Delay dependent dissipative analysis and control for time delay descriptor systems

Scientific Journal of Control Engineering October 2014, Volume 4, Issue 5, PP.114-121

Delay-dependent Dissipative Analysis and Control for Time-delay Descriptor Systems Chunyan Ding Yantai Engineering and Technology College, Yantai Shandong 264006, China Email: dingchunyan@gmail.com

Abstract This paper studies the delay-dependent dissipative analysis and control for descriptor systems with time-delay. A sufficient condition for the system to be admissible and strictly dissipative is presented in terms of linear matrix inequalities (LMIs). Based on this, a state feedback dissipative controller is designed by introducing a slack variable. The illustrative example shows the effectiveness of the proposed method.

Keywords: Dissipative Control; Time-delay Descriptor Systems; LMI

1 INTRODUCTION Dissipative theory plays an important role in the stability research of control systems since it was brought forward in 1970s by Willems[1], the implication is that there exists a non-negative energy function (namely storage function) such that the energy consumption of a control system is always less than the supply rate of the energy. A new kind of design and analytic idea is put forward in the dissipative theory, i.e. from the point of view of energy, and energy is described in the form of input and output. The dissipativity theory is very useful for a wide range of fields such as circuit, network, system and control theory. On the other hand, delay is one of the main reasons to cause the system unstable [2]. Due to many practical systems with time delay is bounded, delay-dependent criteria have less conservativeness relative to the delay-independent criteria. Many significant results on delay-dependent dissipative control have been available in the literatures [3, 4].

In recent years, more attention has been devoted to studying descriptor systems due to the fact that the descriptor system better describes physical systems than the state-space systems. Many important theoretical results have been proposed for time-delay descriptor systems on H∞ control, stability, admissibility in terms of linear matrix inequality [5, 6, 7, 8]. As to the dissipative analysis and control for time-delay descriptor system, there also have been some deep results on recent studies [9, 10, 11, 12]. The problems of admissibility and dissipativity were studied for discrete-time singular systems with mixed time-varying delays in [10]. And a criterion that warrants the admissibility of the considered systems is established by taking advantage of the delay partitioning technique. The problem of αdissipativity analysis for continuous-time singular systems with time-delay were studied in [11], the results presented were not only dependent on the delay but also dependent on the dissipative margin α.

In this paper, delay-dependent dissipative analysis and control for time-delay descriptor systems is studied. By employing new Lyapunov function and linear matrix inequality technique, sufficient condition for the system to be admissible and dissipative is presented. At the same time, a state feedback controller is designed by introducing a slack variable. Finally, numerical example shows that the slack variable λ could reduce conservatism. Moreover, the results on H  performance analysis and passivity analysis of time delay descriptor system are unified in the presented results.

2 PROBLEM FORMULATION Consider the following descriptor systems with time delay: - 114 http://www.sj-ce.org

Ex(t )  Ax(t )  A1 x(t   )  Bu (t )  D (t ) z (t )  Cx(t )  B1u (t )  D1 (t ) y (t )  C1 x(t ) x(t )   (t )

(1)

t  [  0]

where x(t )  Rn  u(t )  Rm  (t )  Rm  z(t )  Rm are the state, control input, external disturbance and control output, respectively.   0 is the delay constant of the system,  (t ) is the continuous vector-valued initial function. rankE  n The matrices in the system are of appropriate dimensions.

The following definitions and lemmas for the systems Ex(t )  Ax(t ) could be used in the rest of the paper.

Definition 1 [13] (i). A pencil sE  A (or a pair ( E A)) is regular if det(sE  A) is not identically zero. (ii). For a regular pencil sE  A , the finite eigenvalues of sE  A are said to be the finite modes of ( E A) .

Suppose that Ev1  0 , then the infinite eigenvalues associated with the generalized principal vectors vk satisfying Evk  Avk 1 k  2 3 4 , are impulse modes of ( E A) .

Definition 2 The system Ex(t )  Ax(t )  Ad x(t   ) is admissible if it is regular, impulse free and stable.

Lemma 1[14] If the pair ( E A) is regular, impulse free, then the solution of the time-delay descriptor system Ex(t )  Ax(t )  Ad x(t   ) is exist and unique.

Lemma 2[15] If the pairs ( E A) and ( E A  Ad ) are regular and impulse free, then the system Ex(t )  Ax(t )  Ad x(t   ) is regular and impulse free.

Definition 3 Given the supply rate r ( (t ) z(t )) , system (1) with u(t )  0 is said to be dissipative if the following dissipative inequality V (t )  r ((t ) z(t ))t  0

(2)

holds. When the inequality holds strictly, the system is said to be strictly dissipative.

In this paper, we consider the quadratic supply rate with the following form: r ((t ) z(t ))  zT Qz  2zT S  T R

where Q and R are the given symmetric matrices, and S is the given matrix with suitable dimension. In general, we suppose Q  QT  0 .

Lemma 3 [16] For any appropriate dimension matrices Q H  M  with Q is symmetrical, then Q  HFM  M T F T H T  0

holds for any F with F T F  I , if and only if there exist a scalar   0 such that Q   HH T   1M T M  0

3 STRICT DISSIPATIVITY ANALYSIS At first, the problem of the delay-dependent strict dissipative analysis for system (1) with u(t )  0 is considered, and the following theorem could be obtained.

Theorem 1. If there exists the nonsingular matrix P , symmetric definite matrices W  N such that the following LMIs hold: ET P  PT E  0 - 115 http://www.sj-ce.org

(3)

1 2 3  AT N     0  AT N  4 1      5  DT N   0   0          N 

(4)

Then system (1) with u(t )  0 is admissible and strictly dissipative. where

1  AT P  PT A  C T QC  W  E T NE 2  PT A1  E T NE 3  PT D  C T QD1  C T S 4  W  E T NE 5   D1T QD1  D1T S  S T D1  R Proof: At first we prove system (1) with u(t )  0 (t )  0 is admissible. Since rankE  n , two nonsingular matrices G H could be found such that I GEH   r 0

0 0

Let  

A1

 

A3

GAH  

 P1 P 2   G T PH    A  P 3 P 4

A2    4 

According to (3), P 2  0 could be easily obtained. Left and right multiplying 1 by H T  H respectively, then A4T P 4  P T4 A4  0

Therefore A4 is nonsingular, matrix pair ( E A) is regular and impulse free. Left and right multiplying (4) by the matrix diag{I  I  0 0 0 } and its transpose, there holds PT ( A  Ad )  ( A  Ad )T P  0

Therefore ( E A  Ad ) is regular and impulse free combing (3). Then (1) with u(t )  0 (t )  0 is regular and impulse free by Lemma 2.2.

For simple, the following notation will be used in the paper: f1 (t )  Ax(t )  A1 x(t   ) f 2 (t )  Ax(t )  A1 x(t   )  D (t )

Choose the Lyapunov function as follows: V1 (t )  xT (t ) ET Px(t )  

t

t 

0

t



t 

xT ( )Wx( )d   

 

f1T ( ) Nf1 ( )d  d

(5)

where ET P  PT E  0W  W T  0 N  N T  0

The derivative of the Lyapunov function along system (1) with u(t )  0 (t )  0 is T T T T T V 1(t )  f1 (t ) Px(t )  x (t ) P f1 (t )  x (t )Wx(t )  x (t   )Wx(t   )

 2 f1T (t ) Nf1 (t )   

t

t 

- 116 http://www.sj-ce.org

f1T ( ) Nf1 ( )d 

(6)

By using of Jensen integral inequality [19] , for any positive-definite matrix N  Rnn , scalar   0 and vector function x(s)  [0 ]  Rnn  the following inequality holds T

 x(t )    ET NE E T NE   x(t )    [ Ex( )] N[ Ex( )]d      T  T t   x(t   )   E NE  E NE   x(t   )  t

T

Therefore it is easily obtained that T V 1(t )  1 (t )11 (t )

where

1T (t )   xT (t ) xT (t   )  

7    8     6

1  

(7)

where

6  AT P  PT A  W  E T NE   2 AT NA 7  PT A1  E T NE   2 AT NA1 8  W  E T NE   2 A1T NA1 According to (4), it is easily obtained that 1  0 , then the system (1) with u(t )  0 (t )  0 is admissible.

Next we will prove that system (1) with u(t )  0 is strictly dissipative． Based on the definition of dissipativity, if system (1) is strictly dissipative, it holds the following inequality V (t )  F (t )  0

(8)

where F (t )  2T (t )22 (t ) , 2T (t )   xT (t ) xT (t   ) T (t )  C T QC 0  C T QD1  C T S   2    0 0  T T T     D QD  D S  S D  R 1 1 1 1  

(9)

By introducing the Lyapunov function: V2 (t )  xT (t ) ET Px(t )  

t

t 

0

t



t 

xT ( )Wx( )d   

 

f 2T ( ) Nf 2 ( )d d

(10)

As to arbitrary nonzero  (t ) , the derivative of Lyapunov function V2 (t ) along system (1) with u(t )  0 is T T T T T V 2(t )  f 2 (t ) Px(t )  x (t ) P f 2 (t )  x (t )Wx(t )  x (t   )Wx(t   )

 2 f 2T (t ) Nf 2 (t )   

t

t 

f 2T ( ) Nf 2 ( )d 

(11)

By using of Jensen integral inequality, V 2(t )  2 (t )32 (t ) T

where 6 7 3    8   

PT D 2 AT ND   2 A1T ND   2 DT ND 

Then - 117 http://www.sj-ce.org

(12)

  C T QC 7 9    8  2 A1T ND  V 2(t )  F (t )     10    6      

(13)

where

9  PT D   2 AT ND  C T QD1  C T S 10   2 DT ND  D1T QD1  D1T S  S T D1  R According to (4) and Schur complement lemma, V 2(t )  F (t )  0 .

The proof is completed.

4 STATE FEEDBACK DISSIPATIVE CONTROL In this section, we will design the controller u(t )  Kx(t )

where K  Rmn is the controller gain matrix such that the closed-loop system Ex(t )  Ac x(t )  A1 x(t   )  D (t )

(14)

z (t )  Cc x(t )  D1 (t )

is admissible and delay-dependent dissipative.

Theorem 1. If there exists nonsingular matrix X , positive definite matrices W , N , matrix K and scalars such that EX  X T ET  0  11                   

A1 X T

12 13

, 2,

3

(15)

0

0

XE T

XAT  KBT

0

0

0

 XE T

X A1 T

0

0 0

0 0

0 0

0 0

1 XE T 2 1  XE T 2 0 0

W

0

0

 

5 

0  DT N I 0







N

0

0

0

N

0









 1I

0

0

0

0









0

1 I

0

0

0













 2I

0

0

1

1

 

 

 

 

 

 

 

















1 2

I

0



 3I





 0   0    0  0   0  0 N   0   0  0   0  1   3 I 

(16)

where

11  XAT  AX T  KBT  B K T  W 12  D  XC T QD1  XC T S  K B1 TQD1  K B1 T S 13  XC T M T  K B1 T M Then u(t )  Kx(t ) is the dissipative controller for system (1), and the controller gain is K  K T X T .

Proof: In order to design dissipative controller for the system, we suppose Q  M T M , then the inequality (4) is - 118 http://www.sj-ce.org

equal to the following inequality by Schur complement lemma, 14 2 3   0 4     5        

CT M T  AT N  0  A1T N  0  DT N   0  I 0    N 

(17)

where

14  AT P  PT A  W  ET NE By substituting Ac  A  BK  Cc  C  B1K for A C in (17), the following inequality holds: 15       

PT A1  ET NE 16 17  AT N  K T BT N   W  E T NE 0 0  A1T N  0  5 0  DT N    I 0      N

(18)

where

15  AT P  PT A  K T BT P  PT BK  W  E T NE 16  PT D  C T QD1  K T B1 TQD1  C T S  K T B1 T S 17  C T M T  K T B1 T M T According to Lemma 3, (18) holds if and only if there exists a scalar  T 15  E NE                 

1

such that

W

0

0

 A1T N



5

0

 DT N

1 T E N 2 1  ET N 2 0





I

0

0







N

0









 1I











PT A1 16 17  AT N  K T BT N

 ET   T  E   0 0 0   0  0   1  1 I 

(19)

Left and right multiplying (19) by the matrix diag{PT  PT  I  I  I  I} and its transpose, meanwhile let X  PT  K  XK T W  XWX T

(20)

Then the following inequality could be obtained:  11            

A1 X T

12

W

0

0

 XA1T N



5

0

 DT N

1 XE T N 2 1  XE T N 2 0





I

0

0







N

0









 1I











XCT M T  K B1 T M  XAT N   KBT N

- 119 http://www.sj-ce.org

 XE T    XE T    0 0 0   0  0   1  1 I 

(21)

According to the Lemma 3, (16) could be easily obtained. Then the controller gain K  K T X T could be presented from (20).

5

NUMERICAL EXAMPLES

To demonstrate the effectiveness and applicability of the proposed method, a simple example is presented in this section. Attention is focused on the controller synthesis for time-delay descriptor systems.

Consider time-delay descriptor system (1) with the following parameters: 1 0  10 4  4 6 128 97  5 1  01 26 E  A  A1    D B   B1         10  35  0 0   3 7   3 4  4 3 2  0 2 3  C  4 10

 4 2 D1     1 5 

 1 0   4 10  Q  S    0 24  2 20

 8 10 R   10 5 

Using cone complementary linearization to solve the convex optimization problem (15), (16), the maximum timedelay   62304 could be obtain, and  01435 00417  X    00417 00300 

9407959 5771829  W   5771829 3547095 

 00002 00006 N    00006 00020 

 3899812 29515 K    2393027 19263

1

 04315

2

 52443

3

 03462

Then the corresponding dissipative controller is: 02834 83738 u (t )  Kx(t )  10e  003    x(t )  00013 00661

6

(22)

CONCLUSION

Delay-dependent strictly dissipative control problem has been studied for a class of time-delay descriptor systems.

By using the linear matrix inequality and integral inequality techniques, sufficient condition for system to be admissible and strictly dissipative has been derived. And the sufficient condition is composed of a set of delay dependent linear matrix inequality. By introducing a slack invariable, the corresponding strictly dissipative state feedback controller is designed. The numerical example illustrates that different  could get different maximum delay upper bound.

ACKNOWLEDGMENT The author thanks the anonymous reviewers and the Editors for their insights and suggestions which helped enhance the quality of the paper.

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