New design switching rule for the robust mean square ... - IEEE Xplore

New design switching rule for the robust mean square stability of uncertain stochastic discrete-time hybrid systems Grienggrai Rajchakit Major of Mathematics Maejo University Chiangmai, Thailand Email: [email protected]

Abstract-This paper is concerned with robust mean square stability of uncertain stochastic switched discrete time-delay systems. The system to be considered is subject to interval time­ varying delays, which allows the delay to be a fast time-varying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for the robust mean square stability for the uncertain stochastic discrete time-delay system is designed via linear matrix inequalities.

I.

INTRO DUCTION

Switched systems constitute an important class of hybrid systems. Such systems can be described by a family of continuous-time subsystems (or discrete-time subsystems) and a rule that orchestrates the switching between them. It is well known that a wide class of physical systems in power systems, chemical process control systems, navigation systems, auto­ mobile speed change system, and so forth may be appropri­ ately described by the switched model [1-7]. In the study of switched systems, most works have been centralized on the problem of stability. In the last two decades, there has been increasing interest in the stability analysis for such switched systems; see, for example, [8, 9] and the references cited therein. Two important methods are used to construct the switching law for the stability analysis of the switched sys­ tems. One is the state-driven switching strategy [9]; the other is the time-driven switching strategy [8]. A switched system is a hybrid dynamical system consisting of a finite number of subsystems and a logical rule that manages switching between these subsystems (see, e.g., [1-10] and the references therein). The main approach for stability analysis relies on the use of Lyapunov-Krasovskii functional and linear matrix inequality (LMI) approach for constructing a common Lyapunov function [11, 12, 13]. Although many important results have been obtained for switched linear continuous-time systems, there are few results concerning the stability of switched linear discrete systems with time-varying delays. In [14, 15], a class of switching signals has been identified for the considered switched discrete-time delay systems to be stable under the average dwell time scheme. This paper studies robust mean square stability problem for uncertain stochastic switched linear discrete-time delay

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with interval time-varying delays. Specifically, our goal is to develop a constructive way to design switching rule to robustly mean square stable the uncertain stochastic linear discrete­ time delay systems. By using improved Lyapunov-Krasovskii functional combined with LMIs technique, we propose new criteria for the robust mean square stability of the uncertain stochastic linear discrete-time delay system. Compared to the existing results, our result has its own advantages. First, the time delay is assumed to be a time-varying function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, the delay function is bounded but not restricted to zero. Second, the approach allows us to design the switching rule for robust mean square stability in terms of of LMIs. The paper is organized as follows: Section II presents def­ initions and some well-known technical propositions needed for the proof of the main results. Switching rule for the robust mean square stability is presented in Section III. II.

PRELIMINARIE S

The following notations will be used throughout this paper. R+ denotes the set of all real non-negative numbers; Rn denotes the n-dimensional space with the scalar product of two vectors (x, y) or xTYi Rnxr denotes the space of all matrices of (n x r ) - dimension. N + denotes the set of all non-negative integers; AT denotes the transpose of A; a matrix A is symmetric if A = AT. Matrix A is semi-positive definite (A 2: 0) if (Ax, x) 2: 0, for all x E Rni A is positive definite (A > 0) if (Ax, x) > ° for all x i= 0; A 2: B means A - B 2: 0. A(A) denotes the set of all eigenvalues of A; Amin(A) = min{ReA : A E A(A)}. Consider a uncertain stochastic discrete systems with inter­ val time-varying delay of the form

x(k + 1) (A/, + �AI'(k))x(k) + (BI' + �BI'(k))x(k - d(k)) + al'(x(k), x(k - d(k)), k)w(k), k E N +, x(k) Vk, k -d2, -d2 + 1, ..., 0, =

=

=

(1) where x(k) E Rn is the state, ')' (. ) : Rn -7 N : = {I, 2, . . . , N} is the switching rule, which is a function

depending on the state at each time and will be designed. A switching function is a rule which determines a switching sequence for a given switching system. Moreover, , (x(k))= i implies that the system realization is chosen as the ith system, i= 1, 2, ...,N. It is seen that the system (1) can be viewed as an autonomous switched system in which the effective subsys­ tem changes when the state x(k) hits predefined boundaries. Ai, Bi, i = 1, 2, ...,N are given constant matrices and the time-varying uncertain matrices �Ai(k) and �Bi(k) are de­ fined by: �Ai(k)= EiaFia(k)Hia, �Bi(k)= EibFib(k)Hib, where Eia, Eib, Hia, Hib are known constant real matrices with appropriate dimensions. Fia(k), Fib(k) are unknown un­ certain matrices satisfying

k= 0, 1, 2, ..., (2) where I is the identity matrix of appropriate dimension, w(k) is a scalar Wiener process (Brownian Motion) on (0, F, P) with

in the mean square if there exists a positive definite scalar function V(k, x(k) : Rn x Rn -+ R such that

E[�V(k, x(k))]= E[V(k + 1, x(k + 1)) - V(k, x(k))] < 0, along any trajectory of solution of the system (1).

is

strictly

CJT(x(k), x(k - d(k)), k)CJi(X(k), x(k - d(k)), k) � Pi1XT(k)x(k) + Pi2XT(k - d(k))x(k - d(k), (4) E Rn, x(k), x(k - d(k) where Pi1 > 0 and Pi2 > 0, i= 1, 2, ...,N are known constant scalars. The time-varying function d(k): N+ -+ N+ satisfies the following condition:

complete

if there

1, 2, . . . ,N, L�l 6i > 0 such that N

L 6iJi < o. i=l If N =

2 then the above condition is also necessary for the strict completeness.

Proposition 2.2. (Cauchy inequality) For any symmetric positive definite marix N E Mnxn and a, b E Rn we have +a

E[w(k)]= 0, E[w 2(k)]= 1, E[w(i)w(j)]= O(i i= j), (3) and CJi: Rn x Rn x R -+ Rn, i= 1, 2, ...,N is the continuous function, and is assumed to satisfy that

{Jd, i = 1, 2, . . . ,N, > exist 6i O, i

Proposition 2.1. [16] The system

Tb � a TNa + bTN -lb.

Proposition 2.3. [16] Let E, H and F be any constant matrices of appropriate dimensions and FT F � I. For any E > 0, we have

III. MAIN Let us set

[:

RESULTS

Will

Wi= Remark 2.1. It is worth noting that the time delay is a time-varying function belonging to a given interval, in which the lower bound of delay is not restricted to zero. Definition 2.1. The uncertain stochastic switched system (1) is robustly stable if there exists a switching function , . such that the zero solution of the uncertain stochastic switched system is robustly stable.

()

{Ji}, i= 1, 2, x

Definition 2.2. The system of matrices . . . ,N, is said to be strictly complete if for every E Rn\{O} there is E {I, . . . ,N} such that < O. It is easy to see that the system { is strictly complete if and only if

i

2,

xTJix Jd

N

U (Xi= Rn\{O},

i=l where

(Xi= {x E Rn: XTJiX < O}, i= 1, 2, ...,N. Definition 2.3. The discrete-time system

(1) is robustly stable

*

where

Will= Q - P, Wi12= Sl - SlAi, Wi13= -SlBi' Wi22= P + Sl + sf + HTaHia + SlEibE�sf, Wi23= -SlBi' Wi33= -Q + 2H�Hib + 2pi2I, Ji= (d2 - ddQ - SlAi - ATsf + 2SlEiaETasf + SlEibE�sf + HTaHia + 2Pi1I, (Xi= {x E Rn: xTJix < O}, i= 1, 2, ...,N, i- I O!l= (Xl, O!i= (Xi \ U O!j, i= 2, 3, . . . ,N. j=l

(5)

The main result of this paper is summarized in the following theorem.

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Theorem 1. The uncertain stochastic switched system ( 1) is robustly stable in the mean square if there exist symmetric positive definite matrices P > 0, Q > ° and matrix SI satisfying the following conditions

Wi < 0,

(ii)

i=

0= -CJTx(k + 1) + CJT(Ai + EiaFia(k)Hia)x(k) +CJT(Bi + EibFib(k)Hib)X(k - d(k)) + CJTCJiw(k), we have

1, 2, ...,N. ,(x(k))

The switching rule is chosen as

x(k) E O'i.

i, whenever x

Proof Consider the following Lyapunov-Krasovskii functional for any ith system (1)

0.5x(k) [-Slx(k + 1) + Sl(Ai + EiaFia(k)Hia)x(k) +Sl(Bi + EibFib(k)Hib)X(k - d(k)) + SlCJiW(k)] °

[-CJTx(k + 1) + CJf(Ai + EiaFia(k)Hia)x(k) +CJT(Bi + EibFib(k)Hib)X(k - d(k)) + CJTCJiw(k)] Therefore, from (7) it follows that

where

k-l V1(k)= xT(k)Px(k), V2(k)= L xT(i)Qx(i), i=k-d(k) k-l L xT(I)Qx(I), j=-d2+21=k+j+l

We can verify that (6) >-Iilx(k)11 2 � V(k). Let us set �(k)= [x(k) x(k + 1) x(k - d(k)) w(k)]T, and

(� f � �} (f � � �) G

H �

�

Then, the difference of VI (k) along the solution of the system (1) and taking the mathematical expectation, we obtained

( r)

E[�Vl(k)]= E[xT(k + I)Px(k + 1) - xT(k)Px(k)] O.5 ) E[(T(k)H«k) _ 2(T(k)GT �

[.

E[�Vl(k)]= E[xT(k)[-P - SIAi - SIEiaFia(k)Hia - ATs[ - HTaFi�(k)EiaSflx(k) + 2XT(k)[SI - SIAi - SIEiaFia(k)Hia]x(k + 1) + 2xT(k)[-SIBi - SIEibFib(k)Hib]X(k - d(k)) + 2XT(k)[-SlCJi - CJTAi - CJTEiaFia(k)Hia]w(k) + x(k + 1)[SI + Sflx(k + 1) + 2x(k + 1)[-SlBi - Sl(EibFib(k)Hib)]x(k - d(k)) + 2x(k + 1)[CJT - SlCJi]W(k) + xT(k - d(k))[-CJTBi - CJTEibFib(k)Hib]W(k) + wT(k)[-2CJTCJi]w(k)], By asumption (3), we have E[�Vl(k)]= E[xT(k)[-P - SIAi - SIEiaFia(k)Hia - ATs[ - HTaFi�(k)EiaSflx(k) + 2xT(k)[SI - SIAi - SIEiaFia(k)Hia]x(k + 1) + 2xT(k)[-SlBi - SlEibFib(k)Hib]X(k - d(k)) + x(k + 1)[SI + Sflx(k + 1) + 2x(k + 1)[-SIBi - SIEibFib(k)Hib]X(k - d(k)) - 2CJTCJi], Applying Proposition 2.2, Proposition 2.3, condition assumption (4), the following estimations hold

(2) and

(7) because of

(�)

e(k)H�(k)= x(k + I)Px(k + 1), 0.5X(k) 2(T(k)d"

xT(k)Px(k). �

Using the expression of system (1)

0= -Slx(k + 1) + Sl(Ai + EiaFia(k)Hia)x(k) +Sl(Bi + EibFib(k)Hib)X(k - d(k)) + SlCJiW(k), 216

-SlEiaFia(k)Hia-HTaFi�(k)ETaS[ � SlEiaETaS[ +HTaHia, -2xT(k)SlEiaFia(k)HiaX(k + 1) � xT(k)SIEiaETaS[ x(k) + x(k + If HTaHiaX(k + 1), _2XT(k)SlEibFib(k)HibX(k - d(k)) � xT(k)SlEibE�S[ x(k) + x(k - d(k)f H�HibX(k - d(k)), _2xT(k + I)SIEibFib(k)HibX(k - d(k)) � xT(k+l)SIEibE�S[ x(k+l)+x(k-d(k)f H�HibX(k-d(k)), -CJT(x(k), x(k - d(k)), k)CJi(X(k), x(k - d(k)), k) �

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V3(k) is given by -d1+l k E[�V3(k)] E[ L L xT(l)Qx(l) j=-d2+2l=k+j -d1+l k-l L L xT(l)Qx(l)] j=-d2+2l=k+j+l -d1+l k-l E[ L [L xT(l)Qx(l) + xT(k)Q(�)x(k)

The difference of =

Therefore, we have

E[�Vl(k)] E[xT(k)[-P - SlAi - Af sf + 2SlEiaE� sf + SlEibE,£ sf + S2EiaE� SJ + H� Hia + 2pilI]X(k) + 2XT(k)[Sl - SlAi]X(k + 1) + 2XT(k)[-SlBi - S2Ai]X(k - d(k)) + x(k + l)[Sl + sf + H� Hia + SlEibE'£ Sfl x ( k + 1) + 2x(k + 1)[S2 - SlBi]X(k - d(k)) + xT(k - d(k))[2H'£Hib + 2pi2I]x(k - d(k))], =

=

k-l - L xT(l)Qx(l) l=k+j - xT(k + j -l)Qx(k + j - 1)]] -d1+l E[ L [xT(k)Qx(k) j=-d2+2 - xT(k + j -l)Qx(k + j - 1)]] E[(d2 - d I)xT(k)Qx(k) k-d1 L xT(j)Qx(j)]. j=k+l-d2

(8)

=

=

The difference of

V2(k) is given by

k xT(i)Qx(i) L i=k+l-d(k+l) k-l L xT(i)Qx(i)] i=k-d(k) k-d1 = E[ L xT(i)Qx(i) i=k+l-d(k+l) + xT(k)Qx(k) - xT(k - d(k))Qx(k - d(k)) k-l + L xT(i)Qx(i) i=k+l-d1 k-l L xT(i)Qx(i)]. i=k+l-d(k)

E[�V2(k)] E[ =

(11)

d(k):::; d2, and k-d1 k-d1 xT(i)Qx(i) L L xT(i)Qx(i):::; 0, i=k+l-d2 i=k=l-d(k+l)

Since

we obtain from (10) and (11) that

E[�V2(k) + �V3(k)]:::; E[(d2 - dl + l)xT(k)Qx(k) xT(k d(k))Qx(k d(k))]. _

E[�V(k)]:::; E[xT(k)Jix(k) + 'lj;T(k)W(Ij;(k)],

[

'Ij;(k) [x(k) x(k + 1) x(k - d(k))]T, Will Wi =

=

* *

k-l k-l L xT(i)Qx(i) - L xT(i)Qx(i) :::; 0, i=k+l-d1 i=k+l-d(k)

Will Wi12 Wi13 Wi22 Wi23 Wi33

and hence from (9) we have

(10)

(13)

where

d(k) ?: d l we have

k-d1 xT(i)Qx(i) E[�V2(k)]:::; E[ L i=k+l-d(k+l) + xT(k)Qx(k) - xT(k - d(k))Qx(k - d(k))].

(12)

_

Therefore, combining the inequalities (8), (12) gives

(9) Since

_

= = = = = =

*

Q - P, Sl - SlAi, -SlBi' P + Sl + Sf + H� Hia + SlEibE'£ Sf , -SlBi' -Q + 2H'£ Hib + 2pi2I,

and

Ji ( d2-d l) Q-SlAi-Af sf +2SlEiaE� sf +SlEibE,£ sf +H� Hia + 2PilI. =

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Therefore, we finally obtain from (13) and the condition (ii) that

E[�V(k)] < E[xT(k)Jix(k)], k 0, 1, 2, ....

Vi

=

1, 2, ....,N,

=

We now apply the condition (i) and Proposition 2.1., the system Ji is strictly complete, and the sets (Xi and ai by (5) are well defined such that N

U (Xi

i=l

=

Rn\{O},

N

U ai

i=l

=

Therefore, for any

Rn\{O},

=

=

0,i i= j.

E Rn, k = 1, 2, ..., there exists i E E ai' By choosing switching i whenever x(k) E ai, from the condition

x(k)

{I, 2, . . . ,N} such that rule as 1'(x(k)) (13) we have

ai n aj

[9] C.H. Lien, K.W. Yu, Y.J. Chung, YF . . Lin, L.Y. Chung and J.D. Chen, Exponential stability analysis for uncertain switched neutral systems with interval-time-varying state delay, Nonlinear Analysis: Hybrid systems, 3(2009),334-342. [10] G. Xie, L. Wang, Quadratic stability and stabilization of discrete-time switched systems with state delay, In: Proc. of the IEEE Conference on Decision and Control, Atlantics, December 2004, 3235-3240. [II] S. Boyd, L.E. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. [12] D.H. Ji, J.H. Park, w.J. Yoo and S.c. Won, Robust memory state feed­ back model predictive control for discrete-time uncertain state delayed systems, Appl. Math. Computation, 215(2009), 2035-2044. [13] G.S. Zhai, B. Hu, K. Yasuda, and A. Michel, Qualitative analysis of discrete- time switched systems. In: Proc. of the American Control Conference, 2002, 1880-1885. [14] W.A. Zhang, Li Yu, Stability analysis for discrete-time switched time­ delay systems, Automatica, 45(2009), 2265-2271. [15] F. Uhlig, A recurring theorem about pairs of quadratic forms and extensions, Linear Algebra Appl., 25(1979), 219-237. [16] R.P. Agarwal, Difference Equations and Inequalities, Second Edition, Marcel Dekker, New York, 2000.

x(k)

E[�V(k)]:::; E[xT(k)Jix(k)] < 0, k

=

1, 2, ...,

which, combining the condition (6), Definition 2.3 and the Lyapunov stability theorem [16], concludes the proof of the theorem in the mean square. IV.

CONCLUSION

This paper has proposed a switching design for the robust stability of uncertain stochastic switched discrete time-delay systems with interval time-varying delays. Based on the dis­ crete Lyapunov functional, a switching rule for the robust stability for the uncertain stochastic switched discrete time­ delay system is designed via linear matrix inequalities. ACKNOWLEDGE MENT

This work was supported by the Office of Agricultural Re­ search and Extension Maejo University Chiangmai Thailand, the Thailand Research Fund Grant, the Higher Education Com­ mission and Faculty of Science, Maejo University, Thailand. REFERENCES

[1] D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems, IEEE Control Syst. Mag., 19(1999), 57-70. [2] A.Y. Savkin and R.J. Evans, Hybrid Dynamical Systems: Controller and Sensor Switching Problems, Springer, New York, 2001. [3] Z. Sun and S.S. Ge, Switched Linear Systems: Control and Design, Springer, London, 2005. [4] VN. Phat, Y. Kongtham, and K. Ratchagit, LMI approach to exponential stability of linear systems with interval time-varying delays, Linear Algebra Appl., 436(2012), 243-251. [5] VN. Phat, K. Ratchagit, Stability and stabilization of switched linear discrete-time systems with interval time-varying delay, Nonlinear Analy­ sis: Hybrid Systems, 5(2011), 605-612. [6] K. Ratchagit, VN. Phat, Stability criterion for discrete-time systems, Journal of Inequalities and Applications, 2010(2010), 1-6. [7] K. Ratchagit, Asymptotic stability of nonlinear delay-difference system via matrix inequalities and application, International Journal of Compu­ tational Methods, 6(2009), 389-397. [8] F. Gao, S. Zhong and X. Gao, Delay-dependent stability of a type of linear switching systems with discrete and distributed time delays, Appl. Math. Computation, 196(2008), 24-39.

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