Further results on H control for discretetime ... - Wiley Online Library

OPTIMAL CONTROL APPLICATIONS AND METHODS Optim. Control Appl. Meth. 2013; 34:328–347 Published online 27 April 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/oca.2023

Further results on H1 control for discrete-time uncertain singular systems with interval time-varying delays in state and input Huanyu Zhu, Xian Zhang* ,† and Shaochun Cui School of Mathematical Science, Heilongjiang University, Harbin 150080, China

SUMMARY This paper investigates the state feedback robust H1 control problem of a class of discrete-time singular systems with norm-bounded uncertainties and interval time-varying delays in state and input. A new bounded real lemma for discrete-time singular systems with a pair of time-varying interval state delays is first investigated. Mathematical comparisons of the new bounded real lemma and two existing ones are presented. Then, on the basis of the bounded real lemma proposed here, a sufficient condition in the form of nonlinear matrix inequality, such that the considered state feedback robust H1 control problem is solvable, is given. In order to solve the nonlinear matrix inequality, a cone complementarity linearization algorithm is offered. Several numerical examples are presented to show the applicability of the proposed approach. Copyright © 2012 John Wiley & Sons, Ltd. Received 3 June 2011; Revised 24 October 2011; Accepted 10 January 2012 KEY WORDS:

discrete-time singular systems; time-varying interval delay; H1 control; bounded real lemma; cone complementarity linearization (CCL)

1. INTRODUCTION The singular system model can better describe a large class of systems than traditional state-space ones do, as it contains not only differential/difference equations but also algebraic ones, where the algebraic equations are usually used to represent constraint conditions among physical quantities in practice. So such systems can preserve the structure of practical systems and have extensive applications in power systems, robotic systems, and networks [1–3]. Also, time delay and uncertainties, which yield instability and poor performance, are encountered in various engineering systems, such as manufacturing system, networked system, economic system, and chemical engineering system. Therefore, in the last decade, much attention has been paid on the analysis and control of uncertain singular time-delay systems (for example, [4–7] and the references therein). One of the important problems related to uncertain singular time-delay systems is robust H1 control, and hence lots of results on robust H1 control problems of uncertain singular time-delay systems have been obtained ([4, 6, 8–14] and the references therein). However, most of these results are related to the continuetime uncertain singular time-delay systems. For the discrete-time case, the research on robust H1 control mainly focus on uncertain singular systems with only state delay [8, 15–19]. To the best of our knowledge, only Ma et al. [9] and Kim [4] investigated the robust H1 control problem for uncertain discrete-time singular systems with state and input delays simultaneously. When the involved state and input delays are the same integer d with 0 6 d 6 dN , Ma et al. [9] first transformed the robust H1 control problem for discrete-time singular time-delay systems with norm-bounded uncertainties to one for standard state-space uncertain time-delay systems by a series of restricted equivalent transformations. Then, a delay-dependent sufficient condition, which *Correspondence to: Xian Zhang, School of Mathematical Science, Heilongjiang University, Harbin 150080, China. † E-mail: [email protected] Copyright © 2012 John Wiley & Sons, Ltd.

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329

guarantees that the robust state feedback H1 control problem is solvable, is given, and the state feedback H1 controller is solved by using the so-called cone complementarity linearization (CCL) technique [20]. As mentioned in [9, Remark 6], a deficiency of the approach proposed in [9] is that the matrix QN 21 has to be preselected. Kim [4] generalized the results of Ma et al. in [9] to the N case of state delay d.k/ with 0 6 d 6 d.k/ 6 dN and input delay h.k/ with 0 6 h 6 d.k/ 6 h. Kim’s approach does not require the preselection of any matrix or doing any restricted equivalent transformation, which overcomes the deficiency of the one by Ma et al. A common characteristic of these two approaches proposed in [4, 9] is that free-weighting matrices are introduced to solve conveniently the state feedback gain. It should be emphasized that, in order to solve the robust H1 control problem considered by Kim, a bounded real lemma for uncertain discrete-time singular systems with a pair of interval time-varying state delays is required, whereas the robust H1 control problem of uncertain singular systems with only single state delay can be solved by using a bounded real lemma for uncertain discrete-time singular systems with single state delay. Therefore, in general, the approaches to robust H1 control for uncertain singular systems with only single state delay is not available for uncertain singular systems with state and input delays simultaneously. The purpose of this paper is to investigate less conservative solvability conditions for the state feedback robust H1 control problem considered by Kim [4], which can yet be regarded as a generalization of the results of Ma et al. [9]. Firstly, a new bounded real lemma (Theorem 4.1) for discrete-time singular systems with a pair of interval time-varying state delays is first obtained by combining Lyapunov stability theory and Jensen inequality technique. Theorem 4.1 removes the free-weighting matrices required in [4, Theorem 1], which reduces the computational complexity. Furthermore, it is mathematically proven in 4.2 that (i) Theorem 4.1 and [4, Theorem 1] have the same conservativeness; and (ii) a special case (Corollary 4.2) of the bounded real lemma proposed here is less conservative than one presented in [9, Lemma 3]. Secondly, on the basis of the new bounded real lemma, a delay-dependent sufficient condition in the form of nonlinear matrix inequality, which guarantees that the considered state feedback H1 control problem is solvable, is given. In order to deal with the nonlinear problem, a CCL algorithm ([20] for the CCL technique) is designed to calculate a state feedback gain. Finally, several numerical examples are presented to compare the approach proposed in this paper with those in [4, 9] and illustrate the effectiveness of the new approach. It should be mentioned that the approach proposed in this paper can also used to solve the robust H1 control problems considered in [16–19]. Notation: The notation used throughout the note is fairly standard. P > 0 .> 0/ means that P is real symmetric positive definite (semi-definite); I and 0 refer to the corresponding dimensional identity matrix and zero matrix, respectively;  stands for the symmetric terms in a symmetric matrix. 2. PROBLEM FORMULATION Consider the uncertain discrete-time singular systems with time-varying delays described by the following: Ex.k C 1/ D .A C A.k//x.k/ C .Ad C Ad .k//x.k  d.k// C .B C B.k//u.k/ C .Bh C Bh .k//u.k  h.k// C .Bw C Bw .k//w.k/, ´.k/ D C x.k/ C Cd x.k  d.k// C Du.k/ C Dh u.k  h.k// C Dw w.k/,

(1)

x.k/ D .k/, k D  ,  C 1,    , 0,  D max.d , h/, where x.k/ is the n-dimensional state vector; u.k/ is the m-dimensional control input; w.k/ is the p-dimensional disturbance input; ´.k/ is the q-dimensional controlled output; E is a known matrix satisfying 0 < rankE D r 6 n; .k/ is a known initial sequence; d.k/ and h.k/ are time-varying Copyright © 2012 John Wiley & Sons, Ltd.

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H. ZHU, X. ZHANG AND S. CUI

interval delays such that 0 6 d 6 d.k/ 6 d , 0 6 h 6 h.k/ 6 hI

(2)

d , h, d > 0, and h > 0 are known integers presenting the bounds of d.k/ and h.k/; A, Ad , B, Bh , Bw , C , Cd , D, Dh , and Dw are known real matrices with appropriate dimensions; A.k/, B.k/, Ad .k/, Bh .k/, and Bw .k/ represent norm-bounded uncertainties and are assumed to be of the following form:     A.k/ B.k/ Ad .k/ Bh .k/ Bw .k/ D MF .k/ H1 H2 H3 H4 H5 , (3) here M , H1 , H2 , H3 , H4 , and H5 are known real matrices with appropriate dimensions; and F .k/ is an unknown matrix satisfying F .k/T F .k/ 6 I for k D 0, 1, 2,    . In order to formulate clearly the problems considered in this paper, we introduce the following discrete-time singular system with a pair of state delays: Ex.k C 1/ D Ax.k/ C Ad x.k  d.k// C Mh x.k  h.k// C Bw w.k/, ´.k/ D C x.k/ C Cd x.k  d.k// C Nh x.k  h.k// C Dw w.k/,

(4)

x.k/ D .k/, k D  ,  C 1,    , 0,  D max.d , h/, where Mh and Nh are known real matrices with appropriate dimensions. When h.k/ D d.k/, system (4) simplifies to the following system: Ex.k C 1/ D Ax.k/ C Gx.k  d.k// C Bw w.k/, ´.k/ D C x.k/ C J x.k  d.k// C Dw w.k/,

(5)

x.k/ D .k/, k D  ,  C 1,    , 0,  D max.d , h/, where G D Ad C Mh and J D Cd C Nh . Definition 2.1 ([1]) The pair .E, A/ is said to be regular if det.sE  A/ is not identically zero, and if, in addition, its degree is equal to rankE, then .E, A/ is further said to be causal. Definition 2.2 (i) System (4) with w.k/ D 0 is said to be regular and causal if the pairs .E, A/, .E, A C Ad /, .E, A C Mh /, and .E, A C Ad C Mh / are regular and causal. (ii) System (4) with w.k/ D 0 is said to be asymptotically stable if (a) for any " > 0, there exists a scalar ı."/ > 0 such that for any compatible initial condition .k/ satisfying sup k.k/k 6 ı."/, the solution x.k/ of system (4) satisfies kx.k/k 6 " for any k > 0;  6k60

and (b) x.k/ ! 0 when k ! 1. (iii) System (4) with w.k/ D 0 is said to be admissible if it is regular, causal, and asymptotically stable. It should be mentioned that if system (4) with w.k/ D 0 is regular and causal, then it has a unique solution for any given compatible initial value .k/. In particular, when both d.k/ and h.k/ are constant delays, system (4) has a unique solution if the pair .E, A/ is regular and causal. This is the reason that Definition 2.2 is different from those in [4, 9]. For system (4) and a given positive scalar , consider the H1 performance measure JD

1 X

.´.k/T ´.k/   2 w.k/T w.k//.

(6)

kD0

Definition 2.3 System (4) is said to be admissible with H1 performance  if it is admissible when w.k/  0 and satisfies J < 0 for all k > 0 under the zero initial condition. Copyright © 2012 John Wiley & Sons, Ltd.

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It should be emphasized that when rankE D n, system (1) is inherently regular and causal. Therefore, in this case, the ‘admissible’ in Definitions 2.2 and 2.3 should be replaced by ‘asymptotically stable’. When the state feedback u.k/ D Kx.k/ is applied to system (1), the resultant closed-loop system is obtained as follows: Q Ex.k C 1/ D Ax.k/ C AQd x.k  d.k// C MQ h x.k  h.k// C BQ w w.k/, ´.k/ D CQ x.k/ C Cd x.k  d.k// C NQ h x.k  h.k// C Dw w.k/,

(7)

x.k/ D .k/, k D  ,  C 1,    , 0,  D max.d , h/, where AQ D A C BK C MF .k/.H1 C H2 K/, AQd D Ad C MF .k/H3 , BQ w D Bw C MF .k/H5 , MQ h D Bh K C MF .k/H4 K, CQ D C C DK, NQ h D Dh K. The first aim of this paper is to design state feedback gain K such that the closed-loop system (7) is admissible with a given H1 performance . As a medium step, a new bounded real lemma for system (4) is investigated. Furthermore, it is mathematically proven that the proposed bounded real lemma has the same conservativeness with [4, Theorem 1]. However, the proposed bounded real lemma removes some free-weighting matrices required in [4, Theorem 1], which reduces the computational complexity. In this special case of h.k/ D d.k/, the closed-loop system (7) becomes the following: Q Q Ex.k C 1/ D Ax.k/ C Gx.k  d.k// C BQ w w.k/, ´.k/ D CQ x.k/ C JQ x.k  d.k// C Dw w.k/,

(8)

x.k/ D .k/, k D  ,  C 1,    , 0,  D max.d , h/, where GQ D AQd C MQ h and JQ D Cd C NQ h . When d.k/ is a constant delay, Ma et al. [9], by the fast and slow subsystems decomposition, offered a CCL algorithm to obtain a state feedback such that system (8) is admissible with a given H1 performance . However, their approach is not available when d.k/ is a time-varying delay. The second aim of this paper is to extend the results of Ma et al. to the time-varying delay case. 3. PRELIMINARY RESULTS The following several lemma will be helpful to present the main results of this paper. Lemma 3.1 ([5]) Let E, A 2 Rnn satisfy 0 < rankE < n, and R be any matrix whose columns form a basis of kerE T . If there exist matrices M and N such that AT RM C M T RT A C E T N C N T E < 0,

(9)

then the matrix pair .E, A/ is regular and causal. Lemma 3.2 ([21]) For matrices P > 0, M , and N with appropriate dimensions, F .t / with F T .t /F .t / 6 I , and a scalar " > 0, one has the following inequalities: (i) .MF .t /N /T P C P .MF .t /N / 6 "1 PMM T P C "N T N . (ii) If P 1  "1 MM T > 0, then  1 .A C MF .t /N /T P .A C MF .t /N / 6 AT P 1  "1 MM T A C "N T N . Copyright © 2012 John Wiley & Sons, Ltd.

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Lemma 3.3 ([9], Lemma 3) For a given scalar  > 0, if there exist symmetric matrices P > 0, W1 > 0, H > 0, S1 > 0, and a matrix V such that 2 3 ˆ11 V 0 AT P d .A  I /T S1 C T 6 V T W1 0 GT P d G T S1 JT 7 6 7 6 T T T 7 0 0 I Bw P d Bw S1 Dw 6 7 < 0, (10) 6 PA PG PBw P 0 0 7 6 7 4 d S .A  I / d S G d S B 0 d S1 0 5 1 1 1 w C J Dw 0 0 I 

H VT

V S1

 > 0,

(11)

where ˆ11 D P C W1 C d H C V C V T and  D  2 , then system (5), with h Ei D I and d.k/  d , is asymptotically stable with H1 performance  for any time-delay d 2 0, d . Lemma 3.4 ([4], Theorem 1) Let d , d , h, and h be defined as previously. For given a scalar  > 0, the discrete-time delayed singular system (4) is admissible with H1 performance  if there exist matrices P > 0, Si > 0, Wi > 0.i D 1, 2/, Z, Nj .j D 1, 2, 3, 4/, and L such that the following linear matrix inequality (LMI)   ƒ1 ƒ2 ƒ WD < 0, (12)  ƒ3 where

2 6 6 ƒ1 D 6 6 4

ƒ11

ƒ12

ƒ13



ƒ22

CdT Nh





ƒ33







3

ƒ14

7 E T L C CdT Dw 7 7, E T L C NhT Dw 7 5 T I C Dw Dw

ƒ11 D AT ˆZ T C ZˆT A C N1T E C E T N1 C N3T E C E T N3  E T PE C d  W1 C h W2 , ƒ12 D ZˆT Ad  N1T E C E T N2 C C T Cd , ƒ13 D ZˆT Mh  N3T E C E T N4 C C T Nh , ƒ14 D ZˆT Bw C 2E T L C C T Dw , ƒ22 D N2T E  E T N2  W1 C CdT Cd , ƒ33 D N4T E  E T N4  W2 C NhT Nh ,  ƒ3 D diag P , d S1 , hS2 , d S1 , hS2 , I , 2

AT P

6 T 6 Ad P ƒ2 D 6 6 4 MhT P BwT P

3

d .A  E/T S1

h.A  E/T S2

d N1T

hN3T

d ATd S1

hATd S2

d N2T

0

d MhT S1

hMhT S2

0

hN4T

7 0 7 7, 7 0 5

d BwT S1

hBwT S2

d LT

hLT

0

CT

d  D d  d C 1, h D h  h C 1, ˆ is any matrix whose columns form a basis of ker E T , and  is defined as previously. Copyright © 2012 John Wiley & Sons, Ltd.

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4. BOUNDED REAL LEMMAS In this section, we will investigate new bounded real lemmas for the considered systems in this paper. Furthermore, mathematical comparisons of the proposed bounded real lemmas and those in [4, 9] will be presented. 4.1. New bounded real lemmas The following theorem presents a bounded real lemma for system (4). Theorem 4.1 Let d , d , h, and h be defined as previously. For given a scalar  > 0, the discrete-time delayed singular system (4) subject to (2) is admissible with H1 performance  if there exist matrices P > 0, Si > 0, Wi > 0.i D 1, 2/, and Z such that the following LMI  WD 1 C 2 C 3 < 0,

(13)

where 1 D e1T .d  W1 C h W2  E T PE/e1  e2T W1 e2  e3T W2 e3 C A T P A C A1T .d S1 C hS2 /A1  d h

1

1

.e1  e2 /T E T S1 E.e1  e2 /

.e1  e3 /T E T S2 E.e1  e3 /, 

A Ad

e1 D



I

0 0 0



,

e4 D



0 0 0 I



,

2 D e1T ZˆT A C A T ˆZ T e1 , 3 D C T C  e4T e4 , A D A1 D A  Ee1 , C D

e2 D



0 I

0 0





C

Cd

Nh



0 0 I

, e3 D



Dw 0

, 

,

Mh

Bw



,

and ˆ, d  , h , and  are defined as previously. Proof We first show the regularity and causality of the delayed singular system (4) with w.k/ D 0. It follows from (13) that 2 3 1 1 Y ZˆT Ad C d E T S1 E ZˆT Mh C h E T S2 E 6 T 7 1 1 4 Ad ˆZ T C d E T S1 E 5 < 0, W1  d E T S1 E 0 MhT ˆZ T C h

1

E T S2 E

0

h

1

E T S2 E

where Y D d  W1 C h W2  E T PE C AT ˆZ T C ZˆT A  d

1

E T S1 E  h

1

E T S2 E,

and hence d

1

E T S1 E  h

1

E T S2 E  E T PE C AT ˆZ T C ZˆT A < 0,

 E T PE C .A C Ad /T ˆZ T C ZˆT .A C Ad / < 0,  E T PE C .A C Mh /T ˆZ T C ZˆT .A C Mh / < 0, and E T PE C .A C Ad C Mh /T ˆZ T C ZˆT .A C Ad C Mh / < 0. Copyright © 2012 John Wiley & Sons, Ltd.

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By Lemma 3.1, one can conclude that the pairs .E, A/, .E, A C Ad /, .E, A C Mh /, and .E, A C Ad C Mh / are regular and causal. Therefore, by Definition 2.2, the delayed singular system (4) with w.k/ D 0 is regular and causal. Next, we will show the asymptotical stability of the delayed singular system (4) with w.k/ D 0. Set .k/ D Œx T .k/ x T .k  d.k// x T .k  h.k// w T .k/ T , y.l/ D x.l C 1/  x.l/. Choose a Lyapunov–Krasovskii functional candidate as V .k/ D V1 .k/ C V2 .k/ C V3 .k/ C V4 .k/ C V5 .k/,

(14)

where 0 X

V1 .k/ D x T .k/E T PEx.k/, V2 .k/ D

k1 X

y T .j /E T S1 Ey.j /,

Dd C1 j Dk1C

0 X

V3 .k/ D

k1 X

y T .j /E T S2 Ey.j /,

DhC1 j Dk1C

d C1

V4 .k/ D

k1 X

X

x T .l/W1 x.l/,

j Dd C1 lDkCj 1

hC1

V5 .k/ D

k1 X

X

x T .l/W2 x.l/.

j DhC1 lDkCj 1

Define V .k/ D V .k C 1/  V .k/. Then V .k/ D

5 X

Vi .k/,

i D1

where V1 .k/ D V1 .k C 1/  V1 .k/ D x T .k C 1/E T PEx.k C 1/  x T .k/E T PEx.k/ D T .k/.A T P A  e1T E T PEe1 / .k/, X0

Œ.y T .k/E T S1 Ey.k/  y T .k  1 C /E T S1 Ey.k  1 C /

Xk1 D d y T .k/E T S1 Ey.k/  y T .j /E T S1 Ey.j / j Dkd Xk1 y T .j /E T S1 Ey.j / 6 d y T .k/E T S1 Ey.k/ 

V2 .k/ D

Dd C1

j Dkd.k/

6

T

.k/.d A1T S1 A1

d

1

.e1  e2 /T E T S1 E.e1  e2 // .k/,

V3 .k/ 6 T .k/.hA1T S2 A1  h Copyright © 2012 John Wiley & Sons, Ltd.

1

.e1  e3 /T E T S2 E.e1  e3 // .k/, Optim. Control Appl. Meth. 2013; 34:328–347 DOI: 10.1002/oca

H1 CONTROL FOR DISCRETE TIME-DELAY UNCERTAIN SINGULAR SYSTEMS

V4 .k/ D 6

Xd C1

Œx T .k/W1 x.k/  x T .k C j  1/W1 x.k j Dd C1 d  x T .k/W1 x.k/  x T .k  d.k//W1 x.k  d.k//

335

C j  1/

  D T .k/ d  e1T W1 e1  e2T W1 e2 .k/,

V5 .k/ 6 h x T .k/W2 x.k/  x T .k  h.k//W2 x.k  h.k//   D T .k/ h e1T W2 e1  e3T W2 e3 .k/. Thus, V .k/ 6 T .k/1 .k/.

(15) T

Noticing that ˆ is any matrix whose columns form a basic of ker E , we can deduce that 2x T .k C 1/E T ˆZ T x.k/ D 0, that is, T .k/2 .k/ D 0. This, together with (13) and (15), implies that V .k/ < 0 when w.k/  0, which guarantees the asymptotical stability of system (4) with w.k/ D 0. Finally, we will show that the delayed singular system (4) has H1 performance  under the zero initial condition. Clearly, X1 JD .´.k/T ´.k/   2 w.k/T w.k// kD0 X1 .´.k/T ´.k/   2 w.k/T w.k// C V .1/  V .0/ 6 kD0 X1 .´.k/T ´.k/   2 w.k/T w.k/ C V .k// D kD0 X1 .k/T  .k/. 6 kD0

Therefore, the LMI (13) guarantees J < 0, that is, system (4) has H1 performance  under the zero initial condition.  Remark 4.1 In order to get the minimum value of H1 performance , Theorem 4.1 can be changed as follows: Minimize  subject to LMI (13). p Here, the optimal H1 performance is obtained from   D . A similar process is available for Corollaries 4.1, 4.2, and 5.1, and Theorems 5.1 and 5.2. Remark 4.2 When d.k/  d (respectively, h.k/  h), that is, d.k/ (respectively, h.k/) is a constant delay, Theorem 4.1 is available by setting d D d D d (respectively, h D h D h). Likewise, Corollaries 4.1 and 5.1 and Theorems 5.1 and 5.2 are available when d.k/  d (respecting, h.k/  h). Similar to Theorem 4.1, a new bounded real lemma for system (5) can be easily obtained as follows: Corollary 4.1 Let d and d be defined as previously. For given a scalar  > 0, the discrete-time delayed singular system (5) subject to 0 6 d 6 d.k/ 6 d is admissible with H1 performance  if there exist matrices P > 0, S1 > 0, W1 > 0, and Z such that the following LMI Q WD  Q1C Q2C Q 3 < 0, 

(16)

where Q 1 D eN1T .d  W1  E T PE/eN1  eN2T W1 eN2 C AT P A C d AT1 S1 A1  d

1

.eN1  eN2 /T E T S1 E.eN1  eN2 /,

Copyright © 2012 John Wiley & Sons, Ltd.

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H. ZHU, X. ZHANG AND S. CUI

Q 2 D eN1T ZˆT A C AT ˆZ T eN1 ,  Q 3 D C T C  eN3T eN3 ,  AD CD



C

J

Dw





, eN1 D

A G 

I

Bw

0 0





, A1 D A  E eN1 ,

, eN2 D



0 I

0



, eN3 D



0 0 I



,



and ˆ, d , and  are defined as previously. In the special case that E D I and d.k/  d , the following corollary follows directly from Corollary 4.1 and Remark 4.2. Corollary 4.2 Let d be defined as previously. For given a scalar  > 0, the discrete-time delayed singular system (5), with E D I and d.k/  d , subject to 0 6 d 6 d , is admissible with H1 performance  if there exist matrices P > 0, S1 > 0, and W1 > 0 such that the following LMI N WD  N1C Q 3 < 0, 

(17)

where N 1 D eN1T .W1  P /eN1  eN2T W1 eN2 C AT P A C d AN T1 S1 AN 1  d

1

.eN1  eN2 /T S1 .eN1  eN2 /, AN 1 D A  eN1 ,

Q 3 , and A are defined as previously. and eNi .i D 1, 2/,  4.2. Mathematical comparisons of several bounded real lemmas The following theorem presents a mathematical comparison of Corollary 4.2 and Lemma 3.3 (i.e., [9, Lemma 3]). Theorem 4.2 Let  and d be defined as previously. If the LMIs (10) and (11) in Lemma 3.3 (i.e., [9, Lemma 3]) are feasible, then the LMI (17) in Corollary 4.2 is feasible. Proof By Schur complement lemma, the inequality (10) is equivalent to ‡ WD ‡1 C ‡2 < 0, where ‡1 D eN1T .W1  P /eN1  eN2T W1 eN2 C AT P A C d AN T1 S1 AN 1 C C T C  eN3T eN3 , ‡2 D eN1T .d H C V C V T /eN1  eN1T V eN2  eN2T V T eN1 , and , eNi , i D 1, 2, 3, A, AN 1 , and C are defined as previously. On the basis of the inequality (11), it can be obtained that H > V S11 V T , and hence, ‡2 > d eN1T V S11 V T eN1 C eN1T V .eN1  eN2 / C .eN1  eN2 /T V T eN1 . Because ‡Q WD d eN1T V S11 V T eN1 C eN1T V .eN1  eN2 / C .eN1  eN2 /T V T eN1 Cd

1

.eN1  eN2 /T S1 .eN1  eN2 / > 0,

it can be obtained that N1C Q 3 D ‡1 C ‡Q  d 1 .eN1  eN2 /T S1 .eN1  eN2 /  ‡1 C ‡2 < 0,  that is, the inequality (17) in Corollary 4.2 is feasible.

Copyright © 2012 John Wiley & Sons, Ltd.



Optim. Control Appl. Meth. 2013; 34:328–347 DOI: 10.1002/oca

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337

Remark 4.3 Theorem 4.2 indicates that if LMIs in [9, Lemma 3] are feasible, then the LMI in Corollary 4.2 is feasible. Therefore, the bounded real lemma investigated in Corollary 4.2 is less conservative than the one in [9]. Also, Corollary 4.2 removes free-weighting matrices H and V , which reduces the computational complexity. The following theorem presents a mathematical comparison of Theorem 4.1 and Lemma 3.4 (i.e., [4, Theorem 1]). Theorem 4.3 Let , d , d , h, and h be defined as previously. Then the LMI (12) in Lemma 3.4 (i.e., [4, Theorem 1]) is feasible if and only if the LMI (13) in Theorem 4.1 is feasible. Proof Let ƒ1 , ƒ2 , and ƒ3 be defined as in Lemma 3.4, and let  be defined as in Theorem 4.1. By a direct computation, it is easy to verify that T ƒ1  ƒ2 ƒ1 3 ƒ2 D  C d

Ch

1

1

.d N1 C S1 E.e1  e2 //T S11 .d N1 C S1 E.e1  e2 //

.hN2 C S2 E.e1  e3 //T S21 .hN2 C S2 E.e1  e3 //,

(18)

where N1 D



N1

N2

0 L



, N2 D



N3

0 N4

L



.

The ‘only if’ part. If the LMI (12) in Lemma 3.4 is feasible, then there exist real symmetric positive-definite matrices P , S1 , S2 , W1 , and W2 , and matrices Z, N1 , N2 , N3 , N4 , and L such that   ƒ1 ƒ 2 < 0. ƒ WD  ƒ3 By Schur complementary lemma and (18), one can easily show that  < 0, that is, the LMI (13) in Theorem 4.1 is feasible. The ‘if’ part. If the LMI (13) in Theorem 4.1 is feasible, then there exist real symmetric positive-definite matrices P , S1 , S2 , W1 , and W2 and a matrix Z such that  < 0. Set " 1 #   1 d S1 E d S1 E N1 N2 D L D 0, . 1 1 N3 N4 h S2 E h S2 E T 1 T Then (18) reduces to ƒ1  ƒ2 ƒ1 3 ƒ2 D , and hence ƒ1  ƒ2 ƒ3 ƒ2 < 0. Using Schur complement lemma and ƒ3 < 0, we obtain that ƒ < 0, that is, the LMI (12) in Lemma 3.4 is feasible. This completes the proof. 

Remark 4.4 Theorem 4.3 indicates that [4, Theorem 1] and Theorem 4.1 of this paper have the same conservativeness. However, Theorem 4.1 of this paper removes free-weighting matrices N1 , N2 , N3 , N4 , and L which are required in [4, Theorem 1]. This reduces the computational complexity. Therefore, the bounded real lemma investigated in Theorem 4.1 is more useful than the one in [4]. 5. H1 CONTROL In this section, we will present how to determine a state feedback such that the closed-loop system (7) is admissible with a given H1 performance . With this in mind, some notations have to be introduced. Clearly, there exist the nonsingular matrices U and V such that   Ir 0 V. (19) E DU 0 0 Copyright © 2012 John Wiley & Sons, Ltd.

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Because all columns of ˆ form a basis of ker E T , one can assume without loss of generality that   0 T . (20) ˆDU Inr Now, the following theorem can be derived from Theorem 4.1, (19) and (20). Theorem 5.1 Let U , V , d , d , h, and h be defined as previously. For given a scalar  > 0, the discrete-time delayed singular system (4) subject to (2) is admissible with H1 performance  if there exist matrices     P1 P2 Si1 Si 2 T 1 T U > 0, Si WD U U 1 > 0, P WD U  P3  Si 3  1  P1 0 1 Q O U T with det ZO 2 6D 0 such that Wi > 0 .i D 1, 2/, and Z D V ZO1 ZO 2    1 < 0, (21)  2 where      D e1T d  WQ 1 C h WQ 2  2U diag P11 , 0 U T e1  e2T WQ 1 e2  e3T WQ 2 e3  e4T e4   1 C e1T AQ C AQT e1  d .e1  e2 /T U diag P11 S11 P11 , 0 U T .e1  e2 /   1  h .e1  e3 /T U diag P11 S21 P11 , 0 U T .e1  e3 /, 1 D AQ D



h

AZO

p d AQ1T

AQ1T

Ad ZO

Mh ZO

p hAQ1T Bw



CQT

i

  , 2 D diag P 1 , S11 , S21 , I ,

O 1 , CQ D , AQ1 D AQ  E Ze



C ZO

Cd ZO

Nh ZO

Dw



,

and ei .i D 1, 2, 3, 4/, d  , h , and  are defined as previously. Proof It follows from (21) and Schur complementary lemma that   1 1 2 1 < 0. Set ‚ D diag.ZO 1 , ZO 1 , ZO 1 , I /, and Wi D ZO T WQ i ZO 1 , i D 1, 2. Then it follows from (19) that   ‚T   1 1 2 1 ‚ D e1T .d  W1 C h W2  2E T PE/e1  e2T W1 e2  e3T W2 e3  e4T e4 C A1T .P C d S1 C hS2 /A1 C e1T ZO T A C A T ZO 1 e1 C C T C d

1

.e1  e2 /T E T S1 E.e1  e2 /  h

1

(22)

.e1  e3 /T E T S2 E.e1  e3 /.

This, together with A1T P A1 C e1T ZO T A C A T ZO 1 e1  e1T E T PEe1

D A T P A C e1T ZˆT A C A T ˆZ T e1   P2  P1 ZO 1T ZO 2T , implies that the LMI (13) holds. By Theorem 4.1, the proof with Z D V T ZO 2T is completed.  Copyright © 2012 John Wiley & Sons, Ltd.

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Remark 5.1 Comparing Theorems 4.1 and 5.1, one can find that the LMI (13) is transformed to the nonlinear inequality (21). However, because of (21), it is found from the numerical examples offered in Section 7 that the approach proposed in this section may be less conservative than those in [4, 9]. Moreover, a CCL algorithm will be designed in the next section to solve the class of nonlinear matrix inequalities of the form (21) (respectively, (23), (24), and (27)). On the basis of Theorem 5.1, the following theorem can be obtained, which offers an approach to design a state feedback controller such that the resultant closed-loop system (7) is admissible with H1 performance . Theorem 5.2 Let U , V , d , d , h, and h be defined as previously. For given a scalar  > 0, the closed-loop discretetime delayed singular system (7)  subject to (2) and (3) is admissible  with H1 performance  if P1 P2 Si1 Si 2 T 1 T U > 0, Si WD U U 1 > 0, WQ i > 0 there exist matrices P WD U  P3  Si 3 " # P11 0 1 O and ZO D V U T with det ZO 2 6D 0, and scalars " > 0 and "j > 0 .i D 1, 2/, K, ZO1 ZO 2 .j D 1, 2, 3/ such that 3 2 „ „1 „2 0 6  2 0 „4 7 < 0, (23) 4   „ 0 5 5    „6 N C e T AO2 C AOT e1 C "e T MM T e1 , where „ D  1 2 1    T  N D e1 d WQ 1 C h WQ 2  2U diag.P11 , 0/U T e1  e2T WQ 1 e2  e3T WQ 2 e3  e4T e4    1  d .e1  e2 /T U diag P11 S11 P11 , 0 U T .e1  e2 /   1  h .e1  e3 /T U diag P11 S21 P11 , 0 U T .e1  e3 /, „1 D

h

AN2T

p d AN2T

p hAN2T

„4 D



COT

MQ T

0

i

T

, „2 D

h

HO T

HO T

p d HO T

i p hHO T ,

, MQ D diag.M , M , M /,

  1 1 „5 D diag "I , "1 1 I , "2 I , "3 I , „6 D diag ."1 I , "2 I , "3 I / , O 1, AO2 D AN C BO k , AN2 D AO2  E Ze AN D



BO k D

AZO 

Ad ZO

B KO

0 Bw

0 Bh KO

0



, HO D



, CO D

 

H1 ZO C H2 KO C ZO C D KO

H3 ZO Cd ZO

H4 KO Dh KO

H5 Dw





,

,

and ei .i D 1, 2, 3, 4/, 2 , d  , h , and  are defined as previously. Furthermore, if the inequality (23) is feasible, a state feedback controller gain is given by K D KO ZO 1 . Proof On the basis of Theorem 5.1 and Schur complementary lemma, it suffices to show that O WD  N C AO1T .P C d S1 C hS2 /AO1 C COT CO C e1T AO C AOT e1 < 0,  Copyright © 2012 John Wiley & Sons, Ltd.

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H. ZHU, X. ZHANG AND S. CUI

where AO1 D AN2 C MF .k/HO and AO D AO2 C MF .k/HO . Using Lemma 3.2, we have that   T 1 N O 6 N C e1T AO2 C AO2T e1 C "e1T MM T e1 C AN2T P 1  "1  A2 1 MM   T 1 N C "1 HO T HO C COT CO C "1 HO T HO C d AN2T S11  "1 A2 2 MM   1 T C d "2 HO T HO C h"3 HO T HO C hAN2T S21  "1 AN2 . 3 MM O < 0. ThereThis, together with Schur complementary lemma and the condition (23), implies that  fore, the closed-loop discrete-time delayed singular system (7) subject to (2) and (3) is admissible with H1 performance .  Following the lines in the proofs of Theorems 5.1 and 5.2, the following corollary can be easily obtained from Corollary 4.1. Corollary 5.1 Let U , V , d , and d be defined as previously. For given a scalar  > 0, the closed-loop discrete-time delayed singular system (8) subject to 0 6 d 6 d.k/ 6 d and (3) is admissiP1 P2 U 1 > 0, S1 WD ble with H1 performance  if there exist matrices P WD U T  P3    1  0 P1 S11 S12 U 1 > 0, WQ 1 > 0, KO and ZO D V 1 U T with det ZO 2 6D 0, and U T  S13 ZO1 ZO 2 scalars " > 0 and "j > 0 .j D 1, 2/ such that 3 2 … …1 …2 0 6  …3 0 …4 7 < 0, (24) 4   …5 0 5    …6 N C "eN T MM T eN1 C eN T AO 2 C AO T eN1 , where … D … 1 1 2  1  T  N D eN1 .d WQ 1  2U diag P1 , 0/U T eN1  eN2T WQ 1 eN2  eN3T eN3 …   1  d .eN1  eN2 /T U diag P11 S11 P11 , 0 U T .eN1  eN2 /, …1 D

h

AN T2

p d AN T2

COT

i

, …2 D

h

HN T

HN T

   …3 D diag P 1 , S11 , I , …4 D MN T

i p d HN T , 0

T

,

  1 MN D diag.M , M /, …5 D diag "I , "1 1 I , "2 I , …6 D diag."1 I , "2 I /, AO 2 D AN C BOk , AN 2 D AO 2  E ZO eN1 , AN D



AZO

BOk D



Ad ZO

B KO

Bw

Bh KO

0



, HN D







, CO D

H1 ZO C H2 KO C ZO C D KO

H3 ZO C H4 KO Cd ZO C Dh KO

H5 Dw





,

,

and eNi .i D 1, 2, 3/, d  , and  are defined as previously. Furthermore, if the inequality (24) is feasible, a state feedback gain is given by K D KO ZO 1 . Considering the special system as follows, Ex.k C 1/ D Ax.k/ C Ad x.k  d / C Bu.k/, Copyright © 2012 John Wiley & Sons, Ltd.

(25)

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341

when the state feedback controller u.k/ D Kx.k/ is applied to system (25), the resultant closed-loop system is obtained as follows: Ex.k C 1/ D .A C BK/x.k/ C Ad x.k  d /,

(26)

the following corollary can be easily obtained from Corollary 5.1. Corollary 5.2 Let U , V , d , and d be defined as previously. The closed-loop discrete-time delayed singular   system P P 1 2 U 1 > 0, (26) subject to 0 6 d 6 d is admissible if there exist matrices P WD U T  P3    1  0 S11 S12 O and ZO D V 1 P1 U 1 > 0, WQ 1 > 0, K, U T with S1 WD U T  S13 ZO1 ZO 2 det ZO 2 6D 0, such that   1 2 < 0, (27)  3 where     O T2 eO1 C eO1T d  WQ 1  2U diag P11 , 0 U T eO1  eO2T WQ 1 eO2 O2CA 1 D eO1T A   1  d .eO1  eO2 /T U diag P11 S11 P11 , 0 U T .eO1  eO2 /, 2 D

h

NT A 2

i p   O 2 D A C Bk , N T , 3 D diag P 1 , S11 , A dA 2

N2DA O 2  E ZO eN1 , A D A eO1 D



I

 0

AZO 

Ad ZO

, eO2 D





, Bk D

0 I





B KO

0



,

,

and d  is defined as previously. Furthermore, if the inequality (27) is feasible, a state feedback gain is given by K D KO ZO 1 . 6. A CONE COMPLEMENTARITY LINEARIZATION ALGORITHM TO DESIGN STATE FEEDBACK GAINS Because the inequalities (23), (24), and (27) are not LMI, the state feedback gains K cannot be solved by the LMI Toolbox of MATLAB. In order to solve the nonlinear inequality (23), in this section we will design a CCL algorithm (note that CCL algorithms can also be designed to solve (21), (24), and (27)). This requires introduction of the real symmetric positive-definite matrices PO , PO1 , SOi , Ti , TOi .i D 1, 2/, and j .j D 1, 2, 3/ satisfying   Si1 P1 > 0, (28)  TOi P PO D I , P1 PO1 D I , Si SOi D I , Ti TOi D I , i D 1, 2, "j j D 1, j D 1, 2, 3. Obviously, the inequality (23) is feasible if (28), (29), and 2 3 O „1 „2 0 6  „ O 3 0 „4 7 6 7 0 for given d > 0, h > 0, h > 0, and  > 0, such that the LMIs (28) and (30) are feasible under the constraint condition (29). Algorithm 6.1 (CCL algorithm) Step 1 Choose a sufficiently small initial d > d , such that there exists a feasible solution to (28), (30), and



P 

P > 0, PO > 0, PO1 > 0, Si > 0, SOi > 0, Ti > 0, TOi > 0, WQ i > 0, " > 0, "j > 0, j > 0,        I P1 I Si I Ti I > 0, > 0, > 0, > 0, PO  PO1  SOi  TOi   "j 1 > 0, i D 1, 2, j D 1, 2, 3.  j

(31)

Set dmax D d . Step 2 Find a feasible set of P0 , PO0 , PO10 , Si 0 , SOi 0 , Ti 0 , TOi 0 , WQ i 0 .i D 1, 2/, "0 "j 0 , j 0 .j D 1, 2, 3/, ZO 10 , ZO 20 with det ZO 2 6D 0 and KO 0 satisfying (28), (30), and (31). Set k D 0. Step 3 Solve the following LMI problem for the variables P , PO , PO1 , Si , SOi , Ti , TOi , WQ i .i D 1, 2/, O ", "j , j .j D 1, 2, 3/, ZO 1 , ZO 2 with det ZO 2 6D 0, and K: min

subject to (30) and (31)

tr‰ C

3 X

."j j k C "j k j / C tr.P1k PO1 C P1 PO1k /,

j D1

where ‰ D Pk PO C P POk C

2 X .Si SOi k C Si k SOi C Ti TOi k C Ti k TOi /. i D1

Set PkC1 D P , POkC1 D PO , PO1 kC1 D PO1 , Si kC1 D Si , SOi kC1 D SOi , Ti TOi kC1 D TOi .i D 1, 2/, "j kC1 D "j , and j kC1 D j .j D 1, 2, 3/. Step 4 If the LMI 2 3 Q1 „ Q2 0 „ „ 6  „ Q4 7 Q3 0 „ 6 7 4   „ Q 5 0 5 < 0,    „6 where Q1D „

h

Copyright © 2012 John Wiley & Sons, Ltd.

AN2T P

p d AN2T S1

p hAN2T S2

COT

i

kC1

D Ti ,

(32)

,

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H1 CONTROL FOR DISCRETE TIME-DELAY UNCERTAIN SINGULAR SYSTEMS

Q2D „

h

HO T

"1 HO T

p "2 d HO T

i p "3 hHO T ,

Q4D Q 3 D diag.P , S1 , S2 , I /, „ „



MN T

0

T

,

Q 5 D diag."I , "1 I , "2 I , "3 I /, MN D diag.P T M , S1T M , S2T M /, „ is feasible for the variables P2 , P3 , ", Si , WQ i .i D 1, 2/, "j .j D 1, 2, 3/, and the matrices P1 , ZO i .i D 1, 2/, and KO obtained in Step 3, then set dmax D d , increase d by a small amount, and return to Step 2. If LMI in (32) is infeasible within a specified number of iteration, then stop. Otherwise, set k D k C 1 and go to Step 3. Remark 6.1 By an approach similar to Algorithm 6.1, one can compute the minimum of  for given d > 0, d > 0, h > 0, and h > 0 such that the LMIs (28) and (30) are feasible under the constraint conditions (29). Remark 6.2 One can also solve the nonlinear inequalities (24) and (27) in Corollaries 5.1 and 5.2, respectively, by an approach similar to Algorithm 6.1. 7. NUMERICAL EXAMPLES In this section, we give several examples to demonstrate the effectiveness of the method proposed in this paper. Example 7.1 ([4] Example 1) Consider system (1) with the following parameters:           1 0 0 0.1 0.02 0 0 0.01 ED , AD , BD , Ad D , Bh D , 0 0 0.2 0 1 0 0.1 0  Bw D

0.1 1



 , CD 

Dw D

0.1 0.1

0.1 0 0 1





 , MD

H3 D

 , Cd D



0.02 0.1

0.01 0.02



0 0.1 0.1 0

 , H1 D





 , DD

0.1 0



1 0



 , Dh D

0.01 0.02

 ,

, H2 D 0.05,

, H4 D 0.02, H5 D 0.01.

Here, the aim is to find the min  for given d , d , h, and h such that the closed-loop system (7) is admissible with the H1 performance . Table I gives more detailed comparison results on min  obtained from Theorem 5.2 and [4, Theorem 2]. This illustrates that Theorem 5.2 may be more usable than [4, Theorem 2]. Example 7.2 Consider system (1) with the following parameters:           1 0 0.1 0 0.02 0 0 0.01 ED , AD , BD , Ad D , Bh D , 2 0 0 0.1 1 0.3 0.1 0  Bw D

0.1 1



 , CD

0.1 0 0 1

Copyright © 2012 John Wiley & Sons, Ltd.



 , Cd D

0 0.1 0.1 0



 , DD

1 0



 , Dh D

0.01 0.02

 ,

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H. ZHU, X. ZHANG AND S. CUI

Table I. Comparison of the minimums of  for different d when h D 4 and d D h D 0. d

min  (Theorem 5.2)

min  ([4, Theorem 2])

0.9424 0.9428 0.9428 0.9428 0.9428 0.9533 0.9617

1.1576 1.1748 1.1897 1.2037 1.2677 Infeasible Infeasible

1 2 3 4 10 50 100

 Dw D

0.1 0.1



H3 D

 , MD 

0.2 0.1

0.01 0.02

 , H1 D 



0.1 0



, H2 D 0.05,

, H4 D 0.02, H5 D 0.01.

Choose  D 1.2 in order to find the max d for given d , h, and h such that system (7) is admissible with the H1 performance . Table II gives more detailed comparison results on max d obtained from Theorem 5.2 and [4, Theorem 2]. This illustrates that Theorem 5.2 may be more usable than [4, Theorem 2]. When h D d D 1 and h D 4, for different d , the min  and the corresponding H1 state feedback gain K, such that the closed-loop system (7) subject to (2) and (3) is admissible with H1 performance , are listed in Table III. It is clearly shown that the min  increases as d increases. Under two situations, the state trajectories of the closed-loop uncertain system in Example 7.2 are given in Figures 1 and 2. The parameter uncertainty and the disturbance input are chosen as 1 F .k/ D sin.k/ and w.k/ D kC1 , respectively. Especially, we choose d.k/ D 5.5 sin.k/ C 6.5 and h.k/ D 1.5 sin.k/ C 2.5 in Figure 1 and d.k/ D 10 sin.k/ C 12 and h.k/ D 2 sin.k/ C 6 in Figure 2. Use the MATLAB function round to approximate d.k/ and h.k/ in numerical experiences. Table II. Comparison of the maximums of d for given , d , h, and h. Methods (h D 1, h D 4)

max d (d D 2)

max d (d D 10)

max d (d D 20)

Theorem 5.2 ( D 1.2) [4, Theorem 2] ( D 1.2) Theorem 5.2 ( D 1) [4, Theorem 2] ( D 1)

19 8 9 Infeasible

28 16 14 Infeasible

32 26 23 Infeasible

Methods (h D 4, h D 8) Theorem 5.2( D 1.2) [4, Theorem 2]( D 1.2) Theorem 5.2( D 1) [4, Theorem 2]( D 1)

max d (d D 2) 22 8 8 Infeasible

max d (d D 10) 25 16 18 Infeasible

max d (d D 20) 31 26 24 Infeasible

Table III. The minimum  and K for different d when h D 4 and h D d D 1. d

min 

1

0.8181

5

0.8695

8

0.9802

12

1.0192

Copyright © 2012 John Wiley & Sons, Ltd.

K  

11.7564

26.8789

76.0387 62.6040   2.4417 7.2683   3.2993 9.1939

 

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345

2 x1(k)

1.5

x (k) 2

1 0.5

x(k)

0 −0.5 −1 −1.5 −2 −2.5 −15

−10

−5

0

5

10

15

20

25

30

35

k

Figure 1. State responses of the closed-loop system in Example 7.2 with  D 1.0192, d D 12, h D 4, and h D d D 1. 120 x1(k)

100

x (k) 2

80 60

k

40 20 0

−20 −40 −60

−20

−10

0

10

20

30

40

x(k)

Figure 2. State responses of the closed-loop system in Example 7.2 with  D 1.2, d D 22, h D 8, h D 4, and d D 2.

Remark 7.1 It should be emphasized that the approach proposed by Ma et al. [9] cannot be applied to Examples 7.1 and 7.2 because their approach is only available for the special case h.k/ D d.k/  d . In order to compare the approaches proposed in [9] and this paper, we offer the following example.

Example 7.3 Consider system (1) when h.k/ D d.k/  d with the following parameters:           1 0 0.1 0 0.02 0 0 0.01 ED , AD , BD , Ad D , Bh D , 2 0 0 0.2 1.2 0.2 0.1 0 Copyright © 2012 John Wiley & Sons, Ltd.

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Table IV. The minimum  and K for different d .

 Bw D

0.1 1

d

min 

1

0.6965



2.2988

5.8680



2

0.6885



0.4751

4.2025



3

0.6141



6.5221

5.2027



5

0.6078



5.6313

4.4911



10

0.6168



5.9667

4.6330



20

0.6280



6.9290

4.9080



50

0.6430



8.7813

5.3336



100

0.6759



13.7885

7.1649



200

0.7094



23.1511

10.8199



500

0.7983



50.9397  21.7736





 , CD 

Dw D

0.1 0.1

0.1 0 0 1





H3 D

 , Cd D 

, MD 

K

0.2 0.1

0.01 0.02

0 0.1 0.1 0

 , H1 D 





 , DD

0.1 0



1 0



 , Dh D

0.01 0.02

 ,

, H2 D 0.05,

, H4 D 0.02, H5 D 0.01.

For different d , the min  and the corresponding H1 state feedback gain K, such that the closedloop system (8) subject to (2) and d.k/  d is admissible with H1 performance , are listed in Table IV, which are obtained from Corollary 5.1. By [9, Theorem 2] and Corollary 5.1, respectively, it is found that system (8) is admissible with the H1 performance  D 0.8 when 2 6 d 6 16 and 1 6 d 6 503, respectively. These illustrate that Corollary 5.1 may be more usable than [9, Theorem 2]. Example 7.4 Consider system (25) with the following parameters:         1 0 1 1 0 1 0.4 , Ad D , BD , AD . ED 1.5 2 0 0 0 1 0 Because the .2, 2/-th entry of A is 0, it follows that the matrix pair .E, A/ must not be causal, and hence the unforced part of the considered system is not admissible for all the delay d . By Corollary 5.2 and the corresponding CCL algorithm, one can obtain that the maximum of d is 16, such that theclosed-loop system (26)  is admissible, and a corresponding state feedback gain is given by K D 10.2277 9.6544 . This demonstrates the effectiveness of the proposed CCL algorithm. 8. CONCLUSIONS This paper studied the robust H1 control problem of a class of discrete-time uncertain singular systems with time-varying interval delays in state and control input. A new bounded real lemma, which removes some free-weighting matrices in [4, Theorem 1], for discrete-time singular systems with a pair of time-varying interval state delays was derived. Furthermore, it is mathematically Copyright © 2012 John Wiley & Sons, Ltd.

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investigated that (i) a special case of the bounded real lemma proposed here is less conservative than [9, Lemma 3]; and (ii) the proposed bounded real lemma and [4, Theorem 1] have the same conservativeness. This indicates the advantage of the proposed bounded real lemma. On the basis of the new bounded real lemma, a sufficient condition, such that the considered H1 control problem is solvable, was given. As the sufficient condition is a nonlinear matrix inequality, a CCL algorithm was developed to solve H1 state feedback gain. Several numerical examples are provided to show the applicability of the proposed approach. Finally, it should be mentioned that the approach proposed in this paper can also be used to solve the robust H1 control problem of discrete-time uncertain singular systems with only single state delay. ACKNOWLEDGEMENT

This work is supported by the fund Heilongjiang Education Committee under Grant No. 12511417, the fund of Heilongjiang University Innovation Team Support Plan under Grant Hdtd2001-03, and the Fund of Heilongjiang University Academic Scientific and Technological Innovation of Students. The authors thank the anonymous referees for their helpful comments and suggestions which greatly improved this note. REFERENCES 1. Dai L. Singular Control Systems, Lect. Notes Control Inform. Sci., Vol. 118. Springer-Verlag: Berlin, 1989. 2. Lewis FL. A survey of linear singular systems. Circuits Syst Signal Process 1986; 5(1):3–36. 3. Duan GR. Analysis and Design of Descriptor Linear Systems, Adv. Mechanics Math., Vol. 23. Springer: New York, 2010. 4. Kim JH. Delay-dependent robust H1 control for discrete-time uncertain singular systems with interval time-varying delays in state and control input. Journal of the Franklin Institute 2010; 347:1704–1722. 5. Zhang X, Zhu HY. Robust stability and stabilization criteria for discrete singular time-delay LPV systems. Asian Journal of Control. DOI: 10.1002/asjc.418 (online). 6. He Y, Ma J, Zhang X, Yang XR. Robust H1 control via memory state feedback for singular systems with time delay and linear fractional parametric uncertainties. In Proceedings of 2010 Chinese Conference Decision Control. IEEE: Piscataway, New Jersey, 2010; 30–34. 7. Li FB, Zhang X. A delay-dependent bounded real lemma for singular LPV systems with time-variant delay. International Journal of Robust and Nonlinear Control. DOI: 10.1002/rnc.1714 (online). 8. Chen N, Zhai GS, Gui WH. Robust decentralized H1 control of multi-channel discrete-time descriptor systems with time-delay. International Journal of Innovative Computing, Information and Control 2009; 5(4):971–979. 9. Ma S, Zhang CH, Cheng ZL. Delay-dependent robust H1 control for uncertain discrete-time singular systems with time-delays. Journal of Computational and Applied Mathematics 2008; 217:194–211. 10. Du ZP, Zhang QL, Li Y. Delay-dependent robust H1 control for uncertain singular systems with multiple state delays. IET Control Theory and Applications 2009; 3(6):731–740. 11. Fang M. Delay-dependent robust H1 control for uncertain singular systems with state delay. Acta Automatica Sinia 2009; 35(1):65–70. 12. Ma SP, Zhang CH, Wu Z. Delay-dependent stability and H1 control for uncertain discrete switched singular systems with time-delay. Applied Mathematics Sinica 2008; 206(1):413–424. 13. Ma SP, Zhang CH. Robust stability and H1 control for uncertain discrete markovian jump singular systems with mode-dependent time-delay. International Journal of Robust and Nonlinear Control 2009; 19:965–985. 14. Du ZP, Zhang QL, Chang GS. Delay-dependent robust H1 control for uncertain descriptor systems with multiple state delays. Optimal Control Applications and Methods 2010; 31:375–387. 15. Chen N, Gui WH, Xie YF, Li JZ. Robust decentralized H1 control of multi-channel discrete-time descriptor systems with time-delays. Proceedings of 46th IEEE Conference Decision Control, New Orleans, LA, 2007; 1233–1238. 16. Wang HJ, Xue AK, Lu RQ, Chen Y. Delay-dependent robust H1 control for uncertain discrete singular timevarying delay systems based on a finite sum inequality. In Proceedings of 26th Chinese Control Conference. IEEE: Piscataway, New Jersey, 2007; 595–599. 17. Wang HJ, Wang JZ, Ge M, Xue AK, Lu RQ. Delay-dependent robust H1 control for uncertain discrete singular systems. Control Theory and Applications 2008; 25(6):1145–1150. 18. Huang D, Ma SP. Robust H1 control of uncertain discrete singular systems with time-varying delays. In Proceedings of 26th Chinese Control Conference. IEEE: Piscataway, New Jersey, 2006; 740–745. 19. Xu S, Lam J, Yang C. Robust H1 control for discrete singular systems with state delay and parameter uncertainty. Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications Algorithms 2002; 9(4):539–554. 20. Ghaoui LE, Oustry F, AitRami M. A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Transactions on Automatic Control 1997; 42(8):1171–1176. 21. Wang Y, Xie L, de Souza CE. Robust control of a class of uncertain nonlinear systems. Systems and Control Letters 1992; 19(2):139–149. Copyright © 2012 John Wiley & Sons, Ltd.

Optim. Control Appl. Meth. 2013; 34:328–347 DOI: 10.1002/oca