- Research
- Open Access
- Published:

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

*Advances in Difference Equations*
**volume 2012**, Article number: 28 (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.

## 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., [1–3] 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 [4–18]. 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 [19–24]. 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; *A* ≥ *B* means *A - B* ≥ 0. *λ*(*A*) denotes the set of all eigenvalues of *A; λ*_{min}(*A*) = min{*Re* λ: λ ∈ λ(*A*)}.

Consider a discrete systems with interval time-varying delay of the form

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:

**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

where

**Proposition 1** [12] *The system* {*J*_{
i
}}, *i =* 1,2,..., *N, is strictly complete if there exist* δ_{
i
}≥ 0,*i* =1,2,...,*N*, {\sum}_{i=1}^{N}{\delta}_{i}>0 *such that*

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

## Main results

Let us set

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*

\begin{array}{l}\left(i\right)\exists {\delta}_{i}\ge 0,i=1,2,\dots ,N,{\sum}_{i=1}^{N}{\delta}_{i}>0:{\sum}_{\text{i=1}}^{N}{\delta}_{\text{i}}{J}_{i}\left(R,Q\right)0.\phantom{\rule{2em}{0ex}}\\ \left(ii\right)\phantom{\rule{0.1em}{0ex}}{\mathcal{W}}_{i}\left(P,Q,R\right)0,\phantom{\rule{1em}{0ex}}i=1,2,\dots ,N.\phantom{\rule{2em}{0ex}}\end{array}

*The switching rule is chosen as* γ(*x*(*k*)) = *i, whenever* x\left(k\right)\in {\stackrel{\u0304}{\alpha}}_{i}.

*Proof*. Consider the following Lyapunov-Krasovskii functional for any *i* th system (1)

where

We can verify that

Let us set *ξ*(*k*) = [*x*(*k*) *x*(*k* + 1) *x*(*k - d*(*k*))]^{T}, and

Then, the difference of *V*_{1}(*k*) along the solution of the system is given by

because of

Using the expression of system (1)

we have

Therefore, from (3) it follows that

where

The difference of *V*_{2}(*k*) is given by

Since *d*(*k*) ≥ *d*_{1} we have

and hence from (6) we have

The difference of *V*_{3}(*k*) is given by

Since *d*(*k*) ≤ *d*_{2}, and

we obtain from (7) and (8) that

Therefore, combining the inequalities (5), (9) gives

where

Therefore, we finally obtain from (10) and the condition (ii) that

We now apply the condition (i) and Proposition 1, the system *J*_{
i
}(*R*, *Q*) is strictly complete, and the sets *α*_{
i
} and {\stackrel{\u0304}{\alpha}}_{i} by (2) are well defined such that

Therefore, for any *x*(*k*) ∈ *R*^{n}*, k* = 1,2,..., there exists *i* ∈ {1,2,..., *N*} such that x\left(k\right)\in {\stackrel{\u0304}{\alpha}}_{i}. By choosing switching rule as γ(*x*(*k*)) = *i* whenever x\left(k\right)\in {\stackrel{\u0304}{\alpha}}_{i}, from the condition (10) we have

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 [4–6] 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 *d*_{1} = 1,*d*_{2} = 4 and

By LMI toolbox of Matlab, we find that the conditions (i), (ii) of Theorem 1 are satisfied with δ_{1} *=* 0.1, *δ*_{2} = 0.2 and

In this case, we have

Moreover, the sum

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 α_{1} and α_{2} in Figure 1 and Figure 2 are given as

Obviously, the union of these sets is equal to *R*^{2} \ {0}. The switching regions are defined as

By Theorem 1 the system is asymptotically stable and the switching rule is chosen as γ(*x*(*k*)) = *i* whenever x\left(k\right)\in {\stackrel{\u0304}{\alpha}}_{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.

## References

Liberzon D, Morse AS: Basic problems in stability and design of switched systems.

*IEEE Control Syst Mag*1999, 19: 57–70.Savkin AV, Evans RJ: Hybrid Dynamical Systems: Controller and Sensor Switching Problems. In

*Springer*. New York; 2001.Ratchagit K: Asymptotic stability of nonlinear delay-difference system via matrix inequalities and application.

*International Journal of Computational Methods*2009, 389–397.Sun Z, Ge SS: Switched Linear Systems: Control and Design. In

*Springer*. London; 2005.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.2011.07.016Gao 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.2007.05.053Lien 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.2009.02.010Phat 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.2011.05.006Ratchagit K, Phat VN: Stability criterion for discrete-time systems.

*Journal of Inequalities and Applications*2010, 10: 1–6.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.Boyd S, Ghaoui LE, Feron E, Balakrishnan V: Linear Matrix Inequalities in System and Control Theory. In

*SIAM*. Philadelphia; 1994.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.2009.07.052Zhai GS, Hu B, Yasuda K, Michel A: Qualitative analysis of discrete-time switched systems.

*In: Proc of the American Control Conference*2002, 1880–1885.Zhang WA, Li Yu: Stability analysis for discrete-time switched time-delay systems.

*Automatica*2009, 45: 2265–2271. 10.1016/j.automatica.2009.05.027Uhlig F: A recurring theorem about pairs of quadratic forms and extensions.

*Linear Algebra Appl*1979, 25: 219–237.Agarwal RP:

*Difference Equations and Inequalities*. Second edition. Marcel Dekker, New York; 2000.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 pagesDiblik 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 pagesDiblik J, Khusainov DYa, Ruzickova : Solutions of Discrete equations with prescrid asymptotic behavior.

*Dynamic Systems and Applications*2004, 4: 395–402.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.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 pagesDiblik 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.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.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.2008.11.027

## Acknowledgements

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

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Competing interests

The author declares that they have no competing interests.

### Authors' contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

## Authors’ original submitted files for images

Below are the links to the authors’ original submitted files for images.

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Sangapate, P. New sufficient conditions for the asymptotic stability of discrete time-delay systems.
*Adv Differ Equ* **2012, **28 (2012). https://doi.org/10.1186/1687-1847-2012-28

Received:

Accepted:

Published:

DOI: https://doi.org/10.1186/1687-1847-2012-28

### Keywords

- Switching design
- discrete system
- asymptotic stability
- Lyapunov function
- linear matrix inequality