Moment Stability of the Critical Case of PWM Feedback Systems with Stochastic Perturbations

Abstract

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.

1. 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, 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 [10] 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 [14] developed a method of determining the mean square value of the output of a PWM system with Gaussian random input. Sun et al. [15] presented a set of theoretical results for mean square exponential stability, asymptotical mean square stability, and the 𝑝th moment exponential stability by using spectrum technique. Besides, in [16], the authors gave some definitions on the 𝑝th moment stable in mean and established several 𝑝th 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 [20], 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 [20], 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 𝑝th 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 𝑝th 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 (Ω,𝐹,𝑃) denote the underlying probability space for all the systems that will be considered, where Ω  is the sample space, 𝐹 is the 𝜎-algebra of subsets of the sample space, and 𝑃 is the probability measure. An 𝑅^𝑛-valued random variable 𝑥 with domain 𝑋 is a measurable function from Ω to 𝑋⊂𝑅^𝑛. A family {𝑥(t),𝑡∈𝐼} of 𝑅^𝑛-valued random variables with domain 𝑋 defined on a probability space (Ω,𝐹,𝑃) is called a stochastic process with index set 𝐼 and state space (𝑋,𝐹^𝑛).

Definition 2.1. Let (𝑋,𝑑) be a metric space, 𝑋⊂𝑅^𝑛,𝐴⊂𝑋, and let 𝑇⊂𝑅⁺. For any fixed 𝑎∈𝐴 (𝑎 is called the initial state), 𝑡₀∈𝑇, a stochastic process {𝑥(𝑡,𝜔,𝑎,𝑡₀),𝑡∈𝑇_𝑎,𝑡0} with domain 𝑋 is called a stochastic motion if 𝑥(𝑡₀,𝜔,𝑎,𝑡₀)=𝑎 for all 𝜔∈Ω, where 𝑇_𝑎,𝑡0=[𝑡₀,𝑡₁)∩𝑇,𝑡₁>𝑡₀, and 𝑡₁ is finite or infinite.

Definition 2.2. Let 𝑆 be a family of stochastic motions with domain 𝑋 given by 𝑆⊂𝑥⋅,⋅,𝑎,𝑡₀∶𝑥𝑡₀,𝜔,𝑎,𝑡₀=𝑎,𝜔∈Ω,𝑎∈𝐴,𝑡₀∈𝑇. (2.1)
We call the four-tuple {𝑇,𝑋,𝐴,𝑆} a stochastic dynamical system.

Definition 2.3. Let {𝑇,𝑋,𝐴,𝑆} be a stochastic dynamical system. A set 𝑀⊂𝐴 is said to be invariant with respect to system 𝑆 (or short, (𝑆,𝑀) is invariant) if 𝑎∈𝑀 implies that 𝑃𝜔∶𝑥𝑡,𝜔,𝑎,𝑡₀∈𝑀∀𝑡∈𝑇_𝑎,𝑡0=1,∀𝑡₀∈𝑇,andall𝑥⋅,⋅,𝑎,𝑡₀∈𝑆. (2.2)

Definition 2.4. 𝑥₀∈𝐴 is called an equilibrium point of a stochastic dynamical system {𝑇,𝑋,𝐴,𝑆} if the set {𝑥₀} is invariant with respect to 𝑆.

Definition 2.5. 𝑆 is said to be the 𝑝th moment uniformly bounded if, for every 𝜂>0 and for every 𝑡₀∈𝑇, there exists a 𝛽=𝛽(𝜂)>0 such that if 𝑑(𝑎,𝑥₀)<𝜂, then for all 𝑥(⋅,⋅,𝑎,𝑡₀)∈𝑆,𝐸𝑑(𝑥(𝑡,𝜔,𝑎,𝑡₀),𝑥₀)^𝑝<𝛽 for all 𝑡∈𝑇_𝑎,𝑡0, where 𝑡₀ is a fixed point in 𝑋.𝑆 is said to be the 𝑝th moment uniformly ultimately bounded in mean if there exists 𝐵>0 and if for every 𝛿>0 and for every 𝑡₀∈𝑇 there exists a 𝜏=𝜏(𝛿) such that for all 𝑥(⋅,⋅,𝑎,𝑡₀)∈𝑆,𝐸𝑑(𝑥(𝑡,𝜔,𝑎,𝑡₀),𝑥₀)^𝑝<𝛽, for all 𝑡>𝑡₀+𝜏, whenever 𝑑(𝑎,𝑥₀)<𝛿, where 𝑥₀ is a fixed point in 𝑋.

Definition 2.6. Let {𝑇,𝑋,𝐴,𝑆} be a stochastic dynamical system, and let 𝑑 be the metric on 𝑋. A set 𝑀⊂𝐴 is said to be the 𝑝th moment stable in mean (i.e., (𝑆,𝑀) is said to be the 𝑝th moment stable in mean) if, for any 𝜀>0,𝑡₀∈𝑇, there exists 𝛿=𝛿(𝑡₀,𝜀) such that 𝐸𝑑(𝑥(𝑡,𝜔,𝑎,𝑡₀),𝑀)^𝑝<𝜀 for any process 𝑥(⋅,⋅,𝑎,𝑡₀)∈𝑆, whenever 𝑑(𝑎,𝑀)<𝛿, where 𝐸 denotes the expectation of a random process. If 𝛿 is independent of 𝑡₀,(𝑆,𝑀) is said to be the 𝑝th moment uniformly stable in mean. (𝑆,𝑀) is said to be the 𝑝th moment uniformly asymptotically stable in mean if it is uniformly stable in the 𝑝th mean and if, for any 𝜀>0, there exists a 𝛿=𝛿(𝜀)>0 and a 𝜏=𝜏(𝜀) such that for any process 𝑥(⋅,⋅,𝑎,𝑡₀)∈𝑆, whenever 𝑑(𝑎,𝑀)<𝛿 implies that 𝐸𝑑(𝑥(𝑡,𝜔,𝑎,𝑡₀),𝑀)^𝑝<𝜀 for all 𝑡>𝑡₀+𝜏.
We can similarly define the 𝑝th moment asymptotic stability in mean, the 𝑝th moment asymptotic stability in mean in the large, the 𝑝th moment uniform asymptotic stability in mean in the large, the𝑝th moment exponential stability in mean, and the 𝑝th moment exponential stability in mean in the large. When 𝑝=2, 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.

Figure 1 

Block diagram of PWM feedback systems subjected to multiplicative disturbances [20].

The pulse-width modulator is described by 𝑢(𝑡)(𝑒(𝑡=(𝑒(=𝑚))𝑀sgn𝑘𝑇))𝑡∈𝑘𝑇,𝑘𝑇+𝑇_𝑘,0𝑡∈𝑘𝑇+𝑇_𝑘,(𝑘+1)𝑇, (3.1) 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 sign are given, respectively, by 𝑇_𝑘=𝛽|𝑒(,|𝑒(≤𝑇𝑘𝑇)|𝑘𝑇)|𝛽;|𝑒(>𝑇𝑇,𝑘𝑇)|𝛽,sgn(𝜎)=1𝜎>00𝜎=0−1𝜎<0. (3.2)(3.3) 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 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)𝑑𝑡+𝐵𝑢(𝑡)𝑑𝑡+𝐺𝑥(𝑡)𝑑𝑊_𝑡,𝑦(𝑡)=𝐶𝑥(𝑡), (3.4) where 𝑥∈𝑅^𝑛,𝑦∈𝑅,𝑢∈𝑅 is output of the pulse-width modulator, 𝐴,𝐵,𝐶, and 𝐺 are matrices of appropriate dimensions, and 𝑊(𝑡) is a scalar wiener process.

Furthermore, we assume that the linear plant is critically stable; that is, the matrix 𝐴 is stable with one and only one eigenvalue equal to zero. Without loss of generality, we assume that 𝐴𝐴=¹0𝐵00,𝐵=¹𝑏,𝐶=𝐶₁1𝐺,𝐺=¹00𝑔, (3.5) where 𝐴₁ is Hurwitz stable, 𝑥₁∈𝑅^𝑛−1 and 𝑥₂∈𝑅, and system (3.4) can be rewritten as 𝑑𝑥₁(𝑡)=𝐴₁𝑥₁(𝑡)𝑑𝑡+𝐵₁𝑢(𝑡)𝑑𝑡+𝐺₁𝑥₁(𝑡)𝑑𝑊_𝑡,𝑑𝑥₂(𝑡)=𝑏𝑢(𝑡)𝑑𝑡+𝑔𝑥₂(𝑡)𝑑𝑊_𝑡,𝑦(𝑡)=𝐶₁𝑥₁(𝑡)+𝑥₂(𝑡). (3.6)

Note that (𝑥^𝑇₁,𝑥₂)=(0^𝑇,0) is an equilibrium point of PWM feedback system (3.6).

Lemma 3.1 (The Itô isometry). Assume that 𝑊(𝑡) is a scalar wiener process, and If 𝜙(𝑡,𝜔) is bounded and elementary, then 𝐸^𝑇_𝑆𝜙(𝑡,𝜔)𝑑𝑊_𝑡²=𝐸^𝑇_𝑆𝜙(𝑡,𝜔)²𝑑𝑡. (3.7)

Proof. Put Δ𝑊_𝑗=𝑊_𝑡𝑗+1−𝑊_𝑡𝑗, then𝐸𝑒_𝑖𝑒_𝑗Δ𝑊_𝑖Δ𝑊_𝑗=𝐸𝑒0if𝑖≠𝑗_2𝑗𝑡_𝑗+1−𝑡_𝑗if𝑖=𝑗. (3.8) Using that 𝑒_𝑖𝑒_𝑗Δ𝑊_𝑖 and Δ𝑊_𝑗 are independent if 𝑖<𝑗. Thus 𝐸^𝑇_𝑆𝜙(𝑡,𝜔)𝑑𝑊_𝑡²=_𝑖,𝑗𝐸𝑒_𝑖𝑒_𝑗Δ𝑊_𝑖Δ𝑊_𝑗=_𝑗𝑒²_𝑗⋅𝑡_𝑖−𝑡_𝑗=𝐸^𝑇_𝑆𝜙(𝑡,𝜔)²𝑑𝑡. (3.9)

Theorem 3.2. Assume that the matrix 𝐴₁ in (3.6) is Hurwitz stable, 𝑏>0, and 𝐶₁≠0. Then the equilibrium 𝑥_𝑒=0 of PWM feedback system (3.6) is the 𝑝th moment uniformly asymptotically stable in mean in the large whenever there exists a positive definite matrix 𝑃=^𝑃10_0𝜉 such that 𝑀𝛽<inf_{𝑇𝑘∈[0,𝑇]}(2𝑏𝜉/(𝐺₂(𝑇_𝑘)+𝑏²𝜉+max(0,𝜃(𝑇_𝑘)))), and there exists a 𝛿>0 (an upper bounded of 𝛿 will be given in the proof ) such that ‖𝐺‖<𝛿, where 𝜃𝑇_𝑘=𝑏²𝜉²𝜆_𝑀𝐶^𝑇₁𝐶₁−𝑏𝜉𝜆_𝑚𝑈𝑇_𝑘𝐶₁+𝑈𝑇_𝑘𝐶₁^𝑇+𝜆_𝑀𝑈𝑇_𝑘𝑈^𝑇𝑇_𝑘,𝑈𝑇_𝑘=𝑃₁−𝐼𝐼−𝑒^{−𝐴1𝑇𝑘}𝑇_𝑘𝐴₁⁻¹𝐵,𝐺₂𝑇_𝑘=𝐼−𝑒^{−𝐴1𝑇𝑘}𝑇_𝑘𝐴₁⁻¹𝐵^𝑇𝑃₁−𝐼𝐼−𝑒^{−𝐴1𝑇𝑘}𝑇_𝑘𝐴₁⁻¹𝐵. (3.10)

Proof. Since (𝐸‖𝑋‖^𝑝)^1/𝑝 is nondecreasing in 𝑝, the equilibrium 𝑥_𝑒=0 is stable (resp., asymptotically stable, etc.) and implies that it is the qth moment stable (resp., asymptotically stable, etc.) in mean for all 𝑞<𝑝. Therefore, it suffices to prove the theorem for even integers.
Integrating (3.6), we have 𝑥₁(𝑡)=𝑒^{𝐴1(𝑡−𝑘𝑇)}𝑥₁(𝑘𝑇)+^𝑡_𝑘𝑇𝑒^{𝐴1(𝑡−𝑠)}𝐵₁𝑢(𝑠)𝑑𝑠+^𝑡_𝑘𝑇𝑒^{𝐴1(𝑡−𝑠)}𝐺₁𝑥₁(𝑠)𝑑𝑊_𝑠,𝑥₂(𝑡)=𝑥₂(𝑘𝑇)+^𝑡_𝑘𝑇𝑏𝑢(𝑠)𝑑𝑠+^𝑡_𝑘𝑇𝑔𝑥₂(𝑠)𝑑𝑊_𝑠. (3.11) Therefore, when 𝑡=𝑘𝑇+𝑇, 𝑥₁(𝑘𝑇+𝑇)=𝑒^𝐴1𝑇𝑥₁(𝑘𝑇)−^𝑇𝑘₀𝑒^−𝐴1𝑠𝐵₁𝑀sgn𝐶₁𝑥₁(𝑘𝑇)+𝑥₂()+𝑘𝑇𝑑𝑠_{𝑘𝑇+𝑇𝑘𝑇}𝑒^{𝐴1(𝑘𝑇+𝑇−𝑠)}𝐺₁𝑥₁(𝑠)𝑑𝑊_𝑠=𝑒^𝐴1𝑇𝐼−𝑀𝛽𝑊𝜏_𝑘𝐶₁𝑥₁(𝑘𝑇)−𝑀𝛽𝑊₁𝜏_𝑘𝑥₂(𝑘𝑇)+^{𝑘𝑇+𝑇}_𝑘𝑇𝑒^{𝐴1(𝑘𝑇+𝑇−𝑠)}𝐺₁𝑥₁(𝑠)𝑑𝑊_𝑠,𝑥₂(𝑘𝑇+𝑇)=𝑥₂(𝑘𝑇)−^𝑇𝑘₀𝑏𝑀sgn𝐶₁𝑥₁(𝑘𝑇)+𝑥₂(𝑘𝑇)𝑑𝑠+^{𝑘𝑇+𝑇}_𝑘𝑇𝑔𝑥₂(𝑠)𝑑𝑊_𝑠𝑇=−𝑀𝛽𝑏^𝑘𝜏_𝑘𝐶₁𝑥₁()+𝑇𝑘𝑇1−𝑀𝛽𝑏^𝑘𝜏_𝑘𝑥₂(𝑘𝑇)+^{𝑘𝑇+𝑇}_𝑘𝑇𝑔𝑥₂(𝑠)𝑑𝑊_𝑠 (3.12)(3.13) with 𝜏_𝑘=𝛽𝐶₁𝑥₁(𝑘𝑇)+𝑥₂(=𝑇𝑘𝑇)_𝑘,𝑇_𝑘<𝑇,≥𝑇,𝑇_𝑘𝑊𝜏=𝑇,_𝑘=0𝜏_𝑘=0,𝐼−𝑒^{−𝐴1𝑇𝑘}𝑇_𝑘𝐴₁⁻¹𝐵𝜏_𝑘<𝑇,𝐼−𝑒^−𝐴1𝑇𝜏_𝑘𝐴₁⁻¹𝑇𝐵=𝜏_𝑘𝑊(𝑇)𝜏_𝑘≥𝑇. (3.14)
To simplify our notations, we let 𝐻=𝐼−𝑀𝛽𝑊𝜏_𝑘−𝑀𝛽𝑊𝜏_𝑘,𝑍(𝑘𝑇)=^{𝑘𝑇+𝑇}_𝑘𝑇𝑒^{𝐴1(𝑘𝑇+𝑇−𝑠)}𝐺₁𝑥₁(𝑠)𝑑𝑊_𝑠,𝑍(𝑘𝑇)=^{𝑘𝑇+𝑇}_𝑘𝑇𝑔𝑥₂(𝑠)𝑑𝑊_𝑠,𝜌𝜏_𝑘=0,𝜏_𝑘𝑇=0,_𝑘𝜏_𝑘,𝜏_𝑘∈(),𝐿𝜏0,∞_𝑘=𝑇−𝑀𝛽𝑏^𝑘𝜏_𝑘𝐶₁𝑇1−𝑀𝛽𝑏^𝑘𝜏_𝑘=−𝑀𝛽𝑏𝜌𝜏_𝑘𝐶₁1−𝑀𝛽𝑏𝜌𝜏_𝑘. (3.15) Then (3.12) and (3.13) are reduced to 𝑥₁(𝑘𝑇+𝑇)=𝑒^𝐴1𝑇()(),𝑥𝐻𝑥𝑘𝑇+𝑍𝑘𝑇₂(𝑘𝑇+𝑇)=𝐿𝜏_𝑘𝑥(𝑘𝑇)+𝑍(𝑘𝑇). (3.16)(3.17)
Since 𝐴₁ is Hurwitz stable, then 𝑒^𝐴1𝑇 is Schur stable. Therefore, there exists a positive definite matrix 𝑃₁=𝑃₁^𝑇 such that 𝑒^{𝐴1𝑇𝑇}𝑃₁𝑒^𝐴1𝑇−𝑃₁=−𝐼. (3.18)
Choosing the quadratic Lyapunov function 𝑉∶𝑅^𝑛→𝑅⁺,𝑉(𝑥)=𝑥^𝑇𝑃𝑥, (3.19) where 𝑃𝑃=¹00𝜉,𝑃₁>0,𝜉>0. (3.20) Then, 𝐸[𝑉(𝑥([𝑉(𝑥(𝑥𝑘𝑇+𝑇))]−𝐸𝑘𝑇))]=𝐸^𝑇(𝑘𝑇+𝑇)𝑃𝑥(𝑘𝑇+𝑇)−𝐸𝑥^𝑇()(𝑥𝑘𝑇𝑃𝑥𝑘𝑇)=𝐸^𝑇₁(𝑘𝑇+𝑇)𝑃₁𝑥₁(𝑘𝑇+𝑇)+𝐸𝑥^𝑇₂(𝑘𝑇+𝑇)𝜉𝑥₂(𝑥𝑘𝑇+𝑇)−𝐸^𝑇()(𝑒𝑘𝑇𝑃𝑥𝑘𝑇)=𝐸^𝐴1𝑇𝐻𝑥(𝑘𝑇)+𝑍(𝑘𝑇)^𝑇𝑃₁𝑒^𝐴1𝑇()()𝐿𝜏𝐻𝑥𝑘𝑇+𝑍𝑘𝑇+𝐸_𝑘𝑥(𝑘𝑇)+𝑍(𝑘𝑇)^𝑇𝜉𝐿𝜏_𝑘𝑥()+𝑍()𝑥𝑘𝑇𝑘𝑇−𝐸^𝑇()(𝑥𝑘𝑇𝑃𝑥𝑘𝑇)=𝐸^𝑇(𝑘𝑇)𝐻^𝑇𝑃₁−𝐼𝐻+𝜉𝐿^𝑇𝜏_𝑘𝐿𝜏_𝑘𝑥(𝑍−𝑃𝑘𝑇)+2𝐸^𝑇(𝑘𝑇)𝑃₁𝑒^𝐴1𝑇𝐻𝑥₁(𝑘𝑇)+2𝐸𝑍^𝑇(𝑘𝑇)𝜉𝑥₂(𝑍𝑘𝑇)+𝐸^𝑇(𝑘𝑇)𝑃₁𝑍(𝑘𝑇)+𝐸𝑍^𝑇()𝜉𝑍(𝑥𝑘𝑇𝑘𝑇)=𝐸^𝑇(𝑘𝑇)𝐻^𝑇𝑃₁−𝐼𝐻+𝜉𝐿^𝑇𝜏_𝑘𝐿𝜏_𝑘𝑥(ℎ−𝑃𝑘𝑇)+2𝐸^𝑇(𝑘𝑇)Φ𝑥(𝑘𝑇)+𝐸ℎ^𝑇(𝑘𝑇)𝑃ℎ(𝑘𝑇), (3.21) where 𝑥()=𝑥𝑘𝑇¹()𝑥𝑘𝑇₂()()=𝑍()𝑍(),𝑃𝑘𝑇,ℎ𝑘𝑇𝑘𝑇𝑘𝑇Φ=¹𝑒^𝐴1𝑇𝐻𝜉𝐿𝜏_𝑘=𝑃₁𝑒^𝐴1𝑇𝐼−𝑀𝛽𝑊𝜏_𝑘𝐶₁−𝑀𝛽𝑃₁𝑒^𝐴1𝑇𝑊𝜏_𝑘−𝑀𝛽𝑏𝜌𝜏_𝑘𝜉𝐶₁𝜉−𝑀𝛽𝑏𝜌𝜏_𝑘𝜉=Φ₁+,000𝜉 (3.22) with Φ₁=𝑃₁𝑒^𝐴1𝑇𝐼−𝑀𝛽𝑊₁𝜏_𝑘𝐶₁−𝑀𝛽𝑃₁𝑒^𝐴1𝑇𝑊₁𝜏_𝑘𝑀𝛽𝑏𝜌𝜏_𝑘𝜉−𝑀𝛽𝑏𝜌𝜏_𝑘𝜉. (3.23) Let 𝑄=𝐻^𝑇𝑃₁−𝐼𝐻+𝜉𝐿^𝑇𝜏_𝑘𝐿𝜏_𝑘=𝑈𝜏−𝑃−𝐼−𝑀𝛽_𝑘𝐶₁+𝑈𝜏_𝑘𝐶₁^𝑇+𝛾+2𝑀𝛽𝑏𝜌𝜏_𝑘𝜉𝐶₁𝐶^𝑇₁𝐷₁₂𝐷^𝑇₁₂𝛾, (3.24) where 𝛾=𝑀²𝛽²𝐺₂𝜏_𝑘+𝑏²𝜌²𝜏_𝑘𝜉−2𝑀𝛽𝑏𝜌𝜏_𝑘𝐷𝜉,₁₂=−𝑀𝛽𝑈𝜏_𝑘+𝛾+𝑀𝛽𝑏𝜌𝜏_𝑘𝜉𝐶^𝑇₁, (3.25) with 𝑈𝜏_𝑘=𝑃₁−𝐼𝑊𝜏_𝑘,𝐺₂𝜏_𝑘=𝑊^𝑇𝜏_𝑘𝑃₁−𝐼𝑊𝜏_𝑘. (3.26) Then (3.21) is reduced to 𝐸[𝑉(𝑥([𝑉(𝑥(𝑥𝑘𝑇+𝑇))]−𝐸𝑘𝑇))]=𝐸^𝑇(𝑘𝑇)𝑄𝑥(𝑘𝑇)+2𝐸ℎ^𝑇(𝑘𝑇)Φ₁𝑥(ℎ𝑘𝑇)+𝐸^𝑇(𝑘𝑇)𝑃ℎ(𝑘𝑇)+2𝜉𝐸𝑍(𝑘𝑇)𝑥₂(𝑘𝑇). (3.27) By elementary transformation of matrix, we have 𝐼𝐷𝑄=𝛾¹²𝐷𝐷01𝐼0_𝑇12𝛾1, (3.28) where 𝐷𝐷=−¹¹00−𝛾 (3.29) with 𝐷₁₁=1𝛾𝛾𝐼+𝑀²𝛽²𝑈𝜏_𝑘𝑈^𝑇𝜏_𝑘−𝑀²𝛽²𝑏𝜌𝜏_𝑘𝜉𝑈𝜏_𝑘𝐶₁+𝑈𝜏_𝑘𝐶₁^𝑇+𝑀²𝛽²𝑏²𝜌²𝜏_𝑘𝜉²𝐶^𝑇₁𝐶₁. (3.30) Let 𝜆_𝑚(⋅) and 𝜆_𝑀(⋅) denote the minimum and maximum eigenvalues of a matrix, respectively. It is obvious that 𝐷 is a negative definite matrix if 𝐷₁₁ is positive definite matrix with 𝛾<0. Note that, with 𝛾<0, we have 𝜆_𝑚𝐷₁₁≥1+𝑀𝛽²𝛾⁻¹𝜆_𝑀𝑈𝜏_𝑘𝑈^𝑇𝜏_𝑘−𝑏𝜌𝜏_𝑘𝜉𝜆_𝑚𝑈𝜏_𝑘𝐶₁+𝑈𝜏_𝑘𝐶₁^𝑇+𝑏²𝜌²𝜏_𝑘𝜉²𝜆_𝑀𝐶^𝑇₁𝐶₁. (3.31) Therefore, the claim that 𝐷₁₁ is positive definite is true if 𝑀𝛽𝐺₂𝜏_𝑘+𝑏²𝜌²𝜏_𝑘𝜉−2𝑏𝜌𝜏_𝑘𝜉+𝑀𝛽𝜆_𝑀𝑈𝜏_𝑘𝑈^𝑇𝜏_𝑘−𝑏𝜌𝜏_𝑘𝜉𝜆_𝑚𝑈𝜏_𝑘𝐶₁+𝑈𝜏_𝑘𝐶₁^𝑇+𝑏²𝜌²𝜏_𝑘𝜉²𝜆_𝑀𝐶^𝑇₁𝐶₁≤0 (3.32) with 𝑀𝛽𝐺₂𝜏_𝑘+𝑏²𝜌²𝜏_𝑘𝜉−2𝑏𝜌𝜏_𝑘𝜉<0. (3.33) That is, 𝑀𝛽<inf_{𝜏𝑘∈[0,∞]}2𝑏𝜌𝜏_𝑘𝜉𝐺₂𝜏_𝑘+𝑏²𝜉+max0,𝜃𝜏_𝑘=inf_{𝑇𝑘∈[]0,𝑇}2𝑏𝜉𝐺₂𝑇_𝑘+𝑏²𝜉+max0,𝜃𝑇_𝑘, (3.34) where 𝜃𝜏_𝑘=𝑏²𝜌²𝜏_𝑘𝜉²𝜆_𝑀𝐶^𝑇₁𝐶₁−𝑏𝜌𝜏_𝑘𝜉𝜆_𝑚𝑈𝜏_𝑘𝐶₁+𝑈𝜏_𝑘𝐶₁^𝑇+𝜆_𝑀𝑈𝜏_𝑘𝑈^𝑇𝜏_𝑘,𝜃𝑇_𝑘=𝑏²𝜉²𝜆_𝑀𝐶^𝑇₁𝐶₁−𝑏𝜉𝜆_𝑚𝑈𝑇_𝑘𝐶₁+𝑈𝑇_𝑘𝐶₁^𝑇+𝜆_𝑀𝑈𝑇_𝑘𝑈^𝑇𝑇_𝑘. (3.35)
Thus, whenever 𝑀𝛽 satisfies (3.34) for some positive constant 𝜉, then 𝐷 is negative definite matrix, and, therefore, 𝑄 is negative definite matrix. By elementary transform of matrix, one observes that Φ₁=𝐼𝑃₁𝑒^𝐴1𝑇𝑊𝜏_𝑘𝑏𝜌𝜏_𝑘𝜉Φ01²𝐶𝐼0₁1, (3.36) where Φ₂=𝑃₁𝑒^𝐴1𝑇00−𝑀𝛽𝜌𝜏_𝑘𝑏𝜉. (3.37) So, we have ‖Φ₁‖≤‖Φ₂‖.
Now we establish an estimate for ℎ(𝑘𝑇). Let 𝜇>0 be arbitrary. We will show that there exists 𝛿>0 such that 𝐸‖ℎ(𝑘𝑇)‖²<𝜇𝐸‖𝑥(𝑘𝑇)‖² whenever ‖𝐺‖<𝛿.
For 𝑡∈[𝑘𝑇,𝑘𝑇+𝑇], we have 𝑥(𝑡)=𝑥(𝑘𝑇)+^𝑡_𝑘𝑇(𝐴𝑥(𝑠)+𝐵𝑢(𝑠))𝑑𝑠+^𝑡_𝑘𝑇𝐺𝑥(𝑠)𝑑𝑊_𝑠. (3.38) Note that 𝐸^𝑡_𝑘𝑇‖𝐵‖|𝑢(𝑠)|𝑑𝑠²≤𝑀²𝛽²‖𝐵‖²‖𝐶‖²𝐸‖𝑥(𝑘𝑇)‖². (3.39) Then, 𝐸‖𝑥(𝑡)‖²4≤𝐸‖𝑥(𝑘𝑇)‖²+𝐸^𝑡_𝑘𝑇𝐴𝑥(𝑠)𝑑𝑠²+𝐸^𝑡_𝑘𝑇‖𝐵‖|𝑢(𝑠)|𝑑𝑠²+𝐸^𝑡_𝑘𝑇𝐺𝑥(𝑠)𝑑𝑊_𝑠^𝑇𝑡_𝑘𝑇𝐺𝑥(𝑠)𝑑𝑊_𝑠. (3.40) By Lemma 3.1, one observes that 𝐸^𝑡_𝑘𝑇𝐺𝑥(𝑠)𝑑𝑊_𝑠^𝑇𝑡_𝑘𝑇𝐺𝑥(𝑠)𝑑𝑊_𝑠=𝐸^𝑡_𝑘𝑇𝑥^𝑇(𝑠)𝐺^𝑇𝐺𝑥(𝑠)𝑑𝑠. (3.41) Then (3.40) is reduced to 𝐸‖𝑥(𝑡)‖²4≤𝐾₀𝐸‖𝑥(𝑘𝑇)‖²+‖𝐴‖²+‖𝐺‖^2𝑡_𝑘𝑇𝐸‖𝑥(𝑠)‖²𝑑𝑠, (3.42) where 𝐾₀=1+𝑀²𝛽²‖𝐵‖²‖𝐶‖².
By Gronwall inequality, we have 𝐸‖𝑥(𝑡)‖²≤4𝐾₀𝑒^{(‖𝐴‖2+‖𝐺‖2)(𝑡−𝑘𝑇)}𝐸‖𝑥(𝑘𝑇)‖²≤4𝐾₀𝑒^{(‖𝐴‖2+‖𝐺‖2)𝑇}𝐸‖𝑥(𝑘𝑇)‖². (3.43) Thus, 𝐸‖ℎ(𝑘𝑇)‖²=𝐸^{𝑘𝑇+𝑇}_𝑘𝑇𝑒^{𝐴1(𝑘𝑇+𝑇−𝑠)}𝐺₁𝑥₁(𝑠)𝑑𝑊_𝑠^{𝑇𝑘𝑇+𝑇}_𝑘𝑇𝑒^{𝐴1(𝑘𝑇+𝑇−𝑠)}𝐺₁𝑥₁(𝑠)𝑑𝑊_𝑠+𝐸^{𝑘𝑇+𝑇}_𝑘𝑇𝑔𝑥₂(𝑠)𝑑𝑊_𝑠^{𝑇𝑘𝑇+𝑇}_𝑘𝑇𝑔𝑥₂(𝑠)𝑑𝑊_𝑠=𝐸^{𝑘𝑇+𝑇}_𝑘𝑇𝑥^𝑇₁(𝑠)𝐺^𝑇₁𝑒^{𝐴𝑇1(𝑘𝑇+𝑇−𝑠)}𝑒^{𝐴1(𝑘𝑇+𝑇−𝑠)}𝐺₁𝑥₁(𝑠)𝑑𝑠+𝐸^{𝑘𝑇+𝑇}_𝑘𝑇𝑔²𝑥²₂(𝑠)𝑑𝑠=𝐸^{𝑘𝑇+𝑇}_𝑘𝑇𝑥^𝑇(𝑠)𝐺^𝑇𝐺𝑥(𝑠)𝑑𝑠≤𝐾₁‖𝐺‖^{2𝑘𝑇+𝑇}_𝑘𝑇𝐸‖𝑥(𝑠)‖²𝑑𝑠≤4𝐾₀𝐾₁‖𝐺‖²𝑒^{(‖𝐴‖2+‖𝐺‖2)𝑇}𝐸‖𝑥(𝑘𝑇)‖², (3.44) where 𝑒𝐺=^{𝐴1(𝑘𝑇+𝑇−𝑠)}0,𝐾0𝑔₁=max_{𝑠∈[0,𝑇]}𝑒^{𝐴𝑇1𝑠}𝑒^𝐴1𝑠,𝐾=4𝐾₀𝐾₁𝑒^{‖𝐴‖2𝑇}. (3.45)
Thus, there exists a 𝛿>0 such that 𝐾‖𝐺‖²𝑒^{‖𝐺‖2𝑇}<𝜇 whenever ‖𝐺‖<𝛿. Choosing 𝜇>0, so that 𝑐(𝜇)=1−2(‖Φ₂‖+𝜉)√𝜇−‖𝑃‖𝜇>0. Then, we obtain that 𝐸[𝑉(𝑥([𝑉(𝑥(𝑐𝜇𝑘𝑇+𝑇))]−𝐸𝑘𝑇))]≤−𝜆_𝑀(𝑃)𝐸[𝑉(𝑥(𝑘𝑇))]. (3.46) Therefore, 𝐸[𝑉(𝑥(𝑘𝑇+𝑇))]−𝐸[𝑉(𝑥(𝑘𝑇))] is negative definite.
If we define 𝑀={0} and 𝑆={𝑥∶𝑥(𝑡,𝑎,𝑡₀)=𝐸[𝑉(𝑥(𝑡,𝑎,𝑡₀))]}, then we have determined in an unambiguous way a deterministic dynamical system {𝑅⁺,𝑅⁺,𝑅⁺,𝑆}, corresponding to the stochastic dynamical system {𝑅⁺,𝑅^𝑛,𝑅^𝑛,𝑆₍₄₎} (𝑆₍₄₎ is the trivial solution set of system (3.6)). In this case 𝑀 is an invariant set for {𝑅⁺,𝑅⁺,𝑅⁺,𝑆}.
In fact, {𝐸𝑉(𝑥(𝑘𝑇)} is uniformly asymptotically stable (see, e.g., [21]). For 𝑡∈[𝑘𝑇,𝑘𝑇+𝑇], we have 𝐸[𝑉(𝑥(𝑡))]≤𝜆_𝑀(𝑃)𝐾𝑒^{‖𝐺‖2𝑇}𝐸‖𝑥(𝑘𝑇)‖²≤𝜆_𝑀(𝑃)𝜆_𝑚(𝑃)𝑉(𝑥(𝑘𝑇)). (3.47)
This implies that 𝐸[𝑉(𝑥(𝑡))] in the interval [𝑘𝑇,𝑘𝑇+𝑇] is bounded by 𝑟𝐸𝑉(𝑥(𝑘𝑇)), where 𝑟>0 is constant. Therefore, 𝐸[𝑉(𝑥(𝑡))] converges to the origin simultaneously with 𝐸𝑉(𝑥(𝑘𝑇)). We conclude that the trivial solution of deterministic system 𝑆 is asymptotically stable. Now recall that (𝑆,𝑀) is uniformly stable if, for every 𝜀>0,𝑡₀∈𝑅⁺, there exists a 𝛿=𝛿(𝜀)>0 such that 𝑥(𝑡,𝑎,𝑡₀)<𝜀 for all 𝑡∈𝑅⁺_𝑎,𝑡0=𝑅⁺_𝑎,𝑡0, and for all 𝑥(⋅,𝑎,𝑡₀)∈𝑆, whenever 𝑎<𝛿.
Since 𝑥(𝑡,𝑎,𝑡₀)=𝐸𝑉(𝑥(𝑡)), thus, (𝑆,𝑀) is uniformly stable means that, for every 𝜀>0,𝑡₀∈𝑅⁺, there exists a 𝛿=𝛿(𝜀)>0 such that 𝐸𝑉(𝑥(𝑡))<𝜀 for 𝑡∈𝑅⁺_𝑎,𝑡0 and for all 𝑥(⋅,𝑎,𝑡₀)∈𝑆, whenever 𝐸𝑉(𝑎)<𝛿, since 𝐸𝑉(𝑥(𝑡))/𝜆_𝑀(𝑃)≤𝐸‖𝑥(𝑡)‖²≤𝐸𝑉(𝑥(𝑡))/𝜆_𝑚(𝑃); hence, for every 𝜀>0,𝑡₀∈𝑅⁺, for any process 𝑥(⋅,⋅,𝑎,𝑡₀)∈𝑆, we have, for 𝑡∈𝑅⁺_𝑎,𝑡0,𝐸‖𝑥(𝑡)‖²≤𝜀/𝜆_𝑚(𝑃), whenever ‖𝑎‖<𝛿₁, where 𝛿₁=𝛿/𝜆_𝑀(𝑃). This is precisely the definition of uniform stability in the mean square of the trivial solution 𝑥_𝑒=0 of 𝑆₍₄₎. Therefore, the uniform stability of (𝑆,𝑀) is equivalent to the uniform stability in the mean square of (𝑆₍₄₎,{𝑥_𝑒}).
We can now conclude that the equilibrium 𝑥_𝑒=0 of PWM feedback system (3.6) is uniformly asymptotically stable in mean square.
For 𝑝=2𝑞,𝑞≥1, we have 𝐸𝑉(𝑥(𝑘𝑇+𝑇))^𝑞−𝐸𝑉(𝑥(𝑘𝑇))^𝑞=𝐸(𝑉(𝑥(𝑘𝑇+𝑇))−𝑉(𝑥(𝑘𝑇)))×𝑉(𝑥(𝑘𝑇+𝑇))^𝑞−1+⋯+𝑉(𝑥(𝑘𝑇))^𝑞−1−1≤𝐸𝜆_𝑀(𝑃)𝑉(𝑥(𝑘𝑇))+2𝐸ℎ^𝑇(𝑘𝑇)Φ𝑥(𝑘𝑇)+𝐸ℎ^𝑇()(×𝑉(𝑥(𝑘𝑇𝑃ℎ𝑘𝑇)𝑘𝑇+𝑇))^𝑞−1+⋯+𝑉(𝑥(𝑘𝑇))^𝑞−11≤−𝜆_𝑀(𝑃)𝐸𝑉(𝑥(𝑘𝑇))^𝑞+𝐸2ℎ^𝑇(𝑘𝑇)Φ𝑥(𝑘𝑇)+2ℎ^𝑇()()×𝑉(𝑥(𝑘𝑇Φℎ𝑘𝑇𝑘𝑇+𝑇))^𝑞−1+⋯+𝑉(𝑥(𝑘𝑇))^𝑞−1. (3.48) It can easily be verified that the second expectation in the above inequality can always be chosen to be less than 𝜇𝐸𝑉(𝑥(𝑘𝑇))^𝑞 for arbitrary 𝜇, when ‖𝐺‖ is sufficiently small. Similarly as in [22], we can show that 𝐸𝑉(𝑥(𝑡))^𝑞 is bounded by 𝐸𝑉(𝑥(𝑘𝑇))^𝑞 when 𝑡∈[𝑘𝑇,𝑘𝑇+𝑇], where 𝑎>0 is a constant.
The rest of the proof of the 𝑝th moment uniform asymptotic stability of the trivial solution of (3.6) with 𝑝=2𝑞 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 𝑝th moment uniformly asymptotically stable in mean for even integers, and, hence, we conclude the proof for all 𝑝>0.

Remark 3.3. The upper bound of 𝑀𝛽 is given by (3.34) that can easily be computed and optimized. A simple procedure is presented in [20] to accomplish that. We will employ this procedure in a specific example in Section 4.

Theorem 3.4. Assume that 𝐴₁ is Hurwitz stable, 𝐶₁=0, and 𝑏>0. Then the trivial solution of the PWM feedback system (3.6) is the 𝑝th moment uniformly asymptotically stable in mean whenever 𝑀𝛽<2/𝑏, and there exists a 𝛿>0 (an upper bounded of 𝛿 will be given in the proof of Theorem 3.2) such that ‖G‖<𝛿.

Proof. When 𝐶₁=0, we have 𝐷₁₁=1𝛾𝛾𝐼+𝑀²𝛽²𝑈𝜏_𝑘𝑈^𝑇𝜏_𝑘. (3.49) If 𝜆_𝑚(𝐷₁₁)>0,𝐷₁₁ is positive definite matrix. And if −𝛾>0, then 𝐷 is positive negative definite matrix. Therefore, whenever 𝜆_𝑚(𝐷₁₁)>0 and 𝛾<0, it can easily be verified that 𝑀𝛽𝑏²−2𝑏𝜉+𝜆_𝑀𝑈𝜏_𝑘𝑈^𝑇𝜏_𝑘+𝐺₂𝜏_𝑘>0. (3.50) If 𝑀𝛽<2/𝑏, there exists 𝜉 such that (3.50) and 𝜆<0 are true for all 𝑀𝛽. Hence, whenever 𝑀𝛽<2/𝑏 for some positive constant 𝜉, then 𝑄 is negative definite matrix.
The rest of the proof of the uniformly 𝑝th 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 (𝑀𝛽) is sufficiently large, and if we choose that 𝑏 is sufficiently small, then the trivial solution of the system (3.6) is the 𝑝th 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 𝑑𝑥₁(𝑡)=𝐴₁𝑥₁(𝑡)𝑑𝑡+𝐵₁𝑢(𝑡)𝑑𝑡+𝜀₁𝑥₁(𝑡)𝑑𝑊_𝑡,𝑑𝑥₂(𝑡)=𝑏𝑢(𝑡)𝑑𝑡+𝜀₂𝑥₂(𝑡)𝑑𝑊_𝑡,𝑦(𝑡)=𝐶₁𝑥₁(𝑡)+𝑥₂(𝑡), (3.51) where 𝜀₁∈𝑅^𝑛−1,𝜀₂∈𝑅 have positive components. We let 𝜀=(𝜀₁𝜀₂)∈𝑅^𝑛 denote the magnitude of the random noise. A block diagram of system (3.51) is shown in Figure 2.

Figure 2 

Block diagram of PWM feedback systems subjected to additive disturbances.

Theorem 3.6. Assume that A₁ is Hurwitz stable, and the solution processes of the PWM feedback system (3.6) are the 𝑝th moment uniformly ultimately bounded provided by 𝑏>0, all 𝑀𝛽∈(0,∞) (for random disturbance). When 𝑏≤0, the solution processes are unbounded.

Proof. The proof of Theorem 3.6 is similar as that of [20].

To end this section, we present an algorithm for computing stability bound.

Stability Bound Algorithm
An upper bound of 𝑀𝛽 that satisfies (3.34) can be computed and optimized in the following manner.(S1) Determine the matrix 𝑃₁ by solving (𝑒^𝐴1𝑇)^𝑇𝑃₁𝑒^𝐴1𝑇−𝑃₁=−𝐼.(S2) Choose a precision level 𝛿>0 and a correspondingly dense partition of the interval [0,𝑇], and say the set {𝑡₀=0,𝑡₁,…,𝑡_𝑁=𝑇}, where 0<𝑡_𝑗+1−𝑡_𝑗<𝛿,𝑗=0,1,2,…,𝑁−1.(S3) For each　𝑖,𝑗=0,1,2,…,𝑁−1, calculate 𝑊𝑡_𝑗=𝐼−𝑒^{−𝐴1𝑡𝑗}𝑡_𝑗𝐴₁⁻¹𝐵,𝐺₂𝑡_𝑗=𝑊𝑡_𝑗^𝑇𝑃₁−𝐼𝑊𝑡_𝑗,𝑈𝑡_𝑗=𝑃₁−𝐼𝑊𝑡_𝑗. (3.52)(S4) Search for the largest　𝑀𝛽　such that 𝑀𝛽<inf_{𝑡𝑗∈[0,∞]}2𝑏𝜌𝑡_𝑗𝜉𝐺₂𝑡_𝑗+𝑏²𝜉+max0,𝜃𝑡_𝑗, (3.53) where 𝜃𝑡_𝑗=𝑏²𝜉²𝜆_𝑀𝐶^𝑇₁𝐶₁−𝑏𝜉𝜆_𝑚𝑈𝑡_𝑗𝐶₁+𝑈𝑡_𝑗𝐶₁^𝑇+𝜆_𝑀𝑈𝑡_𝑗𝑈^𝑇𝑡_𝑗. (3.54)(S5) Repeat steps 1–4, using finer partition of the interval [0,𝑇] until there is no further significant improvement for 𝑀𝛽_opt.

4. Examples

Example 4.1. Consider the system (3.6) with second-order critical stable plant described by transfer function 𝑇(𝑠)=𝑘/𝑠(1+(𝑠/𝑎)),𝑎>0. The state space representation of this system is given by 𝑘𝑘𝐴=−𝑎000,𝐵=,𝐶=11. (4.1) Hence 𝐴₁=−𝑎,𝐵₁=𝑘,𝐶₁=1, and 𝑃₁,𝑊(𝑇_𝑘),𝑈(𝑇_𝑘) in Theorem 3.2 are calculated in this case as 𝑃₁=11−𝑒^−2𝑎𝑇,𝑊𝑇_𝑘=𝑒^𝑎𝑇𝑘−1𝑘𝑎𝑇_𝑘,𝑈𝑇_𝑘=𝑘𝑒^𝑎𝑇𝑘−1𝑎𝑇_𝑘𝑒^2𝑎𝑇−1,𝐺₂𝑇_𝑘=𝑘²𝑒^𝑎𝑇𝑘−1²𝑎²𝑇²_𝑘𝑒^2𝑎𝑇,𝜃𝑇−1_𝑘=𝑘²𝜉²−2𝑘²𝑒^𝑎𝑇𝑘−1𝑎𝑇_𝑘𝑒^2𝑎𝑇𝑘−1𝜉+²𝑒^𝑎𝑇𝑘−1²𝑎²𝑇²_𝑘𝑒^2𝑎𝑇−1²=𝑘²𝑒𝜉−^𝑎𝑇𝑘−1𝑎𝑇_𝑘𝑒^2𝑎𝑇−1²≥0. (4.2) Choosing 𝜉=1/𝑎𝑇(1+𝑒^−𝑎𝑇), it can be seen that the stability bound is given by 1𝑀𝛽<𝑒^𝑎𝑇−1/𝑎𝑇𝑒^𝑎𝑇+()𝑘.+11/2 (4.3) Assume that 𝑀=1, and then the condition for uniformly asymptotic stability in 𝑝th mean is given by 1>1𝑘𝛽−2𝑎𝑇𝑎𝑇𝑒^𝑎𝑇+1+12. (4.4) In Example 4.1, we obtain the stability results depicted in Figure 3 (𝑘𝛽_opt versus 𝑎𝑇).

Figure 3 

Estimates of upper bounds for 𝑘𝛽 for the second-order PWM feedback system of Example 4.1.

Example 4.2. Consider PWM feedback system (3.6) with transfer function 𝑇(𝑠)=1/𝑠(1+𝑠). The state space representation of this system is given by 11𝐴=−1000,𝐵=,𝐶=11. (4.5) Letting 𝛿=0.01, and assuming 𝑇=1, we obtain the estimation of the upper bound of 𝑀𝛽, that is, 𝑀𝛽≤1.0394. For 𝑀𝛽∈(0,1.0394), we compute 𝜇_max such that 𝑐(𝜇)=1−2(‖𝑅₂‖+𝜉)√𝜇−‖𝑃‖𝜇>0 is true for all 𝜇<𝜇_max. Next, we compute 𝛿_max such that 𝐾𝛿²𝑒^𝛿2𝑇<𝜇_max is true for all 𝛿<𝛿_max, where 𝐾=4𝐾₁(1+𝑀²𝛽²‖𝐵‖²‖𝐶‖²)𝑒^{‖𝐴‖2𝑇} and 𝐾₁=max_{𝑠∈[0,𝑇]}‖𝑒^{𝐴𝑇1𝑠}𝑒^𝐴1𝑠‖. In Figure 4, 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, and the output of pulse-width modulator is either +𝑀 or −𝑀. Therefore, as 𝑀 increases (for fixed 𝛽), the maximum ‖𝐺‖ allowable to ensure the 𝑝th moment uniform asymptotic stability in mean will decrease; besides, if disturbance of the feedback system (3.6) is increase (less than 𝛿_max), the trivial solution of system (3.6) is the 𝑝th moment uniformly asymptotically stable in mean by decreasing the value of 𝑀𝛽, as shown in Figure 4.

Figure 4 

Upper bounds for ‖𝐺‖ when 𝑀𝛽∈(0,1.0394).

In Figure 5, we plot the sample response of 𝑥=(𝑥₁𝑥₂)^𝑇 with 𝑀=0.2,𝑇=𝛽=1,𝐺=^0.020_00.006 and 𝑥(0)=(20.5)^𝑇. We generated 100 sample responses of 𝑥(𝑡) and computed the average ‖𝑥(𝑡)‖ of ‖𝑥(𝑡)‖. Figure 6 shows the average ‖𝑥(𝑡)‖ tending to zero as time 𝑡 increases. However, ‖𝑥(𝑡)‖ does not diminish entirely to zero since it is an approximation to the mean 𝐸‖𝑥(𝑡)‖.

Figure 5 

Sample response of PWM feedback system (3.6) in Example 4.2.

Figure 6 

‖𝑥(𝑡)‖ of PWM feedback system (3.6) in Example 4.2.

5. Conclusions

We studied the critical case of PWM feedback systems with random perturbations and establish several Lyapunov and Lagrange criteria for the 𝑝th 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 𝑝th moment globally asymptotically stable in mean provided that the upper bounds of parameters of pulse-width modulator are selected properly.

Acknowledgment

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.