Stability Analysis of Systems With Time-Varying Delays via an Improved Integral Inequality

This paper is concerned with a new stability criterion for systems with time-varying delays. Firstly, a generalized integral inequality is proposed, which includes some existing inequalities as special cases. Then, a new stability criterion is derived by choosing some new Lyapunov-Krasovskii functionals. Finally, the effectiveness of our method is shown by a numerical example.


I. INTRODUCTION
Over the past few decades, stalility analysis has been one of hot issues for many dynamic systems such as delayed differential systems [1], nonlinear stochastic networked control systems [2], time-delay systems [3]. For the stability of time-delay systems, the LKF method has been widely used to get stability results by LMI [4]. Choosing LKF [5] and estimating the derivative [6]- [10] are the main factors in leading to conservatism. Therefore, how to establish effective integral inequality techniques for this estimation becomes an important task to get less conservative results for the systems with time-varying delays. The Jensen inequality [11] has been wildly used to estimate the bound of the single integral term although it may introduce undesirable conservatism. Recently, to overcome the conservatism, the Wirtinger-based inequality was introduced in [12]. Then, further improvements were proposed by using a free matrix-based integral inequality [13]. More recently, based on Legendre polynomials, some new integral inequalities were derived in [14], which include Jensen and Wirtingerbased inequalities and also the recent inequalities [15], [16] as particular cases. However, these new inequalities were mainly used to the case of constant delays [14], [15]. Very recently, various improved inequalities were proposed to obtain stability criteria for systems with time-varying discrete delays, such as improved Jensen inequality [17], second-order Bessel-Legendre inequality [18], generalized reciprocally convex inequality [19], quadratic-partitioning based inequality [20], generalized free-weighting-matrix The associate editor coordinating the review of this manuscript and approving it for publication was Bing Li . based inequality [21], generalized free-matrix-based integral inequality [22], [23]. Among all of the inequalities, the generalized free-matrix-based integral inequality [23] can reduce the conservatism effectively. The relationship between u k y(t)dtdu k · · · du 1 was not considered in [17], which motivates further investigation. This paper presents a generalized integral inequality which includes those in [11], [12], [17], [24] as special cases. Based on the generalized integral inequality, a new stability criterion is proposed. An example is introduced to show the superiority of the proposed criterion. The contributions of our paper are as follows: i=0 T i P i , which includes those in [11], [12], [17], [24] as special cases.
• Both the generalized integral inequality and the new LKF include fourth integrals, which may obtain more general results.
• In this paper, the relationship between   u k y(t)dtdu k · · · du 1 is considered.
Notation: See

II. PRELIMINARY
Consider the systems described by wherex(t) ∈ R n is the system state, A, B are n × n constant matrices. The time-varying delay h(t) satisfies where

Lemma 3 ([24]):
For any matrix P ∈ S n + , and any continuously differentiable function y : [a, d] −→ R n , then we can obtain d aẏ Lemma 4: For a matrix P ∈ S n + , and any continuously differentiable function y : [a, d] −→ R n , then we can obtain d aẏ where i , i = 0, 1, 2, 3 are the same as in Lemma 3 Proof: Based on Lemma 1 and Lemma 3, we obtain where i , i = 0, 1, 2, 3 are the same as in Lemma 3. By Lemma 1, we havē Then, we calculate the values of a 20 and a 21 .

V. CONCLUSION
This paper focus on stability of systems with time-varying delays. By a new augmented LKF and combined with a generalized integral inequality, a new stability criterion is obtained. Both the generalized integral inequality and the new augmented LKF include fourth integrals, which may obtain more general results. A numerical example is given to show the effectiveness of the proposed criterion. In the future work, the proposed stability approach can be applied to other dynamic systems such as a singular system and a neural network system.