Further Stability Analysis of Time-Delay Systems with Nonlinear Perturbations

In this study, we focus on stability analysis for systems with time-varying delay and nonlinear perturbations. In order to cut down the conservatism of the existing stability criteria, we utilize the triple integral forms of Lyapunov-Krasovskii functional (LKF). In addition, by using single and double integral forms of Wirtinger-based inequality, we overcome some conservatism which come from Jensen’s inequality. Three well-known numerical examples are given at the end. Compared with some existing results, our results have less conservatism.


Introduction
Time-delay has attracted a lot of interests because it is widely encountered in communication systems, neural networks, economic systems, biological systems, and networked control systems [1][2][3][4][5][6].Since time-delay will cause serious degradation of system performance, a lot of researchers are making considerable effort for stability analysis of time-delay systems in the last few decades.The LKF approach and LMI method are the efficient instrument to get the delay-dependent stability criterion.However, only a sufficient condition we can get by utilizing these methods.Therefore, we focus on cutting down the conservatism of stability criterion.As we know, the maximum allowable delay bound (MADB) of the time-delay can measure the conservatism of stability criterion.We can get the larger MADB of time-varying delay according to the stability criterion with less conservatism.
In a lot of existing results, Jensen's inequality has played an important role.However, it will induce some conservatism hard to overcome.To cut down the conservatism, Wirtingerbased integral inequality [7], which can be used to obtain much tighter lower bound of single integral terms, was proposed.Very recently, based on Wirtinger-based integral inequality, a Wirtinger-based double integral inequality was proposed to get much tighter lower bound of double integral terms [8].
In a lot of recent literatures [9][10][11], researchers only utilize single and double integral forms of LKF to derive delaydependent stability criterion.We believe that triple integral form of LKF is helpful for improving the performance of former criteria.
Motivated by the previous discussions, we are concerned about the stability analysis for the time-varying delay systems with nonlinear perturbations.We introduce the triple integral forms of LKF to cut down the conservatism.Taking the time derivative of Compared with some existing results, our results have less conservatism.
The organization of this paper is as follows.The system is presented in Section 2, and some useful lemmas are given.Then the new stability criterion and proof are given in Section 3. In Section 4, the effectiveness of our results can be illustrated by three examples.Section 5 is the conclusion of our investigation.

Mathematical Problems in Engineering
Notations.In this paper, the superscript  means the transpose of a matrix; col{⋅} denotes the column vector.

Problem Formulation and Preliminaries
The time-varying delay systems with nonlinear perturbations are considered as follows: with () ∈ R  as the state variate; ,  ∈ R  *  are predefined constant matrices; () ≥ 0 represent a time-varying delay satisfying The nonlinear perturbations ℎ((), ) ∈ R  and (( − ()), ) ∈ R  can be abbreviated as ℎ and , assumed as follows: where  and  are positive scalars and ,  ∈ R  *  are constant matrices.In order to derive improved stability criterion, the following lemmas will be used.

Mathematical Problems in Engineering
When ℎ =  = 0, that means system without perturbation.We can consider the following system: Stability analysis of system ( 26) is studied in a lot of literatures.From Theorem 4, the stability criterion of system ( 26) can be obtained easily.
Remark 7. From the viewpoint of control theory, the changed trend of system in reality is not only decided by the current states but also related to its past states.This is why the study of time-delay system is significant.This paper is interested in the stability analysis of the system with time-delay which has strong background in reality.The stability criterion we proposed can be used to design effective control strategy in a specific engineering file, such as synchronization of coronary artery.In order to compare with exiting results, a lot of papers related to stability analysis are concerned about the MADB which can be calculated by stability criterion.We can see the same particular examples in  as Examples 1-3 in our manuscript.In these examples structure of the matrices is prescribed.The purpose of these examples is to measure the conservatism of proposed criterion.The larger MADB, the less conservatism.

Numerical Examples
The advantages of our results can be illustrated by the following examples.
Example 1. Nonlinear system (1) is subject to (2) and ( 3) with For given   , , , and   , utilizing Theorem 4, we calculate the MADB   which can guarantee the stability of system (1).The result is listed in Table 1.We can see that the stability criterion we proposed has less conservatism than others.It is worth mentioning that when  = 0, Theorem 4   Table 2: Maximal   with given   = 0 for   = 0.
Example 2. Consider system (26) with For given   = 3, 4, 5 and different   = 0.1, 0.3, 0.5, 0.9, we calculate the MADB   which is listed in Table 3. Table 3 illustrates the methods presented in Corollary 5 providing less conservative results than others.It is worth noting that the effect is very obvious when   ≤ 0.5.
Sometimes the information of () is unavailable or () is undifferentiable.We calculate the MADB   under different   for unknown   .From Table 4, it is observed that the results obtained by Corollary 6 are much less conservative than others.Finally, for given   = 0 and different   = 0.1, 0.3, using Corollary 5 we get the MADB   listed in Table 5.It is clear to see that our results have much less conservatism when the lower bound is   = 0.

Conclusion
We have researched the stability analysis problem for the time-varying delay systems.We utilize the new augmented LKF which is constructed based on single and double integral forms of Wirtinger-based inequality.Combining reciprocally convex method, we get an improved delay-dependent stability criterion.Finally, examples illustrate that the stability criterion we obtained is less conservative than some existing results.

Table 3 :
Maximal   with given   for different   .

Table 4 :
Maximal   with given   for unknown   .