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Dynamic output feedback guaranteed cost control for linear systems with interval time-varying delays in states and outputs
Applied Mathematics and Computation 218 (2012) 10697–10707
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Dynamic output feedback guaranteed cost control for linear systems with interval time-varying delays in states and outputs M.V. Thuan a,⇑, V.N. Phat b, H.M. Trinh c a
Department of Mathematics, Thainguyen University, Thainguyen, Viet Nam Institute of Mathematics, VAST, 18 Hoang Quoc Viet Road, Hanoi 10307, Viet Nam c School of Engineering, Deakin University, Geelong, VIC 3217, Australia b
a r t i c l e
i n f o
Keywords: Dynamic output feedback Exponential stabilization Interval time-varying delays Linear matrix inequality Lyapunov function
a b s t r a c t This paper addresses the dynamic output feedback guaranteed cost control problem of a class of time-delay systems where the state and output contain interval nondifferentiable time-varying delays. The proposed controller uses only the delayed output measurement to stabilize the closed-loop system and guarantee an adequate level of system performance. By constructing a set of multiple Lyapunov–Krasovskii functionals which include triple-integral terms, a new criterion for the existence of dynamic output feedback guaranteed cost controllers is established and expressed in terms of linear matrix inequality (LMI). A numerical example is given to illustrate the obtained results. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction In many practical systems, it is desirable to design control systems which are not only asymptotically or exponentially stable but can also guarantee an adequate level of system performance. Guaranteed cost control [1–3] has the advantage of providing an upper bound on a given system performance index and thus the system performance degradation incurred by the uncertainties or time delays is guaranteed to be less than this bound. Based on this idea, dynamic output feedback control for guaranteed cost stabilization has been a subject of considerable research interest in the literature. The Lyapunov–Krasovskii functional approach together with LMI-based design methods have been among the popular and effective tool in the design of dynamic output feedback controllers for time-delay systems. Nevertheless, despite such diversity of results available, most existing work either assumed that the time delays are constant or differentiable and that the controlled outputs of the systems are independent of time delays (see, e.g. [4–10]). Although, in some cases, delay feedback guaranteed cost control for systems with timevarying delays were considered in [11–15], the approach used there can not be applied to systems with interval, non-differentiable time-varying delays. So far, there has been no reported work on the design of dynamic output feedback controllers for linear systems with interval, non-differentiable time-varying delays and where the outputs also subjected to such delays. This paper addresses the problem of designing dynamic output feedback guaranteed cost controllers for a more general class of timedelay systems where the state and output contain interval, nonsmooth time-varying delays. In fact, this problem is difficult to solve in convex programming approach; particularly, when the time-varying delays are interval, non-differentiable and the output is also subjected to such time-varying delay functions. It is clear that the application of any dynamic output feedback controller to such time-delay systems would lead to closed-loop systems with multiple delays. The difficulties then arise when one attempts to derive exponential LMI stabilizability conditions and to extract the controller’s parameters for these systems. To solve the problem, we introduce a new set of Lyapunov–Krasovskii functionals that can avoid from taking any derivative of the delay functions, and due to the fact that the delays are intervals, we use a combination of Lyapunov–Krasovskii functionals associated with the lower and upper bounds of the delays. With this new set of Lyapunov–Krasovskii functionals that includes ⇑ Corresponding author. E-mail address: [email protected] (M.V. Thuan). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.04.039
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triple-integral terms [16,17], we are able to derive a new existence condition, which can be expressed as a delay-dependent LMI condition, therefore leading to a straightforward derivation of all the controller’s parameters. Our newly proposed dynamic output feedback controller can ensure exponential stability of the closed-loop system while also can guarantee an adequate level of system performance, which is expressed in terms of a guaranteed cost function. 2. Problem formulation and preliminaries The following notations will be used throughout this paper: Rþ denotes the set of all real non-negative numbers; Rn denotes the n dimensional space with the scalar product h; i and the vector norm k k; Rnr denotes the space of all matrices of ðn rÞdimensions; AT denotes the transpose of matrix A; A is symmetric if A ¼ AT ; I denotes the identity matrix; kðAÞ denotes the set of all eigenvalues of A; kmax ðAÞ ¼ maxfRek; k 2 kðAÞg; xt :¼ fxðt þ sÞ : s 2 ½h; 0g denotes the segment of the trajectory xðtÞ with the norm kxt k ¼ sups2½h;0 kxðt þ sÞk; C1 ð½0; t; Rn Þ denotes the set of all Rn valued continuous differentiable functions on ½0; t; L2 ð½0; t; Rm Þ denotes the set of all the Rm valued square integrable functions on ½0; t; Matrix A is called semi-positive definite ðA P 0Þ if hAx; xi P 0, for all x 2 Rn ; A is positive definite ðA > 0Þ if hAx; xi > 0 for all x – 0; A > B means A B > 0. The notation diagf. . .g stands for a block-diagonal matrix. The symmetric term in a matrix is denoted by ⁄ Consider a control system with time-varying delays of the form
_ xðtÞ ¼ AxðtÞ þ Dxðt h1 ðtÞÞ þ BuðtÞ; yðtÞ ¼ Cxðt h2 ðtÞÞ; xðtÞ ¼ /ðtÞ; t 2 ½d; 0;
ð2:1Þ
1 ; h 2 g; xðtÞ 2 Rn is the state; uðtÞ 2 Rm is the control, yðtÞ 2 Rr is the observation output, A; D; B; C are given where d ¼ maxfh _ matrices of appropriate dimensions and /ðtÞ 2 C1 ð½d; 0; Rn Þ is the initial with its norm k/kC1 ¼ supt2½d;0 fk/ðtÞk; k/ðtÞkg. The time-varying delay functions h1 ðtÞ and h2 ðtÞ satisfying
; 0 < h1 < h1 ðtÞ 6 h 1
: 0 < h2 < h2 ðtÞ 6 h 2
It is worth noting that the time delay is assumed to be a continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, but the delay function is bounded but not restricted to being zero. In this paper, we design the following dynamic output feedback controller in order to exponentially stabilize system (2.1):
_ nðtÞ ¼ A1 nðtÞ þ B1 yðtÞ; uðtÞ ¼ C 1 nðtÞ;
t P 0;
nðtÞ ¼ 0;
ð2:2Þ
t 2 ½d; 0;
where nðtÞ 2 Rn and A1 ; B1 ; C 1 are constant matrices of appropriate dimensions to be determined later. The performance index associated with the system (2.1) is the following quadratic function
J¼
Z
1
½xT ðtÞQxðtÞ þ uT ðtÞRuðtÞdt;
ð2:3Þ
0
where Q 2 Rnn ; R 2 Rmm are given symmetric positive definite matrices. Applying controller (2.2) to the system (2.1) results in the closed-loop system
z_ ðtÞ ¼ AzðtÞ þ D1 zðt h1 ðtÞÞ þ D2 zðt h2 ðtÞÞ; zðtÞ ¼ /ðtÞ; where zðtÞ ¼ xT ðtÞ
A¼
ð2:4Þ
t 2 ½d; 0; nT ðtÞ
A BC 1 ; 0 A1
T
; /ðtÞ ¼ /ðtÞT
D1 ¼
D 0 ; 0 0
0
T
D2 ¼
and
0
0
B1 C
0
:
The corresponding closed-loop cost function is
J¼
Z
0
1
½xT ðtÞQxðtÞ þ nT ðtÞC T1 RC 1 nðtÞdt:
ð2:5Þ
The objective of this paper is to design a dynamic output feedback controller (2.2) for system (2.1) and cost function (2.3) such that the resulting closed-loop system (2.4) is exponentially stable and the closed-loop value of the cost function (2.3) satisfies J 6 J , where J is some specified constant. Definition 2.1. Given a > 0. The zero solution of system (2.1) is a-stable if there exists a positive number b > 0 such that every solution xðt; /Þ satisfies the condition
kxðt; /Þk 6 beat k/k;
8t P 0:
Definition 2.2. For the observation control system (2.1) and cost function (2.3), if there exist a control law u ðtÞ and a positive constant J such that the zero solution of the closed-loop system (2.4) is exponentially stable and the closed-loop value (2.3) satisfies J 6 J , then J is said to be a guaranteed cost index, u ðtÞ is said to be a guaranteed cost control law of the system and its corresponding cost function.
M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
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We introduce the following technical well-known propositions for the proof of the main result. Proposition 2.1. For any x; y 2 Rn and positive symmetric definite matrix N 2 Rnn , we have
2yT x 6 xT N1 x þ yT Ny: 0 0 and scalar h; h; defined, then
Z
t
xT ðsÞZxðsÞds 6
th
Z
h
Z
h
1 h
Z
t
xðsÞds
T Z Z
th
t
xT ðsÞZxðsÞds ds 6
tþs
xðsÞds
t th
Z
2 2 h2 h
h
h
Z
ð2:6Þ !T
t
xðsÞds ds tþs
Z
Z
h
h
Z
!
t
xðsÞds ds
ð2:7Þ
tþs
Proposition 2.3 (Schur complement Lemma [18]). Given constant matrices X; Y; Z with appropriate dimensions satisfying X ¼ X T ; Y ¼ Y T > 0. Then X þ Z T Y 1 Z < 0 if and only if
!
X
ZT
Z
Y
< 0:
3. Main results In this section, using the Lyapunov function method, we establish a delay-dependent criterion for dynmic output feedback stabilization of system (2.1) with interval time-varying delays. Before introducing main result, the notations of several matrix variables are defined for simplicity. Let us denote k ¼ kmin ðP 1 Þ; 1 2 1 1 2 2 K ¼ kmax ðP 1 Þ þ h1 kmax ðP 1 Q 1 P 1 Þ þ ðh1 h1 Þkmax ðP1 Q 2 P1 Þ þ h1 kmax ðP1 S1 P1 Þ þ ðh21 h1 Þkmax ðP1 S2 P1 Þ þ h1 kmax ðP1 S3 P1 Þ 2 2 2 1 2 1 3 1 3 2 3 1 1 1 1 1 1 1 1 þ ðh1 h1 Þkmax ðP S4 P Þ þ h1 kmax ðP R1 P Þ þ ðh1 h1 Þkmax ðP R2 P Þ þ þh2 kmax ðP U 1 P Þ þ ðh2 h2 Þkmax ðP1 U 2 P 1 Þ 2 6 6 1 2 1 2 1 2 1 2 1 3 2 2 1 1 1 1 1 1 1 1 þ h2 kmax ðP 1 W 1 P1 Þ þ ðh 2 h2 Þkmax ðP W 2 P Þ þ h2 kmax ðP W 3 P Þ þ ðh2 h2 Þkmax ðP W 4 P Þ þ h2 kmax ðP H1 P Þ 2 2 2 2 6 1 3 3 h2 Þkmax ðP 1 H2 P 1 Þ; þ ðh 6 2 N ¼ ðNij Þ1515 ;
W ¼ ½ QP 0 0 0 0 0 0 0 0 0 0 0 0 0 0 T ; N11 ¼ AP þ PAT þ 2aP þ Q 1 þ U 1 þ h1 S3 þ ðh1 h1 ÞS4 þ h2 W 3 þ ðh2 h2 ÞW 4
h Þ 1 2ah2 ðh 2 2 e W 1 2e4ah2 H1 2 e4ah2 H2 ; h2 ðh2 þ h2 Þ
N22 ¼
2
2
h1 h1
e2ah1 S2 ; N33 ¼
N44 ¼ e2ah1 Q 2 e2ah1 Q 1
N55 ¼ e2ah1 Q 2
1
h1 h1
h2 h2
e2ah2 W 2 ;
1 2ah1 1 e S1 e2ah1 S2 ; h1 h1 h1
e2ah1 S2 ;
N66 ¼ e2ah2 U 2 e2ah2 U 1
1 2ah2 1 e W1 e2ah2 W 2 ; h2 h2 h2
1 e2ah2 W 2 ; h2 h2 1 2 N88 ¼ e2ah1 S3 2 e4ah1 R1 ; h1 h1 1 2 N99 ¼ e2ah1 S4 e4ah1 R2 ; 2 h2 h1 h1 h 1 1
N77 ¼ e2ah2 U 2
N10;10 ¼
1 2ah2 2 e W 3 2 e4ah2 H1 ; h2 h2
N11;11 ¼
1
h2 h2
e2ah2 W 4
2 e4ah2 H2 ; 2 h2 h 2 2
1 h1 Þ 1 2ah1 ðh e S1 2e4ah1 R1 2 e4ah1 R2 h1 ðh1 þ h1 Þ
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M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
1 2
1 2
2
1 2
2
2
1 2
2
2 h ÞH2 2P; N12;12 ¼ h1 S1 þ ðh 1 h1 ÞS2 þ h1 R1 þ ðh21 h1 ÞR2 þ h2 W 1 þ ðh 2 h2 ÞW 2 þ h2 H1 þ ðh 2 2 1 2ah1 e S1 ; N15 ¼ 0; h1 1 2ah2 2 4ah1 2 N16 ¼ e W 1 ; N17 ¼ 0; N18 ¼ e R1 ; N19 ¼ e4ah1 R2 ; h2 h1 h1 þ h1 2 2 N1;10 ¼ e4ah2 H1 ; N1;11 ¼ e4ah2 H2 ; N1;12 ¼ PAT ; h2 h2 þ h2
N13;13 ¼ Y þ Y T þ 2aP þ BRBT ; N12 ¼ DP; N13 ¼ 0; N14 ¼
X1;13 ¼ BBT ; N1;14 ¼ N1;15 ¼ 0; N23 ¼ 0; N24 ¼ N25 ¼
1
h1 h1
e2ah1 S2 ; N26 ¼ N27 ¼ N28 ¼ N29 ¼ N2;10 ¼ N2;11 ¼ 0; 1 e2ah2 W 2 ; h2 h2
N2;12 ¼ PDT ; N2;13 ¼ N2;14 ¼ N2;15 ¼ 0; N34 ¼ N35 ¼ 0; N36 ¼ N37 ¼
N38 ¼ N39 ¼ N3;10 ¼ N3;11 ¼ N3;12 ¼ N3;13 ¼ 0; N3;14 ¼ PC T ; N3;15 ¼ 0; N45 ¼ N46 ¼ N47 ¼ N48 ¼ N49 ¼ N4;10 ¼ N4;11 ¼ N4;12 ¼ N4;13 ¼ N4;14 ¼ N4;15 ¼ 0; N56 ¼ N57 ¼ N58 ¼ N59 ¼ N5;10 ¼ N5;11 ¼ N5;12 ¼ N5;13 ¼ N5;14 ¼ N5;15 ¼ 0; N67 ¼ ¼ N6;15 ¼ 0; N78 ¼ ¼ N7;15 ¼ 0; N89 ¼ ¼ N8;15 ¼ 0; N9;10 ¼ ¼ N9;15 ¼ 0; N10;11 ¼ ¼ N10;15 ¼ 0; N11;12 ¼ ¼ N11;15 ¼ 0; N12;13 ¼ BBT ; N12;14 ¼ N12;15 ¼ 0; N13;14 ¼ 0; N13;15 ¼ PC T ; N14;14 ¼ N15;15 ¼ I; N14;15 ¼ 0:
Theorem 3.1. For given Q > 0; R > 0 and a > 0, there exists a dynamic output feedback controller (2.2) for system (2.1) if there exist symmetric positive definite matrices P; Q 1 ; Q 2 ; U 1 ; U 2 ; R1 ; R2 ; H1 ; H2 ; Si ; W i ; i ¼ 1; . . . ; 4, and matrix Y satisfying the following LMI
N WT
W
< 0:
Q
ð3:1Þ
The guaranteed constant of the cost function is
J ¼ Kk/k2 : Proof. Let us denote
Q i ¼ P1 Q i P1 ; 1
1
Sj ¼ P Sj P ;
U i ¼ P1 U i P1 ; 1
Ri ¼ P1 Ri P1 ;
1
W j ¼ P W jP ;
Hi ¼ P1 Hi P1 ;
i ¼ 1; 2;
j ¼ 1; 2; 3; 4:
Construct a Lyapunov–Krasovskii functional as follows:
Vðt; zt Þ ¼
9 X V i ðt; zt Þ; i¼1
where V 1 ¼ xT ðtÞP1 xðtÞ þ nT ðtÞP1 nðtÞ; Z t Z V2 ¼ e2aðstÞ xT ðsÞQ 1 xðsÞds þ V3 ¼
Z
0
h1
V4 ¼
Z
0
h1
V5 ¼
Z
0
h1
V6 ¼
Z
Z
t
Z
Z
t
e2aðstÞ xT ðsÞS3 xðsÞds ds þ
Z
0
Z
t
V8 ¼
Z
0
h2
V9 ¼
Z
0
h2
Z
t
Z
th2
th 2
t
h
0
e2aðstÞ xT ðsÞW 3 xðsÞds ds þ Z
t
tþs
_ sÞds ds; e2aðstÞ x_ T ðsÞS2 xð
Z
h 1
t
e2aðstÞ xT ðsÞS4 xðsÞds ds;
tþs
Z
h1
h 1
Z
0
Z
t
_ sÞds dsdh; e2aðsþstÞ x_ T ðsÞR2 xð
tþs
h
e2aðstÞ xT ðsÞU 2 xðsÞds;
_ sÞds ds þ e2aðstÞ x_ T ðsÞW 1 xð
tþs
Z
h1
t
tþs
_ sÞds dsdh þ e2aðsþstÞ x_ T ðsÞR1 xð
tþs
Z
Z
tþs
h
Z
h1
h 1
tþs
e2aðstÞ xT ðsÞU 1 xðsÞds þ
h2
Z
tþs
t
0
e2aðstÞ xT ðsÞQ 2 xðsÞds;
_ sÞds ds þ e2aðstÞ x_ T ðsÞS1 xð
th2
V7 ¼
th1
th 1
th1
Z
h2
h 2
Z
h2
h 2
Z
t
_ sÞds ds; e2aðstÞ x_ T ðsÞW 2 xð
tþS
Z
t
e2aðstÞ xT ðsÞW 4 xðsÞds ds;
tþs
_ sÞds dsdh þ e2aðsþstÞ x_ T ðsÞH1 xð
Z
h2 h 2
Z h
0
Z
t
tþs
_ sÞds dsdh: e2aðsþstÞ x_ T ðsÞH2 xð
ð3:2Þ
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Taking the time derivative of V i ; i ¼ 1; . . . ; 9 along the trajectory of system (2.4) yields h i h i T T V_ 1 ¼ xT ðtÞ P 1 A þ A P 1 xðtÞ þ 2xT ðtÞP 1 Dxðt h1 ðtÞÞ þ 2xT ðtÞP 1 BC 1 nðtÞ þ nT ðtÞ P 1 A1 þ A1 P1 nðtÞ þ 2nT ðtÞP 1 B1 Cxðt h2 ðtÞÞ; ÞQ xðt h Þ; V_ 2 ¼ 2aV 2 þ xT ðtÞQ 1 xðtÞ e2ah1 xT ðt h1 ÞQ 1 xðt h1 Þ þ e2ah1 xT ðt h1 ÞQ 2 xðt h1 Þ e2ah1 xT ðt h 1 1 2 Z Z h i t th1 h ÞS xðtÞ _ _ V_ 3 6 2aV 3 þ x_ T ðtÞ h1 S1 þ ðh e2ah1 x_ T ðsÞS1 xðsÞds e2ah1 x_ T ðsÞS2 xðsÞds; 1 1 2 _ th1
Z h i h ÞS xðtÞ e2ah1 V_ 4 6 2aV 4 þ xT ðtÞ h1 S3 þ ðh 1 1 4
t
xT ðsÞS3 xðsÞds e
2ah 1
Z
th 1
th1 th 1
th1
xT ðsÞS4 xðsÞds;
Z 0 Z t Z h1 Z t 1 2 1 2 2 4ah1 _ _ _ V_ 5 6 2aV 5 þ x_ T ðtÞ h1 R1 þ ðh x_ T ðsÞR1 xðsÞdsdh e4ah1 x_ T ðsÞR2 xðsÞdsdh; 1 h1 ÞR2 xðtÞ e 2 2 h h1 tþh tþh 1 2 ÞU 2 xðt h 2 Þ; V_ 6 ¼ 2aV 6 þ xT ðtÞU 1 xðtÞ e2ah2 xT ðt h2 ÞU 1 xðt h2 Þ þ e2ah2 xT ðt h2 ÞU 2 xðt h2 Þ e2ah2 xT ðt h Z t Z th2 h ÞW xðtÞ _ _ V_ 7 6 2aV 7 þ x_ T ðtÞ h2 W 1 þ ðh e2ah2 x_ T ðsÞW 1 xðsÞds e2ah2 x_ T ðsÞW 2 xðsÞds; 2 2 2 _ th 2
th2
h ÞW xðtÞ e2ah2 V_ 8 6 2aV 8 þ x ðtÞ h2 W 3 þ ðh 2 2 4 T
Z
t
2ah 2
T
x ðsÞW 3 xðsÞds e
Z
th2
th 2
th2
xT ðsÞW 4 xðsÞds;
Z 0 Z t Z h2 Z t 1 2 1 2 2 4ah2 _ _ _ V_ 9 6 2aV 9 þ x_ T ðtÞ h2 H1 þ ðh x_ T ðsÞH1 xðsÞdsdh e4ah2 x_ T ðsÞH2 xðsÞdsdh: 2 h2 ÞH2 xðtÞ e 2 2 h h2 tþh tþh 2
Therefore,
h i 1 h1 ÞS4 þ h2 W 3 þ ðh 2 h2 ÞW 4 xðtÞ V_ þ 2aV 6 xT ðtÞ P1 A þ AT P1 þ 2aP1 þ Q 1 þ U 1 þ h1 S3 þ ðh h i þ nT ðtÞ P1 A1 þ AT1 P1 þ 2aP 1 nðtÞ þ 2xT ðtÞP1 Dxðt h1 ðtÞÞ þ 2xT ðtÞP1 BC 1 nðtÞ þ 2nT ðtÞP1 B1 Cxðt h2 ðtÞÞ h i h i 1 Þ e2ah1 Q xðt h 1 Þ þ xT ðt h1 Þ e2ah1 Q 2 e2ah1 Q 1 xðt h1 Þ þ xT ðt h 2 h i 2 Þ 2 Þ e2ah2 U 2 xðt h þ xT ðt h2 Þ e2ah2 U 2 e2ah2 U 1 xðt h2 Þ þ xT ðt h 2 h2 ÞR2 þ h2 W 1 þ ðh 2 h2 ÞH2 xðtÞ 1 h1 ÞS2 þ 1 h2 R1 þ 1 ðh 2 h2 ÞW 2 þ 1 h2 H1 þ 1 ðh _ þ x_ T ðtÞ h1 S1 þ ðh 1 2 2 1 2 1 2 2 2 2 Z t Z th1 Z t _ ds e2ah1 _ ds e2ah1 x_ T ðsÞS1 xðsÞ x_ T ðsÞS2 xðsÞ e2ah1 xT ðsÞS3 xðsÞ ds th 1
th1
e2ah1 e2ah2
Z
th1
th 1
Z
t
xT ðsÞS4 xðsÞ ds e4ah1
Z
h1
_ ds e2ah2 x_ T ðsÞW 1 xðsÞ
Z
Z
th2
th 2
th2
th 2
th2
e2ah2
th1
Z
0
Z
xT ðsÞW 4 xðsÞ ds e4ah2
0
t
_ ds dh e4ah1 x_ T ðsÞR1 xðsÞ
Z
Z
Z
h 1
tþh
_ ds e2ah2 x_ T ðsÞW 2 xðsÞ
h1
t
_ ds dh x_ T ðsÞR2 xðsÞ tþh
t
xT ðsÞW 3 xðsÞ ds
th2
Z
t
_ ds dh e4ah2 x_ T ðsÞH1 xðsÞ
Z
tþh
h2
h2
h 2
Z
t
_ ds dh: x_ T ðsÞH2 xðsÞ
Using Proposition 2.2, the following inequalities hold Z t T Z t Z t 1 1 T _ _ _ e2ah1 e2ah1 S1 x_ T ðsÞS1 xðsÞds 6 xðsÞds xðsÞds ¼ ½xðtÞ xðt h1 Þ e2ah1 S1 ½xðtÞ xðt h1 Þ; h1 th1 h1 th1 th1
e2ah1
Z
th1
th 1
_ ds ¼ e2ah1 x_ T ðsÞS2 xðsÞ
Z
th1 ðtÞ
th 1
Z
1
6 h1 h1
!T
th1 ðtÞ
_ ds xðsÞ
Z
e
h1 h1
th1 ðtÞ
!T
th1
_ ds xðsÞ
Z
th1
2ah 1
e
2ah 1
S2
S2
Z
ð3:4Þ
_ ds x_ T ðsÞS2 xðsÞ
th1 ðtÞ
th 1
1
1
_ ds e2ah1 x_ T ðsÞS2 xðsÞ
ð3:3Þ
tþh
!
th1 ðtÞ
_ ds xðsÞ
th 1
Z
!
th1
_ ds xðsÞ
th1 ðtÞ
1 ÞT e2ah1 S2 xðt h1 ðtÞÞ xðt h 1 Þ xðt h1 ðtÞÞ xðt h
¼ h1 h1 1 T ½xðt h1 Þ xðt h1 ðtÞÞ e2ah1 S2 ½xðt h1 Þ xðt h1 ðtÞÞ; h1 h1
ð3:5Þ
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M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
e2ah1
Z
t
xT ðsÞS3 xðsÞ ds 6
th1
¼
e2ah1
Z
th1
1 h1 1 h1
Z Z
T Z xðsÞ ds e2ah1 S3
t
th1
th1
T Z xðhÞ dh e2ah1 S3
t
th1
Z
Z
0
Z
t
2
e
Z
h1
Z
h 1
t
_T
_ x ðsÞR2 xðsÞdsdh 6
tþh
¼
e2ah2
Z
t
_ ds 6 x_ T ðsÞW 1 xðsÞ
th2
1 h2
2 h1
ð3:6Þ
Z
!
th1
xðsÞ ds
th 1
!
th1
xðhÞ dh ;
tþh
Z
0
T
t
_ ds dh xðsÞ
t
tþh
h1
Z
ð3:7Þ
th 1
T Z _ ds dh e4ah1 R1 xðsÞ
t
h1 xðtÞ
xðhÞ dh e4ah1 R1 h1 xðtÞ
Z
th1
Z
2 2 h2 h 1 1 2
h1
Z
h 1
!T
t
_ xðsÞdsdh
4ah 1
e
Z
R2
1 h1 ÞxðtÞ ðh
Z
#T
th1
ð3:8Þ
!
t
_ xðsÞdsdh
tþh
"
th 1
th2
Z
1 h1 ÞxðtÞ e4ah1 R2 ðh
xðhÞdh t
h1
h 1
tþh
T Z _ ds e2ah2 W 1 xðsÞ
t
xðhÞ dh ;
t
th1
"
2 h2 h 1 1
Z
Z
0
h1
2
¼
4ah 1
Z
Z
e2ah1 S4
xðhÞ dh
th 1
h1
tþh
h1
!T
th1
e2ah1 S4
xðsÞ ds th 1
2
_ ds dh 6 x_ T ðsÞR1 xðsÞ
!T
th1
Z
1 ¼ h1 h1 e4ah1
xðhÞ dh ;
t
th1
1 xT ðsÞS4 xðsÞ ds 6 h1 h1
th 1
xðsÞ ds
t
Z
#
th1
xðhÞdh ; th 1
_ ds xðsÞ
th2
1 T ¼ ½xðtÞ xðt h2 Þ e2ah2 W 1 ½xðtÞ xðt h2 Þ; h2 Z th2 Z th2 ðtÞ Z th2 _ ds ¼ e2ah2 _ ds e2ah2 _ ds e2ah2 x_ T ðsÞW 2 xðsÞ x_ T ðsÞW 2 xðsÞ x_ T ðsÞW 2 xðsÞ th 2
th 2
Z
1 6 h2 h2
th2 ðtÞ
!T
th2 ðtÞ
th 2
1 h2 h2
!T
th2
Z
e2ah2 W 2
_ ds xðsÞ
Z
e2ah2 W 2
_ ds xðsÞ
!
th2 ðtÞ
_ ds xðsÞ !
th2
_ ds xðsÞ
th2 ðtÞ
2 ÞT e2ah2 W 2 xðt h2 ðtÞÞ xðt h 2 Þ xðt h2 ðtÞÞ xðt h
¼ h2 h2 1 T ½xðt h2 Þ xðt h2 ðtÞÞ e2ah2 W 2 ½xðt h2 Þ xðt h2 ðtÞÞ; h2 h2 e2ah2
Z
t
xT ðsÞW 3 xðsÞ ds 6
th2
1 h2
1 ¼ h2 2ah 2
e
Z
th2
T
Z Z
T Z xðsÞ ds e2ah2 W 3
t
th2 t
0
h2
Z
t
_ ds dh 6 x_ T ðsÞH1 xðsÞ
tþh
¼
2 2 2
h2
!T
th2
xðsÞ ds
th 2
Z
xðhÞ dh
Z
0
h2
Z
2ah 2
e !T
th2
th 2
2
h2
t
ð3:11Þ
xðsÞ ds xðhÞ dh ;
ð3:12Þ
th2
Z
1
¼ h2 h2 Z
th2
th2
1
e4ah2
t
T Z xðhÞ dh e2ah2 W 3
x ðsÞW 4 xðsÞ ds 6 h2 h2
th 2
ð3:10Þ
th 2
Z
th2 ðtÞ
1
ð3:9Þ
W4
2ah 2
e
W4
Z Z
!
th2
xðhÞ dh ;
th 2
tþh
h2 xðtÞ
xðsÞ ds
th 2
T Z _ ds dh e4ah2 H1 xðsÞ
t
!
th2
0
h2
Z
t
th2
T
Z
ð3:13Þ
t
_ ds dh xðsÞ
tþh
xðhÞ dh e4ah2 H1 h2 xðtÞ
Z
t th2
xðhÞ dh ;
ð3:14Þ
10703
M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
4ah 2
e
Z
h2
Z
h 2
t
_T
_ ds dh 6 x ðsÞH2 xðsÞ
tþh
¼
Z
2 2 h2 h 2 2 2 2 h2 h 2 2
h2
h 2
Z
!T
t
_ ds dh xðsÞ
4ah 2
e
Z
H2
h ÞxðtÞ ðh 2 2
Z
th2
#T xðhÞ dh
Z
h 2
tþh
"
h2
4ah 2
e
th 2
!
t
_ ds dh xðsÞ
tþh
" H2
h ÞxðtÞ ðh 2 2
Z
#
th2
xðhÞ dh : th 2
ð3:15Þ Let us define
B1 ¼ PC T
ð3:16Þ
and using Proposition 2.1, we have
2nT ðtÞP1 B1 Cxðt h2 ðtÞÞ ¼ 2nT ðtÞC T Cxðt h2 ðtÞÞ 6 nT ðtÞC T CnðtÞ þ xT ðt h2 ðtÞÞC T Cxðt h2 ðtÞÞ:
ð3:17Þ
On the other hand, from the identity
_ AxðtÞ þ Dxðt h1 ðtÞÞ þ BC 1 nðtÞ xðtÞ ¼ 0; we have
_ ¼ 0: 2x_ T ðtÞP1 ½AxðtÞ þ Dxðt h1 ðtÞÞ þ BC 1 nðtÞ xðtÞ
ð3:18Þ
From (3.3)–(3.18), we have
h i _ zt Þ þ 2aVðt; zt Þ 6 gT ðtÞXgðtÞ xT ðtÞQxðtÞ þ nT ðtÞC T RC 1 nðtÞ ; Vðt; 1
ð3:19Þ
where
X ¼ ðXij Þ1313 ; 2
gðtÞ ¼ 4xT ðtÞxT ðt h1 ðtÞÞxT ðt h2 ðtÞÞxT ðt h1 ÞxT ðt h1 ÞxT ðt h2 ÞxT ðt h 2 Þ Z
T Z xðhÞ dh
t
th 2
th2
t
T Z xðhÞ dh
th1
th1
th 1
!T xðhÞ dh
3T
!T
th2
Z
x_ T ðtÞ nT ðtÞ5 ;
xðhÞ dh
with
X11 ¼ P1 A þ AT P1 þ 2aP1 þ Q 1 þ U 1 þ h1 S3 þ ðh1 h1 ÞS4 þ h2 W 3 þ ðh2 h2 ÞW 4
X22 X44
1 h1 Þ 2 h2 Þ ðh 1 ðh 2 e4ah1 R2 e2ah2 W 1 2e4ah2 H1 2 e4ah2 H2 þ Q ; h2 ðh1 þ h1 Þ ðh2 þ h2 Þ 2 2 ¼ e2ah1 S2 ; X33 ¼ e2ah2 W 2 þ C T C; h1 h1 h2 h2 1 2ah1 1 2ah1 2ah1 ¼e Q2 e Q1 e S1 e2ah1 S2 ; h1 h1 h1
X55 ¼ e2ah1 Q 2
1
h1 h1
e2ah1 S2 ;
X66 ¼ e2ah2 U 2 e2ah2 U 1
1 2ah2 1 e W1 e2ah2 W 2 ; h2 h2 h2
1 e2ah2 W 2 ; h2 h2 1 2 ¼ e2ah1 S3 2 e4ah1 R1 ; h1 h1 1 2 ¼ e2ah1 S4 e4ah1 R2 ; 2 2 h1 h1 h h
X77 ¼ e2ah2 U 2 X88 X99
1
X10;10
1
1 2 ¼ e2ah2 W 3 2 e4ah2 H1 ; h2 h2
X11;11 ¼
1
h2 h2
e2ah2 W 4
2 e4ah2 H2 ; 2 2 h2 h2
1 2ah1 e S1 2e4ah1 R1 h1
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M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
1 2
1 2
1 2
1 2
X12;12 ¼ h1 S1 þ ðh1 h1 ÞS2 þ h21 R1 þ ðh21 h21 ÞR2 þ h2 W 1 þ ðh2 h2 ÞW 2 þ h22 H1 þ ðh22 h22 ÞH2 2P1 ; X13;13 ¼ P1 A1 þ AT1 P1 þ 2aP1 þ C T C þ C T1 RC 1 ; X14 ¼
1 2ah1 e S1 ; h1
X16 ¼
1 2ah2 e W 1; h2
X1;10 ¼
X17 ¼ 0;
X18 ¼
2 4ah1 e R1 ; h1
2
X1;11 ¼ e4ah2 H2 ; h2 þ h2 1
X24 ¼ X25 ¼ e2ah1 S2 ; h1 h1
X2;12 ¼ DT P 1 ;
X13 ¼ 0;
X15 ¼ 0;
2 4ah2 e H1 ; h2
X23 ¼ 0;
X12 ¼ P1 D;
X2;13 ¼ 0;
2
X19 ¼ e4ah1 R2 ; h1 þ h1
X1;12 ¼ AT P1 ;
X1;13 ¼ P1 BC 1 ;
X26 ¼ X27 ¼ X28 ¼ X29 ¼ X2;10 ¼ X2;11 ¼ 0;
X34 ¼ X35 ¼ 0;
1
X36 ¼ X37 ¼ e2ah2 W 2 ; h2 h2
X38 ¼ X39 ¼ X3;10 ¼ X3;11 ¼ X3;12 ¼ X3;13 ¼ 0; X45 ¼ X46 ¼ X47 ¼ X48 ¼ X49 ¼ X4;10 ¼ X4;11 ¼ X4;12 ¼ X4;13 ¼ 0; X56 ¼ X57 ¼ X58 ¼ X59 ¼ X5;10 ¼ X5;11 ¼ X5;12 ¼ X5;13 ¼ 0; X67 ¼ ¼ X6;13 ¼ 0;
X78 ¼ ¼ X7;13 ¼ 0;
X10;11 ¼ X10;12 ¼ X10;13 ¼ 0;
X89 ¼ ¼ X8;13 ¼ 0;
X11;12 ¼ X11;13 ¼ 0;
X9;10 ¼ ¼ X9;13 ¼ 0;
1
X12;13 ¼ P BC 1 :
Now, pre- and post-multiply both sides of X with
diagfP; P; P; P; P; P; P; P; P; P; P; P; Pg and letting
A1 ¼ YP1 ;
C 1 ¼ BT P1
ð3:20Þ
and using Schur complement Lemma we have X < 0 is equivalent to condition (3.1). From condition (3.1), we have
h i _ zt Þ þ 2aVðt; zt Þ 6 xT ðtÞQxðtÞ þ nT ðtÞC T RC 1 nðtÞ : Vðt; 1 h
T
Since xT ðtÞQxðtÞ þ n
i
ðtÞC T1 RC 1 nðtÞ
_ zt Þ þ 2aVðt; zt Þ 6 0; Vðt;
ð3:21Þ
P 0, we have
t 2 Rþ :
ð3:22Þ
Integrating both sides of (3.22) from 0 to t gives
Vðt; zt Þ 6 Vð0; z0 Þe2at ;
8t P 0:
By simple computation, we have
Vðt; zt Þ P kmin ðP1 ÞkxðtÞk2 þ kmin ðP1 ÞknðtÞk2 P kmin ðP1 ÞkxðtÞk2 ¼ kkxðtÞk2 ;
8t P 0:
and
Vð0; z0 Þ 6 Kk/k2C1 : Hence
kkxðt; /Þk2 6 Vðt; zt Þ 6 Vð0; z0 Þe2at 6 Ke2at k/k2C1 ; then the solution xðt; /Þ of the system satisfies
kxðt; /Þk 6
rffiffiffiffi K at e k/kC1 ; k
8t P 0;
which implies the closed-loop system is astable. To find the guaranteed cost control value, from (3.21) we have
h i _ zt Þ 6 xT ðtÞQxðtÞ þ nT ðtÞC T RC 1 nðtÞ : Vðt; 1 Integrating both sides of (3.23) from 0 to s implies
Vðs; zs Þ Vð0; z0 Þ 6
Z sh 0
i xT ðtÞQxðtÞ þ nT ðtÞC T1 RC 1 nðtÞ dt:
ð3:23Þ
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M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
Hence
Z sh 0
i xT ðtÞQxðtÞ þ nT ðtÞC T1 RC 1 nðtÞ dt 6 Vð0; z0 Þ ¼ Kk/k2C1 :
Letting s ! þ1, we obtain
J¼
Z
þ1
h
0
i xT ðtÞQxðtÞ þ nT ðtÞC T1 RC 1 nðtÞ dt 6 Kk/k2C 1 :¼ J :
This completes the proof of the theorem.
h
Remark 3.1. Based on constructing a set of new Lyapunov–Krasovskii functionals (3.2), Theorem 3.1 provides sufficient condition for the closed-loop system to be exponentially stable with a prescribed decay rate a, while the existing methods of [6,7,9,10] provide asymptotic stability. Remark 3.2. Given any solution of the LMI (3.1) in Theorem 3.1, a corresponding stabilizing controller of the form (2.2) can be constructed as follows: Step 1. Solve LMI (3.1) and obtain matrices P and Y. Step 2. Compute the invertible matrix P1 . Step 3. Using the solution P1 ; Y of LMI (3.1), define A1 ; B1 ; C 1 by (3.16) and (3.20):
B1 ¼ PC T ;
A1 ¼ YP1 ;
C 1 ¼ BT P1 :
ð3:24Þ
To illustrate the design procedure, the following numerical example is given. Example 3.1. Consider a linear observation control system with time-varying delays (2.1), where
A¼
0:1
0
0:1 0:9
;
D¼
0
2
0:5 0:5
;
B¼
0:9 0:8
;
C ¼ ½ 1 0:5 ;
with
(
(
2
h1 ðtÞ ¼ 0:1 þ 0:3 sin t
if
t 2 I ¼ [kP0 ½2kp; ð2k þ 1Þp
h1 ðtÞ ¼ 0:1
if
t 2 Rþ n I ;
h2 ðtÞ ¼ 0:1 þ 0:3 cos2 t
if
t 2 I ¼ [kP0 ½2kp; ð2k þ 1Þp
h2 ðtÞ ¼ 0:1
if
t 2 Rþ n I ;
Note that h1 ðtÞ; h2 ðtÞ are non-differentiable, therefore, the stability criteria proposed in [10,11,14] are not applicable to this system. Given
a ¼ 1; Q ¼
0:1
0
0
0:2
;
R ¼ 1:5:
By using the LMI Toolbox in MATLAB 1 ¼ 0:4; h2 ¼ 0:1; h 2 ¼ 0:4, and h1 ¼ 0:1; h
P¼
1:1694 0:9763
;
Q1 ¼
17:2453
[19],
the
15:8160
;
LMI
(3.1)
Q2 ¼
in
Theorem
15:4517 14:4884
; 0:9763 8:8587 15:8160 100:6733 14:4884 75:5063 0:5235 2:5245 7:5832 9:5533 0:6904 0:0228 ; U2 ¼ ; R1 ¼ ; U1 ¼ 2:5245 52:3380 9:5533 161:6656 0:0228 0:9457 0:1569 1:1703 0:4091 0:1564 0:1569 1:1703 R2 ¼ ; H1 ¼ ; H2 ¼ ; 1:1703 15:3105 0:1564 2:6638 1:1703 15:3105 1:5361 1:6940 2:6633 2:5927 4:5852 0:5598 S1 ¼ ; S2 ¼ ; S3 ¼ ; 1:6940 7:7595 2:5927 11:4131 0:5598 13:0816 2:2391 3:1311 2:6749 3:2836 0:2623 0:2928 S4 ¼ ; W1 ¼ ; W2 ¼ ; 3:1311 50:7949 3:2836 35:4138 0:2928 12:8799 6:7110 0:5143 2:2391 3:1311 94:1305 55:3449 W3 ¼ ; W4 ¼ ; Y¼ : 0:5143 14:1273 3:1311 50:7949 35:6782 96:5021
3.1
is
satisfied
with
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M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
10
x1 x2
8 6 4 2 0 −2 −4
0
2
4
6
8
10
Time (sec) Fig. 1. Closed-loop trajectories of x1 ðtÞ and x2 ðtÞ.
Therefore, the dynamic output feedback controller for the system (2.1) is given as follows
94:3933 16:6507 A1 ¼ YP1 ¼ ; 43:6168 15:7005 0:6813 B1 ¼ PC T ¼ ; 3:4530
ð3:25Þ
C 1 ¼ BT P1 ¼ ½ 0:9306 0:1929 : This implies that the system (2.1) is 1 stable by the dynamic output feedback controller (3.25). Moreover, the solution of the closed-loop system satisfies
kxðt; /Þk 6 8:8068et k/kC1 and the optimal guaranteed cost of the closed-loop system is as follows:
J ¼ 8:6323k/k2C1 : Fig. 1 shows the simulation of the solutions x1 ðtÞ and x2 ðtÞ with the initial condition /ðtÞ ¼ ½10 5T ; t 2 ½0:4; 0. 4. Conclusion In this paper, we have investigated the dynamic output feedback guaranteed cost control for linear systems with interval time-varying delays in both states and outputs The designed dynamic output feedback stabilizing controller guarantees the closed-loop system to be exponential stable with a prescribed decay rate. By constructing a set of Lyapunov–Krasovskii functionals which contains triple-integral terms, a sufficient condition for the existence of such controllers is established in terms of LMI. A numerical example has been presented to illustrate the efficiency of the result. Acknowledgements This work is supported by the National Foundation for Science and Technology Development, Vietnam and the Faculty Strategic Fund, Deakin University, Australia. References [1] S.S.L. Chang, T.K. Peng, Adaptive guaranteed cost control of systems with uncertain parameters, IEEE Trans. Aut. Contr. 17 (1972) 474–483. [2] L. Yu, J. Chu, An LMI approach to guaranteed cost control of linear uncertain time-delay systems, Automatica 35 (1999) 1155–1159. [3] S.H. Esfahani, I.R. Petersen, An LMI approach to output-feedback-guaranteed cost control for uncertain time-delay systems, Int. J. Robust Nonl. Contr. 10 (2000) 157–174. [4] M.C. De Oliveira, J.C. Geromel, J. Bernusson, Design of dynamic output feedback decentralized controllers via a separation procedure, Int. J. Control 73 (2000) 371–381. [5] E.F. Costa, V.A. Oliveira, On the design of guaranteed cost controllers for a class of uncertain linear systems, Syst. Contr. Lett. 46 (2002) 17–29.
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[6] J.H. Park, On design of dynamic output feedback controller for GCS of large-scale systems with delays in interconnections: LMI optimization approach, Appl. Math. Comput. 161 (2005) 423–432. [7] J.H. Park, H.Y. Jung, J.I. Park, S.G. Lee, Decentralized dynamic output feedback controller design for guaranteed cost stabilization of large-scale discretedelay systems, Appl. Math. Comput. 156 (2004) 307–320. [8] J.H. Park, Dynamic output guaranteed cost controller for neutral systems with input delay, Chaos Solitons Fract. 23 (2005) 1819–1828. [9] U. Baser, B. Kizilsac, Dynamic output feedback control problem for linear neutral systems, IEEE Trans. Aut. Contr. 52 (2007) 1113–1118. [10] S.W. Yun, Y.J. Choi, P.G. Park, Dynamic output-feedback guaranteed cost control for linear systems with uniform input quantization, Nonlinear Dyn. 62 (2010) 95–104. [11] C.H. Lien, Non-fragile guaranteed cost control for uncertain neutral dynamic systems with time-varying delays in state and control input, Chaos Solitons Fract. 31 (2007) 889–899. [12] T. Weihua, Z. Huaguang, Optimal guaranteed cost control for fuzzy descriptor systems with time-varying delay, J. Syst. Eng. Electron. 19 (2008) 584–59. [13] E. Fridman, Y. Orlov, Exponential stability of linear distributed parameter systems with time-varying delays, Automatica 45 (2009) 194–201. [14] P. Niamsup, K. Mukdasai, V.N. Phat, Improved exponential stability for time-varying systems with nonlinear delayed perturbations, Appl. Math. Comput. 204 (2008) 490–495. [15] H. Gao, X. Meng, T. Chen, J. Lam, Stabilization of networked control systems via dynamic output-feedback controllers, SIAM J. Control Optim. 48 (2010) 3643–3658. [16] J. Sun, G.P. Liu, J. Chen, D. Rees, Improved delay-range-dependent stability criteria for linear systems with time-varying dealys, Automatica 46 (2010) 466–470. [17] R. Rakkiyappan, P. Balasubramaniam, R. Krishnasamy, Delay dependent stability analysis of neural systems with mixed time-varying delays and nonlinear perturbations, J. Comput. Appl. Math. 235 (2011) 2147–2156. [18] S. Boyd, L. El. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. [19] P. Gahinet, A. Nemirovskii, A.J. Laub, M. Chilali, LMI Control Toolbox for Use with MATLAB, The MathWorks, Inc., 1995.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Dynamic output feedback guaranteed cost control for linear systems with interval time-varying delays in states and outputs M.V. Thuan a,⇑, V.N. Phat b, H.M. Trinh c a
Department of Mathematics, Thainguyen University, Thainguyen, Viet Nam Institute of Mathematics, VAST, 18 Hoang Quoc Viet Road, Hanoi 10307, Viet Nam c School of Engineering, Deakin University, Geelong, VIC 3217, Australia b
a r t i c l e
i n f o
Keywords: Dynamic output feedback Exponential stabilization Interval time-varying delays Linear matrix inequality Lyapunov function
a b s t r a c t This paper addresses the dynamic output feedback guaranteed cost control problem of a class of time-delay systems where the state and output contain interval nondifferentiable time-varying delays. The proposed controller uses only the delayed output measurement to stabilize the closed-loop system and guarantee an adequate level of system performance. By constructing a set of multiple Lyapunov–Krasovskii functionals which include triple-integral terms, a new criterion for the existence of dynamic output feedback guaranteed cost controllers is established and expressed in terms of linear matrix inequality (LMI). A numerical example is given to illustrate the obtained results. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction In many practical systems, it is desirable to design control systems which are not only asymptotically or exponentially stable but can also guarantee an adequate level of system performance. Guaranteed cost control [1–3] has the advantage of providing an upper bound on a given system performance index and thus the system performance degradation incurred by the uncertainties or time delays is guaranteed to be less than this bound. Based on this idea, dynamic output feedback control for guaranteed cost stabilization has been a subject of considerable research interest in the literature. The Lyapunov–Krasovskii functional approach together with LMI-based design methods have been among the popular and effective tool in the design of dynamic output feedback controllers for time-delay systems. Nevertheless, despite such diversity of results available, most existing work either assumed that the time delays are constant or differentiable and that the controlled outputs of the systems are independent of time delays (see, e.g. [4–10]). Although, in some cases, delay feedback guaranteed cost control for systems with timevarying delays were considered in [11–15], the approach used there can not be applied to systems with interval, non-differentiable time-varying delays. So far, there has been no reported work on the design of dynamic output feedback controllers for linear systems with interval, non-differentiable time-varying delays and where the outputs also subjected to such delays. This paper addresses the problem of designing dynamic output feedback guaranteed cost controllers for a more general class of timedelay systems where the state and output contain interval, nonsmooth time-varying delays. In fact, this problem is difficult to solve in convex programming approach; particularly, when the time-varying delays are interval, non-differentiable and the output is also subjected to such time-varying delay functions. It is clear that the application of any dynamic output feedback controller to such time-delay systems would lead to closed-loop systems with multiple delays. The difficulties then arise when one attempts to derive exponential LMI stabilizability conditions and to extract the controller’s parameters for these systems. To solve the problem, we introduce a new set of Lyapunov–Krasovskii functionals that can avoid from taking any derivative of the delay functions, and due to the fact that the delays are intervals, we use a combination of Lyapunov–Krasovskii functionals associated with the lower and upper bounds of the delays. With this new set of Lyapunov–Krasovskii functionals that includes ⇑ Corresponding author. E-mail address: [email protected] (M.V. Thuan). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.04.039
10698
M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
triple-integral terms [16,17], we are able to derive a new existence condition, which can be expressed as a delay-dependent LMI condition, therefore leading to a straightforward derivation of all the controller’s parameters. Our newly proposed dynamic output feedback controller can ensure exponential stability of the closed-loop system while also can guarantee an adequate level of system performance, which is expressed in terms of a guaranteed cost function. 2. Problem formulation and preliminaries The following notations will be used throughout this paper: Rþ denotes the set of all real non-negative numbers; Rn denotes the n dimensional space with the scalar product h; i and the vector norm k k; Rnr denotes the space of all matrices of ðn rÞdimensions; AT denotes the transpose of matrix A; A is symmetric if A ¼ AT ; I denotes the identity matrix; kðAÞ denotes the set of all eigenvalues of A; kmax ðAÞ ¼ maxfRek; k 2 kðAÞg; xt :¼ fxðt þ sÞ : s 2 ½h; 0g denotes the segment of the trajectory xðtÞ with the norm kxt k ¼ sups2½h;0 kxðt þ sÞk; C1 ð½0; t; Rn Þ denotes the set of all Rn valued continuous differentiable functions on ½0; t; L2 ð½0; t; Rm Þ denotes the set of all the Rm valued square integrable functions on ½0; t; Matrix A is called semi-positive definite ðA P 0Þ if hAx; xi P 0, for all x 2 Rn ; A is positive definite ðA > 0Þ if hAx; xi > 0 for all x – 0; A > B means A B > 0. The notation diagf. . .g stands for a block-diagonal matrix. The symmetric term in a matrix is denoted by ⁄ Consider a control system with time-varying delays of the form
_ xðtÞ ¼ AxðtÞ þ Dxðt h1 ðtÞÞ þ BuðtÞ; yðtÞ ¼ Cxðt h2 ðtÞÞ; xðtÞ ¼ /ðtÞ; t 2 ½d; 0;
ð2:1Þ
1 ; h 2 g; xðtÞ 2 Rn is the state; uðtÞ 2 Rm is the control, yðtÞ 2 Rr is the observation output, A; D; B; C are given where d ¼ maxfh _ matrices of appropriate dimensions and /ðtÞ 2 C1 ð½d; 0; Rn Þ is the initial with its norm k/kC1 ¼ supt2½d;0 fk/ðtÞk; k/ðtÞkg. The time-varying delay functions h1 ðtÞ and h2 ðtÞ satisfying
; 0 < h1 < h1 ðtÞ 6 h 1
: 0 < h2 < h2 ðtÞ 6 h 2
It is worth noting that the time delay is assumed to be a continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, but the delay function is bounded but not restricted to being zero. In this paper, we design the following dynamic output feedback controller in order to exponentially stabilize system (2.1):
_ nðtÞ ¼ A1 nðtÞ þ B1 yðtÞ; uðtÞ ¼ C 1 nðtÞ;
t P 0;
nðtÞ ¼ 0;
ð2:2Þ
t 2 ½d; 0;
where nðtÞ 2 Rn and A1 ; B1 ; C 1 are constant matrices of appropriate dimensions to be determined later. The performance index associated with the system (2.1) is the following quadratic function
J¼
Z
1
½xT ðtÞQxðtÞ þ uT ðtÞRuðtÞdt;
ð2:3Þ
0
where Q 2 Rnn ; R 2 Rmm are given symmetric positive definite matrices. Applying controller (2.2) to the system (2.1) results in the closed-loop system
z_ ðtÞ ¼ AzðtÞ þ D1 zðt h1 ðtÞÞ þ D2 zðt h2 ðtÞÞ; zðtÞ ¼ /ðtÞ; where zðtÞ ¼ xT ðtÞ
A¼
ð2:4Þ
t 2 ½d; 0; nT ðtÞ
A BC 1 ; 0 A1
T
; /ðtÞ ¼ /ðtÞT
D1 ¼
D 0 ; 0 0
0
T
D2 ¼
and
0
0
B1 C
0
:
The corresponding closed-loop cost function is
J¼
Z
0
1
½xT ðtÞQxðtÞ þ nT ðtÞC T1 RC 1 nðtÞdt:
ð2:5Þ
The objective of this paper is to design a dynamic output feedback controller (2.2) for system (2.1) and cost function (2.3) such that the resulting closed-loop system (2.4) is exponentially stable and the closed-loop value of the cost function (2.3) satisfies J 6 J , where J is some specified constant. Definition 2.1. Given a > 0. The zero solution of system (2.1) is a-stable if there exists a positive number b > 0 such that every solution xðt; /Þ satisfies the condition
kxðt; /Þk 6 beat k/k;
8t P 0:
Definition 2.2. For the observation control system (2.1) and cost function (2.3), if there exist a control law u ðtÞ and a positive constant J such that the zero solution of the closed-loop system (2.4) is exponentially stable and the closed-loop value (2.3) satisfies J 6 J , then J is said to be a guaranteed cost index, u ðtÞ is said to be a guaranteed cost control law of the system and its corresponding cost function.
M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
10699
We introduce the following technical well-known propositions for the proof of the main result. Proposition 2.1. For any x; y 2 Rn and positive symmetric definite matrix N 2 Rnn , we have
2yT x 6 xT N1 x þ yT Ny: 0
Z
t
xT ðsÞZxðsÞds 6
th
Z
h
Z
h
1 h
Z
t
xðsÞds
T Z Z
th
t
xT ðsÞZxðsÞds ds 6
tþs
xðsÞds
t th
Z
2 2 h2 h
h
h
Z
ð2:6Þ !T
t
xðsÞds ds tþs
Z
Z
h
h
Z
!
t
xðsÞds ds
ð2:7Þ
tþs
Proposition 2.3 (Schur complement Lemma [18]). Given constant matrices X; Y; Z with appropriate dimensions satisfying X ¼ X T ; Y ¼ Y T > 0. Then X þ Z T Y 1 Z < 0 if and only if
!
X
ZT
Z
Y
< 0:
3. Main results In this section, using the Lyapunov function method, we establish a delay-dependent criterion for dynmic output feedback stabilization of system (2.1) with interval time-varying delays. Before introducing main result, the notations of several matrix variables are defined for simplicity. Let us denote k ¼ kmin ðP 1 Þ; 1 2 1 1 2 2 K ¼ kmax ðP 1 Þ þ h1 kmax ðP 1 Q 1 P 1 Þ þ ðh1 h1 Þkmax ðP1 Q 2 P1 Þ þ h1 kmax ðP1 S1 P1 Þ þ ðh21 h1 Þkmax ðP1 S2 P1 Þ þ h1 kmax ðP1 S3 P1 Þ 2 2 2 1 2 1 3 1 3 2 3 1 1 1 1 1 1 1 1 þ ðh1 h1 Þkmax ðP S4 P Þ þ h1 kmax ðP R1 P Þ þ ðh1 h1 Þkmax ðP R2 P Þ þ þh2 kmax ðP U 1 P Þ þ ðh2 h2 Þkmax ðP1 U 2 P 1 Þ 2 6 6 1 2 1 2 1 2 1 2 1 3 2 2 1 1 1 1 1 1 1 1 þ h2 kmax ðP 1 W 1 P1 Þ þ ðh 2 h2 Þkmax ðP W 2 P Þ þ h2 kmax ðP W 3 P Þ þ ðh2 h2 Þkmax ðP W 4 P Þ þ h2 kmax ðP H1 P Þ 2 2 2 2 6 1 3 3 h2 Þkmax ðP 1 H2 P 1 Þ; þ ðh 6 2 N ¼ ðNij Þ1515 ;
W ¼ ½ QP 0 0 0 0 0 0 0 0 0 0 0 0 0 0 T ; N11 ¼ AP þ PAT þ 2aP þ Q 1 þ U 1 þ h1 S3 þ ðh1 h1 ÞS4 þ h2 W 3 þ ðh2 h2 ÞW 4
h Þ 1 2ah2 ðh 2 2 e W 1 2e4ah2 H1 2 e4ah2 H2 ; h2 ðh2 þ h2 Þ
N22 ¼
2
2
h1 h1
e2ah1 S2 ; N33 ¼
N44 ¼ e2ah1 Q 2 e2ah1 Q 1
N55 ¼ e2ah1 Q 2
1
h1 h1
h2 h2
e2ah2 W 2 ;
1 2ah1 1 e S1 e2ah1 S2 ; h1 h1 h1
e2ah1 S2 ;
N66 ¼ e2ah2 U 2 e2ah2 U 1
1 2ah2 1 e W1 e2ah2 W 2 ; h2 h2 h2
1 e2ah2 W 2 ; h2 h2 1 2 N88 ¼ e2ah1 S3 2 e4ah1 R1 ; h1 h1 1 2 N99 ¼ e2ah1 S4 e4ah1 R2 ; 2 h2 h1 h1 h 1 1
N77 ¼ e2ah2 U 2
N10;10 ¼
1 2ah2 2 e W 3 2 e4ah2 H1 ; h2 h2
N11;11 ¼
1
h2 h2
e2ah2 W 4
2 e4ah2 H2 ; 2 h2 h 2 2
1 h1 Þ 1 2ah1 ðh e S1 2e4ah1 R1 2 e4ah1 R2 h1 ðh1 þ h1 Þ
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M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
1 2
1 2
2
1 2
2
2
1 2
2
2 h ÞH2 2P; N12;12 ¼ h1 S1 þ ðh 1 h1 ÞS2 þ h1 R1 þ ðh21 h1 ÞR2 þ h2 W 1 þ ðh 2 h2 ÞW 2 þ h2 H1 þ ðh 2 2 1 2ah1 e S1 ; N15 ¼ 0; h1 1 2ah2 2 4ah1 2 N16 ¼ e W 1 ; N17 ¼ 0; N18 ¼ e R1 ; N19 ¼ e4ah1 R2 ; h2 h1 h1 þ h1 2 2 N1;10 ¼ e4ah2 H1 ; N1;11 ¼ e4ah2 H2 ; N1;12 ¼ PAT ; h2 h2 þ h2
N13;13 ¼ Y þ Y T þ 2aP þ BRBT ; N12 ¼ DP; N13 ¼ 0; N14 ¼
X1;13 ¼ BBT ; N1;14 ¼ N1;15 ¼ 0; N23 ¼ 0; N24 ¼ N25 ¼
1
h1 h1
e2ah1 S2 ; N26 ¼ N27 ¼ N28 ¼ N29 ¼ N2;10 ¼ N2;11 ¼ 0; 1 e2ah2 W 2 ; h2 h2
N2;12 ¼ PDT ; N2;13 ¼ N2;14 ¼ N2;15 ¼ 0; N34 ¼ N35 ¼ 0; N36 ¼ N37 ¼
N38 ¼ N39 ¼ N3;10 ¼ N3;11 ¼ N3;12 ¼ N3;13 ¼ 0; N3;14 ¼ PC T ; N3;15 ¼ 0; N45 ¼ N46 ¼ N47 ¼ N48 ¼ N49 ¼ N4;10 ¼ N4;11 ¼ N4;12 ¼ N4;13 ¼ N4;14 ¼ N4;15 ¼ 0; N56 ¼ N57 ¼ N58 ¼ N59 ¼ N5;10 ¼ N5;11 ¼ N5;12 ¼ N5;13 ¼ N5;14 ¼ N5;15 ¼ 0; N67 ¼ ¼ N6;15 ¼ 0; N78 ¼ ¼ N7;15 ¼ 0; N89 ¼ ¼ N8;15 ¼ 0; N9;10 ¼ ¼ N9;15 ¼ 0; N10;11 ¼ ¼ N10;15 ¼ 0; N11;12 ¼ ¼ N11;15 ¼ 0; N12;13 ¼ BBT ; N12;14 ¼ N12;15 ¼ 0; N13;14 ¼ 0; N13;15 ¼ PC T ; N14;14 ¼ N15;15 ¼ I; N14;15 ¼ 0:
Theorem 3.1. For given Q > 0; R > 0 and a > 0, there exists a dynamic output feedback controller (2.2) for system (2.1) if there exist symmetric positive definite matrices P; Q 1 ; Q 2 ; U 1 ; U 2 ; R1 ; R2 ; H1 ; H2 ; Si ; W i ; i ¼ 1; . . . ; 4, and matrix Y satisfying the following LMI
N WT
W
< 0:
Q
ð3:1Þ
The guaranteed constant of the cost function is
J ¼ Kk/k2 : Proof. Let us denote
Q i ¼ P1 Q i P1 ; 1
1
Sj ¼ P Sj P ;
U i ¼ P1 U i P1 ; 1
Ri ¼ P1 Ri P1 ;
1
W j ¼ P W jP ;
Hi ¼ P1 Hi P1 ;
i ¼ 1; 2;
j ¼ 1; 2; 3; 4:
Construct a Lyapunov–Krasovskii functional as follows:
Vðt; zt Þ ¼
9 X V i ðt; zt Þ; i¼1
where V 1 ¼ xT ðtÞP1 xðtÞ þ nT ðtÞP1 nðtÞ; Z t Z V2 ¼ e2aðstÞ xT ðsÞQ 1 xðsÞds þ V3 ¼
Z
0
h1
V4 ¼
Z
0
h1
V5 ¼
Z
0
h1
V6 ¼
Z
Z
t
Z
Z
t
e2aðstÞ xT ðsÞS3 xðsÞds ds þ
Z
0
Z
t
V8 ¼
Z
0
h2
V9 ¼
Z
0
h2
Z
t
Z
th2
th 2
t
h
0
e2aðstÞ xT ðsÞW 3 xðsÞds ds þ Z
t
tþs
_ sÞds ds; e2aðstÞ x_ T ðsÞS2 xð
Z
h 1
t
e2aðstÞ xT ðsÞS4 xðsÞds ds;
tþs
Z
h1
h 1
Z
0
Z
t
_ sÞds dsdh; e2aðsþstÞ x_ T ðsÞR2 xð
tþs
h
e2aðstÞ xT ðsÞU 2 xðsÞds;
_ sÞds ds þ e2aðstÞ x_ T ðsÞW 1 xð
tþs
Z
h1
t
tþs
_ sÞds dsdh þ e2aðsþstÞ x_ T ðsÞR1 xð
tþs
Z
Z
tþs
h
Z
h1
h 1
tþs
e2aðstÞ xT ðsÞU 1 xðsÞds þ
h2
Z
tþs
t
0
e2aðstÞ xT ðsÞQ 2 xðsÞds;
_ sÞds ds þ e2aðstÞ x_ T ðsÞS1 xð
th2
V7 ¼
th1
th 1
th1
Z
h2
h 2
Z
h2
h 2
Z
t
_ sÞds ds; e2aðstÞ x_ T ðsÞW 2 xð
tþS
Z
t
e2aðstÞ xT ðsÞW 4 xðsÞds ds;
tþs
_ sÞds dsdh þ e2aðsþstÞ x_ T ðsÞH1 xð
Z
h2 h 2
Z h
0
Z
t
tþs
_ sÞds dsdh: e2aðsþstÞ x_ T ðsÞH2 xð
ð3:2Þ
10701
M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
Taking the time derivative of V i ; i ¼ 1; . . . ; 9 along the trajectory of system (2.4) yields h i h i T T V_ 1 ¼ xT ðtÞ P 1 A þ A P 1 xðtÞ þ 2xT ðtÞP 1 Dxðt h1 ðtÞÞ þ 2xT ðtÞP 1 BC 1 nðtÞ þ nT ðtÞ P 1 A1 þ A1 P1 nðtÞ þ 2nT ðtÞP 1 B1 Cxðt h2 ðtÞÞ; ÞQ xðt h Þ; V_ 2 ¼ 2aV 2 þ xT ðtÞQ 1 xðtÞ e2ah1 xT ðt h1 ÞQ 1 xðt h1 Þ þ e2ah1 xT ðt h1 ÞQ 2 xðt h1 Þ e2ah1 xT ðt h 1 1 2 Z Z h i t th1 h ÞS xðtÞ _ _ V_ 3 6 2aV 3 þ x_ T ðtÞ h1 S1 þ ðh e2ah1 x_ T ðsÞS1 xðsÞds e2ah1 x_ T ðsÞS2 xðsÞds; 1 1 2 _ th1
Z h i h ÞS xðtÞ e2ah1 V_ 4 6 2aV 4 þ xT ðtÞ h1 S3 þ ðh 1 1 4
t
xT ðsÞS3 xðsÞds e
2ah 1
Z
th 1
th1 th 1
th1
xT ðsÞS4 xðsÞds;
Z 0 Z t Z h1 Z t 1 2 1 2 2 4ah1 _ _ _ V_ 5 6 2aV 5 þ x_ T ðtÞ h1 R1 þ ðh x_ T ðsÞR1 xðsÞdsdh e4ah1 x_ T ðsÞR2 xðsÞdsdh; 1 h1 ÞR2 xðtÞ e 2 2 h h1 tþh tþh 1 2 ÞU 2 xðt h 2 Þ; V_ 6 ¼ 2aV 6 þ xT ðtÞU 1 xðtÞ e2ah2 xT ðt h2 ÞU 1 xðt h2 Þ þ e2ah2 xT ðt h2 ÞU 2 xðt h2 Þ e2ah2 xT ðt h Z t Z th2 h ÞW xðtÞ _ _ V_ 7 6 2aV 7 þ x_ T ðtÞ h2 W 1 þ ðh e2ah2 x_ T ðsÞW 1 xðsÞds e2ah2 x_ T ðsÞW 2 xðsÞds; 2 2 2 _ th 2
th2
h ÞW xðtÞ e2ah2 V_ 8 6 2aV 8 þ x ðtÞ h2 W 3 þ ðh 2 2 4 T
Z
t
2ah 2
T
x ðsÞW 3 xðsÞds e
Z
th2
th 2
th2
xT ðsÞW 4 xðsÞds;
Z 0 Z t Z h2 Z t 1 2 1 2 2 4ah2 _ _ _ V_ 9 6 2aV 9 þ x_ T ðtÞ h2 H1 þ ðh x_ T ðsÞH1 xðsÞdsdh e4ah2 x_ T ðsÞH2 xðsÞdsdh: 2 h2 ÞH2 xðtÞ e 2 2 h h2 tþh tþh 2
Therefore,
h i 1 h1 ÞS4 þ h2 W 3 þ ðh 2 h2 ÞW 4 xðtÞ V_ þ 2aV 6 xT ðtÞ P1 A þ AT P1 þ 2aP1 þ Q 1 þ U 1 þ h1 S3 þ ðh h i þ nT ðtÞ P1 A1 þ AT1 P1 þ 2aP 1 nðtÞ þ 2xT ðtÞP1 Dxðt h1 ðtÞÞ þ 2xT ðtÞP1 BC 1 nðtÞ þ 2nT ðtÞP1 B1 Cxðt h2 ðtÞÞ h i h i 1 Þ e2ah1 Q xðt h 1 Þ þ xT ðt h1 Þ e2ah1 Q 2 e2ah1 Q 1 xðt h1 Þ þ xT ðt h 2 h i 2 Þ 2 Þ e2ah2 U 2 xðt h þ xT ðt h2 Þ e2ah2 U 2 e2ah2 U 1 xðt h2 Þ þ xT ðt h 2 h2 ÞR2 þ h2 W 1 þ ðh 2 h2 ÞH2 xðtÞ 1 h1 ÞS2 þ 1 h2 R1 þ 1 ðh 2 h2 ÞW 2 þ 1 h2 H1 þ 1 ðh _ þ x_ T ðtÞ h1 S1 þ ðh 1 2 2 1 2 1 2 2 2 2 Z t Z th1 Z t _ ds e2ah1 _ ds e2ah1 x_ T ðsÞS1 xðsÞ x_ T ðsÞS2 xðsÞ e2ah1 xT ðsÞS3 xðsÞ ds th 1
th1
e2ah1 e2ah2
Z
th1
th 1
Z
t
xT ðsÞS4 xðsÞ ds e4ah1
Z
h1
_ ds e2ah2 x_ T ðsÞW 1 xðsÞ
Z
Z
th2
th 2
th2
th 2
th2
e2ah2
th1
Z
0
Z
xT ðsÞW 4 xðsÞ ds e4ah2
0
t
_ ds dh e4ah1 x_ T ðsÞR1 xðsÞ
Z
Z
Z
h 1
tþh
_ ds e2ah2 x_ T ðsÞW 2 xðsÞ
h1
t
_ ds dh x_ T ðsÞR2 xðsÞ tþh
t
xT ðsÞW 3 xðsÞ ds
th2
Z
t
_ ds dh e4ah2 x_ T ðsÞH1 xðsÞ
Z
tþh
h2
h2
h 2
Z
t
_ ds dh: x_ T ðsÞH2 xðsÞ
Using Proposition 2.2, the following inequalities hold Z t T Z t Z t 1 1 T _ _ _ e2ah1 e2ah1 S1 x_ T ðsÞS1 xðsÞds 6 xðsÞds xðsÞds ¼ ½xðtÞ xðt h1 Þ e2ah1 S1 ½xðtÞ xðt h1 Þ; h1 th1 h1 th1 th1
e2ah1
Z
th1
th 1
_ ds ¼ e2ah1 x_ T ðsÞS2 xðsÞ
Z
th1 ðtÞ
th 1
Z
1
6 h1 h1
!T
th1 ðtÞ
_ ds xðsÞ
Z
e
h1 h1
th1 ðtÞ
!T
th1
_ ds xðsÞ
Z
th1
2ah 1
e
2ah 1
S2
S2
Z
ð3:4Þ
_ ds x_ T ðsÞS2 xðsÞ
th1 ðtÞ
th 1
1
1
_ ds e2ah1 x_ T ðsÞS2 xðsÞ
ð3:3Þ
tþh
!
th1 ðtÞ
_ ds xðsÞ
th 1
Z
!
th1
_ ds xðsÞ
th1 ðtÞ
1 ÞT e2ah1 S2 xðt h1 ðtÞÞ xðt h 1 Þ xðt h1 ðtÞÞ xðt h
¼ h1 h1 1 T ½xðt h1 Þ xðt h1 ðtÞÞ e2ah1 S2 ½xðt h1 Þ xðt h1 ðtÞÞ; h1 h1
ð3:5Þ
10702
M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
e2ah1
Z
t
xT ðsÞS3 xðsÞ ds 6
th1
¼
e2ah1
Z
th1
1 h1 1 h1
Z Z
T Z xðsÞ ds e2ah1 S3
t
th1
th1
T Z xðhÞ dh e2ah1 S3
t
th1
Z
Z
0
Z
t
2
e
Z
h1
Z
h 1
t
_T
_ x ðsÞR2 xðsÞdsdh 6
tþh
¼
e2ah2
Z
t
_ ds 6 x_ T ðsÞW 1 xðsÞ
th2
1 h2
2 h1
ð3:6Þ
Z
!
th1
xðsÞ ds
th 1
!
th1
xðhÞ dh ;
tþh
Z
0
T
t
_ ds dh xðsÞ
t
tþh
h1
Z
ð3:7Þ
th 1
T Z _ ds dh e4ah1 R1 xðsÞ
t
h1 xðtÞ
xðhÞ dh e4ah1 R1 h1 xðtÞ
Z
th1
Z
2 2 h2 h 1 1 2
h1
Z
h 1
!T
t
_ xðsÞdsdh
4ah 1
e
Z
R2
1 h1 ÞxðtÞ ðh
Z
#T
th1
ð3:8Þ
!
t
_ xðsÞdsdh
tþh
"
th 1
th2
Z
1 h1 ÞxðtÞ e4ah1 R2 ðh
xðhÞdh t
h1
h 1
tþh
T Z _ ds e2ah2 W 1 xðsÞ
t
xðhÞ dh ;
t
th1
"
2 h2 h 1 1
Z
Z
0
h1
2
¼
4ah 1
Z
Z
e2ah1 S4
xðhÞ dh
th 1
h1
tþh
h1
!T
th1
e2ah1 S4
xðsÞ ds th 1
2
_ ds dh 6 x_ T ðsÞR1 xðsÞ
!T
th1
Z
1 ¼ h1 h1 e4ah1
xðhÞ dh ;
t
th1
1 xT ðsÞS4 xðsÞ ds 6 h1 h1
th 1
xðsÞ ds
t
Z
#
th1
xðhÞdh ; th 1
_ ds xðsÞ
th2
1 T ¼ ½xðtÞ xðt h2 Þ e2ah2 W 1 ½xðtÞ xðt h2 Þ; h2 Z th2 Z th2 ðtÞ Z th2 _ ds ¼ e2ah2 _ ds e2ah2 _ ds e2ah2 x_ T ðsÞW 2 xðsÞ x_ T ðsÞW 2 xðsÞ x_ T ðsÞW 2 xðsÞ th 2
th 2
Z
1 6 h2 h2
th2 ðtÞ
!T
th2 ðtÞ
th 2
1 h2 h2
!T
th2
Z
e2ah2 W 2
_ ds xðsÞ
Z
e2ah2 W 2
_ ds xðsÞ
!
th2 ðtÞ
_ ds xðsÞ !
th2
_ ds xðsÞ
th2 ðtÞ
2 ÞT e2ah2 W 2 xðt h2 ðtÞÞ xðt h 2 Þ xðt h2 ðtÞÞ xðt h
¼ h2 h2 1 T ½xðt h2 Þ xðt h2 ðtÞÞ e2ah2 W 2 ½xðt h2 Þ xðt h2 ðtÞÞ; h2 h2 e2ah2
Z
t
xT ðsÞW 3 xðsÞ ds 6
th2
1 h2
1 ¼ h2 2ah 2
e
Z
th2
T
Z Z
T Z xðsÞ ds e2ah2 W 3
t
th2 t
0
h2
Z
t
_ ds dh 6 x_ T ðsÞH1 xðsÞ
tþh
¼
2 2 2
h2
!T
th2
xðsÞ ds
th 2
Z
xðhÞ dh
Z
0
h2
Z
2ah 2
e !T
th2
th 2
2
h2
t
ð3:11Þ
xðsÞ ds xðhÞ dh ;
ð3:12Þ
th2
Z
1
¼ h2 h2 Z
th2
th2
1
e4ah2
t
T Z xðhÞ dh e2ah2 W 3
x ðsÞW 4 xðsÞ ds 6 h2 h2
th 2
ð3:10Þ
th 2
Z
th2 ðtÞ
1
ð3:9Þ
W4
2ah 2
e
W4
Z Z
!
th2
xðhÞ dh ;
th 2
tþh
h2 xðtÞ
xðsÞ ds
th 2
T Z _ ds dh e4ah2 H1 xðsÞ
t
!
th2
0
h2
Z
t
th2
T
Z
ð3:13Þ
t
_ ds dh xðsÞ
tþh
xðhÞ dh e4ah2 H1 h2 xðtÞ
Z
t th2
xðhÞ dh ;
ð3:14Þ
10703
M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
4ah 2
e
Z
h2
Z
h 2
t
_T
_ ds dh 6 x ðsÞH2 xðsÞ
tþh
¼
Z
2 2 h2 h 2 2 2 2 h2 h 2 2
h2
h 2
Z
!T
t
_ ds dh xðsÞ
4ah 2
e
Z
H2
h ÞxðtÞ ðh 2 2
Z
th2
#T xðhÞ dh
Z
h 2
tþh
"
h2
4ah 2
e
th 2
!
t
_ ds dh xðsÞ
tþh
" H2
h ÞxðtÞ ðh 2 2
Z
#
th2
xðhÞ dh : th 2
ð3:15Þ Let us define
B1 ¼ PC T
ð3:16Þ
and using Proposition 2.1, we have
2nT ðtÞP1 B1 Cxðt h2 ðtÞÞ ¼ 2nT ðtÞC T Cxðt h2 ðtÞÞ 6 nT ðtÞC T CnðtÞ þ xT ðt h2 ðtÞÞC T Cxðt h2 ðtÞÞ:
ð3:17Þ
On the other hand, from the identity
_ AxðtÞ þ Dxðt h1 ðtÞÞ þ BC 1 nðtÞ xðtÞ ¼ 0; we have
_ ¼ 0: 2x_ T ðtÞP1 ½AxðtÞ þ Dxðt h1 ðtÞÞ þ BC 1 nðtÞ xðtÞ
ð3:18Þ
From (3.3)–(3.18), we have
h i _ zt Þ þ 2aVðt; zt Þ 6 gT ðtÞXgðtÞ xT ðtÞQxðtÞ þ nT ðtÞC T RC 1 nðtÞ ; Vðt; 1
ð3:19Þ
where
X ¼ ðXij Þ1313 ; 2
gðtÞ ¼ 4xT ðtÞxT ðt h1 ðtÞÞxT ðt h2 ðtÞÞxT ðt h1 ÞxT ðt h1 ÞxT ðt h2 ÞxT ðt h 2 Þ Z
T Z xðhÞ dh
t
th 2
th2
t
T Z xðhÞ dh
th1
th1
th 1
!T xðhÞ dh
3T
!T
th2
Z
x_ T ðtÞ nT ðtÞ5 ;
xðhÞ dh
with
X11 ¼ P1 A þ AT P1 þ 2aP1 þ Q 1 þ U 1 þ h1 S3 þ ðh1 h1 ÞS4 þ h2 W 3 þ ðh2 h2 ÞW 4
X22 X44
1 h1 Þ 2 h2 Þ ðh 1 ðh 2 e4ah1 R2 e2ah2 W 1 2e4ah2 H1 2 e4ah2 H2 þ Q ; h2 ðh1 þ h1 Þ ðh2 þ h2 Þ 2 2 ¼ e2ah1 S2 ; X33 ¼ e2ah2 W 2 þ C T C; h1 h1 h2 h2 1 2ah1 1 2ah1 2ah1 ¼e Q2 e Q1 e S1 e2ah1 S2 ; h1 h1 h1
X55 ¼ e2ah1 Q 2
1
h1 h1
e2ah1 S2 ;
X66 ¼ e2ah2 U 2 e2ah2 U 1
1 2ah2 1 e W1 e2ah2 W 2 ; h2 h2 h2
1 e2ah2 W 2 ; h2 h2 1 2 ¼ e2ah1 S3 2 e4ah1 R1 ; h1 h1 1 2 ¼ e2ah1 S4 e4ah1 R2 ; 2 2 h1 h1 h h
X77 ¼ e2ah2 U 2 X88 X99
1
X10;10
1
1 2 ¼ e2ah2 W 3 2 e4ah2 H1 ; h2 h2
X11;11 ¼
1
h2 h2
e2ah2 W 4
2 e4ah2 H2 ; 2 2 h2 h2
1 2ah1 e S1 2e4ah1 R1 h1
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M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
1 2
1 2
1 2
1 2
X12;12 ¼ h1 S1 þ ðh1 h1 ÞS2 þ h21 R1 þ ðh21 h21 ÞR2 þ h2 W 1 þ ðh2 h2 ÞW 2 þ h22 H1 þ ðh22 h22 ÞH2 2P1 ; X13;13 ¼ P1 A1 þ AT1 P1 þ 2aP1 þ C T C þ C T1 RC 1 ; X14 ¼
1 2ah1 e S1 ; h1
X16 ¼
1 2ah2 e W 1; h2
X1;10 ¼
X17 ¼ 0;
X18 ¼
2 4ah1 e R1 ; h1
2
X1;11 ¼ e4ah2 H2 ; h2 þ h2 1
X24 ¼ X25 ¼ e2ah1 S2 ; h1 h1
X2;12 ¼ DT P 1 ;
X13 ¼ 0;
X15 ¼ 0;
2 4ah2 e H1 ; h2
X23 ¼ 0;
X12 ¼ P1 D;
X2;13 ¼ 0;
2
X19 ¼ e4ah1 R2 ; h1 þ h1
X1;12 ¼ AT P1 ;
X1;13 ¼ P1 BC 1 ;
X26 ¼ X27 ¼ X28 ¼ X29 ¼ X2;10 ¼ X2;11 ¼ 0;
X34 ¼ X35 ¼ 0;
1
X36 ¼ X37 ¼ e2ah2 W 2 ; h2 h2
X38 ¼ X39 ¼ X3;10 ¼ X3;11 ¼ X3;12 ¼ X3;13 ¼ 0; X45 ¼ X46 ¼ X47 ¼ X48 ¼ X49 ¼ X4;10 ¼ X4;11 ¼ X4;12 ¼ X4;13 ¼ 0; X56 ¼ X57 ¼ X58 ¼ X59 ¼ X5;10 ¼ X5;11 ¼ X5;12 ¼ X5;13 ¼ 0; X67 ¼ ¼ X6;13 ¼ 0;
X78 ¼ ¼ X7;13 ¼ 0;
X10;11 ¼ X10;12 ¼ X10;13 ¼ 0;
X89 ¼ ¼ X8;13 ¼ 0;
X11;12 ¼ X11;13 ¼ 0;
X9;10 ¼ ¼ X9;13 ¼ 0;
1
X12;13 ¼ P BC 1 :
Now, pre- and post-multiply both sides of X with
diagfP; P; P; P; P; P; P; P; P; P; P; P; Pg and letting
A1 ¼ YP1 ;
C 1 ¼ BT P1
ð3:20Þ
and using Schur complement Lemma we have X < 0 is equivalent to condition (3.1). From condition (3.1), we have
h i _ zt Þ þ 2aVðt; zt Þ 6 xT ðtÞQxðtÞ þ nT ðtÞC T RC 1 nðtÞ : Vðt; 1 h
T
Since xT ðtÞQxðtÞ þ n
i
ðtÞC T1 RC 1 nðtÞ
_ zt Þ þ 2aVðt; zt Þ 6 0; Vðt;
ð3:21Þ
P 0, we have
t 2 Rþ :
ð3:22Þ
Integrating both sides of (3.22) from 0 to t gives
Vðt; zt Þ 6 Vð0; z0 Þe2at ;
8t P 0:
By simple computation, we have
Vðt; zt Þ P kmin ðP1 ÞkxðtÞk2 þ kmin ðP1 ÞknðtÞk2 P kmin ðP1 ÞkxðtÞk2 ¼ kkxðtÞk2 ;
8t P 0:
and
Vð0; z0 Þ 6 Kk/k2C1 : Hence
kkxðt; /Þk2 6 Vðt; zt Þ 6 Vð0; z0 Þe2at 6 Ke2at k/k2C1 ; then the solution xðt; /Þ of the system satisfies
kxðt; /Þk 6
rffiffiffiffi K at e k/kC1 ; k
8t P 0;
which implies the closed-loop system is astable. To find the guaranteed cost control value, from (3.21) we have
h i _ zt Þ 6 xT ðtÞQxðtÞ þ nT ðtÞC T RC 1 nðtÞ : Vðt; 1 Integrating both sides of (3.23) from 0 to s implies
Vðs; zs Þ Vð0; z0 Þ 6
Z sh 0
i xT ðtÞQxðtÞ þ nT ðtÞC T1 RC 1 nðtÞ dt:
ð3:23Þ
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M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
Hence
Z sh 0
i xT ðtÞQxðtÞ þ nT ðtÞC T1 RC 1 nðtÞ dt 6 Vð0; z0 Þ ¼ Kk/k2C1 :
Letting s ! þ1, we obtain
J¼
Z
þ1
h
0
i xT ðtÞQxðtÞ þ nT ðtÞC T1 RC 1 nðtÞ dt 6 Kk/k2C 1 :¼ J :
This completes the proof of the theorem.
h
Remark 3.1. Based on constructing a set of new Lyapunov–Krasovskii functionals (3.2), Theorem 3.1 provides sufficient condition for the closed-loop system to be exponentially stable with a prescribed decay rate a, while the existing methods of [6,7,9,10] provide asymptotic stability. Remark 3.2. Given any solution of the LMI (3.1) in Theorem 3.1, a corresponding stabilizing controller of the form (2.2) can be constructed as follows: Step 1. Solve LMI (3.1) and obtain matrices P and Y. Step 2. Compute the invertible matrix P1 . Step 3. Using the solution P1 ; Y of LMI (3.1), define A1 ; B1 ; C 1 by (3.16) and (3.20):
B1 ¼ PC T ;
A1 ¼ YP1 ;
C 1 ¼ BT P1 :
ð3:24Þ
To illustrate the design procedure, the following numerical example is given. Example 3.1. Consider a linear observation control system with time-varying delays (2.1), where
A¼
0:1
0
0:1 0:9
;
D¼
0
2
0:5 0:5
;
B¼
0:9 0:8
;
C ¼ ½ 1 0:5 ;
with
(
(
2
h1 ðtÞ ¼ 0:1 þ 0:3 sin t
if
t 2 I ¼ [kP0 ½2kp; ð2k þ 1Þp
h1 ðtÞ ¼ 0:1
if
t 2 Rþ n I ;
h2 ðtÞ ¼ 0:1 þ 0:3 cos2 t
if
t 2 I ¼ [kP0 ½2kp; ð2k þ 1Þp
h2 ðtÞ ¼ 0:1
if
t 2 Rþ n I ;
Note that h1 ðtÞ; h2 ðtÞ are non-differentiable, therefore, the stability criteria proposed in [10,11,14] are not applicable to this system. Given
a ¼ 1; Q ¼
0:1
0
0
0:2
;
R ¼ 1:5:
By using the LMI Toolbox in MATLAB 1 ¼ 0:4; h2 ¼ 0:1; h 2 ¼ 0:4, and h1 ¼ 0:1; h
P¼
1:1694 0:9763
;
Q1 ¼
17:2453
[19],
the
15:8160
;
LMI
(3.1)
Q2 ¼
in
Theorem
15:4517 14:4884
; 0:9763 8:8587 15:8160 100:6733 14:4884 75:5063 0:5235 2:5245 7:5832 9:5533 0:6904 0:0228 ; U2 ¼ ; R1 ¼ ; U1 ¼ 2:5245 52:3380 9:5533 161:6656 0:0228 0:9457 0:1569 1:1703 0:4091 0:1564 0:1569 1:1703 R2 ¼ ; H1 ¼ ; H2 ¼ ; 1:1703 15:3105 0:1564 2:6638 1:1703 15:3105 1:5361 1:6940 2:6633 2:5927 4:5852 0:5598 S1 ¼ ; S2 ¼ ; S3 ¼ ; 1:6940 7:7595 2:5927 11:4131 0:5598 13:0816 2:2391 3:1311 2:6749 3:2836 0:2623 0:2928 S4 ¼ ; W1 ¼ ; W2 ¼ ; 3:1311 50:7949 3:2836 35:4138 0:2928 12:8799 6:7110 0:5143 2:2391 3:1311 94:1305 55:3449 W3 ¼ ; W4 ¼ ; Y¼ : 0:5143 14:1273 3:1311 50:7949 35:6782 96:5021
3.1
is
satisfied
with
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M.V. Thuan et al. / Applied Mathematics and Computation 218 (2012) 10697–10707
10
x1 x2
8 6 4 2 0 −2 −4
0
2
4
6
8
10
Time (sec) Fig. 1. Closed-loop trajectories of x1 ðtÞ and x2 ðtÞ.
Therefore, the dynamic output feedback controller for the system (2.1) is given as follows
94:3933 16:6507 A1 ¼ YP1 ¼ ; 43:6168 15:7005 0:6813 B1 ¼ PC T ¼ ; 3:4530
ð3:25Þ
C 1 ¼ BT P1 ¼ ½ 0:9306 0:1929 : This implies that the system (2.1) is 1 stable by the dynamic output feedback controller (3.25). Moreover, the solution of the closed-loop system satisfies
kxðt; /Þk 6 8:8068et k/kC1 and the optimal guaranteed cost of the closed-loop system is as follows:
J ¼ 8:6323k/k2C1 : Fig. 1 shows the simulation of the solutions x1 ðtÞ and x2 ðtÞ with the initial condition /ðtÞ ¼ ½10 5T ; t 2 ½0:4; 0. 4. Conclusion In this paper, we have investigated the dynamic output feedback guaranteed cost control for linear systems with interval time-varying delays in both states and outputs The designed dynamic output feedback stabilizing controller guarantees the closed-loop system to be exponential stable with a prescribed decay rate. By constructing a set of Lyapunov–Krasovskii functionals which contains triple-integral terms, a sufficient condition for the existence of such controllers is established in terms of LMI. A numerical example has been presented to illustrate the efficiency of the result. Acknowledgements This work is supported by the National Foundation for Science and Technology Development, Vietnam and the Faculty Strategic Fund, Deakin University, Australia. References [1] S.S.L. Chang, T.K. Peng, Adaptive guaranteed cost control of systems with uncertain parameters, IEEE Trans. Aut. Contr. 17 (1972) 474–483. [2] L. Yu, J. Chu, An LMI approach to guaranteed cost control of linear uncertain time-delay systems, Automatica 35 (1999) 1155–1159. [3] S.H. Esfahani, I.R. Petersen, An LMI approach to output-feedback-guaranteed cost control for uncertain time-delay systems, Int. J. Robust Nonl. Contr. 10 (2000) 157–174. [4] M.C. De Oliveira, J.C. Geromel, J. Bernusson, Design of dynamic output feedback decentralized controllers via a separation procedure, Int. J. Control 73 (2000) 371–381. [5] E.F. Costa, V.A. Oliveira, On the design of guaranteed cost controllers for a class of uncertain linear systems, Syst. Contr. Lett. 46 (2002) 17–29.
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