Delay-Dependent Robust H ∞ Performance for Uncertain Neutral Systems with Mixed Time-Varying Delays and Nonlinear Perturbations

This paper deals with the problems of delay-dependent stability and H∞ performance for uncertain neutral systems with timevarying delays, and nonlinear perturbations.The time-varying delays are neutral, discrete, and distributed time-varying delays that the upper bounds for the delays are available.The restrictions on the derivatives of the discrete and distributed time-varying delays are removed, which mean that a fast discrete time-varying delay is allowed. The uncertainties under consideration are nonlinear time-varying parameter perturbations and norm-bounded uncertainties, respectively. Firstly, by applying a novel LyapunovKrasovskii functional approach, Wirtinger-based integral inequality, Peng-Park’s integral inequality, decomposition technique of constant matrix, descriptor model transformation, Leibniz Newton formula and utilization of zero equation, and improved delaydependent bounded real lemmas (BRL) for systems are established in terms of linearmatrix inequalities (LMIs).Then, based on the obtainedBRL, some less conservative delay-dependent stability criteria of uncertain neutral systemswithmixed time-varyingdelays and nonlinear perturbations are obtained and improved H∞ performance criterion with the framework of LMIs is introduced. Finally, some numerical examples are given to illustrate that the presented method is effective.


Introduction
Time delay is frequently a source of instability and a source of generation of oscillation in many dynamic systems such as hybrid systems (and practical application) [1,2], networked control systems, biological systems, mechanical systems, and chemical or process control systems [3].Thus, analysis and synthesis problem for systems with time-varying delay have become an important issue and large varieties of problems have been researched since the nineties by several researchers [4][5][6][7][8],  ∞ performance [9][10][11].
In some physical system, the system models can be described by functional differential equation of neutral type, in which the models depend on the state delay but also depend on the state derivatives, are often encountered in various fields, such as population ecology [12], distributed networks containing lossless transmission lines [13], heat exchangers, and robots in contact with rigid environments [14].For interesting research methods, stability criteria for application neutral stochastic systems and neural networks have been discussed in [15][16][17][18][19]. On the one hand, some LMI criteria on robust stability for uncertain ones have been deeply derived in [20][21][22][23].Very recently, improved  ∞ performance analysis and stability for uncertain systems with time-varying delays were proposed in [10,11,24].However, there are rooms for further improvements in the feasible region of criteria for  ∞ performance and stability.
Stability criteria for time-delay systems are generally divided into two classes: delay-independent one and delaydependent one.Delay-independent stability criteria tend to be more conservative, especially for small size delay, and such criteria do not give any information on the size of the 2 Mathematical Problems in Engineering delay.On the other hand, delay-dependent stability criteria are concerned with the size of the delay and usually provide a maximal delay size.Generally speaking, the latter ones are less conservative than the former ones when the timedelay values are small.Much time and efforts have been put into the development of some techniques and new Lyapunv-Krasovskii functional because how to choose Lyapunov-Krasovskii functional and estimate an upper bound of timederivative of Lyapunov-Krasovskii functional play key roles to improve the feasible region of stability criteria.Delaydependent stability criteria for these systems are established in terms of linear matrix inequalities (LMIs).
With above motivations, based on Lyapunov stability theory, improved  ∞ performance criteria and stability analysis for uncertain neutral systems with mixed timevarying delays and nonlinear perturbations are derived by the framework of LMIs which will be introduced in Theorem 11.Some numerical examples are given to illustrate that the presented method is effective.

Main Results
In this section, a new  ∞ performance and stability analysis for the system (1) will be introduced by using the Lyapunov functional method combining with linear matrix inequality technique.Firstly, we will establish a new version of delaydependent BRL for the nominal system (1).We can rewrite the nominal system (1) as follows: We introduce the following notations for later use: where and other terms are 0.

Numerical Examples
Example .Consider the following uncertain neutral system with mixed time-varying delays (26).We consider robust asymptotic stability with  ∞ performance  of system (26) by using Theorem 11.The system ( 26) is specified as follows: It is easy to see that  = 0.  (80) Table 1 lists the comparison of the upper bounds delays for asymptotic stability of system (79) by different methods.And the derivative of the discrete time-varying delay is unknown, because we removed the derivative of the discrete timevarying delay.We can see from Table 1 that our results are superior to those in [6,[30][31][32][33].
Example .Consider system (26) with  = 0, () ≡ 0 and   (84) Table 2 lists the comparison of the upper bounds delays for asymptotic stability of system (82) by different methods.And the derivative of the discrete time-varying delay is unknown, because we removed the derivative of the discrete timevarying delay.It is clear that our results are superior to those in [22,32,34,35].
Example .We consider system (82) with the parameters (85)  (86) Table 3 shows the comparison of the upper bounds delay allowed obtained for asymptotic stability of system (82) by other methods.And the derivative of the discrete timevarying delay is unknown, because we removed the derivative of the discrete time-varying delay.It can be found from Table 3 that our results are significantly better than those in [22,34].

Conclusions
The problem of robust stability and  ∞ performance for uncertain neutral systems with time-varying delays was studied.The restriction on the derivative of the discrete timevarying delay is removed.The uncertainties under consideration are nonlinear time-varying parameter perturbations and norm-bounded uncertainties, respectively.By applying a novel Lyapunov-Krasovskii functional approach, Wirtingerbased integral inequality and Peng-Park's integral inequality, decomposition technique of constant matrix, descriptor model transformation, Leibniz-Newton formula and utilization of zero equation, and improved delay-dependent bounded real lemmas (BRL) for systems are established in terms of linear matrix inequalities (LMIs).Then, based on the obtained BRL, some less conservative delay-dependent stability criteria of uncertain neutral systems with mixed timevarying delays and nonlinear perturbations are obtained and improved  ∞ performance criterion with the framework of LMIs is introduced.Numerical examples have shown significant improvements over some existing results.

Table 2 :
Maximum allowable upper bounds ℎ  with   = 1 and given   for different   in Example 3.