Robust Stability of Discrete Large-Scale Interval Time-Delay Systems

Robust Stability of Discrete Large-Scale Interval Time-Delay Systems Chien-Hua Lee*1, Ping-Sung Liao2 Department of Electrical Engineering, Cheng-Shiu University, Kaohsiung 833, Taiwan, R.O.C. chienhua@csu.edu.tw; 2bsliao@csu.edu.tw

*1

Abstract This paper addresses the robust stability analysis for discrete large-scale interval systems subjected to time delays. We establish a very simple upper solution bound of the discrete algebraic Lyapunov equation (DALE). Then, using the Lyapunov equation approach associated with this upper bound and some linear algebraic techniques, several simple criteria are presented to guarantee the robust stability of the overall systems. The featuresof these present results arethat they are independent of any Lyapunov equation although the Lyapunov equation approach is adopted.

Keywords Interval Matrix; Robust Stability; Large-Scale System; Time-Delay; Uncertainty

Introduction The so-called interval matrices are caused by unavoidable parametric variaty, changes in operating conditions, aging, and so on. Some or all entries of the mentioned matrices are known only to the extent that each belongs to a special closed interval. When matrices of a state equation are interval matrix, the system is called an interval system. In the recent literature, considerable researches for the mentioned systems have been proposed. In practice, time delay(s) exist(s) in real-life systems and should be integrated into system model. However, surveying the literature, there are only few works have been devoted to the research of stability analysis and/or stabilization controller design for interval time-delay systems. Compare to single systems, large-scale systems have high dimensionality of system equations and hence the analysis and design problems of large-scale systems are more complicated. In literature, many contributions have been devoted to the research for these kinds of systems during the past decades [Guan et. Al., 2002, Lee and Hsien, 1997, Lin and Zhang, 2008, Oark, 2002, Stojanovic and Debeljkovic, 2005, Wang and Mau, 1997, Wu et. Al., 1993, Xinzhang and Yongqing, 1998, Xu, 1995]. For large-scale interval time-delay system, it is seen that a sufficient criterion has been developed by Xinzhang and Yongqing (1998) for the robust stabilization for discrete nonlinear large-scale interval systems with non-integral delays. The stability analysis for continuous and discrete large-scale systems has been discussed by Stojanovic and Debeljkovic (2005). In Stojanovic and Debeljkovic’s work (2005), two sufficient conditions for the mentioned systems have been derived by using the Gersgorin theorem. Surveying literature, it seems that Stojanovic and Debeljkovic’s work (2005) is the only one that treats the stability test problem for discrete large-scale interval time-delay systems.

However, the systems treated by Stojanovic and Debeljkovic (2005) contain only two interconnected subsystems. In general, a large-scale system should not only possess two subsystems. Therefore, to study more general cases, this paper addresses the same problem for discrete large-scale interval time-delay systems that contain more than two interconnected subsystems. By using the Lyapunov equation approach associated with a simple upper bound of the solution of the discrete Lyapunov equation, several delay-independent stability criteria are derived for the aforementioned systems. An interesting consequence is that these obtained criteria do not involve any Lyapunov equation. Finally, we give a numerical example to demonstrate the merits and verify the correctness of the presented schemes. It is believed that the present schemes are helpful for the controller design of large-scale interval time-delay systems.

Main Results Consider the discrete composite interval time-delay system S which is described as an interconnection of N subsystems S1 , S2 , , S N which are represented by 30

International Journal of Automation and Control Engineering, Vol. 4, No. 1—April 2015 2325-7407/15/01 030-5 © 2015 DEStech Publications, Inc. doi:10.12783/ijace.2015.0401.08

Robust Stability of Discrete Large-Scale Interval Time-Delay Systems

N

Si : xi (k= + 1) AiI xi (k ) + ∑ AijI x j (k − d= ij ), i 1, 2, , N j =1

31

(1)

where x(⋅) ∈ ℜn is the state vector, integer dij > 0 denotes the delay, AijI = AiI for i = j , and AiI and AijI are interval matrices with appropriate dimensions and have the properties: AiI= { Ai Ai= [aipq ], uipq ≤ aipq ≤ vipq , p, q= 1, 2, , n} ,

(2)

AijI= { Aij Aij= [aijpq ], eipjq ≤ aijpq ≤ fijpq , p, q= 1, 2, , n} .

(3)

Define matrices U i , Vi , Eij and Fij , respectively, as

= U i [u= [v= [eipq ], and Fij = [ fipq ] . ipq ], Vi ipq ], Eij

(4)

From the above properties and definitions, we derive the following simple condition for the robust stability of the system (1).

Theorem 1. If for i = 1, 2,  , N , N

N

( H i + ∑ Gij )( H i + ∑ G ji ) < 1 ,

(5)

=j 1 =j 1

then the composite system (1) is robustly stable where matrices H i = [hipq ] and Gij = [ gijpq ] are defined, respectively, as

{

hipq = max uipq , vipq

} and g

ijpq

{

}

= max eijpq , f .

(6)

Proof. In light of the fact A ≤ A for any matrix A, the condition (5) implies AiI < 1 . This means that the matrix AiI is stable. Thus, by the Lyapunov stability theorem, the following Lyapunov equation is satisfied.

(7)

AiIT Pi AiI − Pi = −Qi

where i = 1, 2,  , N , Qi is any given positive definite matrix, and Pi denotes the unique positive definite matrix.

Note that this Lyapunov equation is unsolvable. Here, we choose Qi as  N Gij N N ATjiI A jiI T  Qi α ∑ AiI AiI + ∑ ( H i + ∑ Gij ) =  j 1 Hi = j 1 =j 1 = G ji 

   

(8)

where α is an arbitrarily positive constant. Now, the Lyapunov equation (7) can be rewrite as T   N N A jiI A jiI AT A AiIT (α I − Pi ) AiI − (α I − Pi ) = Qi + α AiIT AiI − α I = α  ( H i + ∑ Gij )( iI iI + ∑ )−I.   Hi = j 1= j 1 G ji  

(9)

Since AiIT AiI ≤ AiI ATjiI A jiI ≤ A jiI

2

I ≤ AiI 2

2

I ≤ A jiI

I ≤ Hi 2

2

I ≤ G ji

I, 2

(10) I,

(11)

we have T N AiIT AiI N A jiI A jiI +∑ ≤ ( H i + ∑ G ji ) I .

Hi = j 1= j 1 G ji

(12)

Therefore, it is seen that if the condition (5) holds, then the right-hand side of (9) is a negative definite matrix and

32

Chien-Hua Lee, Ping-Sung Liao

the solution α I − Pi is positive definite. That is, the unique positive solution Pi has the upper bound Pi < α I .

(13)

For system (1), a Lyapunov function is constructed as follows.

N

N

i =1

i =1

N

dij

N

V ( xi (k )) = ∑ Vi ( xi (k )) = ∑ xiT (k ) Pi xi (k ) + ∑ [ H i + ∑ Gik ] ∑ xTj (k − n) =j 1

T AijI AijI

k 1= n 1 =

Gij

x j ( k − n)

(14)

We use V , Vi , xi , and x jd to replace V ( xi (k )), Vi ( xi (k )), xi (k ) and x j (k − dij ) , respectively, in the following and later descriptions. Taking the forward difference for the Lyapunov function (12) results in N

N

i =1

i =1

N

N

∆V = ∑= ∆Vi ∑ {xiT (k + 1) Pi xi (k + 1) − xiT Pi xi + ∑ [ H i + ∑ Gik ][ xTj

T AijI Pi AijI

k 1 =j 1 =

N

N

N

x j − xTjd

Gij

N

T AijI Pi AijI

Gij

N

N

x jd ]}

N

T T = ∑ {xiT ( AiIT Pi AiI − Pi ) xi + xiT AiIT Pi ∑ AijI x jd + ∑ xTjd AijI Pi AiI xi + ∑ xTjd AijI Pi ∑ AijI x jd + ∑ [ H i + ∑ Gik ] i 1 =j 1 =j 1

×[ xTj

Since AT B + BT A ≤ η AT A +

1

η

T AijI Pi AijI

Gij

=j 1 =j 1 T AijI Pi AijI

x j − xTjd

Gij

k 1 =j 1 =

x jd ]} .

(15)

BT B for any positive constant η [29], we have the following relations.

N

N

N

Gij

j =1

Hi

T xiT AiIT Pi ∑ AijI x jd + ∑ xTjd AijI Pi AiI xi ≤ ∑ j =1

j =1

N

Hi

j =1

Gij

xiT AiIT Pi AiI x jd + ∑

T xTjd AijI Pi AijI x jd

(16)

N

N

N

T T ∑ x jd AijI Pi ∑ AijI x jd =j 1 =j 1

N

≤∑

∑ Gik

k =1

j =1

(17)

T xTjd AijI Pi x jd

Gij

Substituting the above inequalities into (15) leads to N

N

∆V ≤ ∑ {xiT [−Qi + ∑

=i 1 =j 1

Gij Hi

N

N

AiIT Pi AiI + ∑ [ H i + ∑ Gik ] k 1 =j 1 =

ATjiI Pj A jiI G ji

]xi } .

(18)

It is seen that if the condition (5) is satisfied, then we can apply the upper bound (13) into (18) and obtain N

Gij

j =1

Hi

∆V < xiT [−Qi + α ( ∑

N

N

AiIT AiI + ∑ [ H i + ∑ Gik ] × k 1 =j 1 =

ATjiI Pj A jiI G ji

)]xi = 0

(19)

where (8) and the following relation is used.

N N

∑∑

=i 1 =j 1

xTj

T N N N AijI Pi AijI ATjiI Pj A jiI T ([ H i + ∑ Gik )= xj xi . ∑ xi ∑ ([ H i + ∑ Gik ) k 1 k 1 = =i 1 =j 1 = Gij G ji N

(20)

Therefore, the system (1) is robustly stable and the proof is completed.

Remark 1. Notice that the Lyapunov equation (7) is unsolvable. However, it is seen that we do not need to solve this unsolvable equation. By utilizing the upper solution bound (13) associated with the Lyapunov equation approach, a simple condition that assure the robust stability of system (1) can be obtained Remark 2. In fact, for the application of the Lyapunov equation, the positive matrix Q in most existing researches is considered as a free matrix or is selected as cI for simplification. According to the presented result, Theorem 1, it is shown that by choosing properly the matrix Q, we can develop a simple upper bound of the solution matrix P. Utilizing this upper bound, a robust stability condition that do not involve any Lyapunov equation then can be

Robust Stability of Discrete Large-Scale Interval Time-Delay Systems

33

established.

A Numerical Example Demonstration is given as below to show the applicability of the obtained results.

Example 1. Consider the following large-scale interval time-delay system:  −0.2 x1 (k + 1) =  [ 0.3 0.34] [ 0.25 0.35] x2 (k + 1) =  0.15   0.3 x3 (k + 1) =  [ 0 0.1]

[0.25 [0.25

[ 0.06 0.1] 0.3]   x1 (k ) +  0.35]  −0.1

 [ 0.06 0.1] 0.2  x2 (k ) +  0.4 0.46 [ ]  0.02

[0.1 0.14] x (k ) [0.14 0.3] 3

[ −0.08 0.04] x (k − d ) + [ −0.1 0] 0.05  x (k − d )  12 13 0 [ −0.04 0]  2 [ −0.1 0] 3  [0.06 [ −0.1

0.1 0.1]   x (k − d 21 ) +  − 0.01 0.03] 0]  1 [

[ 0.04 0.08]  [ −0.1 0] 0  +  x1 (k − d31 ) +  − 0.1 0 − 0.1 ]  [  [ −0.1 −0.04]

0.05  x (k − d 23 ) 0 [ 0.1] 3

 0.04 x (k − d32 ) .

0.08 0.12 [ ] 2

For this case and from (6), we have H1

 0.2 0.3  0.1 0.08  0.1 0.05 , , G12  , G13 =  =   0.1  0 0.34 0.35  0.1 0.04 

0.35 0.2   0.1 0.1  0.1 0.05 H2 =  = , G121 = , G23    , 0.15 0.46  0.02 0.1 0.03 0.1 

0.3 0.14  0.08 0  0.1 0.04  = H 3 =  , G31 =  , G32 0.1 0.12  .

0.1 0.3 0.1 0.1      

In light of Theorem 1, we have 3

3

( H1 + ∑ G1 j )( H1 + ∑ G j1 ) =0.8316 < 1 ,

=j 1 =j 1

3

3

3

( H 2 + ∑ G2 j )( H1 + ∑ G j 2 ) = 0.8431 < 1 ,

=j 1 =j 1 3

( H 3 + ∑ G3 j )( H1 + ∑ G j 3 ) =0.5234 < 1 .

=j 1 =j 1

which indicates this large-scale system is robustly stable.

Conclusions In this paper, a new approach of the robust stability analysis for discrete large-scale interval time-delay systems has been proposed. It is found that, by choosing properly the positive definite matrix Q, the DALE can has a simple upper solution bound. By utilizing the Lyapunov equation approach associated with this upper bound and linear algebraic techniques, a concise criterion that assures the robust stability of the aforementioned systems has been derived. It is found that this obtained result does not involve any Lyapunov equation although the Lyapunov approach is utilized. Furthermore, the obtained result does not involve any inverse matrix and free variation.

Therefore, it is easy to be checked. Finally, an illustrative example has been given to demonstrate the applicability of the presented schemes. We believe that this work is useful for stabilization controller design of discrete largescale interval time-delay systems.

ACKNOWLEDGMENT

The author would like to thank the National Science Council, the Republic of China, for financial support of this research under the grant NSC 102-2221-E-230-014.

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Chien-Hua Lee received the B.S. and the M. S. degrees from Ta-Tung Institute of Technology, Taipei, Taiwan, R.O.C., in 1984 and 1991, respectively, and the Ph.D. degree in 1993 from National Cheng Kung University, Tainan, Taiwan, R.O.C., both in electrical engineering.

From 1984-1988, he was an associate engineer of Chun Shan Institute of Science and technology, Taoyuan, Taiwan, R.O.C. From 1994 to 2000, he was an associate professor of the Department of Electrical Engineering, Kung Shan Institute of Technology, Tainan, Taiwan, R.O.C. He is now a professor of the Department of Electrical Engineering, Cheng Shiu University, Kaohsiung, Taiwan, R.O.C. His research interests include robust control, time-delay systems, large-scale systems and bilinear systems.

Ping-Sung Liao was born in Taiwan 1958. He received the B.S. degree from the department of Engineering and Science, Cheng Kung University, Taiwan, 1980, the M.S. degree in electrical engineering from National Tsing Hua University Taiwan, 1985, and the Ph.D. degree in the department of Engineering and Science, Cheng Kung University, 1999.

He has been with Cheng Shiu University of Science and Technology since 1986. He has been associate professor since 1999. His previous research interests are image processing and graph theory. His current research is microcontroller applications.