Generalized guaranteed cost control with D-stability and multiple output constraints

Applied Mathematics and Computation 218 (2012) 12013–12027

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Generalized guaranteed cost control with D-stability and multiple output constraints Minqing Xiao a,b,⇑, Hongye Su a, Weihua Xu a a b

National Laboratory of Industrial Control Technology, Institute of Cyber-Systems and Control, Zhejiang University, Hangzhou 310027, PR China School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, PR China

a r t i c l e

i n f o

Keywords: Guaranteed cost control D-stability Output constraint Linear system Delta domain Linear matrix inequality

a b s t r a c t This paper is concerned with the problem of robust generalized guaranteed cost control with D-stability and multiple output constraints for a class of linear uncertain systems. Being a combination of output performance indices, a generalized cost function is considered to the linear polytopic uncertain systems described in a unified framework. The aim is to design a state feedback controller, such that the closed-loop system is robust D-stable, and the upper bound of the generalized cost function is as small as possible subject to multiple output constraints. Based on parameter-dependent Lyapunov functions, convex conditions for the existence of such controllers are presented in terms of linear matrix inequality. The proposed approach shows a unified treatment of the linear systems in the differential, shift, and delta domains. Numerical examples are provided to illustrate the effectiveness of the design method. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Guaranteed cost control is a robust control approach for uncertain systems, which aims to design a feedback controller such that the closed-loop system asymptotic stability and a certain level of a quadratic performance are ensured for all model uncertainties of a given class. During the past three decades, guaranteed cost control has been investigated for various control systems by many researchers (see, for example, [1–13] and the references therein). Among these researches, guaranteed cost control with D-stability constraint has been considered to linear uncertain systems [1,5,7,12]. The purpose of guaranteed cost control with D-stability constraint (i.e. guaranteed cost D-stabilization) is not only to guarantee the level of quadratic performance, but also to ensure that the poles of the control system lie inside a specified disk in the complex plane. It is well known that the poles of linear time-invariant system are closely related with the transient response characteristics of the system. In practical applications, one cannot place all the poles in precise locations due to parametric uncertainties originating from various sources, such as identification errors, aging of devices, and so on. Therefore, it is reasonable to assign all the poles of the closed-loop system in a desired region rather than exact assignment. This has brought about the study of pole assignment in a specified region [14–22]. In [5], an LMI-based algorithm is proposed to design robust guaranteed cost controllers that ensure closed-loop eigenvalue placement and a certain level of closed-loop performance for linear systems with parametric uncertainties. Based on a so-called satisfactory control strategy, [7] investigated the problem of reliable H1 guaranteed cost control with D-stability and input constraints for a class of uncertain discrete-time systems. In [12], a design method of guaranteed cost control with D-stability constraint had been developed to a power system. ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (M. Xiao). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.06.010

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On the other hand, in many practical systems, there are various constraints imposed on input and output variables. The problems of control with input and (or) output constraints have been the focus of much research literature (see, e.g. [23–29] and the references therein). Recently, Balandin and Kogan [28] discussed the suboptimal guaranteed performance control with multiple output constraints for a class of linear uncertain continuous-time system. Instead of quadratic function of the state and control variables, they introduced a new performance index, which is a combination of integral criteria and maxima in time of norm of linear functions of state and control variables, to be the cost function. Their results were used to synthesize a suboptimal linear shock isolator. Unfortunately, D-stability of the control system was not considered in [28]. As we know, in practice, many engineered systems must meet multiple objectives and multiple constraints. But to the author’s best knowledge, the guaranteed cost control with simultaneous D-stability constraint and multiple output constraints has not been explored. Motivate by this, the paper investigate the robust generalized guaranteed cost control with D-stability and multiple output constraints for a class of linear polytopic uncertain systems. The generalized cost function in [28] and two classes of output performance index constraints are adopted. The objective is to design a state feedback controller, such that the closed-loop system is robust D-stable, and the upper bound of the generalized cost function is as small as possible subject to multiple output constraints. Based on parameter-dependent Lyapunov function approach, convex conditions for the existence of such controllers are obtained by means of linear matrix inequalities. The results are presented in a unified framework of continuous-time systems and discrete-time systems in both the shift and delta domains [30–35]. It is worth noting that, although the results deal with the control problem under output constraints, it can be utilized to the corresponding problem under input constraints. The paper is organized as follows: in the following section, the problem statement and basic notions are given. In Section 3, based on parameter-dependent Lyapunov function approach and linear matrix inequality technique, the sufficient condition of robust D-stability under multiple output constraints is discussed, then the design methods of c-suboptimal generalized guaranteed cost controller and optimal generalized guaranteed cost controller are proposed, respectively. Numerical examples are given to illustrate the effectiveness of the proposed approach in Section 4 and, finally, Section 5 provides concluding remarks. Notation: R, C denote the real number field and the complex number field, respectively; Zþ denotes the set of nonnegative integer; Rn stands for the n-dimensional Euclidean space; I denotes the identity matrix; for symmetric matrices X and Y, the notation X P Y(respectively, X > Y) means that the matrix X  Y is positive semi-definite (respectively, positive definite); Rez and jzj stand for the real part and modulus of a complex number z; the superscripts ’T’ denotes the transpose; ‘⁄’ in a symmetric matrix represents the term that is induced by symmetry. 2. Preliminaries Consider the following linear uncertain system described in a unified framework

DxðtÞ ¼ AxðtÞ þ BuðtÞ;

xð0Þ ¼ x0 ;

yi ðtÞ ¼ C i xðtÞ þ Di uðtÞ;

i ¼ 1; 2; . . . ; M;

ð1Þ

where xðtÞ 2 Rn is the system state, uðtÞ 2 Rp is the system input, yi ðtÞ 2 Rqi , i ¼ 1; 2; . . . ; M, are the system outputs. The symbol D denotes the differential operator d=dt, the forward shift operator q, or the delta operator d ¼ ðq  1Þ=h, that is,

DxðtÞ ¼

8 dxðtÞ > > > dt ; <

for continuous time;

xðt þ 1Þ; for discrete time with the shift operator; > > > :1 ½xðt þ 1Þ  xðtÞ; for discrete time with the delta operator; h

where h is the sampling period. The set T of time arguments t depends on the operator D, specifically,

8 ½0; 1Þ; D ¼ d=dt; > > < T ¼ Zþ ; D ¼ q; > > : þ Z ; D ¼ d: It is worth mentioning that, for discrete time with the delta operator, the time arguments t represent for the sampling instant th, i.e., integer t is a simplification of th. In the system (1), A; B are uncertain real matrices with appropriate dimensions, and belong to a convex polytopic set defined as

( X¼ where ½ Aj

½A B : ½A B ¼

N X

N X

j¼1

j¼1

xj ½ Aj Bj ; xj P 0;

Bj , j ¼ 1; . . . ; N, are the vertices of X.

xj ¼ 1

) ð2Þ

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For each output vector yi , i ¼ 1; 2; . . . ; M, we define two classes of output performance indices as follows

J ð1;iÞ ¼ maxfyTi ðtÞyi ðtÞg; tP0 8R1 T yi ðtÞyi ðtÞdt; D ¼ d=dt; > 0 > > 1 > X > > < yTi ðtÞyi ðtÞ; D ¼ q; J ð2;iÞ ¼ t¼0 > > 1 > X > > > yTi ðtÞyi ðtÞ; D ¼ d: : t¼0

In this paper, the following combined performance index (see [28]) will be adopted for the system (1),

J ¼

M X ki J i ;

ð3Þ

i¼1

where ki , i ¼ 1; . . . ; M, are given non-negative real numbers (not all equal 0) representing weighting factors, J i ¼ J ð1;iÞ , or J ð2;iÞ . The combined performance index J is named a generalized cost function. We assume that the states of the system (1) are all measurable, then associated with the state feedback controller

uðtÞ ¼ KxðtÞ;

ð4Þ

the closed-loop delta operator system is

e DxðtÞ ¼ AxðtÞ; xð0Þ ¼ x0 ; e yi ðtÞ ¼ C i xðtÞ; i ¼ 1; 2; . . . ; M;

ð5Þ

e ¼ A þ BK, C~i ¼ C i þ Di K. Denote A e j ¼ Aj þ Bj K, j ¼ 1; . . . ; N, then where A

e¼ A

N X

xj Ae j ; xj P 0;

j¼1

N X

xj ¼ 1:

ð6Þ

j¼1

It is known that the stable region of a linear time-invariant system is as follows

8 fz 2 C : Rez < 0g; D ¼ d=dt; > > < fz 2 C : jzj < 1g; D ¼ q; S, n o > > 2 h : z 2 C : jzj þ Rez < 0 ; D ¼ d: 2

ð7Þ

Let Dða; rÞ denotes a disk region centered at a þ 0j and radius r on the complex plane. We know that, for discrete time system with the shift operator, S ¼ Dð0; 1Þ, and for discrete time system with the delta operator, S ¼ Dð1=h; 1=hÞ. If all poles of a linear time-invariant system lie inside an expected disk region Dða; rÞ # S, then the linear system is called Dða; rÞ-stable, or sometimes for brevity, D-stable [14]. It is worth of mentioning that, Dða; rÞ # S means that, a and r must satisfy

8 D ¼ d=dt; > < a < 0; r < a; 1 < a < 1; r < 1  jaj; D ¼ q; >   : 2  h < a < 0; 0 < r < min a; 2h þ a ; D ¼ d: The main objective of the work is to solve the following two problems: Problem 1. Robust Dða; rÞ-stability analysis under multiple output constraints: Investigate the condition of robust Dða; rÞstability for the closed-loop system (5) with the following multiple output constraints

J k 6 ck ;

k ¼ 1; . . . ; M;

ð8Þ

where ck (k ¼ 1; . . . ; M) are given positive real numbers, J k ¼ J ð1;kÞ or J ð2;kÞ , k ¼ 1; . . . ; M. Problem 2. Robust generalized guaranteed cost control with Dða; rÞ-stability and multiple output constraints: Determine a state feedback controller, such that for all admissible uncertainties the closed-loop system (5) is robust Dða; rÞ-stable, and the upper bound of the value of the generalized cost function

J ¼

m X ki J i ;

1 6 m 6 M; ki > 0; i ¼ 1; . . . ; m;

ð9Þ

i¼1

is as small as possible subject to the output constraints (if any)

J k 6 ck ;

k ¼ m þ 1; . . . ; M;

ð10Þ

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where ck (k ¼ m þ 1; . . . ; M) are given positive numbers, J l ¼ J ð1;lÞ or J ð2;lÞ , l ¼ 1; . . . ; M. Remark 1. The generalized cost function (3) can be regarded as an extension of the ordinary quadratic cost function which is often adopted in the problem of guaranteed cost control. For example, in continuous time case, the system is as follows

_ xðtÞ ¼ AxðtÞ þ BuðtÞ;

xð0Þ ¼ x0 ;

ð11Þ

the ordinary quadratic cost function is

J0 ¼

Z

1

½xT ðtÞRxðtÞ þ uT ðtÞQuðtÞdt

ð12Þ

0

with weighting matrices R > 0, Q > 0. We can construct an output signal y1 ðtÞ ¼ C 1 xðtÞ þ D1 uðtÞ, such that

J 1 ¼ J ð2;1Þ ¼

Z

1

0

yT1 ðtÞy1 ðtÞdt ¼ J 0 :

ð13Þ

In fact, because of R > 0, Q > 0, it can be written that R ¼ R21 , Q ¼ Q 21 , where R1 , Q 1 are symmetric positive definite matrices, now let

C1 ¼



R1



0

D1 ¼

;



0 Q1

 ;

then (13) holds. So we can regard (3) as an extension of the ordinary cost function (12). Before tackling the presented control problems, we need introduce the following lemma which will be used subsequently.

Lemma 1 [14]. For a normal linear time-invariant system DxðtÞ ¼ A0 xðtÞ, given Dða; rÞ # S, then the system is Dða; rÞ-stable if and only if there exists symmetric matrix P > 0, such that

ðA0  aIÞT PðA0  aIÞ  r2 P < 0: Though the discrete time system with the delta operator was not involved in [14], the result can be naturally extended to the delta operator system.

3. Main results 3.1. Robust Dða; rÞ-stability with multiple output constraints In order to solve the problem of robust generalized guaranteed cost control with Dða; rÞ-stability and multiple output constraints, we should discuss robust Dða; rÞ-stability with multiple output constraints for the closed-loop system (5) firstly. Taking advantage of parameter-dependent Lyapunov function and linear matrix inequality approach, we present the following three theorems for different cases of output constraints. 3.1.1. Multiple output constraints: J ð2;iÞ 6 ci (i ¼ 1; . . . ; M) Suppose that the multiple output constraints of the the system (5) are

J i ¼ J ð2;iÞ 6 ci ;

i ¼ 1; . . . ; M;

ð14Þ

where ci , i ¼ 1; . . . ; M, are given positive numbers. The following result gives a sufficient condition of robust Dða; rÞ-stability with multiple output constraints (14). Theorem 1. Given Dða; rÞ # S, if there exists symmetric matrices X j > 0, j ¼ 1; . . . ; N, matrix G such that for each j 2 f1; 2; . . . ; Ng, the following linear matrix inequalities hold,

2 6 6 6 4 "

rX j  rG  rGT



e j G  aG A

rX j

e iG C

0

X j

x0

xT0

1



3

7 7  7 < 0; 5

i ¼ 1; . . . ; M;

ð15Þ

ci I

# 6 0;

ð16Þ

then for all admissible uncertainties the system (5) is robust Dða; rÞ-stable, and meet the output performance index constraints (14).

M. Xiao et al. / Applied Mathematics and Computation 218 (2012) 12013–12027

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Proof. Denote the left side of (15) by Cij . From (15) and (6), we have

2

rX  rG  rGT 6 e 4 AG  aG e iG C

3  N 7 X xj Cij < 0;  5¼ j¼1 ci I

 rX 0

i ¼ 1; . . . ; M;

ð17Þ

where X ¼ x1 X 1 þ    þ xN X N . Matrix inequality (17) implies that G þ GT > X > 0, so G is invertible. Now pre- and post-multiplying block diagonal matrix diagfGT ; X 1 ; Ig and diagfG1 ; X 1 ; Ig to the left-hand side of the inequality (17) respectively, we obtain

2

rGT ðX  G  GT ÞG1 6 6 e  aIÞ X 1 ð A 4 ei C





3

7  7 5 < 0:

rX 1

ð18Þ

ci I

0

On the other hand, by X > 0, we know ðG1  X 1 ÞT XðG1  X 1 Þ P 0, that is,

X 1 6 GT ðX  G  GT ÞG1 : Now let P ¼ X 1 . From (18), we have

2

rP  6 e 6 Pð A  aIÞ rP 4 ei 0 C

3



7  7 5 < 0: ci I

ð19Þ

Denote

1 r

e  aIÞT Pð A e  aIÞ  rP; U ¼ ðA then by the inequality (19) and the Schur complement [36], we know that U < 0. According to Lemma 1, for all admissible uncertainties, the system (5) is Dða; rÞ-stable. By the inequality (16), we get



X xT0

x0 1

 ¼

N X j¼1

xj



X j

x0

xT0

1

 6 0;

then using the Schur complement, one obtains

xT0 Px0 6 1:

ð20Þ

To prove the output performance index constraints (14) satisfying, we will discuss in three cases depended on the operator D. Case 1 D ¼ d=dt. In this case, a < 0, and r < a. It follows that,

1 a2  r 2 P r r 2 2 2 1 e e  ða þ rÞIT P½ A e  ða þ rÞI þ ða þ rÞ þ a  r P ¼ 1 ½ A e  ða þ rÞI  ða þ rÞIT P½ A ¼ ½A r r r þ 2ða  rÞP > 0;

eT PA e T P þ P AÞ e ¼ ½A e  ða þ rÞ A e T P  ða þ rÞP A e þ U  ðA

eT P þ PA e < U. On the other hand, by (19) and the Schur complement, we have U þ c1 C eT C e i < 0, Thus, the following that is, A i i inequality holds

e þ c1 C eT P þ PA eT C e i < 0; A i i

ð21Þ

Now choose the Lyapunov function

VðtÞ ¼ xT ðtÞPxðtÞ; then the derivative of VðtÞ along the trajectory of the closed-loop system (5) is T e T P þ P AÞxðtÞ e eT e _ VðtÞ ¼ xT ðtÞð A < c1 i x ðtÞ C i C i xðtÞ;

so, we have

eT C e i xðtÞ < c VðtÞ: _ xT ðtÞ C i i

ð22Þ

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Integrating two sides of the above inequality from 0 to 1, since the closed-loop system (5) is Dða; rÞ-stable, and certainly, asymptotically stable, we have

Ji ¼

Z

1

0

eT C e i xðtÞ 6 c Vð0Þ ¼ c xT Px0 ; xT ðtÞ C i i 0 i

now by (20), the constraints (14) are satisfied. Case 2. D ¼ q. In this case, r < 1 and jaj < 1  r. Then,

 2 2 1  r eT e a eT a e þa r þrP A PA  A P PA r 1r 1r r 1  r e a T  e a  a2 a2  r 2 þ r A I P A I  P Pþ ¼ r 1r 1r r rð1  rÞ 1  r e a T  e a  ð1  rÞ2  a2 ¼ A I P A I þ P > 0: r 1r 1r 1r

eTPA e  PÞ ¼ U  ðA

Similar to case 1, we obtain

eT PA e  P þ c1 C eT C e i < 0; A i i

ð23Þ

Choose the Lyapunov function

VðtÞ ¼ xT ðtÞPxðtÞ; the difference of VðtÞ along the trajectory of the closed-loop system (5) is

eTPA e  PÞxðtÞ < c1 xT ðtÞ C e i xðtÞ; eT C DVðtÞ ¼ xT ðtÞð A i i so, we have

e i xðtÞ < c DVðtÞ: eT C xT ðtÞ C i i Summing both sides of the above inequality from 0 to 1, and by the closed-loop system (5) is Dða; rÞ-stable, we get

Ji ¼

1 X e i xðtÞ 6 c Vð0Þ ¼ c xT Px0 ; eT C xT ðtÞ C i i 0 i t¼0

also by (20), the constraints (14) are met. Case 3. D ¼ d. In this case, 0 < r < minfa; 2h þ ag, so 0 < r < 1h, then 1  rh > 0, a þ r < 0, and r  a  2h < 0. Thus, we have

 2 2 1 eTPA e T P þ P AÞ e  a þ r ðA e þa r P h A r r r 1  rh  e a þ r T  e a þ r  a2  r 2 ða þ rÞ2 A I P A I þ½  P ¼ r 1  rh 1  rh r rð1  rhÞ  1  rh  e a þ r T  e aþr  h 2 P > 0: A I P A I þ ða þ rÞ r  a  ¼ r 1  rh 1  rh 1  rh h

eT PA eþA e T P þ P AÞ e ¼ U  ðh A

Also similar to case 1, we obtain

eT PA eþA eT P þ PA e þ c1 C eT C e i < 0; hA i i

ð24Þ

Choose the Lyapunov function

VðtÞ ¼ xT ðtÞPxðtÞ: Since dxðtÞ ¼ tem (5) is

xðtþ1ÞxðtÞ , h

that is, xðk þ 1Þ ¼ xðkÞ þ hdxðkÞ, then the difference of VðtÞ along the trajectory of the closed-loop sys-

e T PðI þ h AÞ e  PxðtÞ ¼ xT ðtÞðh2 A e þ hA e T P þ hP AÞxðtÞ e eTPA DVðtÞ ¼ xT ðt þ 1ÞPxðt þ 1Þ  xT ðtÞPxðtÞ ¼ xT ðtÞ½ðI þ h AÞ eT e < c1 i hx ðtÞ C i C i xðtÞ; T

so, we have

e i xðtÞ < c DVðtÞ: eT C hx ðtÞ C i i T

ð25Þ

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Summing both sides of the above inequality from 0 to 1, and by the closed-loop system (5) is Dða; rÞ-stable, we get

Ji ¼

1 X

e i xðtÞ 6 c Vð0Þ ¼ c xT Px0 ; eT C xT ðtÞ C i i 0 i

t¼0

also by (20), the constraints (14) are met. This completes the proof. h

Remark 2. In fact, it is easy to find that the symmetric positive definite matrix Xj in (15) may vary by i. But it will be difficult to obtain a gain matrix of the desired controller under different X j . The requirement of the same X j will help us to get the control law. The method is often used in multi-objective control, while it cause the potential conservatism of the results. 3.1.2. Multiple output constraints: J ð1;iÞ 6 ci (i ¼ 1; . . . ; M) Suppose that the multiple output constraints of the the system (5) are

J i ¼ J ð1;iÞ 6 ci ;

i ¼ 1; . . . ; M;

ð26Þ

where ci , i ¼ 1; . . . ; M, are given positive numbers. The following result gives a sufficient condition of robust Dða; rÞ-stability with multiple output constraints (26). Theorem 2. Given Dða; rÞ # S, if there exists symmetric matrices X j > 0, j ¼ 1; . . . ; N, matrix G such that for each j 2 f1; 2; . . . ; Ng, the following inequalities hold,

" "

rX j  rG  rGT e j G  aG A

#

 rX j #

X j  G  GT  e ci I C iG   X j x0 6 0; xT0 1

< 0;

6 0;

ð27Þ

i ¼ 1; . . . ; M;

ð28Þ ð29Þ

then the closed-loop system (5) is robust Dða; rÞ-stable, and the output performance index constraints (26) are satisfied. Proof. Similar to the proof of Theorem 1, denote X ¼



PN

j¼1

 rP  e  aIÞ rP < 0: Pð A

xj X j , P ¼ X 1 , then from (27), we have ð30Þ

So the closed-loop system (5) is robust Dða; rÞ-stable for all admissible uncertainties. From (29), we know xT0 Px0 6 1. And by _ (30), the function VðtÞ ¼ xT ðtÞPxðtÞ is a Lyapunov function of the closed-loop system (5), that is, VðtÞ < 0 (for D ¼ d=dt), or DVðtÞ < 0 (for D ¼ q, or d). This means that VðtÞ is monotonously convergent to zero. Therefore, for all tð–0Þ 2 T , VðtÞ < Vð0Þ ¼ xT0 Px0 6 1. From (28), for any i 2 f1; . . . ; Mg,

"

X  G  GT e iG C



#

ci I

"

X j  G  GT ¼ xj e iG C j¼1 N X

 ci I

# 6 0:

Analogize to the analysis in Theorem 1, the following inequality holds



P ei C

 ci I



6 0:

By the Schur complement, it follows that

eT C e i 6 c P: C i i Hence,

eT C e i xðtÞg 6 c maxfxT ðtÞPxðtÞg 6 c : J i ¼ J ð1;iÞ ¼ maxfyTi ðtÞyi ðtÞg ¼ maxfxT ðtÞ C i i i tP0

The proof is complete. h

tP0

tP0

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3.1.3. Multiple output constraints: J ð2;kÞ 6 ck (k ¼ 1; . . . ; s), J ð1;lÞ 6 cl (l ¼ s þ 1; . . . ; M) Suppose that the multiple output constraints of the the system (5) are

J k ¼ J ð2;kÞ 6 ck ;

k ¼ 1; . . . ; s;

J l ¼ J ð1;lÞ 6 cl ;

l ¼ s þ 1; . . . ; M;

ð31Þ

where s is a given positive integer and 1 6 s < M, ci are given positive numbers, i ¼ 1; . . . ; M. Now following from the proofs of the above two theorems, we can obtain a sufficient condition of robust Dða; rÞ-stability for the system (5) subject to the multiple output constraints (31). Theorem 3. Given Dða; rÞ # S, if there exists symmetric matrices X j > 0; j ¼ 1; . . . ; N, and matrix G, such that for each j 2 f1; 2; . . . ; Ng, the following inequalities hold,

2 6 6 6 6 4 2 4 "

rX j  rG  rGT



e j G  aG A

rX j

e kG C

0 3

X j  G  GT



e lG C

cl I

X j

x0

xT0

1

3



7 7  7 7 < 0; 5

k ¼ 1; . . . ; s;

ð32Þ

ck I

5 6 0;

l ¼ s þ 1; . . . ; M;

ð33Þ

# 6 0;

ð34Þ

then the closed-loop system (5) is robust Dða; rÞ-stable for all admissible uncertainties, and satisfies the output performance index constraints (31). Proof. According to Theorem 1, from (32) and (34) it follows that the system (5) is robust Dða; rÞ-stable and J k ¼ J ð2;kÞ 6 ck , k ¼ 1; . . . ; s. On the other hand, the matrix inequality (32) implies (27). So associating with (33) and (34), it obtains from Theorem 2 that J l ¼ J ð1;lÞ 6 cl , l ¼ s þ 1; . . . ; M. h 3.2. Generalized guaranteed cost control with Dða; rÞ-stability and multiple output constraints Taking advantage of the above three theorems, we can cope with Problem 2. For convenience of discussion, we consider the generalized cost function (9) specifically,

J ¼

m1 m X X ki J ð2;iÞ þ ki J ð1;iÞ ; i¼1

ð35Þ

i¼m1 þ1

where 0 6 m1 6 m, and the output constraints (8) are

J ð2;kÞ 6 ck ;

k ¼ m þ 1; . . . ; m2 ;

J ð1;lÞ 6 cl ;

l ¼ m2 þ 1; . . . ; M;

ð36Þ

where m 6 m2 6 M. Some special cases of the three given integers m, m1 and m2 should be noted. When m ¼ M, the constraints (8) are not present, that is, Problem 2 becomes the corresponding control problem without output constraint. The two extreme cases, m1 ¼ 0 and m1 ¼ m, mean that only the second term and only the first term appear in the right side of (35), respectively. Besides, m2 ¼ m means that the first line in (36) does not exist, and m2 ¼ M means that the second line does not exist. Before deal with the problem of optimal generalized guaranteed cost control, we firstly discuss the problem of c-suboptimal generalized guaranteed cost control, which is to find a state feedback controller for a given c > 0, such that the closed-loop system is robust Dða; rÞ-stable, and the value of the generalized cost function (35) satisfies J 6 c, while the output constraints (36) are met. For the general case of m, m1 and m2 , i.e., 1 6 m1 < m < m2 < M, we give the following result by using Theorem 3. For the special cases of m, m1 and m2 , one can get the similar results by the same way. Theorem 4. For the unified linear system (1) with the generalized cost function (35) and the output constraints (36), given Dða; rÞ # S and c > 0, if there exists symmetric matrices X j > 0, j ¼ 1; . . . ; N, matrix G and Y, scalar ci > 0, i ¼ 1; . . . ; m, such that for each j 2 f1; 2; . . . ; Ng, the following inequalities hold,

M. Xiao et al. / Applied Mathematics and Computation 218 (2012) 12013–12027

k1 c1 þ k2 c2 þ    þ km cm 6 c; 2 3   rX j  rG  rGT 6 7 4 Aj G þ Bj Y  aG rX j  5 < 0; " 

C k G þ Dk Y X j  G  GT



C l G þ Dl Y  X j x0

cl I

xT0

1

12021

ð37Þ k ¼ 1; . . . ; m1 ; m þ 1; . . . ; m2 ;

ð38Þ

ck I

0 #

6 0;

l ¼ m1 þ 1; . . . ; m; m2 þ 1; . . . ; M;

6 0;

ð39Þ ð40Þ

then with the state feedback control law K ¼ YG1 , for all admissible uncertainties, the closed-loop system (5) is robust Dða; rÞ-stable, the output constraints (36) are met, and the value of the generalized cost function (35) satisfies J 6 c. Proof. If the condition of the theorem holds, then from (38) or (39), we know that G þ GT > 0, so G is invertible. Now let K ¼ YG1 , i.e., Y ¼ KG, and substitute it into (38) and (39), it follows from Theorem 3 that the closed-system (5) is robust Dða; rÞ-stable, and the output indices satisfy

J ð2;kÞ 6 ck ;

k ¼ 1; . . . ; m1 ; m þ 1; . . . ; m2 ;

J ð1;lÞ 6 cl ;

l ¼ m1 þ 1; . . . ; m; m2 þ 1; . . . ; M:

Thus, the output constraints (36) are met. And from (37), the value of the generalized cost function

J ¼

m1 m X X ki J ð2;iÞ þ ki J ð1;iÞ 6 k1 c1 þ k2 c2 þ    þ km cm 6 c: i¼1

ð41Þ

i¼m1 þ1

Hence, the proof is completed. h Now, we are in a position to present the solution for Problem 2. From (41) in Theorem 4, we know that the minimization of k1 c1 þ k2 c2 þ    þ km cm implies the minimization of the upper bound of the generalized cost function (35), so the following result is apparent. Theorem 5. For the unified linear system (1) with the generalized cost function (35) and the output constraints (36), given Dða; rÞ # S, if the following optimization problem

min J ¼ k1 c1 þ k2 c2 þ    þ km cm 2 3   rX j  rG  rGT 6 7 s:t: ðiÞ4 Aj G þ Bj Y  aG rX j  5 < 0;

ð42Þ

X 1 ;...;X N ;G;Y;c1 ;...;cm

" ðiiÞ ðiiiÞ

C k G þ Dk Y X j  G  GT



C l G þ Dl Y   X j x0

cl I

xT0

1

k ¼ 1; . . . ; m1 ; m þ 1; . . . ; m2 ; j ¼ 1; 2; . . . ; N;

ck I

0 #

6 0;

l ¼ m1 þ 1; . . . ; m; m2 þ 1; . . . ; M; j ¼ 1; 2; . . . ; N;

6 0; j ¼ 1; 2; . . . ; N;

b Y b 1 is the optimal robust genb; c bG ^1 ; . . . ; c ^m ; bJÞ, then the state feedback controller (4) with K ¼ Y has a solution ðX^1 ; . . . ; X^N ; G; eralized guaranteed cost controller for the unified linear system (1) with Dða; rÞ-stability and multiple output constraints (36), and the optimal upper bound of the generalized cost function (35) is bJ. The above theorem presents a design method of the optimal robust generalized guaranteed cost controller for the unified linear system (1) with Dða; rÞ-stability and multiple output performance index constraints. It is clearly that the optimization problem (42) is a convex optimization problem with linear matrix inequality constraints, it can be effectively solved by existing software such as the LMI Toolbox in Matlab [37]. The convexity of the optimization problem ensures that a global optimum, when it exists, is reachable.

Remark 3. It is need to point out that, when m1 ¼ 0 and m2 ¼ m, according to Theorem 2, the inequality (38) in Theorem 4 and the constrained condition (i) in the optimization problem (42) must be replaced by

"

rX j  rG  rGT



Aj G þ Bj Y  aG rX j

# < 0:

The proposed results can be used to cope with the problem of optimal robust guaranteed cost Dða; rÞ-stabilization without performance constraints. For example, consider the problem of the ordinary robust optimal guaranteed cost Dða; rÞ-stabiliza-

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M. Xiao et al. / Applied Mathematics and Computation 218 (2012) 12013–12027

tion for the continuous time system (11) with the cost function (12) (see Remark 1). The parameter uncertainties of the system (11) are also (2). Making use of Theorem 5, we have Corollary 1. For the linear continuous time polytopic uncertain system (11), given Dða; rÞ # S, if the following optimization problem

c

min

ð43Þ

X 1 ;...;X N ;G;Y;c

2

 rX j  rG  rGT 6 6 Aj G þ Bj Y  aG rX j s:t: ðiÞ6 6 G 0 4  ðiiÞ

Y  x0

X j xT0

1

 

 

cR1



0

cQ 1

0 6 0;

3 7 7 7 < 0; 7 5

j ¼ 1; 2; . . . ; N;

j ¼ 1; 2; . . . ; N;

b Y b 1 is an optimal robust guaranteed cost b; c bG ^Þ, then the state feedback controller (4) with K ¼ Y has a solution ðX^1 ; . . . ; X^N ; G; ^. Furthermore, these reDða; rÞ-stabilization controller of the system (11), and the minimal upper bound of cost function (12) is c sults can be exploited to deal with the corresponding control problem with saturation of actuators. In actual application, the magnitudes of control inputs are often restricted into a given limit [25–27]. Now we consider the optimal robust generalized guaranteed cost control for the system (1) with control input constraints. The generalized cost function is also (35), and the output constraints are also (36). But taking the saturation of actuators and energy limit into account, the control input uðtÞ is restricted by

jui ðtÞj 6 ai ;

i ¼ 1; . . . ; p;

ð44Þ

where ui ðtÞ is the ith component of the input vector uðtÞ, and ai > 0 is a known constant, i ¼ 1; . . . ; p. We can regard ui ðtÞ (i ¼ 1; . . . ; p) as a group of new outputs of the system (1)

zi ðtÞ ¼ 0xðtÞ þ E i uðtÞ;

i ¼ 1; . . . ; p;

ð45Þ

where E i is the ith row of the p  p identity matrix. Then, the input constraints (44) can be converted into output constraints

maxfzTi ðtÞzi ðtÞg 6 a2i ; tP0

i ¼ 1; . . . ; p:

ð46Þ

Thus, we have the following result. Corollary 2. For the unified linear system (1), given Dða; rÞ # S, ai > 0, i ¼ 1; . . . ; p, if the following optimization problem

J ¼ k1 c1 þ k2 c2 þ    þ km cm 3   rX j  rG  rGT 6 7 s:t: ðiÞ4 Aj G þ Bj Y  aG rX j  5 < 0; min 2

ð47Þ

X 1 ;...;X N ;G;Y;c1 ;...;cm

" ðiiÞ

C k G þ Dk Y X j  G  GT



C l G þ Dl Y

cl I

X j  G  GT



EiY

a2i

" ðiiiÞ

0 #

 X j ðivÞ xT0

x0 1



6 0;

k ¼ 1; . . . ; m1 ; m þ 1; . . . ; m2 ; j ¼ 1; 2; . . . ; N;

ck I

6 0;

l ¼ m1 þ 1; . . . ; m; m2 þ 1; . . . ; M; j ¼ 1; 2; . . . ; N;

6 0;

i ¼ 1; . . . ; p; j ¼ 1; 2; . . . ; N;

#

j ¼ 1; 2; . . . ; N;

b Y b 1 is the optimal robust generalb; c bG ^1 ; . . . ; c ^m ; bJÞ, then the state feedback controller (4) with K ¼ Y has a solution ðX^1 ; . . . ; X^N ; G; ized guaranteed cost controller which meets the magnitudes constraints (44), and guarantees the closed-loop system (5) to be robust Dða; rÞ-stable and the output constraints (36) to be satisfied. The optimal upper bound of the generalized cost (35) is bJ. Remark 4. From the above theorems and corollaries, we know that, the corresponding controller existing or not, depends on the initial state x0 . To remove this dependence, a deterministic approach can be adopted [2]. It is assumed that the initial state of the system (1) is arbitrary but belongs to the set P ¼ fb 2 Rn : b ¼ U g; gT g 6 1g, where U is a given matrix. In this situation, we need replace the matrix inequality



X j

x0

xT0

1



60

ð48Þ

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M. Xiao et al. / Applied Mathematics and Computation 218 (2012) 12013–12027

in the above theorems and corollaries by

"

X j

U

UT

I

# 6 0:

ð49Þ

In fact, the matrix inequality (49) is equivalent to U T X 1 it j U 6 I, T T 1 T xT0 X 1 x ¼ g U X U g 6 g g 6 1, which implies that the inequality (48) holds. 0 j j

shows

that

for

any

x0 ¼ U g 2 P,

4. Numerical examples In the section, we present two examples to illustrate the effectiveness of our results. Example 1. A shock isolation investigated in [28] is described by

_ xðtÞ ¼ AxðtÞ þ BuðtÞ; xð0Þ ¼ ½0; 1T ; yi ðtÞ ¼ C i xðtÞ þ Di uðtÞ; i ¼ 1; 2;

ð50Þ

where x ¼ ½x1 ; x2 T , x1 represents for the displacement, x2 ¼ x_ 1 , and

 A¼

0 1 0 0

 ;

B¼

  0 ; 1

C 1 ¼ ½ 1 0 ;

D1 ¼ 0;

C 2 ¼ ½ 0 0 ;

D2 ¼ 1:

In fact, y1 ðtÞ ¼ x1 ðtÞ, y2 ðtÞ ¼ uðtÞ. Our objective is to synthesize a state feedback controller u ¼ Kx, where K ¼ ½c k, such that J 1 ¼ maxtP0 jy1 ðtÞj is as small as possible subject to the constraint J 2 ¼ maxtP0 jy2 ðtÞj 6 1 and Dð2; 1:6Þ-stability of the closed-loop system, while the same problem without Dð2; 1:6Þ-stability constraint is solved in [28]. Actually, the parameters k and c stand for the damping and stiffness coefficients of the shock isolator, respectively. By applying Theorem 5 and solving the corresponding optimization problem via the LMI sovler MINCX of Matlab, we obtain an

uðtÞ ¼ ½ 0:5483 0:9949 xðtÞ; and the minimal upper bound of the generalized cost J 1 is bJ ¼ 1:82. With the controller, the closed-loop system is

_ xðtÞ ¼



0



1

0:5483 0:9949

xðtÞ;

xð0Þ ¼

  0 1

;

ð51Þ

and the performance indices are

maxjy1 ðtÞj ¼ maxjx1 ðtÞj ¼ 0:6337; tP0

tP0

maxjy2 ðtÞj ¼ 0:9949 < 1; tP0

and the closed-loop poles are 0:4974  0:5485i in the disk Dð2; 1:6Þ. In [28], without Dð2; 1:6Þ-stability constraint, the optimal parameters of the shock isolator is c ¼ 1:333, k ¼ 0:817, i.e., the closed-loop system is

_ xðtÞ ¼



0

1

1:333 0:817

 xðtÞ;

xð0Þ ¼

  0 1

;

ð52Þ

the minimal upper bound of J 1 is 0:650, the poles are 0:4085  1:0799i outside the disk Dð2; 1:6Þ, and the performance indices are

maxjy1 ðtÞj ¼ maxjx1 ðtÞj ¼ 0:5490; tP0

tP0

maxjy2 ðtÞj ¼ 0:9609 < 1: tP0

Fig. 1 shows that the state trajectories of the two closed-loop systems with initial state xð0Þ. From Fig. 1, we can find that, comparing with the system (52), the oscillation of the trajectory of the system (51) is alleviated and the negative amplitude is reduced, which means that the transient performance of the system (51) is improved, although the maximal value of x1 , i.e. the displacement magnifies a little. If consider a ¼ 1000, r ¼ 1000, then we obtain the following optimal generalized guaranteed cost controller via solving the optimization problem in Theorem 5,

uðtÞ ¼ ½ 1:3308 0:8160 xðtÞ; and the minimal upper bound of J 1 is 0:6510. In this case, c ¼ 1:3308, k ¼ 0:8160. We find that the result is very closed to that one in [28] (see the system (52)). So the problem of optimal generalized guaranteed cost control with D-stability constraint includes the corresponding control problem without regional pole constraint as special cases. To tackle the control problem without regional pole constraint by our results, we should let r to be large enough and jaj ¼ r for continuous time systems, a ¼ 0, r ¼ 1 for discrete time with the shift operator, and a ¼ 1=h, r ¼ 1=h for discrete time with the delta operator.

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M. Xiao et al. / Applied Mathematics and Computation 218 (2012) 12013–12027

0.8

system (51) system (52)

0.6

x1(t)

0.4 0.2 0 −0.2

0

2

4

6

8

10

12

14

16

18

20

t 1

system (51) system (52)

0.75

x2(t)

0.5 0.25 0 −0.25 −0.5

0

2

4

6

8

10

12

14

16

18

20

t Fig. 1. The trajectories of the closed-loop system (51) and (52).

Example 2. Consider a delta operator uncertain linear discrete systems (1) with sampling period h ¼ 0:1 s and parameters as follows

2

0:3761 þ e1

6 A ¼ 4 2:1858 þ e2 0:2046

0:1550

0:0039

1:1030 þ e2 0:0550

C 1 ¼ ½ 1:10 0:44 1:30 ; C 2 ¼ ½ 0:30 0:25 0:47 ;

3

2

7 0:5348 5;

1:2886

3

6 7 B ¼ 4 2:0083 þ e2 5;

0:3059

0:9315

D1 ¼ 0:45; D2 ¼ 0:33;

C 3 ¼ ½ 0:51 0:72 0:22 ; C 4 ¼ ½ 0:20 0:10 0:80 ;

D3 ¼ 0:72; D4 ¼ 0:89;

x0 ¼ ½ 0:44 0:13 0:21 T ; where e1 and e2 are uncertain parameters satisfying 0:1 6 e1 6 0:1, 0:2 6 e2 6 0:2. This set of uncertainties defines a polytope with four vertices, obtained by combination of the extremum values of uncertain parameters. The stable region of the delta operator system is S ¼ Dð10; 10Þ. The performance indices are

J1 ¼

1 X jy1 ðtÞj2 ; t¼0

J 2 ¼ maxjy2 ðtÞj2 ; tP0

J3 ¼

1 X t¼0

jy3 ðtÞj2 ;

J 4 ¼ maxjy4 ðtÞj2 ; tP0

and the generalized cost function is

J ¼ J 1 þ J 2: The control problem is to design an optimal generalized guaranteed cost state feedback controller (4), such that the closedloop system is robust Dð5; 4Þ-stable, and the upper bound of the value of the generalized cost is as small as possible subject to the performance index constraints

J 3 6 1; J 4 6 1: Using Theorem 5 and solving the optimization problem (42), we obtain the optimal solution as follows

2

0:2864 0:0796 0:1411

3

7 b ¼6 G 4 0:1119 0:1118 0:0762 5; 0:1478 0:0620 0:0785 bJ ¼ 0:4797;

b ¼ ½ 0:5678 0:3365 0:3205 ; Y

M. Xiao et al. / Applied Mathematics and Computation 218 (2012) 12013–12027

12025

so the optimal generalized guaranteed cost controller is

b 1 xðtÞ ¼ ½ 16:8132 12:0242 37:8411 xðtÞ; bG uðtÞ ¼ Y

ð53Þ

and the least upper bound of the generalized closed-loop cost function J is 0:4797. Now we take some specific values of the admissible uncertainties to verify the result, for example, consider e2 ¼ 0:1. We calculate and find that the output performance of the corresponding closed-loop system are

J 1 ¼ 0:2081;

J 2 ¼ 0:0053;

J 3 ¼ 0:7601;

e1 ¼ 0:05,

J 4 ¼ 0:3322;

indeed, J ¼ J 1 þ J 2 ¼ 0:2134 is less than the upper bound 0:4797, and the output constraints J 3 6 1, J 4 6 1 are met. The poles of the closed-loop system are 1:1575, 2:6608  2:8264i inside the disk Dð5; 4Þ. Fig. 2 shows the state trajectories of the closed-loop system. Fig. 3 shows that the closed-loop poles with all admissible uncertainties distribute in the desired disk region Dð5; 4Þ. The four output signals of the closed-loop system are shown in Fig. 4. At each subplot of Fig. 4 is shown four curves, one for each vertex of the polytope describing the uncertain closed-loop system. From (53), we calculate and find that maxtP0 juðtÞj ¼ 1:0143. If we want the input juðtÞj < 1 for all t, we can use Corollary 2 to cope with the additional input constraint. Solving the optimization problem (47) for the system, we get the optimal solution, and obtain the state feedback controller as

x1(t)

0.4

x2(t) x (t)

0.3

3

0.2 0.1 0 −0.1 0

0.5

1

1.5

2

2.5

3

3.5

4

t Fig. 2. The state trajectories of the closed-loop system (e1 ¼ 0:05,

e2 ¼ 0:1).

10 8

imaginary axis

6 4 D(−10,10) 2

D(−5,4)

0 −2 −4 −6 −8 −10 −20

−15

−10

−5

0

real axis Fig. 3. Pole distribution of the closed-loop systems with all admissible uncertainties.

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M. Xiao et al. / Applied Mathematics and Computation 218 (2012) 12013–12027

0.4

0.2

0.3

0.1

1

2

y (t)

y (t)

0.2 0.1

−0.1

0 0

1

2 t

3

−0.2

4

0.2

0

0

−0.2

−0.2

y (t)

0.2

0

1

2 t

3

4

0

1

2 t

3

4

4

3

y (t)

−0.1

0

−0.4 −0.6 −0.8

−0.4 −0.6

0

1

2 t

3

4

−0.8

Fig. 4. The output responses of the uncertain closed-loop systems with the initial state x0 .

uðtÞ ¼ ½ 13:8304 10:1906 30:9023 xðtÞ:

ð54Þ

The minimal upper bound of the generalized cost J obtained by (47) is bJ ¼ 1:5247. Now by calculating, maxtP0 juðtÞj ¼ 0:9207 meets the input constraint juðtÞj < 1. 5. Conclusion In this paper, we have discussed the problem of robust generalized guaranteed cost control under D-stability and multiple output constraints for a class of linear polytopic uncertain systems. A new output performance index called generalized cost function is adopted. The sufficient conditions for Dða; rÞ-stability with multiple output constraints are given via LMI approach, and the design methods of c-suboptimal and optimal generalized guaranteed cost controller are proposed respectively, subject to Dða; rÞ-stability and multiple output constraints. The results can be exploited to cope with a class of control input constraints. The approach is expressed in a unified framework involved the differential, shift, and delta domains. Two numerical examples showed the feasibility and effectiveness of the proposed approach. Acknowledgements This work is supported by the National Basic Research Program of China (973 Program: 2007CB714006), the National Natural Science Foundation of China (NSFC: 61134007), and the Project of Fujian’s Universities Serving Construction for ’Haixi’: Technology of Informationization Based on Mathematics. References [1] O.R. Moheimani, I.R. Petersen, Quadratic guaranteed cost control with robust pole placement in a disk, IEE Proceedings - Control Theory and Applications 143 (1996) 37–43.. [2] I.R. Petersen, D.C. McFarlane, M.A. Rotea, Optimal guaranteed cost control of discrete-time uncertain linear systems, International Journal of Robust and Nonlinear Control 8 (1998) 649–657. [3] P. Shi, R. Agarwal, E.K. Boukas, Y. Shi, Guaranteed cost control of uncertain discrete time-delay systems, Journal of Computational and Applied Mathematics 157 (2) (2003) 435–451. [4] E. Fridman, U. Shaked, Stability and guaranteed cost control of uncertain discrete delay systems, International Journal of Control 78 (2005) 235–246. [5] O.I. Kosmidou, Robust control with pole shifting via performance index modification, Applied Mathematics and Computation 182 (2006) 596–606. [6] J.H. Park, Guaranteed cost control for uncertain large scale systems with time-delays via delayed feedback, Chaos, Solitons & Fractals 27 (2006) 800– 812. [7] D. Zhang, H. Su, J. Chu, Z. Wang, Satisfactory reliable H1 guaranteed cost control with D-stability and control input constraints, IET Control Theory and Applications 2 (2008) 643–653. [8] J. Wang, J. Wang, W. Yuan, P. Shi, Gain-scheduled guaranteed cost control of LPV systems with time-varying state and input delays, International Journal of Innovative Computing, Information and Control 5 (10) (2009) 3377–3390. [9] S. Xu, J. Lam, P. Shi, E.K. Boukas, Y. Zou, Guaranteed cost control for uncertain neutral stochastic systems via dynamic output feedback controllers, Journal of Optimization Theory and Applications 143 (1) (2009) 207–223. [10] J. Zhang, P. Shi, J. Qiu, Non-fragile guaranteed cost control for uncertain stochastic nonlinear time-delay systems, Journal of the Franklin Institute 346 (2009) 676–690. [11] J. Wu, S.K. Nguang, J. Shen, G.J. Liu, Y.G. Li, Fuzzy guaranteed cost tracking control for boiler-turbines via TS fuzzy model, International Journal of Innovative Computing, Information and Control 6 (12) (2010) 5575–5586.

M. Xiao et al. / Applied Mathematics and Computation 218 (2012) 12013–12027

12027

[12] H.M. Soliman, A. Dabroum, M.S. Mahmoud, M. Soliman, Guaranteed-cost reliable control with regional pole placement of a power system, Journal of the Franklin Institute 348 (2011) 884–898. [13] Y. Wang, Z. Zuo, New results on guaranteed cost control of uncertain discrete systems with both state and input delays, International Journal of Innovative Computing, Information and Control 7 (4) (2011) 1713–1724. [14] K. Furuta, S.B. Kim, Pole assignment in a specified disk, IEEE Transactions on Automatic Control 32 (5) (1987) 423–427. [15] G. Garcia, J. Bemussou, Pole assignment for uncertain systems in a specified disk by state feedback, IEEE Transactions on Automatic Control 40 (1995) 184–190. [16] M. Chilali, P. Gahinet, P. Apkarian, Robust pole placement in LMI regions, IEEE Transactions on Automatic Control 44 (11) (1999) 2257–2270. [17] S. Xu, J. Lam, L. Zheng, Robust D-stability analysis for uncertain discrete singular systems with state delay, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 49 (4) (2002) 551–555. [18] V.J.S. Leite, P.L.D. Peres, An improved LMI condition for robust D-stability of uncertain polytopic systems, IEEE Transactions on Automatic Control 48 (3) (2003) 500–504. [19] S.H. Chen, J.H. Chou, Robust eigenvalue-clustering in a specified circular region for linear uncertain discrete singular systems with state delay, Applied Mathematics and Computation 186 (2007) 1660–1670. [20] W. Zhou, H. Su, Theory and Applications of Robust Control with D-stability Constraints, Science Press, Beijing, 2009. [21] W.J. Mao, An LMI approach to D-stability and D-stabilization of linear discrete singular systems with state delayq, Applied Mathematics and Computation 218 (2011) 1694–1704. [22] S.K. Yang, Robust pole-clustering of structured uncertain systems in union of disjointed circular regions, Applied Mathematics and Computation 217 (2011) 7488–7495. [23] E.G. Gilbert, K.T. Tan, Linear systems with output constraints: The theory and application of maximal output admissible sets, in: Proceedings of American Control Conference, 1991, pp. 351–359. [24] A. Megretski, A. Rantzer, System analysis via integral quadratic constraints, IEEE Transactions on Automatic Control 42 (1997) 819–830. [25] T. Hu, Z. Lin, Control Systems With Actuator Saturation: Analisys and Design, Boston, MA, Birkhauser, 2001. [26] B.M. Chen, T.H. Lee, K. Peng, V. Venkataramanan, Composite nonlinear feedback control for linear systems with input saturation: theory and an application, IEEE Transactions on Automatic Control 48 (2003) 427–439. [27] J.M.G. da Silva, D. Limon, T. Alamo, E.F. Camacho, Dynamic output feedback for discrete-time systems under amplitude and rate actuator constraints, IEEE Transactions on Automatic Control 53 (2008) 2367–2372. [28] D.V. Balandin, M.M. Kogan, LMI based multi-objective control under multiple integral and output constraints, International Journal of Control 83 (2010) 227–232. [29] B. Zhou, Z. Lin, G.R. Duan, L1 and L2 low-gain feedback: their properties, characterizations and applications in constrained control, IEEE Transactions on Automatic Control 56 (2011) 1030–1045. [30] R.H. Middleton, G.C. Goodwin, Improved finite word length characteristics in digital control using delta operators, IEEE Transactions on Automatic Control 31 (1986) 1015–1021. [31] G.C. Goodwin, R.H. Middleton, H.V. Poor, High-speed digital signal processing and control, Proceedings of the IEEE 80 (1992) 240–259. [32] C. Neuman, Properties of the delta operator model of dynamical physical systems IEEE Transactions on Systems, Man and Cybernetics 23 (1993) 296– 301. [33] H. Yang, Y. Xia, P. Shi, M. Fu, A novel delta operator Kalman filter design and convergence analysis, IEEE Trans on Circuits and Systems I: Regular Papers 58 (10) (2011) 2458–2468. [34] G. Maione, High-speed digital realizations of fractional operators in the delta domain, IEEE Transactions On Automatic Control 56 (2011) 607–702. [35] H. Yang, P. Shi, J. Zhang, J. Qiu, Robust H1 control for a class of discrete time fuzzy systems via delta operator approach, Information Sciences 184 (1) (2012) 230–245. [36] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities in system and control theory, SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, 1994. [37] P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Control Toolbox, Mathworks Inc., Cambridge, 1995.