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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 and the delayed state but also with , where is a continuous function defined on . 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.
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, [1–25] and the references cited therein.
Recently, Zhang et al.  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  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 . Wang et al.  introduced a new parameter in the Lyapunov functional for the time-varying 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.  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 time-varying delay systems with nonlinear perturbations. Thuan et al.  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.
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 but also with , where is a continuous function satisfying 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 . Furthermore, the stabilization conditions are shown to be less conservative than those reported in Zhang et al.  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 means that is symmetric and positive (semipositive) definite. and 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.
2. Problem Description and Preliminaries
Consider the following system with a nonlinear perturbation: where is the state vector, is the control input vector, , and , with , where is the Banach space of continuous functions. The delay is time-varying and satisfies where is a positive constant and and are constants representing the lower and upper bounds of the delay, respectively. is a nonlinear perturbation satisfying where and are positive scalars and satisfies .
Remark 1. In [3, 8, 16], the authors assumed that the nonlinear terms satisfy 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.
The following definitions and lemmas will be used for providing the main results in the sequel.
Definition 2. Given a scalar : system (1) with is -stable if there exists a positive number such that every solution of the system satisfies
Definition 3. Given a scalar : system (1) is -stabilizable if there exists a state feedback control such that the closed-loop system is -stable.
Lemma 4 (see ). For any and a positive symmetric definite matrix ,
Lemma 5 (lower bound lemma for reciprocal convexity; see ). Let have positive values in an open subset of . Then, the reciprocally convex combination of over satisfies subject to
3. Main Results
We use the following notation for the convenience:
The following theorem presents an exponential stabilization condition for (1).
Theorem 6. Let and assume that conditions (2) and (3) are satisfied. If there exist matrices , , , , , , , and such that the following LMIs hold where then system (1) is robustly -stabilizable, the state feedback control , and the solution of the closed-loop 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.
(i) Assume first that . From Jensen’s inequality  and Lemma 5, and so On the other hand, if or , then or , respectively. Hence, inequality (27) holds.
Using Jensen’s inequality , it is not difficult to arrive at the inequalities It follows now from (6) that On the other hand, Therefore, formulas (19)–(31) imply that where If we pre- and postmultiply by and let then the condition is equivalent to condition (11) by using Schur Complement Lemma.
(ii) Assume now that . Applications of Jensen’s inequality  and Lemma 5 yield and hence Note that when or , one can obtain or , respectively. Therefore, inequality (36) is satisfied.
Using Jensen’s inequality , we deduce that Combining (19)–(25), (28), and (30)–(37), we obtain where Pre- and postmultiplying by and letting , the condition is equivalent to condition (12) 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 and condition (2) is satisfied. If there exist matrices , , , , , , , and such that where then system (43) is robustly -stabilizable, the state feedback control , and the solution of the closed-loop system satisfies
4. Numerical Examples
In this section, two numerical examples are given to illustrate the effectiveness of the results obtained in this paper.
Example 1. Consider the system with a nonlinear perturbation where , and the nonlinear perturbation satisfies Note that , , , and . It is not difficult to check that and are Hurwitz unstable.
Let . Then . Using the LMI Toolbox in MATLAB, LMIs (11)–(13) in Theorem 6 are satisfied with Furthermore, the solution of the system satisfies and the stabilizing feedback control Observe that the results reported in [3, 8, 16] cannot be applied to (47) since the nonlinear perturbation is related with the term .
Example 2. Consider a linear system with an interval time-varying delay where and . Note that , and . It is easy to check that and are Hurwitz unstable. Given , using the LMI Toolbox in MATLAB, LMIs (44) in Corollary 7 are satisfied with Moreover, the solution of the system satisfies and the stabilizing feedback control
Figure 1 shows the trajectories of and of the open-loop system with the initial condition , . Figure 2 shows the trajectories of and of the closed-loop system with the state feedback and the initial condition , .
Letting the lower and upper bounds of the time delay be the same as in Zhang et al. , our results also ensure exponential stability with an -convergence rate as given in Table 1. Note that Zhang et al.  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.  by letting and be the same as in . For selected and