Stability Analysis of Systems with Interval Time-Varying Delays via a New Integral Inequality

Over the past decades, providing less conservative stability conditions for linear systems with time-varying delays has attracted considerable attention. -e difficulty relies on the handling of the integral terms arising in the derivative of the LKF. -e 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


Introduction
Consider the following systems with interval time-varying delays: where x(t) ∈ R n is the state vector and A, B, ∈ R n×n are constant matrices. e time-varying delay h(t) is continuous and satisfies Over the past decades, providing less conservative stability conditions for linear systems with time-varying delays has attracted considerable attention. e difficulty relies on the handling of the integral terms arising in the derivative of the LKF. e 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][4][5], Wirtinger-based inequality [6][7][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 x(s)ds du 1 , x(s)ds du k . . . du 1 was not considered in [12], which may yield conservative results. en, 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 timevarying 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 V(x t ), the 3 RΩ 3 , which may yield conservative results. (2) e assumption on the derivative of time-varying delay h(t) is included in the stability criterion [14]. us, 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. e highlights of our paper are as follows. (1) e 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 h(t). (3) Upper bound of h 2 in our paper is quite close to the analytical bound. e advantage of the proposed criterion has been illustrated via two numerical examples. roughout the paper, the set S n + denotes the set of n × n symmetric positive definite matrices and the set S n denotes the set of n × n symmetric matrices. For any square matrix A, define He(A) � A + A T .

Main Result
Based on the following lemmas, a less conservative stability criterion for systems with interval time-varying delays is established.

Proof. Consider a LKF candidate given by
Calculate the derivative of V(x t ) along the solution of system (1) as follows: en, it can be rewritten as where

Complexity
Let α � (h(t) − h 1 /h 12 ), and applying Lemma 3 yields , for any matrices N 1 , N 2 ∈ R 13n×4n , and applying Lemma 1 yields where According to Lemma 2, if LMI (11) is verified for α � 0, 1 { }, then the inequality Φ(α) + Ξ(α) < 0 holds for all α ∈ (0, 1). is completes the proof. e 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

Numerical Examples
In this section, two numerical examples are introduced to illustrate the advantage of the proposed criterion.  [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 h 2 � 3.45, initial state (− 0.02, 0.02) T , the simulation of the state trajectories of system (1) is given in Figure 1.   [6,9,10,12]. Table 2 shows that our method is Complexity more effective than those in [6,9,10,12]. For h 2 � 4.16, initial state (− 0.01, 0.01) T , the simulation of the state trajectories of system (1) is given in Figure 2.

Conclusions
An improved stability criterion of systems with interval time-varying delays has been proposed in our paper. e 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
e authors declare that there are no conflicts of interest regarding the publication of this paper.