Discrete inequalities based on multiple auxiliary functions and their applications to stability analysis of time-delay systems

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Abstract

This paper presents new discrete inequalities for single summation and double summation. These inequalities are based on multiple auxiliary functions and include the Jensen discrete inequality and the discrete Wirtinger-based inequality as special cases. An application of these discrete inequalities to analyze stability of linear discrete systems with an interval time-varying delay is studied and a less conservative stability condition is obtained. Three numerical examples are given to show the effectiveness of the obtained stability condition.

Introduction

Recently, there is a growing interest in extending the Jensen inequality and its applications to stability analysis and stabilization of time-delay systems (see, [1], [2], [3], [4], [5], [6], [7] and the references therein). For continuous-time systems, Liu and Fridman [1] first introduced a new integral inequality which encompasses the Jensen inequality [8] for a single integral and they referred to it as the Wirtinger inequality. Later, Seuret and Gouaisbaut reported further developed inequalities, namely Wirtinger-based inequality and Bessel–Legendre inequality, for a single integral [2], [3], [4]. Very recently, Park et al. proposed some generalized inequalities, which are based on some auxiliary functions, for both single integral and double integral [5]. On the other hand, for discrete-time systems, so far, there is only one discrete version of the Wirtinger-based inequality available for a single summation. This inequality was recently and simultaneously reported in [6], [7], [16]. However, to the best our knowledge, there has been no result reported on any extended Jensen discrete inequality for a double summation. This is therefore one of the main motivations of our paper.

In this paper, our main objectives are: (i) to derive new discrete inequalities for both single summation and double summation; and (ii) to apply the newly derived inequalities to analyze stability of linear discrete systems with an interval time-varying delay. Firstly, by adopting the method recently reported in [5] and using the convexity of a quadratic function, we derive new discrete inequalities, which are based on multiple auxiliary functions, for both single summation and double summation. We show that some existing discrete inequalities including various Jensen discrete inequalities [9], [10], [11] and the discrete Wirtinger-based inequality reported recently in [6], [7], [16] are the special cases of our newly derived discrete inequalities. Secondly, by combining these derived inequalities with the Lyapunov method, we obtain a new stability criterion for linear discrete-time systems with an interval time-varying delay. Lastly, three numerical examples with comparison to the most improved and recent results in the literature are given to show the effectiveness of the obtained stability condition.

Section snippets

New discrete inequalities

In this section, we derive new discrete inequalities for both single summation and double summation. The following lemma will be used in the derivation of our new inequalities.

Lemma 1

Boyd and Vandenberghe [12, Chapter 3]

For a given function f:RnR, the following statements are equivalent:

  • (i)

    f is convex,

  • (ii)

    f(y)f(x)+f(x)T(yx),x,yRn,

  • (iii)

    2f(x)0,xRn.

We now give a discrete inequality for a single summation as follows.

Lemma 2

For a given positive-definite n×n-matrix R, three given non-negative integers a, b, k satisfying a<bk, a vector function x(·)Rn

Applications to stability analysis of time-delay systems

In this section, we combine the newly derived inequalities (48), (49), (50), (52) with the Lyapunov–Krasovskii method to derive a new result for stability of the following discrete-time system:x(k+1)=Ax(k)+A1x(kh(k)),k0,x(k)=?(k),k=hM,,0,where x(k)Rn is the state vector, A, A1 are known constant n×n-matrices, the time-varying delay h(k) is assumed to satisfy 0<hmh(k)hM, where hm and hM are known integers.

The reciprocally convex combination inequality [2], [13] and Finsler׳s Lemma [14]

Numerical examples

In this section, to show the effectiveness of our newly derived stability condition, we consider three numerical examples and compare our result with the most improved and the recent results in the literature.

Example 1

Consider system (53) with A=[10.010.021.001],A1=[000.010],and delay h(k) is considered for the following two cases: (i) h(k)=h is a constant; and (ii) h(k) varies within an interval [hm,hM].

For case (i), the delay ranges obtained by Theorem 1, Remark 6 and the most recent methods are

Conclusion

This paper has presented three new discrete inequalities based on multiple auxiliary functionals for single summation and double summation. These inequalities have vast potential applications in stability analysis and stabilization of discrete time-delay systems. A new stability criterion obtained by utilizing these inequalities for linear discrete systems with an interval time-varying delay has been derived. Three numerical examples with comparison to the most recent results have been studied

Acknowledgments

The authors would like to thank the Chief Editor, Associate Editor and the anonymous Reviewers for their careful reading and useful comments that help us improve the paper. The authors would like to thank Professors Chen Peng, Jin Zhang, Zhiguang Feng, Engang Tian for providing their Matlab codes, which have helped us to create Table 2, Table 3, Table 4. This work was supported by the Australian Research Council under the Discovery Grant DP130101532 and NAFOSTED, Vietnam under Grant

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