p th Moment Exponential Stability of Stochastic PWM Feedback Systems with Time-Varying Delays

This paper further studies the p th moment exponential stability of stochastic pulse-width-modulated (cid:2) PWM (cid:3) feedback systems with distributed time-varying delays. We establish several globally exponential stability criteria for such PWM feedback systems by using Lyapunov-Krasovskii functional and then present an upper bound of the parameter of PWM when the system is stable and such system has stronger anti-interference performance than the system without time-varying delays. Furthermore, we present two examples to show the e ﬀ ectiveness and conservativeness of the theoretical results. Abstract stochastic systems with time-varying delays. Besides, Sun and Cao (cid:5) 17 (cid:6) give some deﬁnitions on the p th moment exponential stability in mean and established several p th moment globally stability criteria in mean. In (cid:5) 12, 13 (cid:6) , Hou and Michel established new Lyapunov and Lagrange stability results for pulse-width-modulation (cid:2) PWM (cid:3) feedback systems subjected to random disturbance. To the best of the authors’ knowledge, there are few (cid:2) if any (cid:3) results for the stability analysis of stochastic PWM systems with time-varying delays. Based on the pulse-width-modulation feedback system uniqueness, obviously, such system subjected to random disturbance and time-varying delays is somewhat complex in comparison with most of the systems in the literature. It is noted that the linear plant considered herein is Hurwitz stable , that is, all the eigenvalues of the linear plant are in the left side of the complex plane. In the present paper, we try to make the contribution on this issue. By choosing reasonable Lyapunov-Krasovskii functional, combined with linear matrix inequalities and It (cid:2) o integration method, we will establish several Lyapunov and Lagrange criteria for p th moment exponential stability in mean and then present an algorithm to compute the upper bound for the parameters of PWM. We will characterize the relationship among the parameters of pulse-width modulation, time-varying delays, and the coe ﬃ cient of state vectors of the feedback systems. It will be shown that when the random disturbance is su ﬃ ciently small such PWM feedback system is p th moment exponentially stable in mean provided that the upper bounds of parameters of pulse-width modulator are selected properly. We also demonstrate that such system has the stronger anti-interference performance and tending to the equilibrium point speed more quickly by means of two numerical examples.


Introduction
Pulse-width modulation has extensively been used in attitude control systems, adaptive control systems, signal processing, power control systems, modeling of neuron behavior, and the like e.g., see 1-5 . In many areas, especially engineering applications, how to keep the scheduled operation or work of the state counts for much. Therefore, there has been a growing research interest on the stability analysis for PWM feedback systems, and a set of stability results have been established by a variety of methods 6-22 . In the actual process, however, it is always operated in all kinds of accidental or continued disturbances. Time delay will inevitably occur in electronic neural networks owing to the unavoidable finite switching speed of amplifiers. In recent years, the stability analysis of stochastic systems, especially the systems with time delay, is interesting to many investigators, and many results of stability criteria of these systems have been reported 15-22 . There are, however, only a few results concerning the qualitative properties of stochastic impulsive systems with time-varying delays. In 15 , the authors investigated robust exponential stability and delayed-state-feedback stabilization of uncertain impulsive A set M ⊂ A is said to be the pth moment exponentially stable in mean i.e., S, M is said to be the pth moment exponentially stable in mean if for any t 0 ∈ T , there exists a δ δ t 0 , ε and constants β > 0, k > 1 such that E d x t, ω, a, t 0 , M p < k a p e −β t−t 0 for any process x ·, ·, a, t 0 ∈ S, whenever d a, M < δ, where a is called the initial state and E · denotes the expectation of a random process. If δ is independent of t 0 , S, M is said to be the pth moment uniformly exponentially stable in mean. S, M is said to be the pth moment uniformly asymptotically exponentially stable in mean if it is uniformly stable in the pth mean and if there exists δ > 0, τ > 0 and constants β > 0, k > 1 such that for any process x ·, ·, a, t 0 ∈ S, whenever d a, M < δ implies that E d x t, ω, a, t 0 , M p < k a p e −β t−t 0 for all t > t 0 τ.

Main Results
The PWM feedback system to be considered in this paper is shown in Figure 1.
The pulse-width modulator is described by where e t r t − y t with r t being the external input and y t the system output, and, for k 0, 1, 2, . . ., the pulse-width T k and the sign function sgn are given, respectively, by

4 Abstract and Applied Analysis
The sampling period T , the amplitude of the pulse M, and β are all assumed to be constants. And throughout this paper, we always assume that r t ≡ 0. Under these assumptions, the PWM feedback system with the output function can be described by where x ∈ R n , y ∈ R, u ∈ R, is output of the pulse-width modulator, A, A d , B, C 1 , C 2 , and G are matrices of appropriate dimensions, d d i ∈ R n , 0 < d i ≤ T − T k , i 1, 2, . . . , n, and W t is a scalar wiener process.
Note that x T 1 , x 2 0 T , 0 T is an equilibrium point of PWM feedback system 3.3 .

Lemma 3.2 the It o isometry . Assume W t is a scalar wiener process. If φ t, ω is bounded and elementary, then
using that e i e j ΔW i and ΔW j are independent if i < j. Thus

3.6
Theorem 3.3. Assume that the matrix A in 3.3 is Hurwitz stable. Then the equilibrium x e 0 of PWM feedback system 3.3 is the pth p ∈ Z moment uniformly exponentially stable in mean in the large provided that the following conditions are satisfied:

3.11
iii there exists a constant δ > 0, such that, whenever G < δ, where μ is scalar satisfying Abstract and Applied Analysis

3.15
Proof. Since E X p 1/p is nondecreasing in p, the equilibrium x e 0 is stable resp., asymptotically stable, etc. implying that it is the qth moment stable resp., asymptotically stable, etc. in mean for all q < p. Firstly, we will provide to prove the theorem for even integers.

3.16
Therefore, when t kT T ,

3.17
Abstract and Applied Analysis 7 For t kT T − d, we have

3.19
To simplify our notations, let 3.20 Then, 3.17 and 3.18 are reduced to

3.23
Abstract and Applied Analysis

3.24
Then 3.23 is reduced to

3.25
Let λ m · and λ M · denote the minimum and maximum eigenvalues of a matrix, respectively. By Lemma 3.1, it is obvious that H is a negative definite matrix if and only if ϕ 11 < 0 and ϕ 22 − ϕ 12 ϕ −1 11 ϕ T 12 < 0. Noting that

3.26
From condition i it follows that Θ Mβ < 0, and hence the claim that ϕ 11 < 0 is true. From condition ii we then conclude that H is a negative definite matrix based on Lemma 3.1.
Now we establish an estimation on Z kT and Z kT . Let μ > 0 be arbitrary. We will show that there exists δ > 0 whenever G < δ, such that E Z kT 2 < μE X kT 2

3.30
Similarly, we have

3.31
Noting that By the Gronwall inequality, we have

3.35
Thus, we have Thus, when condition iii is satisfied, we have Then, we obtain

3.39
Therefore, Noticing that V X 0 where a x 0 is initial state. For t ∈ kT, kT T , we have 3.42

Abstract and Applied Analysis
Noticing that E V x t ≥ λ m P E x t 2 , one obtains that Thereforce, by virtue of Definition 2.5 with p 2, we know that the equilibrium point of system 3.3 is uniformly exponentially stable in mean square. Now we will proof the pth moment uniformly exponentially stable in mean of system 3.3 .
For p 2q, q ≥ 1, we have

3.45
It is obvious that

3.46
Abstract and Applied Analysis

15
For t ∈ kT, kT T , we have < e t ln Θ/ρ P q x 0 2q P q a p e t ln Θ/ρ .

3.47
Similarly, we have
Therefore, we have shown that the trivial solution of system 3.3 is the pth moment uniformly exponentially stable in mean for even integers. In the same way, the theorem is satisfied for odd integers. Hence, we conclude the proof for all p > 0. Remark 3.4. The upper of PWM is given by that can easily be computed and optimized. We will employ a simple procedure in a specific example in Section 4.
Remark 3.5. The pth moment exponential stability considered, in this paper, the system tending to equilibrium speeds more quickly than others. The change of the status vectors as time increases will be showed in Figure 3. Corollary 3.6. Assume that A is Hurwitz stable, C 1 / 0, C 2 0, namely, the output of PWM feedback system y t linear dependences on current status vectors x t . If the parameter of PWM Mβ satisfies whenever G is sufficiently small (an upper bound of G has been given in the proof of Theorem 3.3),

Examples
Example 4.1. Consider the system 3.3 with one order Hurwitz stable plant described by transfer function T s 1/ s 1 . The state space representation of this system is given by  A −1, A d 0.1, B 1, C 1 1, C 2 −1, assuming the period T 1, the time delay d 0.1. Hence W T k , P i , G i T k , U i T k , i 1, 2, in Theorem 3.3 are calculated in this case as where the pulse width T k satisfies 0 < T k ≤ 0.9, and we obtain the estimation of the upper bound of Mβ, that is, Mβ ≤ 0.9398. For Mβ ∈ 0, 0.9398 , we compute μ max 0.7 such that for all μ < μ max . c μ 1 − 2 Φ √ μ − 4 P μ > 0 is true. Next, we compute δ max 0.22 such that K · f A d , G < μ max is true for all δ < δ max , where K 5K 1 1 M 2 β 2 B 2 C 2 e A 2 T and K 1 max s∈ 0,T e A T s e As . In Figure 2, we depict the estimates of the upper bound δ max of G versus Mβ. We observe that δ max decreases as Mβ increases. When the states are sufficiently far away from the origin so that T k T − d, the curvature of the curve reduces more slowly than without time-varying delays. Namely, as Mβ increases, the anti-interference performance of the system 3.3 is stronger than the stochastic PWM feedback system without time-varying delays. Furthermore, as M increases for fixed β , the maximum G allowable to ensure the pth moment uniform exponential stability in mean will decrease, besides, if disturbance of the feedback system 3.3 is increased less than δ max , the trivial solution of system 3.3 is the pth moment uniformly exponentially stable in mean by decreasing the value of Mβ, as shown in Figure 2. In Figure 3, we plot the sample response of x x 1 x 2 T with M 0.2, T β 1, d 0.1, and x 0 2 0.5 T . We observe the system 3.3 tending to the equilibrium point speed quickly.

Conclusions
We studied the stochastic PWM feedback systems with time-varying delays and established several Lyapunov and Lagrange criteria for the pth moment exponential stability in mean, then presented an algorithm to compute the upper bound for the parameters of PWM, and finally given two numerical examples to verify the effectiveness of theoretical results. We characterized the relationship among the parameters of pulse-width modulation, time delay, and the coefficient of state vectors of the feedback systems and showed that when the random disturbance is sufficiently small such PWM feedback system is the pth moment uniformly exponentially stable in mean provided that the upper bounds of parameters of pulse-width modulator are selected properly.