with those of the 2-QCSMC algorithm since that algorithm was designed underthe assumption of having only matched uncertainty [18]
Note that the controller gains and the controllersparameters in both the experiments are listed in Table 1
6 Conclusions
In this work an output feedback dynamic sliding modecontroller is presented which is capable of dealing with a classof SISO nonlinear systems operating under both matchedand unmatched state dependent uncertaintiesThe uncertainsystem output trajectories are asymptotically regulated tozero inspite of the presence of the uncertainties while asliding mode is enforced in finite time along an integralmanifold The use of the integral sliding manifold allowsone to subdivide the control design procedure into twosteps First a linear control component is designed by poleplacement and then a discontinuous control component isadded so as to cope with the uncertainty presenceThe designprocedure is relying on a suitably transformed system which
10 Journal of Nonlinear Dynamics
generally appears in a canonical form and the control inputappears with 119896 time derivatives As a consequence the controlacting on the original system is obtained as the output ofa chain of integrators and is accordingly continuous Thiscan be a clear benefit in many applications such as thoseof mechanical nature where a discontinuous control actioncould be nonappropriate or even disruptive for the actuatorsand systemrsquos health
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[2] M Fliess ldquoGeneralized controller canonical forms for linearand nonlinear dynamicsrdquo IEEE Transactions on AutomaticControl vol 35 no 9 pp 994ndash1001 1990
[3] H S Ramirez ldquoOn the dynamical sliding mode control of non-linear systemsrdquo International Journal of Control vol 57 no 5pp 1039ndash1061 1993
[4] X-Y Lu and S K Spurgeon ldquoOutput feedback stabilization ofSISO nonlinear systems via dynamic sliding modesrdquo Interna-tional Journal of Control vol 70 no 5 pp 735ndash759 1998
[5] V I Utkin Sliding Modes in Control Optimization SpringerBerlin Germany 1992
[6] J C Scarratt A Zinober R E Mills M Rios-Bolıvar A Fer-rara and L Giacomini ldquoDynamical adaptive first and second-order sliding backstepping control of nonlinear nontriangularuncertain systemsrdquo Journal of Dynamic Systems Measurementand Control vol 122 no 4 pp 746ndash752 2000
[7] D Swaroop J K Hedrick P P Yip and J C Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo IEEE Transac-tions on Automatic Control vol 45 no 10 pp 1893ndash1899 2000
[8] A Ferrara and L Giacomini ldquoOnmodular backstepping designwith second order sliding modesrdquo Automatica vol 37 no 1 pp129ndash135 2001
[9] S Tong and H-X Li ldquoFuzzy adaptive sliding-mode control forMIMOnonlinear systemsrdquo IEEETransactions on Fuzzy Systemsvol 11 no 3 pp 354ndash360 2003
[10] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[11] S Tong C Liu andY Li ldquoFuzzy-adaptive decentralized output-feedback control for large-scale nonlinear systems with dynam-ical uncertaintiesrdquo IEEE Transactions on Fuzzy Systems vol 18no 5 pp 845ndash861 2010
[12] V I Utkin J Guldner and J Shi SlidingModeControl in Electro-mechanical Systems Taylor amp Francis London UK 1999
[13] A Levant and L Alelishvili ldquoIntegral high-order slidingmodesrdquo IEEE Transactions on Automatic Control vol 52 no 7pp 1278ndash1282 2007
[14] G Bartolini A Ferrara and E Usai ldquoOutput tracking controlof uncertain nonlinear second-order systemsrdquo Automatica vol33 no 12 pp 2203ndash2212 1997
[15] G Bartolini A Ferrara and E Usai ldquoChattering avoidanceby second-order sliding mode controlrdquo IEEE Transactions onAutomatic Control vol 43 no 2 pp 241ndash246 1998
[16] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[17] I Boiko L Fridman andM I Castellanos ldquoAnalysis of second-order sliding-mode algorithms in the frequency domainrdquo IEEETransactions on Automatic Control vol 49 no 6 pp 946ndash9502004
[18] A Levant ldquoQuasi-continuous high-order sliding mode con-trollersrdquo IEEE Transaction on Automatic Control vol 50 no 11pp 1812ndash1816 2005
[19] F Dinuzzo and A Ferrara ldquoHigher order sliding mode con-trollers with optimal reachingrdquo IEEE Transactions onAutomaticControl vol 54 no 9 pp 2126ndash2136 2009
[20] H H Choi ldquoVariable structure output feedback control designfor a class of uncertain dynamic systemsrdquo Automatica vol 38no 2 pp 335ndash341 2002
[21] P Park D J Choi and S G Kong ldquoOutput feedback variablestructure control for linear systems with uncertainties anddisturbancesrdquo Automatica vol 43 no 1 pp 72ndash79 2007
[22] J Xiang WWei and H Su ldquoAn ILMI approach to robust staticoutput feedback sliding mode controlrdquo International Journal ofControl vol 79 no 8 pp 959ndash967 2006
[23] J M A da Silva C Edwards and S K Spurgeon ldquoSliding-mode output-feedback control based on LMIs for plants withmismatched uncertaintiesrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3675ndash3683 2009
[24] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[25] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[26] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[27] J-L Chang ldquoDynamic output integral sliding-mode controlwith disturbance attenuationrdquo IEEE Transactions on AutomaticControl vol 54 no 11 pp 2653ndash2658 2009
[28] Q Khan A I Bhatti and Q Ahmed ldquoDynamic integral slidingmode control of nonlinear SISO systems with states dependentmatched and mismatched uncertaintiesrdquo in Proceedings of the18thWorld Congress of the International Federation of AutomaticControl (IFAC rsquo11) pp 3932ndash3937 Milan Italy September 2011
[29] A Isidori Nonlinear Control Systems Springer 3rd edition1995
[30] M Fliess ldquoNonlinear control theory and differential algebrardquo inModeling and Adaptive Control C Byrness and A KurzhanskiEds vol 105 of Lecture Notes in Control and Information Sci-ence pp 134ndash145 Springer New York NY USA 1988
[31] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamic integralsliding mode for MIMO uncertain nonlinear systemsrdquo Interna-tional Journal of Control Automation and Systems vol 9 no 1pp 151ndash160 2011
[32] G Bartolini A Pisano E Punta and E Usai ldquoA survey of appli-cations of second-order sliding mode control to mechanicalsystemsrdquo International Journal of Control vol 76 no 9-10 pp875ndash892 2003
[23] developed an algorithm in which the existence and thereachability problems have been formulated using a polytopicdescription in order to tackle unmatched uncertainties withreduced chattering Cao and Xu [24] proposed a nonlinearintegral-type sliding surface for the system in the presenceof both matched and unmatched uncertainties The stabil-ity of the controlled system with unmatched uncertaintiesdepends on the controlled nominal system and on the natureand size of the equivalent unmatched uncertainties In theaforementioned approaches robustness is ensured but with acompromise on chattering alleviation Castanos and Fridman[25] analyzed the robust features of the integral sliding modeand used119867
infintheory to overtake the undesirable effects of the
uncertainties Rubagotti et al [26] extends thework presentedin [25] providing a control law which minimizes the effectof the uncertainties Chang [27] proposed a dynamic outputfeedback controller design according to an integral slidingmode approach for linear systems Note that in the aforemen-tioned papers it is assumed that all the states of the system areavailable since they are explicitly used to construct the controllaw
The main contribution in this work is that the uncertainnonlinear system operating under a class of states dependentmatched and unmatched uncertainties is transformed to ageneralized controllable canonical form which is analogousto that of Lu and Spurgeon [4] in some terms In additionan integral manifold based control law is developed whichestablishes sliding mode from the very beginning of theprocessThe control acting on the original system is obtainedas the output of a chain of integrators and is accordinglycontinuous thus attaining the aim of chattering attenuationThis can be a clear benefit in many applications such asthose of mechanical nature where a discontinuous controlaction could be nonappropriate Furthermore the controllerrobustness is analyzed in the presence of both matched andunmatched uncertaintiesThe claim is verified by consideringa very simple academic example Note that the outputfeedback control of nonlinear systems which can be put inLGCC form was previously faced in a preliminary versionin Khan et al [28]
The rest of the paper is organized as follows In Section 2the problem formulation is presented and in Section 3the design of the proposed control law is outlined InSection 4 the stability analysis in the presence of matchedand unmatched uncertainties is carried out A numericalexample is discussed in Section 5which is relevant to a systemwith relative degree two and is considered to be affectedby matched and unmatched uncertainties Some concludingremarks are reported in Section 6
2 Problem Formulation
Consider a nonlinear SISO dynamic system represented bythe state equation analogous to that considered in [24]
where 119909 isin 119877119899 is the state vector 119906 isin 119877 is scalar control input
119891(119909 119905) and 119892(119909 119905) are smooth vector fields 120575119898and Δ119892
119898(119909 119905)
are matched uncertainties 119891119906(119909 119905) is the unmatched uncer-
tainty vector and119910 = ℎ(119909) isin 119877 is a sufficiently smooth outputfunction The following assumption is introduced
Assumption 1 120575119898 Δ119892119898(119909 119905) and 119891
119906(119909 119905) are continuous and
bounded with continuous bounded time derivatives for all(119909 119905) isin 119877
119899
times 119877+ that is |Δ119892
119898(119909 119905)| le 120588
119898 |120575119898| le 1 minus 120576
119898
where 120576119898is some positive constant and 119891
119906(119909 119905) le 120588
119906
The problem we want to solve (Problem 1) is that ofsteering the output 119910 to zero asymptotically that is an outputregulation problem is considered in the presence of matchedand unmatched uncertainties
In order to design the proposed controller system (1) issuitably transformed To this end we denote with
The relative degree ldquo119903rdquo of the systemwith respect to the outputis the 119903th derivative of the output function in which the input119906 appears explicitly [29] One has
is the matched uncertainty term and 119865119906(119910 119905) is the
unmatched uncertainty term
Note 1 To transform the system (6) into a suitable form itis first assumed that there are no uncertainties in the systemTherefore by defining the transformation 119910
(119894minus1)
= 120585119894[3] 120585 =
(1205851 1205852 120585
119899) for the nominal system the uncertainties can
be represented in the transformed variables like 120577119894(120585 119905) =
119871119894
119891119906
ℎ(119909) for 119894 = 1 2 119899 and 119910 = 120585 Finally system (6) canbe written as follows
The representation in (8) is analogous to the so-called Localgeneralized controllable canonical (LGCC) form [2] in thesense that it differs from the basic LGCC form since it isalso affected by uncertainties With reference to system (8)the following assumption (which is an alternative form ofAssumption 1) is introduced
Assumption 2 Assume that |120593(120585 )| le 119862 |120574(120585)| le 119870119872
|Δ119866119898(119910 119905)| le 120573
1 |119865119906(120585 119905)| le 120573
2 and |120577
119894(120585 119905)| le
120583119894 119894 = 1 2 119899 minus 1 where 120573
1 1205732and 120583
119894are positive
constants Furthermore consider that 1205771(120585 119905) + 120577
2(120585 119905) +
sdot sdot sdot + 119865119906(120585 119905) cong ΔΦ(120585 119905) and |ΔΦ(120585 119905)| le 120591
Now note that the nominal system corresponding tosystem (8) is given by
Definition 3 The differential input output (I-O) form (orLGCC form) is termed as proper if [4]
(1) it is single input single output(2) 120595(120585 119906(119896)) isin 119862
1(3) the following regularity condition is satisfied
det[
[
120597120595 (120585 119906(119896)
)
120597119906(119896)]
]
= 0 (10)
Definition 4 The zero dynamics of the system in (9) aredefined as [4]
120595 (0 119906(119896)
) = 0 (11)
The system in (9) is called minimum phase if the zerodynamics defined in (11) are uniformly asymptotically stableNote that the zero dynamics in the I-O form are the dynamicsof the control and is the generalization of the definitionin Fliess [30] They are different from the zero dynamicsmentioned in Isidori [29] which are the dynamics of theuncontrollable states
Assumption 5 System (9) is proper and minimum phaseaccording to Definitions 3 and 4 respectively
Now the original control problem (Problem 1) can bereformulated with reference to system (8) under Assump-tion 2 and to the nominal system in (9) subject to Assump-tion 5 The new problem (Problem 2) is that of steeringthe state vector 120585 = [120585
1 1205852 120585
119899]119879 of system (8) to
zero asymptotically inspite of the presence of matched andunmatched uncertainties that is a state regulation problemis now considered Clearly the solution to Problem 2 impliesthe solution to Problem 1 since 120585
1= 119910 = ℎ(119909)
3 The Proposed Control Law Design
In analogy with Khan et al [31] where only the presence ofmatched uncertainties was considered we propose a controllaw of dynamic nature which can be expressed as
119906(119896)
= 119906(119896)
0+ 119906(119896)
1 (12)
4 Journal of Nonlinear Dynamics
The first part 119906(119896)0
isin 119877 is continuous and stabilizes the systemat the equilibrium point while the second part 119906(119896)
1isin 119877 is
discontinuous in nature and can be classified as an integralSMC Its role is to reject uncertainties In the next subsectionsthe design of 119906(119896)
0isin 119877 and 119906
(119896)
1isin 119877 will be discussed Starting
from the nominal case and then moving to the case in whichthe presence of matched and unmatched uncertainties arealso considered
31 The Nominal Case
311 Design of 119906(119896)0 The nominal system in (9) in alternative
In the design of 119906(119896)0 system (13) is first considered to be
independent of nonlinearities that is 120594(120585 119906(119896)) = 0 and itis also supposed to operate under 119906(119896)
0only Then system (13)
becomes1205851= 1205852
1205852= 1205853
120585119899= 119906(119896)
0
(15)
This is a linear system so it can be written as
120585 = 119860120585 + 119861119906(119896)
0 (16)
where
119860 = [0(119899minus1)times1
119868(119899minus1)(119899minus1)
01times1
01times(119899minus1)
]
119861 = [0(119899minus1)times1
1]
(17)
For the sake of simplicity the input 119906(119896)0
is designed via poleplacement that is
119906(119896)
0= minus119870119879
0120585 (18)
312 Design of 119906(119896)1 Now in order to achieve the desired
performance robust compensation of the uncertainties isneeded To this end we select the following sliding manifoldof integral type [12]
120590 (120585) = 1205900(120585) + 119911 (19)
where 1205900(120585) is a conventional sliding surface which is math-
ematically defined by 1205900(120585) = sum
119899
119894=1119888119894120585119894 with 119888
119899= 1 and 119911 is
the integral term The time derivative of (19) along (9) yields
(120585) =
119899minus1
sum
119894=1
119888119894120585119894+1
+ 120594 (120585 119906(119896)
) + 119906(119896)
0+ 119906(119896)
1+ (20)
Now choose z with the following expression
= minus(
119899minus1
sum
119894=1
119888119894120585119894+1
+ 119906(119896)
0)
119911 (0) = minus1205900(120585 (0))
(21)
Then (20) becomes
(120585) = 120594 (120585 119906(119896)
) + 119906(119896)
1
= 120593 (120585 ) + (120574 (120585) minus 1) 119906(119896)
0+ 120574 (120585) 119906
119896
1
(22)
This initial condition 119911(0) is adjusted in such a way to meetthe requirement 120590(0) = 0
Taking into account the reachability condition defined asfollows [5]
= minus119870 sign120590 (23)
and comparing (22) with (23) the expression of the discon-tinuous control component 119906(119896)
1becomes
119906(119896)
1=
1
120574 (120585)
(minus120593 (120585 ) minus (120574 (120585) minus 1) 119906(119896)
0minus 119870 sign120590) (24)
This control law enforces sliding mode along the slidingmanifold defined in (19) The constant 119870 can be selectedaccording to the subsequent stability analysis
Note that this control law can be implemented by integratingthe derivative of the control 119906(119896) ldquo119896rdquo times so that the controlinput actually applied to the system is continuous This canbe a benefit for various class of systems such as those ofmechanical type for which a discontinuous control actioncould be disruptive
Remark 6 Thecoefficients of the conventional sliding surfaceare chosen by tacking into the dynamic response of thesystem However in real applications these constants can alsobe optimized using LMIs methods
Remark 7 The proposed methodology needs the availabilityof the system output and of its derivatives for the controllerimplementation In case the output derivatives are not avail-able for measurements one can use for instance a finite timesliding mode differentiator like the one proposed in [16] toreconstruct them
Journal of Nonlinear Dynamics 5
4 Stability Analysis
In this section the proposed control law when applied tothe uncertain nonlinear system in question is theoreticallyanalyzed First the case in which only matched uncertaintiesare present will be discussed and then the more general caseof matched and unmatched uncertainties will be considered
41 The System Operating under Matched Uncertainties Nowwe assume that the system operates only under matcheduncertainties Thus system (8) with matched uncertaintiesbecomes
To show that this system is stabilized in finite time in thepresence ofmatched uncertainties the following theorem canbe stated
Theorem 8 Consider that Assumptions 2 and 5 are satisfiedThe sliding surface is chosen as 120590(120585) = 0 where 120590(120585) is definedin (19) and the control law 119906
(119896) is selected according to (25) Ifthe gain 119870 is chosen according to the following condition
10038161003816100381610038161003816+ (1 minus 120576
119898) 119862 + 119870
1198721205731+ 1205781]
(33)
as in (27) Note that (32) can also be written as
+ radic21205781
radic119881 lt 0 (34)
This implies that 120590(120585) converges to zero in a finite time 119905119904[1]
such that
119905119904le radic2120578
minus1
1
radic119881 (120590 (0)) (35)
which completes the proof
Corollary 9 Thedynamics of the system (26) with control law(25) and sliding manifold 120590(120585) = 0 with 120590(120585) defined in (19)in sliding mode are governed by the linear control law (18)
Proof The nonlinear system (26) can be written in thefollowing alternate form
1205851= 1205852
1205852= 1205853
120585119899= 120593 (120585 ) + (120574 (120585) minus 1) 119906
(119896)
+ 119906(119896)
0
+ 119906(119896)
1+ 120574 (120585) 119906
(119896)
120575119898+ Δ119866119898(120585 119905)
(36)
The time derivative of (19) along (36) yields
(120585) =
119899minus1
sum
119894=1
119888119894120585119894+1
+ 120593 (120585 ) + (120574 (120585) minus 1) 119906(119896)
+ 119906(119896)
0+ 119906(119896)
1+ 120574 (120585) 119906
(119896)
120575119898+ Δ119866119898(120585 119905) +
(37)
Substituting (21) into (37) posing (120585) = 0 and solving withrespect to the control variable 119906(119896) one obtains the so-calledequivalent control [5] as
where 120585119904is the state of system (26) while in sliding mode
Thus it is proved that the system in sliding mode operatesunder the continuous control law and the eigenvalues of thecontrolled transformed system in sliding mode are those of119860 minus 119861119870
119879
0
42 The System Operating under Both Matched andUnmatched Uncertainties In this subsection it is nowassumed that the considered system operates under bothmatched and unmatched uncertainties and the controlobjective is to regulate the output of the system in thepresence of these uncertainties To prove that the proposedcontrol law is capable of compensating for these uncertainterms the following theorem can be stated
Theorem 10 Consider that Assumptions 2 and 5 are satisfiedThe sliding surface is chosen as 120590(120585) = 0 where 120590(120585) is definedin (19) and the control law 119906
(119896) is selected according to (25) Ifthe gain 119870 is chosen according to the following condition
10038161003816100381610038161003816+ (1 minus 120576
119898) 119862 + 119870
1198721205731+1205782+120591]
(46)
The expression in (45) can be placed in the same formatlike that of (34) Note that the finite time 119905
119904in this case is
given by the formula in (35) with 1205782instead of 120578
1 Thus it
is confirmed that when the gain of the control law (25) isselected according to (40) the finite time enforcement of thesliding mode is guaranteed in the presence of matched andunmatched uncertainties which proves the theorem
Corollary 11 The dynamics of system (8) with control law(25) and an integral manifold 120590(120585) = 0 with 120590(120585) defined in(19) in slidingmode are governed by the linear control law (18)
Proof The proof can be performed by following the sameprocedure as in the proof of Corollary 9 with the onlydifference that in this case the equivalent control is equal to
119906(119896)
eq = minus1
120574 (120585) (1 + 120575119898)
times [120593 (120585 ) minus 119906(119896)
0+120574 (120585) Δ119866
119898(120585 119905)+ΔΦ (120585 119905)]
(47)
5 Illustrative Example (System with RelativeDegree 2)
Consider the following uncertain nonlinear system [4]
1= 1199092+ 1198911(119909 119905)
2= 1199092
1+ (1199092
2+ 1) 119906 (1 + 120575
119898) + Δ119892
119898(119909 119905) + 119909
3+ 1198912(119909 119905)
3= minus1199093+ 11990921199092
3+ 1198913(119909 119905)
(48)
where 1199091 1199092 and 119909
3are the states of the nonlinear system
The terms 120575119898 Δ119892119898(119909 119905) are matched uncertainties and
119891119894(119909 119905) are components of themismatched uncertainty which
satisfy Assumptions 1 and 2 and these terms contributeto the system uncertainty with the following mathematicalexpressions
1198911(119909 119905) = minus 119909
3+ 11990921199092
3+ (minus119909
3+ 11990921199092
3)2
+ 025 sin (119905) cos (31199092) + 026
1198912(119909 119905) = 025 sin (119905) cos (3119909
2) + 01
Journal of Nonlinear Dynamics 7
minus02
minus5
minus100 5 10 15 20 25 30
0
5
10
Time (s) Time (s)
Time (s)0 5 10 15 20 25 30
0
02
04
06
08O
utpu
tx1
regu
latio
n
x
u
1
Con
trol i
nput
u
0 5 10 15 20 25 30
0
5
10
0 5 10 15 20 25 30
0
5
10
15
minus5
minus5
minus10
minus15
Time (s)
120585 i
12058531205852
1205851
i=1
23
regu
latio
nSl
idin
g va
riabl
e120590
120590
conv
erge
nce
Figure 1 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched uncertainty
via the proposed control law
1198913(119909 119905) = minus 119909
3+ 11990921199092
3+ 3(minus119909
3+ 11990921199092
3)2
+ 025 sin (119905) cos (31199092) + 01
Δ119892119898(119909 119905) = 3 (minus119909
3+ 11990921199092
3)
120575119898= 03 cos (120587119905119909
2)
(49)
The output of interest is the state variable 1199091 The relative
degree of which is 2 Consequently the system in (48) canbe expressed in LGCC form as follows
120593 (120585 ) = 212058511205852+ 212058521205853119906 minus (120585
3minus 1205852
1minus (1205852
2+ 1) 119906)
+ 1205852(1205853minus 1205852
1minus (1205852
2+ 1) 119906)
(52)
The regularity condition mentioned in Definition 3 holdsand the zero dynamics of this system express according toDefinition 4 becomes
+ 119906 = 0 (53)
This confirms that the nominal system is minimum phaseThe corresponding linear system becomes
120585 = 119860120585 + 1198610 (54)
where
119860 = [
[
0 1 0
0 0 1
0 0 0
]
]
119861 = [
[
0
0
1
]
]
(55)
8 Journal of Nonlinear Dynamics
0 5 10 15 20 25 30
0
02
04
06
08
0
02
04
06
08
0 5 10 15 20 25 30
Time (s)
0 5 10 15 20 25 30
0
05
1
Time (s)
x1 120590
1205852
1205851u
Out
putx
1re
gula
tion
120585 ii
=1
2 re
gula
tion
Slid
ing
varia
ble120590
conv
erge
nce
minus02minus02
minus05
minus1
0
5
10
minus5
minus10
Con
trol i
nput
u
0 5 10 15 20 25 30
Figure 2 Output regulation control effort sliding variable convergence and [1205851 1205852]119879 regulation in the presence of matched uncertainty via
and the final expression of the control law takes the form
= 0+
1
120574 (120585)
[minus120593 (120585 ) minus (120574 (120585) minus 1) 0minus 119870 sign (120590)]
(59)
In this study we compare the results of the proposed controllaw with that of quasicontinuous high order sliding mode
controller proposed by Levant in [18] To apply such anapproach we denote
119904 = 1199091
119904 = 1199092
(60)
So that the expression of the quasicontinuous sliding modecontroller in case of relative degree (2-QCSMC) takes thefollowing form
119906 = minus
120572 ( 119904 + |119904|12 sign (119904))
| 119904| + |119904|12
(61)
where120572 is the controller gainwhich can be selected accordingto Bartolini et al [32] As proved in Levant [18] the controllaw (61) provides a finite time slidingmode of the systemwitha control law which is continuous everywhere except on thesecond order sliding manifold 119904 = 119904 = 0
Note 2 It is not necessary that every system whose output isavailable can be put in the form appearing in (8) and (9)
Case 1 (system operated with matched uncertainty) In thisstudy the system with matched uncertainties (ie with
Journal of Nonlinear Dynamics 9
x1 120590
u12058521205851
1205853
120585 1i
=1
23
regu
latio
n
0
0
5 10 15 20 25 30
02
04
06
08
0 5 10 15 20 25 30
0
2
4
6
8
0 5 10 15 20 25 30
05
101520
Time (s)0
0
5 10 15 20 25 30
10
20
Time (s)
Slid
ing
varia
ble120590
conv
erge
nce
minus2
minus4
minus20
minus10
minus15
minus10
minus5Con
trol i
nput
uO
utpu
tx1
regu
latio
n
minus20
Figure 3 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched and
unmatched uncertainty via the proposed control law
Table 1 The parameters values of the control law used in the simulations
Constants 1198961
1198962
1198963
1198881
1198882
1198883
119870 or 1205722-QCSMC mdash mdash mdash mdash mdash mdash 4Proposed control law with 119903 = 2 4902 1807 59 6 5 mdash 230
119891119894(119909 119905) = 0 for 119894 = 1 2 3) is simulated to confirm the afore-
mentioned claim of the compensation of uncertain termsThis test with matched uncertainty is also performed with2-QCSMC previously mentioned The results are reportedin Figures 1 and 2 In these Figures it can be seen that theoutput system with state vector [120585
1 1205852 1205853]119879 is regulated in the
presence of uncertainties It is noticeable that the proposedmethodology provides a satisfactory regulation of the systemoutput via a continuous control law The 2-QCSMC alsoprovides excellent performance yet with a control law whichbecomes discontinuous when the output regulation objectiveis attained Apart from that both the controllers need to use adifferentiator [16 18] to construct the derivatives of the outputvariable necessary in the control laws
Case 2 (system operated under matched and unmatcheduncertainties) In this section the test with both matchedand unmatched uncertainty is performed The results withthe proposed control law are depicted in Figure 3These sim-ulation results confirm the robust and chattering free natureof the proposed controller as well as its capability of efficientlysolving the regulation problem even in this particularly
critical case In view of the nature of the uncertainty now con-sidered we cannot compare our results with those of the 2-QCSMC algorithm since that algorithm was designed underthe assumption of having only matched uncertainty [18]
Note that the controller gains and the controllersparameters in both the experiments are listed in Table 1
6 Conclusions
In this work an output feedback dynamic sliding modecontroller is presented which is capable of dealing with a classof SISO nonlinear systems operating under both matchedand unmatched state dependent uncertaintiesThe uncertainsystem output trajectories are asymptotically regulated tozero inspite of the presence of the uncertainties while asliding mode is enforced in finite time along an integralmanifold The use of the integral sliding manifold allowsone to subdivide the control design procedure into twosteps First a linear control component is designed by poleplacement and then a discontinuous control component isadded so as to cope with the uncertainty presenceThe designprocedure is relying on a suitably transformed system which
10 Journal of Nonlinear Dynamics
generally appears in a canonical form and the control inputappears with 119896 time derivatives As a consequence the controlacting on the original system is obtained as the output ofa chain of integrators and is accordingly continuous Thiscan be a clear benefit in many applications such as thoseof mechanical nature where a discontinuous control actioncould be nonappropriate or even disruptive for the actuatorsand systemrsquos health
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[2] M Fliess ldquoGeneralized controller canonical forms for linearand nonlinear dynamicsrdquo IEEE Transactions on AutomaticControl vol 35 no 9 pp 994ndash1001 1990
[3] H S Ramirez ldquoOn the dynamical sliding mode control of non-linear systemsrdquo International Journal of Control vol 57 no 5pp 1039ndash1061 1993
[4] X-Y Lu and S K Spurgeon ldquoOutput feedback stabilization ofSISO nonlinear systems via dynamic sliding modesrdquo Interna-tional Journal of Control vol 70 no 5 pp 735ndash759 1998
[5] V I Utkin Sliding Modes in Control Optimization SpringerBerlin Germany 1992
[6] J C Scarratt A Zinober R E Mills M Rios-Bolıvar A Fer-rara and L Giacomini ldquoDynamical adaptive first and second-order sliding backstepping control of nonlinear nontriangularuncertain systemsrdquo Journal of Dynamic Systems Measurementand Control vol 122 no 4 pp 746ndash752 2000
[7] D Swaroop J K Hedrick P P Yip and J C Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo IEEE Transac-tions on Automatic Control vol 45 no 10 pp 1893ndash1899 2000
[8] A Ferrara and L Giacomini ldquoOnmodular backstepping designwith second order sliding modesrdquo Automatica vol 37 no 1 pp129ndash135 2001
[9] S Tong and H-X Li ldquoFuzzy adaptive sliding-mode control forMIMOnonlinear systemsrdquo IEEETransactions on Fuzzy Systemsvol 11 no 3 pp 354ndash360 2003
[10] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[11] S Tong C Liu andY Li ldquoFuzzy-adaptive decentralized output-feedback control for large-scale nonlinear systems with dynam-ical uncertaintiesrdquo IEEE Transactions on Fuzzy Systems vol 18no 5 pp 845ndash861 2010
[12] V I Utkin J Guldner and J Shi SlidingModeControl in Electro-mechanical Systems Taylor amp Francis London UK 1999
[13] A Levant and L Alelishvili ldquoIntegral high-order slidingmodesrdquo IEEE Transactions on Automatic Control vol 52 no 7pp 1278ndash1282 2007
[14] G Bartolini A Ferrara and E Usai ldquoOutput tracking controlof uncertain nonlinear second-order systemsrdquo Automatica vol33 no 12 pp 2203ndash2212 1997
[15] G Bartolini A Ferrara and E Usai ldquoChattering avoidanceby second-order sliding mode controlrdquo IEEE Transactions onAutomatic Control vol 43 no 2 pp 241ndash246 1998
[16] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[17] I Boiko L Fridman andM I Castellanos ldquoAnalysis of second-order sliding-mode algorithms in the frequency domainrdquo IEEETransactions on Automatic Control vol 49 no 6 pp 946ndash9502004
[18] A Levant ldquoQuasi-continuous high-order sliding mode con-trollersrdquo IEEE Transaction on Automatic Control vol 50 no 11pp 1812ndash1816 2005
[19] F Dinuzzo and A Ferrara ldquoHigher order sliding mode con-trollers with optimal reachingrdquo IEEE Transactions onAutomaticControl vol 54 no 9 pp 2126ndash2136 2009
[20] H H Choi ldquoVariable structure output feedback control designfor a class of uncertain dynamic systemsrdquo Automatica vol 38no 2 pp 335ndash341 2002
[21] P Park D J Choi and S G Kong ldquoOutput feedback variablestructure control for linear systems with uncertainties anddisturbancesrdquo Automatica vol 43 no 1 pp 72ndash79 2007
[22] J Xiang WWei and H Su ldquoAn ILMI approach to robust staticoutput feedback sliding mode controlrdquo International Journal ofControl vol 79 no 8 pp 959ndash967 2006
[23] J M A da Silva C Edwards and S K Spurgeon ldquoSliding-mode output-feedback control based on LMIs for plants withmismatched uncertaintiesrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3675ndash3683 2009
[24] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[25] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[26] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[27] J-L Chang ldquoDynamic output integral sliding-mode controlwith disturbance attenuationrdquo IEEE Transactions on AutomaticControl vol 54 no 11 pp 2653ndash2658 2009
[28] Q Khan A I Bhatti and Q Ahmed ldquoDynamic integral slidingmode control of nonlinear SISO systems with states dependentmatched and mismatched uncertaintiesrdquo in Proceedings of the18thWorld Congress of the International Federation of AutomaticControl (IFAC rsquo11) pp 3932ndash3937 Milan Italy September 2011
[29] A Isidori Nonlinear Control Systems Springer 3rd edition1995
[30] M Fliess ldquoNonlinear control theory and differential algebrardquo inModeling and Adaptive Control C Byrness and A KurzhanskiEds vol 105 of Lecture Notes in Control and Information Sci-ence pp 134ndash145 Springer New York NY USA 1988
[31] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamic integralsliding mode for MIMO uncertain nonlinear systemsrdquo Interna-tional Journal of Control Automation and Systems vol 9 no 1pp 151ndash160 2011
[32] G Bartolini A Pisano E Punta and E Usai ldquoA survey of appli-cations of second-order sliding mode control to mechanicalsystemsrdquo International Journal of Control vol 76 no 9-10 pp875ndash892 2003
is the matched uncertainty term and 119865119906(119910 119905) is the
unmatched uncertainty term
Note 1 To transform the system (6) into a suitable form itis first assumed that there are no uncertainties in the systemTherefore by defining the transformation 119910
(119894minus1)
= 120585119894[3] 120585 =
(1205851 1205852 120585
119899) for the nominal system the uncertainties can
be represented in the transformed variables like 120577119894(120585 119905) =
119871119894
119891119906
ℎ(119909) for 119894 = 1 2 119899 and 119910 = 120585 Finally system (6) canbe written as follows
The representation in (8) is analogous to the so-called Localgeneralized controllable canonical (LGCC) form [2] in thesense that it differs from the basic LGCC form since it isalso affected by uncertainties With reference to system (8)the following assumption (which is an alternative form ofAssumption 1) is introduced
Assumption 2 Assume that |120593(120585 )| le 119862 |120574(120585)| le 119870119872
|Δ119866119898(119910 119905)| le 120573
1 |119865119906(120585 119905)| le 120573
2 and |120577
119894(120585 119905)| le
120583119894 119894 = 1 2 119899 minus 1 where 120573
1 1205732and 120583
119894are positive
constants Furthermore consider that 1205771(120585 119905) + 120577
2(120585 119905) +
sdot sdot sdot + 119865119906(120585 119905) cong ΔΦ(120585 119905) and |ΔΦ(120585 119905)| le 120591
Now note that the nominal system corresponding tosystem (8) is given by
Definition 3 The differential input output (I-O) form (orLGCC form) is termed as proper if [4]
(1) it is single input single output(2) 120595(120585 119906(119896)) isin 119862
1(3) the following regularity condition is satisfied
det[
[
120597120595 (120585 119906(119896)
)
120597119906(119896)]
]
= 0 (10)
Definition 4 The zero dynamics of the system in (9) aredefined as [4]
120595 (0 119906(119896)
) = 0 (11)
The system in (9) is called minimum phase if the zerodynamics defined in (11) are uniformly asymptotically stableNote that the zero dynamics in the I-O form are the dynamicsof the control and is the generalization of the definitionin Fliess [30] They are different from the zero dynamicsmentioned in Isidori [29] which are the dynamics of theuncontrollable states
Assumption 5 System (9) is proper and minimum phaseaccording to Definitions 3 and 4 respectively
Now the original control problem (Problem 1) can bereformulated with reference to system (8) under Assump-tion 2 and to the nominal system in (9) subject to Assump-tion 5 The new problem (Problem 2) is that of steeringthe state vector 120585 = [120585
1 1205852 120585
119899]119879 of system (8) to
zero asymptotically inspite of the presence of matched andunmatched uncertainties that is a state regulation problemis now considered Clearly the solution to Problem 2 impliesthe solution to Problem 1 since 120585
1= 119910 = ℎ(119909)
3 The Proposed Control Law Design
In analogy with Khan et al [31] where only the presence ofmatched uncertainties was considered we propose a controllaw of dynamic nature which can be expressed as
119906(119896)
= 119906(119896)
0+ 119906(119896)
1 (12)
4 Journal of Nonlinear Dynamics
The first part 119906(119896)0
isin 119877 is continuous and stabilizes the systemat the equilibrium point while the second part 119906(119896)
1isin 119877 is
discontinuous in nature and can be classified as an integralSMC Its role is to reject uncertainties In the next subsectionsthe design of 119906(119896)
0isin 119877 and 119906
(119896)
1isin 119877 will be discussed Starting
from the nominal case and then moving to the case in whichthe presence of matched and unmatched uncertainties arealso considered
31 The Nominal Case
311 Design of 119906(119896)0 The nominal system in (9) in alternative
In the design of 119906(119896)0 system (13) is first considered to be
independent of nonlinearities that is 120594(120585 119906(119896)) = 0 and itis also supposed to operate under 119906(119896)
0only Then system (13)
becomes1205851= 1205852
1205852= 1205853
120585119899= 119906(119896)
0
(15)
This is a linear system so it can be written as
120585 = 119860120585 + 119861119906(119896)
0 (16)
where
119860 = [0(119899minus1)times1
119868(119899minus1)(119899minus1)
01times1
01times(119899minus1)
]
119861 = [0(119899minus1)times1
1]
(17)
For the sake of simplicity the input 119906(119896)0
is designed via poleplacement that is
119906(119896)
0= minus119870119879
0120585 (18)
312 Design of 119906(119896)1 Now in order to achieve the desired
performance robust compensation of the uncertainties isneeded To this end we select the following sliding manifoldof integral type [12]
120590 (120585) = 1205900(120585) + 119911 (19)
where 1205900(120585) is a conventional sliding surface which is math-
ematically defined by 1205900(120585) = sum
119899
119894=1119888119894120585119894 with 119888
119899= 1 and 119911 is
the integral term The time derivative of (19) along (9) yields
(120585) =
119899minus1
sum
119894=1
119888119894120585119894+1
+ 120594 (120585 119906(119896)
) + 119906(119896)
0+ 119906(119896)
1+ (20)
Now choose z with the following expression
= minus(
119899minus1
sum
119894=1
119888119894120585119894+1
+ 119906(119896)
0)
119911 (0) = minus1205900(120585 (0))
(21)
Then (20) becomes
(120585) = 120594 (120585 119906(119896)
) + 119906(119896)
1
= 120593 (120585 ) + (120574 (120585) minus 1) 119906(119896)
0+ 120574 (120585) 119906
119896
1
(22)
This initial condition 119911(0) is adjusted in such a way to meetthe requirement 120590(0) = 0
Taking into account the reachability condition defined asfollows [5]
= minus119870 sign120590 (23)
and comparing (22) with (23) the expression of the discon-tinuous control component 119906(119896)
1becomes
119906(119896)
1=
1
120574 (120585)
(minus120593 (120585 ) minus (120574 (120585) minus 1) 119906(119896)
0minus 119870 sign120590) (24)
This control law enforces sliding mode along the slidingmanifold defined in (19) The constant 119870 can be selectedaccording to the subsequent stability analysis
Note that this control law can be implemented by integratingthe derivative of the control 119906(119896) ldquo119896rdquo times so that the controlinput actually applied to the system is continuous This canbe a benefit for various class of systems such as those ofmechanical type for which a discontinuous control actioncould be disruptive
Remark 6 Thecoefficients of the conventional sliding surfaceare chosen by tacking into the dynamic response of thesystem However in real applications these constants can alsobe optimized using LMIs methods
Remark 7 The proposed methodology needs the availabilityof the system output and of its derivatives for the controllerimplementation In case the output derivatives are not avail-able for measurements one can use for instance a finite timesliding mode differentiator like the one proposed in [16] toreconstruct them
Journal of Nonlinear Dynamics 5
4 Stability Analysis
In this section the proposed control law when applied tothe uncertain nonlinear system in question is theoreticallyanalyzed First the case in which only matched uncertaintiesare present will be discussed and then the more general caseof matched and unmatched uncertainties will be considered
41 The System Operating under Matched Uncertainties Nowwe assume that the system operates only under matcheduncertainties Thus system (8) with matched uncertaintiesbecomes
To show that this system is stabilized in finite time in thepresence ofmatched uncertainties the following theorem canbe stated
Theorem 8 Consider that Assumptions 2 and 5 are satisfiedThe sliding surface is chosen as 120590(120585) = 0 where 120590(120585) is definedin (19) and the control law 119906
(119896) is selected according to (25) Ifthe gain 119870 is chosen according to the following condition
10038161003816100381610038161003816+ (1 minus 120576
119898) 119862 + 119870
1198721205731+ 1205781]
(33)
as in (27) Note that (32) can also be written as
+ radic21205781
radic119881 lt 0 (34)
This implies that 120590(120585) converges to zero in a finite time 119905119904[1]
such that
119905119904le radic2120578
minus1
1
radic119881 (120590 (0)) (35)
which completes the proof
Corollary 9 Thedynamics of the system (26) with control law(25) and sliding manifold 120590(120585) = 0 with 120590(120585) defined in (19)in sliding mode are governed by the linear control law (18)
Proof The nonlinear system (26) can be written in thefollowing alternate form
1205851= 1205852
1205852= 1205853
120585119899= 120593 (120585 ) + (120574 (120585) minus 1) 119906
(119896)
+ 119906(119896)
0
+ 119906(119896)
1+ 120574 (120585) 119906
(119896)
120575119898+ Δ119866119898(120585 119905)
(36)
The time derivative of (19) along (36) yields
(120585) =
119899minus1
sum
119894=1
119888119894120585119894+1
+ 120593 (120585 ) + (120574 (120585) minus 1) 119906(119896)
+ 119906(119896)
0+ 119906(119896)
1+ 120574 (120585) 119906
(119896)
120575119898+ Δ119866119898(120585 119905) +
(37)
Substituting (21) into (37) posing (120585) = 0 and solving withrespect to the control variable 119906(119896) one obtains the so-calledequivalent control [5] as
where 120585119904is the state of system (26) while in sliding mode
Thus it is proved that the system in sliding mode operatesunder the continuous control law and the eigenvalues of thecontrolled transformed system in sliding mode are those of119860 minus 119861119870
119879
0
42 The System Operating under Both Matched andUnmatched Uncertainties In this subsection it is nowassumed that the considered system operates under bothmatched and unmatched uncertainties and the controlobjective is to regulate the output of the system in thepresence of these uncertainties To prove that the proposedcontrol law is capable of compensating for these uncertainterms the following theorem can be stated
Theorem 10 Consider that Assumptions 2 and 5 are satisfiedThe sliding surface is chosen as 120590(120585) = 0 where 120590(120585) is definedin (19) and the control law 119906
(119896) is selected according to (25) Ifthe gain 119870 is chosen according to the following condition
10038161003816100381610038161003816+ (1 minus 120576
119898) 119862 + 119870
1198721205731+1205782+120591]
(46)
The expression in (45) can be placed in the same formatlike that of (34) Note that the finite time 119905
119904in this case is
given by the formula in (35) with 1205782instead of 120578
1 Thus it
is confirmed that when the gain of the control law (25) isselected according to (40) the finite time enforcement of thesliding mode is guaranteed in the presence of matched andunmatched uncertainties which proves the theorem
Corollary 11 The dynamics of system (8) with control law(25) and an integral manifold 120590(120585) = 0 with 120590(120585) defined in(19) in slidingmode are governed by the linear control law (18)
Proof The proof can be performed by following the sameprocedure as in the proof of Corollary 9 with the onlydifference that in this case the equivalent control is equal to
119906(119896)
eq = minus1
120574 (120585) (1 + 120575119898)
times [120593 (120585 ) minus 119906(119896)
0+120574 (120585) Δ119866
119898(120585 119905)+ΔΦ (120585 119905)]
(47)
5 Illustrative Example (System with RelativeDegree 2)
Consider the following uncertain nonlinear system [4]
1= 1199092+ 1198911(119909 119905)
2= 1199092
1+ (1199092
2+ 1) 119906 (1 + 120575
119898) + Δ119892
119898(119909 119905) + 119909
3+ 1198912(119909 119905)
3= minus1199093+ 11990921199092
3+ 1198913(119909 119905)
(48)
where 1199091 1199092 and 119909
3are the states of the nonlinear system
The terms 120575119898 Δ119892119898(119909 119905) are matched uncertainties and
119891119894(119909 119905) are components of themismatched uncertainty which
satisfy Assumptions 1 and 2 and these terms contributeto the system uncertainty with the following mathematicalexpressions
1198911(119909 119905) = minus 119909
3+ 11990921199092
3+ (minus119909
3+ 11990921199092
3)2
+ 025 sin (119905) cos (31199092) + 026
1198912(119909 119905) = 025 sin (119905) cos (3119909
2) + 01
Journal of Nonlinear Dynamics 7
minus02
minus5
minus100 5 10 15 20 25 30
0
5
10
Time (s) Time (s)
Time (s)0 5 10 15 20 25 30
0
02
04
06
08O
utpu
tx1
regu
latio
n
x
u
1
Con
trol i
nput
u
0 5 10 15 20 25 30
0
5
10
0 5 10 15 20 25 30
0
5
10
15
minus5
minus5
minus10
minus15
Time (s)
120585 i
12058531205852
1205851
i=1
23
regu
latio
nSl
idin
g va
riabl
e120590
120590
conv
erge
nce
Figure 1 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched uncertainty
via the proposed control law
1198913(119909 119905) = minus 119909
3+ 11990921199092
3+ 3(minus119909
3+ 11990921199092
3)2
+ 025 sin (119905) cos (31199092) + 01
Δ119892119898(119909 119905) = 3 (minus119909
3+ 11990921199092
3)
120575119898= 03 cos (120587119905119909
2)
(49)
The output of interest is the state variable 1199091 The relative
degree of which is 2 Consequently the system in (48) canbe expressed in LGCC form as follows
120593 (120585 ) = 212058511205852+ 212058521205853119906 minus (120585
3minus 1205852
1minus (1205852
2+ 1) 119906)
+ 1205852(1205853minus 1205852
1minus (1205852
2+ 1) 119906)
(52)
The regularity condition mentioned in Definition 3 holdsand the zero dynamics of this system express according toDefinition 4 becomes
+ 119906 = 0 (53)
This confirms that the nominal system is minimum phaseThe corresponding linear system becomes
120585 = 119860120585 + 1198610 (54)
where
119860 = [
[
0 1 0
0 0 1
0 0 0
]
]
119861 = [
[
0
0
1
]
]
(55)
8 Journal of Nonlinear Dynamics
0 5 10 15 20 25 30
0
02
04
06
08
0
02
04
06
08
0 5 10 15 20 25 30
Time (s)
0 5 10 15 20 25 30
0
05
1
Time (s)
x1 120590
1205852
1205851u
Out
putx
1re
gula
tion
120585 ii
=1
2 re
gula
tion
Slid
ing
varia
ble120590
conv
erge
nce
minus02minus02
minus05
minus1
0
5
10
minus5
minus10
Con
trol i
nput
u
0 5 10 15 20 25 30
Figure 2 Output regulation control effort sliding variable convergence and [1205851 1205852]119879 regulation in the presence of matched uncertainty via
and the final expression of the control law takes the form
= 0+
1
120574 (120585)
[minus120593 (120585 ) minus (120574 (120585) minus 1) 0minus 119870 sign (120590)]
(59)
In this study we compare the results of the proposed controllaw with that of quasicontinuous high order sliding mode
controller proposed by Levant in [18] To apply such anapproach we denote
119904 = 1199091
119904 = 1199092
(60)
So that the expression of the quasicontinuous sliding modecontroller in case of relative degree (2-QCSMC) takes thefollowing form
119906 = minus
120572 ( 119904 + |119904|12 sign (119904))
| 119904| + |119904|12
(61)
where120572 is the controller gainwhich can be selected accordingto Bartolini et al [32] As proved in Levant [18] the controllaw (61) provides a finite time slidingmode of the systemwitha control law which is continuous everywhere except on thesecond order sliding manifold 119904 = 119904 = 0
Note 2 It is not necessary that every system whose output isavailable can be put in the form appearing in (8) and (9)
Case 1 (system operated with matched uncertainty) In thisstudy the system with matched uncertainties (ie with
Journal of Nonlinear Dynamics 9
x1 120590
u12058521205851
1205853
120585 1i
=1
23
regu
latio
n
0
0
5 10 15 20 25 30
02
04
06
08
0 5 10 15 20 25 30
0
2
4
6
8
0 5 10 15 20 25 30
05
101520
Time (s)0
0
5 10 15 20 25 30
10
20
Time (s)
Slid
ing
varia
ble120590
conv
erge
nce
minus2
minus4
minus20
minus10
minus15
minus10
minus5Con
trol i
nput
uO
utpu
tx1
regu
latio
n
minus20
Figure 3 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched and
unmatched uncertainty via the proposed control law
Table 1 The parameters values of the control law used in the simulations
Constants 1198961
1198962
1198963
1198881
1198882
1198883
119870 or 1205722-QCSMC mdash mdash mdash mdash mdash mdash 4Proposed control law with 119903 = 2 4902 1807 59 6 5 mdash 230
119891119894(119909 119905) = 0 for 119894 = 1 2 3) is simulated to confirm the afore-
mentioned claim of the compensation of uncertain termsThis test with matched uncertainty is also performed with2-QCSMC previously mentioned The results are reportedin Figures 1 and 2 In these Figures it can be seen that theoutput system with state vector [120585
1 1205852 1205853]119879 is regulated in the
presence of uncertainties It is noticeable that the proposedmethodology provides a satisfactory regulation of the systemoutput via a continuous control law The 2-QCSMC alsoprovides excellent performance yet with a control law whichbecomes discontinuous when the output regulation objectiveis attained Apart from that both the controllers need to use adifferentiator [16 18] to construct the derivatives of the outputvariable necessary in the control laws
Case 2 (system operated under matched and unmatcheduncertainties) In this section the test with both matchedand unmatched uncertainty is performed The results withthe proposed control law are depicted in Figure 3These sim-ulation results confirm the robust and chattering free natureof the proposed controller as well as its capability of efficientlysolving the regulation problem even in this particularly
critical case In view of the nature of the uncertainty now con-sidered we cannot compare our results with those of the 2-QCSMC algorithm since that algorithm was designed underthe assumption of having only matched uncertainty [18]
Note that the controller gains and the controllersparameters in both the experiments are listed in Table 1
6 Conclusions
In this work an output feedback dynamic sliding modecontroller is presented which is capable of dealing with a classof SISO nonlinear systems operating under both matchedand unmatched state dependent uncertaintiesThe uncertainsystem output trajectories are asymptotically regulated tozero inspite of the presence of the uncertainties while asliding mode is enforced in finite time along an integralmanifold The use of the integral sliding manifold allowsone to subdivide the control design procedure into twosteps First a linear control component is designed by poleplacement and then a discontinuous control component isadded so as to cope with the uncertainty presenceThe designprocedure is relying on a suitably transformed system which
10 Journal of Nonlinear Dynamics
generally appears in a canonical form and the control inputappears with 119896 time derivatives As a consequence the controlacting on the original system is obtained as the output ofa chain of integrators and is accordingly continuous Thiscan be a clear benefit in many applications such as thoseof mechanical nature where a discontinuous control actioncould be nonappropriate or even disruptive for the actuatorsand systemrsquos health
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[2] M Fliess ldquoGeneralized controller canonical forms for linearand nonlinear dynamicsrdquo IEEE Transactions on AutomaticControl vol 35 no 9 pp 994ndash1001 1990
[3] H S Ramirez ldquoOn the dynamical sliding mode control of non-linear systemsrdquo International Journal of Control vol 57 no 5pp 1039ndash1061 1993
[4] X-Y Lu and S K Spurgeon ldquoOutput feedback stabilization ofSISO nonlinear systems via dynamic sliding modesrdquo Interna-tional Journal of Control vol 70 no 5 pp 735ndash759 1998
[5] V I Utkin Sliding Modes in Control Optimization SpringerBerlin Germany 1992
[6] J C Scarratt A Zinober R E Mills M Rios-Bolıvar A Fer-rara and L Giacomini ldquoDynamical adaptive first and second-order sliding backstepping control of nonlinear nontriangularuncertain systemsrdquo Journal of Dynamic Systems Measurementand Control vol 122 no 4 pp 746ndash752 2000
[7] D Swaroop J K Hedrick P P Yip and J C Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo IEEE Transac-tions on Automatic Control vol 45 no 10 pp 1893ndash1899 2000
[8] A Ferrara and L Giacomini ldquoOnmodular backstepping designwith second order sliding modesrdquo Automatica vol 37 no 1 pp129ndash135 2001
[9] S Tong and H-X Li ldquoFuzzy adaptive sliding-mode control forMIMOnonlinear systemsrdquo IEEETransactions on Fuzzy Systemsvol 11 no 3 pp 354ndash360 2003
[10] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[11] S Tong C Liu andY Li ldquoFuzzy-adaptive decentralized output-feedback control for large-scale nonlinear systems with dynam-ical uncertaintiesrdquo IEEE Transactions on Fuzzy Systems vol 18no 5 pp 845ndash861 2010
[12] V I Utkin J Guldner and J Shi SlidingModeControl in Electro-mechanical Systems Taylor amp Francis London UK 1999
[13] A Levant and L Alelishvili ldquoIntegral high-order slidingmodesrdquo IEEE Transactions on Automatic Control vol 52 no 7pp 1278ndash1282 2007
[14] G Bartolini A Ferrara and E Usai ldquoOutput tracking controlof uncertain nonlinear second-order systemsrdquo Automatica vol33 no 12 pp 2203ndash2212 1997
[15] G Bartolini A Ferrara and E Usai ldquoChattering avoidanceby second-order sliding mode controlrdquo IEEE Transactions onAutomatic Control vol 43 no 2 pp 241ndash246 1998
[16] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[17] I Boiko L Fridman andM I Castellanos ldquoAnalysis of second-order sliding-mode algorithms in the frequency domainrdquo IEEETransactions on Automatic Control vol 49 no 6 pp 946ndash9502004
[18] A Levant ldquoQuasi-continuous high-order sliding mode con-trollersrdquo IEEE Transaction on Automatic Control vol 50 no 11pp 1812ndash1816 2005
[19] F Dinuzzo and A Ferrara ldquoHigher order sliding mode con-trollers with optimal reachingrdquo IEEE Transactions onAutomaticControl vol 54 no 9 pp 2126ndash2136 2009
[20] H H Choi ldquoVariable structure output feedback control designfor a class of uncertain dynamic systemsrdquo Automatica vol 38no 2 pp 335ndash341 2002
[21] P Park D J Choi and S G Kong ldquoOutput feedback variablestructure control for linear systems with uncertainties anddisturbancesrdquo Automatica vol 43 no 1 pp 72ndash79 2007
[22] J Xiang WWei and H Su ldquoAn ILMI approach to robust staticoutput feedback sliding mode controlrdquo International Journal ofControl vol 79 no 8 pp 959ndash967 2006
[23] J M A da Silva C Edwards and S K Spurgeon ldquoSliding-mode output-feedback control based on LMIs for plants withmismatched uncertaintiesrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3675ndash3683 2009
[24] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[25] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[26] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[27] J-L Chang ldquoDynamic output integral sliding-mode controlwith disturbance attenuationrdquo IEEE Transactions on AutomaticControl vol 54 no 11 pp 2653ndash2658 2009
[28] Q Khan A I Bhatti and Q Ahmed ldquoDynamic integral slidingmode control of nonlinear SISO systems with states dependentmatched and mismatched uncertaintiesrdquo in Proceedings of the18thWorld Congress of the International Federation of AutomaticControl (IFAC rsquo11) pp 3932ndash3937 Milan Italy September 2011
[29] A Isidori Nonlinear Control Systems Springer 3rd edition1995
[30] M Fliess ldquoNonlinear control theory and differential algebrardquo inModeling and Adaptive Control C Byrness and A KurzhanskiEds vol 105 of Lecture Notes in Control and Information Sci-ence pp 134ndash145 Springer New York NY USA 1988
[31] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamic integralsliding mode for MIMO uncertain nonlinear systemsrdquo Interna-tional Journal of Control Automation and Systems vol 9 no 1pp 151ndash160 2011
[32] G Bartolini A Pisano E Punta and E Usai ldquoA survey of appli-cations of second-order sliding mode control to mechanicalsystemsrdquo International Journal of Control vol 76 no 9-10 pp875ndash892 2003
isin 119877 is continuous and stabilizes the systemat the equilibrium point while the second part 119906(119896)
1isin 119877 is
discontinuous in nature and can be classified as an integralSMC Its role is to reject uncertainties In the next subsectionsthe design of 119906(119896)
0isin 119877 and 119906
(119896)
1isin 119877 will be discussed Starting
from the nominal case and then moving to the case in whichthe presence of matched and unmatched uncertainties arealso considered
31 The Nominal Case
311 Design of 119906(119896)0 The nominal system in (9) in alternative
In the design of 119906(119896)0 system (13) is first considered to be
independent of nonlinearities that is 120594(120585 119906(119896)) = 0 and itis also supposed to operate under 119906(119896)
0only Then system (13)
becomes1205851= 1205852
1205852= 1205853
120585119899= 119906(119896)
0
(15)
This is a linear system so it can be written as
120585 = 119860120585 + 119861119906(119896)
0 (16)
where
119860 = [0(119899minus1)times1
119868(119899minus1)(119899minus1)
01times1
01times(119899minus1)
]
119861 = [0(119899minus1)times1
1]
(17)
For the sake of simplicity the input 119906(119896)0
is designed via poleplacement that is
119906(119896)
0= minus119870119879
0120585 (18)
312 Design of 119906(119896)1 Now in order to achieve the desired
performance robust compensation of the uncertainties isneeded To this end we select the following sliding manifoldof integral type [12]
120590 (120585) = 1205900(120585) + 119911 (19)
where 1205900(120585) is a conventional sliding surface which is math-
ematically defined by 1205900(120585) = sum
119899
119894=1119888119894120585119894 with 119888
119899= 1 and 119911 is
the integral term The time derivative of (19) along (9) yields
(120585) =
119899minus1
sum
119894=1
119888119894120585119894+1
+ 120594 (120585 119906(119896)
) + 119906(119896)
0+ 119906(119896)
1+ (20)
Now choose z with the following expression
= minus(
119899minus1
sum
119894=1
119888119894120585119894+1
+ 119906(119896)
0)
119911 (0) = minus1205900(120585 (0))
(21)
Then (20) becomes
(120585) = 120594 (120585 119906(119896)
) + 119906(119896)
1
= 120593 (120585 ) + (120574 (120585) minus 1) 119906(119896)
0+ 120574 (120585) 119906
119896
1
(22)
This initial condition 119911(0) is adjusted in such a way to meetthe requirement 120590(0) = 0
Taking into account the reachability condition defined asfollows [5]
= minus119870 sign120590 (23)
and comparing (22) with (23) the expression of the discon-tinuous control component 119906(119896)
1becomes
119906(119896)
1=
1
120574 (120585)
(minus120593 (120585 ) minus (120574 (120585) minus 1) 119906(119896)
0minus 119870 sign120590) (24)
This control law enforces sliding mode along the slidingmanifold defined in (19) The constant 119870 can be selectedaccording to the subsequent stability analysis
Note that this control law can be implemented by integratingthe derivative of the control 119906(119896) ldquo119896rdquo times so that the controlinput actually applied to the system is continuous This canbe a benefit for various class of systems such as those ofmechanical type for which a discontinuous control actioncould be disruptive
Remark 6 Thecoefficients of the conventional sliding surfaceare chosen by tacking into the dynamic response of thesystem However in real applications these constants can alsobe optimized using LMIs methods
Remark 7 The proposed methodology needs the availabilityof the system output and of its derivatives for the controllerimplementation In case the output derivatives are not avail-able for measurements one can use for instance a finite timesliding mode differentiator like the one proposed in [16] toreconstruct them
Journal of Nonlinear Dynamics 5
4 Stability Analysis
In this section the proposed control law when applied tothe uncertain nonlinear system in question is theoreticallyanalyzed First the case in which only matched uncertaintiesare present will be discussed and then the more general caseof matched and unmatched uncertainties will be considered
41 The System Operating under Matched Uncertainties Nowwe assume that the system operates only under matcheduncertainties Thus system (8) with matched uncertaintiesbecomes
To show that this system is stabilized in finite time in thepresence ofmatched uncertainties the following theorem canbe stated
Theorem 8 Consider that Assumptions 2 and 5 are satisfiedThe sliding surface is chosen as 120590(120585) = 0 where 120590(120585) is definedin (19) and the control law 119906
(119896) is selected according to (25) Ifthe gain 119870 is chosen according to the following condition
10038161003816100381610038161003816+ (1 minus 120576
119898) 119862 + 119870
1198721205731+ 1205781]
(33)
as in (27) Note that (32) can also be written as
+ radic21205781
radic119881 lt 0 (34)
This implies that 120590(120585) converges to zero in a finite time 119905119904[1]
such that
119905119904le radic2120578
minus1
1
radic119881 (120590 (0)) (35)
which completes the proof
Corollary 9 Thedynamics of the system (26) with control law(25) and sliding manifold 120590(120585) = 0 with 120590(120585) defined in (19)in sliding mode are governed by the linear control law (18)
Proof The nonlinear system (26) can be written in thefollowing alternate form
1205851= 1205852
1205852= 1205853
120585119899= 120593 (120585 ) + (120574 (120585) minus 1) 119906
(119896)
+ 119906(119896)
0
+ 119906(119896)
1+ 120574 (120585) 119906
(119896)
120575119898+ Δ119866119898(120585 119905)
(36)
The time derivative of (19) along (36) yields
(120585) =
119899minus1
sum
119894=1
119888119894120585119894+1
+ 120593 (120585 ) + (120574 (120585) minus 1) 119906(119896)
+ 119906(119896)
0+ 119906(119896)
1+ 120574 (120585) 119906
(119896)
120575119898+ Δ119866119898(120585 119905) +
(37)
Substituting (21) into (37) posing (120585) = 0 and solving withrespect to the control variable 119906(119896) one obtains the so-calledequivalent control [5] as
where 120585119904is the state of system (26) while in sliding mode
Thus it is proved that the system in sliding mode operatesunder the continuous control law and the eigenvalues of thecontrolled transformed system in sliding mode are those of119860 minus 119861119870
119879
0
42 The System Operating under Both Matched andUnmatched Uncertainties In this subsection it is nowassumed that the considered system operates under bothmatched and unmatched uncertainties and the controlobjective is to regulate the output of the system in thepresence of these uncertainties To prove that the proposedcontrol law is capable of compensating for these uncertainterms the following theorem can be stated
Theorem 10 Consider that Assumptions 2 and 5 are satisfiedThe sliding surface is chosen as 120590(120585) = 0 where 120590(120585) is definedin (19) and the control law 119906
(119896) is selected according to (25) Ifthe gain 119870 is chosen according to the following condition
10038161003816100381610038161003816+ (1 minus 120576
119898) 119862 + 119870
1198721205731+1205782+120591]
(46)
The expression in (45) can be placed in the same formatlike that of (34) Note that the finite time 119905
119904in this case is
given by the formula in (35) with 1205782instead of 120578
1 Thus it
is confirmed that when the gain of the control law (25) isselected according to (40) the finite time enforcement of thesliding mode is guaranteed in the presence of matched andunmatched uncertainties which proves the theorem
Corollary 11 The dynamics of system (8) with control law(25) and an integral manifold 120590(120585) = 0 with 120590(120585) defined in(19) in slidingmode are governed by the linear control law (18)
Proof The proof can be performed by following the sameprocedure as in the proof of Corollary 9 with the onlydifference that in this case the equivalent control is equal to
119906(119896)
eq = minus1
120574 (120585) (1 + 120575119898)
times [120593 (120585 ) minus 119906(119896)
0+120574 (120585) Δ119866
119898(120585 119905)+ΔΦ (120585 119905)]
(47)
5 Illustrative Example (System with RelativeDegree 2)
Consider the following uncertain nonlinear system [4]
1= 1199092+ 1198911(119909 119905)
2= 1199092
1+ (1199092
2+ 1) 119906 (1 + 120575
119898) + Δ119892
119898(119909 119905) + 119909
3+ 1198912(119909 119905)
3= minus1199093+ 11990921199092
3+ 1198913(119909 119905)
(48)
where 1199091 1199092 and 119909
3are the states of the nonlinear system
The terms 120575119898 Δ119892119898(119909 119905) are matched uncertainties and
119891119894(119909 119905) are components of themismatched uncertainty which
satisfy Assumptions 1 and 2 and these terms contributeto the system uncertainty with the following mathematicalexpressions
1198911(119909 119905) = minus 119909
3+ 11990921199092
3+ (minus119909
3+ 11990921199092
3)2
+ 025 sin (119905) cos (31199092) + 026
1198912(119909 119905) = 025 sin (119905) cos (3119909
2) + 01
Journal of Nonlinear Dynamics 7
minus02
minus5
minus100 5 10 15 20 25 30
0
5
10
Time (s) Time (s)
Time (s)0 5 10 15 20 25 30
0
02
04
06
08O
utpu
tx1
regu
latio
n
x
u
1
Con
trol i
nput
u
0 5 10 15 20 25 30
0
5
10
0 5 10 15 20 25 30
0
5
10
15
minus5
minus5
minus10
minus15
Time (s)
120585 i
12058531205852
1205851
i=1
23
regu
latio
nSl
idin
g va
riabl
e120590
120590
conv
erge
nce
Figure 1 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched uncertainty
via the proposed control law
1198913(119909 119905) = minus 119909
3+ 11990921199092
3+ 3(minus119909
3+ 11990921199092
3)2
+ 025 sin (119905) cos (31199092) + 01
Δ119892119898(119909 119905) = 3 (minus119909
3+ 11990921199092
3)
120575119898= 03 cos (120587119905119909
2)
(49)
The output of interest is the state variable 1199091 The relative
degree of which is 2 Consequently the system in (48) canbe expressed in LGCC form as follows
120593 (120585 ) = 212058511205852+ 212058521205853119906 minus (120585
3minus 1205852
1minus (1205852
2+ 1) 119906)
+ 1205852(1205853minus 1205852
1minus (1205852
2+ 1) 119906)
(52)
The regularity condition mentioned in Definition 3 holdsand the zero dynamics of this system express according toDefinition 4 becomes
+ 119906 = 0 (53)
This confirms that the nominal system is minimum phaseThe corresponding linear system becomes
120585 = 119860120585 + 1198610 (54)
where
119860 = [
[
0 1 0
0 0 1
0 0 0
]
]
119861 = [
[
0
0
1
]
]
(55)
8 Journal of Nonlinear Dynamics
0 5 10 15 20 25 30
0
02
04
06
08
0
02
04
06
08
0 5 10 15 20 25 30
Time (s)
0 5 10 15 20 25 30
0
05
1
Time (s)
x1 120590
1205852
1205851u
Out
putx
1re
gula
tion
120585 ii
=1
2 re
gula
tion
Slid
ing
varia
ble120590
conv
erge
nce
minus02minus02
minus05
minus1
0
5
10
minus5
minus10
Con
trol i
nput
u
0 5 10 15 20 25 30
Figure 2 Output regulation control effort sliding variable convergence and [1205851 1205852]119879 regulation in the presence of matched uncertainty via
and the final expression of the control law takes the form
= 0+
1
120574 (120585)
[minus120593 (120585 ) minus (120574 (120585) minus 1) 0minus 119870 sign (120590)]
(59)
In this study we compare the results of the proposed controllaw with that of quasicontinuous high order sliding mode
controller proposed by Levant in [18] To apply such anapproach we denote
119904 = 1199091
119904 = 1199092
(60)
So that the expression of the quasicontinuous sliding modecontroller in case of relative degree (2-QCSMC) takes thefollowing form
119906 = minus
120572 ( 119904 + |119904|12 sign (119904))
| 119904| + |119904|12
(61)
where120572 is the controller gainwhich can be selected accordingto Bartolini et al [32] As proved in Levant [18] the controllaw (61) provides a finite time slidingmode of the systemwitha control law which is continuous everywhere except on thesecond order sliding manifold 119904 = 119904 = 0
Note 2 It is not necessary that every system whose output isavailable can be put in the form appearing in (8) and (9)
Case 1 (system operated with matched uncertainty) In thisstudy the system with matched uncertainties (ie with
Journal of Nonlinear Dynamics 9
x1 120590
u12058521205851
1205853
120585 1i
=1
23
regu
latio
n
0
0
5 10 15 20 25 30
02
04
06
08
0 5 10 15 20 25 30
0
2
4
6
8
0 5 10 15 20 25 30
05
101520
Time (s)0
0
5 10 15 20 25 30
10
20
Time (s)
Slid
ing
varia
ble120590
conv
erge
nce
minus2
minus4
minus20
minus10
minus15
minus10
minus5Con
trol i
nput
uO
utpu
tx1
regu
latio
n
minus20
Figure 3 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched and
unmatched uncertainty via the proposed control law
Table 1 The parameters values of the control law used in the simulations
Constants 1198961
1198962
1198963
1198881
1198882
1198883
119870 or 1205722-QCSMC mdash mdash mdash mdash mdash mdash 4Proposed control law with 119903 = 2 4902 1807 59 6 5 mdash 230
119891119894(119909 119905) = 0 for 119894 = 1 2 3) is simulated to confirm the afore-
mentioned claim of the compensation of uncertain termsThis test with matched uncertainty is also performed with2-QCSMC previously mentioned The results are reportedin Figures 1 and 2 In these Figures it can be seen that theoutput system with state vector [120585
1 1205852 1205853]119879 is regulated in the
presence of uncertainties It is noticeable that the proposedmethodology provides a satisfactory regulation of the systemoutput via a continuous control law The 2-QCSMC alsoprovides excellent performance yet with a control law whichbecomes discontinuous when the output regulation objectiveis attained Apart from that both the controllers need to use adifferentiator [16 18] to construct the derivatives of the outputvariable necessary in the control laws
Case 2 (system operated under matched and unmatcheduncertainties) In this section the test with both matchedand unmatched uncertainty is performed The results withthe proposed control law are depicted in Figure 3These sim-ulation results confirm the robust and chattering free natureof the proposed controller as well as its capability of efficientlysolving the regulation problem even in this particularly
critical case In view of the nature of the uncertainty now con-sidered we cannot compare our results with those of the 2-QCSMC algorithm since that algorithm was designed underthe assumption of having only matched uncertainty [18]
Note that the controller gains and the controllersparameters in both the experiments are listed in Table 1
6 Conclusions
In this work an output feedback dynamic sliding modecontroller is presented which is capable of dealing with a classof SISO nonlinear systems operating under both matchedand unmatched state dependent uncertaintiesThe uncertainsystem output trajectories are asymptotically regulated tozero inspite of the presence of the uncertainties while asliding mode is enforced in finite time along an integralmanifold The use of the integral sliding manifold allowsone to subdivide the control design procedure into twosteps First a linear control component is designed by poleplacement and then a discontinuous control component isadded so as to cope with the uncertainty presenceThe designprocedure is relying on a suitably transformed system which
10 Journal of Nonlinear Dynamics
generally appears in a canonical form and the control inputappears with 119896 time derivatives As a consequence the controlacting on the original system is obtained as the output ofa chain of integrators and is accordingly continuous Thiscan be a clear benefit in many applications such as thoseof mechanical nature where a discontinuous control actioncould be nonappropriate or even disruptive for the actuatorsand systemrsquos health
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[2] M Fliess ldquoGeneralized controller canonical forms for linearand nonlinear dynamicsrdquo IEEE Transactions on AutomaticControl vol 35 no 9 pp 994ndash1001 1990
[3] H S Ramirez ldquoOn the dynamical sliding mode control of non-linear systemsrdquo International Journal of Control vol 57 no 5pp 1039ndash1061 1993
[4] X-Y Lu and S K Spurgeon ldquoOutput feedback stabilization ofSISO nonlinear systems via dynamic sliding modesrdquo Interna-tional Journal of Control vol 70 no 5 pp 735ndash759 1998
[5] V I Utkin Sliding Modes in Control Optimization SpringerBerlin Germany 1992
[6] J C Scarratt A Zinober R E Mills M Rios-Bolıvar A Fer-rara and L Giacomini ldquoDynamical adaptive first and second-order sliding backstepping control of nonlinear nontriangularuncertain systemsrdquo Journal of Dynamic Systems Measurementand Control vol 122 no 4 pp 746ndash752 2000
[7] D Swaroop J K Hedrick P P Yip and J C Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo IEEE Transac-tions on Automatic Control vol 45 no 10 pp 1893ndash1899 2000
[8] A Ferrara and L Giacomini ldquoOnmodular backstepping designwith second order sliding modesrdquo Automatica vol 37 no 1 pp129ndash135 2001
[9] S Tong and H-X Li ldquoFuzzy adaptive sliding-mode control forMIMOnonlinear systemsrdquo IEEETransactions on Fuzzy Systemsvol 11 no 3 pp 354ndash360 2003
[10] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[11] S Tong C Liu andY Li ldquoFuzzy-adaptive decentralized output-feedback control for large-scale nonlinear systems with dynam-ical uncertaintiesrdquo IEEE Transactions on Fuzzy Systems vol 18no 5 pp 845ndash861 2010
[12] V I Utkin J Guldner and J Shi SlidingModeControl in Electro-mechanical Systems Taylor amp Francis London UK 1999
[13] A Levant and L Alelishvili ldquoIntegral high-order slidingmodesrdquo IEEE Transactions on Automatic Control vol 52 no 7pp 1278ndash1282 2007
[14] G Bartolini A Ferrara and E Usai ldquoOutput tracking controlof uncertain nonlinear second-order systemsrdquo Automatica vol33 no 12 pp 2203ndash2212 1997
[15] G Bartolini A Ferrara and E Usai ldquoChattering avoidanceby second-order sliding mode controlrdquo IEEE Transactions onAutomatic Control vol 43 no 2 pp 241ndash246 1998
[16] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[17] I Boiko L Fridman andM I Castellanos ldquoAnalysis of second-order sliding-mode algorithms in the frequency domainrdquo IEEETransactions on Automatic Control vol 49 no 6 pp 946ndash9502004
[18] A Levant ldquoQuasi-continuous high-order sliding mode con-trollersrdquo IEEE Transaction on Automatic Control vol 50 no 11pp 1812ndash1816 2005
[19] F Dinuzzo and A Ferrara ldquoHigher order sliding mode con-trollers with optimal reachingrdquo IEEE Transactions onAutomaticControl vol 54 no 9 pp 2126ndash2136 2009
[20] H H Choi ldquoVariable structure output feedback control designfor a class of uncertain dynamic systemsrdquo Automatica vol 38no 2 pp 335ndash341 2002
[21] P Park D J Choi and S G Kong ldquoOutput feedback variablestructure control for linear systems with uncertainties anddisturbancesrdquo Automatica vol 43 no 1 pp 72ndash79 2007
[22] J Xiang WWei and H Su ldquoAn ILMI approach to robust staticoutput feedback sliding mode controlrdquo International Journal ofControl vol 79 no 8 pp 959ndash967 2006
[23] J M A da Silva C Edwards and S K Spurgeon ldquoSliding-mode output-feedback control based on LMIs for plants withmismatched uncertaintiesrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3675ndash3683 2009
[24] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[25] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[26] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[27] J-L Chang ldquoDynamic output integral sliding-mode controlwith disturbance attenuationrdquo IEEE Transactions on AutomaticControl vol 54 no 11 pp 2653ndash2658 2009
[28] Q Khan A I Bhatti and Q Ahmed ldquoDynamic integral slidingmode control of nonlinear SISO systems with states dependentmatched and mismatched uncertaintiesrdquo in Proceedings of the18thWorld Congress of the International Federation of AutomaticControl (IFAC rsquo11) pp 3932ndash3937 Milan Italy September 2011
[29] A Isidori Nonlinear Control Systems Springer 3rd edition1995
[30] M Fliess ldquoNonlinear control theory and differential algebrardquo inModeling and Adaptive Control C Byrness and A KurzhanskiEds vol 105 of Lecture Notes in Control and Information Sci-ence pp 134ndash145 Springer New York NY USA 1988
[31] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamic integralsliding mode for MIMO uncertain nonlinear systemsrdquo Interna-tional Journal of Control Automation and Systems vol 9 no 1pp 151ndash160 2011
[32] G Bartolini A Pisano E Punta and E Usai ldquoA survey of appli-cations of second-order sliding mode control to mechanicalsystemsrdquo International Journal of Control vol 76 no 9-10 pp875ndash892 2003
In this section the proposed control law when applied tothe uncertain nonlinear system in question is theoreticallyanalyzed First the case in which only matched uncertaintiesare present will be discussed and then the more general caseof matched and unmatched uncertainties will be considered
41 The System Operating under Matched Uncertainties Nowwe assume that the system operates only under matcheduncertainties Thus system (8) with matched uncertaintiesbecomes
To show that this system is stabilized in finite time in thepresence ofmatched uncertainties the following theorem canbe stated
Theorem 8 Consider that Assumptions 2 and 5 are satisfiedThe sliding surface is chosen as 120590(120585) = 0 where 120590(120585) is definedin (19) and the control law 119906
(119896) is selected according to (25) Ifthe gain 119870 is chosen according to the following condition
10038161003816100381610038161003816+ (1 minus 120576
119898) 119862 + 119870
1198721205731+ 1205781]
(33)
as in (27) Note that (32) can also be written as
+ radic21205781
radic119881 lt 0 (34)
This implies that 120590(120585) converges to zero in a finite time 119905119904[1]
such that
119905119904le radic2120578
minus1
1
radic119881 (120590 (0)) (35)
which completes the proof
Corollary 9 Thedynamics of the system (26) with control law(25) and sliding manifold 120590(120585) = 0 with 120590(120585) defined in (19)in sliding mode are governed by the linear control law (18)
Proof The nonlinear system (26) can be written in thefollowing alternate form
1205851= 1205852
1205852= 1205853
120585119899= 120593 (120585 ) + (120574 (120585) minus 1) 119906
(119896)
+ 119906(119896)
0
+ 119906(119896)
1+ 120574 (120585) 119906
(119896)
120575119898+ Δ119866119898(120585 119905)
(36)
The time derivative of (19) along (36) yields
(120585) =
119899minus1
sum
119894=1
119888119894120585119894+1
+ 120593 (120585 ) + (120574 (120585) minus 1) 119906(119896)
+ 119906(119896)
0+ 119906(119896)
1+ 120574 (120585) 119906
(119896)
120575119898+ Δ119866119898(120585 119905) +
(37)
Substituting (21) into (37) posing (120585) = 0 and solving withrespect to the control variable 119906(119896) one obtains the so-calledequivalent control [5] as
where 120585119904is the state of system (26) while in sliding mode
Thus it is proved that the system in sliding mode operatesunder the continuous control law and the eigenvalues of thecontrolled transformed system in sliding mode are those of119860 minus 119861119870
119879
0
42 The System Operating under Both Matched andUnmatched Uncertainties In this subsection it is nowassumed that the considered system operates under bothmatched and unmatched uncertainties and the controlobjective is to regulate the output of the system in thepresence of these uncertainties To prove that the proposedcontrol law is capable of compensating for these uncertainterms the following theorem can be stated
Theorem 10 Consider that Assumptions 2 and 5 are satisfiedThe sliding surface is chosen as 120590(120585) = 0 where 120590(120585) is definedin (19) and the control law 119906
(119896) is selected according to (25) Ifthe gain 119870 is chosen according to the following condition
10038161003816100381610038161003816+ (1 minus 120576
119898) 119862 + 119870
1198721205731+1205782+120591]
(46)
The expression in (45) can be placed in the same formatlike that of (34) Note that the finite time 119905
119904in this case is
given by the formula in (35) with 1205782instead of 120578
1 Thus it
is confirmed that when the gain of the control law (25) isselected according to (40) the finite time enforcement of thesliding mode is guaranteed in the presence of matched andunmatched uncertainties which proves the theorem
Corollary 11 The dynamics of system (8) with control law(25) and an integral manifold 120590(120585) = 0 with 120590(120585) defined in(19) in slidingmode are governed by the linear control law (18)
Proof The proof can be performed by following the sameprocedure as in the proof of Corollary 9 with the onlydifference that in this case the equivalent control is equal to
119906(119896)
eq = minus1
120574 (120585) (1 + 120575119898)
times [120593 (120585 ) minus 119906(119896)
0+120574 (120585) Δ119866
119898(120585 119905)+ΔΦ (120585 119905)]
(47)
5 Illustrative Example (System with RelativeDegree 2)
Consider the following uncertain nonlinear system [4]
1= 1199092+ 1198911(119909 119905)
2= 1199092
1+ (1199092
2+ 1) 119906 (1 + 120575
119898) + Δ119892
119898(119909 119905) + 119909
3+ 1198912(119909 119905)
3= minus1199093+ 11990921199092
3+ 1198913(119909 119905)
(48)
where 1199091 1199092 and 119909
3are the states of the nonlinear system
The terms 120575119898 Δ119892119898(119909 119905) are matched uncertainties and
119891119894(119909 119905) are components of themismatched uncertainty which
satisfy Assumptions 1 and 2 and these terms contributeto the system uncertainty with the following mathematicalexpressions
1198911(119909 119905) = minus 119909
3+ 11990921199092
3+ (minus119909
3+ 11990921199092
3)2
+ 025 sin (119905) cos (31199092) + 026
1198912(119909 119905) = 025 sin (119905) cos (3119909
2) + 01
Journal of Nonlinear Dynamics 7
minus02
minus5
minus100 5 10 15 20 25 30
0
5
10
Time (s) Time (s)
Time (s)0 5 10 15 20 25 30
0
02
04
06
08O
utpu
tx1
regu
latio
n
x
u
1
Con
trol i
nput
u
0 5 10 15 20 25 30
0
5
10
0 5 10 15 20 25 30
0
5
10
15
minus5
minus5
minus10
minus15
Time (s)
120585 i
12058531205852
1205851
i=1
23
regu
latio
nSl
idin
g va
riabl
e120590
120590
conv
erge
nce
Figure 1 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched uncertainty
via the proposed control law
1198913(119909 119905) = minus 119909
3+ 11990921199092
3+ 3(minus119909
3+ 11990921199092
3)2
+ 025 sin (119905) cos (31199092) + 01
Δ119892119898(119909 119905) = 3 (minus119909
3+ 11990921199092
3)
120575119898= 03 cos (120587119905119909
2)
(49)
The output of interest is the state variable 1199091 The relative
degree of which is 2 Consequently the system in (48) canbe expressed in LGCC form as follows
120593 (120585 ) = 212058511205852+ 212058521205853119906 minus (120585
3minus 1205852
1minus (1205852
2+ 1) 119906)
+ 1205852(1205853minus 1205852
1minus (1205852
2+ 1) 119906)
(52)
The regularity condition mentioned in Definition 3 holdsand the zero dynamics of this system express according toDefinition 4 becomes
+ 119906 = 0 (53)
This confirms that the nominal system is minimum phaseThe corresponding linear system becomes
120585 = 119860120585 + 1198610 (54)
where
119860 = [
[
0 1 0
0 0 1
0 0 0
]
]
119861 = [
[
0
0
1
]
]
(55)
8 Journal of Nonlinear Dynamics
0 5 10 15 20 25 30
0
02
04
06
08
0
02
04
06
08
0 5 10 15 20 25 30
Time (s)
0 5 10 15 20 25 30
0
05
1
Time (s)
x1 120590
1205852
1205851u
Out
putx
1re
gula
tion
120585 ii
=1
2 re
gula
tion
Slid
ing
varia
ble120590
conv
erge
nce
minus02minus02
minus05
minus1
0
5
10
minus5
minus10
Con
trol i
nput
u
0 5 10 15 20 25 30
Figure 2 Output regulation control effort sliding variable convergence and [1205851 1205852]119879 regulation in the presence of matched uncertainty via
and the final expression of the control law takes the form
= 0+
1
120574 (120585)
[minus120593 (120585 ) minus (120574 (120585) minus 1) 0minus 119870 sign (120590)]
(59)
In this study we compare the results of the proposed controllaw with that of quasicontinuous high order sliding mode
controller proposed by Levant in [18] To apply such anapproach we denote
119904 = 1199091
119904 = 1199092
(60)
So that the expression of the quasicontinuous sliding modecontroller in case of relative degree (2-QCSMC) takes thefollowing form
119906 = minus
120572 ( 119904 + |119904|12 sign (119904))
| 119904| + |119904|12
(61)
where120572 is the controller gainwhich can be selected accordingto Bartolini et al [32] As proved in Levant [18] the controllaw (61) provides a finite time slidingmode of the systemwitha control law which is continuous everywhere except on thesecond order sliding manifold 119904 = 119904 = 0
Note 2 It is not necessary that every system whose output isavailable can be put in the form appearing in (8) and (9)
Case 1 (system operated with matched uncertainty) In thisstudy the system with matched uncertainties (ie with
Journal of Nonlinear Dynamics 9
x1 120590
u12058521205851
1205853
120585 1i
=1
23
regu
latio
n
0
0
5 10 15 20 25 30
02
04
06
08
0 5 10 15 20 25 30
0
2
4
6
8
0 5 10 15 20 25 30
05
101520
Time (s)0
0
5 10 15 20 25 30
10
20
Time (s)
Slid
ing
varia
ble120590
conv
erge
nce
minus2
minus4
minus20
minus10
minus15
minus10
minus5Con
trol i
nput
uO
utpu
tx1
regu
latio
n
minus20
Figure 3 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched and
unmatched uncertainty via the proposed control law
Table 1 The parameters values of the control law used in the simulations
Constants 1198961
1198962
1198963
1198881
1198882
1198883
119870 or 1205722-QCSMC mdash mdash mdash mdash mdash mdash 4Proposed control law with 119903 = 2 4902 1807 59 6 5 mdash 230
119891119894(119909 119905) = 0 for 119894 = 1 2 3) is simulated to confirm the afore-
mentioned claim of the compensation of uncertain termsThis test with matched uncertainty is also performed with2-QCSMC previously mentioned The results are reportedin Figures 1 and 2 In these Figures it can be seen that theoutput system with state vector [120585
1 1205852 1205853]119879 is regulated in the
presence of uncertainties It is noticeable that the proposedmethodology provides a satisfactory regulation of the systemoutput via a continuous control law The 2-QCSMC alsoprovides excellent performance yet with a control law whichbecomes discontinuous when the output regulation objectiveis attained Apart from that both the controllers need to use adifferentiator [16 18] to construct the derivatives of the outputvariable necessary in the control laws
Case 2 (system operated under matched and unmatcheduncertainties) In this section the test with both matchedand unmatched uncertainty is performed The results withthe proposed control law are depicted in Figure 3These sim-ulation results confirm the robust and chattering free natureof the proposed controller as well as its capability of efficientlysolving the regulation problem even in this particularly
critical case In view of the nature of the uncertainty now con-sidered we cannot compare our results with those of the 2-QCSMC algorithm since that algorithm was designed underthe assumption of having only matched uncertainty [18]
Note that the controller gains and the controllersparameters in both the experiments are listed in Table 1
6 Conclusions
In this work an output feedback dynamic sliding modecontroller is presented which is capable of dealing with a classof SISO nonlinear systems operating under both matchedand unmatched state dependent uncertaintiesThe uncertainsystem output trajectories are asymptotically regulated tozero inspite of the presence of the uncertainties while asliding mode is enforced in finite time along an integralmanifold The use of the integral sliding manifold allowsone to subdivide the control design procedure into twosteps First a linear control component is designed by poleplacement and then a discontinuous control component isadded so as to cope with the uncertainty presenceThe designprocedure is relying on a suitably transformed system which
10 Journal of Nonlinear Dynamics
generally appears in a canonical form and the control inputappears with 119896 time derivatives As a consequence the controlacting on the original system is obtained as the output ofa chain of integrators and is accordingly continuous Thiscan be a clear benefit in many applications such as thoseof mechanical nature where a discontinuous control actioncould be nonappropriate or even disruptive for the actuatorsand systemrsquos health
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[2] M Fliess ldquoGeneralized controller canonical forms for linearand nonlinear dynamicsrdquo IEEE Transactions on AutomaticControl vol 35 no 9 pp 994ndash1001 1990
[3] H S Ramirez ldquoOn the dynamical sliding mode control of non-linear systemsrdquo International Journal of Control vol 57 no 5pp 1039ndash1061 1993
[4] X-Y Lu and S K Spurgeon ldquoOutput feedback stabilization ofSISO nonlinear systems via dynamic sliding modesrdquo Interna-tional Journal of Control vol 70 no 5 pp 735ndash759 1998
[5] V I Utkin Sliding Modes in Control Optimization SpringerBerlin Germany 1992
[6] J C Scarratt A Zinober R E Mills M Rios-Bolıvar A Fer-rara and L Giacomini ldquoDynamical adaptive first and second-order sliding backstepping control of nonlinear nontriangularuncertain systemsrdquo Journal of Dynamic Systems Measurementand Control vol 122 no 4 pp 746ndash752 2000
[7] D Swaroop J K Hedrick P P Yip and J C Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo IEEE Transac-tions on Automatic Control vol 45 no 10 pp 1893ndash1899 2000
[8] A Ferrara and L Giacomini ldquoOnmodular backstepping designwith second order sliding modesrdquo Automatica vol 37 no 1 pp129ndash135 2001
[9] S Tong and H-X Li ldquoFuzzy adaptive sliding-mode control forMIMOnonlinear systemsrdquo IEEETransactions on Fuzzy Systemsvol 11 no 3 pp 354ndash360 2003
[10] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[11] S Tong C Liu andY Li ldquoFuzzy-adaptive decentralized output-feedback control for large-scale nonlinear systems with dynam-ical uncertaintiesrdquo IEEE Transactions on Fuzzy Systems vol 18no 5 pp 845ndash861 2010
[12] V I Utkin J Guldner and J Shi SlidingModeControl in Electro-mechanical Systems Taylor amp Francis London UK 1999
[13] A Levant and L Alelishvili ldquoIntegral high-order slidingmodesrdquo IEEE Transactions on Automatic Control vol 52 no 7pp 1278ndash1282 2007
[14] G Bartolini A Ferrara and E Usai ldquoOutput tracking controlof uncertain nonlinear second-order systemsrdquo Automatica vol33 no 12 pp 2203ndash2212 1997
[15] G Bartolini A Ferrara and E Usai ldquoChattering avoidanceby second-order sliding mode controlrdquo IEEE Transactions onAutomatic Control vol 43 no 2 pp 241ndash246 1998
[16] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[17] I Boiko L Fridman andM I Castellanos ldquoAnalysis of second-order sliding-mode algorithms in the frequency domainrdquo IEEETransactions on Automatic Control vol 49 no 6 pp 946ndash9502004
[18] A Levant ldquoQuasi-continuous high-order sliding mode con-trollersrdquo IEEE Transaction on Automatic Control vol 50 no 11pp 1812ndash1816 2005
[19] F Dinuzzo and A Ferrara ldquoHigher order sliding mode con-trollers with optimal reachingrdquo IEEE Transactions onAutomaticControl vol 54 no 9 pp 2126ndash2136 2009
[20] H H Choi ldquoVariable structure output feedback control designfor a class of uncertain dynamic systemsrdquo Automatica vol 38no 2 pp 335ndash341 2002
[21] P Park D J Choi and S G Kong ldquoOutput feedback variablestructure control for linear systems with uncertainties anddisturbancesrdquo Automatica vol 43 no 1 pp 72ndash79 2007
[22] J Xiang WWei and H Su ldquoAn ILMI approach to robust staticoutput feedback sliding mode controlrdquo International Journal ofControl vol 79 no 8 pp 959ndash967 2006
[23] J M A da Silva C Edwards and S K Spurgeon ldquoSliding-mode output-feedback control based on LMIs for plants withmismatched uncertaintiesrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3675ndash3683 2009
[24] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[25] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[26] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[27] J-L Chang ldquoDynamic output integral sliding-mode controlwith disturbance attenuationrdquo IEEE Transactions on AutomaticControl vol 54 no 11 pp 2653ndash2658 2009
[28] Q Khan A I Bhatti and Q Ahmed ldquoDynamic integral slidingmode control of nonlinear SISO systems with states dependentmatched and mismatched uncertaintiesrdquo in Proceedings of the18thWorld Congress of the International Federation of AutomaticControl (IFAC rsquo11) pp 3932ndash3937 Milan Italy September 2011
[29] A Isidori Nonlinear Control Systems Springer 3rd edition1995
[30] M Fliess ldquoNonlinear control theory and differential algebrardquo inModeling and Adaptive Control C Byrness and A KurzhanskiEds vol 105 of Lecture Notes in Control and Information Sci-ence pp 134ndash145 Springer New York NY USA 1988
[31] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamic integralsliding mode for MIMO uncertain nonlinear systemsrdquo Interna-tional Journal of Control Automation and Systems vol 9 no 1pp 151ndash160 2011
[32] G Bartolini A Pisano E Punta and E Usai ldquoA survey of appli-cations of second-order sliding mode control to mechanicalsystemsrdquo International Journal of Control vol 76 no 9-10 pp875ndash892 2003
where 120585119904is the state of system (26) while in sliding mode
Thus it is proved that the system in sliding mode operatesunder the continuous control law and the eigenvalues of thecontrolled transformed system in sliding mode are those of119860 minus 119861119870
119879
0
42 The System Operating under Both Matched andUnmatched Uncertainties In this subsection it is nowassumed that the considered system operates under bothmatched and unmatched uncertainties and the controlobjective is to regulate the output of the system in thepresence of these uncertainties To prove that the proposedcontrol law is capable of compensating for these uncertainterms the following theorem can be stated
Theorem 10 Consider that Assumptions 2 and 5 are satisfiedThe sliding surface is chosen as 120590(120585) = 0 where 120590(120585) is definedin (19) and the control law 119906
(119896) is selected according to (25) Ifthe gain 119870 is chosen according to the following condition
10038161003816100381610038161003816+ (1 minus 120576
119898) 119862 + 119870
1198721205731+1205782+120591]
(46)
The expression in (45) can be placed in the same formatlike that of (34) Note that the finite time 119905
119904in this case is
given by the formula in (35) with 1205782instead of 120578
1 Thus it
is confirmed that when the gain of the control law (25) isselected according to (40) the finite time enforcement of thesliding mode is guaranteed in the presence of matched andunmatched uncertainties which proves the theorem
Corollary 11 The dynamics of system (8) with control law(25) and an integral manifold 120590(120585) = 0 with 120590(120585) defined in(19) in slidingmode are governed by the linear control law (18)
Proof The proof can be performed by following the sameprocedure as in the proof of Corollary 9 with the onlydifference that in this case the equivalent control is equal to
119906(119896)
eq = minus1
120574 (120585) (1 + 120575119898)
times [120593 (120585 ) minus 119906(119896)
0+120574 (120585) Δ119866
119898(120585 119905)+ΔΦ (120585 119905)]
(47)
5 Illustrative Example (System with RelativeDegree 2)
Consider the following uncertain nonlinear system [4]
1= 1199092+ 1198911(119909 119905)
2= 1199092
1+ (1199092
2+ 1) 119906 (1 + 120575
119898) + Δ119892
119898(119909 119905) + 119909
3+ 1198912(119909 119905)
3= minus1199093+ 11990921199092
3+ 1198913(119909 119905)
(48)
where 1199091 1199092 and 119909
3are the states of the nonlinear system
The terms 120575119898 Δ119892119898(119909 119905) are matched uncertainties and
119891119894(119909 119905) are components of themismatched uncertainty which
satisfy Assumptions 1 and 2 and these terms contributeto the system uncertainty with the following mathematicalexpressions
1198911(119909 119905) = minus 119909
3+ 11990921199092
3+ (minus119909
3+ 11990921199092
3)2
+ 025 sin (119905) cos (31199092) + 026
1198912(119909 119905) = 025 sin (119905) cos (3119909
2) + 01
Journal of Nonlinear Dynamics 7
minus02
minus5
minus100 5 10 15 20 25 30
0
5
10
Time (s) Time (s)
Time (s)0 5 10 15 20 25 30
0
02
04
06
08O
utpu
tx1
regu
latio
n
x
u
1
Con
trol i
nput
u
0 5 10 15 20 25 30
0
5
10
0 5 10 15 20 25 30
0
5
10
15
minus5
minus5
minus10
minus15
Time (s)
120585 i
12058531205852
1205851
i=1
23
regu
latio
nSl
idin
g va
riabl
e120590
120590
conv
erge
nce
Figure 1 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched uncertainty
via the proposed control law
1198913(119909 119905) = minus 119909
3+ 11990921199092
3+ 3(minus119909
3+ 11990921199092
3)2
+ 025 sin (119905) cos (31199092) + 01
Δ119892119898(119909 119905) = 3 (minus119909
3+ 11990921199092
3)
120575119898= 03 cos (120587119905119909
2)
(49)
The output of interest is the state variable 1199091 The relative
degree of which is 2 Consequently the system in (48) canbe expressed in LGCC form as follows
120593 (120585 ) = 212058511205852+ 212058521205853119906 minus (120585
3minus 1205852
1minus (1205852
2+ 1) 119906)
+ 1205852(1205853minus 1205852
1minus (1205852
2+ 1) 119906)
(52)
The regularity condition mentioned in Definition 3 holdsand the zero dynamics of this system express according toDefinition 4 becomes
+ 119906 = 0 (53)
This confirms that the nominal system is minimum phaseThe corresponding linear system becomes
120585 = 119860120585 + 1198610 (54)
where
119860 = [
[
0 1 0
0 0 1
0 0 0
]
]
119861 = [
[
0
0
1
]
]
(55)
8 Journal of Nonlinear Dynamics
0 5 10 15 20 25 30
0
02
04
06
08
0
02
04
06
08
0 5 10 15 20 25 30
Time (s)
0 5 10 15 20 25 30
0
05
1
Time (s)
x1 120590
1205852
1205851u
Out
putx
1re
gula
tion
120585 ii
=1
2 re
gula
tion
Slid
ing
varia
ble120590
conv
erge
nce
minus02minus02
minus05
minus1
0
5
10
minus5
minus10
Con
trol i
nput
u
0 5 10 15 20 25 30
Figure 2 Output regulation control effort sliding variable convergence and [1205851 1205852]119879 regulation in the presence of matched uncertainty via
and the final expression of the control law takes the form
= 0+
1
120574 (120585)
[minus120593 (120585 ) minus (120574 (120585) minus 1) 0minus 119870 sign (120590)]
(59)
In this study we compare the results of the proposed controllaw with that of quasicontinuous high order sliding mode
controller proposed by Levant in [18] To apply such anapproach we denote
119904 = 1199091
119904 = 1199092
(60)
So that the expression of the quasicontinuous sliding modecontroller in case of relative degree (2-QCSMC) takes thefollowing form
119906 = minus
120572 ( 119904 + |119904|12 sign (119904))
| 119904| + |119904|12
(61)
where120572 is the controller gainwhich can be selected accordingto Bartolini et al [32] As proved in Levant [18] the controllaw (61) provides a finite time slidingmode of the systemwitha control law which is continuous everywhere except on thesecond order sliding manifold 119904 = 119904 = 0
Note 2 It is not necessary that every system whose output isavailable can be put in the form appearing in (8) and (9)
Case 1 (system operated with matched uncertainty) In thisstudy the system with matched uncertainties (ie with
Journal of Nonlinear Dynamics 9
x1 120590
u12058521205851
1205853
120585 1i
=1
23
regu
latio
n
0
0
5 10 15 20 25 30
02
04
06
08
0 5 10 15 20 25 30
0
2
4
6
8
0 5 10 15 20 25 30
05
101520
Time (s)0
0
5 10 15 20 25 30
10
20
Time (s)
Slid
ing
varia
ble120590
conv
erge
nce
minus2
minus4
minus20
minus10
minus15
minus10
minus5Con
trol i
nput
uO
utpu
tx1
regu
latio
n
minus20
Figure 3 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched and
unmatched uncertainty via the proposed control law
Table 1 The parameters values of the control law used in the simulations
Constants 1198961
1198962
1198963
1198881
1198882
1198883
119870 or 1205722-QCSMC mdash mdash mdash mdash mdash mdash 4Proposed control law with 119903 = 2 4902 1807 59 6 5 mdash 230
119891119894(119909 119905) = 0 for 119894 = 1 2 3) is simulated to confirm the afore-
mentioned claim of the compensation of uncertain termsThis test with matched uncertainty is also performed with2-QCSMC previously mentioned The results are reportedin Figures 1 and 2 In these Figures it can be seen that theoutput system with state vector [120585
1 1205852 1205853]119879 is regulated in the
presence of uncertainties It is noticeable that the proposedmethodology provides a satisfactory regulation of the systemoutput via a continuous control law The 2-QCSMC alsoprovides excellent performance yet with a control law whichbecomes discontinuous when the output regulation objectiveis attained Apart from that both the controllers need to use adifferentiator [16 18] to construct the derivatives of the outputvariable necessary in the control laws
Case 2 (system operated under matched and unmatcheduncertainties) In this section the test with both matchedand unmatched uncertainty is performed The results withthe proposed control law are depicted in Figure 3These sim-ulation results confirm the robust and chattering free natureof the proposed controller as well as its capability of efficientlysolving the regulation problem even in this particularly
critical case In view of the nature of the uncertainty now con-sidered we cannot compare our results with those of the 2-QCSMC algorithm since that algorithm was designed underthe assumption of having only matched uncertainty [18]
Note that the controller gains and the controllersparameters in both the experiments are listed in Table 1
6 Conclusions
In this work an output feedback dynamic sliding modecontroller is presented which is capable of dealing with a classof SISO nonlinear systems operating under both matchedand unmatched state dependent uncertaintiesThe uncertainsystem output trajectories are asymptotically regulated tozero inspite of the presence of the uncertainties while asliding mode is enforced in finite time along an integralmanifold The use of the integral sliding manifold allowsone to subdivide the control design procedure into twosteps First a linear control component is designed by poleplacement and then a discontinuous control component isadded so as to cope with the uncertainty presenceThe designprocedure is relying on a suitably transformed system which
10 Journal of Nonlinear Dynamics
generally appears in a canonical form and the control inputappears with 119896 time derivatives As a consequence the controlacting on the original system is obtained as the output ofa chain of integrators and is accordingly continuous Thiscan be a clear benefit in many applications such as thoseof mechanical nature where a discontinuous control actioncould be nonappropriate or even disruptive for the actuatorsand systemrsquos health
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[2] M Fliess ldquoGeneralized controller canonical forms for linearand nonlinear dynamicsrdquo IEEE Transactions on AutomaticControl vol 35 no 9 pp 994ndash1001 1990
[3] H S Ramirez ldquoOn the dynamical sliding mode control of non-linear systemsrdquo International Journal of Control vol 57 no 5pp 1039ndash1061 1993
[4] X-Y Lu and S K Spurgeon ldquoOutput feedback stabilization ofSISO nonlinear systems via dynamic sliding modesrdquo Interna-tional Journal of Control vol 70 no 5 pp 735ndash759 1998
[5] V I Utkin Sliding Modes in Control Optimization SpringerBerlin Germany 1992
[6] J C Scarratt A Zinober R E Mills M Rios-Bolıvar A Fer-rara and L Giacomini ldquoDynamical adaptive first and second-order sliding backstepping control of nonlinear nontriangularuncertain systemsrdquo Journal of Dynamic Systems Measurementand Control vol 122 no 4 pp 746ndash752 2000
[7] D Swaroop J K Hedrick P P Yip and J C Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo IEEE Transac-tions on Automatic Control vol 45 no 10 pp 1893ndash1899 2000
[8] A Ferrara and L Giacomini ldquoOnmodular backstepping designwith second order sliding modesrdquo Automatica vol 37 no 1 pp129ndash135 2001
[9] S Tong and H-X Li ldquoFuzzy adaptive sliding-mode control forMIMOnonlinear systemsrdquo IEEETransactions on Fuzzy Systemsvol 11 no 3 pp 354ndash360 2003
[10] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[11] S Tong C Liu andY Li ldquoFuzzy-adaptive decentralized output-feedback control for large-scale nonlinear systems with dynam-ical uncertaintiesrdquo IEEE Transactions on Fuzzy Systems vol 18no 5 pp 845ndash861 2010
[12] V I Utkin J Guldner and J Shi SlidingModeControl in Electro-mechanical Systems Taylor amp Francis London UK 1999
[13] A Levant and L Alelishvili ldquoIntegral high-order slidingmodesrdquo IEEE Transactions on Automatic Control vol 52 no 7pp 1278ndash1282 2007
[14] G Bartolini A Ferrara and E Usai ldquoOutput tracking controlof uncertain nonlinear second-order systemsrdquo Automatica vol33 no 12 pp 2203ndash2212 1997
[15] G Bartolini A Ferrara and E Usai ldquoChattering avoidanceby second-order sliding mode controlrdquo IEEE Transactions onAutomatic Control vol 43 no 2 pp 241ndash246 1998
[16] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[17] I Boiko L Fridman andM I Castellanos ldquoAnalysis of second-order sliding-mode algorithms in the frequency domainrdquo IEEETransactions on Automatic Control vol 49 no 6 pp 946ndash9502004
[18] A Levant ldquoQuasi-continuous high-order sliding mode con-trollersrdquo IEEE Transaction on Automatic Control vol 50 no 11pp 1812ndash1816 2005
[19] F Dinuzzo and A Ferrara ldquoHigher order sliding mode con-trollers with optimal reachingrdquo IEEE Transactions onAutomaticControl vol 54 no 9 pp 2126ndash2136 2009
[20] H H Choi ldquoVariable structure output feedback control designfor a class of uncertain dynamic systemsrdquo Automatica vol 38no 2 pp 335ndash341 2002
[21] P Park D J Choi and S G Kong ldquoOutput feedback variablestructure control for linear systems with uncertainties anddisturbancesrdquo Automatica vol 43 no 1 pp 72ndash79 2007
[22] J Xiang WWei and H Su ldquoAn ILMI approach to robust staticoutput feedback sliding mode controlrdquo International Journal ofControl vol 79 no 8 pp 959ndash967 2006
[23] J M A da Silva C Edwards and S K Spurgeon ldquoSliding-mode output-feedback control based on LMIs for plants withmismatched uncertaintiesrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3675ndash3683 2009
[24] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[25] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[26] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[27] J-L Chang ldquoDynamic output integral sliding-mode controlwith disturbance attenuationrdquo IEEE Transactions on AutomaticControl vol 54 no 11 pp 2653ndash2658 2009
[28] Q Khan A I Bhatti and Q Ahmed ldquoDynamic integral slidingmode control of nonlinear SISO systems with states dependentmatched and mismatched uncertaintiesrdquo in Proceedings of the18thWorld Congress of the International Federation of AutomaticControl (IFAC rsquo11) pp 3932ndash3937 Milan Italy September 2011
[29] A Isidori Nonlinear Control Systems Springer 3rd edition1995
[30] M Fliess ldquoNonlinear control theory and differential algebrardquo inModeling and Adaptive Control C Byrness and A KurzhanskiEds vol 105 of Lecture Notes in Control and Information Sci-ence pp 134ndash145 Springer New York NY USA 1988
[31] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamic integralsliding mode for MIMO uncertain nonlinear systemsrdquo Interna-tional Journal of Control Automation and Systems vol 9 no 1pp 151ndash160 2011
[32] G Bartolini A Pisano E Punta and E Usai ldquoA survey of appli-cations of second-order sliding mode control to mechanicalsystemsrdquo International Journal of Control vol 76 no 9-10 pp875ndash892 2003
Figure 1 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched uncertainty
via the proposed control law
1198913(119909 119905) = minus 119909
3+ 11990921199092
3+ 3(minus119909
3+ 11990921199092
3)2
+ 025 sin (119905) cos (31199092) + 01
Δ119892119898(119909 119905) = 3 (minus119909
3+ 11990921199092
3)
120575119898= 03 cos (120587119905119909
2)
(49)
The output of interest is the state variable 1199091 The relative
degree of which is 2 Consequently the system in (48) canbe expressed in LGCC form as follows
120593 (120585 ) = 212058511205852+ 212058521205853119906 minus (120585
3minus 1205852
1minus (1205852
2+ 1) 119906)
+ 1205852(1205853minus 1205852
1minus (1205852
2+ 1) 119906)
(52)
The regularity condition mentioned in Definition 3 holdsand the zero dynamics of this system express according toDefinition 4 becomes
+ 119906 = 0 (53)
This confirms that the nominal system is minimum phaseThe corresponding linear system becomes
120585 = 119860120585 + 1198610 (54)
where
119860 = [
[
0 1 0
0 0 1
0 0 0
]
]
119861 = [
[
0
0
1
]
]
(55)
8 Journal of Nonlinear Dynamics
0 5 10 15 20 25 30
0
02
04
06
08
0
02
04
06
08
0 5 10 15 20 25 30
Time (s)
0 5 10 15 20 25 30
0
05
1
Time (s)
x1 120590
1205852
1205851u
Out
putx
1re
gula
tion
120585 ii
=1
2 re
gula
tion
Slid
ing
varia
ble120590
conv
erge
nce
minus02minus02
minus05
minus1
0
5
10
minus5
minus10
Con
trol i
nput
u
0 5 10 15 20 25 30
Figure 2 Output regulation control effort sliding variable convergence and [1205851 1205852]119879 regulation in the presence of matched uncertainty via
and the final expression of the control law takes the form
= 0+
1
120574 (120585)
[minus120593 (120585 ) minus (120574 (120585) minus 1) 0minus 119870 sign (120590)]
(59)
In this study we compare the results of the proposed controllaw with that of quasicontinuous high order sliding mode
controller proposed by Levant in [18] To apply such anapproach we denote
119904 = 1199091
119904 = 1199092
(60)
So that the expression of the quasicontinuous sliding modecontroller in case of relative degree (2-QCSMC) takes thefollowing form
119906 = minus
120572 ( 119904 + |119904|12 sign (119904))
| 119904| + |119904|12
(61)
where120572 is the controller gainwhich can be selected accordingto Bartolini et al [32] As proved in Levant [18] the controllaw (61) provides a finite time slidingmode of the systemwitha control law which is continuous everywhere except on thesecond order sliding manifold 119904 = 119904 = 0
Note 2 It is not necessary that every system whose output isavailable can be put in the form appearing in (8) and (9)
Case 1 (system operated with matched uncertainty) In thisstudy the system with matched uncertainties (ie with
Journal of Nonlinear Dynamics 9
x1 120590
u12058521205851
1205853
120585 1i
=1
23
regu
latio
n
0
0
5 10 15 20 25 30
02
04
06
08
0 5 10 15 20 25 30
0
2
4
6
8
0 5 10 15 20 25 30
05
101520
Time (s)0
0
5 10 15 20 25 30
10
20
Time (s)
Slid
ing
varia
ble120590
conv
erge
nce
minus2
minus4
minus20
minus10
minus15
minus10
minus5Con
trol i
nput
uO
utpu
tx1
regu
latio
n
minus20
Figure 3 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched and
unmatched uncertainty via the proposed control law
Table 1 The parameters values of the control law used in the simulations
Constants 1198961
1198962
1198963
1198881
1198882
1198883
119870 or 1205722-QCSMC mdash mdash mdash mdash mdash mdash 4Proposed control law with 119903 = 2 4902 1807 59 6 5 mdash 230
119891119894(119909 119905) = 0 for 119894 = 1 2 3) is simulated to confirm the afore-
mentioned claim of the compensation of uncertain termsThis test with matched uncertainty is also performed with2-QCSMC previously mentioned The results are reportedin Figures 1 and 2 In these Figures it can be seen that theoutput system with state vector [120585
1 1205852 1205853]119879 is regulated in the
presence of uncertainties It is noticeable that the proposedmethodology provides a satisfactory regulation of the systemoutput via a continuous control law The 2-QCSMC alsoprovides excellent performance yet with a control law whichbecomes discontinuous when the output regulation objectiveis attained Apart from that both the controllers need to use adifferentiator [16 18] to construct the derivatives of the outputvariable necessary in the control laws
Case 2 (system operated under matched and unmatcheduncertainties) In this section the test with both matchedand unmatched uncertainty is performed The results withthe proposed control law are depicted in Figure 3These sim-ulation results confirm the robust and chattering free natureof the proposed controller as well as its capability of efficientlysolving the regulation problem even in this particularly
critical case In view of the nature of the uncertainty now con-sidered we cannot compare our results with those of the 2-QCSMC algorithm since that algorithm was designed underthe assumption of having only matched uncertainty [18]
Note that the controller gains and the controllersparameters in both the experiments are listed in Table 1
6 Conclusions
In this work an output feedback dynamic sliding modecontroller is presented which is capable of dealing with a classof SISO nonlinear systems operating under both matchedand unmatched state dependent uncertaintiesThe uncertainsystem output trajectories are asymptotically regulated tozero inspite of the presence of the uncertainties while asliding mode is enforced in finite time along an integralmanifold The use of the integral sliding manifold allowsone to subdivide the control design procedure into twosteps First a linear control component is designed by poleplacement and then a discontinuous control component isadded so as to cope with the uncertainty presenceThe designprocedure is relying on a suitably transformed system which
10 Journal of Nonlinear Dynamics
generally appears in a canonical form and the control inputappears with 119896 time derivatives As a consequence the controlacting on the original system is obtained as the output ofa chain of integrators and is accordingly continuous Thiscan be a clear benefit in many applications such as thoseof mechanical nature where a discontinuous control actioncould be nonappropriate or even disruptive for the actuatorsand systemrsquos health
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[2] M Fliess ldquoGeneralized controller canonical forms for linearand nonlinear dynamicsrdquo IEEE Transactions on AutomaticControl vol 35 no 9 pp 994ndash1001 1990
[3] H S Ramirez ldquoOn the dynamical sliding mode control of non-linear systemsrdquo International Journal of Control vol 57 no 5pp 1039ndash1061 1993
[4] X-Y Lu and S K Spurgeon ldquoOutput feedback stabilization ofSISO nonlinear systems via dynamic sliding modesrdquo Interna-tional Journal of Control vol 70 no 5 pp 735ndash759 1998
[5] V I Utkin Sliding Modes in Control Optimization SpringerBerlin Germany 1992
[6] J C Scarratt A Zinober R E Mills M Rios-Bolıvar A Fer-rara and L Giacomini ldquoDynamical adaptive first and second-order sliding backstepping control of nonlinear nontriangularuncertain systemsrdquo Journal of Dynamic Systems Measurementand Control vol 122 no 4 pp 746ndash752 2000
[7] D Swaroop J K Hedrick P P Yip and J C Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo IEEE Transac-tions on Automatic Control vol 45 no 10 pp 1893ndash1899 2000
[8] A Ferrara and L Giacomini ldquoOnmodular backstepping designwith second order sliding modesrdquo Automatica vol 37 no 1 pp129ndash135 2001
[9] S Tong and H-X Li ldquoFuzzy adaptive sliding-mode control forMIMOnonlinear systemsrdquo IEEETransactions on Fuzzy Systemsvol 11 no 3 pp 354ndash360 2003
[10] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[11] S Tong C Liu andY Li ldquoFuzzy-adaptive decentralized output-feedback control for large-scale nonlinear systems with dynam-ical uncertaintiesrdquo IEEE Transactions on Fuzzy Systems vol 18no 5 pp 845ndash861 2010
[12] V I Utkin J Guldner and J Shi SlidingModeControl in Electro-mechanical Systems Taylor amp Francis London UK 1999
[13] A Levant and L Alelishvili ldquoIntegral high-order slidingmodesrdquo IEEE Transactions on Automatic Control vol 52 no 7pp 1278ndash1282 2007
[14] G Bartolini A Ferrara and E Usai ldquoOutput tracking controlof uncertain nonlinear second-order systemsrdquo Automatica vol33 no 12 pp 2203ndash2212 1997
[15] G Bartolini A Ferrara and E Usai ldquoChattering avoidanceby second-order sliding mode controlrdquo IEEE Transactions onAutomatic Control vol 43 no 2 pp 241ndash246 1998
[16] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[17] I Boiko L Fridman andM I Castellanos ldquoAnalysis of second-order sliding-mode algorithms in the frequency domainrdquo IEEETransactions on Automatic Control vol 49 no 6 pp 946ndash9502004
[18] A Levant ldquoQuasi-continuous high-order sliding mode con-trollersrdquo IEEE Transaction on Automatic Control vol 50 no 11pp 1812ndash1816 2005
[19] F Dinuzzo and A Ferrara ldquoHigher order sliding mode con-trollers with optimal reachingrdquo IEEE Transactions onAutomaticControl vol 54 no 9 pp 2126ndash2136 2009
[20] H H Choi ldquoVariable structure output feedback control designfor a class of uncertain dynamic systemsrdquo Automatica vol 38no 2 pp 335ndash341 2002
[21] P Park D J Choi and S G Kong ldquoOutput feedback variablestructure control for linear systems with uncertainties anddisturbancesrdquo Automatica vol 43 no 1 pp 72ndash79 2007
[22] J Xiang WWei and H Su ldquoAn ILMI approach to robust staticoutput feedback sliding mode controlrdquo International Journal ofControl vol 79 no 8 pp 959ndash967 2006
[23] J M A da Silva C Edwards and S K Spurgeon ldquoSliding-mode output-feedback control based on LMIs for plants withmismatched uncertaintiesrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3675ndash3683 2009
[24] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[25] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[26] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[27] J-L Chang ldquoDynamic output integral sliding-mode controlwith disturbance attenuationrdquo IEEE Transactions on AutomaticControl vol 54 no 11 pp 2653ndash2658 2009
[28] Q Khan A I Bhatti and Q Ahmed ldquoDynamic integral slidingmode control of nonlinear SISO systems with states dependentmatched and mismatched uncertaintiesrdquo in Proceedings of the18thWorld Congress of the International Federation of AutomaticControl (IFAC rsquo11) pp 3932ndash3937 Milan Italy September 2011
[29] A Isidori Nonlinear Control Systems Springer 3rd edition1995
[30] M Fliess ldquoNonlinear control theory and differential algebrardquo inModeling and Adaptive Control C Byrness and A KurzhanskiEds vol 105 of Lecture Notes in Control and Information Sci-ence pp 134ndash145 Springer New York NY USA 1988
[31] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamic integralsliding mode for MIMO uncertain nonlinear systemsrdquo Interna-tional Journal of Control Automation and Systems vol 9 no 1pp 151ndash160 2011
[32] G Bartolini A Pisano E Punta and E Usai ldquoA survey of appli-cations of second-order sliding mode control to mechanicalsystemsrdquo International Journal of Control vol 76 no 9-10 pp875ndash892 2003
Figure 2 Output regulation control effort sliding variable convergence and [1205851 1205852]119879 regulation in the presence of matched uncertainty via
and the final expression of the control law takes the form
= 0+
1
120574 (120585)
[minus120593 (120585 ) minus (120574 (120585) minus 1) 0minus 119870 sign (120590)]
(59)
In this study we compare the results of the proposed controllaw with that of quasicontinuous high order sliding mode
controller proposed by Levant in [18] To apply such anapproach we denote
119904 = 1199091
119904 = 1199092
(60)
So that the expression of the quasicontinuous sliding modecontroller in case of relative degree (2-QCSMC) takes thefollowing form
119906 = minus
120572 ( 119904 + |119904|12 sign (119904))
| 119904| + |119904|12
(61)
where120572 is the controller gainwhich can be selected accordingto Bartolini et al [32] As proved in Levant [18] the controllaw (61) provides a finite time slidingmode of the systemwitha control law which is continuous everywhere except on thesecond order sliding manifold 119904 = 119904 = 0
Note 2 It is not necessary that every system whose output isavailable can be put in the form appearing in (8) and (9)
Case 1 (system operated with matched uncertainty) In thisstudy the system with matched uncertainties (ie with
Journal of Nonlinear Dynamics 9
x1 120590
u12058521205851
1205853
120585 1i
=1
23
regu
latio
n
0
0
5 10 15 20 25 30
02
04
06
08
0 5 10 15 20 25 30
0
2
4
6
8
0 5 10 15 20 25 30
05
101520
Time (s)0
0
5 10 15 20 25 30
10
20
Time (s)
Slid
ing
varia
ble120590
conv
erge
nce
minus2
minus4
minus20
minus10
minus15
minus10
minus5Con
trol i
nput
uO
utpu
tx1
regu
latio
n
minus20
Figure 3 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched and
unmatched uncertainty via the proposed control law
Table 1 The parameters values of the control law used in the simulations
Constants 1198961
1198962
1198963
1198881
1198882
1198883
119870 or 1205722-QCSMC mdash mdash mdash mdash mdash mdash 4Proposed control law with 119903 = 2 4902 1807 59 6 5 mdash 230
119891119894(119909 119905) = 0 for 119894 = 1 2 3) is simulated to confirm the afore-
mentioned claim of the compensation of uncertain termsThis test with matched uncertainty is also performed with2-QCSMC previously mentioned The results are reportedin Figures 1 and 2 In these Figures it can be seen that theoutput system with state vector [120585
1 1205852 1205853]119879 is regulated in the
presence of uncertainties It is noticeable that the proposedmethodology provides a satisfactory regulation of the systemoutput via a continuous control law The 2-QCSMC alsoprovides excellent performance yet with a control law whichbecomes discontinuous when the output regulation objectiveis attained Apart from that both the controllers need to use adifferentiator [16 18] to construct the derivatives of the outputvariable necessary in the control laws
Case 2 (system operated under matched and unmatcheduncertainties) In this section the test with both matchedand unmatched uncertainty is performed The results withthe proposed control law are depicted in Figure 3These sim-ulation results confirm the robust and chattering free natureof the proposed controller as well as its capability of efficientlysolving the regulation problem even in this particularly
critical case In view of the nature of the uncertainty now con-sidered we cannot compare our results with those of the 2-QCSMC algorithm since that algorithm was designed underthe assumption of having only matched uncertainty [18]
Note that the controller gains and the controllersparameters in both the experiments are listed in Table 1
6 Conclusions
In this work an output feedback dynamic sliding modecontroller is presented which is capable of dealing with a classof SISO nonlinear systems operating under both matchedand unmatched state dependent uncertaintiesThe uncertainsystem output trajectories are asymptotically regulated tozero inspite of the presence of the uncertainties while asliding mode is enforced in finite time along an integralmanifold The use of the integral sliding manifold allowsone to subdivide the control design procedure into twosteps First a linear control component is designed by poleplacement and then a discontinuous control component isadded so as to cope with the uncertainty presenceThe designprocedure is relying on a suitably transformed system which
10 Journal of Nonlinear Dynamics
generally appears in a canonical form and the control inputappears with 119896 time derivatives As a consequence the controlacting on the original system is obtained as the output ofa chain of integrators and is accordingly continuous Thiscan be a clear benefit in many applications such as thoseof mechanical nature where a discontinuous control actioncould be nonappropriate or even disruptive for the actuatorsand systemrsquos health
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[2] M Fliess ldquoGeneralized controller canonical forms for linearand nonlinear dynamicsrdquo IEEE Transactions on AutomaticControl vol 35 no 9 pp 994ndash1001 1990
[3] H S Ramirez ldquoOn the dynamical sliding mode control of non-linear systemsrdquo International Journal of Control vol 57 no 5pp 1039ndash1061 1993
[4] X-Y Lu and S K Spurgeon ldquoOutput feedback stabilization ofSISO nonlinear systems via dynamic sliding modesrdquo Interna-tional Journal of Control vol 70 no 5 pp 735ndash759 1998
[5] V I Utkin Sliding Modes in Control Optimization SpringerBerlin Germany 1992
[6] J C Scarratt A Zinober R E Mills M Rios-Bolıvar A Fer-rara and L Giacomini ldquoDynamical adaptive first and second-order sliding backstepping control of nonlinear nontriangularuncertain systemsrdquo Journal of Dynamic Systems Measurementand Control vol 122 no 4 pp 746ndash752 2000
[7] D Swaroop J K Hedrick P P Yip and J C Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo IEEE Transac-tions on Automatic Control vol 45 no 10 pp 1893ndash1899 2000
[8] A Ferrara and L Giacomini ldquoOnmodular backstepping designwith second order sliding modesrdquo Automatica vol 37 no 1 pp129ndash135 2001
[9] S Tong and H-X Li ldquoFuzzy adaptive sliding-mode control forMIMOnonlinear systemsrdquo IEEETransactions on Fuzzy Systemsvol 11 no 3 pp 354ndash360 2003
[10] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[11] S Tong C Liu andY Li ldquoFuzzy-adaptive decentralized output-feedback control for large-scale nonlinear systems with dynam-ical uncertaintiesrdquo IEEE Transactions on Fuzzy Systems vol 18no 5 pp 845ndash861 2010
[12] V I Utkin J Guldner and J Shi SlidingModeControl in Electro-mechanical Systems Taylor amp Francis London UK 1999
[13] A Levant and L Alelishvili ldquoIntegral high-order slidingmodesrdquo IEEE Transactions on Automatic Control vol 52 no 7pp 1278ndash1282 2007
[14] G Bartolini A Ferrara and E Usai ldquoOutput tracking controlof uncertain nonlinear second-order systemsrdquo Automatica vol33 no 12 pp 2203ndash2212 1997
[15] G Bartolini A Ferrara and E Usai ldquoChattering avoidanceby second-order sliding mode controlrdquo IEEE Transactions onAutomatic Control vol 43 no 2 pp 241ndash246 1998
[16] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[17] I Boiko L Fridman andM I Castellanos ldquoAnalysis of second-order sliding-mode algorithms in the frequency domainrdquo IEEETransactions on Automatic Control vol 49 no 6 pp 946ndash9502004
[18] A Levant ldquoQuasi-continuous high-order sliding mode con-trollersrdquo IEEE Transaction on Automatic Control vol 50 no 11pp 1812ndash1816 2005
[19] F Dinuzzo and A Ferrara ldquoHigher order sliding mode con-trollers with optimal reachingrdquo IEEE Transactions onAutomaticControl vol 54 no 9 pp 2126ndash2136 2009
[20] H H Choi ldquoVariable structure output feedback control designfor a class of uncertain dynamic systemsrdquo Automatica vol 38no 2 pp 335ndash341 2002
[21] P Park D J Choi and S G Kong ldquoOutput feedback variablestructure control for linear systems with uncertainties anddisturbancesrdquo Automatica vol 43 no 1 pp 72ndash79 2007
[22] J Xiang WWei and H Su ldquoAn ILMI approach to robust staticoutput feedback sliding mode controlrdquo International Journal ofControl vol 79 no 8 pp 959ndash967 2006
[23] J M A da Silva C Edwards and S K Spurgeon ldquoSliding-mode output-feedback control based on LMIs for plants withmismatched uncertaintiesrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3675ndash3683 2009
[24] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[25] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[26] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[27] J-L Chang ldquoDynamic output integral sliding-mode controlwith disturbance attenuationrdquo IEEE Transactions on AutomaticControl vol 54 no 11 pp 2653ndash2658 2009
[28] Q Khan A I Bhatti and Q Ahmed ldquoDynamic integral slidingmode control of nonlinear SISO systems with states dependentmatched and mismatched uncertaintiesrdquo in Proceedings of the18thWorld Congress of the International Federation of AutomaticControl (IFAC rsquo11) pp 3932ndash3937 Milan Italy September 2011
[29] A Isidori Nonlinear Control Systems Springer 3rd edition1995
[30] M Fliess ldquoNonlinear control theory and differential algebrardquo inModeling and Adaptive Control C Byrness and A KurzhanskiEds vol 105 of Lecture Notes in Control and Information Sci-ence pp 134ndash145 Springer New York NY USA 1988
[31] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamic integralsliding mode for MIMO uncertain nonlinear systemsrdquo Interna-tional Journal of Control Automation and Systems vol 9 no 1pp 151ndash160 2011
[32] G Bartolini A Pisano E Punta and E Usai ldquoA survey of appli-cations of second-order sliding mode control to mechanicalsystemsrdquo International Journal of Control vol 76 no 9-10 pp875ndash892 2003
Figure 3 Output regulation control effort sliding variable convergence and [1205851 1205852 1205853]119879 regulation in the presence of matched and
unmatched uncertainty via the proposed control law
Table 1 The parameters values of the control law used in the simulations
Constants 1198961
1198962
1198963
1198881
1198882
1198883
119870 or 1205722-QCSMC mdash mdash mdash mdash mdash mdash 4Proposed control law with 119903 = 2 4902 1807 59 6 5 mdash 230
119891119894(119909 119905) = 0 for 119894 = 1 2 3) is simulated to confirm the afore-
mentioned claim of the compensation of uncertain termsThis test with matched uncertainty is also performed with2-QCSMC previously mentioned The results are reportedin Figures 1 and 2 In these Figures it can be seen that theoutput system with state vector [120585
1 1205852 1205853]119879 is regulated in the
presence of uncertainties It is noticeable that the proposedmethodology provides a satisfactory regulation of the systemoutput via a continuous control law The 2-QCSMC alsoprovides excellent performance yet with a control law whichbecomes discontinuous when the output regulation objectiveis attained Apart from that both the controllers need to use adifferentiator [16 18] to construct the derivatives of the outputvariable necessary in the control laws
Case 2 (system operated under matched and unmatcheduncertainties) In this section the test with both matchedand unmatched uncertainty is performed The results withthe proposed control law are depicted in Figure 3These sim-ulation results confirm the robust and chattering free natureof the proposed controller as well as its capability of efficientlysolving the regulation problem even in this particularly
critical case In view of the nature of the uncertainty now con-sidered we cannot compare our results with those of the 2-QCSMC algorithm since that algorithm was designed underthe assumption of having only matched uncertainty [18]
Note that the controller gains and the controllersparameters in both the experiments are listed in Table 1
6 Conclusions
In this work an output feedback dynamic sliding modecontroller is presented which is capable of dealing with a classof SISO nonlinear systems operating under both matchedand unmatched state dependent uncertaintiesThe uncertainsystem output trajectories are asymptotically regulated tozero inspite of the presence of the uncertainties while asliding mode is enforced in finite time along an integralmanifold The use of the integral sliding manifold allowsone to subdivide the control design procedure into twosteps First a linear control component is designed by poleplacement and then a discontinuous control component isadded so as to cope with the uncertainty presenceThe designprocedure is relying on a suitably transformed system which
10 Journal of Nonlinear Dynamics
generally appears in a canonical form and the control inputappears with 119896 time derivatives As a consequence the controlacting on the original system is obtained as the output ofa chain of integrators and is accordingly continuous Thiscan be a clear benefit in many applications such as thoseof mechanical nature where a discontinuous control actioncould be nonappropriate or even disruptive for the actuatorsand systemrsquos health
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[2] M Fliess ldquoGeneralized controller canonical forms for linearand nonlinear dynamicsrdquo IEEE Transactions on AutomaticControl vol 35 no 9 pp 994ndash1001 1990
[3] H S Ramirez ldquoOn the dynamical sliding mode control of non-linear systemsrdquo International Journal of Control vol 57 no 5pp 1039ndash1061 1993
[4] X-Y Lu and S K Spurgeon ldquoOutput feedback stabilization ofSISO nonlinear systems via dynamic sliding modesrdquo Interna-tional Journal of Control vol 70 no 5 pp 735ndash759 1998
[5] V I Utkin Sliding Modes in Control Optimization SpringerBerlin Germany 1992
[6] J C Scarratt A Zinober R E Mills M Rios-Bolıvar A Fer-rara and L Giacomini ldquoDynamical adaptive first and second-order sliding backstepping control of nonlinear nontriangularuncertain systemsrdquo Journal of Dynamic Systems Measurementand Control vol 122 no 4 pp 746ndash752 2000
[7] D Swaroop J K Hedrick P P Yip and J C Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo IEEE Transac-tions on Automatic Control vol 45 no 10 pp 1893ndash1899 2000
[8] A Ferrara and L Giacomini ldquoOnmodular backstepping designwith second order sliding modesrdquo Automatica vol 37 no 1 pp129ndash135 2001
[9] S Tong and H-X Li ldquoFuzzy adaptive sliding-mode control forMIMOnonlinear systemsrdquo IEEETransactions on Fuzzy Systemsvol 11 no 3 pp 354ndash360 2003
[10] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[11] S Tong C Liu andY Li ldquoFuzzy-adaptive decentralized output-feedback control for large-scale nonlinear systems with dynam-ical uncertaintiesrdquo IEEE Transactions on Fuzzy Systems vol 18no 5 pp 845ndash861 2010
[12] V I Utkin J Guldner and J Shi SlidingModeControl in Electro-mechanical Systems Taylor amp Francis London UK 1999
[13] A Levant and L Alelishvili ldquoIntegral high-order slidingmodesrdquo IEEE Transactions on Automatic Control vol 52 no 7pp 1278ndash1282 2007
[14] G Bartolini A Ferrara and E Usai ldquoOutput tracking controlof uncertain nonlinear second-order systemsrdquo Automatica vol33 no 12 pp 2203ndash2212 1997
[15] G Bartolini A Ferrara and E Usai ldquoChattering avoidanceby second-order sliding mode controlrdquo IEEE Transactions onAutomatic Control vol 43 no 2 pp 241ndash246 1998
[16] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[17] I Boiko L Fridman andM I Castellanos ldquoAnalysis of second-order sliding-mode algorithms in the frequency domainrdquo IEEETransactions on Automatic Control vol 49 no 6 pp 946ndash9502004
[18] A Levant ldquoQuasi-continuous high-order sliding mode con-trollersrdquo IEEE Transaction on Automatic Control vol 50 no 11pp 1812ndash1816 2005
[19] F Dinuzzo and A Ferrara ldquoHigher order sliding mode con-trollers with optimal reachingrdquo IEEE Transactions onAutomaticControl vol 54 no 9 pp 2126ndash2136 2009
[20] H H Choi ldquoVariable structure output feedback control designfor a class of uncertain dynamic systemsrdquo Automatica vol 38no 2 pp 335ndash341 2002
[21] P Park D J Choi and S G Kong ldquoOutput feedback variablestructure control for linear systems with uncertainties anddisturbancesrdquo Automatica vol 43 no 1 pp 72ndash79 2007
[22] J Xiang WWei and H Su ldquoAn ILMI approach to robust staticoutput feedback sliding mode controlrdquo International Journal ofControl vol 79 no 8 pp 959ndash967 2006
[23] J M A da Silva C Edwards and S K Spurgeon ldquoSliding-mode output-feedback control based on LMIs for plants withmismatched uncertaintiesrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3675ndash3683 2009
[24] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[25] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[26] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[27] J-L Chang ldquoDynamic output integral sliding-mode controlwith disturbance attenuationrdquo IEEE Transactions on AutomaticControl vol 54 no 11 pp 2653ndash2658 2009
[28] Q Khan A I Bhatti and Q Ahmed ldquoDynamic integral slidingmode control of nonlinear SISO systems with states dependentmatched and mismatched uncertaintiesrdquo in Proceedings of the18thWorld Congress of the International Federation of AutomaticControl (IFAC rsquo11) pp 3932ndash3937 Milan Italy September 2011
[29] A Isidori Nonlinear Control Systems Springer 3rd edition1995
[30] M Fliess ldquoNonlinear control theory and differential algebrardquo inModeling and Adaptive Control C Byrness and A KurzhanskiEds vol 105 of Lecture Notes in Control and Information Sci-ence pp 134ndash145 Springer New York NY USA 1988
[31] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamic integralsliding mode for MIMO uncertain nonlinear systemsrdquo Interna-tional Journal of Control Automation and Systems vol 9 no 1pp 151ndash160 2011
[32] G Bartolini A Pisano E Punta and E Usai ldquoA survey of appli-cations of second-order sliding mode control to mechanicalsystemsrdquo International Journal of Control vol 76 no 9-10 pp875ndash892 2003
generally appears in a canonical form and the control inputappears with 119896 time derivatives As a consequence the controlacting on the original system is obtained as the output ofa chain of integrators and is accordingly continuous Thiscan be a clear benefit in many applications such as thoseof mechanical nature where a discontinuous control actioncould be nonappropriate or even disruptive for the actuatorsand systemrsquos health
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[2] M Fliess ldquoGeneralized controller canonical forms for linearand nonlinear dynamicsrdquo IEEE Transactions on AutomaticControl vol 35 no 9 pp 994ndash1001 1990
[3] H S Ramirez ldquoOn the dynamical sliding mode control of non-linear systemsrdquo International Journal of Control vol 57 no 5pp 1039ndash1061 1993
[4] X-Y Lu and S K Spurgeon ldquoOutput feedback stabilization ofSISO nonlinear systems via dynamic sliding modesrdquo Interna-tional Journal of Control vol 70 no 5 pp 735ndash759 1998
[5] V I Utkin Sliding Modes in Control Optimization SpringerBerlin Germany 1992
[6] J C Scarratt A Zinober R E Mills M Rios-Bolıvar A Fer-rara and L Giacomini ldquoDynamical adaptive first and second-order sliding backstepping control of nonlinear nontriangularuncertain systemsrdquo Journal of Dynamic Systems Measurementand Control vol 122 no 4 pp 746ndash752 2000
[7] D Swaroop J K Hedrick P P Yip and J C Gerdes ldquoDynamicsurface control for a class of nonlinear systemsrdquo IEEE Transac-tions on Automatic Control vol 45 no 10 pp 1893ndash1899 2000
[8] A Ferrara and L Giacomini ldquoOnmodular backstepping designwith second order sliding modesrdquo Automatica vol 37 no 1 pp129ndash135 2001
[9] S Tong and H-X Li ldquoFuzzy adaptive sliding-mode control forMIMOnonlinear systemsrdquo IEEETransactions on Fuzzy Systemsvol 11 no 3 pp 354ndash360 2003
[10] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009
[11] S Tong C Liu andY Li ldquoFuzzy-adaptive decentralized output-feedback control for large-scale nonlinear systems with dynam-ical uncertaintiesrdquo IEEE Transactions on Fuzzy Systems vol 18no 5 pp 845ndash861 2010
[12] V I Utkin J Guldner and J Shi SlidingModeControl in Electro-mechanical Systems Taylor amp Francis London UK 1999
[13] A Levant and L Alelishvili ldquoIntegral high-order slidingmodesrdquo IEEE Transactions on Automatic Control vol 52 no 7pp 1278ndash1282 2007
[14] G Bartolini A Ferrara and E Usai ldquoOutput tracking controlof uncertain nonlinear second-order systemsrdquo Automatica vol33 no 12 pp 2203ndash2212 1997
[15] G Bartolini A Ferrara and E Usai ldquoChattering avoidanceby second-order sliding mode controlrdquo IEEE Transactions onAutomatic Control vol 43 no 2 pp 241ndash246 1998
[16] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003
[17] I Boiko L Fridman andM I Castellanos ldquoAnalysis of second-order sliding-mode algorithms in the frequency domainrdquo IEEETransactions on Automatic Control vol 49 no 6 pp 946ndash9502004
[18] A Levant ldquoQuasi-continuous high-order sliding mode con-trollersrdquo IEEE Transaction on Automatic Control vol 50 no 11pp 1812ndash1816 2005
[19] F Dinuzzo and A Ferrara ldquoHigher order sliding mode con-trollers with optimal reachingrdquo IEEE Transactions onAutomaticControl vol 54 no 9 pp 2126ndash2136 2009
[20] H H Choi ldquoVariable structure output feedback control designfor a class of uncertain dynamic systemsrdquo Automatica vol 38no 2 pp 335ndash341 2002
[21] P Park D J Choi and S G Kong ldquoOutput feedback variablestructure control for linear systems with uncertainties anddisturbancesrdquo Automatica vol 43 no 1 pp 72ndash79 2007
[22] J Xiang WWei and H Su ldquoAn ILMI approach to robust staticoutput feedback sliding mode controlrdquo International Journal ofControl vol 79 no 8 pp 959ndash967 2006
[23] J M A da Silva C Edwards and S K Spurgeon ldquoSliding-mode output-feedback control based on LMIs for plants withmismatched uncertaintiesrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3675ndash3683 2009
[24] W-J Cao and J-X Xu ldquoNonlinear integral-type sliding surfacefor both matched and unmatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 49 no 8 pp 1355ndash13602004
[25] F Castanos and L Fridman ldquoAnalysis and design of integralsliding manifolds for systems with unmatched perturbationsrdquoIEEE Transactions on Automatic Control vol 51 no 5 pp 853ndash858 2006
[26] M Rubagotti A Estrada F Castanos A Ferrara and L Frid-man ldquoIntegral sliding mode control for nonlinear systems withmatched and unmatched perturbationsrdquo IEEE Transactions onAutomatic Control vol 56 no 11 pp 2699ndash2704 2011
[27] J-L Chang ldquoDynamic output integral sliding-mode controlwith disturbance attenuationrdquo IEEE Transactions on AutomaticControl vol 54 no 11 pp 2653ndash2658 2009
[28] Q Khan A I Bhatti and Q Ahmed ldquoDynamic integral slidingmode control of nonlinear SISO systems with states dependentmatched and mismatched uncertaintiesrdquo in Proceedings of the18thWorld Congress of the International Federation of AutomaticControl (IFAC rsquo11) pp 3932ndash3937 Milan Italy September 2011
[29] A Isidori Nonlinear Control Systems Springer 3rd edition1995
[30] M Fliess ldquoNonlinear control theory and differential algebrardquo inModeling and Adaptive Control C Byrness and A KurzhanskiEds vol 105 of Lecture Notes in Control and Information Sci-ence pp 134ndash145 Springer New York NY USA 1988
[31] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamic integralsliding mode for MIMO uncertain nonlinear systemsrdquo Interna-tional Journal of Control Automation and Systems vol 9 no 1pp 151ndash160 2011
[32] G Bartolini A Pisano E Punta and E Usai ldquoA survey of appli-cations of second-order sliding mode control to mechanicalsystemsrdquo International Journal of Control vol 76 no 9-10 pp875ndash892 2003
