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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 736408, 19 pages
Moment Stability of the Critical Case of PWM Feedback Systems with Stochastic Perturbations
College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China
Received 13 August 2012; Accepted 20 September 2012
Academic Editor: Chuandong Li
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.
This paper further studies the moment stability of pulse-width-modulated (PWM) feedback system which is subjected to multiplicative and additive random disturbance modeled by the derivative of Wiener process. Different from the existing investigation, we focus on its critical case. The linear plant considered herein is assumed to be critically stable; that is, the plant has one and only one pole at the origin, and the rest of the poles are left half of complex plane. We establish several globally asymptotically stability criteria for such PWM feedback systems and then propose an algorithm to calculate the stability bound effectively. Furthermore, we present two numerical examples to show the effectiveness 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, it is always operated in all kinds of accidental or continued disturbances. It is of prime importance whether or not keeping the scheduled operation or work of the state. 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 [6–20]. By the method of positive kernels of integral operator, Halanaĭ in  proposed two versions of direct Lyapunov and Lagrange stability results for PWM feedback system with Hurwitz stable plant. In [12, 13], the authors studied the stability properties of nonlinear PWM feedback systems.
There are, however, only a few results concerning the qualitative properties of PWM feedback system subjected to random disturbance. Gupta and Jury  developed a method of determining the mean square value of the output of a PWM system with Gaussian random input. Sun et al.  presented a set of theoretical results for mean square exponential stability, asymptotical mean square stability, and the th moment exponential stability by using spectrum technique. Besides, in , the authors gave some definitions on the th moment stable in mean and established several th moment globally stability criteria in mean. Recently, the authors in [17, 18] investigated the dynamical systems subjected to noise disturbance by linear matrix inequalities (LMIs) and control Lyapunov methods. In , the authors established new Lyapunov and Lagrange stability results for pulse-width-modulated (PWM) feedback systems subjected to random disturbance. The linear plant considered therein was assumed to be Hurwitz stable. An optimization procedure was also presented in , which is expected to improve the analysis performance significantly. And then they have shown that when the parameters of PWM are within a certain computable range, the PWM feedback system is stable.
To the best of the authors’ knowledge, there are a few (if any) results for the stability analysis of the critical case of PWM systems with stochastic perturbations. In the present paper, we try to make the contribution on this issue. It is noted that the linear plant considered herein is critically stable; that is, the linear plant has one and only one pole at the origin, and the rest of the poles are left half of complex plane. Obviously, such systems are somewhat complex in comparison with most of the systems in literature. We will establish several Lyapunov and Lagrange criteria for the th moment uniform asymptotical stability in mean and then present an algorithm to compute the upper bound for the parameters of PWM. We will characterize the relationship between the parameters of PWM and the coefficients of state vectors of the feedback systems. It will be shown that when the random disturbance is sufficiently small, such PWM feedback system is the th moment globally asymptotically stable in mean provided that the upper bounds of parameters of pulse-width modulator are selected properly. We also demonstrate the effectiveness of our results by means of two numerical examples.
2. Notations and Some Definitions
We 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
Definition 2.4. is called an equilibrium point of a stochastic dynamical system if the set is invariant with respect to .
Definition 2.5. is said to be the th moment uniformly bounded if, for every and for every , there exists a such that if , then for all for all , where is a fixed point in is said to be the th moment uniformly ultimately bounded in mean if there exists and if for every and for every there exists a such that for all , for all , whenever , where is a fixed point in .
Definition 2.6. Let be a stochastic dynamical system, and let be the metric on . A set is said to be the th moment stable in mean (i.e., is said to be the th moment stable in mean) if, for any , there exists such that for any process , whenever , where denotes the expectation of a random process. If is independent of is said to be the th moment uniformly stable in mean. is said to be the th moment uniformly asymptotically stable in mean if it is uniformly stable in the th mean and if, for any , there exists a and a such that for any process , whenever implies that for all .
We can similarly define the th moment asymptotic stability in mean, the th moment asymptotic stability in mean in the large, the th moment uniform asymptotic stability in mean in the large, theth moment exponential stability in mean, and the th moment exponential stability in mean in the large. When , we speak of various stability and bound concepts in the mean square.
3. Stability Analysis of PWM Feedback System: The Critical Case
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 sign 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.
Furthermore, we assume that the linear plant is critically stable; that is, the matrix is stable with one and only one eigenvalue equal to zero. Without loss of generality, we assume that where is Hurwitz stable, and , and system (3.4) can be rewritten as
Note that is an equilibrium point of PWM feedback system (3.6).
Lemma 3.1 (The Itô isometry). Assume that is a scalar wiener process, and If is bounded and elementary, then
Proof. Put , then Using that and are independent if . Thus
Theorem 3.2. Assume that the matrix in (3.6) is Hurwitz stable, , and . Then the equilibrium of PWM feedback system (3.6) is the th moment uniformly asymptotically stable in mean in the large whenever there exists a positive definite matrix such that , and there exists a (an upper bounded of will be given in the proof ) such that , where
Proof. Since is nondecreasing in , the equilibrium is stable (resp., asymptotically stable, etc.) and implies that it is the qth moment stable (resp., asymptotically stable, etc.) in mean for all . Therefore, it suffices to prove the theorem for even integers.
Integrating (3.6), we have Therefore, when , with
To simplify our notations, we let Then (3.12) and (3.13) are reduced to
Since is Hurwitz stable, then is Schur stable. Therefore, there exists a positive definite matrix such that
Choosing the quadratic Lyapunov function where Then, where with Let where with Then (3.21) is reduced to By elementary transformation of matrix, we have where with Let and denote the minimum and maximum eigenvalues of a matrix, respectively. It is obvious that is a negative definite matrix if is positive definite matrix with . Note that, with , we have Therefore, the claim that is positive definite is true if with That is, where
Thus, whenever satisfies (3.34) for some positive constant , then is negative definite matrix, and, therefore, is negative definite matrix. By elementary transform of matrix, one observes that where So, we have .
Now we establish an estimate for . Let be arbitrary. We will show that there exists such that whenever .
For , we have Note that Then, By Lemma 3.1, one observes that Then (3.40) is reduced to where .
By Gronwall inequality, we have Thus, where
Thus, there exists a such that whenever . Choosing , so that . Then, we obtain that Therefore, is negative definite.
If we define and , then we have determined in an unambiguous way a deterministic dynamical system , corresponding to the stochastic dynamical system ( is the trivial solution set of system (3.6)). In this case is an invariant set for .
In fact, is uniformly asymptotically stable (see, e.g., ). For , we have
This implies that in the interval is bounded by , where is constant. Therefore, converges to the origin simultaneously with . We conclude that the trivial solution of deterministic system is asymptotically stable. Now recall that is uniformly stable if, for every , there exists a such that for all , and for all , whenever .
Since , thus, is uniformly stable means that, for every , there exists a such that for and for all , whenever , since ; hence, for every , for any process , we have, for , whenever , where . This is precisely the definition of uniform stability in the mean square of the trivial solution of . Therefore, the uniform stability of is equivalent to the uniform stability in the mean square of .
We can now conclude that the equilibrium of PWM feedback system (3.6) is uniformly asymptotically stable in mean square.
For , we have It can easily be verified that the second expectation in the above inequality can always be chosen to be less than for arbitrary , when is sufficiently small. Similarly as in , we can show that is bounded by when , where is a constant.
The rest of the proof of the th moment uniform asymptotic stability of the trivial solution of (3.6) with proceeds similarly as the proof of uniform asymptotic stability in the mean square of the trivial solution of (3.6).
Therefore, we have shown that the trivial solution of system (3.6) is the th moment uniformly asymptotically stable in mean for even integers, and, hence, we conclude the proof for all .
Remark 3.3. The upper bound of is given by (3.34) that can easily be computed and optimized. A simple procedure is presented in  to accomplish that. We will employ this procedure in a specific example in Section 4.
Theorem 3.4. Assume that is Hurwitz stable, , and . Then the trivial solution of the PWM feedback system (3.6) is the th moment uniformly asymptotically stable in mean whenever , and there exists a (an upper bounded of will be given in the proof of Theorem 3.2) such that .
Proof. When , we have
If is positive definite matrix. And if , then is positive negative definite matrix. Therefore, whenever and , it can easily be verified that
If , there exists such that (3.50) and are true for all . Hence, whenever for some positive constant , then is negative definite matrix.
The rest of the proof of the uniformly th moment asymptotic stability in mean of the trivial solution of (3.6) proceeds similarly as that of Theorem 3.2.
Remark 3.5. From stability results of Theorem 3.4, we can obtain when upper bound of the parameters of PWM feedback system () is sufficiently large, and if we choose that is sufficiently small, then the trivial solution of the system (3.6) is the th moment uniformly asymptotic stability in mean.
Next, we consider PWM feedback system with additive noise in the plant, described by equations of the form. We have where have positive components. We let denote the magnitude of the random noise. A block diagram of system (3.51) is shown in Figure 2.
Theorem 3.6. Assume that is Hurwitz stable, and the solution processes of the PWM feedback system (3.6) are the th moment uniformly ultimately bounded provided by , all (for random disturbance). When , the solution processes are unbounded.
To end this section, we present an algorithm for computing stability bound.
Stability Bound Algorithm
An upper bound of that satisfies (3.34) can be computed and optimized in the following manner.(S1) Determine the matrix by solving .(S2) Choose a precision level and a correspondingly dense partition of the interval , and say the set , where .(S3) For each , calculate (S4) Search for the largest such that where (S5) Repeat steps 1–4, using finer partition of the interval until there is no further significant improvement for .
Example 4.1. Consider the system (3.6) with second-order critical stable plant described by transfer function . The state space representation of this system is given by Hence , and in Theorem 3.2 are calculated in this case as Choosing , it can be seen that the stability bound is given by Assume that , and then the condition for uniformly asymptotic stability in th mean is given by In Example 4.1, we obtain the stability results depicted in Figure 3 ( versus ).
Example 4.2. Consider PWM feedback system (3.6) with transfer function . The state space representation of this system is given by
Letting , and assuming , we obtain the estimation of the upper bound of , that is, . For , we compute such that is true for all . Next, we compute such that is true for all , where and . In Figure 4, 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 , and the output of pulse-width modulator is either or . Therefore, as increases (for fixed ), the maximum allowable to ensure the th moment uniform asymptotic stability in mean will decrease; besides, if disturbance of the feedback system (3.6) is increase (less than ), the trivial solution of system (3.6) is the th moment uniformly asymptotically stable in mean by decreasing the value of , as shown in Figure 4.
In Figure 5, we plot the sample response of with and . We generated 100 sample responses of and computed the average of . Figure 6 shows the average tending to zero as time increases. However, does not diminish entirely to zero since it is an approximation to the mean .
We studied the critical case of PWM feedback systems with random perturbations and establish several Lyapunov and Lagrange criteria for the th moment uniform asymptotical stability in mean, and then we presented an algorithm to compute the upper bound for the parameters of PWM and finally gave two numerical examples to verify the effectiveness of theoretical results. We characterized the relationship between the parameters of PWM and the coefficients of state vectors of the feedback systems and showed that when the random disturbance is sufficiently small such that PWM feedback system is the th moment globally asymptotically stable in mean provided that the upper bounds of parameters of pulse-width modulator are selected properly.
This work was supported by the National Natural Science Foundation of China under Grant no. 60974020 and the Fundamental Research Funds for the Central Universities of China under Grant nos. CDJZR10100015 and CDJZR10185501.
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