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



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

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RobustExponentialStabilityforUncertainDiscreteͲTimeSwitchedSystemswithIntervalTimeͲVaryingDelaythroughaSwitchingSignal, JenqͲDerChenetal./1187Ͳ1197

switched discrete systems with interval timevarying 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 T

a matrix A, we denote the transpose by A , spectral norm by A , symmetric positive (negative) definite by A ! 0 ( A  0 ), maximal eigenvalue by Omax ( A) , minimal eigenvalue by Omin ( A) , n u m dimension by A num . A d B means that matrix

B  A 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

N ! 1, xk

s

max

T rM , rM 1,", 0

x . Define N ^1, 2,", N` ,

xk  T .

0  rm d r k d rM

(2)

where rm and rM are two given positive integers. The

matrices

Ai

Ai  'Ai k ,

Ai , Ad i  ƒ

nu n

are

Ai , Adi

assumed

to

Ad i  'Ad i k ,

Adi

be

where

i 1,2,", N , are some given constant matrices. 'Ai k , 'Ad i k , i 1,2, " , N , are some perturbed matrices and satisfy the following conditions

>'A k i

,

'Ad i k @

M i Fi k >N1i

where M i and N ji ,

N 2i @

(3)

j 1,2 i 1,2, " , N , are

some given constant matrices of system with appropriate dimensions, and Fi k , i 1, 2, " , N , are unknown matrices representing the parameter perturbations which satisfy

Fi T k Fi k d I

(4)

2. Problem statement and preliminaries Consider the following uncertain discrete switched system with interval time-varying delay of the form

x(k  1)

AV ( k ) xk  AdV ( k ) xk  r k

xT I T , T where x k  ƒ

rM ,  rM  1, " , 0 , n

(1)

is the state vector, xk is the

state at time k defined by xk (T ) : x ( k  T ), T  ^ rM ,  rM  1, " , 0`, switching

signal law V (k , x(k )) is a piecewise constant function depending on the state in each time. The switching signal V (k , x(k )) takes its value in the

N and will be designed. Moreover, the V (k ) i implied that the i  th subsystem Ai , Adi is activated, and I k  ƒ n is a vector-

finite set

valued initial state function. The time-varying delay r k is a function from ^0, 1, 2, 3, "` to

^0, 1, 2, 3, "`, and satisfies the following condition:

1188

Vol.12,December2014

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 T V T U T  0 for all V T V d I ,if and only if there exists a scalar H ! 0 such that T M  H 1 ˜ UUT  H ˜ W TW M  H 1 ˜ UUT  H 1 ˜ HW HW  0 . Lemma 2. (Schur complement of [18]). For a given ªW11 W12 º with T matrix W W11 W11T , W22 W22 , «W T W » 22 ¼ ¬ 12 then the following conditions are equivalent: (1) W  0, (2) W22  0, W11  W12W221W12T  0. Define the some sets

:i D, P,U, Ai

^xƒ : x (D n

T

2

˜ AiT PAi  P U)x d 0`,

(5)

where 0  D  1 , P ! 0 , and Q ! 0 , then we obtain

RobustExponentialStabilityforUncertainDiscreteͲTimeSwitchedSystemswithIntervalTimeͲVaryingDelaythroughaSwitchingSignal, JenqͲDerChenetal./1187Ͳ1197

:1

:1 , : 2

:i

:i  ‰ : j ,", : N

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

:1  :1 ,",

i 1

N 1

:N  ‰ : j

j 1

j 1

(6)

From the above sets definition, then the switching law is designed as follows:

V

i , if x  :i , i  N ,

(7)

x T k (D 2 ˜ AqT PAq  P  U ) xk  0 ,

'V  x k  0

x(k):q ,

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

where :i is defined in (6).

3. Main results

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

Theorem 1. System (1) is globally exponentially stable with convergence rate 0  D  1 for any timevarying delay r k satisfying (2), if there exist some

Lemma 3. If there exist some scalars 0 d D d 1 ,

0 d D i d 1 , i  N , and

N

¦D

i

1 , some matrices

i 1

P ! 0 and Q ! 0 , such that

N

ª PU AiT P º » 0 D2 ˜ P¼ ¬

¦Di ˜ « i1

N

M,

Obviously,-we-can-obtain- ‰ : i

ƒ n -and- :i ˆ :j

i z j , i, j 1,2,", N ,where

is an empty set and

i 1

M

:i is defined in (6). Remark 1. Consider a discrete switched system: x(k  1) AV ( k ) xk .

n u n matrices P ! 0 , S ! 0 , T ! 0 , Q ! 0 , U ! 0 , Ri ! 0 , i 1, 2, 3, 4 , R j11  ƒ 4 nu4 n , R j12  ƒ 4 nun , q 1,2, " , N , such that the following LMI conditions are feasible:

ª Ri11 « * ¬

where matrix P ! 0 . We can obtain

>

; jq

'V  xk

N

ª P  U

i 1

¬

¦Di ˜ «

@

'V xk D 2k ˜ D 2 ˜ xT k 1 Pxk 1  xT k Pxk . Assume

V (k , x(k )) q  N , then we have D ˜ xT k D 2 ˜ AqT PAq  P  U  U xk 2 k

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

(9)

AiT P º »0  D 2 ˜ P¼

(10)

(11)

^

diag  D 2 ˜ AqT PAq  U  S  rMm  1 ˜ Q  T

;111q

D

2 rm

˜S

D

2 rm

˜ rm ˜ R111  R112W1  W R

D

2 rM

5

(8)

 D 2 rM ˜ Q



 D 2 rM ˜ T T 1

`

T 112



˜ >rM ˜ R211  rM ˜ R311  rMm ˜ R511





 ¦ R j12W j  W jT R Tj12 , j 2

AqT P º »0  D 2 ˜ P¼

R5 ,

where

where the matrix U ! 0 .

ª P  U « ¬

Ri12 º ! 0 , i 1,2,3,4,5 , R4 Ri »¼

ª;11j q ;12q ;13q º » , j 1,2, q 1,2,", N , « « ;22q ;23q »  0 «

;33q »¼ ¬

Now we can choose the Lyapunov function as

V  xk D 2 k ˜ x T k Px k

Hq ,

1, 2, 3, 4,5 , and the positive numbers

j

2 ;11 q

^

diag  D 2 ˜ AqT PAq  U  S  rMm  1 ˜ Q  T

 D 2 rm ˜ S

 D 2 rM ˜ Q  D 2 rM ˜ T

`

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RobustExponentialStabilityforUncertainDiscreteͲTimeSwitchedSystemswithIntervalTimeͲVaryingDelaythroughaSwitchingSignal, JenqͲDerChenetal./1187Ͳ1197



T  D 2 rm ˜ rm ˜ R111  R112W1  W1T R112

D

˜ >rM ˜ R211  rM ˜ R311  rMm ˜ R411

2 rM

5





 ¦ R j12W j  W jT R Tj12



j 2

;12 q

ª AqT P ( Aq  I )T R1 « 0 « 0 T T « Adq P Adq R1 « 0 0 ¬«

0 T Adq R2

0 T Adq R3

0

0

 diag ^H q ˜ I

; 33 q

rˆMm

( Aq  I )T R4 º » 0 » T Adq R4 » » 0 »¼

R1 rm

ª0 H q ˜ N1Tq º « » 0 », «0 «0 H q ˜ N 2Tq » ; 23q « » 0 ¼» ¬«0

;13q

>I

W4

>0

ª PM q «R M « 1 q « R2 M q « « R3 M q « R4 M q ¬

H q ˜ I `, rMm

0  I 0@nu4n , W3 0 I  I @nu4n , W5

8

¦V x i

k

V6 xk

R4 ½ ¾ rMm ¿

>I >I >0

 rm

V7  xk

V8  xk

rM  rm ,

 2T

˜ x T T Tx T

(15)

k  rM

k 1

¦D T

˜ xT T QxT 

2T

k r ( k )

 rm

k 1

¦ T¦ D

l  rM 1

 2T

˜ xT T QxT

k l

V1  xk D 2 k ˜ x T k Px k

k 1

0

¦ T ¦D

 2T

l  rm 1

k 1l

0

¦D T

k 1

¦

l  r ( k ) 1 k 1

 2T

2T

k 1l

k 1

0

l  rM 1

k 1l

k 1

¦ T¦D

l  rM

 2T

k l

 2T

(17)

˜K T T R2KT 

˜ (T  k  l 1) ˜K T T R2K T

¦ T ¦D

 rm 1

˜K T T R1K T

˜K T T R3K T

˜K T T R4K T

(18)

(19)

(20)

P ! 0 , S ! 0 , Q ! 0 , T ! 0 , Ri ! 0 , i 1,2,3,4 , and K T x T  1  x T . Taking the difference of Lyapunov functionals V j ( xk ) ,

where

 I 0 0@nu4n , 0 0  I @nu4n , I  I 0@nu4n .

j 1,2,3,4,5,6,7,8 , along the solutions of system (1) has the form

>

@

'V1 xk D 2 k ˜ D 2 ˜ xT k  1 Pxk  1  xT k Pxk

(21)

functionals

>

'V2 xk D 2k ˜ xT k Sxk D

2rm

(12)

@

˜ xT k  rm Sxk  rm

(22)

>

'V3 xk D 2k ˜ xT k Txk  D 2r ˜ xT k  rM Txk  rM @

where

Vol.12,December2014

k 1

¦ ¦D

i 1

1190

(14)

k  rm

l  rM 1 T k l

0º 0»» , 0» » 0» 0»¼

Proof._The Lyapunov-Krasovskii candidate is given by

V xk

¦D T

V5  xk

R3 rM

R2 rˆMm

1 (rMm 1) ˜ (rM  rm) , W1 2

W2

V3  xk

˜ x T T SxT

 2T

(16)

( Aq  I )T R3

­  diag ®D 2 ˜ P ¯

¦D T

V4  xk

( Aq  I )T R2

; 22 q

k 1

V2  xk

(13)

M

(23)

RobustExponentialStabilityforUncertainDiscreteͲTimeSwitchedSystemswithIntervalTimeͲVaryingDelaythroughaSwitchingSignal, JenqͲDerChenetal./1187Ͳ1197

k

'V4  xk

¦D

By some simple derivations, we have

˜ xT T QxT

 2T

T k  r ( k 1) 1

0

¦ D  2T ˜ x T T QxT



D

˜ rM  rm ˜ x k Qx k T

k  rm

¦D



k

¦D

2T

˜ x T QxT  T

T k r ( k 1)1

k 1

¦D T

2T

D

¦D



˜ x T QxT

¦D 2T ˜ xT T QxT

k 1

¦D

 2T



˜ xT T QxT

¦D

D

˜ x k  r k Qx k  r k k  rm

¦D

¦D T

 2T

˜ x T T Qx T

'V5  xk D 

k 1

¦D T

 2T

k  rm

l  r ( k 1) 1

k 1

l  r ( k ) 1



 rm



¦D T

¦D T

 rm

k 1

¦ T¦D

l rM 1

 2T

k  l 1

k l

2T

¦D T k l

 2T

k 1 l

k

¦

l  rM 1

k

¦

0

¦

(25)

¦

k 1

k

l  rM 1

 rm

k 1

 2T

k  l 1

¦ T¦ D

l  rM 1

¦D

 2T

 2T



˜K T R2K T T

˜K T T R2K T



 rm

¦ T¦D

l rM 1

˜ (T  k  l ) ˜ K T T R2K T



k 1

l  rM 1  rm

k 1

l rm 1

¦D T

˜ (T  k  l  1) ˜K T T R2KT

(27)

kl

k 1

¦D

 2T

T k r ( k )

˜KT T R2KT

˜K T T R2KT

 2T

˜ (T  k  l ) ˜K T T R2K T

 2T

˜ (T  k  l ) ˜K T T R2K T

k  l 1

k 1

¦ T¦ D

l  rM 1

2T

(28)

˜ (T  k  l  1) ˜K T T R2K T

k  l 1

 rm

¦ T¦D

˜KT T R2KT

˜ (l ) ˜K k R2K k

¦D T

l  rM 1

kl

k1

0

2T

˜ (T  k  l ) ˜K T T R2K T

 2T

¦

¦

k1

0

˜K T T R2K T

k l

 rm

(26)

 2T

k l

¦D T



˜ rm ˜K k R1K k

˜K T T R1K T

 rm

¦ T¦ D

 rm

T

0

˜K T T R2K T

˜KT T R2KT 

l  rM 1

'V6 xk



˜ x k  r k Qx k  r k T

2 k

2T

l  rM 1

T k  rM 1

D

k1



T

d D  2 k ˜ x T k Qx k  2 ( k  r ( k ))

 2T

d D 2k ˜ rM ˜K T k R2Kk 

˜ x T QxT

2 ( k  r ( k ))

˜K T k  1  l R2K k  1  l

k l

kr(k)

T

T k  rm 1

k 1

0

˜K T T R2K T

dD2k ˜ rM ˜KT k R2Kk 

T k  rM 1



 2  k 1 l

l  r ( k ) 1

 D 2 ( k  r ( k )) ˜ x T k  r k Qx k  r k

 2T

 2T

k l

¦ T¦ D



T k  r ( k ) 1

k 1

¦D T

l  r ( k ) 1

k 1

d D 2k ˜ xT k Qxk 

0

¦D



˜ xT T QxT

 2T

k 1

0

¦

T k  r ( k 1) 1 k 1

k 1 l

l  r ( k 1) 1

T

˜K T T R2K T

 2T

˜ r ( k  1) ˜K T k R2K k

2 k



k r ( k )

D 2k ˜ xT k Qxk 

¦D T

l  r ( k ) 1

T k  rM 1

By some simple derivations, we have

k 1

¦

(24)

˜K T T R2K T

 2T

k l

0



˜ x T T QxT

 2T

¦D T

l  r ( k 1) 1

T k r (k )

2 k

k

¦

k 1

k l

 2T

˜K T T R2K T

§ (rM  rm  1) ˜ rMm · ¸ ˜K k R2K k 2 ¹ ©

D 2k ˜ ¨

JournalofAppliedResearchandTechnology

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RobustExponentialStabilityforUncertainDiscreteͲTimeSwitchedSystemswithIntervalTimeͲVaryingDelaythroughaSwitchingSignal, JenqͲDerChenetal./1187Ͳ1197



 rm

k 1

¦D 2T ˜K T T R2K T

¦

(29)

l  rM 1 T k  l

ª E k º ª R111 ¦ « » « T k  rm ¬K T ¼ ¬ *

rm ˜ E k R111E k  2E k k R112

'V7  xk D 2 k ˜ rM ˜K T k R3K k 

k 1

¦D

T

˜K T R3K T

 2T

(30)

T

T k  rM



¦ D 2T ˜K T T R4K T

k  rm 1

¦D

 2T

T k r ( k )

˜K T R4K T T

Then we can obtain the following result from (21)-(31) 'V  x k

^

i 1



 D 2 rM ˜ x T k  rM Txk  rM

k 1

 D 2 rM



ª ˜« ¬T

k 1

¦ K T R K T  T ¦K T R K T T

2

T

k  rm 1

T k  rM

T k r (k )



3

k  rM

k  r ( k ) 1

¦K T T R4K T 

¦K T R K T T

4

(32)

R412 º ª E k º R4 »¼ «¬K T »¼

k r  k 1

k  r  k 1

T k rM

¦K T R K T t 0

(36)

T

4

ª E k º ª R511 ¦ » « « k  r k ¬K T ¼ ¬ * T

R512 º ª E k º R4 »¼ «¬K T »¼ k rm 1

ET k >xT (k) xT k  rm xT k  rk xT k  rM @ k 1

¦K T xk  xk  r k T k r k

and the similar equations, we have

Vol.12,December2014

T

k  rm 1

T

1192

(35)

3

T k  rM

Define

From LMIs in (9) with

¦K T

T k  rM

rM  rk ˜ E T k R411E k  2E T k R412 ¦KT

1

k r ( k )

k 1

T

ª E k º ª R411 ¦ « » « T k  rM ¬K T ¼ ¬ *

T

k 1

R312 º ª E k º R3 »¼ «¬K T »¼

¦K T R K T t 0

k  r k 1

¦K T R K T T k  rm

k 1

(34)

2

T

T k  rM

 D 2 rM ˜ x T k  r k Qx k  r k  K T (k )>rm ˜ R1  rˆMm ˜ R2  rM ˜ R3  rMm ˜ R4 @K (k )

¦K T

T k r (k )

T

rM ˜ E T k R311E k  2E T k R312

 D 2 rm ˜ x T k  rm Sx k  rm

k 1

¦K T R K T t 0

T k r (k )

d D 2 k ˜ D 2 ˜ x T k  1 Pxk  1  x T k > P  S  rMm  1 ˜ Q  T @x k

 D 2 rm ˜

k 1



ª E k º ª R311 ¦ « » « T k  rM ¬K T ¼ ¬ *

¦ 'Vi xk

R212 º ª E k º R2 »¼ «¬K T »¼

T

k 1

8

(33)

1

r (k ) ˜ E T k R211E k  2 E T k R212

(31)

k  rm

T

ª E k º ª R211 ¦ « » « k  r ( k ) ¬K T ¼ ¬ *

T

k 1

¦K T T

¦K T R K T t 0

k 1

k  r ( k ) 1

T k  rM



k 1

T

T k  rm

'V8  x k D 2 k ˜ ( rM  rm ) ˜K T k R4K k 

R112 º ª E k º R1 »¼ «¬K T »¼

T

k 1

r(k)  rm ˜ E T k R511E k  2E T k R512 ¦KT T k r  k



k  rm 1

¦K T R K T t 0 T

T k r k

4

Assume the particular case V (k , x(k )) we can obtain the following result

(37)

q , then

RobustExponentialStabilityforUncertainDiscreteͲTimeSwitchedSystemswithIntervalTimeͲVaryingDelaythroughaSwitchingSignal, JenqͲDerChenetal./1187Ͳ1197

^

>

 2 E T k R112 >xk  xk  rm @ 

D

2 k

^

˜D

2 rM

>

º ½°

T k  rm

¼ °¿

¦K T T R1K T » ¾

˜ r (k ) ˜ E k R211E k

>

^

k 1

T

 2E T k R212>xk  xk  r(k ) @ 

2

¼¿

º½°  2E k R312 >xk  xk  rM @  ¦K T R3K T »¾ T k  rM ¼°¿

D

2k

^

˜D

2 rM

>

˜ rM  r k ˜ E

 2E T k R412>xk  r(k)  xk  rM @ 

D

2 k

^

˜D

2 rM

>

˜ r ( k )  rm ˜ E

T

 2E T k R512>xk  rm  xk  rk @ 

^

k R411 E k

º½°

¦K T R KT »¾° T 4

k rM

k R511 E k k rm 1

2

¼¿

º½°

¦K T R KT »¾° T T

4

k r k

d D ˜ D ˜ x k  1 Pxk  1  x T k  P  S  rMm  1 ˜ Q  T x k 2 k

¼¿

T

 D 2 rm ˜ x T k  rm Sx k  rm

 D 2 rM ˜ xT k  r k Qxk  r k  K T ( k )>rm ˜ R1  rˆMm ˜ R2  rM ˜ R3  rMm ˜ R4 @K ( k )

>

@

 2 E T k R112 >xk  xk  rm @

D

>

˜ rM ˜ E k R311E k

m

2

T q

q

i ;11 q

2q

(38)

r(k )  rm rMm

ª AqT P º « » 0 » 2 «  T D ˜P « Adq P » « » ¬« 0 ¼»



ª AqT P º « » 1 « 0 » « AdqT P » « » ¬« 0 ¼»

T



ªAq I T º ªAq I T º « » « » 0 0 « T »rm ˜ R1 rˆMm˜ R2 rM ˜ R3 rMm˜ R4 « T » « A » « A » « dq » « dq » «¬ 0 »¼ «¬ 0 »¼

T

with

(39)

i ;11 q , W j , i 1,2 , j 1,2,3,4,5 , q 1,2, " , N ,

are defined in (10).

0 d H (k ) d 1, the H (k) ˜ ;1q  (1H (k))˜ ;2q is a

convex combination of

;1q and ;2q . Hence

H (k) ˜ ;1q  (1H (k))˜ ;2q  0 will imply ;1q  0 and ;2q  0 Define

ª ª r (k )  rm  rm º T  D 2 rM ˜ « « » ˜ rMm ˜ E k R211 E k r Mm ¼ ¬« ¬ T  2 E k R212 >x k  x k  r ( k ) @ 2 rM

;iq

Since

 D 2 rM ˜ x T k  rM Tx k  rM

 D 2 rm ˜ rm ˜ E T k R111 E k

Where H (k )

T

k r k 1 T

T

T

½

T

 D 2 k ˜ D 2 rM ˜ rM ˜ E T k R311 E k k 1

512

2 k

1q

k r ( k )

T

T

T

º° K T R KT »¾ ¦ ° T k 1

`@  2 E k R >xk  r  x k  r k @`@ D ˜ >x (k )D ˜ A PA  P  U x(k )  E k ˜ H (k ) ˜ ;  (1  H (k )) ˜ ; E k @  2 E T k R512 >xk  rm  xk  r k @

'V  xk  D 2 k ˜ D 2 rm ˜ rm ˜ E T k R111E k

@

T

@

 2 E T k R312 >x k  x k  rM @

ªª r  rk  rm  rm º T  D 2rM ˜ «« M » ˜ rMm ˜ E k R411E k rMm «¬¬ ¼ T  2 E k R412 >xk  r (k )  xk  rM @

@

ª ª r ( k )  rm º º T  D 2 rM ˜ « « » ˜ rMm ˜ E k R511 E k » rMm ¼ ¬« ¬ ¼»

;ˆ iq

i ª;11 ;12q º q T T T « »  /q Fq k *q  *q Fq k /q * ; 22q » ¬« ¼

(40)

where

/Tq

*

T q

>0

>N

0 0 0 MqT P MqT R1 MqT R2 MqT R3 MqT R4

1q

@

@

0 N2q 0 0 0 0 0 0 , and ;

i 11q ,

;12 q , ; 22 q are defined in (10). By the condition (11) with Lemma 3 and the switching signal in (8), we can obtain the following result

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xT k (D 2 ˜ AqT PAq  P U)xk  0 , x(k):q

(41)

By Lemmas 1-3 with above conditions, the conditions

; iq  0 , i 1,2 , in (10) will imply ;ˆ iq  0 , i 1,2 , ˆ 0, i in (40), the conditions ; iq

1,2 , in (40) will

also imply ;iq  0 in (39).Therefore, the conditions

;iq  0 ,

i 1,2 ,

in

(39)

are

equivalent

to

H (k ) ˜ ;1q  (1 H (k )) ˜ ;2q  0 in (38). From the condition

(41)

and the matrix in (28) with (9), we have H (k) ˜ ;1q  (1 H (k))˜ ;2q  0

'V  x k V x k 1  V  x k d 0 , k

V  xk 1 d V  xk , k

0, 1, 2, 3, "

0, 1, 2, 3, "

This implies

V  x k d V  x0 , k

0, 1, 2, 3, " ,

D 2 k ˜ Omin ( P) ˜ x(k ) d V xk d V x0 d G1 ˜ x0 s , 2

i 1,2,3,4,5 , in (9), are determined appropriately by LMI toolbox of MATLAB. 4. Illustrative examples Example 1. Consider the system (1) with no uncertainties and the following parameters: (Example 3.1 of [7])

0.54

1.02

(42)

In order to show the obtained results, the allowable delay upper bound and switching sets that guarantees the global exponential stability for system (1) with (42) is provided in Table 1. The delay upper bound and switching sets of switched system (1) with (42)

2

D

1 , rm

1 , rM

5 globally

asymptotically stable (choose D1 D2 0.5)

where

G1 Omax ( P)  rm ˜ Omax ( S )  rM ˜ Omax (T )

[7]

 rM2  rM rm  rm ˜ Omax (Q)  rm2 ˜ Omax ( R1 ) 1 § ·  ¨ rM2  rM  rm  1 ˜ rM  rm ˜ rM  1 ¸ 2 © ¹ 2 ˜ Omax(R2 )  rM ˜ Omax(R3 )  rM  rm ˜ rM ˜ Omax(R4 )

xk d G1 OminP ˜D ˜ x0 s , k k

0, 1, 2, 3, " .

Therefore, we conclude that the system (1) is exponentially stable with convergence rate 0  D  1. This completes this proof. Remark 2. By setting D 1 in Theorem 1, the global asymptotic stability of system (1) can be achieved. Remark 3. In Theorem 1, some additional nonnegative inequalities in (33) - (37) are proposed. 4 nu 4 n , Ri12  ƒ 4 nun , Rj ! 0 , j 1,2,3,4 , Ri11  ƒ

Vol.12,December2014

­°>x1 x2 @T  ƒ 2 : 12.0213 x12 ½° ® ¾ °¯ 38.8254 x1 x2  31.6356 x 2  0°¿

:1

, :2 D

1 , rm

ƒ 2  :1 .

1 , rM

187 globally

asymptotically stable ( D1 0.1and D2 0.9)

­°>x x @T  ƒ2 : 2.9225x12 ½° :1 ® 1 2 ¾ 2 °¯ 10.1092x1 x2  7.7291x  0°¿

By some simple derivations, we have

1194

 0.01  0.06

º ª º , Ad1 « N 2, A1 ª« » », ¬ 0.17  0.31¼ ¬ 0.01 0.04 ¼ ª 0.01  0.06º ª 0.11 0.18 º , Ad2 « A2 « » ». ¬ 0.01 0.04 ¼ ¬0.03 0.04¼

Our results

D

, : 2 ƒ 2  :1 . 0.85 , rm 1 , rM 7 globally

exponentially stable ( D1 0.1andD2 0.9)

:1

­°>x1 x2 @T  ƒ 2 : 1.5344 x12 ½° ® ¾ °¯ 5.7566 x1 x2  5.0051x 2  0°¿

, :2

ƒ 2  :1 .

Table 1. Some obtained results for our proposed results in this paper.

Example 2. Consider the system (1) with the following parameters: (Example 3.1 of [10])

RobustExponentialStabilityforUncertainDiscreteͲTimeSwitchedSystemswithIntervalTimeͲVaryingDelaythroughaSwitchingSignal, JenqͲDerChenetal./1187Ͳ1197

N A2 NA1

2,

A1

ª 0 .8 « 0 ¬

0 .2 º , 0 . 91 »¼

Ad 1

ª0.3 0 º , ª0.12 0 º « 0 0.58» Ad 2 «0.11 0.11» , M1 ¼ ¬ ¬ ¼ ª0.01 0 º NA2 NB1 NB2 « » ¬ 0 0.01¼

0 º, ª 0.1 « 0.1  0.1» ¬ ¼ ª0.1 0 º M2 « » ¬ 0 0.1¼

The delay upper bound and switching sets of switched system (1) with (44)

rm 1, rM 6 (under arbitrary switching signal)

Our result

1 , rM

:1

­°>x1 x2 @T  ƒ 2 : 18.6759 x12 ½° ® ¾ °¯ 0.6803 x1 x2  0.4395 x 2  0°¿

, :2

D1

:1

N A2

ª0.12 0 º , M1 M2 « » NA1 NA2 ¬ 0 0.12¼

0.1 and D 2

, :2

rm

D1

0 .9 )

ƒ 2  :1 .

2 , rM

0.1 and D 2

4

(

0 .9 )

­°>x1 x2 @T  ƒ 2 : 1.2306 x12 ½° ® ¾ °¯ 1.5704 x1 x2  0.7528x 2  0°¿ , : 2 ƒ 2  :1 .

Example 4. Consider the system (1) with the following parameters: (Example of [9])

2 A1

0 º ª 0.15 « 0.1  0.1» , ¬ ¼ ª0.14 0 º ª 0 . 05 º « 0.1  0.1» , M 1 « 0 » , ¬ ¼ ¬ ¼

0 º ª 0.7 «0.08 0.95» , Ad 1 ¬ ¼

0º ª 0.7 «0.08 0.9» , Ad 2 ¬ ¼ ª 0.05º N11 >0.2 0.3@, M2 « », ¬0.02¼ N21 >0  0.1@ , N22 >0.3 0.2@ . A2

In order to show the obtained results, the allowable delay bounds that guarantee the global asymptotic stability for system (1) with (44) is provided in Table 3.

(

Table 3. Some delay bounds to guarantee the global asymptotic stability ( D 1 ),of system.

N

(44)

4

­°>x1 x2 @  ƒ 2 : 3.4451x12 ½° ® ¾ °¯ 3.4325 x1 x2  1.7489 x 2  0°¿

Our result

Table 2. Some delay bounds to guarantee the global asymptotic stability ( D 1 ) of system.

ª  0. 2  0. 2 º «  0.1  0.2» ¼ ¬ 0 . 2 0 . 1 º ª « 0.1 0.1» ¼ ¬ ª0.01 0 º NB1 NB2 « » ¬ 0 0.01¼

3

T

ƒ 2  :1 .

2 A1 ª«0.7 0.2 º» , Ad 1 ¬0.1 0.71¼  0.01º ª 0.4 , « 0.01 0.58 » Ad 2 ¬ ¼

1 , rM

rm

:1

Example 3. Consider the system (1) with the following parameters:

2 , rM

rm

11 ( D1 0.1 andD2 0.9)

rm

3

(under arbitrary switching signal)

The delay upper bound and switching sets of switched system (1) with (43)

[4]

1 , rM

(under arbitrary switching signal)

[4]

In order to show the obtained results, the allowable delay upper bound that guarantees the global asymptotic stability for system (1) with (43) is provided in Table 2.

rm 1, rM 5 (under arbitrary switching signal)

2

(under arbitrary switching signal)

rm

(43)

[10]

1 , rM

rm

[10]

N12 >0.1 0.1@ (45)

In order to show the obtained results, the allowable delay bounds that guarantee the global asymptotic stability for system (1) with (45) is provided in Table 4.

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RobustExponentialStabilityforUncertainDiscreteͲTimeSwitchedSystemswithIntervalTimeͲVaryingDelaythroughaSwitchingSignal, JenqͲDerChenetal./1187Ͳ1197

The delay upper bound and switching sets of switched system (1) with (45) rm 2 , rM 6 [9] (under arbitrary switching signal) rm 1 , rM 10 (under arbitrary switching signal)

2 , rM

rm

10

(under arbitrary switching signal) rm 4 , rM 11

[4]

(under arbitrary switching signal)

8 , rM

rm

12

1 , rM

11 ( D1 0.1andD2 0.9)

­°>x1 x2 @  ƒ : 0.7576 x ½° ® ¾ 2 °¯ 18.2930 x1 x2  50.3305x  0°¿ T

:1

, :2

rm

Our result

2 , rM

2

2 1

ƒ 2  :1 .

11 ( D1 0.1 and D 2

0.9 )

­°>x1 x2 @T  ƒ 2 : 0.1590 x12 ½° :1 ® ¾ °¯ 1.4113x1 x2  4.4661x 2  0°¿ , : 2 ƒ 2  :1 . rm 4 , rM 13 ( D1 0.1 and D2 0.9 ) :1

rm

:1

­°>x1 x2 @T  ƒ 2 : 0.7476 x12 ½° ® ¾ °¯ 9.8448 x1 x2  29.7229 x 2  0°¿

, :2 ƒ2  :1 . 8 , rM 16 ( D1 0.1 and D2 0.9 ) ­°>x1 x2 @T  ƒ 2 : 0.4987 x12 ½° ® ¾ °¯ 2.8498x1 x2  8.0945x 2  0°¿ , :2

ƒ 2  :1 .

Table 4. Some delay bounds to guarantee the global asymptotic stability ( D 1 ) of system.

Note that comparing the proposed examples, we can design a new switching signal to guarantee the global exponential stability for system (1). At the same time, less conservative results are reached without tuning any parameters.

1196

Vol.12,December2014

In this paper, the global exponential stability for uncertain discrete switched systems with interval time-varying delay has been considered. Some additional nonnegative inequalities and LMI approach are used to guarantee the proposed results of the system. The obtained results have shown to be less conservative via some numerical examples. In [19-20], the practical applications have been successfully designed in the different control fields. Acknowledgments

(under arbitrary switching signal)

rm

5. Conclusions

The research reported here was supported by the National Science Council of Taiwan, R.O.C. under grant no. NSC102-2221-E-507-006.

RobustExponentialStabilityforUncertainDiscreteͲTimeSwitchedSystemswithIntervalTimeͲVaryingDelaythroughaSwitchingSignal, JenqͲDerChenetal./1187Ͳ1197

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[13] L. Zhang et al., “Robust stability and stabilisation of uncertain switched linear discrete time-delay systems,” IET Control Theory Applications, Vol. 2, pp. 606-614, 2008. [14] W. A. Zhang and L. Yu., Stability analysis for discrete-time switched time-delay systems, Automatica, Vol. 45, pp. 2265-2271, 2009. [15] R. S. Gau et al., “Novel stability conditions for interval delayed neural networks with multiple timevarying delays,” International Journal of Innovative Computing, Information and Control , Vol. 7, pp. 433444, 2011. [16] T. Li et al., “Robust stability for neural networks with time-varying delays and linear fractional uncertainties,” Neurocomputing, Vol. 71, pp. 421-427, 2007. [17] K. Gu et al., “Stability of Time-Delay Systems,” Birkhauser, Boston, Massachusetts, 2003. [18] S. P. Boyd et al., “Linear Matrix Inequalities in System and Control Theory,” SIAM, Philadelphia, 1994. [19] S. L. Chu, M. J. Lo., “A New Design Methodology for Composing Complex Digital Systems,” Journal of Applied Research and Technology, vol. 11, pp. 195205, 2013. [20] Y. L. Chen and Z. R. Chen., “A PID Positioning Controller with a Curve Fitting Model Based on RFID Technology,” Journal of Applied Research and Technology, vol. 11, pp. 301-310, 2013.

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