Stability analysis for time delay systems via a generalized double integral inequality

A time delay widely exists in many fields such as chemistry, biology, industry, and so on. Since a time delay arising in a system may cause instability, the stability analysis of time-delay systems has been wildly studied in the past few decades [1,2]. The main purpose is paid to determine the admissible delay, for which the systems remain stable. It is well known that the LKF method has been widely used to obtain stability conditions for time-delay systems [3, 4]. The main purpose of the LKF method is to estimate the integral term arising in the time derivative of double integral term in the LKF. Therefore, to get less conservative stability criteria, many integral inequality methods are derived. Those inequality methods include Jensen inequality [5, 6], Wirtinger inequality [7–9], double integral inequlity [10–12], various improved integral inequalities [13–26]. The Jensen inequality expressed as Vab(ẏ) = ∫ b


Introduction
A time delay widely exists in many fields such as chemistry, biology, industry, and so on. Since a time delay arising in a system may cause instability, the stability analysis of time-delay systems has been wildly studied in the past few decades [1,2]. The main purpose is paid to determine the admissible delay, for which the systems remain stable.
The Jensen inequality expressed as where The further improved inequality expressed asV ab (ẏ) [13][14][15], respectively, where Ω 2 , Ω 3 , Ω 4 are defined in Lemma 4 [15]. However, these results only estimate the integral term arising in the time derivative of double integral term in the LKF. This paper presents a generalized double integral inequality which includes those in [10][11][12] as special cases. A new stability criterion is proposed by choosing a new LKF and using the generalized double integral inequality. Both the generalized integral inequality and the new LKF include fourth integrals, which may yield less conservative results. Two examples are introduced to show the effectiveness of the proposed criterion. The contributions of our paper are as follows: where ω and ζ are defined in Lemma 3. The above inequality includes those in [10][11][12] as special cases.
• Both the new double integral inequality and the new LKF include fourth integrals, which may obtain more general results. Notation: See Table 1.

Preliminary
Consider the time delay systems aṡ where y(t) ∈ R n is the state vector, h > 0 is constant time-delay and the initial condition φ(t) is a continuous function. Lemma 1.
[15] For a matrix P ∈ S n + , and any continuously differentiable function x : where This completes the proof. Lemma 3. For a differential function x : [a, b] → R n , a matrix P ∈ R n + , a vector ζ ∈ R k , and any matrices M i ∈ R k×n (i = 1, 2, 3, 4), then the following inequality holds: The result can be easily obtained by choosing n = 3, f 1 (s) = b−a λ T 3 R, and M 4 = 0. In addition, the inequality (12) of Lemma 5 in [12] is a special case of Lemma 3 by set ting
Proof. Introduce a LKF as Then, the time derivative of V(y t ) along the trajectories of system (1) as followṡ where Thus, according to (3.2)-(3.5), we haveV(y t ) ≤ η T (t)Ψη(t). Thus, if (3.1)holds, then, for a sufficient small scalar ε > 0,V(y t ) ≤ −ε y(t) 2 holds, which ensures system (1) is asymptotically stable. The proof is completed. Remark 2. Both the double integral inequality and the new LKF include fourth integrals, which may yield novel stability results. Furthermore, in order to fully consider relevant information of the double integral inequality in Lemma 3, the y T (s)dsdu 3 du 2 du 1 is added as a state vector.

Conclusion
This paper focus on a new stability condition for a class of time delay systems. By using two generalized integral inequalities and a new augmented LKF, a new stability criterion is obtained. Both the double integral inequality and the new LKF include fourth integrals, which may yield more general results. Two numerical examples are proposed to show the effectiveness of the proposed criterion.