Observer based output feedback control of linear systems with inputand output delays
Bin Zhou a,1, Zhao-Yan Li b, Zongli Lin c
a Center for Control Theory and Guidance Technology, Harbin Institute of Technology, P.O. Box 416, Harbin, 150001, Chinab Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, Chinac Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, P.O. Box 400743-4743, Charlottesville, VA 22904-4743, USA
a r t i c l e i n f o
Article history:Received 21 August 2012Received in revised form20 February 2013Accepted 18 March 2013Available online 4 May 2013
This paper is concerned with observer based output feedback control of linear systems with both (mul-tiple) input and output delays. Our recently developed truncated predictor feedback (TPF) approach forstate feedback stabilization of time-delay systems is generalized to the design of observers. By imposingsome restrictions on the open-loop system, two classes of observer based output feedback controllers, onebeing finite dimensional and the other infinite dimensional, are constructed. It is further shown that, theinfinite dimensional observer based output feedback controllers can be generalized to linear systemswithboth time-varying input and output delays. It is also shown that the separation principle holds for the infi-nite dimensional observer based output feedback controllers, but does not hold for the finite dimensionalones. Numerical examples are worked out to illustrate the effectiveness of the proposed approaches.
Delay differential equations,which are also knownas functionaldifferential equations, can be utilized to model many practicalphysical systems, especially those systems influenced by the effectof transmission, transportation and inertia phenomena. The con-trol of time-delay systems is challenging since these systems areinherently infinite dimensional. Existing methods that have beenwell developed for conventional control systems modeled by or-dinary differential equations are generally not directly applicable.As a result, control of time delay systems has receivedmuch atten-tion for several decades and a large number of research results havebeen reported in the literature that deal with various analysis anddesign problems (see, for instance, Chen, Fu, Niculescu, and Guan(2010), Foias, Ozbay, and Tannenbaum (1996), Fridman, Shaked,
This work was supported in part by the National Natural Science Foundationof China under grant numbers 61104124 and 61273028, by the FundamentalResearch Funds for the Central Universities under Grant HIT.BRETIII.201210, andby the National Science Foundation of the United States under grant number CMMI-1129752. The material in this paper was presented at the 51st IEEE Conference onDecision and Control (CDC) December 10–13, 2012, Maui, Hawaii, USA. This paperwas recommended for publication in revised form by Associate Editor Akira Kojimaunder the direction of Editor Ian R. Petersen.
and Liu (2009), Gu, Kharitonov, and Chen (2003), He, Wang, Lin,and Wu (2007), Karafyllis (0000), Lam, Gao, and Wang (2007) andZhang, Zhang, and Xie (2004) and the references therein).
State feedback control is very powerful for both ordinary dif-ferential equations and functional differential equations as the fullinformation of the state vectors is assumed to be accessible forfeedback. Therefore, if the state vectors aremeasurable, state feed-back is the best choice. It is under the assumption of accessibil-ity of state vectors that numerous results on control of time delaysystems have been reported in the literature (see Chen and Zheng(2007), He et al. (2007), Xie, Fridman, and Shaked (2001), Xu, Lam,and Yang (2001), Yakoubi and Chitour (2007) and Zhou, Gao, Lin,and Duan (2012) and the references cited there). State feedbackis particularly efficient for linear systems with input delay by us-ing the predictor feedback (Furukawa & Shimemura, 1983; Misaki,Uchida, Azuma, & Fujita, 2004). However, in many real world con-trol systems, only the measured output information, rather thanthe full state information, is available for feedback. As the abilityof static output feedback is generally limited (see Cao, Lam, andSun (1998) for more detailed introduction), it is more realistic touse an observer based output feedback controller, which is a dy-namic output feedback controller that estimates the system stateson-line. Therefore, from the practical point of view, observer basedoutput feedback design is important.
Several results on the observer based output feedback control oftime delay systems are available in the literature (see, for example,Bhat and Koivo (1976), Krstic (2009), Leyva-Ramos and Pearson
2040 B. Zhou et al. / Automatica 49 (2013) 2039–2052
(2000) and Sun (2002) and the references therein). One of themost remarkable results is on the observer based output feedbackcontrol of linear systems with both multiple input and multipleoutput delays (Watanabe & Ito, 1981). The underlying observersare constructed by using the predictor approach (Bekiaris-Liberis& Krstic, 2011; Kojima, Uchida, Shimemura, & Ishijima, 1994;Krstic, 2010a; Mazenc, Niculescu, & Krstic, 2012; Olbrot, 1978) andhave been generalized to more general cases in Klamka (1982).These observers, as designed in, Klamka (1982), Kojima et al.(1994) andWatanabe and Ito (1981), are naturally the dual resultsof the predictor feedback for linear systems with input delays(Olbrot, 1978). The predictor feedback can also be interpretedas model reduction based controllers (Artstein, 1982) and finitespectrum assignment (Manitius & Olbrot, 1979). However, dual tothe predictor feedback, the predictor based observer designed inKlamka (1982) and Watanabe and Ito (1981) involves distributedterms that are integration of the past output signals (Artstein,1982; Krstic, 2010a; Manitius & Olbrot, 1979). As a result, if theopen-loop system is not exponentially stable, the predictor basedobserver designed in Klamka (1982) and Watanabe and Ito (1981)can only be implemented via approximating the integral termswith a sum of point-wise delays by numerical quadrature rulesuch as rectangular, trapezoidal and Simpson’s rules (Manitius &Olbrot, 1979; Mondie & Michiels, 2003). However, even in thecase of state feedback, the effect of such a semi-discretizationon the asymptotic stability of the closed-loop system is verycomplicated (Van Assche, Dambrine, Lafay, & Richard, 1999). Moredetailed exposition of the problems encountered in implementingthe distributed terms in the predictor based controllers canbe found in Mirkin (2004), Mondie and Michiels (2003) andZhou, Lin, and Duan (2011a, 2012). We mention that for linearsystems with input and output delays, the standard four-blockoutput feedback H∞ control problem was solved in Meinsma andMirkin (2005) by treating the multiple delay operator as a specialseries connection of the adobe delay operators.
To avoid the implementation problem for the predictor basedcontrollers, we recently proposed in Zhou, Lin, Duan (2012) (seealso Zhou et al. (2011a)) a new approach named truncated pre-dictor feedback (TPF). The idea of the TPF is that, provided theopen-loop system satisfies some assumptions, and with the pa-rameterization of the nominal feedback gain in the predictor basedfeedback controllers by a positive scalar γ , the distributed term inthe predictor based controller is a high order infinitesimal with re-spect to γ and can thus be safely neglected when the value of γ issmall enough (Lin & Fang, 2007; Zhou et al., 2011a; Zhou, Lin, Duan,2012). This TPF approach has been proven to be very effective eventhe delay in the system are time-varying (Zhou et al., 2011a; Zhou,Lin, Duan, 2012) and distributed (Zhou, Gao et al., 2012).
The aim of the present paper is to generalize the idea of theTPF approach in Zhou et al. (2011a); Zhou, Lin, Duan (2012)to the design of observer based output feedback controllers forlinear systems with both input and output delays. As a result, theimplementation difficulty inherent in the predictor based outputfeedback controllers proposed inKlamka (1982) andWatanabe andIto (1981) is completely avoided. In particular, we will proposetwo classes of observer based output feedback controllers byusing the TPF approach. The first type of observer based outputfeedback controllers can be regarded as the generalization of theTPF designed in Zhou et al. (2011a); Zhou, Lin, Duan (2012) tothe design of observers with the help of the separation principle.However, this class of observer based output feedback controllersare infinite dimensional and may still be hard to implement ifthe open-loop system contains distributed delays. To avoid thisproblem, another class of finite dimensional observer based outputfeedback controllers are proposed. This class of finite dimensionalcontrollers are very easy to implement as only the input and
output vectors themselves are required to be used. The derivationof this class of finite dimensional observer based output feedbackcontrollers is highly nontrivial as the separation principle no longerholds. Indeed, an involved stability analysis should be carried out.A detailed numerical example shows that a finite dimensionalobserver based output feedback controller outperforms the infinitedimensional one.
The remainder of this paper is organized as follows. The prob-lem formulation and some preliminary results are given in Sec-tion 2. The infinite dimensional and finite dimensional observerbased output feedback controllers by the TPF approach are thenrespectively studied in Sections 3 and 4. A numerical example isgiven in Section 5 to show the effectiveness of the proposed de-sign. Finally, Section 6 concludes the paper.
Notation. The notation used in this paper is fairly standard. For avector u ∈ Rm, we use ∥u∥∞ to denote the ∞-norm of u. For amatrix A ∈ Rn×n, AT and ∥A∥ are respectively its transpose and 2-norm. For a positive scalar τ , let Cn,τ = C ([−τ , 0] ,Rn) denote theBanach space of continuous vector functions mapping the interval[−τ , 0] into Rn with the topology of uniform convergence, and letxt ∈ Cn,τ denote the restriction of x (t) to the interval [t − τ , t]translated to [−τ , 0], that is, xt (θ) = x (t + θ) , θ ∈ [−τ , 0]. Fi-nally, for a linear system characterized by thematrix pair (A, B), wesay that it is asymptotically null controllable with bounded con-trols (ANCBC) if (A, B) is stabilizable in the ordinary linear systemstheory sense and all the eigenvalues of A are located on the closedleft-half plane.
2. Problem formulation and preliminary results
2.1. Problem formulation
Consider the following linear system with multiple delays inboth the inputs and the outputs
x (t) = Ax (t)+p
i=1
Biu (t − hi) ,
y (t) =q
j=1
Cjxt − lj
,
(1)
where A ∈ Rn×n, Bi ∈ Rn×m, i ∈ I [1, p], and Cj ∈ Rr×n, j ∈ I [1, q],are constant matrices, and hi, i ∈ I [1, p], and lj, j ∈ I [1, q], areknown nonnegative constant scalars representing respectively theinput delays and the output delays. Without loss of generality, weassume that
0 ≤ h1 < h2 < · · · < hp = h, (2)
0 ≤ l1 < l2 < · · · < lq = l. (3)
The initial conditions for system (1) are assumed to be x0 (θ) ,∀θ ∈[−l, 0], and u0 (θ) ,∀θ ∈ [−h− l, 0) (Watanabe & Ito, 1981).
We suppose that u and y are measurable, but x and the initialfunctions x0 (θ) and u0 (θ) are unknown and unmeasurable, whichare the standard assumptions on observer based output feedbackscheme (Watanabe & Ito, 1981). In this paper, we are interestedin the design of observer based output feedback stabilizing con-trollers that are easy to implement for the time-delay system in (1).
Associated with the time-delay system in (1), we define twoconstant matrices
B. Zhou et al. / Automatica 49 (2013) 2039–2052 2041
It is well known that, the time-delay system in (1) is stabilizableand detectable if and only if (A, B) is stabilizable and (A, C) isdetectable in the ordinary linear systems theory sense (Olbrot,1972, 1978; Watanabe & Ito, 1981). These two conditions arenecessary and sufficient for the existence of an output feedbackstabilizing controller (Watanabe & Ito, 1981) and thus are assumedto be satisfied in this paper.
The time-delay system in (1) is a special case of the followinggeneral time-delay system with both point delays and distributeddelays in its inputs and outputs (Klamka, 1982)
x (t) = Ax (t)+p
i=1
Biu (t − hi)+
0
−h0B0 (s) u (t + s) ds,
y (t) =q
j=1
Cjxt − lj
+
0
−l0C0 (s) x (t + s) ds,
(5)
where A ∈ Rn×n, Bi ∈ Rn×m, i ∈ I [1, p] , Cj ∈ Rr×n, j ∈ I [1, q] ,hi, i ∈ I [1, p], and lj, j ∈ I [1, q], are as defined as in system (1),and B0 (t) : [−h0, 0] → Rn×m and C0 (t) : [−l0, 0] → Rr×n arepiece-wise continuous matrix functions with integrable elements.In this case, the matrices B and C defined in (4) are respectivelyreplaced by the following ones (Klamka, 1982)
B = Bhi
pi=0
=
pi=1
e−AhiBi +
0
−h0eAsB0 (s) ds,
C = C
ljqj=0
=
qj=1
Cje−Alj + 0
−l0C0 (s) eAsds.
(6)
Similarly to system (1), if we are interested in output feedback sta-bilization of system (5), it should be assumed that (A, B) is stabiliz-able and (A, C) is detectable in the ordinary linear systems theorysense. In this paper, for notational simplicity, we will mainly con-sider the delay system in (1) and will point out the possibility ofgeneralizing the obtained results to the general case (5).
2.2. State feedback and observer based output feedback by prediction
It is well known that, this class of delay systems can be reducedto the linear system
χ (t) = Aχ (t)+ Bu (t) , ∀t ≥ h, (7)
where χ (t) is a new state vector defined by
χ (t) = x (t)+p
i=1
0
−hie−A(hi+s)Biu (t + s) ds, ∀t ≥ h. (8)
This technique is known as prediction (Krstic, 2010a), model re-duction (Artstein, 1982), or finite spectrum assignment (Manitius& Olbrot, 1979). Hence, if the state vector x is available for feed-back, by designing a nominal feedback for the transformed system(7) as
u (t) = −Fχ (t)
= −F
x (t)+
pi=1
0
−hie−A(hi+s)Biu (t + s) ds
, (9)
where t ≥ h and F is such that A− BF is asymptotically stable, theclosed-loop system is equivalent to a finite dimensional linear sys-tem and thus possesses only a finite number of spectrum (Manitius& Olbrot, 1979).
If the state of system (1) is not available for feedback, anobserver should be constructed to estimate the state vector byusing the output signal of the system. By noticing that the predictorbased feedback (9) does not use the state x (t) directly but theauxiliary state χ (t), reference (Watanabe & Ito, 1981) proposed a
novel observer to construct the auxiliary stateχ (t) instead of x (t).To introduce this result, we denote an auxiliary output as
ω (t) =q
j=1
Cje−Aljp
i=1
0
−(lj+hi)e−A(hi+s)Biu (t + s) ds+ y (t) ,
∀t ≥ l+ h. (10)
Since the input and output signals of system (1) are assumed to bemeasurable, such signal ω (t) can be computed accordingly.
Lemma 1 (Watanabe and Ito (1981)). The linear time-delay systemin (1) is reduced to the following delay-free systemχ (t) = Aχ (t)+ Bu (t) ,ω (t) = Cχ (t) , ∀t ≥ l+ h. (11)
Based on this lemma, the conventional observer based outputfeedback controller for the original time-delay system (1) can beconstructed easily asz (t) = Az (t)+ Bu (t)+ L (ω (t)− Cz (t)) ,u (t) = −Fz (t) , z (0) = z0, ∀t ≥ 0, (12)
where F and L are such that A − BF and A − LC are bothasymptotically stable, and
ω (τ) =
qj=1
Cje−Aljp
i=1
0
−(lj+hi)e−A(hi+s)Biu0 (τ + s) ds
+ y (τ ) , ∀τ ∈ [0, l+ h). (13)
However, both the state feedback in (9) and the observer basedoutput feedback in (12) are infinite dimensional and are gener-ally very hard to implement since the control u (t) and the aux-iliary output ω (t) involve the past information of u (t), namely,u (t + θ) , θ ∈ [−l− h, 0]. In fact, it has been shown in Olbrot(1978) and Watanabe and Ito (1981) that if A contains poles thatare not in the open left-half plane, obtaining u (t) and ω (t) as anoutput of a process model makes the overall system internally un-stable since the unstable modes of A are uncontrollable and un-observable. As a result, such controllers can only be implementedby approximating the integral terms involving u (t + θ) , θ ∈[−l− h, 0], with a numerical quadrature rule such as the rectangu-lar, trapezoidal or Simpson’s rule (Manitius & Olbrot, 1979). How-ever, even for the state feedback case, it has been demonstrated inVan Assche et al. (1999) with a scalar example that, for some pre-scribed system parameters, the control law approximated by nu-merical quadrature methods such as the Simpson rule may lose itsability to stabilize system (1) no matter how precise the approxi-mation is. A careful study of this phenomenon has been carried outin Mirkin (2004).
2.3. An assumption
Recently, we have proposed a truncated version of the predic-tor based controller (9) under the assumption that (A, B) is asymp-totically null controllable with bounded controls (ANCBC). A linearsystem (A, B) is ANCBC if it is stabilizable in the usual linear sys-tems theory sense and all the eigenvalues of A are located in theclosed left-half plane. As the stable modes of A does not affect thestabilizability of system (1), such a condition can be replaced by thefollowing one (Zhou, Lin, & Duan, 2010; Zhou et al., 2011a; Zhou,Lin, Duan, 2012):
Assumption 1. Thematrix pair (A, B) ∈Rn×n× Rn×m
is control-
lable with all the eigenvalues of A being on the imaginary axis.
2042 B. Zhou et al. / Automatica 49 (2013) 2039–2052
Under this condition, if the feedback gain F = F (γ ) : (0, 1] →Rm×n is properly designed such that
limγ→0+
F (γ ) = 0, limγ→0+
1γ∥F (γ )∥ <∞, (14)
the distributed terms in (9) are dominated by the linear term−Fx (t) and thus may be safely neglected in u (t) when γ issufficiently small (Zhou et al., 2011a; Zhou, Lin, Duan, 2012). Asa result, the predictor based feedback law (9) can be truncated asu (t) = −F (γ ) x (t). The following result has been proven in Zhouet al. (2010) regarding the stability of the closed-loop systemunderthe truncated predictor feedback (TPF).
Lemma 2. Assume that (A, B) satisfies Assumption 1. Then thereexists a number γ ∗ = γ ∗
hi
pi=1
such that the following TPF
u (t) = −F (γ ) x (t) = −BTP (γ ) x (t) , (15)
asymptotically stabilizes system (1) for all γ ∈ (0, γ ∗]. Here P (γ ) isthe unique positive definite solution to the following algebraic Riccatiequation (ARE)
ATP + PA− PBBTP = −γ P. (16)
The main advantage of the TPF (15) over the predictor feedback(9) is that the numerical problem encountered in the implementa-tion of the integral (distributed) terms in (9) is entirely avoided. Inthis paper, under Assumption 1, by utilizing the predictor basedobserver in (12) and by following the TPF approach, we willpresent observer based output feedback controllers that are easy toimplement.
Assumption 1 is necessary for achieving stabilization in thepresence of delays that are arbitrarily large but bounded. Indeed, asimple first order system with a single input delay and a positiveopen loop pole can be found in Lin and Fang (2007), which wasshown to be not stabilizable by any feedback law if the delay in theinput exceeds a certain value.
3. Infinite dimensional observer based output feedback
3.1. Multiple output delays
Motivated by the results obtained in Zhou et al. (2010), namely,Lemma 2, we propose the following observer based output feed-back controller (see Fig. 1)
z (t) = Az (t)+p
i=1
Biu (t − hi)
+L
y (t)−
qj=1
Cjzt − lj
,
u (t) = −Fz (t) , ∀t ≥ 0,
(17)
where F and L are gains to be specified. The initial conditions arez0 (θ) ,∀θ ∈ [−l, 0] , and u0 (θ) ,∀θ ∈ [−h, 0). Notice that thefirst two terms in the first equation of (17) are copies of the originaltime-delay system (1), which helps us to get a neat error system.Notice also that, though this observer is also infinite dimensional,it is easier to implement than the infinite dimensional observerin (12) since only the past information of u (t) at the time pointst − hi, i ∈ I [1, p], and the past information of z (t) at the timepoints t − lj, j ∈ I [1, q], are required.
Fig. 1. Infinite dimensional observer based output feedback.
Theorem 1. Assume that (A, B) satisfies Assumption 1 and (A, C) isobservable. Let F = F (γ ) be defined in (15) and L = L (ρ) = QCT
with Q being the unique positive definite solution to the following ARE
AQ + QAT− QCTCQ = −ρQ , (18)
where ρ > 0. Then for any given arbitrarily large yet bounded inputdelays hi
pi=1 and output delays
ljqj=1, there exist two scalars
γ ∗ = γ ∗hi
pi=1
, ρ∗ = ρ∗
ljqj=1
, (19)
such that the observer based output feedback controller (17) asymp-totically stabilizes system (1) for all γ ∈ (0, γ ∗] and ρ ∈ (0, ρ∗].
Proof. Denote e (t) = x (t) − z (t). Then it follows from (1) and(17) that
e (t) = Ae (t)− L
q
j=1
Cjet − lj
. (20)
Moreover, with the control u (t) defined in (17), the closed-loopsystem reads
x (t) = Ax (t)−p
i=1
BiFz (t − hi)
=
Ax (t)−
pi=1
BiFx (t − hi)
−
pi=1
BiFe (t − hi) . (21)
The characteristic polynomial of the closed-loop system consistingof (20) and (21) is
∆ (s) = ∆b (s, γ )∆c (s, ρ) , (22)
where∆b (s, γ ) and∆c (s, ρ) are respectively defined by
∆b (s, γ ) , det
sIn −
A−
pi=1
Bie−hisF (γ )
, (23)
∆c (s, ρ) , det
sIn −
AT−
qj=1
CTj e−ljsLT (ρ)
. (24)
Since (A, B) satisfies Assumption 1, it follows from Lemma 2 thatthere exists a scalar γ ∗ = γ ∗
hi
pi=1
such that all of the zeros of
the characteristic quasi-polynomial∆b (s, γ ) are on the open left-half plane for all γ ∈ (0, γ ∗]. Similarly, by the duality principle,there exists aρ∗ = ρ∗(
ljqj=1) such that all the zeros of∆c (s, ρ) =
0 are on the open left-half plane for all ρ ∈ (0, ρ∗]. The proof iscompleted.
B. Zhou et al. / Automatica 49 (2013) 2039–2052 2043
It follows that a separation principle exists in the design ofthe observer based output feedback controller (17), namely, thefeedback gains F and L can be designed separately. Moreover, boththe gains L and F fall into the category of lowgain feedback, namely,larger values of hi
pi=1 (
ljqj=1) allow only smaller values of γ (ρ),
and consequently, smaller values of ∥F∥ (∥L∥) (Zhou et al., 2010).
Remark 1. In (17), not the observer state z (t) but the delayedobserver states z
t − lj
, j ∈ I [1, q], are fed back, which is not
desirable since, on the one hand, the delay effect will degrade theperformances of the observer (this is why L should be a low gain);and on the other hand, it makes the implementation of this ob-server expensive.
Remark 2. The observer based output feedback (17) can begeneralized to the general delay system (5). However, in such case,this observer is even harder to implement as it needs to implementthe terms
0−h0
B0 (s) u (t + s) ds and 0−l0
C0 (s) z (t + s) ds, whichcan only be obtained via numerical approximation.
3.2. A single output delay
If there is only a single delay in the output of system (1), namely,q = 1, an alternative observer can be designed. To introduce thisresult, we rewrite the delay system in (1) as followsx (t) = Ax (t)+
pi=1
Biu (t − hi) ,
y (t) = C1x (t − l1) ,(25)
where C1 ∈ Rr×n is a constant matrix, and l1 > 0 is a known non-negative scalar representing the output delay. The observer for thissystem is constructed as follows:
z (t) = Az (t)+p
i=1
Biu (t − l1 − hi)
+L (y (t)− C1z (t)) ,u (t) = −FeAl1z (t) ,
(26)
where L is a given constant matrix such that A − LC1 is Hurwitz,F = F (γ ) is as defined in (15), and the initial conditions are as-sumed to be z (0) and u′0 (θ) ,∀θ ∈ [− (l1 + h) , 0).
Theorem 2. Assume that (A, B) satisfies Assumption 1 and (A, C1)is observable. Then for any given arbitrarily large yet bounded inputdelays hi
pi=1 and output delay l1, there exists a scalar γ ∗ = γ ∗
hipi=1 , l1
such that the observer based output feedback con-
troller (26) asymptotically stabilizes system (25) for all γ ∈ (0, γ ∗].
Proof. By denoting a new state vector
ξ (t) = x (t − l1) , ∀t ≥ l1, (27)
system (25) can be rewritten asξ (t) = Aξ (t)+p
i=1
Biu (t − l1 − hi) ,
y (t) = C1ξ (t) , ∀t ≥ l1,(28)
which is a linear system having only input delays. Let e (t) =ξ (t) − z (t) = x (t − l1) − z (t) ,∀t ≥ l1. Then it follows from(26) and (28) that
e (t) = (A− LC1) e (t) , ∀t ≥ l1, (29)
and similarly, by letting u (t) = −F ′z (t) ,
x (t) = Ax (t)−p
i=1
BiF ′z (t − hi)
= Ax (t)−p
i=1
BiF ′x (t − l1 − hi)
+
pi=1
BiF ′e (t − hi) , ∀t ≥ l1. (30)
Since A − LC1 is Hurwitz, it follows from (29) that system (30) isasymptotically stable if and only if
x′ (t) = Ax′ (t)−p
i=1
BiF ′x′ (t − l1 − hi) , ∀t ≥ l1, (31)
is. According to Lemma 2, if F ′ =B′T P ′, where
B′ ,p
i=1
e−A(hi+l1)Bi = e−Al1B, (32)
and P ′ is the unique positive definite solution to the following ARE
ATP ′ + P ′A− P ′B′B′T P ′ = −γ P ′, (33)
then there exists a γ ∗ = γ ∗hi
pi=1 , l1
such that (31) is asymp-
totically stable for all γ ∈ (0, γ ∗]. Notice that (33) is equivalent to(16) by denoting P = e−A
T l1P ′e−Al1 . Consequently,
F ′ =B′T P ′ = BTe−A
T l1eAT l1PeAl1 = BTPeAl1 = FeAl1 . (34)
The proof is completed.
The separation principle clearly holds for this class of observerbased controllers. In fact, it follows from (29) and (31) that theobserver gain L and the feedback gain F can be designed separately,and, moreover, the zero set of the closed-loop system is the unionof the zero set of A−LC1 and the zero set of the characteristic quasi-polynomial of system (31), which is independent of L. Thus, theadvantage of the observer in (26) over the observer in (17) is thatthe observer gain L in the former no longer needs to be of the lowgain type.
3.3. Single input and single output time-varying delays
In this subsection, the state transformation technique impliedby (27) is utilized to study observer based output feedback controlof the following linear system with time-varying delaysx (t) = Ax (t)+ B1u (φ (t)) ,y (t) = C1x (ϕ (t)) ,
(35)
where A ∈ Rn×n, B1 ∈ Rn×m, C1 ∈ Rr×n are constant matrices, andφ (t) , ϕ (t) : R+ → R are continuously differentiable functionsthat incorporate the input and output delays, respectively (Krstic,2010b). The functions φ (t) and ϕ (t) can be defined in a morestandard form
φ (t) = t − h (t) , ϕ (t) = t − l (t) , (36)
where h (t) , l (t) : R+ → R+ are the time-varying delays. Similarto Krstic (2010b) and Zhou et al. (2011a); Zhou, Lin, Duan (2012),we assume that φ (t) and ϕ (t) satisfy the following assumption.
2044 B. Zhou et al. / Automatica 49 (2013) 2039–2052
Assumption 2. The functions φ, ϕ : R+ → R are continuouslydifferentiable, invertible and exactly known functions and suchthat, for all t ∈ R+,
0 < φ ≤ φ (t) ≤ φ <∞, (37)
0 < ϕ ≤ ϕ (t) ≤ ϕ <∞, (38)
and the delays h (t) and l (t) are bounded, namely, there exist twofinite, yet arbitrarily large, numbers h and l such that
0 ≤ h (t) ≤ h, 0 ≤ l (t) ≤ l, ∀t ∈ R. (39)
It follows that the inverse functions of φ (t) and ϕ (t), namely,φ−1 (t) andϕ−1 (t) exist and are such thatφ−1 (t)−t andϕ−1 (t)−t are bounded (Zhou et al., 2011a; Zhou, Lin, Duan, 2012), namely,there exist two finite positive constants αφ and αϕ , such that
0 ≤ φ−1 (t)− t ≤ αφ, 0 ≤ ϕ−1 (t)− t ≤ αϕ, ∀t ∈ R. (40)
Define a new function as
ψ (t) = ϕ (φ (t)) . (41)
Then it is not difficult to verify that ψ (t) has all the properties ofφ (t), particularly, ψ−1 (t)− t is bounded for all t and
|ψ (t)− t| = |t − h (t)− l (t − h (t))− t| ≤ h+ l,∀t ∈ R. (42)
Our observer based output feedback controller for system (35)is designed asz (t) = ϕ (t) (Az (t)+ B1u (ψ (t))+ L (y (t)− C1z (t))) ,u (t) = −F (t, γ ) z (t) = −BT
1PeA(ψ−1(t)−t)z (t) ,
(43)
where the initial condition is assumed to be z0 (θ) ,∀θ ∈−h, 0
,
u0 (θ) ,∀θ ∈ [−h+ l
, 0), L is a constantmatrix such that A−LC1
is Hurwitz and P = P (γ ) is the unique positive definite solutionto the following ARE
ATP + PA− PB1BT1P = −γ P. (44)
We notice that, since ψ−1 (t) − t is bounded for all t, the time-varying gain matrix F (t, γ ) is also bounded for all t.
To prove that controller (43) indeed stabilizes system (35),we first recall the following result regarding state feedbackstabilization of (35) by TPF.
Lemma 3 (Zhou et al. (2011a); Zhou, Lin, Duan (2012)). Let φ (t)satisfy Assumption 2. Assume that (A, B1) is controllable and all of theeigenvalues of A are on the imaginary axis. Then there exists a numberγ ∗ = γ ∗
hsuch that the following TPF
u (t) = −BT1P (γ ) e
A(φ−1(t)−t)x (t) , ∀γ ∈ (0, γ ∗], (45)
asymptotically stabilizes system (35), where P is the unique positivedefinite solution to the ARE (44).
Then we can prove the following result.
Theorem 3. Let Assumption 2 be satisfied. Assume that (A, B1) iscontrollable, (A, C1) is observable, and all of the eigenvalues of A areon the imaginary axis. Then there exists a number γ ∗ = γ ∗
h+ l
such that the observer based output feedback (43) asymptotically sta-bilizes system (35).
Proof. Similar to (27), we define a new state vector
ξ (t) = x (ϕ (t)) . (46)
We notice that such a technique has also been used in Krstic(2010b) to design distributed observers for a class of linear systemswith time-varying output delay but no input delay. Then, by using(35), we have
ξ (t) = ϕ (t) (Aξ (t)+ B1u (ψ (t))) , (47)
and y (t) = C1ξ (t). Let e (t) = ξ (t) − z (t). Then it follows from(43) and (47) that
e (t) = ϕ (t) (A− LC1) e (t) . (48)
Let V (e (t)) = eT (t)Qe (t)where Q > 0 solves
(A− LC1)T Q + Q (A− LC1) = −In. (49)
We can compute
V (e (t)) = −ϕ (t) ∥e (t)∥2 ≤ −ϕ ∥e (t)∥2 , (50)
which implies that system (48) is asymptotically stable. On theother hand, we can obtain
x (t) = Ax (t)− B1F (t, γ ) x (ψ (t))+ B1F (t, γ ) e (φ (t)) . (51)
As the dynamics of e (t) is exponentially stable and F (t, γ ) isbounded for all t , system (51) is asymptotically stable if and only if
x′ (t) = Ax′ (t)− B1F (t, γ ) x′ (ψ (t)) , (52)
is. However, the stability of (52) is guaranteed by Lemma 3.
It is not difficult to verify that (43) reduces to (26), andTheorem 3 reduces to Theorem 2, if
p = 1, φ (t) = t − h1, ϕ (t) = t − l1. (53)
Remark 3. We can see that in (43) the observer gain L and thefeedback gain F (t, γ ) are designed independent of each other.Moreover, from (52) we see that the determination of the valueof the parameter γ in F (t, γ ) is also independent of L. Theseindicate that the separation principle holds for the observer basedcontroller (43).
3.4. Determinations of the design parameters γ and ρ
In this subsection, we discuss how to choose the parametersγ in (17) and (26), and the parameter ρ in (17) such that theclosed-loop system is asymptotically stable. For simplicity, we onlyconsider the determination of γ in (17) since the others can beconsidered similarly.
From the proof of Theorem1we see that the closed-loop system(21) is stable if and only if all the zeros of∆b (s, γ ) = 0 are locatedin the open left-half plane. For a given γ ≥ 0, denote the real partsof the right-most zeros of∆b (s, γ ) = 0 by
λbmax (γ ) = max Re s : ∆b (s, γ ) = 0 . (54)
Then the closed-loop system is asymptotically stable if and onlyif λbmax (γ ) < 0. Moreover, it is well known that the convergencerate of the closed-loop system (21) is completely determined byλbmax (γ ), namely, the smaller the value of λbmax (γ ) is, the fasterthe state converges to the origin (Hale, 1977).
According to the results in Zhou et al. (2010), there is a valueγsup > 0 such that λbmax (γ ) < 0,∀γ ∈
0, γsup
, namely,
0, γsup
is themaximal interval around 0 such that∆b (s, γ ) = 0has no un-stable zeros. Therefore, by continuity of zeros of quasi-polynomials(Ruan & Wei, 2003), there exists a value γopt ∈
0, γsup
such that
B. Zhou et al. / Automatica 49 (2013) 2039–2052 2045
λbmax (γ ) is minimized at γ = γopt. Denote the minimal value byλbmaxmin, namely,
λbmaxmin = minγ∈(0,γsup)
λbmax (γ )
= min
γ∈(0,γsup)max Re s : ∆b (s, γ ) = 0 . (55)
Then we can see that λbmaxmin is themaximal convergence rate thatthe observer based output feedback (17) can achieve. From thispoint of view, we should choose γ = γopt in the observer basedoutput feedback controller (17).
Regarding the computation of λbmax (γ ), we can adopt theefficient software package DDE-BIFTOOL (Engelborghs, Luzyanina,& Samaey, 2001). In practice, we can choose
γ = k∆γ , k = 0, 1, . . . ,N, (56)
where∆γ is a sufficiently small number denoting the step size andN is chosen as the minimal number such that λbmax
N∆γ
= 0.
According the computational results of λbmax (γ ), the optimal valueγopt and the corresponding λbmaxmin can be obtained accordingly(Zhou, Gao et al., 2012; Zhou et al., 2011a; Zhou, Lin, Duan, 2012).
Remark 4. In practice, it may be difficult to find γopt. However, itis relatively easy to compute γsup by applying the trial-and-errormethod. This can be done in the following steps: (1) Choose γa = 0.(2) Choose a large enough number γ b such that λbmax (γb) definedin (54) satisfies λbmax (γb) > 0. (3) Let γt = 1
2 (γa + γb) and com-pute λbmax (γt) defined in (54). If λbmax (γt) < 0, then γa ← 1
2(γa + γb); otherwise, γb ← 1
2 (γa + γb). (4) Repeat the third stepuntil |γa − γb| ≤ ε, where ε is a prescribed small number (for ex-ample, ε = 10−6). (5) Set γsup = γa.
4. Finite dimensional observer based output feedback
In this section, based on observer (12), we will propose a novelfinite dimensional observer based output feedback controller whichonly utilizes the current information of the output and input ofsystem (1). The basic idea can be explained as follows.We considerthe distributed term in ω (t), namely,
η (t) =q
j=1
Cje−Aljp
i=1
0
−(lj+hi)e−A(hi+s)Biu (t + s) ds, (57)
where t ≥ l + h. If F = F (γ ) : (0, 1] → Rm×n is such that (14) issatisfied, as u (t) = −Fz (t), it is possible to reduce the value of γsuch that ∥η (t)∥ can be reduced to a sufficiently ‘‘small’’ level, and,consequently, the effect of η (t) on the first equation of (12) can beignored safely. As a result, the predictor based infinite dimensionalobserver (12) reduces to the following finite dimensional one (seeFig. 2)z (t) = Az (t)+ Bu (t)+ L (y (t)− Cz (t)) ,u (t) = −Fz (t) , ∀t ≥ 0, (58)
where the initial condition is assumed to be z (0). It is clear thatthis observer is very easy to implement.
As we have explained in Section 2.3, a necessary and sufficientcondition for the existence of a stabilizing feedback gain F satisfy-ing (14) is that (A, B) is ANCBC, which, without loss of generality,is equivalent to Assumption 1.
The remaining question is whether the finite dimensionalobserver based output feedback controller (58) can indeed stabilizethe original time-delay system (1). The answer to this question ishighly nontrivial since the closed-loop system consisting of (1) and(58) is infinite dimensional and coupled.
Fig. 2. Finite dimensional observer based output feedback.
4.1. Stability of the closed-loop system
Regarding the stability of the closed-loop system by applyingthe finite dimensional observer based output feedback controller(58), we can prove the following result.
Theorem 4. Assume that (A, B) satisfies Assumption 1 and (A, C) isobservable. Let L be such that A− LC is Hurwitz and F be as designedin (15). Then for any given arbitrarily large yet bounded delays hi
pi=1
andljqj=1, there exists a
γ ∗ = γ ∗L, hi
pi=1 ,
ljqj=1
, (59)
such that the finite dimensional observer based output feedback(58) stabilizes system (1).
Proof. Let χ (t) , t ≥ h, be defined as in (8) and e (t) = χ (t) −z (t) , t ≥ l + h. Hence from Lemma 1 we know that χ (t) , t ≥l+h, satisfies (11). Since the closed-loop system is linear and time-invariant, without loss of generality, we prove the stability withinitial time t0 = l+ h.
It follows from (8), (10), (11) and (58) that
e (t) = Ae (t)− L (y (t)− Cz (t))= Ae (t)− L (Cχ (t)− η (t)− Cz (t))= Ae (t)− LC (χ (t)− z (t))+ Lη (t)= (A− LC) e (t)+ Lη (t) , ∀t ≥ l+ h. (60)
By using the control law in (58), the closed-loop system can beobtained from the open-loop system (11) as follows
Associatedwith the unique positive definite solution to the ARE(16), we define, for all t ≥ l+ h,
V1 (χ (t)) = χT (t) Pχ (t) , V2 (e (t)) = eT (t)Qe (t) , (62)
where Q > 0 solves the following Lyapunov equation
(A− LC)T Q + Q (A− LC) = −In. (63)
Then the time-derivative of V1 (χ (t)) along the trajectories ofsystem (61) is given by, for all t ≥ l+ h,
V1 (χ (t)) = χT (t)(A− BF)T P + P (A− BF)
χ (t)
+χT (t) PBFe (t)+ eT (t) F TBTPχ (t)= −γχT (t) Pχ (t)− χT (t) PBBTPχ (t)
2046 B. Zhou et al. / Automatica 49 (2013) 2039–2052
+χT (t) PBFe (t)+ eT (t) F TBTPχ (t)≤ −γχT (t) Pχ (t)− χT (t) PBBTPχ (t)+χT (t) PBBTPχ (t)+ eT (t) F TFe (t)
= −γχT (t) Pχ (t)+ eT (t) F TFe (t)
≤ −γχT (t) Pχ (t)+ nγ eT (t) Pe (t) , (64)
wherewe have used Lemma 4 in the Appendix. Similarly, the time-derivative of V2 (e (t)) along the trajectories of system (60) can becomputed as, for all t ≥ l+ h,
V2 (e (t)) = eT (t)(A− LC)T Q + Q (A− LC)
e (t)
+ eT (t)QLη (t)+ ηT (t) LTQe (t)= −∥e (t)∥2 + eT (t)QLη (t)+ ηT (t) LTQe (t)
≤ −∥e (t)∥2 +12eT (t) e (t)+ 2ηT (QL)T QLη
≤ −12∥e (t)∥2 + 2 ∥QL∥2 ηT (t) η (t) . (65)
In the following, we will simplify the term ηT (t) η (t) in (65).By Lemma 6, we have
ηT (t) η (t) ≤ q
q
j=1
ηTj (t) ηj (t)
, ∀t ≥ l+ h, (66)
where ηj (t) , j ∈ I [1, q] , t ≥ l+ h, are defined by
ηj (t) = Cje−Aj ljp
i=1
0
−(lj+hi)e−A(hi+s)Biu (t + s) ds. (67)
By using the continuous-time and discrete-time Jensen inequali-ties in Lemmas 5 and 6 again, we have
ηTj (t) ηj (t) ≤ cj
0
−(l+h)uT (t + s) u (t + s) ds, (68)
where we have defined
cj = supθ∈[0,lj]
e−Aθ2 p2 Cje−Aj lj2 ∥Bi∥
2 (l+ h) , (69)
which is a constant independent of L and γ . In view of u (t) =−Fz (t) , t ≥ l + h, and by using Lemma 6, we further have, forall t ≥ 2 (l+ h) ,
ηTj ηj ≤ cj
0
−(l+h)zT (t + s) F TFz (t + s) ds
≤ cjnγ 0
−(l+h)zT (t + s) Pz (t + s) ds
= cjnγ 0
−(l+h)V1 (χ (t + s)− e (t + s)) ds
≤ 2cjnγ 0
−(l+h)(V1 (χ (t + s))+ V1 (e (t + s))) ds
≤ 2cnγ 0
−(l+h)(V1 (χ (t + s))+ V1 (e (t + s))) ds, (70)
where c = maxj∈I[1,q]cj. Inserting (70) into (66) gives, for allt ≥ 2 (l+ h)
ηTη ≤ 2cq2nγ 0
−(l+h)(V1 (χ (t + s))+ V1 (e (t + s))) ds
, (71)
by which (65) can be written as, for all t ≥ 2 (l+ h) ,
V2 (e (t)) ≤ 4 ∥QL∥2 cq2nγ 0
−(l+h)V1 (χ (t + s)) ds
+ 4 ∥QL∥2 cq2nγ 0
−(l+h)V1 (e (t + s)) ds
−12∥e (t)∥2 . (72)
Consider the nonnegative functional
V3 (χt) = 4 ∥QL∥2 cq2nγ l+h
0
t
t−sχT (l) Pχ (l) dl
ds, (73)
where t ≥ 2 (l+ h). Its time derivative is equal to, for all t ≥2 (l+ h) ,
V3 (χt) = 4 ∥QL∥2 cq2nγ (l+ h) χT (t) Pχ (t)
− 4 ∥QL∥2 cq2nγ 0
−(l+h)V1 (χ (t + s)) ds. (74)
Similarly, the time derivative of the following functional
V4 (et) = 4 ∥QL∥2 cq2nγ l+h
0
t
t−seT (l) Pe (l) dl
ds, (75)
where t ≥ 2 (l+ h), can be computed as
V4 (et) = 4 ∥QL∥2 cq2nγ (l+ h) eT (t) Pe (t)
− 4 ∥QL∥2 cq2nγ 0
−(l+h)V1 (e (t + s)) ds. (76)
It follows from (72), (74) and (76) that
V2 (e (t))+ V3 (χt)+ V4 (et)
≤ −12∥e (t)∥2 + 4 ∥QL∥2 cq2nγ (l+ h) V1 (χ (t))
+ 4 ∥QL∥2 cq2nγ (l+ h) eT (t) Pe (t) , ∀t ≥ 2 (l+ h) . (77)
Now, for t ≥ 2 (l+ h), we choose the following Lyapunovfunctional
V (χt , et) = V1 (χ (t))+ γ (V2 (e (t))+ V3 (χt)+ V4 (et)) , (78)
whose time derivative, in view of (64) and (77), satisfies, for allt ≥ 2 (l+ h) ,
V (χt , et) = V1 (χ (t))+ γV2 (e (t))+ V3 (χt)+ V4 (et)
≤ −γ eT (t)
12In − nP − 4 ∥QL∥2 cq2nγ (l+ h) P
e (t)
− γ1− 4 ∥QL∥2 cq2nγ (l+ h)
χT (t) Pχ (t) . (79)
There clearly exists a γ ∗ = γ ∗(L, hipi=1, lj
qj=1) such that
12In − nP − 4 ∥QL∥2 cq2nγ (l+ h) P ≥
14In, (80)
1− 4 ∥QL∥2 cq2nγ (l+ h) ≥14, (81)
are satisfied for all γ ∈ [0, γ ∗]. With these inequalities, it followsfrom (79) that
V (χt , et) ≤ −14γ∥e (t)∥2 + χT (t) Pχ (t)
, (82)
where γ ∈ (0, γ ∗], and t ≥ 2 (l+ h) . Now, by Lyapunov stabilitytheorem, we know from (82) that the state (χ (t) , e (t)) for sys-tems (60) and (61) is asymptotically stable, which further implies
B. Zhou et al. / Automatica 49 (2013) 2039–2052 2047
that the dynamics of z (t) is asymptotically stable. Consequently,we get from (8) and (58) that
∥x (t)∥ ≤ ∥χ (t)∥ + sups∈[−h,0]
e−Ahi pi=1
∥BiF∥
·
0
−hi(∥χ (t + s)∥ + ∥e (t + s)∥) ds, (83)
which implies that the dynamics of x (t) is also asymptotically sta-ble. The proof is finished.
From the proof of Theorem 4 we can see that, differently fromthe infinite dimensional observer in (17), the observer gain L inthe finite dimensional observer (58) is not necessarily of the lowgain type. Moreover, although the design of the observer gain Lis independent of F , the design of F is however dependent on Lsince γ ∗ is dependent on L. This is because the error system (60)and the plant dynamics (61) are coupled with each other. Hence,the separation principle no longer holds true in the observer basedcontroller (58).
Remark 5. The finite dimensional observer (58) can be general-ized to the general time-delay system (5). In this case,we need onlyto replace the matrices B and C defined in (4) with (6).
Remark 6. The extension of the obtained results to the case of ex-ponentially unstable open loop systems is possible. A result simi-lar to Theorem 4 can be obtained without significant modificationof the proof by assuming that the delays in the systems are suffi-ciently small. However, such a result is not significant since it isa consequence of the fact that any linear controller can toleratesmall delays in the systems. What is significant is to obtain tightestimates of the upper bounds of the delays in terms of the param-eters of the system and the observer, which is highly nontrivial andentails a comprehensive further study.
4.2. Semi-global stabilization
In this subsection, we further show that the finite dimensionalobserver based output feedback (58) can even achieve semi-globalstabilization if the input of the open-loop system (1) is subjectto magnitude saturation (energy saturation), namely, the controlsignal u (t) is such that Zhou, Gao et al. (2012); Zhou, Lin, and Duan(2011b); Zhou, Lin, and Lam (2012)
supt∈[−(h+l),∞)
∥u(t)∥∞ ≤ 1,
∞
−(h+l)∥u (t)∥2 dt ≤ 1
. (84)
Theorem 5. Assume that (A, B) satisfies Assumption 1 and (A, C) isobservable. Let L be such that A− LC is Hurwitz and F be as designedin (15). Then, for any given arbitrarily large yet bounded delays h andl, any given arbitrarily large yet bounded setsΩx ⊂ Cn,l andΩz ⊂ Rn,and any initial control signals u0 (θ) ,∀θ ∈ [− (h+ l) , 0) such that
supθ∈[−(h+l),0)
∥u0(θ)∥∞ ≤ 1, 0
−(h+l)∥u0 (θ)∥
2 dθ ≤12
,
(85)
there exists a γ Ď such that the closed-loop system consisting ofsystem (1) and the observer based output feedback controller (58),where γ ∈ (0, γ Ď
], is asymptotically stable withΩx ×Ωz containedin the domain of attraction, and the control signal u (t) satisfies themagnitude saturation constraint (energy saturation constraint) (84).
Proof. We first consider the magnitude saturation case. By usingLemma 4, we get from (58) that, for all t ≥ 2 (l+ h),
uT (t) u (t) = zT (t) PBBTPz (t)≤ nγ (χ (t)− e (t))T P (χ (t)− e (t))≤ 2nγχT (t) Pχ (t)+ 2nγ eT (t) Pe (t)
≤ 2nγχT (t) Pχ (t)+ 2nγ ∥P∥ ∥e (t)∥2 (86)
≤ 2nγχT (t) Pχ (t)+ 2nγτγ
λmin (Q )eT (t)Qe (t)
≤ δγχT (t) Pχ (t)+ γ eT (t)Qe (t)
= δγ (V1 (χ (t))+ γ V2 (e (t)))≤ δγ V (χt , et)
≤ δγ Vχ2(l+h), e2(l+h)
, ∀γ ∈ (0, γ ∗], (87)
where V (χt , et) is defined in (78), τ = τ (γ ∗) and
δ = max2n, 2n
τ
λmin (Q )
, (88)
and we have used (82).Now consider the state vector (χ (t) , e (t)) with t ∈ [0, 2
(l+ h)]. Since the closed-loop system consisting of (1) and (58) is alinear time-delay system and its coefficient matrices are uniformlybounded for all γ ∈ [0, γ ∗], there exist two continuous functionsc (γ ) andα (γ ) ,∀γ ∈ [0, γ ∗] such that (see, for example, Theorem3.1 in Hale (1977))xtzt
c
, supθ∈[−(h+l),0]
x (t + θ)z (t + θ)
≤ c (γ ) eα(γ )tX0, (89)
where t ≥ 0 and
X0 = supθ∈[−l,0]
∥x0 (θ)∥ + supθ∈[−(h+l),0]
∥u0 (θ)∥ + ∥z (0)∥ . (90)
As Ωx and Ωz are bounded and u0 (θ) , θ ∈ [−(h + l), 0) satisfies(85), it follows from (90) that X0 is bounded. On the other hand,as α (γ ) and c (γ ) are bounded for γ ∈ [0, γ ∗], it follows from(89) that x (t) and z (t) are bounded for all t ∈ [0, 2 (l+ h)] andγ ∈ [0, γ ∗]. By (8) and noting that u (t) = −Fz (t) , t ≥ 0, we havethat χ (t) , t ∈ [l+ h, 2 (l+ h)] and e (t) = χ (t)− z (t) , t ≥ l+hare all bounded for all γ ∈ [0, γ ∗]. Consequently, by definition ofV (χt , et), we see that V
χ2(l+h), e2(l+h)
satisfies
limγ→0+
Vχ2(l+h), e2(l+h)
= 0. (91)
Hence it follows from (87) that there exists a γ ∗1 ∈ (0, γ∗] such
that, for all γ ∈ (0, γ ∗1 ],
supt∈[2(l+h),∞)
∥u(t)∥∞ ≤ supt∈[2(l+h),∞)
∥u(t)∥ ≤ 1. (92)
Now consider the control signals u (t) , t ∈ (0, 2 (l+ h)]. It followsfrom (89) that
supt∈[0,2(l+h))
∥u (t)∥ ≤ ∥F∥ supt∈[0,2(l+h))
∥z (t)∥
≤ ∥F∥ supt∈[0,2(l+h))
x (t)z (t)
≤ ∥F∥ c (γ ) eα(γ )2(l+h)X0, (93)
where X0 is defined as in (90). Since α (γ ) and c (γ ) are boundedfor all γ ∈ [0, γ ∗] and limγ→0+ ∥F∥ = 0, we get from (93) thatthere exists a γ Ď
∈ (0, γ ∗1 ] such that, for all γ ∈ (0, γ Ď],
supt∈[0,2(l+h))
∥u (t)∥∞ ≤ supt∈[0,2(l+h))
∥u (t)∥ ≤ 1. (94)
This together with (85) and (92) implies the desired result.
2048 B. Zhou et al. / Automatica 49 (2013) 2039–2052
We next consider the energy saturation case. Let γ ∗2 ∈ (0, γ∗]
be such that ∥P∥ ≤ 1,∀γ ∈ (0, γ ∗2 ]. Then by (86) we get
uT (t) u (t) ≤ 2nγχT (t) Pχ (t)+ ∥P∥ ∥e (t)∥2
≤ 2nγ
χT (t) Pχ (t)+ ∥e (t)∥2
, ∀t ≥ 0. (95)
With this, it follows from (82) that
uT (t) u (t) ≤ −8nV (χt , et) , ∀γ ∈ (0, γ ∗3 ],
∀t ≥ 2 (l+ h) . (96)
The above inequality is similar to (115) in Zhou, Lin, Duan (2012)and therefore the remaining proof is similar to the proof ofTheorem 4 in Zhou, Lin, Duan (2012) and is thus omitted forbrevity.
By using a quite similar technique, we can also show that all theinfinite dimensional observer based output feedback consideredin Section 3 can stabilize the corresponding delay systems semi-globally in the presence of magnitude saturation and energysaturation. The details are omitted for brevity.
4.3. Determination of the design parameter γ
Similar to the development in Section 3.4, we can also considerthe determination of γ in the finite dimensional observer (58).Notice that, differently from the infinite dimensional observersstudied in Section 3, the separation principle does not hold forthe finite dimensional observers (58), and, consequently, theparameter γ in (58) is dependent on L.
By denoting ϖ (t) =xT (t) , zT (t)
T, the closed-loop systemconsisting of (1) and (58) can be written as, for all t ≥ l+ h,
ϖ (t) =A 00 A− BF − LC
ϖ (t)−
pi=1
0 BiF0 0
ϖ (t − hi)
+
qj=1
0 0LCj 0
ϖt − lj
. (97)
The stability of (97) is totally determined by the right-most zerosof its characteristic quasi-polynomial (Hale, 1977),
∆ (s, γ , L) = det
sI2n −
A 00 A− BF − LC
+
pi=1
0 BiF0 0
e−his +
qj=1
0 0LCj 0
e−ljs
. (98)
For a given L, from Theorem 4 we see that there exists a maximalvalue of γ , denoted by γsup (L), such that∆ (s, γ , L) = 0 has all itszeros on the open left-half plane for all γ ∈ (0, γsup (L)) and itsright most zeros on the imaginary axis with γ = γsup (L) (it shouldbe noticed that γsup (L) can be∞). Similarly to the discussion inSection 3.4, for a given L and a γ ≥ 0, we denote
λmax (γ , L) = max Re s : ∆ (s, γ , L) = 0 . (99)The computation of λmax (γ , L) can be carried out similarly to thecomputation of λbmax (γ ) in Section 3.4. Again, by continuity of ze-ros of quasi-polynomials, there exists a γopt ∈
0, γsup (L)
such
that λmax (γ , L) is minimized at γ = γopt (L) with the minimalvalue λmaxmin (L), namely,λmaxmin (L) = min
γ∈(0,γsup(L))max Re s : ∆ (s, γ , L) = 0 . (100)
It follows that, for a given L, the maximal convergence rate of theclosed-loop system is λmaxmin (L). Since λmaxmin (L) is dependenton L which can be designed freely, it is quite hard to find an opti-mal observer gain L such that λmaxmin (L) is minimized. In practice,we may compute λmaxmin (L) for a series of L and then pick the op-timal one among them.
Remark 7. For a given L, a procedure similar to the one given inRemark 4 can also be provided for the computation of γsup (L) forthe closed-loop system. The details are omitted for brevity.
5. Numerical examples
We consider a linear system with constant input and outputdelays as in the form of (1) with
A =
−1 2 −1 0−1 2 −1 10 1 0 11 −2 1 −1
, B1 =
−1 0−1 00 01 0
,
B2 =
0 −10 00 00 −1
,(101)
C1 =1 −2 1 −1
, C2 =
0 0 −1 0
,
and h1 = 0, h2 = 1, l1 = 0, l2 = 1. For the simulation purpose,we choose the initial condition for this system as x0 (θ) = [−4, 4,5, 4]T ,∀θ ∈ [−1, 0] and u0 (θ) = [−2, 1)T, ∀θ ∈ [−2, 0).
For this system, we have λ (A) = 0, 0,±i. It can be verifiedthat neither (A, Bi) , i = 1, 2, is controllable nor
A, Cj
, j = 1, 2,
is observable. However, according to (4), we compute
B =
−1 −2−1 00 11 −1
, C =1 −1 0 0
, (102)
and it follows that (A, B) is controllable and (A, C) is observable.Consequently, Assumption 1 is fulfilled.
By solving the parametric ARE in (16) we get (see Box I),consequently, the feedback gain can be obtained as
F (γ ) =−γ 2 2γ 2
− 2γ −γ 2 γ 2
0 −γ 2− 2γ −γ 2
−γ 2− 2γ
. (104)
Similarly, the unique positive definite solution to the parametricARE (18) can be computed as Q =
qijwhere qij = qji with
q11 = ρ7− 4ρ5
− 2ρ4+ 9ρ3
+ 10ρ2+ 6ρ,
q12 = ρ7− 4ρ5
− ρ4+ 9ρ3
+ 5ρ2+ 2ρ,
q13 = ρ7+ 3ρ6
− ρ5− 8ρ4
− 6ρ3− 5ρ2
− 2ρ,q14 = 3ρ5
− 11ρ3− 6ρ2
− 2ρ,q22 = ρ7
− 4ρ5+ 9ρ3
+ 2ρ,q23 = ρ7
+ 3ρ6− ρ5
− 7ρ4− 2ρ3
− 4ρ2,
q24 = 3ρ5− 11ρ3
− 2ρ,q33 = ρ7
+ 6ρ6+ 12ρ5
+ 8ρ4+ 9ρ3
+ 2ρ2+ 2ρ,
q34 = 3ρ5+ 11ρ4
+ 3ρ3+ 5ρ2,
q44 = 14ρ3+ 2ρ,
(105)
by which the observer gain can be obtained as
L (ρ) =
−ρ4+ 5ρ2
+ 4ρ−ρ4+ 5ρ2
−ρ4− 4ρ3
− ρ2− 2ρ
−6ρ2
. (106)
We point out that L (ρ) is the unique vector such that λ (A− L(ρ) C) = −ρ,−ρ,−ρ ± i.
Wedesign two classes of observers, namely, the infinite dimen-sional observer given by (17) and the finite dimensional observergiven by (58).
B. Zhou et al. / Automatica 49 (2013) 2039–2052 2049
3)
P =
2γ + γ 3
−2γ 3+ γ 2
− 4γ 2γ + γ 3−2γ − γ 3
−2γ 3+ γ 2
− 4γ 5γ 3− 2γ 2
+ 12γ −γ 3+ 2γ 2
− 4γ 3γ 3+ γ 2
+ 6γ2γ + γ 3
−γ 3+ 2γ 2
− 4γ 2γ 3+ 2γ γ 2
− 2γ−2γ − γ 3 3γ 3
+ γ 2+ 6γ γ 2
− 2γ 2γ 3+ 2γ 2
+ 4γ
, (10
Box I.
Fig. 3. The functionsλbmax (γ ) andλcmax (ρ) associatedwith the infinite dimensional
observer (107).
Fig. 4. The state x and the observer error e = x − z associated with the infinitedimensional observer (107).
• The infinite dimensional observer takes the formz (t) = Az (t)+ B0u (t)+ B1u (t − 1)+L (ρ) (y (t)− C1z (t)− C2z (t − 1)) ,
u (t) = −F (γ ) z (t) ,(107)
where F (γ ) and L (ρ) are respectively given by (104) and (106).For the simulation purpose, the initial condition is chosen asz0 (θ) = −x0 (θ) = [4,−4,−5,−4]T ,∀θ ∈ [−1, 0].• The finite dimensional observer takes the form
z (t) = Az (t)+ Bu (t)+ L (y (t)− Cz (t)) ,u (t) = −Fz (t) , (108)
where the feedback gain F = F (γ ) and the observer gain L =L (ρ) are also respectively chosen as (104) and (106). We em-phasize that the observer gain L in (108) can be any vector suchthat A− LC is Hurwitz. Here we choose L (ρ) as in (106) for thecomparison purpose. For simulation and comparison purposes,we choose the initial condition as z (0) = [4,−4,−5,−4]T .
We first consider the determination of the parameters in theinfinite dimensional observer (107). According to the discussion
Fig. 5. The function λmaxmin (ρ) for the finite dimensional observer (108).
Fig. 6. The function λmax (γ , ρ) associated with the finite dimensional observer(108) for ρ = ρopt = 1.96 and ρ = 0.5.
in Section 3.4, the functions λbmax (γ ) and λcmax (ρ) can be com-puted and are all recorded in Fig. 3. Consequently, from the fig-ure we get
γsup, γopt, λ
bmaxmin
= (0.550, 0.432,−0.2123) and
ρsup, ρopt, λcmaxmin
= (0.4495, 0.348,−0.1732). Hence we will
choose γ = γopt = 0.432 and ρ = ρopt = 0.348. As a result, themaximal achievable convergence rate of the closed-loop system ismax
λcmaxmin, λ
bmaxmin
= −0.1732. The state x and the observer
error e = x − z are shown in Fig. 4 from which we see that theclosed-loop system is asymptotically stable.
We next consider the determination of the parameters in thefinite dimensional observer (108). Based on the discussion inSection 4.3, we will compute the function λmaxmin (L) by choos-ing different observer gain L. Since L is parameterized by ρ asshown in (106), we can write, for notation simplicity, λmaxmin (L)as λmaxmin (ρ) and λmax (γ , L) as λmax (γ , ρ), respectively. Then,by using the method as discussed in Section 3.4, the functionλmaxmin (ρ) is shown in Fig. 5 from which we find that it is min-imized with ρ = ρopt = 1.96. Consequently, by computing thefunction λmax
γ , ρopt
as shown on the top of Fig. 6, we obtain
γopt = 0.2889 and λmaxγopt, ρopt
= λmaxmin
ρopt
= −0.1879,
2050 B. Zhou et al. / Automatica 49 (2013) 2039–2052
Fig. 7. The state x and the observer error e = x − z associated with the finitedimensional observer (108).
Fig. 8. A comparison of ∥x; z∥ among the infinite dimensional observer (107) andthe finite dimensional observer (108).
which is the maximal convergence rate that the finite dimensionalobserver (108) can achieve.We then chooseρ = ρopt and γ = γopt.The state signal x and the observer error e = x− z are recorded inFig. 7 fromwhichwe see that the closed-loop system is also asymp-totically stable.
In computing the functionλmaxmin (ρ) associatedwith the finitedimensional observer (108), we observe that if ρ is small enough,for example, ρ ≤ 0.5, then γsup (L) = ∞, i.e., the closed-loop sys-tem is asymptotically stable for all γ > 0. This can be observedfrom the bottom figure of Fig. 6, which corresponds to the par-ticular case that ρ = 0.5. This is quite different from the infinitedimensional observer (107) inwhich γsup is independent of the ob-server gain L and is always finite as long as themaximal input delay,i.e., h, is not zero. However, if we choose ρ = 1, then the closed-loop system is asymptotically stable if andonly ifγ < 2.225,whichis a finite number.
For the comparison purpose, the 2-norms of the overall statevectors [xT, zT]T associated with the infinite dimensional observer(107) and the finite dimensional observer (108) are recorded inFig. 8. From this figure, we can see that the finite dimensionalobserver outperforms the infinite dimensional one.
Finally, to illustrate that the finite dimensional observer (108)can achieve semi-global stabilization of the open-loop system (58)if the actuator is subject to magnitude saturation, we choose theinitial control signals as u0 =
12 ,
12
T,∀θ ∈ [−2, 0), which satis-
fies (85). Then, for a given ρ = ρopt = 1.96, if γ is small enough,for example, γ ≤ 0.016, the closed-loop system is asymptotically
Fig. 9. The control signals generated by the finite dimensional observer (108) withγ = 0.016 and ρ = ρopt = 1.96.
stable and such that u (t) satisfies (85). This can be observed fromFig. 9, in which the control signals associated with γ = 0.016 arerecorded.
6. Conclusions
This paper considered observer based output feedback controlof linear systems with both input and output delays. Our recentlydeveloped truncated predictor feedback (TPF) approach wasgeneralized to the design of observers and two classes of observerswere constructed by using this approach. The first class of observerbased output feedback controllers are infinite dimensional andcan even be generalized to linear systems with both time-varyinginput and output delays. The second class of observers were finitedimensional and are thus easier to implement than the first ones.Moreover, it was shown that the separation principle holds forinfinite dimensional observer based output feedback controllers,but does not hold for the finite dimensional ones. Numericalexamples have been worked out to illustrate the effectiveness ofthe proposed design approach.
The results in this paper can be generalized in at least two as-pects. The first aspect is to extend the results to open-loop expo-nentially unstable systems. However, it is no longer expected thatthe allowed delays in the systems can be arbitrarily large. In thiscase, it would be interesting to design observer based output feed-back controllerswith the help of prediction technique such that theallowable delays are maximized. This problem is highly nontrivialand entails a further study. The second aspect is to use the reduced-order observer instead of the full-order observer. However, someinitial studies indicate that the resulting closed-loop system maybe of neutral type whose stability is very difficult to prove.
Appendix
In this Appendix, we recall some existing basic results that areneeded in establishing the results of this paper. We first recall thefollowing results from Zhou et al. (2010) regarding properties ofsolutions to the ARE (16).
Lemma 4. Assume that the matrix pair (A, B) ∈Rn×n,Rn×m
is
controllable and all the poles of A are on the imaginary axis. Then theparametric ARE
ATP + PA− PBBTP = −γ P, (109)
has a unique positive definite solution P (γ ) = W−1(γ ), whereW (γ ) is the unique positive definite solution to the followingLyapunov equation
B. Zhou et al. / Automatica 49 (2013) 2039–2052 2051
Moreover, limγ→0+ P (γ ) = 0, trBTP (γ ) B
= nγ and P (γ ) BBT
P (γ ) ≤ nγ P (γ ). Furthermore,
∥P (γ )∥ ≥n
m ∥B∥2γ , ∀γ ≥ 0, (111)
and, for any given constant c > 0, there exists a constant τ = τ (c) ≥n
m∥B∥2such that
∥P (γ )∥ ≤ τ (c) γ , ∀γ ∈ [0, c] . (112)
Proof. We need only to prove (111) and (112) since the otherproperties can be found in Zhou et al. (2010). Notice that
nmγ =
1m
trBTP (γ ) B
≤BTP (γ ) B
≤ ∥B∥2 ∥P (γ )∥ , (113)
which implies (111). As P (γ ) is the unique positive definitesolution to the Lyapunov equation (110), we know that P (γ ) is arationalmatrix of γ . Hence, it follows from limγ→0+ P (γ ) = 0 thatlimγ→0+
1γP (γ ) exists and is finite. Therefore,
∥P (γ )∥ = γ 1γP (γ )
≤ γ max
γ∈[0,c]
1γP (γ )
= τ (c) γ , ∀γ ∈ [0, c] , (114)
where
τ (c) , maxγ∈[0,c]
1γP (γ )
, (115)
is a finite number. The proof is completed.
The second technical lemma is the so-called Jensen Inequality.
Lemma 5 (Gu (2000)). For any positive definite matrix Q > 0, twoscalars γ2 and γ1 with γ2 ≥ γ1, and a vector valued function ω :[γ1, γ2] → Rn such that the integrals in the following are well-defined, then γ2
γ1
ωT (β) dβQ γ2
γ1
ω (β) dβ
≤ (γ2 − γ1)
γ2
γ1
ωT (β)Qω (β) dβ. (116)
The final technical lemma can be regarded as the discrete-timeversion of the above Jensen Inequality.
Lemma 6. Let xi ∈ Rn, i ∈ I [1,m] ,m ≥ 1, be a series of vectorsand Q > 0 be given. Then
mi=1
xi
T
Q
mi=1
xi
≤ m
mi=1
xTi Qxi
. (117)
Particularly, if m = 2, then the above inequality reduces to
xT1Qx2 + xT2Qx1 ≤ xT1Qx1 + xT2Qx2. (118)
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Bin Zhou was born in Luotian County, Huanggang, HubeiProvince, PR China on July 28, 1981. He received the Bach-elor’s degree, the Master’s Degree and the Ph.D. degreefrom the Department of Control Science and Engineeringat Harbin Institute of Technology, Harbin, China in 2004,2006 and 2010, respectively. He was a Research Associateat the Department of Mechanical Engineering, Universityof Hong Kong from December 2007 to March 2008, a Vis-iting Fellow at the School of Computing and Mathematics,University of Western Sydney from May 2009 to August2009, and a Visiting Research Scientist at the Department
of Electrical and Computer Engineering, University of Virginia from July 2012 to
August 2013. In February 2009, he joined the School of Astronautics, Harbin Insti-tute of Technology, where he has been a Professor since December 2012.
His current research interests include constrained control, time-delay systems,nonlinear control, and control applications in astronautic engineering. In these ar-eas, he has published about 100 papers, over 70 of which are in archival journals.
He is a reviewer for AmericanMathematical Reviewand is an active reviewer fora number of journals and conferences. He was selected as the ‘‘New Century Excel-lent Talents in University’’, the Ministry of Education of China in 2011. He receivedthe ‘‘National Excellent Doctoral Dissertation Award’’ in 2012 from the AcademicDegrees Committee of the State Council and theMinistry of Education of P.R. China.
Zhao-Yan Li was born in Hebei Province, PR China, onAugust 13, 1982. She received her B.Sc. Degree from theDepartment of Information Engineering at North ChinaUniversity of Water Conservancy and Electric Power,Zhengzhou, PR China, in 2005, and her M.Sc. and Ph.D.Degrees in Department of Mathematics, Harbin Instituteof Technology, PR China, in 2007 and 2010, respectively.She is a Research Associate at the Department of Electri-cal and Computer Engineering, University of Virginia fromJuly 2012 to August 2013. She is now a lecturer in theDepartment of Mathematics at Harbin Institute of Tech-
nology, PR China. Her research interest includes stochastic system theory and time-delay systems.
Zongli Lin is a Professor of Electrical and ComputerEngineering at University of Virginia. He received hisB.S. degree in Mathematics and Computer Science fromXiamen University, Xiamen, China, in 1983, his Masterof Engineering degree in Automatic Control from ChineseAcademy of Space Technology, Beijing, China, in 1989, andhis Ph.D. degree in Electrical and Computer Engineeringfrom Washington State University, Pullman, Washington,USA, in 1994. His current research interests includenonlinear control, robust control, and control applications.He was an Associate Editor of the IEEE Transactions
on Automatic Control (2001–2003), IEEE/ASME Transactions on Mechatronics(2006–2009) and IEEE Control Systems Magazine (2005–2012). He has served onthe operating committees and program committees of several conferences andwasan elected member of the Board of Governors of the IEEE Control Systems Society(2008–2010). He currently serves on the editorial boards of several journals andbook series, including Automatica, Systems & Control Letters, and Science China:Information Science. He is a Fellow of the Institute of Electrical and ElectronicsEngineers (IEEE), International Federation of Automatic Control (IFAC) and theAmerican Association for the Advancement of Science (AAAS).
