New Delay-Range-Dependent Robust Exponential Stability Criteria of Uncertain Impulsive Switched Linear Systems with Mixed Interval Nondifferentiable Time-Varying Delays and Nonlinear Perturbations

We investigate the problem of robust exponential stability analysis for uncertain impulsive switched linear systems with timevarying delays and nonlinear perturbations.The time delays are continuous functions belonging to the given interval delays, which mean that the lower and upper bounds for the time-varying delays are available, but the delay functions are not necessary to be differentiable. The uncertainties under consideration are nonlinear time-varying parameter uncertainties and norm-bounded uncertainties, respectively. Based on the combination of mixed model transformation, Halanay inequality, utilization of zero equations, decomposition technique of coefficient matrices, and a common Lyapunov functional, new delay-range-dependent robust exponential stability criteria are established for the systems in terms of linear matrix inequalities (LMIs). A numerical example is presented to illustrate the effectiveness of the proposed method.


Introduction
The problem of stability analysis for dynamical systems with time delays and uncertainties has been intensively studied since these systems often occur in many industrial systems such as chemical processes, biological systems, population dynamics, neural networks, large-scale systems, and network control systems.The occurrence of the time delays and uncertainties may cause frequently the source of instability or poor performances in various systems.Thus, there has been growing interest in stability analysis and controller design for time-delay systems.However, authors investigated the robust synchronization of coupled fuzzy cellular neural networks with differentiable time-varying delay in [1,2].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; such criteria do not give any information on the size of the delay.On the other hand, delay-dependent stability criteria are concerned with the size of the delay and usually provide a maximal delay size.Most of the existing delaydependent stability criteria are presented by using Lyapunov-Krasovskii approach or Lyapunov-Razumikhin approach.In recent years, much attention has been paid to stability analysis of the uncertain linear systems with interval timevarying delay [3][4][5][6].In [5], the authors studied the delaydependent stability problem for uncertain linear systems with interval time-varying delay.The restriction on the derivative of the interval time-varying delay was removed.Moreover, robust stability analysis of uncertain linear systems with time-varying delays and nonlinear perturbations has received the attention of a lot of theoreticians and engineers in this field over the last few decades [7][8][9][10][11][12][13][14].Furthermore, authors studied the delay-dependent robust stability criteria for linear systems with discrete interval time-varying delay, discrete constant delay, and nonlinear perturbations in [15].However, a descriptor model transformation and a corresponding Lyapunov-Krasovskii functional have been introduced for stability analysis of systems with delays in [16].In [17], the authors studied the problem of stability for linear switching system with time-varying delays.
Over the past decades, the problem of stability analysis for dynamic systems with impulsive effects and switching has arisen in a wide range of disciplines, such as physics, chemical engineering, and biology [18][19][20][21][22][23][24][25][26][27][28].These systems are usually called impulsive switched systems.In [24], the authors studied the asymptotic stability problem for a class of impulsive switched systems with time-invariant delays based on LMI approach.Stability criteria of uncertain impulsive switched systems with time-invariant delays are introduced in [25].Most of the existing delay-dependent stability criteria for time-delay systems are obtained as the upper bounds on the derivative time-varying delays by using Lyapunov-Krasovskii functional.However, it appears that few results are available for stability analysis for impulsive switched systems with time-varying delays.In consequence, it is important and interesting to study the problem of robust stability analysis for uncertain impulsive switched systems with interval nondifferentiable time-varying delays and nonlinear perturbations by using a common Lyapunov functional and Halanay lemma.
In this paper, we present the delay-range-dependent robust exponential stability criteria for uncertain impulsive switched linear systems with mixed interval nondifferentiable time-varying delays and nonlinear perturbations.Based on Halanay inequality, mixed model transformation, utilization of zero equations, decomposition technique of coefficient matrices, and a common Lyapunov functional, some new delay-range-dependent robust exponential stability criteria are derived in terms of LMIs for the systems.In order to reduce the complexity of stability criteria for calculation and finding solutions, mixed model transformation [13,16] and Halanay inequality [29][30][31] are used.Finally, an illustrative example is given to show the effectiveness and advantages of the developed method.
Proof.Consider inequality (17); we have Equivalently, By using Lemma 3 (Schur complement lemma) in the above inequality, we get Premultiplying ( 21) by diag{, , } and postmultiplying by diag{, , }, we obtain the result.The proof of the lemma is complete.
Remark 7. Conditions ( 6) and ( 7) guarantee that  −    () is invertible.It is easy to show that when  = 0, the parametric uncertainty of linear fractional form reduces to a normbounded one.
The objectives of this paper are (i) to establish new delay-range-dependent sufficient conditions for exponential stability of nominal system (1) and (ii) to establish new delayrange-dependent sufficient conditions for robust exponential stability of system (1).

Main Results
In this section, we first present the exponential stability criteria with delays dependence for nominal system (1) via LMI approach.Rewrite the nominal system (1) in the following descriptor system: Let us decompose the constant matrices    and    as where  1   ,  2   ,  1   , and  2   are given real constant matrices with appropriate dimensions.By Leibniz-Newton formula, we have By utilizing the following zero equations, we get where  1 and  2 are real constant matrices with appropriate dimensions which will be chosen to guarantee the exponential stability of the nominal system (1).By ( 23)- (25), system (22) can be represented by the form We now introduce the following notations for later use: where max where   =   / min (),  ∈ , and  is the unique positive root of the equation  −  + ( + ) ℎ = 0.
Proof.Consider a common Lyapunov functional for  ∈ [ −1 ,   ) and a symmetric positive definite matrix .
Next, we now present the new delay-range-dependent robust exponential stability criteria for system (1).We introduce the following notations for later use:

Numerical Example
Example 1.Consider the following uncertain impulsive switched linear system with mixed interval time-varying

Conclusions
We have presented the problem of robust exponential stability criteria for uncertain impulsive switched linear systems with mixed interval nondifferentiable time-varying delays and nonlinear perturbations.By using a common Lyapunov functional, mixed model transformation, Halanay inequality, utilization of zero equations, and LMI approach, new delayrange-dependent robust exponential stability criteria for the systems are established in terms of LMIs.Finally, the theoretical result is illustrated well with a simulation example.