Stability Analysis of Time-Varying Delay Systems with an Improved Type of LKF

School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611 731, China Yangtze Delta Region Institute (Huzhou), University of Electronic Science and Technology of China, Huzhou 313 001, China School of Electronic Information and Electrical Engineering, Chengdu University, Chengdu 610106, China Engineering Research Center of Power Quality of Ministry of Education, Anhui University, Hefei 230 601, China School of Mathematical Science, University of Electronic Science and Technology of China, Chengdu 611 731, China


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]. erefore, in recent decades, researchers have paid so much attention to analyzing the stability and accomplishing the stabilization of timedelayed systems.
e 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][7][8][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], freematrix-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.
is paper consists of the following parts: Section 2 states the description and definition of the system, as well as the lemmas used later. en, the achievements of this paper are proved in eorem 1. Two numerical examples are provided to confirm the results at the end of this paper. roughout this paper, R n denotes the n-dimensional Euclidean space. P > 0 means that P is a real symmetric and positive definite matrix. e superscripts "− 1" and "T" indicate the inverse and the transpose of a matrix, respectively.

Problem Statement
Firstly, we take the following system model with a timevarying delay into account: where x(t) ∈ R n represents the state vector. ϕ(t) is the initial condition with t ∈ [− d, 0], d > 0.A, and A d are system matrices with compatible dimensions. e time delay function d(t) is continuous and defined as e scalar μ 1 and μ 2 , 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.
if the following conditions hold: Lemma 4 (See [29]). Given positive integers m,n and a scalar β ∈ (0, 1), given matrices R 1 > 0, R 2 > 0 and H 1 , H 2 ∈ R n×m . For all vecotrs ξ ∈ R m , the function Θ(β, R 1 , R 2 ) is given by en, if there exists a matrix X ∈ R n×n such that R 1 X X T R 2 > 0, the following equality holds:

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 μ 1 , μ 2 < 1 and d > 0, system (1) is asymptotically stable if there exist appropriately dimensional symmetrical matrices � col e 1 , e 2 , 0, 0 Π 9b � col 0, 0, e 6 , e 8 , Π 11 � col e 2 , e 3 , (d − d(t))e 7 , (d − d(t))e 9 Π 11a � col e 2 , e 3 , 0, 0 Π 11b � col 0, 0, e 7 , e 9 , Proof. We design the functional below for system with each part of W(t) as e positive of W(t) can be proved with matrices and _ V 3 (t) could be divided into two parts: Based on the formula of integration by parts, the following zero equalities hold for symmetric matrices U 1 and U 2 with appropriate dimension.

Mathematical Problems in Engineering
For equation (19), the expressions of η 3 (t) and η 4 (t) are divided into two parts, respectively, i.e., erefore, all the quadratic terms of _ W(t) exist in equations (24), (26), and (27), and the coefficient Ω d 2 (t) of the quadratic terms is Combining equations (16)- (18) and (24)- (28) and referring Lemma 4 simultaneously, _ W(t) 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 w(t) is introduced into this with flexibal coefficients d(t) and d − d(t). In addition, the vector η i (t) 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 d(t)ξ T 3 (t)V 1 ξ 3 (t) 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 ξ 1 (t) are required to translate into triple or multiple integrals in order to reduce the conservatism.
us, these state vectors may satisfy the demand of a complex system.

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 A and A d as e example is aimed at searching the maximum value for the upper bounds of the delay (MVUBD) with the derivative of known delay. We set μ 1 � − μ 2 � μ and assign a value to μ, and the results of the MVUBD computed by eorem 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 μ 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 A and A d have been set as In this example, the value of μ 1 is still set as μ 2 , and the results of the MVUBD which is computed by eorem 1 are shown in Table 2. Figures 4-6 also show the state trajectory with different μ and MVUBD. e results of MVUBD with different μ in Table 2 are much better, which means the method proposed in not only valid but also advanced. It is Mathematical Problems in Engineering     Mathematical Problems in Engineering 9 worth noticing that the MVUBD reaches almost 18.97% than the data published lately when μ is 0.5.

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 e 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
e authors declare no conflicts of interest.