Open Access

New sufficient conditions for the asymptotic stability of discrete time-delay systems

Advances in Difference Equations20122012:28

DOI: 10.1186/1687-1847-2012-28

Received: 26 September 2011

Accepted: 8 March 2012

Published: 8 March 2012

Abstract

This paper is concerned with asymptotic stability of 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 asymptotic stability for the system is designed via linear matrix inequalities. Numerical example is included to illustrate the effectiveness of the result.

Keywords

Switching design discrete system asymptotic stability Lyapunov function linear matrix inequality

Introduction

As an important class of hybrid systems, switched systems arise in many practical processes that cannot be described by exclusively continuous or exclusively discrete models, such as manufacturing, communication networks, automotive engineering control and chemical processes (see, e.g., [13] and the references therein). On the other hand, time-delay phenomena are very common in practical systems. A switched system with time-delay individual subsystems is called a switched time-delay system; in particular, when the subsystems are linear, it is then called a switched time-delay linear system. During the last decades, the stability analysis of switched linear continuous/discrete time-delay systems has attracted a lot of attention [418]. The main approach for stability analysis relies on the use of Lyapunov-Krasovskii functionals and linear matrix inequlity (LMI) approach for constructing a common Lyapunov function [1924]. 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. It was shown in [5, 7, 11] that when all subsystems are asymptotically stable, the switching system is asymptotically stable under an arbitrary switching rule. The asymptotic stability for switching linear discrete time-delay systems has been studied in [10], but the result was limited to constant delays. In [11], 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 asymptotic stability problem for switched linear discrete systems with interval time-varying delays. Specifically, our goal is to develop a constructive way to design switching rule to asymptotically stabilize the system. By using improved Lyapunov-Krasovskii functionals combined with LMIs technique, we propose new criteria for the asymptotic stability of the 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 stbility in terms of of LMIs, which can be solvable by utilizing Matlab's LMI Control Toolbox available in the literature to date.

The paper is organized as follows: Section 2 presents definitions and some well-known technical propositions needed for the proof of the main results. Switching rule for the asymptotic stability is presented in Section 3. Numerical example of the result is given in Section 4.

Preliminaries

The following notations will be used throughout this paper. R+ denotes the set of all real non-negative numbers; R n denotes the n-dimensional space with the scalar product of two vectors 〈x,y〉 or x T y; R n×r denotes the space of all matrices of (n × r)- dimension. A T denotes the transpose of A; a matrix A is symmetric if A = A T .

Matrix A is semi-positive definite (A ≥ 0) if 〈Ax,x〉 ≥ 0, for all x R n ; A is positive definite (A > 0) if 〈Ax, x〉 > 0 for all x ≠ 0; AB means A - B ≥ 0. λ(A) denotes the set of all eigenvalues of A; λmin(A) = min{Re λ: λ λ(A)}.

Consider a discrete systems with interval time-varying delay of the form
x ( k + 1 ) = A γ x ( k ) + B γ x ( k - d ( k ) ) , k = 0 , 1 , 2 , x ( k ) = υ k , k = - d 2 , - d 2 + 1 , , 0 ,
(1)
where x(k) R n is the state, γ(.): R n N:= {1,2,...,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, γ(x(k)) = i implies that the system realization is chosen as the i th system, i = 1,2,..., N. It is seen that the system (1) can be viewed as an autonomous switched system in which the effective subsystem changes when the state x(k) hits predefined boundaries. A i , B i , i = 1,2, ..., N are given constant matrices. The time-varying function d(k) satisfies the following condition:
0 < d 1 d ( k ) d 2 , k = 0 , 1 , 2 ,

Remark 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 1 The switched system (1) is asymptotically stable if there exists a switching function γ(.) such that the zero solution of the system is asymptotically stable.

Definition 2 The system of matrices {J i }, i = 1,2,..., N, is said to be strictly complete if for every x R n \{0} there is i {1, 2,..., N} such that x T J i x < 0.

It is easy to see that the system {J i } is strictly complete if and only if
i = 1 N α i = R n \ { 0 } ,
where
α i = { x R n : x T J i x < 0 } , i = 1 , 2 , , N .
Proposition 1 [12] The system {J i }, i = 1,2,..., N, is strictly complete if there exist δ i ≥ 0,i =1,2,...,N, i = 1 N δ i > 0 such that
i = 1 N δ i J i < 0 .

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

Main results

Let us set
W i ( P , Q , R ) = Q - P R T - A i T R - R T B i R - R T A i P + R + R T - R T B i - B i T R - B i T R - Q , J i ( P , Q ) = ( d 2 - d 1 ) Q - R T A i - A i T R , λ 1 = λ min ( P ) , α i = { x R n : x T J i ( R , Q ) x < 0 } , i = 1 , 2 , , N , α ̄ 1 = α 1 , α ̄ i = α i \ j = 1 i - 1 α ̄ j , i = 2 , 3 , , N .
(2)

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

Theorem 1 The switched system (1) is asymptotically stable if there exist symmetric positive definite matrices P > 0, Q > 0 and matrix R satisfying the following conditions

( i ) δ i 0 , i = 1 , 2 , , N , i = 1 N δ i > 0 : i = 1 N δ i J i ( R , Q ) < 0 . ( i i ) W i ( P , Q , R ) < 0 , i = 1 , 2 , , N .

The switching rule is chosen as γ(x(k)) = i, whenever x ( k ) α ̄ i .

Proof. Consider the following Lyapunov-Krasovskii functional for any i th system (1)
V ( k ) = V 1 ( k ) + V 2 ( k ) + V 3 ( k ) ,
where
V 1 ( k ) = x T ( k ) P x ( k ) , V 2 ( k ) = i = k - d ( k ) k - 1 x T ( i ) Q x ( i ) , V 3 ( k ) = j = - d 2 + 2 - d 1 + 1 l = k + j k - 1 x T ( l ) Q x ( l ) ,
We can verify that
λ 1 x ( k ) 2 V ( k ) .
(3)
Let us set ξ(k) = [x(k) x(k + 1) x(k - d(k))] T , and
H = 0 0 0 0 P 0 0 0 0 , G = P 0 0 R R 0 R 0 I .
Then, the difference of V1(k) along the solution of the system is given by
Δ V 1 ( k ) = x T ( k + 1 ) P x ( k + 1 ) - x T ( k ) P x ( k ) = ξ T ( k ) H ξ ( k ) - 2 ξ T ( k ) G T 0 . 5 x ( k ) 0 0 .
(4)
because of
ξ T ( k ) H ξ ( k ) = x ( k + 1 ) P x ( k + 1 ) .
Using the expression of system (1)
0 = - x ( k + 1 ) + A i ( k ) + B i x ( k - d ( k ) ) ,
we have
2 ξ T ( k ) G T ( x ( k + 1 ) + A i x ( k ) + 0 0.5 x ( k ) B i x ( k d ( k ) ) ) ξ ( k ) = ξ T ( k ) G T ( 0.5 I 0 0 A i I B i 0 0 0 ) ξ ( k ) ξ T ( k ) ( 0.5 I A i T 0 0 I 0 0 B i T 0 ) G ξ ( k ) .
Therefore, from (3) it follows that
Δ V 1 ( k ) = ξ T ( k ) W i ξ ( k ) ,
(5)
where
W i = 0 0 0 0 P 0 0 0 0 - G T 0 . 5 I 0 0 A i - I B i 0 0 0 - 0 . 5 I A i T 0 0 - I 0 0 B i T 0 G .
The difference of V2(k) is given by
Δ V 2 ( k ) = i = k + 1 - d ( k + 1 ) k x T ( i ) Q x ( i ) - i = k - d ( k ) k - 1 x T ( i ) Q x ( i ) = i = k + 1 - d ( k + 1 ) k - d 1 x T ( i ) Q x ( i ) + x T ( k ) Q x ( k ) - x T ( k - d ( k ) ) Q x ( k - d ( k ) ) + i = k + 1 - d 1 k - 1 x T ( i ) Q x ( i ) - i = k + 1 - d ( k ) k - 1 x T ( i ) Q x ( i ) .
(6)
Since d(k) ≥ d1 we have
i = k + 1 - d 1 k - 1 x T ( i ) Q x ( i ) - i = k + 1 - d ( k ) k - 1 x T ( i ) Q x ( i ) 0 ,
and hence from (6) we have
Δ V 2 ( k ) i = k + 1 - d ( k + 1 ) k - d 1 x T ( i ) Q x ( i ) + x T ( k ) Q x ( k ) - x T ( k - d ( k ) ) Q x ( k - d ( k ) ) .
(7)
The difference of V3(k) is given by
Δ V 3 ( k ) = j = d 2 + 2 d 1 + 1 l = k + j + 1 k x T ( l ) Q x ( l ) j = d 2 + 2 d 1 + 1 l = k + j k 1 x T ( l ) Q x ( l ) = j = d 2 + 2 d 1 + 1 [ l = k + j k 1 x T ( l ) Q x ( l ) + x T ( k ) Q ( ξ ) x ( k ) l = k + j k 1 x T ( l ) Q x ( l ) x T ( k + j 1 ) Q x ( k + j 1 ) ] = j = d 2 + 2 d 1 + 1 [ x T ( k ) Q x ( k ) x T ( k + j 1 ) Q x ( k + j 1 ) ] = ( d 2 d 1 ) x T ( k ) Q x ( k ) j = k + 1 d 2 k d 1 x T ( j ) Q x ( j ) .
(8)
Since d(k) ≤ d2, and
i = k + 1 - d ( k + 1 ) k - d 1 x T ( i ) Q x ( i ) - i = k + 1 - d 2 k - d 1 x T ( i ) Q x ( i ) 0 ,
we obtain from (7) and (8) that
Δ V 2 ( k ) + Δ V 3 ( k ) ( d 2 - d 1 + 1 ) x T ( k ) Q x ( k ) - x T ( k - d ( k ) ) Q x ( k - d ( k ) ) .
(9)
Therefore, combining the inequalities (5), (9) gives
Δ V ( k ) x T ( k ) J i ( P , Q ) x ( k ) + ξ T ( k ) W i ( P , Q , R ) ξ ( k ) ,
(10)
where
W i ( P , Q , R ) = Q - P R T - A i T R - R T B i R - R T A i P + R + R T - R T B i - B i T R - B i T R - Q .
Therefore, we finally obtain from (10) and the condition (ii) that
Δ V ( k ) < x T ( k ) J i ( R , Q ) x ( k ) , i = 1 , 2 , , N , k = 0 , 1 , 2 , .
We now apply the condition (i) and Proposition 1, the system J i (R, Q) is strictly complete, and the sets α i and α ̄ i by (2) are well defined such that
i = 1 N α i = R n \ { 0 } , i = 1 N α ̄ i = R n \ { 0 } , α ̄ i α ̄ j = , i j .
Therefore, for any x(k) R n , k = 1,2,..., there exists i {1,2,..., N} such that x ( k ) α ̄ i . By choosing switching rule as γ(x(k)) = i whenever x ( k ) α ̄ i , from the condition (10) we have
Δ V ( k ) x T ( k ) J i ( R , Q ) x ( k ) < 0 , k = 1 , 2 , ,

which, combining the condition (3) and the Lyapunov stability theorem [12], concludes the proof of the theorem.

Remark 2 Note that the resuts proposed in [46] for switching systems to be asymptotically stable under an arbitrary switching rule. The asymptotic stability for switching linear discrete time-delay systems studied in [9] was limited to constant delays. In [10], a class of switching signals has been identified for the considered switched discrete-time delay systems to be stable under the averaged well time scheme.

Numerical example

Example 1 Consider the switched discrete-time system (1), where d1 = 1,d2 = 4 and
( A 1 , B 1 ) = - 0 . 1 0 . 01 0 . 02 - 0 . 2 , - 0 . 1 0 . 01 0 . 02 - 0 . 3 , ( A 2 , B 2 ) = 1 0 . 2 0 . 1 2 , 0 . 1 0 . 02 0 . 01 0 . 2 .
By LMI toolbox of Matlab, we find that the conditions (i), (ii) of Theorem 1 are satisfied with δ1 = 0.1, δ2 = 0.2 and
P = 0 . 7339 0 . 0006 0 . 0006 0 . 8383 , Q = 0 . 2817 0 . 0138 0 . 0138 0 . 3773 , R = - 0 . 9091 - 0 . 1261 0 . 1311 - 0 . 9935 .
In this case, we have
( J 1 ( R , Q ) , J 2 ( R , Q ) ) = - 0 . 9918 - 0 . 0175 - 0 . 0175 - 1 . 4672 , - 0 . 6228 - 0 . 1910 - 0 . 1910 - 2 . 2217 .
Moreover, the sum
δ 1 J 1 ( R , Q ) + δ 2 J 2 ( R , Q ) = - 0 . 2238 - 0 . 0400 - 0 . 0400 - 0 . 5911
is negative definite; i.e. the first entry in the first row and the first column -0.2238 0 is negative and the determinant of the matrix is positive. The sets α1 and α2 in Figure 1 and Figure 2 are given as
α 1 = { ( x 1 , x 2 ) : - 0 . 9918 x 1 2 - 0 . 035 x 1 x 2 - 0 . 1 . 4672 x 2 2 < 0 } , α 2 = { ( x 1 , x 2 ) : 0 . 6228 x 1 2 + 0 . 382 x 1 x 2 + 2 . 2217 x 2 2 > 0 } .
https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-28/MediaObjects/13662_2011_Article_138_Fig1_HTML.jpg
Figure 1

Region α 1 .

https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-28/MediaObjects/13662_2011_Article_138_Fig2_HTML.jpg
Figure 2

Region α 2 .

Obviously, the union of these sets is equal to R2 \ {0}. The switching regions are defined as
α ̄ 1 = { ( x 1 , x 2 ) : - 0 . 9918 x 1 2 - 0 . 035 x 1 x 2 - 0 . 1 . 4672 x 2 2 < 0 } α ̄ 2 = α 2 \ α ̄ 1

By Theorem 1 the system is asymptotically stable and the switching rule is chosen as γ(x(k)) = i whenever x ( k ) α ̄ i .

Conclusion

This paper has proposed a switching design for the asymptotic stability of switched linear discrete-time systems with interval time-varying delays. Based on the discrete Lyapunov functional, a switching rule for the asymptotic stability for the system is designed via linear matrix inequalities.

Declarations

Acknowledgements

This work was supported by the Thai Research Fund Grant, the Higher Education Commission and Faculty of Science, Maejo University, Thailand.

Authors’ Affiliations

(1)
Department of Mathematics, Maejo University

References

  1. Liberzon D, Morse AS: Basic problems in stability and design of switched systems. IEEE Control Syst Mag 1999, 19: 57–70.View ArticleGoogle Scholar
  2. Savkin AV, Evans RJ: Hybrid Dynamical Systems: Controller and Sensor Switching Problems. In Springer. New York; 2001.Google Scholar
  3. Ratchagit K: Asymptotic stability of nonlinear delay-difference system via matrix inequalities and application. International Journal of Computational Methods 2009, 389–397.Google Scholar
  4. Sun Z, Ge SS: Switched Linear Systems: Control and Design. In Springer. London; 2005.View ArticleGoogle Scholar
  5. Phat VN, Kongtham Y, Ratchagit K: LMI approach to exponential stability of linear systems with interval time-varying delays. Linear Algebra Appl 2012, 436: 243–251. 10.1016/j.laa.2011.07.016MATHMathSciNetView ArticleGoogle Scholar
  6. Gao F, Zhong S, Gao X: Delay-dependent stability of a type of linear switching systems with discrete and distributed time delays. Appl Math Computation 2008, 196: 24–39. 10.1016/j.amc.2007.05.053MATHMathSciNetView ArticleGoogle Scholar
  7. Lien CH, Yu KW, Chung YJ, Lin YF, Chung LY, Chen JD: Exponential stability analysis for uncertain switched neutral systems with interval-time-varying state delay. Nonlinear Analysis: Hybrid systems 2009, 3: 334–342. 10.1016/j.nahs.2009.02.010MATHMathSciNetGoogle Scholar
  8. Phat VN, Ratchagit K: Stability and stabilization of switched linear discrete-time systems with interval time-varying delay. Nonlinear Analysis: Hybrid Systems 2011, 5: 605–612. 10.1016/j.nahs.2011.05.006MATHMathSciNetGoogle Scholar
  9. Ratchagit K, Phat VN: Stability criterion for discrete-time systems. Journal of Inequalities and Applications 2010, 10: 1–6.MathSciNetView ArticleGoogle Scholar
  10. Xie G, Wang L: Quadratic stability and stabilization of discrete-time switched systems with state delay. In Atlantics. Proc of the IEEE Conference on Decision and Control; 2004:3235–3240.Google Scholar
  11. Boyd S, Ghaoui LE, Feron E, Balakrishnan V: Linear Matrix Inequalities in System and Control Theory. In SIAM. Philadelphia; 1994.View ArticleGoogle Scholar
  12. Ji DH, Park JH, Yoo WJ, Won SC: Robust memory state feedback model predictive control for discrete-time uncertain state delayed systems. Appl Math Computation 2009, 215: 2035–2044. 10.1016/j.amc.2009.07.052MATHMathSciNetView ArticleGoogle Scholar
  13. Zhai GS, Hu B, Yasuda K, Michel A: Qualitative analysis of discrete-time switched systems. In: Proc of the American Control Conference 2002, 1880–1885.Google Scholar
  14. Zhang WA, Li Yu: Stability analysis for discrete-time switched time-delay systems. Automatica 2009, 45: 2265–2271. 10.1016/j.automatica.2009.05.027MATHView ArticleGoogle Scholar
  15. Uhlig F: A recurring theorem about pairs of quadratic forms and extensions. Linear Algebra Appl 1979, 25: 219–237.MATHMathSciNetView ArticleGoogle Scholar
  16. Agarwal RP: Difference Equations and Inequalities. Second edition. Marcel Dekker, New York; 2000.MATHGoogle Scholar
  17. Khusainov DYa, Diblik J, Svoboda Z, Smarda Z: Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone. Abstract and Applied Analysis 2011. Article ID 154916, 23 pagesGoogle Scholar
  18. Diblik J, Khusainov DYa, Grytsay IV, Smarda Z: Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case. Discrete Dynamics in Nature and Societe 2010. Article ID 539087, 23 pagesGoogle Scholar
  19. Diblik J, Khusainov DYa, Ruzickova : Solutions of Discrete equations with prescrid asymptotic behavior. Dynamic Systems and Applications 2004, 4: 395–402.MathSciNetGoogle Scholar
  20. Diblik J, Khusainov DYa, Grytsay Irina V: Stability Investigation of Nonlinear Quadratic Discrete Dynamics Systems in the Critical Case. In International Symposium on Nonlinear Dynamics, Journal of Physics: Conference Series 96. IOP Publishing; 2008.Google Scholar
  21. Bastinec J, Diblik J, Khusainov DYa, Ryvolova A: Exponential Stability and Estimation of Solutions of Linear Differential Systems of Neutral Type with Constant Coefficients. Boundary Value Problems 2010. Article ID 956121, 20 pagesGoogle Scholar
  22. Diblik J, Hlavickova I: Combination of Liapunov and Retract Methods in the Investigation of the Asymptotic Behavior of Solutions of Systems of Discrete Equations. Dynamic Systems and Applications 2009, 18: 507–538.MATHMathSciNetGoogle Scholar
  23. Bastinec J, Diblik J, Smarda Z: Existence of positive solutions of discrete linear equations with a single delay. Journal of Diference Equations and Applications 2010, 9: 1047–1056.MathSciNetView ArticleGoogle Scholar
  24. Diblik J, Ruzickova M, Smarda Z: Wazewskis method for systems of dynamic equations on time scales. Nonlinear Analysis, Theory, Methods and Applications 2009, 71: 1124–1131. 10.1016/j.na.2008.11.027MathSciNetView ArticleGoogle Scholar

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