Abstract
This paper further investigates the problem of stability for a general linear system with time-varying delays. Firstly an improved type of Lyapunov–Krasovskii functional is introduced with integral and nonintegral terms and time-correlation terms. Referring a few existing papers, some valid inequalities mathematical analysis techniques are used in this paper in order to reduce the conservatism of the system. Finally, two examples are presented to demonstrate the advantages of the proposed tactics in this paper.
1. Introduction
It is widely acknowledged that, in practical systems such as switched systems [1], T-S fuzzy systems [2], and neural networks [3], time delays inevitably occur and affect the system performance [4, 5]. Therefore, in recent decades, researchers have paid so much attention to analyzing the stability and accomplishing the stabilization of time-delayed systems. The Lyapunov–Krasovskii functional (LKF), recognized as one of the most efficient tools for these questions, offers both the methods as well as the challenges for dealing with such problems. On one hand, like in a previous work [6–9], the way to construct LKF with augmented vector is commonly accepted. Meanwhile, adding more multi-integral seems also will be helpful to reduce the conservatism in some cases [10]. On the other hand, many innovative integral inequalities have been proposed to deal with the derivative of LKF, for example, Jensen inequality [11, 12], Wirtinger-based inequality [13], Bessel–Legendre inequality [14], free-matrix-based integral inequality [15, 16], and many new refined types of these integral inequalities such as shown in [17, 18]. Combining the recent studies of LMIs discussed above [13, 19] and improved quadratic integral inequalities such as in [20, 21], a general type of LKF may be employed for stability analysis of time-delayed systems [22, 23], with both the construction of LKF and the way to cope with the derivative of LKF been considered [24, 25].
Motivated by the contents above, an attempt to construct the LKF has been made for the time-varying delay systems in this paper. Consulting the method of LMIs in previous articles with an augmented quadratic term, analysis of a time-varying delay system with less conservatism is revealed.
This paper consists of the following parts: Section 2 states the description and definition of the system, as well as the lemmas used later. Then, the achievements of this paper are proved in Theorem 1. Two numerical examples are provided to confirm the results at the end of this paper.
Throughout this paper, denotes the n-dimensional Euclidean space. means that is a real symmetric and positive definite matrix. The superscripts “−1” and “T” indicate the inverse and the transpose of a matrix, respectively. , diag{…} represents the block diagonal matrix.
2. Problem Statement
Firstly, we take the following system model with a time-varying delay into account:where represents the state vector. is the initial condition with , and are system matrices with compatible dimensions. The time delay function is continuous and defined as
The scalar and , respectively, represent the lower and upper bounds of the time-varying delay.
In order to deduce the results of this paper, some lemmas are introduced before the main results.
Lemma 1 (See [26]).. Let and x be a continuous and differentiable function: . Matrix . The following inequality holds:with
Lemma 2 (See [27]).. Let , where and ,if the following conditions hold:(i)(ii)(iii)
Lemma 3 (See [28]).. For any symmetric positive definite matrix , scalar , and vector function such that the integrations concerned are well defined, the following inequality holds:
Lemma 4 (See [29]).. Given positive integers m,n and a scalar , given matrices and . For all vecotrs , the function is given by
Then, if there exists a matrix such that , the following equality holds:
3. Main Result
In this section, the stability of the system is analyzed. For brevity, the following nomenclature is used to simplify vector and matrix representations:
Theorem 1. In this section, the stability of system (1) is analyzed under the condition of the new LKF. With known scalars and , system (1) is asymptotically stable if there exist appropriately dimensional symmetrical matrices , and satisfying the following LMIS:
Proof. We design the functional below for systemwith each part of asThe positive of can be proved with matrices , respectively. The derivation of each part in isand could be divided into two parts:Based on the formula of integration by parts, the following zero equalities hold for symmetric matrices and with appropriate dimension.Adding equations (21) and (22) to (20), one hasThen, through applying Lemmas 3 and 4, equation (23) would yieldAccording to Lemma 1, with , the integration in is expressed asFor equation (19), the expressions of and are divided into two parts, respectively, i.e.,Therefore, all the quadratic terms of exist in equations (24), (26), and (27), and the coefficient of the quadratic terms isCombining equations (16)–(18) and (24)–(28) and referring Lemma 4 simultaneously, would be negative if LMIs (10)–(12) hold through utilizing Schur complement. Based on Lyapunov stable analysis of dynamic system, the system discussed in this paper is stable.
Remark 1. To consider both the conservatism and computational complexity of the system, a nonintegral term of is introduced into this with flexibal coefficients and . In addition, the vector is not fixed as well, which makes the LKF more general for some conditions of practical system. If the upper bound is much more high than the lower bound, the part could be ignored and so the same is in reverse.
Remark 2. Depending on the definition of Lemma 1, it is not difficult to get tighter bounds of the system through adjusting the number of order. With the increase of the order N, the double integral state vectors in are required to translate into triple or multiple integrals in order to reduce the conservatism. Thus, these state vectors may satisfy the demand of a complex system.
4. Numerical Example
In this section, two numerical examples are provided while comparing other results existed in previous research to demonstrate the proposed methods.
We define the system matrices and as
The example is aimed at searching the maximum value for the upper bounds of the delay (MVUBD) with the derivative of known delay. We set and assign a value to , and the results of the MVUBD computed by Theorem 1 are shown in Table 1. Some other results from former research are also listed in Table 1 for the sake of comparison. Figures 1–3 show the state trajectories with different and MVUBD. From the simulation results, it is concluded that the achieved improvements in this paper are less conservative than other results.
In this example, the state matrices and have been set as
In this example, the value of is still set as , and the results of the MVUBD which is computed by Theorem 1 are shown in Table 2. Figures 4–6 also show the state trajectory with different and MVUBD. The results of MVUBD with different in Table 2 are much better, which means the method proposed in not only valid but also advanced. It is worth noticing that the MVUBD reaches almost 18.97% than the data published lately when is 0.5.
5. Conclusions
A further investigation of the stability for system with time-varying delays has been figured out in this paper. With different methods of solving inequalities, results of less-conservative stability criteria have been achieved. At the end of the paper, two examples, compared with results from other papers, are represented in order to show the availability of the such methods. Based on the construction of LKF in this paper, some other problem such as sampled data control may also be considered in the same way. In the future work, the improved type of LKF proposed in this paper will be extended to a chaotic Lurie system, T-S fuzzy systems, Markov jump systems, quaternion-valued or memristor-based neural networks, and complex dynamical networks.
Data Availability
The raw/processed data required to reproduce these findings cannot be shared at this time as the data also form part of an ongoing study.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant nos. 61 703 060, 12 061 088, 61 802 036, and 61 873 305), the Sichuan Science and Technology Program under Grant no. 21YYJC0469, the Project funded by the China Postdoctoral Science Foundation under Grant nos. 2020M683274 and 2021T140092, Supported by the Open Research Project of the State Key Laboratory of Industrial Control Technology, Zhejiang University, China (no. ICT2021B38), Guangdong Basic and Applied Basic Research Foundation (2021A1515011692), Engineering Research Center of Power Quality of Ministry of Education, and Anhui University (no. KFKT202102).