Robust Exponential Stability for Uncertain Discrete-Time Switched Systems with Interval Time-Varying Delay through a Switching Signal

Journal of Applied Research and Technology 1187  

Robust Exponential Stability for Uncertain Discrete-Time Switched Systems with Interval Time-Varying Delay through a Switching Signal

Jenq-Der Chen¹*, I-Te Wu², Chang-Hua Lien³, Chin-Tan Lee⁴, Ruey-Shin Chen⁵ and Ker-Wei Yu⁶

1,5 Department of Computer Science and Information Engineering National Quemoy University, Kinmen,

Taiwan, R.O.C.

*[email protected].

2 Department of Life Science-National Taiwan Normal University and Department of Sports and Leisure National Quemoy University Kinmen, Taiwan , R.O.C.

3,6 Department of Marine Engineering, National Kaohsiung Marine University Kaohsiung, Taiwan , R.O.C.

4 Department of Electronic Engineering, National Quemoy University Kinmen, Taiwan , R.O.C.

ABSTRACT

This paper deals with the switching signal design to robust exponential stability for uncertain discrete-time switched systems with interval time-varying delay. The lower and upper bounds of the time-varying delay are assumed to be known. By construction of a new Lyapunov-Krasovskii functional and employing linear matrix inequality, some novel sufficient conditions are proposed to guarantee the global exponential stability for such system with parametric perturbations by using a switching signal. In addition, some nonnegative inequalities are used to provide additional degrees of freedom and reduce the conservativeness of systems. Finally, some numerical examples are given to illustrate performance of the proposed design methods.

Keywords: Switching signal, exponential stability, discrete switched systems, interval time-varying delay, additional nonnegative inequality.

   

1. Introduction

A switched system is composed of a family of subsystems and a switching signal that specifies which subsystem is activated to the system trajectories at each instant of time [1]. Some real examples for switched systems are automated highway systems, automotive engine control system, chemical process, constrained robotics, power systems and power electronics, robot manufacture, and stepper motors [2-14]. Switching among linear systems may produce many complicate system behaviors, such as multiple limit cycles and chaos [1]. It is also well known that the existence of delay in a system may cause instability or bad performance in closed control systems [15-17]. Time-delay phenomena are usually appeared in many practical systems, such as AIDS epidemic, chemical engineering systems, hydraulic systems, inferred grinding model, neural network, nuclear reactor, population dynamic model, and rolling mill. Hence stability analysis and

stabilization for discrete switched systems with time delay have been researched in recent years [2-4, 7, 9-10, 12-14].

It is interesting to note that the stable property for each subsystem cannot imply that the overall system is also stable under arbitrary switching signal [4, 6, 8-9]. Another interesting fact is that the stability of a switched system can be achieved by choosing the switching signal even when each subsystem is unstable [5, 7-8]. In [7], a switching signal design technique is proposed to guarantee the asymptotic stability of discrete switched systems with interval time-varying delay. In [14], the switching signal is identified to guarantee the stability of discrete switched time-delay system.

Additional nonnegative inequality terms had been used to improve the conservativeness for the obtained results in our past results [4-5]. Hence the global exponential stability problems for uncertain

Robust Exponential Stability for Uncertain Discrete‐Time Switched Systems with Interval Time‐Varying Delay through a Switching Signal, Jenq‐Der Chen et al. / 1187‐1197

Vol. 12, December 2014  1188 

switched discrete systems with interval time- varying delay under some certain switching signal will be considered in this paper. Some numerical examples are provided to demonstrate the use of obtained results. From the simulation results, our proposed approach provides some less conservative results.

The notations are used throughout this paper. For a matrix A, we denote the transpose by A^T, spectral norm by A , symmetric positive (negative) definite by A0 (A

 0

), maximal eigenvalue by

)

max(A



, minimal eigenvalue by _min(A), n



m dimension by

A

_nm. AB means that matrix

A

B is symmetric positive semi-definite.

0

and

I

denote the zero matrix and identity matrix with compatible dimensions. diag

 

 stands for a block-diagonal matrix. For a vector x, we denote the Euclidean norm by

x

_{. Define} N



,1 2, ,N



,

1

N ,







 

    xk x

M M r s r

k max, 1, ,0

 .

2. Problem statement and preliminaries

Consider the following uncertain discrete switched system with interval time-varying delay of the form

  ^{k A x}  ^{k r}   ^k 

x A k

x (  1 ) 

_(k)



_d(k)



       

x ,

  

r_M,



r_M

 1

,



,

0

, (1) where x

 

k

 

ⁿ is the state vector, x_k is the state at time k defined by



, 1, ,0



), ( : )

(      _M  _M  

k x k r r

x    , switching

signal law

 ( k , x ( k ))

is a piecewise constant function depending on the state in each time. The switching signal

 ( k , x ( k ))

takes its value in the finite set

N

and will be designed. Moreover, the

i k ) 

 (

implied that the i



th subsystem



A ,i Adi



is activated, and

  

k ⁿ is a vector- valued initial state function. The time-varying delay

 

^k

r is a function from



^0,1,2,3,



^to



^0,1,2,3,



, and satisfies the following condition:

 

M

m r k r

r

 



0

(2)

where r_m and r_M are two given positive integers.

The matrices A ,_i A_di are assumed to be

 

k A A

A_i _i _i ^, A_diA_diA_di

 

k ^{, where}

n n di i A

A, ^ , i1 ,2, ,N, are some given constant matrices. A_i

 

k , A_di

 

k , i1 ,2, ,N, are some perturbed matrices and satisfy the following conditions

   



Ai k Adi k



MiFi

 

k



N₁i N₂i



(3) where M and _i N_ji, j1,2 i1 ,2, ,N, are some given constant matrices of system with appropriate dimensions, and F_i

 

k , i1,2,,N, are unknown matrices representing the parameter perturbations which satisfy

   

k F k I

F_i^T _i  (4) In order to derive our main results, the Lemmas are introduced as follows:

Lemma 1. [4] Let U, V, W and M be real matrices of appropriate dimensions with M satisfying MM^T, then MUVWW^TV^TU^T 0 for all V^TV  ,if and I only if there exists a scalar



0 such that

   

⁰

1 1

1        

 ^ UU WW M ^ UU ^ W W

M  ^T  ^T  ^T   ^T 

.

Lemma 2. (Schur complement of [18]). For a given

matrix 



 





22 12

12 11

W W

W

W W_T with _W11W11^T, _W22_W22^T,

then the following conditions are equivalent:

(1) W0, (2) W₂₂0, W₁₁W₁₂W₂₂^1W₁₂^T 0. Define the some sets



^{, , ,}



^



^{: ( 2   ) 0}



_iPUA_i x ⁿ x^T^ A_i^TPA_i P Ux , (5) where 01, P0, and Q0, then we obtain

Journal of Applied Research and Technology 1189 1

1

 



,



₂

 

₁

 

₁

, ,

j i i j

i

    



^

 1

1

, ,

^N _j

N j

N

    



^

 1

1 (6) From the above sets definition, then the switching law is designed as follows:

 i



, if

x  

_i,

i  N

, (7)

where



_i is defined in (6).

The key lemma can be easily proved by the similar way in [7].

Lemma 3. If there exist some scalars

0    1

, 1

0



_i , iN , and



 N 

i i

1

 1, some matrices

0

P and Q0, such that 0

1 2 



 













 



 N i

T i

i P

P A U P

 

Obviously,-we-can-obtain- ⁿ

i N i  

1 -and-_i_j, j

i , i,j1,2,,N,where



is an empty set and



i is defined in (6).

Remark 1. Consider a discrete switched system:

 

k x A k

x( 1) _{(k )} ^.

Now we can choose the Lyapunov function as

 

x x

   

k Px k V k 



^2k ^T

where matrix P0. We can obtain

 

^x



^x

       

^{k Pxk x kPxk}



V _k  ^k  ^T    ^T

 ^2 ^2 1 1 .

Assume



(k,x(k))qN, then we have

 

^{x x}

 

^k



^{A PA P U U}



^x

 

^k

V _k  ^k ^T  ^T_{q q}  





^2



^2

where the matrix U 0.

If the condition in Lemma 3 is satisfied, then we can get

2 0



 















P P A U

P ^T_q

 (8)

By Lemma 2 with (8), then we can have the following result:

 

k

(

^2



A PA



P



U

)

x

 

k

 0

x^T



_q^T _q ,  )x(k _q,

 

0

V x_k

To achieve the exponential stability of discrete switched system, the condition in Lemma 3 is a reasonable choice and feasible setting.

3. Main results

Theorem 1. System (1) is globally exponentially stable with convergence rate

0    1

for any time- varying delay r

 

k satisfying (2), if there exist some

n

n



matrices P0, S

 0

, T0, Q0, U 0,

0

Ri , i1,2,3,4, R_j11^4n4n, R_j12

 

^4nn, 5

, 4 , 3 , 2 ,

1

j , and the positive numbers



_q, N

q1 ,2, , , such that the following LMI conditions are feasible:

* 0

12

11

 



 





i i i

R R

R , i1,2,3,4,5,



R₄R₅



, (9)

0

33 23 22

13 12 11









































q q q

q q j

q

jq ,j,12,q1 ,2, ,N, (10)

0

1 2 



 













 



 N i

T i

i P

P A U P

  (11) where

 



^{A PA U S r Q T}

diag _q^T _{q Mm}

q         

¹₁₁ ^2 1



T Q

S ^{M M}

m r r

r     





²



²



²



m ^{T T}



r^m r R₁₁₁ R₁₁₂W₁ W₁ R₁₁₂

2

   

 



211 311 511

2^rM



r_M



R



r_M



R



r_Mm



R

 

 







 ⁵

2

12 12

j

T j T j j

j W W R

R ,

 



^{A PA U S r Q T}

diag _q^T _{q Mm}

q        

₁₁² ^2 1



T Q

S ^{M M}

m r r

r     





²



²



²

Robust Exponential Stability for Uncertain Discrete‐Time Switched Systems with Interval Time‐Varying Delay through a Switching Signal, Jenq‐Der Chen et al. / 1187‐1197

Vol. 12, December 2014  1190 



m ^{T T}



r^m r R₁₁₁ R₁₁₂W₁ W₁ R₁₁₂

2    







211 311 411

2^rM  r_M R r_M R r_MmR





 







 ⁵

2 12 12

j

T j T j j

j W W R

R









 





0 0

0 0

) (

1 1

12 A P A R

R I A P A

T dq T

dq

T q T q q









 





0 0

0

0 0

0

) ( ) ( ) (

4 3

2

4 3

2

R A R

A R

A

R I A R I A R I A

T dq T

dq T

dq

T q T q T

q







 







Mm M Mm m

q r

R r R r

R r P R

diag ^{2 1 2 3 4}

22  ˆ





























0 0 0

0 0 0

2 1

13 T

q q

T q q

q N

N





,

























0 0 0 0 0

4 3 2 1 23

q q q q q

q

M R

M R

M R

M R

PM ,

 ^{I I} 

diag

_{q q}

q

   



₃₃

 

, r_Mm



r_M



r_m,

) ( ) 1 2 (

ˆ

_Mm

1 r

_Mm

r

_M

r

_m

r    

,

W

₁

  I  I 0 0 

_n4n,

 I I 

_{n n}

W

₂

 0  0

_4 ,

W

₃

  I 0 0  I 

_n4n,

 I I 

_{n n}

W

₄

 0 0 

_4 ,

W

₅

  0 I  I 0 

_n4n. Proof._The Lyapunov-Krasovskii functionals candidate is given by

    



 ⁸

1 i

k i

k V x

x

V (12)

where

 

x x

   

k Px k

V_{1 k} ^2k ^T (13)

  

^

   





 

 ^{1 2}

2

k r k

T k

m

Sx x x

V



  

 (14)

  

^

   





 

 ^{1 2}

3

k r k

T k

M

Tx x x

V



  

 (15)

  

^

   





 

 ¹

) (

2 4

k k r k

T

k x Qx

x V



  

 

   

 



 x Qx

m

M

r r l

k l k

 

^ T













 



1

1 2

(16)

 

 

   

 





1 0

1 1 1

2

5 x R

V

rm l

k l k

T

k

 















 

 (17)

 



 



   

















   







2 0

1 ) (

1 1

2

6 x R

V

k r l

k l k

T k

   

 











2 1

1 2 ( k l 1) R

m

M r r l

k l k

 

^ T













     

 

 

   

 





3 0

1 1 1

2

7 x R

V

rM l

k l k

T

k

 















 

 (19)

 

 

   

 





4

1 1

2

8 x R

V ^m

M r

r l

k l k

T

k ^

 

^











 

 (20)

where P0, S0, Q0, T 0, R_i 0, 4

, 3 , 2 ,

1

i , and 

  

 x1

  

x . Taking the difference of Lyapunov functionals V_j(x_k),

8 , 7 , 6 , 5 , 4 , 3 , 2 ,

1

j , along the solutions of system (1) has the form

 

²



²



¹

 

¹



1     

V x_k



^k



^ x^T k Pxk ^xT

   

^{k Pxk}



(21)

 

^x



^x

   

^{k Sxk}

V _k  ^k ^T

 ₂ ^2



m

 

m

 

T

r^m



x k



r Sxk



r

 

²

(22)

 

^x



^x

   

^kTxk

V _k  ^k ^T

 ₃ ^2



M

 

M

 

T

r^Mx kr Txkr

²

(23) (18)

Journal of Applied Research and Technology 1191

      









 



 ^k

k r k

T

k x Qx

x V

1 ) 1 (

2 4



  



   



^





 

 ¹

) ( k 2

k r k

T Qx

x



  





r_M r_m

    

x^T k Qxk

k  





^2

   



^







 

 ^m

M

r k

r k

T Qx

x

1 2



  

 (24)

By some simple derivations, we have

        



^















   ¹ 

) (

2 1

) 1 (

2 k

k r k k T

k r k

T Qx x Qx

x







     



    

^

   











   

 ¹

1 ) 1 (

2

2 k

k r k

T T

k x kQxk x Qx



  





   



^







 

 ¹

1 ) ( k 2

k r k

T Qx

x



  



 



k r k



Qx



k r

 

k



x^T

k r

k   





^{2(  ( ))}

    

^

   









   

 ¹

1 2

2 k

r k

T T

k

M

Qx x k

Qx k x





 





   



^







 

 ¹

1 k 2

r k

T

m

Qx x



  



 



k r k



Qx



k r

 

k



x^T

k r

k   





^{2(  ( ))}

    

^

   









   

 ^m

M

r k

r k

T T

k x k Qxk x Qx

1 2 2



  





 



k r k



Qx



k r

 

k



x^T

k r

k   





^{2(  ( ))} (25)

 

x r

   

k R k V_{5 k} 



^2k _m



^T ₁



 ^

   



^





 

 ^{k 1 2} ₁

r k

T

m

R



   

 (26)

 

 

   

 





2 0

1 ) 1 (

2

6 x R

V

k r l

k l k

T

k

 







 

 





      









2 0

1 ) (

1

1

2 R

k r l

k l k

 

T















 



   

 











2

1 1

2 ( k l) R

m

M

r r l

k l k

 

^ T





  

    



    ^{  }











2 1

1 2 ( k l 1) R

m

M

r r l

k l k

 

^ T













     

 (27)

By some simple derivations, we have

    ^{  }









2 0

1 ) 1 (

2 R

k r l

k l k

 

T







  

 

      









2 0

1 ) (

1

1

2 R

k r l

k l k

 

T















 



   

k R k k

r ^T

k

 



²

 (  1 ) 

₂



^

      









2 0

1 ) 1 (

1 2 R

k r l

k l k

 

T















 



 

   













     

 ⁰

1 ) (

2 1

2 1 1

k r l

T l

k



k l R



k l



      









2 0

1 ) (

1 2 R

k r l

k l k

 

T













 



             









2 0

1

1 2

2

2 r kR k R

rM

l k

l k

T T

M

k

 

















   



               













2 0

1

1 2

1

)

( 2

2 R R

rm

l k

l k k T

k r k

T

 



 







 





   



    

^

   







    

 ¹

) (

2 2

2

2 k

k r k

T T

M

k r k R k R





   









   

  









2 1

1 2 R

m

M

r r l

k l k

 

^ T













 

 (28)

   

 











2

1 1

2 ( k l) R

m

M

r r l

k l k

 

^ T





  

    

      











2 1

1

2 ( k l 1) R

m

M

r

r l

k l k

 

^ T













     



   



^









  



^m

M

r r l

k R k l

1

2

2

( )  



^

      











2 1

1

1

2 ( k l) R

m

M r r l

k l k

 

^ T















    



      











2 1

1

1

2 ( k l) R

m

M r r l

k l k

 

^ T















    



      









2 1

1 2 R

m

M

r r l

k l k

 

^ T



















   

^{k R k}

r r

r_{M m Mm}

k

 



² ₂

2 ) 1

(  



 



   





^