Impulsive control of nonlinear systems with impulse time window and bounded gain error

Limin Zou, Yang Peng, Yuming Feng, Zhengwen Tu School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China limin-zou@163.com School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, 404100, China peng_yang2011@163.com; tuzhengwen@163.com Key Laboratory of Intelligent Information Processing and Control, Chongqing, 404100, China yumingfeng25928@163.com


Introduction
In this paper, we mainly adopt the notation and terminology in [8]. Within the last three decades, impulsive control theory had been intensively studied because impulsive control can be applied in many fields, such as chaotic systems [9,15,24,30,34], HIV prevention modles [5], complex dynamical systems [2-4, 11, 13, 14, 16, 18, 22, 23, 25, 31, 35-38]. The nonlinear impulsive control systems with impulses at fixed times is given bẏ where x ∈ R n is the state variable, t 0 < τ 1 < τ 2 < · · · , lim k→∞ τ k = ∞ denote the moments when impulsive control occurs, A is an n × n constant matrix, φ : R n → R n is a continuous nonlinear function satisfying φ(t, 0) = 0 and φ(x) 2 x T Lx, where L is a diagonal matrix, and U (k, x) is the impulsive control law. Many researchers have studied impulsive control system (1) [1,17,32,33]. But we cannot guarantee the impulses without any error due to the limit of equipment and technology.
Recently, Feng, Li, and Huang [8] discussed the following nonlinear impulsive control systems with impulse time window: where T > 0 denotes the control period, τ m is unknown within impulse time window (mT, (m + 1)T ), and J m ∈ R n×n is impulsive control gain. The impulsive effects can be stochastically occurred in a impulse time window in system (2), which is more general than ones impulses occurred at fixed times. Some results related to impulse time window can be found in [6,7,12,[26][27][28][29]39].
In many practical applications, the impulsive control gain J m may also contain errors, so we should take into account the influence of impulsive control gain errors on the systems. In this paper, we consider a class of impulsive control systems with impulse time window and bounded gain error as follows: where ∆J m is gain error, which is often time-varying and bounded. As pointed out in [17], we can assume that ∆J m = mF (t)J m , m > 0, and F T (t)F (t) I.
The purpose of this paper is to find some conditions on control gain J m and impulse time window T such that the origin of impulsive control system (3) is asymptotically stable. We establish a new sufficient condition for the stability of system (3). Compared with the results shown in [8,17,19], our result is more general and more applicable. Finally, a numerical example is given to show the effectiveness of our result.

Main results
In this section, we will give the main results. To do this, we need the following lemmas. Lemma 1. (See [21].) Let x, y ∈ R n and ε > 0. Then Lemma 2. Let A, P, B ∈ R n×n such that P is a symmetric and positive definite matrix and µ > 0. Then Proof. Note that for any X ∈ R n×n , the matrix X T X is positive semidefinite. It follows that Small calculations show that the result holds. This completes the proof.
Lemma 3. (See [10].) Let H be a real symmetrical matrix, and λ max (H) λ min (H) be the largest and the smallest eigenvalues of H, respectively. Then for any x ∈ R n , we have Theorem 1. Let P ∈ R n×n be a symmetric and positive definite matrix. If there exit g, µ, ε > 0 such that the following hold: , then the origin of system (3) is asymptotically stable.
Proof. Let us construct the following Lyapunov function: If mT t < mT + τ m , then by Lemma 1 and condition (i) we have Similarly, if mT + τ m < t < (m + 1)T , we also have If t = mT + τ m , then by Lemmas 2 and 3 we have It follows from (5) and (6) that where mT + τ m t < (m + 1)T . By using inequalities (4) and (7) we can derive the following results.
If we choose P = I in Theorem 1, then the condition of Theorem 1 can be simplified as follows.
The condition of Corollary 1 is similar to Theorem 1 shown in [19]. Since impulsive effects can be stochastically occurred in a impulse time window in system (3), Corollary 1 is more general than Theorem 1 shown in [19].
Sometimes, for the sake of convenience, the impulsive control gain J m is always selected as a constant matrix J, then we have the following. Corollary 2. Let P ∈ R n×n be a symmetric and positive definite matrix. If there exit g, µ, ε > 0 such that the following hold: where λ = λ max (P )λ max (P −1 J T J)(1 + µ + m 2 (1 + 1/µ)), then the origin of system (3) is asymptotically stable.
The condition of Corollary 2 is similar to Theorem 1 shown in [8]. Since we take into account the influence of impulsive control gain errors on the systems, Corollary 2 is more practical than Theorem 1 shown in [8].
In many practical applications, the parameters of impulsive control of nonlinear systems contain errors. In what follows, we will consider system (3) with parameter uncertainty. The corresponding system can be described aṡ where ∆A is the parametric uncertainty and has the following form: ∆A = GF (t)H, where F T (t)F (t) I, while G and H are appropriate known matrices.
Theorem 2. Let P ∈ R n×n be a symmetric and positive definite matrix. If there exit g, µ, ε > 0 such that the following hold: , then the origin of system (10) is asymptotically stable. Proof. Let us construct the following Lyapunov function: If mT t < mT + τ m , then by Lemma 1 and condition (i) we have The rest of proof is same as that of Theorem 1, so we omit it here for simplicity. This completes the proof.

A numerical example
In this section, we will illustrate the effectiveness of our result by showing simulation results employing the Chua's system. Throughout this section, we assume that x = [x, y, z] T . The original and dimensionless form of a Chua's oscillator [20] is given bẏ where α and β are parameters, and g(x) is the piecewise linear characteristics of the Chua's diode, which is defined by where a < b < 0 are two constants. By decomposing the linear and nonlinear parts of the system in (11), we rewrite it aṡ Simple calculations show that Thus, we can choose L = diag(α 2 (a − b) 2 , 0, 0). In this example, we set the system parameters as For the sake of simplicity, the impulsive control gain error ∆J m is specified as ∆J m = ∆J = 0.05 sin tJ, and so λ = 0.5012. In order to satisfy condition (i) of Theorem 1, we can choose g = 31.
From the following inequality gT + ln λ < 0 we have T < 0.0233. Thus, by Theorem 1 we know that the origin of system (3) is asymptotically stable. The simulation results with T = 0.0200 are shown in Fig. 2.