Abstract
This paper focuses on delay-dependent stability analysis for systems with interval time-varying delays. Based on a new integral inequality and a generalized reciprocally convex combination matrix inequality, a new delay-dependent stability criterion is obtained in terms of a linear matrix inequality (LMI). Finally, the merits of the proposed criterion are shown by two numerical examples.
1. Introduction
Consider the following systems with interval time-varying delays:where is the state vector and are constant matrices. The time-varying delay is continuous and satisfies
Over the past decades, providing less conservative stability conditions for linear systems with time-varying delays has attracted considerable attention. The difficulty relies on the handling of the integral terms arising in the derivative of the LKF. The free-weighting-matrix approach [1, 2] was applied to handle the integral terms in the early literature. In recent years, estimating integral terms directly via integral inequalities gradually becomes more popular. Various integral inequalities have been proposed, such as Jensen inequality [3–5], Wirtinger-based inequality [6–8], auxiliary function based inequalities [9], free-matrix-based inequalities [10], and relaxed integral inequalities [11]. Very recently, Bessel–Legendre inequality [12] is proposed to handle the stability of system (1), and some less conservative criteria are obtained. However, the relationship between and , , , was not considered in [12], which may yield conservative results. Then, a new integral inequality was proposed in [13] to consider the relationship fully. But the integral inequality was only used to handle the constant time delay. A new integral inequality for dealing with delays is introduced in [14]. A less conservative stability criterion for linear systems with a time-varying delay is proposed by using the new integral inequality. However, there are two aspects which need to be improved. (1) When estimating the derivative of , the term is only estimated as , which may yield conservative results. (2) The assumption on the derivative of time-varying delay is included in the stability criterion [14]. Thus, there is still some room for further investigation.
In this paper, a new delay-dependent stability criterion for linear systems with interval time-varying delays is developed by using a new integral inequality and a generalized reciprocally convex combination matrix inequality. The highlights of our paper are as follows. (1) The features of the new integral inequality in [13] are fully integrated into the construction of the LKF. (2) A less conservative stability criterion is proposed in terms of an LMI without the assumption on the derivative of time-varying delay . (3) Upper bound of in our paper is quite close to the analytical bound. The advantage of the proposed criterion has been illustrated via two numerical examples.
Throughout the paper, the set denotes the set of symmetric positive definite matrices and the set denotes the set of symmetric matrices. For any square matrix A, define .
2. Main Result
Based on the following lemmas, a less conservative stability criterion for systems with interval time-varying delays is established.
Lemma 1. (see [8]). For any matrices , , , , the following inequality holds:where
Lemma 2. (see [8]). Consider a parameter dependent matrix , such that the convex inequalityholds for all . If there exist a matrix and two matrices , such that the inequalityholds for , then the following inequality holds:
Lemma 3. (see [13]). For a matrix and any continuously differentiable function , the following inequality holds:where
Theorem 1. For given scalars , system (1) with time-varying delays satisfying (2) is asymptotically stable if there exist matrices , , , such that the LMI holds for , whereand is defined as for
Proof. Consider a LKF candidate given bywhereCalculate the derivative of along the solution of system (1) as follows:Then, it can be rewritten aswhereLet , and applying Lemma 3 yields Let , then , for any matrices , and applying Lemma 1 yieldswhereFrom (16)–(20), the can be estimated asAccording to Lemma 2, if LMI (11) is verified for , then the inequality holds for all . This completes the proof.
Remark 1. The integral inequality in Lemma 3 is proposed to derive less conservative results because of the term of a triple integral being included. In order to fully consider the features of the integral inequality in Lemma 3, the LKF of our paper includes the terms of a triple integral , which may yield less conservative results.
3. Numerical Examples
In this section, two numerical examples are introduced to illustrate the advantage of the proposed criterion.
Example 1. Consider system (1) withTable 1 presents the maximum upper bounds of with respect to different calculated by Theorem 1 and several criteria from [4, 6, 9, 10, 12]. Table 1 shows that our method is more effective than those in [4, 6, 9, 10, 12]. It should be pointed out that our result is quite close to the analytical bound. For , initial state , the simulation of the state trajectories of system (1) is given in Figure 1.
Example 2. Consider system (1) withTable 2 presents the maximum upper bounds of with respect to different calculated by Theorem 1 and several criteria from [6, 9, 10, 12]. Table 2 shows that our method is more effective than those in [6, 9, 10, 12]. For , initial state , the simulation of the state trajectories of system (1) is given in Figure 2.
4. Conclusions
An improved stability criterion of systems with interval time-varying delays has been proposed in our paper. The features of the new integral inequality are fully integrated into the construction of the LKF. Finally, the merits of the proposed criterion are shown by two numerical examples.
Data Availability
No additional data are available for this paper.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the High-Level Innovative Talents in Guizhou Province (Zun shi ke he rencai[2016]13); Youth Science and Technology Talents Development Project of Education Department of Guizhou Province (Qian jiao he KY zi [2017]256, KY zi [2017]255); Major Research Projects of Innovative Groups of Education Department of Guizhou Province (Qian jiao he KY [2016]046); New Academic Talents and Innovation Exploration Project of Zunyi Normal College (Qian ke he pingtai rencai[2017]5727-19); and Innovation and Entrepreneurship Training Program for College Students of Guizhou Province (2018520891).