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th Moment Exponential Stability of Stochastic PWM Feedback Systems with Time-Varying Delays
This paper further studies the th moment exponential stability of stochastic pulse-width-modulated (PWM) 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 effectiveness and conservativeness of the theoretical results.
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 , the authors investigated robust exponential stability and delayed-state-feedback stabilization of uncertain impulsive stochastic systems with time-varying delays. Besides, Sun and Cao  give some definitions on the th moment exponential stability in mean and established several th moment globally stability criteria in mean. In [12, 13], Hou and Michel established new Lyapunov and Lagrange stability results for pulse-width-modulation (PWM) feedback systems subjected to random disturbance.
To the best of the authors’ knowledge, there are few (if any) 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 integration method, we will establish several Lyapunov and Lagrange criteria for 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 coefficient of state vectors of the feedback systems. It will be shown that when the random disturbance is sufficiently small such PWM feedback system is 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.
2. Notations and Some Definitions
Let denote the underlying probability space for all the systems that will be considered, where is the sample space, is the -algebra of subsets of the sample space, and is the probability measure. An -valued random variable with domain is a measurable function from to . A family of valued random variables with domain defined on a probability space is called a stochastic process with index set and state space .
Definition 2.1. Let be a metric space, , , and let . For any fixed ( is called the initial state), , a stochastic process with domain is called a stochastic motion if for all , where , , and is finite or infinite.
Definition 2.2. Let be a family of stochastic motions with domain given by We call the four-tuple a stochastic dynamical system.
Definition 2.3. Let be a stochastic dynamical system. A set is said to be invariant with respect to system (or short, is invariant) if implies that for all and all.
Definition 2.4. is called an equilibrium point of a stochastic dynamical system if the set is invariant with respect to .
Definition 2.5. Let be a stochastic dynamical system, and let be the metric on . A set is said to be the th moment exponentially stable in mean (i.e., is said to be the th moment exponentially stable in mean) if for any , there exists a and constants , such that for any process , whenever , where is called the initial state and denotes the expectation of a random process. If is independent of , is said to be the th moment uniformly exponentially stable in mean. is said to be the th moment uniformly asymptotically exponentially stable in mean if it is uniformly stable in the th mean and if there exists , and constants , such that for any process , whenever implies that for all .
3. 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 with being the external input and the system output, and, for , the pulse-width and the sign function sgn are given, respectively, by The sampling period , the amplitude of the pulse , and are all assumed to be constants. And throughout this paper, we always assume that . Under these assumptions, the PWM feedback system with the output function can be described by where , , , is output of the pulse-width modulator, , and are matrices of appropriate dimensions, , , , and is a scalar wiener process.
Note that is an equilibrium point of PWM feedback system (3.3).
Lemma 3.1 (Schur complement). Given the matrix . Then is equivalent to any one of the following conditions:(i), ,(ii), .
Lemma 3.2 (the It isometry). Assume is a scalar wiener process. If is bounded and elementary, then
Proof. Put , then using that and are independent if . Thus
Theorem 3.3. Assume that the matrix in (3.3) is Hurwitz stable. Then the equilibrium of PWM feedback system (3.3) is the th moment uniformly exponentially stable in mean in the large provided that the following conditions are satisfied:(i) where with (ii) where (iii)there exists a constant , such that, whenever , where is scalar satisfying with where satisfying
Proof. Since is nondecreasing in , the equilibrium is stable (resp., asymptotically stable, etc.) implying that it is the th moment stable (resp., asymptotically stable, etc.) in mean for all . Firstly, we will provide to prove the theorem for even integers.
Integrating (3.3), we have Therefore, when , For , we have with
To simplify our notations, let Then, (3.17) and (3.18) are reduced to
Choosing the quadratic Lyapunov-Krasovskii functional , Then, where Then (3.23) is reduced to Let and denote the minimum and maximum eigenvalues of a matrix, respectively. By Lemma 3.1, it is obvious that is a negative definite matrix if and only if and . Noting that
From condition (i) it follows that , and hence the claim that is true. From condition (ii) we then conclude that is a negative definite matrix based on Lemma 3.1.
Now we establish an estimation on and . Let be arbitrary. We will show that there exists whenever , such that For , we have Then, we have where Similarly, we have Noting that By Lemma 3.2, one observes that Then the inequality (3.32) is reduced to where .
By the Gronwall inequality, we have Thus, we have where Thus, when condition (iii) is satisfied, we have Then, we obtain Therefore, Noticing that , where is initial state.
For , we have where Noticing that , one obtains that where
Thereforce, by virtue of Definition 2.5 with , we know that the equilibrium point of system (3.3) is uniformly exponentially stable in mean square.
Now we will proof the th moment uniformly exponentially stable in mean of system (3.3).
For , , we have It is obvious that For , we have Similarly, we have where is constant.
Therefore, we have shown that the trivial solution of system (3.3) is the th 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 .
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 th 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 is Hurwitz stable, , , namely, the output of PWM feedback system linear dependences on current status vectors . If the parameter of PWM satisfies whenever is sufficiently small (an upper bound of has been given in the proof of Theorem 3.3), where , with , by choosing satisfied .
Example 4.1. Consider the system (3.3) with one order Hurwitz stable plant described by transfer function . The state space representation of this system is given by , , , , , assuming the period , the time delay . Hence , in Theorem 3.3 are calculated in this case as
where the pulse width satisfies , and we obtain the estimation of the upper bound of , that is, . For , we compute such that for all . is true. Next, we compute such that is true for all , where and . In Figure 2, we depict the estimates of the upper bound of versus .
We observe that decreases as increases. When the states are sufficiently far away from the origin so that , the curvature of the curve reduces more slowly than without time-varying delays. Namely, as increases, the anti-interference performance of the system (3.3) is stronger than the stochastic PWM feedback system without time-varying delays. Furthermore, as increases (for fixed ), the maximum allowable to ensure the th moment uniform exponential stability in mean will decrease, besides, if disturbance of the feedback system (3.3) is increased (less than ), the trivial solution of system (3.3) is the th moment uniformly exponentially stable in mean by decreasing the value of , as shown in Figure 2.
Example 4.2. Consider PWM feedback system (3.3) with transfer function .
The state space representation of this system is given by In Figure 3, we plot the sample response of with , and . We observe the system (3.3) tending to the equilibrium point speed quickly.
We studied the stochastic PWM feedback systems with time-varying delays and established several Lyapunov and Lagrange criteria for the th 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 th moment uniformly exponentially stable in mean provided that the upper bounds of parameters of pulse-width modulator are selected properly.
The work was supported by the Fundamental Research Funds for the Central Universities of China under Grants CDJZR10100015.
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Copyright © 2012 Zhong Zhang and Lixia Ye. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.