Stabilization of Discrete-Time Linear Systems Subject to Input ...
2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013
Stabilization of discrete-time linear systems subject to input saturation and multiple unknown constant delays Xu Wang1 , Ali Saberi2 , Anton A. Stoorvogel3 Abstract— In this paper, we study the semi-global stabilization of general linear discrete-time critically unstable systems subject to input saturation and multiple unknown input delays. Based on a simple frequency-domain stability criterion, we find upper bounds for delays that are inversely proportional to the argument of open-loop eigenvalues on the unit circle. For delays satisfying these upper bounds, linear low-gain state and finite dimensional dynamic measurement feedbacks are constructed to solve the semi-global stabilization problems.
case and at 1 in the discrete-time case. In a recent paper [18], the authors consider general continuous-time critically unstable systems subject to input saturation and multiple unknown input delays and propose upper bounds for delays that are inversely proportional to the modulus of open-loop eigenvalues on the imaginary axis. For tolerable delays, linear low-gain feedbacks can be constructed which solve the semi-global stabilization problem.
I. I NTRODUCTION
The aim of this paper is to extend the results in [18] to discrete-time case. The analysis and design is based on a simple frequency-domain stability criterion for discrete linear time-delay systems. It turns out that the results of discretetime systems are in a strict parallel with those of continuoustime systems. The upper bound of tolerable delays found here is also inversely proportional to the argument of eigenvalues on the unit circle. If all the delays satisfy the proposed upper bounds, linear state and finite dimensional dynamic measurement feedback can be constructed using the H2 low-gain design technique to achieve the semi-global stabilization.
The ubiquitous presence of time delay in a variety of engineering applications has invoked intense research enthusiasm in the study of time-delay systems. Voluminous results have been reported. It is not a goal of this paper to provide a complete review of the enormous literature in this context, nor is it possible. A brief coverage of recent research progress on time-delay systems can be found in [13], [5], [12], [8], [4], [2] and references therein. Input saturation is also an important issue that is inevitable in virtually any controller design for physical systems. Neglect of saturation effect can result in grave consequence of performance deterioration even instability. As such, it has attracted and sustained attention from researchers for decades. Some important previous work is summarized in [1], [14], [17], [15], [6], [7]. In this paper, we study the stabilization of discrete-time linear system subject to both input saturation and delay. A discrete-time system with known input delays can be converted to a delay-free system by state augmentation and hence is easier to deal with. We are interested in the case where the delay is unknown. This problem has been previously studied for both continuous- and discrete-time systems. In [10], a nested-saturation type controller is developed for stabilization of an integrator-chain system. This type of controller is also used for discrete-time systems in [20]. A linear delay-independent low-gain feedback was first constructed in [11] to achieve semi-global stabilization for a chain of integrators. In [23], [22], a different low-gain design is utilized to solve the semi-global stabilization problem for a broader class of critically unstable linear systems that has all the eigenvalues at the origin in the continuous-time
The paper is organized as follows. Two stabilization problems are formulated in Section II. Some preliminary results, including a stability criterion for linear discrete time-delay systems and some key properties of H2 low-gain feedback, are presented in Section III. The main results of this paper are developed in Section IV. In this part, we first stabilize the linearized system without saturation using the low-gain feedback and then show that by proper selection of a tuning parameter, the same controller will solve the semi-global stabilization problems in the presence of saturation. Proofs of several technical lemmas are not included due to space limitation.
A. Notations In this paper, standard notations are used. For any open set G C, @G and G denote its boundary and closure. For z0 2 C and r > 0, D.z0 ; r/ denotes an open disc centered at z0 with radius r. Among all, the unit open disc centered at the origin is of particular importance and will be used very often, as such we denote specially D0 WD D.0; 1/ and C# WD @D.0; 1/. For any K1 ; K2 2 N and K1 K2 , ŒK1 ; K2 WD fk 2 N j K1 k K2 g. Let `n1 .K/ denote the Banach space of finite sequences fy1 ; :::; yK g Cn with norm k k1 D maxi fkyi kg. For column vectors x1 ; : : : xm , we simply use x D Œx1 I : : : I xm to denote the stack vector.
1 Courant Institute of Mathematical Science, New York University, New York, NY 10002, USA. E-mail: [email protected] 2 School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164-2752, USA. E-mail: [email protected]. 3 Department of Electrical Engineering, Mathematics, and Computing Science, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. E-mail: [email protected] The work of Xu Wang and Ali Saberi is partially supported by NAVY grants ONR KKK777SB001 and ONR KKK760SB0012.
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II. P ROBLEM FORMULATION
A. Stability of discrete linear time-delay systems Consider system
Consider a discrete-time linear system subject to input saturation and delay 8 Pm < x.k C 1/ D Ax.k/ C i D1 Bi .ui .k i // ; y.k/ D C x.k/; (1) : x. / D CK ; 2 ΠK; 0
x.k C 1/ D Ax.k/ C
m X
Ai x.k
i /;
(3)
i D1
P where x.k/ 2 Rn and i 2 N. Suppose A C m i D1 Ai is Schur stable. The next lemma is a standard result. Lemma 1: System (3) is asymptotically stable if and only if # " m X det zI A z i Ai ¤ 0;
where x 2 Rn , ui 2 R, i 2 Œ0; Ki , Ki 2 N and K D maxfKi g. The initial condition 2 `n1 .K/. Let 2 3 u1 6 :: 7 u D 4 : 5 ; B D B1 Bm :
iD1
8z … D0 ;
um
Define for ˛ 2 Œ0; 1 "
We can formulate two semi-global stabilization problems as follows: Problem 1: The semi-global asymptotic stabilization via state feedback problem for system (1) is to find, for any set of positive integers Ki > 0 and a priori given bounded set of initial conditions W `n1 .K/ with K D maxfKi g, a delayindependent linear state feedback controller u D F x such that the zero solution of the closed-loop system is locally asymptotically stable for any i 2 Œ0; Ki with W contained in its domain of attraction, i.e. the following properties hold for all i 2 Œ0; Ki , i D 1; :::; m:
F˛ .z/ D det zI
A
.1
˛/
m X
Ai
8i 2 Œ0; Ki :
˛
m X
i D1
(4) #
z
i
Ai :
iD1
(5) The results of this paper are all based on the following lemma. Lemma 2: The system (3) is asymptotically stable if det .F˛ .z// ¤ 0; 8z 2 C# ; 8˛ 2 Œ0; 1: (6) Proof : Suppose (6) holds but (4) does not P hold, that is, F1 .z/ has zeros in C=D0 . However, since AC m iD1 Bi Fi is Schur stable, all the zeros of F0 .z/ must be in the open unit disc D0 . Note that the zeros of F˛ .z/ move continuously as ˛ varies. Hence, there exists ˛0 2 .0; 1 such that det.F˛0 .z0 // D 0 for some z0 2 C# , which contradicts (6).
1) 8" > 0, 9 ı such that if k k1 ı, kx.k/k " for all k 0; 2) 8 2 W , x.k/ ! 0 as k ! 1. Problem 2: The semi-global asymptotic stabilization via measurement feedback problem for system (1) is to find an integer q, and find, for any set of positive integers Ki > 0 and any a priori given bounded set W `nCq 1 .K/ with K D maxfKi g, a delay-independent linear finite dimensional measurement feedback controller .k C 1/ D Ak .k/ C Bk y.k/; .k/ 2 Rq ; (2) u.k/ D Ck .k/ C Dk y.k/;
Consider a special case of (3) where Ai D Bi Fi , that is, x.k C 1/ D Ax.k/ C
m X
Bi Fi x.k
i /:
(7)
i D1
P Assume that AN D A C m iD1 Bi Fi is Schur stable. The following variation of Lemma 2 is more convenient to use. Lemma 3: The system (7) is asymptotically stable if
such that the zero solution of the closed-loop system is locally asymptotically stable for all i 2 Œ0; Ki with W contained in its domain of attraction, i.e. the following properties hold for all i 2 Œ0; Ki : 1) 8" > 0, 9 ı such that if k. I /k1 ı, kx.k/k " for all k 0; 2) 8. I / 2 W , .x.k/; .k// ! 0 as k ! 1. Since the input of (1) is bounded, it is well known that the following assumption is necessary for semi-global stabilization. Assumption 1: .A; B/ is stabilizable, .A; C / is detectable and A has all its eigenvalues in the closed unit disc Cˇ .
det ŒI C ˛G.z/.I
D.z// ¤ 0;
where G.z/ D F .zI and
A BF /
B D B1
8z 2 C# ; 8˛ 2 Œ0; 1; (8)
1
B, D.z/ D diagfz 2 3 F1 6 7 Bm ; F D 4 ::: 5 :
Fm Proof : The proof is straightforward. Note that " m m X X det zI A .1 ˛/ Bi Fi ˛ z i D1
D det ŒzI D det ŒzI
III. P RELIMINARIES
A
BF C ˛B.I BF
i m giD1
# i
Bi Fi
i D1
D.z//F
A det I C ˛.zI A BF / 1 B.I D.z//F D det ŒzI A BF det I C ˛F .zI A BF / 1 B.I D.z// :
In this section, we shall present stability criteria for discrete time-delay system which are the basic of this paper and recall the standard low-gain feedback design and some of its properties. 961
The low-gain state feedback u D F" x can be realized with an observer, which we refer to as low-gain compensator .k C 1/ D A .k/ C BF" .k/ K .y.k/ C .k// ; u.k/ D F" .k/: (16)
Since A C BF is Schur stable, (6) holds if and only if (8) holds. Remark 1: Lemma 2 and 3 are discrete-time counterparts of Lemma 2 and 3 in [21] (see also the work in [3], [9]). However, the conditions (6) and (8) are only sufficient for discrete-time system. B. H2 low-gain state feedback and compensator Consider a discrete-time linear system 8 < x.k C 1/ D Ax.k/ C Bu.k/; x.0/ D x0 y.k/ D C x.k/; : z.k/ D u.k/:
with .0/ D 0 , where K is such that A C KC is Schur stable. It can be shown that (16) is a generalized H2 low-gain “sequence” as it satisfies the aforementioned two properties of an H2 low-gain. First, it is easy to see A C BF" is Schur stable, where A 0 AD ; KC A C KC C BF" 0 BD and F" D 0 F" : B
(9)
Let Assumption 1 hold. Recall the following definition from [19]. An H2 low-gain sequence is a family of parameterized matrices F" with " 2 .0; 1 such that the following properties hold 1) A C BF" is Schur stable for any " 2 .0; 1; 2) the closed-loop system of (9) and u D F" x satisfies lim kzk2 D 0; "#0
8x0 2 Rn :
The next lemma proves property (10) for the closed-loop of (9) and (16). Lemma 5: The closed-loop of (9) and (16) satisfies
(10)
lim kzk2 D 0; "#0
The H2 low-gain sequence can be constructed as F" D
.B 0 P" B C I /
1
B 0 P" A
IV. M AIN RESULT
(11)
Now we are in good position to solve the two stabilization problems formulated in Section II. We will develop the results for a linearized system ignoring saturation utilizing the low-gain state feedback and compensator. Then it can be shown that the input of the resulting closed-loop systems can be made sufficient small to avoid saturation for a compact set of initial conditions, which will lead to the solution of Problem 1 and 2.
where for " 2 .0; 1, P" is the positive definite solution of H2 Algebraic Riccati Equation P" D A0 P" A C "I
A0 P" B.B 0 P" B C I /
1
8x0 ; 0 2 Rn :
B 0 P" A: (12)
It is known that under Assumption 1, P" ! 0, and thus F" ! 0, as " ! 0. Moreover, we also have the following lemma Lemma 4: Define transfer function G" .z/ D F" .zI A BF" / 1 B. We have p (13) kI C G" k1 1 C max .B 0 P" B/; Proof : Define R.z/ D I F" .zI A/ 1 B. R.z/ satisfies the following return difference equality ([16]):
A. Global stabilization of linear discrete-time system with input delay
We first consider the stabilization problem for system (1) in the absence of saturation P x.k C 1/ D Ax.k/ C m i /; iD1 Bi u.k 1 0 0 0 1 0 1 1 (17) R.z / .I CB P" B/R.z/ D I C"B .z I A / .zI A/ B: y.k/ D C x.k/ This implies for z 2 C# , Since the system (17) is linear, it is possible to achieve the global stabilization via linear feedback. We shall show that 1 C max .B 0 P" B/ R.z/ R.z/ I; this in fact can be achieved by a low-gain feedback u D F" x and hence with F" given by (11). 1 Define I F" .zI A/ 1 B/ p ; z 2 C# : 1 C max .B 0 P" B/ i n 0 j! 0 (14) !max D maxf! 2 Œ0; j 9v 2 C ; A v D e v; v Bi ¤ 0g: i Clearly, !max is the largest argument of eigenvalues that are, By matrix inversion lemma, at least partially, controllable via input ui . It will be made i N I C F" .zI A BF" / 1 B clear in the following theorem and its proof that this !max p dictates the delay tolerance in the channel ui . 1 C max .B 0 P" B/; z 2 C# ; Theorem 1: Consider system (17). Let Assumption 1 hold which yields (13). and F D F" be given by (11) and (12) with " 2 .0; 1. Remark 2: An immediate consequence of Lemma 4 is the For any Ki < i , there exists " 2 .0; 1 such that 3!max following relations which will be useful in our analysis. the system(17) where u D F" x with F" given by (11) is p asymptotically stable for " 2 .0; " and i 2 Œ0; Ki . kI C G" k1 WD 1 C max .BP1 B/; In a special case where A has all the eigenvalues equal to 1, kG" k1 1 C ; 8" 2 .0; 1 (15) Theorem 1 immediately implies that any bounded delay can 962
be tolerated with using low-gain feedback u D F" x. This is stated in the following corollary. Corollary 1: Consider system (17). Let Assumption 1 hold and F D F" be given by (11) and (12) with " 2 .0; 1. Suppose A has all the eigenvalues equal to 1. For any given positive integers Ki , there exists " 2 .0; 1 such that the system(17) where u D F" x with F" given by (11) is asymptotically stable for " 2 .0; " and i 2 Œ0; Ki . Proof of Theorem 1 : Consider the closed-loop system x.k C 1/ D Ax.k/ C
m X
Bi F";i x.k
i /;
Note that,
Therefore .1
8! 2 Œ ; ; 8˛ 2 Œ0; 1; (19) where D.z/ D diagfz g. Due to symmetry, we only need to consider the ! 2 Œ0; . Assume A has r eigenvalues on the unit circle which are denoted by e j!q , q D 1; :::; r with !q 2 Œ0; . Given Ki < 3!i for i D 1; :::; m, there exists max a ı > 0 such that 1) The neighborhoods Eq WD Œ!q ı; !q Cı\Œ0; , q D 1; :::; r, around these eigenfrequencies are mutually disjoint; 2) If e j!q is at least partially controllable through input i, 1 i !Ki < Ki !max ; 8! 2 Eq : (20) 3 2 3 Lemma 6: The following properties hold: 1) If e j!q is not controllable via input ui for some i , then BF" /
1
Bi D 0;
It is well known that system (24) without delay is asymptotically stable. Define
uniformly in ! for ! 2 Eq . 2) There exists " such that for " 2 .0; " , kF" .e j! I
A
BF" /
1
(23)
By (20), for any q D 1; :::; r, there exists q 2 .0; 1/ solely depending on Ki such that we get N .I DQ q .e j! // < 1 q for ! 2 Eq . According to (13), there exists a "2 "1 such that for " 2 .0; "2 , N .I C G" .e j! // < 1=.1 q / for any q D 1; :::; r. Therefore, condition (19) is satisfied. The next theorem is concerned with measurement feedback. Theorem 2: Consider system (17). Let Assumption 1 , there exists hold. For any positive integers Ki < i 3!max an " such that for " 2 .0; " , the closed-loop system of (17) and low-gain compensator (16) is asymptotically stable for i 2 Œ0; Ki . Corollary 2: Consider system (17). Let Assumption 1 hold and A has all its eigenvalues equal to 1. For any positive integers Ki , there exists an " such that for " 2 .0; " , the closed-loop system of (17) and low-gain compensator (16) is asymptotically stable for i 2 Œ0; Ki . Proof of Theorem 2 : The closed-loop system is given by P x.k C 1/ D Ax.k/ C m i / i D1 Bi Fi .k (24) .k C 1/ D .A C BF" C KC / .k/ KC x.k/:
i
A
DQ q .e j! / < 1
N .I C G" .e j! // I
where F";i is the ith row of F" . Let G" .z/ D F" .zI A BF" / 1 B. It follows from Lemma 3 that the system (18) is asymptotically stable if det I C ˛G" .e j! /.I D.e j! / ¤ 0;
"#0
p ˛/I C ˛ DQ q .e j! / ˛ ˛:
This together with (22) imply that (21) holds if
(18)
i D1
lim F" .e j! I
˛/I C ˛ DQ q .e j! / ˛/ Re.DQ q .e j! // ˛I:
˛/I C ˛ DQ q .e j! / .1 D .1 ˛/2 C ˛ 2 I C 2˛.1
.1
Bk 31 ; 8! 2 ˝;
G"m .z/ D
[rkD1
where ˝ WD Œ0; n Eq Owing to Lemma 6, we find that there exists an "1 such that (19) is satisfied if for all q D 1; :::; r detŒI C ˛G" .e j! / I DQ q .e j! / ¤ 0;
F" .zI
A
BF" /
1
KC.zI
A
KC /
1
B:
Obviously, G"m .z/ is stable. It follows from Lemma 3 that (24) is global asymptotically stable if detŒI C ˛G"m .e j! / I
D.e j! / ¤ 0;
8! 2 Eq ; 8i 2 Œ0; Ki ; ˛ 2 Œ0; 1 (21)
8! 2 Œ ; ; 8i 2 Œ0; Ki 8˛ 2 Œ0; 1; (25)
provided " "1 where DQ q .e j! / equals D.e j! / with i D 0 for all i ’s such that the eigenvalue e j!q is not controllable via input channel i. Clearly, DQ q .e j! / is still unitary. Moreover, by (20), we find that
where D.z/ D diagfz i gm i D1 . We have the following lemma Lemma 7: Let G" .z/ D F" .zI A BF" / 1 B. Then
Re.DQ q .e j! // > 12 I;
lim G"m .e j! /
8! 2 Eq :
"#0
Let’s consider (21). We can write I C ˛G" .e j! / I DQ q .e j! / D .1 C ˛.I C G" .e
j!
G" .e j! / D 0
uniformly !. If, by Theorem 1, there exists an "1 such that for all " 2 .0; "1 we have (19) satisfied with G" .j!/, then we can find an "2 "1 such that (25) holds for all " 2 .0; "2 .
˛/I C ˛ DQ q .e j! / // I DQ q .e j! / : (22) 963
for all " "1 and this only depends on Ki provided that " "1 . This implies that
B. Semi-global stabilization subject to input saturation In this subsection, we shall show that the low-gain state feedback and compensator which stabilize the linearized systems (17) also solve the semi-global stabilization problem for the same linear system with input saturation (1) by a proper selection of the tuning parameter " with respect to a set of initial conditions. Theorem 3: Consider the system (1). Let Assumption 1 hold. The semi-global asymptotic stabilization via state feedback problem can be solved by the low-gain feedback (11). Specifically, for a set of non-negative integers Ki < i , 3!max i D 1; :::; m and any a priori given compact set of initial conditions W `n1 .K/ where K D maxfKi g, there exists an " such that for any " 2 .0; " , the low-gain feedback (11) achieves local asymptotic stability of the closed-loop system with the domain of attraction containing W for any i 2 Œ0; Ki , i D 1; :::; m. In the special case where all the eigenvalues of A are 1, the low-gain feedback allows any bounded but arbitrarily large input delays. This recovers the partial results in [22]. Corollary 3: Consider the system (1). Let Assumption 1 hold and A has all its eigenvalues equal to 1. For any given set of non-negative integers Ki , i D 1; :::; m and any a priori given compact set of initial conditions W `n1 .K/ where K D maxfKi g, there exists an " such that for any " 2 .0; " , the low-gain feedback (11) achieves local asymptotic stability of the closed-loop system with the domain of attraction containing W for any i 2 Œ0; Ki , i D 1; :::; m. Proof of Theorem 3 : The closed-loop system can be written as m X x.k/ D Ax.k/ C Bi .Fi x.k i // (26)
x.k C 1/ D .A C BF" /x.k/ 3 v1 .k/ 6 7 v" .k/ D 4 ::: 5 ;
.z/ D I
D.z/ D diagf1
1
Fi .k 0;
i /;
k < i ; k i :
F" x.k/ D F" .A C BF" /k x.0/ .g" ı ı/.F" x/.k/ C g" .v" /.k/ and hence F" x.k/ D .1 C g" ı ı/
1
h i F" .A C BF" /k x.0/ C g" .v" /.k/ : (27)
Let w" .k/ D g" .v" /.k/. By the definition of g" , we have kw" k2 kG" .z/k1 kv" k2 .1 C /kv" k2 where is given by (15). Hence for any given initial condition , kw" k2 ! 0 as " ! 0. Then from (27), we get kF" xk2 k.1 C G" .z/.z//
1
k1 kF" .A C BF" /k x.0/k2
C k.1 C G" .z/.z// 1 kF" .A
k
1
k1 kw" k2
C BF" / x.0/k2 C
1 kw" k2 :
Since, by (10), kF" .A C BF" /k x.0/k2 ! 0 and v" ! 0 as " ! 0 and is independent of " (provided " is smaller than "1 ), there exists an "2 such that kF" xk2 1 for " 2 .0; "1 and 2 W . This implies that kF" x.k/k kF" xk2 1 for k 0. At last, since 2 W , there exists " "2 such that kF" x.k/k 1 for k K. The next theorem solves Problem 2. Theorem 4: Consider the system (1). Let Assumption 1 hold. The semi-global asymptotic stabilization via measurement feedback problem can be solved by the low-gain compensator (16). Specifically, for any a priori given compact set of initial conditions W `2n 1 .K/ and a set of positive integers Ki < 3!i , i D 1; :::; m, there exists an " such max that for any " 2 .0; " , the origin of the closed-loop system of (1) and (16) is local asymptotic stable for any i 2 Œ0; Ki , i D 1; :::; m with the domain of attraction containing W . Corollary 4: Consider the system (1). Let Assumption 1 hold and A has all its eigenvalues equal to 1. For any a priori given compact set of initial conditions W `2n 1 .K/ and any given set of positive integers Ki , i D 1; :::; m, there exists an " such that for any " 2 .0; " , the origin of the closedloop system of (1) and (16) is local asymptotic stable for any i 2 Œ0; Ki , i D 1; :::; m with the domain of attraction containing W .
i m g1D1 :
Note that the operators g" and ı have zero initial conditions. From the proof of Theorem 1, we know that (19) is satisfied which guarantees that there exists a > 0 such that .I C G" .z/.z// > ;
vi .k/ D
Since v" .k/ vanishes for k K, 2 W is bounded and F" ! 0, we have for any 2 W , kv" k1 ! 0 and kv" k2 ! 0 as " ! 0. We have
B z
(
vm .k/
This will imply that for system (26) no saturation will be active for all k 0, and hence, the system is linear and stable for " "1 . This will complete the proof. Define two linear time invariant operators g" and ı with the following transfer matrices: BF" /
Bı.F" x/.k/ C Bv" .k/;
2
K/k 1; 8k 0:
A
1 :
where
i Since Ki < 3 !max , the local Lyapunov stability of the origin for sufficiently small " follows from Theorem 1, that is, there exists "1 2 .0; 1 such that for " 2 .0; "1 , the origin of (26) is locally stable. It remains to show the attractivity. It suffices to prove that for system (18) with initial condition in W , there exists "2 "1 such that for " 2 .0; "2 , we shall have that
G" .z/ D F" .zI
k1
Note that for k 0
i D1
kF" x.k
1
k.I C G" .z/.z//
8z 2 C# ; 8i 2 Œ0; Ki ; 964
Proof of Theorem 4 : The closed-loop system can be written as P 8 x.k C 1/ D Ax.k/ C m Bi .Fi .k i // ˆ ˆ < .k C 1/ D .A C BF CiD1 KC / .k/ KC x.k/ " (28) x. / D . /; 8 2 Œ Ki ; 0 ˆ ˆ : . / D . /; 8 2 Œ Ki ; 0:
[7] V. K APILA AND G. G RIGORIADIS, Eds., Actuator saturation control, Marcel Dekker, 2002. [8] V.L. K HARITONOV, S.-I. N ICULESCU , J. M ORENO , AND W. M ICHIELS, “Static output feedback stabilization: necessary conditions for multiple delay controllers”, IEEE Trans. Aut. Contr., 50(1), 2005, pp. 82–86. [9] N. L EHTOMAKI , N.R.S ANDELL J R ., AND M. ATHANS, “Robustness results in linear-quadratic Gaussian based multivariable control designs”, ieee, 26(1), 1981, pp. 75–93. [10] F. M AZENC , S. M ONDIE , AND S.-I. N ICULESCU, “Global asymptotic stabilization for chains of integrators with a delay in the input”, IEEE Trans. Aut. Contr., 48(1), 2003, pp. 57–63. [11] W. M ICHIELS AND D. ROOSE, “Global stabilization of multiple integrators with time delay and input constraints”, in Proceedings of the 3th Workshop on Time-Delay Systems (TDS2001), T. Abdallah, K. Gu, and S.-I. Niculescu, eds., Santa Fe, NM, 2001, Pergamon, pp. 243–248. [12] S.I. N ICULESCU AND W. M ICHIELS, “Stabilizing a chain of integrators using multiple delays”, IEEE Trans. Aut. Contr., 49(5), 2004, pp. 802–807. [13] S.-I. N ICULESCU, “On delay-dependent stability under model transformations of some neutral linear systems”, Int. J. Contr., 74(6), 2001, pp. 609–617. [14] A. S ABERI AND A.A. S TOORVOGEL, “Special issue on control problems with constraints”, Int. J. Robust & Nonlinear Control, 9(10), 1999, pp. 583–734. [15] A. S ABERI , A.A. S TOORVOGEL , AND P. S ANNUTI, Control of linear systems with regulation and input constraints, Communication and Control Engineering Series, Springer Verlag, Berlin, 2000. [16] U. S HAKED, “Guaranteed stability margins for the discrete time linear quadratic optimal regulator”, IEEE Trans. Aut. Contr., 31(2), 1986, pp. 162–165. [17] S. TARBOURIECH AND G. G ARCIA, Eds., Control of uncertain systems with bounded inputs, vol. 227 of Lecture notes in control and information sciences, Springer Verlag, 1997. [18] X. WANG , A. S ABERI , AND A.A. S TOORVOGEL, “Stabilization of linear system with input saturation and unknown delay”, Submitted for publication, available at https://files.nyu.edu/xw665/ public/paper9.pdf, 2012. [19] X. WANG , A.A. S TOORVOGEL , A. S ABERI , AND P. S ANNUTI, “Discrete-time H2 and H1 low-gain theory”, Int. J. Robust & Nonlinear Control, 22(7), 2012, pp. 743–762. [20] K. YAKOUBI AND Y. C HITOUR, “On the Stabilization of Linear Discrete-Time Delay Systems Subject to Input Saturation”, in Advanced Strategies in Control Systems with Input and Output Constraints, Sophie Tarbouriech, Germain Garcia, and Adolf Glattfelder, eds., vol. 346 of Lecture Notes in Control and Information Sciences, Springer Berlin / Heidelberg, 2007, pp. 421–455. [21] J. Z HANG , C. K NOSPE , AND P. T SIOTRAS, “New results for the analysis of linear systems with time-invariant delays”, Int. J. Robust & Nonlinear Control, 13(12), 2003, pp. 1149–1175. [22] B IN Z HOU AND Z ONGLI L IN, “Parametric Lyapunov Equation Approach to Stabilization of Discrete-Time Systems With Input Delay and Saturation”, IEEE Trans. Circ. & Syst.-I Regular papers, 58(11), 2011, pp. 2741 –2754. [23] B. Z HOU , Z. L IN , AND G.R. D UAN, “Global and semi-global stabilization of linear systems with multiple delays and saturations in the input”, SIAM J. Contr. & Opt., 53(8), 2010, pp. 5294–5332.
Suppose Ki ’s satisfy the bound Ki < 3!i . By Theorem 2, max there exists an "1 such that for " 2 .0; "1 , the closed-loop system without saturation is asymptotically stable. Then the local stability of (28) for " "1 follows. Define two linear time invariant operators g"m and ı with z transform G"m .z/ D .z/ D I
F" .zI
A
BF" /
D.z/ D diagf1
1
KC.zI
z
A
KC /
1
B
i m gi D1 :
From the proof of Theorem 2, we know that (25) holds for " "1 . There exists a > 0 such that .I C G"m .z/.z// > ; 8z 2 C# ; 8i 2 Œ0; Ki ;
(29)
where is independent of " provided that " "1 . It follows from Lemma 7 that G"m .z/ ! G" .z/ uniformly on C# where G" .z/ D F" .zI A BF" / 1 B. Since kG" k1 1 C for any " 2 .0; 1 with given by (15), there exists an "2 such that kG"m k1 2.1 C /:
(30)
Given (29), (30) and Lemma 5 hold, we can use exactly the same argument as in the proof of Theorem 3 to prove that there exists an " "1 such that for " 2 .0; " , kF" .k
K/k 1; 8k 0; . ; / 2 W :
V. C ONCLUSION In this paper, the semi-global stabilization problems for general uncritically unstable systems subject to input saturation and multiple unknown delays are solved. We propose upper bounds on delays based on a frequency-domain stability criterion for linear discrete time-delay system and constructed a low-gain state feedback and compensator to achieve the semi-global stabilization with feasible delays. R EFERENCES [1] D.S. B ERNSTEIN AND A.N. M ICHEL, “Special Issue on saturating actuators”, Int. J. Robust & Nonlinear Control, 5(5), 1995, pp. 375– 540. [2] H O -L IM C HOI AND J ONG -TAE L IM, “Stabilization of a chain of integrators with an unknown delay in the input by adaptive output feedback”, IEEE Trans. Aut. Contr., 51(8), 2006, pp. 1359–1363. [3] R. DATKO, “A procedure for determination of the exponential stability of certain differential-difference equations”, Quart. Appl. Math., 36(3), 1978, pp. 279–292. [4] E. F RIDMAN, “New Lyapunov-Kasovskii functionals for stability of linear retarded and neutral type systems”, Syst. & Contr. Letters, 43(4), 2001, pp. 309–319. [5] K. G U , V. L. K HARITONOV, AND J. C HEN, Stability of time-delay systems, Birkhäuser, Boston, MA, 2003. [6] T. H U AND Z. L IN, Control systems with actuator saturation: analysis and design, Birkhäuser, 2001.
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Stabilization of discrete-time linear systems subject to input saturation and multiple unknown constant delays Xu Wang1 , Ali Saberi2 , Anton A. Stoorvogel3 Abstract— In this paper, we study the semi-global stabilization of general linear discrete-time critically unstable systems subject to input saturation and multiple unknown input delays. Based on a simple frequency-domain stability criterion, we find upper bounds for delays that are inversely proportional to the argument of open-loop eigenvalues on the unit circle. For delays satisfying these upper bounds, linear low-gain state and finite dimensional dynamic measurement feedbacks are constructed to solve the semi-global stabilization problems.
case and at 1 in the discrete-time case. In a recent paper [18], the authors consider general continuous-time critically unstable systems subject to input saturation and multiple unknown input delays and propose upper bounds for delays that are inversely proportional to the modulus of open-loop eigenvalues on the imaginary axis. For tolerable delays, linear low-gain feedbacks can be constructed which solve the semi-global stabilization problem.
I. I NTRODUCTION
The aim of this paper is to extend the results in [18] to discrete-time case. The analysis and design is based on a simple frequency-domain stability criterion for discrete linear time-delay systems. It turns out that the results of discretetime systems are in a strict parallel with those of continuoustime systems. The upper bound of tolerable delays found here is also inversely proportional to the argument of eigenvalues on the unit circle. If all the delays satisfy the proposed upper bounds, linear state and finite dimensional dynamic measurement feedback can be constructed using the H2 low-gain design technique to achieve the semi-global stabilization.
The ubiquitous presence of time delay in a variety of engineering applications has invoked intense research enthusiasm in the study of time-delay systems. Voluminous results have been reported. It is not a goal of this paper to provide a complete review of the enormous literature in this context, nor is it possible. A brief coverage of recent research progress on time-delay systems can be found in [13], [5], [12], [8], [4], [2] and references therein. Input saturation is also an important issue that is inevitable in virtually any controller design for physical systems. Neglect of saturation effect can result in grave consequence of performance deterioration even instability. As such, it has attracted and sustained attention from researchers for decades. Some important previous work is summarized in [1], [14], [17], [15], [6], [7]. In this paper, we study the stabilization of discrete-time linear system subject to both input saturation and delay. A discrete-time system with known input delays can be converted to a delay-free system by state augmentation and hence is easier to deal with. We are interested in the case where the delay is unknown. This problem has been previously studied for both continuous- and discrete-time systems. In [10], a nested-saturation type controller is developed for stabilization of an integrator-chain system. This type of controller is also used for discrete-time systems in [20]. A linear delay-independent low-gain feedback was first constructed in [11] to achieve semi-global stabilization for a chain of integrators. In [23], [22], a different low-gain design is utilized to solve the semi-global stabilization problem for a broader class of critically unstable linear systems that has all the eigenvalues at the origin in the continuous-time
The paper is organized as follows. Two stabilization problems are formulated in Section II. Some preliminary results, including a stability criterion for linear discrete time-delay systems and some key properties of H2 low-gain feedback, are presented in Section III. The main results of this paper are developed in Section IV. In this part, we first stabilize the linearized system without saturation using the low-gain feedback and then show that by proper selection of a tuning parameter, the same controller will solve the semi-global stabilization problems in the presence of saturation. Proofs of several technical lemmas are not included due to space limitation.
A. Notations In this paper, standard notations are used. For any open set G C, @G and G denote its boundary and closure. For z0 2 C and r > 0, D.z0 ; r/ denotes an open disc centered at z0 with radius r. Among all, the unit open disc centered at the origin is of particular importance and will be used very often, as such we denote specially D0 WD D.0; 1/ and C# WD @D.0; 1/. For any K1 ; K2 2 N and K1 K2 , ŒK1 ; K2 WD fk 2 N j K1 k K2 g. Let `n1 .K/ denote the Banach space of finite sequences fy1 ; :::; yK g Cn with norm k k1 D maxi fkyi kg. For column vectors x1 ; : : : xm , we simply use x D Œx1 I : : : I xm to denote the stack vector.
1 Courant Institute of Mathematical Science, New York University, New York, NY 10002, USA. E-mail: [email protected] 2 School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164-2752, USA. E-mail: [email protected]. 3 Department of Electrical Engineering, Mathematics, and Computing Science, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. E-mail: [email protected] The work of Xu Wang and Ali Saberi is partially supported by NAVY grants ONR KKK777SB001 and ONR KKK760SB0012.
978-1-4799-0176-0/$31.00 ©2013 AACC
960
II. P ROBLEM FORMULATION
A. Stability of discrete linear time-delay systems Consider system
Consider a discrete-time linear system subject to input saturation and delay 8 Pm < x.k C 1/ D Ax.k/ C i D1 Bi .ui .k i // ; y.k/ D C x.k/; (1) : x. / D CK ; 2 ΠK; 0
x.k C 1/ D Ax.k/ C
m X
Ai x.k
i /;
(3)
i D1
P where x.k/ 2 Rn and i 2 N. Suppose A C m i D1 Ai is Schur stable. The next lemma is a standard result. Lemma 1: System (3) is asymptotically stable if and only if # " m X det zI A z i Ai ¤ 0;
where x 2 Rn , ui 2 R, i 2 Œ0; Ki , Ki 2 N and K D maxfKi g. The initial condition 2 `n1 .K/. Let 2 3 u1 6 :: 7 u D 4 : 5 ; B D B1 Bm :
iD1
8z … D0 ;
um
Define for ˛ 2 Œ0; 1 "
We can formulate two semi-global stabilization problems as follows: Problem 1: The semi-global asymptotic stabilization via state feedback problem for system (1) is to find, for any set of positive integers Ki > 0 and a priori given bounded set of initial conditions W `n1 .K/ with K D maxfKi g, a delayindependent linear state feedback controller u D F x such that the zero solution of the closed-loop system is locally asymptotically stable for any i 2 Œ0; Ki with W contained in its domain of attraction, i.e. the following properties hold for all i 2 Œ0; Ki , i D 1; :::; m:
F˛ .z/ D det zI
A
.1
˛/
m X
Ai
8i 2 Œ0; Ki :
˛
m X
i D1
(4) #
z
i
Ai :
iD1
(5) The results of this paper are all based on the following lemma. Lemma 2: The system (3) is asymptotically stable if det .F˛ .z// ¤ 0; 8z 2 C# ; 8˛ 2 Œ0; 1: (6) Proof : Suppose (6) holds but (4) does not P hold, that is, F1 .z/ has zeros in C=D0 . However, since AC m iD1 Bi Fi is Schur stable, all the zeros of F0 .z/ must be in the open unit disc D0 . Note that the zeros of F˛ .z/ move continuously as ˛ varies. Hence, there exists ˛0 2 .0; 1 such that det.F˛0 .z0 // D 0 for some z0 2 C# , which contradicts (6).
1) 8" > 0, 9 ı such that if k k1 ı, kx.k/k " for all k 0; 2) 8 2 W , x.k/ ! 0 as k ! 1. Problem 2: The semi-global asymptotic stabilization via measurement feedback problem for system (1) is to find an integer q, and find, for any set of positive integers Ki > 0 and any a priori given bounded set W `nCq 1 .K/ with K D maxfKi g, a delay-independent linear finite dimensional measurement feedback controller .k C 1/ D Ak .k/ C Bk y.k/; .k/ 2 Rq ; (2) u.k/ D Ck .k/ C Dk y.k/;
Consider a special case of (3) where Ai D Bi Fi , that is, x.k C 1/ D Ax.k/ C
m X
Bi Fi x.k
i /:
(7)
i D1
P Assume that AN D A C m iD1 Bi Fi is Schur stable. The following variation of Lemma 2 is more convenient to use. Lemma 3: The system (7) is asymptotically stable if
such that the zero solution of the closed-loop system is locally asymptotically stable for all i 2 Œ0; Ki with W contained in its domain of attraction, i.e. the following properties hold for all i 2 Œ0; Ki : 1) 8" > 0, 9 ı such that if k. I /k1 ı, kx.k/k " for all k 0; 2) 8. I / 2 W , .x.k/; .k// ! 0 as k ! 1. Since the input of (1) is bounded, it is well known that the following assumption is necessary for semi-global stabilization. Assumption 1: .A; B/ is stabilizable, .A; C / is detectable and A has all its eigenvalues in the closed unit disc Cˇ .
det ŒI C ˛G.z/.I
D.z// ¤ 0;
where G.z/ D F .zI and
A BF /
B D B1
8z 2 C# ; 8˛ 2 Œ0; 1; (8)
1
B, D.z/ D diagfz 2 3 F1 6 7 Bm ; F D 4 ::: 5 :
Fm Proof : The proof is straightforward. Note that " m m X X det zI A .1 ˛/ Bi Fi ˛ z i D1
D det ŒzI D det ŒzI
III. P RELIMINARIES
A
BF C ˛B.I BF
i m giD1
# i
Bi Fi
i D1
D.z//F
A det I C ˛.zI A BF / 1 B.I D.z//F D det ŒzI A BF det I C ˛F .zI A BF / 1 B.I D.z// :
In this section, we shall present stability criteria for discrete time-delay system which are the basic of this paper and recall the standard low-gain feedback design and some of its properties. 961
The low-gain state feedback u D F" x can be realized with an observer, which we refer to as low-gain compensator .k C 1/ D A .k/ C BF" .k/ K .y.k/ C .k// ; u.k/ D F" .k/: (16)
Since A C BF is Schur stable, (6) holds if and only if (8) holds. Remark 1: Lemma 2 and 3 are discrete-time counterparts of Lemma 2 and 3 in [21] (see also the work in [3], [9]). However, the conditions (6) and (8) are only sufficient for discrete-time system. B. H2 low-gain state feedback and compensator Consider a discrete-time linear system 8 < x.k C 1/ D Ax.k/ C Bu.k/; x.0/ D x0 y.k/ D C x.k/; : z.k/ D u.k/:
with .0/ D 0 , where K is such that A C KC is Schur stable. It can be shown that (16) is a generalized H2 low-gain “sequence” as it satisfies the aforementioned two properties of an H2 low-gain. First, it is easy to see A C BF" is Schur stable, where A 0 AD ; KC A C KC C BF" 0 BD and F" D 0 F" : B
(9)
Let Assumption 1 hold. Recall the following definition from [19]. An H2 low-gain sequence is a family of parameterized matrices F" with " 2 .0; 1 such that the following properties hold 1) A C BF" is Schur stable for any " 2 .0; 1; 2) the closed-loop system of (9) and u D F" x satisfies lim kzk2 D 0; "#0
8x0 2 Rn :
The next lemma proves property (10) for the closed-loop of (9) and (16). Lemma 5: The closed-loop of (9) and (16) satisfies
(10)
lim kzk2 D 0; "#0
The H2 low-gain sequence can be constructed as F" D
.B 0 P" B C I /
1
B 0 P" A
IV. M AIN RESULT
(11)
Now we are in good position to solve the two stabilization problems formulated in Section II. We will develop the results for a linearized system ignoring saturation utilizing the low-gain state feedback and compensator. Then it can be shown that the input of the resulting closed-loop systems can be made sufficient small to avoid saturation for a compact set of initial conditions, which will lead to the solution of Problem 1 and 2.
where for " 2 .0; 1, P" is the positive definite solution of H2 Algebraic Riccati Equation P" D A0 P" A C "I
A0 P" B.B 0 P" B C I /
1
8x0 ; 0 2 Rn :
B 0 P" A: (12)
It is known that under Assumption 1, P" ! 0, and thus F" ! 0, as " ! 0. Moreover, we also have the following lemma Lemma 4: Define transfer function G" .z/ D F" .zI A BF" / 1 B. We have p (13) kI C G" k1 1 C max .B 0 P" B/; Proof : Define R.z/ D I F" .zI A/ 1 B. R.z/ satisfies the following return difference equality ([16]):
A. Global stabilization of linear discrete-time system with input delay
We first consider the stabilization problem for system (1) in the absence of saturation P x.k C 1/ D Ax.k/ C m i /; iD1 Bi u.k 1 0 0 0 1 0 1 1 (17) R.z / .I CB P" B/R.z/ D I C"B .z I A / .zI A/ B: y.k/ D C x.k/ This implies for z 2 C# , Since the system (17) is linear, it is possible to achieve the global stabilization via linear feedback. We shall show that 1 C max .B 0 P" B/ R.z/ R.z/ I; this in fact can be achieved by a low-gain feedback u D F" x and hence with F" given by (11). 1 Define I F" .zI A/ 1 B/ p ; z 2 C# : 1 C max .B 0 P" B/ i n 0 j! 0 (14) !max D maxf! 2 Œ0; j 9v 2 C ; A v D e v; v Bi ¤ 0g: i Clearly, !max is the largest argument of eigenvalues that are, By matrix inversion lemma, at least partially, controllable via input ui . It will be made i N I C F" .zI A BF" / 1 B clear in the following theorem and its proof that this !max p dictates the delay tolerance in the channel ui . 1 C max .B 0 P" B/; z 2 C# ; Theorem 1: Consider system (17). Let Assumption 1 hold which yields (13). and F D F" be given by (11) and (12) with " 2 .0; 1. Remark 2: An immediate consequence of Lemma 4 is the For any Ki < i , there exists " 2 .0; 1 such that 3!max following relations which will be useful in our analysis. the system(17) where u D F" x with F" given by (11) is p asymptotically stable for " 2 .0; " and i 2 Œ0; Ki . kI C G" k1 WD 1 C max .BP1 B/; In a special case where A has all the eigenvalues equal to 1, kG" k1 1 C ; 8" 2 .0; 1 (15) Theorem 1 immediately implies that any bounded delay can 962
be tolerated with using low-gain feedback u D F" x. This is stated in the following corollary. Corollary 1: Consider system (17). Let Assumption 1 hold and F D F" be given by (11) and (12) with " 2 .0; 1. Suppose A has all the eigenvalues equal to 1. For any given positive integers Ki , there exists " 2 .0; 1 such that the system(17) where u D F" x with F" given by (11) is asymptotically stable for " 2 .0; " and i 2 Œ0; Ki . Proof of Theorem 1 : Consider the closed-loop system x.k C 1/ D Ax.k/ C
m X
Bi F";i x.k
i /;
Note that,
Therefore .1
8! 2 Œ ; ; 8˛ 2 Œ0; 1; (19) where D.z/ D diagfz g. Due to symmetry, we only need to consider the ! 2 Œ0; . Assume A has r eigenvalues on the unit circle which are denoted by e j!q , q D 1; :::; r with !q 2 Œ0; . Given Ki < 3!i for i D 1; :::; m, there exists max a ı > 0 such that 1) The neighborhoods Eq WD Œ!q ı; !q Cı\Œ0; , q D 1; :::; r, around these eigenfrequencies are mutually disjoint; 2) If e j!q is at least partially controllable through input i, 1 i !Ki < Ki !max ; 8! 2 Eq : (20) 3 2 3 Lemma 6: The following properties hold: 1) If e j!q is not controllable via input ui for some i , then BF" /
1
Bi D 0;
It is well known that system (24) without delay is asymptotically stable. Define
uniformly in ! for ! 2 Eq . 2) There exists " such that for " 2 .0; " , kF" .e j! I
A
BF" /
1
(23)
By (20), for any q D 1; :::; r, there exists q 2 .0; 1/ solely depending on Ki such that we get N .I DQ q .e j! // < 1 q for ! 2 Eq . According to (13), there exists a "2 "1 such that for " 2 .0; "2 , N .I C G" .e j! // < 1=.1 q / for any q D 1; :::; r. Therefore, condition (19) is satisfied. The next theorem is concerned with measurement feedback. Theorem 2: Consider system (17). Let Assumption 1 , there exists hold. For any positive integers Ki < i 3!max an " such that for " 2 .0; " , the closed-loop system of (17) and low-gain compensator (16) is asymptotically stable for i 2 Œ0; Ki . Corollary 2: Consider system (17). Let Assumption 1 hold and A has all its eigenvalues equal to 1. For any positive integers Ki , there exists an " such that for " 2 .0; " , the closed-loop system of (17) and low-gain compensator (16) is asymptotically stable for i 2 Œ0; Ki . Proof of Theorem 2 : The closed-loop system is given by P x.k C 1/ D Ax.k/ C m i / i D1 Bi Fi .k (24) .k C 1/ D .A C BF" C KC / .k/ KC x.k/:
i
A
DQ q .e j! / < 1
N .I C G" .e j! // I
where F";i is the ith row of F" . Let G" .z/ D F" .zI A BF" / 1 B. It follows from Lemma 3 that the system (18) is asymptotically stable if det I C ˛G" .e j! /.I D.e j! / ¤ 0;
"#0
p ˛/I C ˛ DQ q .e j! / ˛ ˛:
This together with (22) imply that (21) holds if
(18)
i D1
lim F" .e j! I
˛/I C ˛ DQ q .e j! / ˛/ Re.DQ q .e j! // ˛I:
˛/I C ˛ DQ q .e j! / .1 D .1 ˛/2 C ˛ 2 I C 2˛.1
.1
Bk 31 ; 8! 2 ˝;
G"m .z/ D
[rkD1
where ˝ WD Œ0; n Eq Owing to Lemma 6, we find that there exists an "1 such that (19) is satisfied if for all q D 1; :::; r detŒI C ˛G" .e j! / I DQ q .e j! / ¤ 0;
F" .zI
A
BF" /
1
KC.zI
A
KC /
1
B:
Obviously, G"m .z/ is stable. It follows from Lemma 3 that (24) is global asymptotically stable if detŒI C ˛G"m .e j! / I
D.e j! / ¤ 0;
8! 2 Eq ; 8i 2 Œ0; Ki ; ˛ 2 Œ0; 1 (21)
8! 2 Œ ; ; 8i 2 Œ0; Ki 8˛ 2 Œ0; 1; (25)
provided " "1 where DQ q .e j! / equals D.e j! / with i D 0 for all i ’s such that the eigenvalue e j!q is not controllable via input channel i. Clearly, DQ q .e j! / is still unitary. Moreover, by (20), we find that
where D.z/ D diagfz i gm i D1 . We have the following lemma Lemma 7: Let G" .z/ D F" .zI A BF" / 1 B. Then
Re.DQ q .e j! // > 12 I;
lim G"m .e j! /
8! 2 Eq :
"#0
Let’s consider (21). We can write I C ˛G" .e j! / I DQ q .e j! / D .1 C ˛.I C G" .e
j!
G" .e j! / D 0
uniformly !. If, by Theorem 1, there exists an "1 such that for all " 2 .0; "1 we have (19) satisfied with G" .j!/, then we can find an "2 "1 such that (25) holds for all " 2 .0; "2 .
˛/I C ˛ DQ q .e j! / // I DQ q .e j! / : (22) 963
for all " "1 and this only depends on Ki provided that " "1 . This implies that
B. Semi-global stabilization subject to input saturation In this subsection, we shall show that the low-gain state feedback and compensator which stabilize the linearized systems (17) also solve the semi-global stabilization problem for the same linear system with input saturation (1) by a proper selection of the tuning parameter " with respect to a set of initial conditions. Theorem 3: Consider the system (1). Let Assumption 1 hold. The semi-global asymptotic stabilization via state feedback problem can be solved by the low-gain feedback (11). Specifically, for a set of non-negative integers Ki < i , 3!max i D 1; :::; m and any a priori given compact set of initial conditions W `n1 .K/ where K D maxfKi g, there exists an " such that for any " 2 .0; " , the low-gain feedback (11) achieves local asymptotic stability of the closed-loop system with the domain of attraction containing W for any i 2 Œ0; Ki , i D 1; :::; m. In the special case where all the eigenvalues of A are 1, the low-gain feedback allows any bounded but arbitrarily large input delays. This recovers the partial results in [22]. Corollary 3: Consider the system (1). Let Assumption 1 hold and A has all its eigenvalues equal to 1. For any given set of non-negative integers Ki , i D 1; :::; m and any a priori given compact set of initial conditions W `n1 .K/ where K D maxfKi g, there exists an " such that for any " 2 .0; " , the low-gain feedback (11) achieves local asymptotic stability of the closed-loop system with the domain of attraction containing W for any i 2 Œ0; Ki , i D 1; :::; m. Proof of Theorem 3 : The closed-loop system can be written as m X x.k/ D Ax.k/ C Bi .Fi x.k i // (26)
x.k C 1/ D .A C BF" /x.k/ 3 v1 .k/ 6 7 v" .k/ D 4 ::: 5 ;
.z/ D I
D.z/ D diagf1
1
Fi .k 0;
i /;
k < i ; k i :
F" x.k/ D F" .A C BF" /k x.0/ .g" ı ı/.F" x/.k/ C g" .v" /.k/ and hence F" x.k/ D .1 C g" ı ı/
1
h i F" .A C BF" /k x.0/ C g" .v" /.k/ : (27)
Let w" .k/ D g" .v" /.k/. By the definition of g" , we have kw" k2 kG" .z/k1 kv" k2 .1 C /kv" k2 where is given by (15). Hence for any given initial condition , kw" k2 ! 0 as " ! 0. Then from (27), we get kF" xk2 k.1 C G" .z/.z//
1
k1 kF" .A C BF" /k x.0/k2
C k.1 C G" .z/.z// 1 kF" .A
k
1
k1 kw" k2
C BF" / x.0/k2 C
1 kw" k2 :
Since, by (10), kF" .A C BF" /k x.0/k2 ! 0 and v" ! 0 as " ! 0 and is independent of " (provided " is smaller than "1 ), there exists an "2 such that kF" xk2 1 for " 2 .0; "1 and 2 W . This implies that kF" x.k/k kF" xk2 1 for k 0. At last, since 2 W , there exists " "2 such that kF" x.k/k 1 for k K. The next theorem solves Problem 2. Theorem 4: Consider the system (1). Let Assumption 1 hold. The semi-global asymptotic stabilization via measurement feedback problem can be solved by the low-gain compensator (16). Specifically, for any a priori given compact set of initial conditions W `2n 1 .K/ and a set of positive integers Ki < 3!i , i D 1; :::; m, there exists an " such max that for any " 2 .0; " , the origin of the closed-loop system of (1) and (16) is local asymptotic stable for any i 2 Œ0; Ki , i D 1; :::; m with the domain of attraction containing W . Corollary 4: Consider the system (1). Let Assumption 1 hold and A has all its eigenvalues equal to 1. For any a priori given compact set of initial conditions W `2n 1 .K/ and any given set of positive integers Ki , i D 1; :::; m, there exists an " such that for any " 2 .0; " , the origin of the closedloop system of (1) and (16) is local asymptotic stable for any i 2 Œ0; Ki , i D 1; :::; m with the domain of attraction containing W .
i m g1D1 :
Note that the operators g" and ı have zero initial conditions. From the proof of Theorem 1, we know that (19) is satisfied which guarantees that there exists a > 0 such that .I C G" .z/.z// > ;
vi .k/ D
Since v" .k/ vanishes for k K, 2 W is bounded and F" ! 0, we have for any 2 W , kv" k1 ! 0 and kv" k2 ! 0 as " ! 0. We have
B z
(
vm .k/
This will imply that for system (26) no saturation will be active for all k 0, and hence, the system is linear and stable for " "1 . This will complete the proof. Define two linear time invariant operators g" and ı with the following transfer matrices: BF" /
Bı.F" x/.k/ C Bv" .k/;
2
K/k 1; 8k 0:
A
1 :
where
i Since Ki < 3 !max , the local Lyapunov stability of the origin for sufficiently small " follows from Theorem 1, that is, there exists "1 2 .0; 1 such that for " 2 .0; "1 , the origin of (26) is locally stable. It remains to show the attractivity. It suffices to prove that for system (18) with initial condition in W , there exists "2 "1 such that for " 2 .0; "2 , we shall have that
G" .z/ D F" .zI
k1
Note that for k 0
i D1
kF" x.k
1
k.I C G" .z/.z//
8z 2 C# ; 8i 2 Œ0; Ki ; 964
Proof of Theorem 4 : The closed-loop system can be written as P 8 x.k C 1/ D Ax.k/ C m Bi .Fi .k i // ˆ ˆ < .k C 1/ D .A C BF CiD1 KC / .k/ KC x.k/ " (28) x. / D . /; 8 2 Œ Ki ; 0 ˆ ˆ : . / D . /; 8 2 Œ Ki ; 0:
[7] V. K APILA AND G. G RIGORIADIS, Eds., Actuator saturation control, Marcel Dekker, 2002. [8] V.L. K HARITONOV, S.-I. N ICULESCU , J. M ORENO , AND W. M ICHIELS, “Static output feedback stabilization: necessary conditions for multiple delay controllers”, IEEE Trans. Aut. Contr., 50(1), 2005, pp. 82–86. [9] N. L EHTOMAKI , N.R.S ANDELL J R ., AND M. ATHANS, “Robustness results in linear-quadratic Gaussian based multivariable control designs”, ieee, 26(1), 1981, pp. 75–93. [10] F. M AZENC , S. M ONDIE , AND S.-I. N ICULESCU, “Global asymptotic stabilization for chains of integrators with a delay in the input”, IEEE Trans. Aut. Contr., 48(1), 2003, pp. 57–63. [11] W. M ICHIELS AND D. ROOSE, “Global stabilization of multiple integrators with time delay and input constraints”, in Proceedings of the 3th Workshop on Time-Delay Systems (TDS2001), T. Abdallah, K. Gu, and S.-I. Niculescu, eds., Santa Fe, NM, 2001, Pergamon, pp. 243–248. [12] S.I. N ICULESCU AND W. M ICHIELS, “Stabilizing a chain of integrators using multiple delays”, IEEE Trans. Aut. Contr., 49(5), 2004, pp. 802–807. [13] S.-I. N ICULESCU, “On delay-dependent stability under model transformations of some neutral linear systems”, Int. J. Contr., 74(6), 2001, pp. 609–617. [14] A. S ABERI AND A.A. S TOORVOGEL, “Special issue on control problems with constraints”, Int. J. Robust & Nonlinear Control, 9(10), 1999, pp. 583–734. [15] A. S ABERI , A.A. S TOORVOGEL , AND P. S ANNUTI, Control of linear systems with regulation and input constraints, Communication and Control Engineering Series, Springer Verlag, Berlin, 2000. [16] U. S HAKED, “Guaranteed stability margins for the discrete time linear quadratic optimal regulator”, IEEE Trans. Aut. Contr., 31(2), 1986, pp. 162–165. [17] S. TARBOURIECH AND G. G ARCIA, Eds., Control of uncertain systems with bounded inputs, vol. 227 of Lecture notes in control and information sciences, Springer Verlag, 1997. [18] X. WANG , A. S ABERI , AND A.A. S TOORVOGEL, “Stabilization of linear system with input saturation and unknown delay”, Submitted for publication, available at https://files.nyu.edu/xw665/ public/paper9.pdf, 2012. [19] X. WANG , A.A. S TOORVOGEL , A. S ABERI , AND P. S ANNUTI, “Discrete-time H2 and H1 low-gain theory”, Int. J. Robust & Nonlinear Control, 22(7), 2012, pp. 743–762. [20] K. YAKOUBI AND Y. C HITOUR, “On the Stabilization of Linear Discrete-Time Delay Systems Subject to Input Saturation”, in Advanced Strategies in Control Systems with Input and Output Constraints, Sophie Tarbouriech, Germain Garcia, and Adolf Glattfelder, eds., vol. 346 of Lecture Notes in Control and Information Sciences, Springer Berlin / Heidelberg, 2007, pp. 421–455. [21] J. Z HANG , C. K NOSPE , AND P. T SIOTRAS, “New results for the analysis of linear systems with time-invariant delays”, Int. J. Robust & Nonlinear Control, 13(12), 2003, pp. 1149–1175. [22] B IN Z HOU AND Z ONGLI L IN, “Parametric Lyapunov Equation Approach to Stabilization of Discrete-Time Systems With Input Delay and Saturation”, IEEE Trans. Circ. & Syst.-I Regular papers, 58(11), 2011, pp. 2741 –2754. [23] B. Z HOU , Z. L IN , AND G.R. D UAN, “Global and semi-global stabilization of linear systems with multiple delays and saturations in the input”, SIAM J. Contr. & Opt., 53(8), 2010, pp. 5294–5332.
Suppose Ki ’s satisfy the bound Ki < 3!i . By Theorem 2, max there exists an "1 such that for " 2 .0; "1 , the closed-loop system without saturation is asymptotically stable. Then the local stability of (28) for " "1 follows. Define two linear time invariant operators g"m and ı with z transform G"m .z/ D .z/ D I
F" .zI
A
BF" /
D.z/ D diagf1
1
KC.zI
z
A
KC /
1
B
i m gi D1 :
From the proof of Theorem 2, we know that (25) holds for " "1 . There exists a > 0 such that .I C G"m .z/.z// > ; 8z 2 C# ; 8i 2 Œ0; Ki ;
(29)
where is independent of " provided that " "1 . It follows from Lemma 7 that G"m .z/ ! G" .z/ uniformly on C# where G" .z/ D F" .zI A BF" / 1 B. Since kG" k1 1 C for any " 2 .0; 1 with given by (15), there exists an "2 such that kG"m k1 2.1 C /:
(30)
Given (29), (30) and Lemma 5 hold, we can use exactly the same argument as in the proof of Theorem 3 to prove that there exists an " "1 such that for " 2 .0; " , kF" .k
K/k 1; 8k 0; . ; / 2 W :
V. C ONCLUSION In this paper, the semi-global stabilization problems for general uncritically unstable systems subject to input saturation and multiple unknown delays are solved. We propose upper bounds on delays based on a frequency-domain stability criterion for linear discrete time-delay system and constructed a low-gain state feedback and compensator to achieve the semi-global stabilization with feasible delays. R EFERENCES [1] D.S. B ERNSTEIN AND A.N. M ICHEL, “Special Issue on saturating actuators”, Int. J. Robust & Nonlinear Control, 5(5), 1995, pp. 375– 540. [2] H O -L IM C HOI AND J ONG -TAE L IM, “Stabilization of a chain of integrators with an unknown delay in the input by adaptive output feedback”, IEEE Trans. Aut. Contr., 51(8), 2006, pp. 1359–1363. [3] R. DATKO, “A procedure for determination of the exponential stability of certain differential-difference equations”, Quart. Appl. Math., 36(3), 1978, pp. 279–292. [4] E. F RIDMAN, “New Lyapunov-Kasovskii functionals for stability of linear retarded and neutral type systems”, Syst. & Contr. Letters, 43(4), 2001, pp. 309–319. [5] K. G U , V. L. K HARITONOV, AND J. C HEN, Stability of time-delay systems, Birkhäuser, Boston, MA, 2003. [6] T. H U AND Z. L IN, Control systems with actuator saturation: analysis and design, Birkhäuser, 2001.
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