Exponential Stabilization of a Class of Time-Varying Delay Systems with Nonlinear Perturbations

This paper addresses the problem of exponential stabilization of a class of time-varying delay systems with nonlinear perturbations. These perturbations are related not only with current state x(t) and the delayed state x(t − h(t)) but also with β(t), where β(t) is a continuous function defined on [0, +∞). With the delay interval divided into two equidistant subintervals, a novel Lyapunov functional is introduced, and several new exponential stabilization criteria are derived in terms of linear matrix inequalities (LMIs) by employing reciprocally convex approach. Two examples are given to illustrate the effectiveness of the main results.


Introduction
Time delay is commonly encountered in various physical and engineering systems such as aircraft, biological systems, and networked control systems.Since the existence of time delays causes poor performance, oscillation, or even instability, it is very important to investigate stability analysis for systems with time delays before designing control systems.On the other hand, the systems almost present some uncertainties because it is not easy to obtain an exact mathematical model due to environmental noise, uncertain or slowly varying parameters, and so forth.Therefore, considerable amounts of efforts have been done to the stability and stabilization of time-delay systems and time-delay systems with nonlinear perturbations; see, for example,  and the references cited therein.
Recently, Zhang et al. [12] considered interval time-varying delay systems and obtained some delay-dependent conditions by employing Finsler's lemma.Combining the descriptor model transformation and the integral inequality method, Han [3] investigated the robust stability of linear systems with time-varying delay and nonlinear perturbations and obtained several improved stability conditions.On the basis of free weighting matrices technique, robust stabilization criteria for neutral systems with nonlinear perturbations were reported in [9].Wang et al. [6] introduced a new parameter in the Lyapunov functional for the timevarying delay systems with nonlinear perturbations and obtained less conservative results, whereas the range of the time delays considered in the paper was assumed from zero to an upper bound.Note that the stability investigated in the above-mentioned papers was primarily focused on asymptotic stability.Using delay decomposition method and Finsler's lemma, Liu et al. [24] studied the exponential stability of neutral systems with interval time-varying delays and nonlinear perturbations.So far, there are few articles concerning the problem of exponential stabilization of timevarying delay systems with nonlinear perturbations.Thuan et al. [16] provided a detailed analysis for the problem of designing state feedback controllers to exponential stabilization of time-delay systems with nonlinear perturbations by using the integral inequality method and constructing a Lyapunov functional containing the triple integral terms.However, there still exists a gap for reducing both the conservatism and the number of decision variables.

Mathematical Problems in Engineering
In this paper, we study the exponential stabilization of a class of time-delay systems with nonlinear perturbations.The main contributions of this paper can be summarized as follows: (i) a novel Lyapunov functional containing the center point of time-delay interval is constructed; (ii) compared with the systems studied in [3,8,16], the nonlinear perturbations of (1) are related not only with the current state () and the delayed state ( − ℎ()) but also with (), where () is a continuous function satisfying ∫ +∞ 0  2 () 2  < +∞ and  is a positive constant; new sufficient conditions are obtained that ensure the stability of a closed-loop system, which extend and improve the main results of [16].Furthermore, the stabilization conditions are shown to be less conservative than those reported in Zhang et al. [12] when there are no nonlinear perturbations in the system.Finally, two numerical examples are presented to demonstrate the effectiveness and advantages of the main results.
Notation.Throughout the paper,   denotes the -dimensional Euclidean space with vector norm ‖ ⋅ ‖, and  × is the set of all  × -dimensional real matrices. denotes the identity matrix of appropriate dimensions, and the superscript "" stands for matrix transposition.The notation  > 0 (≥ 0) means that  is symmetric and positive (semipositive) definite. min () and  max () denote the minimum and maximum eigenvalues of , respectively.In addition, in symmetric block matrices or long matrix expressions, we use an asterisk ( * ) to represent a term that is induced by symmetry.

Problem Description and Preliminaries
Consider the following system with a nonlinear perturbation: where () ∈   is the state vector, () ∈   is the control input vector, ,  ∈  × , and is the Banach space of continuous functions.The delay ℎ() is time-varying and satisfies where  is a positive constant and ℎ where  and  are positive scalars and () satisfies Remark 1.In [3,8,16], the authors assumed that the nonlinear terms satisfy   (,  ())  (,  ()) ≤  2   ()    () ,   (,  ( − ℎ ()))  (,  ( − ℎ ())) where  and  are constant matrices and  and  are positive scalars.It is obvious that the assumptions on the nonlinear terms given in ( 2) and ( 3) are more general.

Main Results
We use the following notation for the convenience: The following theorem presents an exponential stabilization condition for (1).Theorem 6.Let  > 0 and assume that conditions ( 2) and ( 3) are satisfied.If there exist matrices  > 0, , and  such that the following LMIs hold where then system ( 1) is robustly -stabilizable, the state feedback control () =  −1 (), and the solution (, ) of the closedloop system satisfies Proof.Let us denote Define a Lyapunov functional by where Calculating the time derivative of (  ) along the trajectories of ( 6), we conclude that By virtue of Lemma 4, we have Using (3) and the inequality where , , and  are constants, we obtain It follows from ( 22) and ( 23) that Combining ( 19) and ( 24), we get Now, we estimate the upper bounds of the last three integral terms in inequality (21) as follows.
Using Jensen's inequality [19], it is not difficult to arrive at the inequalities Therefore, formulas ( 19)-(31) imply that where If we pre-and postmultiply Φ 1 by diag{, , , , , } and let then the condition Φ 1 < 0 is equivalent to condition (11) by using Schur Complement Lemma.
From the above discussion, if conditions ( 11)-( 13) are satisfied, then By virtue of (40) and the definition of (()), Hence, we have which implies that the closed-loop system is -stable.The proof is completed.
Corollary 7. Assume that  > 0 and condition ( 2) is satisfied.If there exist matrices  > 0, , and  such that where

Numerical Examples
In this section, two numerical examples are given to illustrate the effectiveness of the results obtained in this paper.
Example 2. Consider a linear system with an interval timevarying delay ẋ () =  () +  ( − ℎ ()) +  () ,  ≥ 0, where and ℎ() = 0.5 + 1.28sin  Letting the lower and upper bounds of the time delay be the same as in Zhang et al. [12], our results also ensure exponential stability with an -convergence rate as given in Table 1.Note that Zhang et al. [12] discussed asymptotic stability, whereas the controller derived in this paper provides exponential stability for the closed-loop system.Furthermore, the maximum bound for  is better than Thuan et al. [16] by letting ℎ 1 and ℎ 2 be the same as in [16].For selected ℎ 1 and  = 0.5, using Corollary 7, one can easily observe that the maximum allowable delay bounds for ℎ 2 are better than those reported in the papers by Zhang et al. [12] and Thuan et al. [16].

Conclusions
In this paper, exponential stabilization of a class of timevarying delay systems with nonlinear perturbations has been investigated.By using the delay decomposition approach and constructing a novel Lyapunov functional, some new delay-dependent stabilization criteria are obtained in order to ensure closed-loop stability of the system with any prescribed -convergence rate.Numerical examples are given to illustrate that the results obtained are much less conservative than some existing results in the literature.Exponential stabilization of impulsive switched delay systems with nonlinear perturbations will be further investigated in the future.

Figure 1
Figure 1 shows the trajectories of  1 () and  2 () of the open-loop system with the initial condition () = [30 10]  ,  ∈ [−1.78, 0]. Figure 2 shows the trajectories of  1 () and  2 () of the closed-loop system with the state feedback () = [−0.8704− 2.9758]() and the initial condition () = [30 10]  ,  ∈ [−1.78, 0].Letting the lower and upper bounds of the time delay be the same as in Zhang et al.[12], our results also ensure exponential stability with an -convergence rate as given in Table1.Note that Zhang et al.[12] discussed asymptotic stability, whereas the controller derived in this paper provides exponential stability for the closed-loop system.Furthermore, the maximum bound for  is better than Thuan et al.[16] by letting ℎ 1 and ℎ 2 be the same as in[16].For selected ℎ 1 and  = 0.5, using Corollary 7, one can easily observe that the maximum allowable delay bounds for ℎ 2 are better than those reported in the papers by Zhang et al.[12] and Thuan et al.[16].