# pth Moment Exponential Stability of Stochastic PWM Feedback Systems with Time-Varying Delays

Authors: Zhong Zhang; Lixia Ye
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper further studies thepth 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.


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## Body

## 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, 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 stochastic systems with time-varying delays. Besides, Sun and Cao [17] give some definitions on the pth moment exponential stability in mean and established several pth 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 isHurwitz 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 Ito^ integration method, we will establish several Lyapunov and Lagrange criteria for pth 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 pth 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(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 Rn-valued random variable x with domain X is a measurable function from Ω to X⊂Rn. A family {x(t),t∈I} of Rn 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,Fn).Definition 2.1.
Let(X,d) be a metric space, X⊂Rn, 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∈Ta,t0} with domain X is called a stochastic motion if x(t0,ω,a,t0)=a for all ω∈Ω, where Ta,t0=[t0,t1)∩T, t1>t0, and t1 is finite or infinite.Definition 2.2.
LetS be a family of stochastic motions with domain X given by
(2.1)S⊂{x(·,·,a,t0):x(t0,ω,a,t0)=a,ω∈Ω,a∈A,t0∈T}.
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 P{ω:x(t,ω,a,t0)∈Mforallt∈Ta,t0}=1, for  all t0∈Tand  allx(·,·,a,t0)∈S.Definition 2.4.
x
0
∈
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.
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 exponentially stable in mean (i.e., (S,M) is said to be the pth moment exponentially stable in mean) if for any t0∈T, there exists a δ=δ(t0,ε) and constants β>0, k>1 such that E[d(x(t,ω,a,t0),M)p]<k∥a∥pe-β(t-t0) for any process x(·,·,a,t0)∈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 t0, (S,M) is said to be thepth moment uniformly exponentially stable in mean. (S,M) is said to be thepth 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,t0)∈S, whenever d(a,M)<δ implies that E[d(x(t,ω,a,t0),M)p]<k∥a∥pe-β(t-t0) for all t>t0+τ.


## 3. Main Results

The PWM feedback system to be considered in this paper is shown in Figure1.Figure 1
Block diagram of stochastic PWM feedback systems with time-varying delays.The pulse-width modulator is described by(3.1)u(t)=m(e(t))={Msgn(e(kT))t∈[kT,kT+Tk),0t∈[kT+Tk,kT+T),
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 Tk and the sign function sgn are given, respectively, by
(3.2)Tk={β|e(kT)|,|e(kT)|≤Tβ,T,|e(kT)|>Tβ,sgn(σ)={1σ>0,0σ=0,-1σ<0.
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
(3.3)dx(t)=Ax(t)dt+Adx(t-d)dt+Bu(t)dt+Gx(t)dWt,y(t)=C1x(t)+C2x(t-d),
where x∈Rn, y∈R, u∈R, is output of the pulse-width modulator, A,Ad,B,C1,C2, and G are matrices of appropriate dimensions, d=(di)∈Rn, 0<di≤T-Tk, i=1,2,…,n, and W(t) is a scalar wiener process.Note that(x1T,x2)=(0T,0)T is an equilibrium point of PWM feedback system (3.3).Lemma 3.1 (Schur complement).
Given the matrixS=(S11S12S21S22). Then S<0 is equivalent to any one of the following conditions:(i)
S11<0, S22-S21S11-1S12<0,(ii)
S22<0, S11-S12S22-1S21<0.Lemma 3.2 (the Ito^ isometry).
AssumeW(t) is a scalar wiener process. If ϕ(t,ω) is bounded and elementary, then
(3.4)E[(∫STϕ(t,ω)dWt)2]=E[∫STϕ(t,ω)2dt].Proof.
PutΔWj=Wtj+1-Wtj, then
(3.5)E[eiejΔWiΔWj]={0ifi≠j,E[ej2](tj+1-tj)ifi=j,
using that eiejΔWi and ΔWj are independent if i<j. Thus
(3.6)E[(∫STϕ(t,ω)dWt)2]=∑i,jE[eiejΔWiΔWj]=∑jej2·(ti-tj)=E[∫STϕ(t,ω)2dt].Theorem 3.3.
Assume that the matrixA in (3.3) is Hurwitz stable. Then the equilibrium xe=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:(i)
(3.7)Mβ<infτk∈[0,∞)λm(Γ1)+λm2(Γ1)+4λM(Γ2)2λM(Γ2),
where
(3.8)Γ1=U1(τk)C1+(U1(τk)C1)T,Γ2=C1T(G1(τk)+G2(τk))C1
with
(3.9)Gi(Tk)=W(Tk)T(Pi-I)W(Tk)Ui(Tk)=(Pi-I)W(Tk),i=1,2,(ii)
(3.10)φ22-φ12φ11-1φ12T<0,
where
(3.11)φ11=-I-MβΓ1+M2β2Γ2,φ12=-Mβ[U1(τk)C2+(U2(τk)C1)TeAd]φ12=+M2β2C1T(G1(τk)+G2(τk))C2,φ22=(eAT)TP2eAT-P2-Mβ[(U2(τk)C2)TeAd+(eAd)TU2(τk)C2]φ22=+M2β2C2T(G1(τk)+G2(τk))C2,(iii)
there exists a constantδ>0, such that, whenever ∥G∥<δ,
(3.12)max{∥Ad∥2,∥G∥2}·e(∥Ad∥2+∥G∥2)T<μ,
where μ is scalar satisfying
(3.13)c(μ)=1-2∥Φ∥μ-4∥P∥μ>0
with
(3.14)Φ=(P1eAT(I-MβW(τk)C1)-MβP1eATW(τk)C2-MβP2eA(T-d)W(τk)C1P2eA(T-d)(eAd-MβW(τk)C2)),
where P1=P1T>0,P2=P2T>0 satisfying
(3.15)(eAT)TP1eAT-P1=-I,(eA(T-d))TP2eA(T-d)-P2=-I.Proof.
Since(E∥X∥p)1/p is nondecreasing in p, the equilibrium xe=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.
Integrating (3.3), we have
(3.16)x(t)=eA(t-kT)x(kT)+∫kTteA(t-s)Adx(s-d)ds+∫kTteA(t-s)Bu(s)ds+∫kTteA(t-s)Gx(s)dWs.
Therefore, when t=kT+T,
(3.17)x(kT+T)=eATx(kT)+∫kTkT+TeA(kT+T-s)Bu(s)ds+∫kTkT+TeA(kT+T-s)Adx(s-d)ds+∫kTkT+TeA(kT+T-s)Gx(s)dWs=eATx(kT)-∫kTkT+TkeA(kT+T-s)BMsgn(C1x(kT)+C2x(kT-d))ds+∫kTkT+TeA(kT+T-s)Adx(s-d)ds+∫kTkT+TeA(kT+T-s)Gx(s)dWs=eAT[(I-MβW(τk)C1)x(kT)-MβW(τk)C2x(kT-d)]+∫kT-dkT+T-deA(kT+T-d-s)Adx(s)ds+∫kTkT+TeA(kT+T-s)Gx(s)dWs.
For t=kT+T-d, we have
(3.18)x(kT+T-d)=eATx(kT-d)+∫kT-dkT+T-deA(kT+T-d-s)Bu(s)ds+∫kT-dkT+T-deA(kT+T-d-s)Adx(s-d)ds+∫kT-dkT+T-deA(kT+T-d-s)Gx(s)dWs=eATx(kT-d)-∫kTkT+TkeA(kT+T-d-s)BMsgn(C1x(kT)+C2x(kT-d))ds+∫kT-dkT+T-deA(kT+T-d-s)Adx(s-d)ds+∫kT-dkT+T-deA(kT+T-d-s)Gx(s)dWs=eA(T-d)[-MβW(τk)C1x(kT)+(eAd-MβW(τk)C2)x(kT-d)]+∫kT-dkT+T-deA(kT+T-d-s)Adx(s-d)ds+∫kTkT+TeA(kT+T-s)Gx(s-d)dWs
with
(3.19)τk=β|C1x(kT)+C2x(kT-d)|={Tk,Tk<T,≥T,Tk=T,W(τk)={0,τk=0,I-e-ATkTkA-1B,τk<T,I-e-ATτkA-1B=TτkW(T),τk≥T.
To simplify our notations, let(3.20)H1=[I-MβW(τk)C1-MβW(τk)C2],H2=[-MβW(τk)C1eAd-MβW(τk)C2],h(kT)=∫kT-dkT+T-deA(kT+T-d-s)Adx(s)ds,h(kT-d)=∫kT-dkT+T-deA(kT+T-d-s)Adx(s-d)ds,h~(kT)=∫kTkT+TeA(kT+T-s)Gx(s)dWs,h~(kT-d)=∫kTkT+TeA(kT+T-s)Gx(s-d)dWs,X(kT)=[xT(kT),xT(kT-d)]T.
Then, (3.17) and (3.18) are reduced to
(3.21)x(kT+T)=eATH1X(kT)+h(kT)+h~(kT),x(kT+T-d)=eA(T-d)H2X(kT-d)+h(kT-d)+h~(kT-d).
Choosing the quadratic Lyapunov-Krasovskii functionalV:Rn→R+,
(3.22)V(x)=xT(t)P1x(t)+xT(t-d)P2x(t-d).
Then,
(3.23)∇E[V(x)]=E[V(x(kT+T))]-E[V(x(kT))]-E[V(x(kT))]-E[V(x(kT-d))]=E[xT(kT+T)P1x(kT+T)]+E[xT(kT+T-d)P2x(kT+T-d)]-E[xT(kT)P1x(kT)]-E[x(kT-d)P2x(kT-d)]=E[XT(kT)(H1T(P1-I)H1+H2T(P2-I)H2-P)X(kT)]+2E[(h(kT)+h~(kT))TP1eATH1X(kT)]+2E[(h(kT-d)+h~(kT-d))TP2eA(T-d)H2X(kT)]+E[(h(kT)+h~(kT))TP1(h(kT)+h~(kT))]+E[(h(kT-d)+h~(kT-d))TP2(h(kT-d)+h~(kT-d))]=E[XT(kT)(H1T(P1-I)H1+H2T(P2-I)H2-P)X(kT)]+2E[ZT(kT)ΦX(kT)]+2E[Z~T(kT)ΦX(kT)]+E[ZT(kT)PZ(kT)]+2E[ZT(kT)PZ~(kT)]+E[Z~T(kT)PZ~(kT)],
where
(3.24)Z(kT)=(hT(kT)hT(kT-d))T,Z~(kT)=(h~T(kT)h~T(kT-d))T,Φ=(P1eATH1P2eA(T-d)H2)=(P1eAT(I-MβW(τk)C1)-MβP1eATW(τk)C2-MβP2eA(T-d)W(τk)C1P2eA(T-d)(eAd-MβW(τk)C2)),H=H1T(P1-I)H1+H2T(P2-I)H2-P=(φ11φ12φ12Tφ22).
Then (3.23) is reduced to
(3.25)∇E[V(x)]=E[XT(kT)HX(kT)]+2E[ZT(kT)ΦX(kT)]+2E[Z~T(kT)ΦX(kT)]+E[ZT(kT)PZ(kT)]+2E[ZT(kT)PZ~(kT)]+E[Z~T(kT)PZ~(kT)].
Let λm(·) and λM(·) 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 φ11<0 and φ22-φ12φ11-1φ12T<0. Noting that
(3.26)λM(φ11)≤-1-Mβλm[U1(τk)C1+(U1(τk)C1)T]+M2β2λM[C1T(G1(τk)+G2(τk))C1]≜ΘMβ.
From condition (i) it follows thatΘMβ<0, and hence the claim that φ11<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 onZ(kT) and Z~(kT). Let μ>0 be arbitrary. We will show that there exists δ>0 whenever ∥G∥<δ, such that
(3.27)E[∥Z(kT)∥2]<μE[∥X(kT)∥2],E[∥Z~(kT)∥2]<μE[∥X(kT)∥2].
For t∈[kT,kT+T], we have
(3.28)x(t)=x(kT)+∫kTtAx(s)ds+∫kTtAdx(s-d)ds+∫kTtBu(s)ds+∫kTtGx(s)dWs=x(kT)+∫kTtAx(s)ds+∫kT-dt-dAdx(s)ds+∫kTtBu(s)ds+∫kTtGx(s)dWs,x(t-d)=x(kT-d)+∫kT-dt-dAx(s)ds+∫kT-dt-dAdx(s-d)ds+∫kT-dt-dBu(s)ds+∫kT-dt-dGx(s)dWs=x(kT-d)+∫kTtAx(s-d)ds+∫kT-dt-dAdx(s-d)ds+∫kT-dt-dBu(s)ds+∫kTtGx(s-d)dWs.
Then, we have
(3.29)X(t)=X(kT)+L(kT)+∫kTtA~X(s)ds+∫kT-dt-dA~dX(s)ds+∫kTtG~X(s)dWs,
where
(3.30)A~=(A00A),A~d=(Ad00Ad),G~=(G00G),L(kT)=(∫kTtBu(s)ds00∫kT-dt-dBu(s)ds).
Similarly, we have
(3.31)Z(kT)=(hT(kT)hT(kT-d))T=(∫kT-dkT+T-deA(kT+T-d-s)Adx(s)ds00∫kT-dkT+T-deA(kT+T-d-s)Adx(s-d)ds)=∫kT-dkT+T-deA(kT+T-d-s)A~dX(s)ds,Z~(kT)=(h~T(kT)h~T(kT-d))T=(∫kTkT+TeA(kT+T-s)Gx(s)dWs00∫kTkT+TeA(kT+T-s)Gx(s-d)dWs)=∫kTkT+TeA(kT+T-s)G~X(s)dWs.
Noting that
(3.32)E[∥L(kT)∥2]≤M2β2∥B∥2∥C∥2E[∥X(kT)∥2],∥A~∥=∥A∥,∥A~d∥=∥Ad∥,∥G~∥=∥G∥,C=(C1,C2),E[∥X(t)∥2]5≤E[∥X(kT)∥2]+E[∥∫kTtA~X(s)ds∥2]+E[∥L(kT)∥2]+E[∥∫kT-dt-dA~dX(s)ds∥2]+E[(∫kTtG~x(s)dWs)T(∫kTtG~x(s)dWs)].
By Lemma 3.2, one observes that
(3.33)E[(∫kTtGx(s)dWs)T(∫kTtGx(s)dWs)]=E[∫kTtxT(s)GTGx(s)ds].
Then the inequality (3.32) is reduced to
(3.34)E[∥X(t)∥2]5≤K0E[∥X(kT)∥2]+∥A~d∥2∫kT-dt-dE[∥X(s)∥2]ds+(∥A~∥2+∥G~∥2)∫kTtE[∥X(s)∥2]ds≤K0E[∥X(kT)∥2]+(∥A~∥2+∥G~∥2+∥A~d∥2)∫kT-dtE[∥X(s)∥2]ds=K0E[∥X(kT)∥2]+(∥A∥2+∥G∥2+∥Ad∥2)∫kT-dtE[∥X(s)∥2]ds,
where K0=1+M2β2∥B∥2∥C∥2.
By the Gronwall inequality, we have(3.35)E[∥X(t)∥2]≤5K0E[∥X(kT)∥2]e(∥A∥2+∥G∥2+∥A~d∥2)(t-kT)≤5K0e(∥A∥2+∥G∥2+∥Ad∥2)TE[∥X(kT)∥2].
Thus, we have
(3.36)E[∥Z(kT)∥2]=E[(∫kT-dkT+T-deA(kT+T-d-s)A~dX(s)ds)T(∫kT-dkT+T-deA(kT+T-d-s)A~dX(s)ds)]≤∥A~d∥2E[∫kT-dkT+T-d(eA(kT+T-d-s))TeA(kT+T-d-s)∥X(s)∥2ds]≤5K0K1∥Ad∥2e(∥A∥2+∥G∥2+∥Ad∥2)TE[∥X(kT)∥2]=K∥Ad∥2e(∥G∥2+∥Ad∥2)TE[∥X(kT)∥2],E[∥Z~(kT)∥2]=E[(∫kTkT+TeA(kT+T-s)G~X(s)dWs)T∫kTkT+TeA(kT+T-s)G~X(s)dWs]=E[∫kTkT+TXT(s)G~T(eA(kT+T-s))TeA(kT+T-s)G~X(s)ds]≤∥G~∥2E[∫kTkT+T(eA(kT+T-s))TeA(kT+T-s)∥X(s)∥2ds]≤K∥G∥2e(∥G∥2+∥Ad∥2)TE[∥X(kT)∥2],
where
(3.37)K1=maxs∈[0,T]∥eATseAs∥,K=5K0K1e∥A∥2T.
Thus, when condition (iii) is satisfied, we have
(3.38)E[∥Z(kT)∥2]<μE[∥X(kT)∥2],E[∥Z~(kT)∥2]<μE[∥X(kT)∥2].
Then, we obtain
(3.39)∇E[V(x)]=E[V(x(kT+T))]-E[V(x(kT))]-E[V(x(kT))]+E[V(x(kT-d))]≤-c(μ)λM(P)E[V(X(kT))]<0.
Therefore,
(3.40)E[V(X(kT+T))]<(1-c(μ)λM(P))E[V(X(kT))].
Noticing that V(X(0))=xT(0)P1x(0)≤∥P∥∥x(0)∥2=∥P∥∥a∥2, where a=x(0) is initial state.
Fort∈[kT,kT+T], we have
(3.41)E[V(x(t))]≤E[V(X(kT))]≤(1-c(μ)λM(P))E[V(X(kT-T))]≤⋯≤(1-c(μ)λM(P))kE[V(x(0))]≤(1-c(μ)λM(P))k∥P1∥∥x(0)∥2<etln(Θ/ρ)∥P1∥∥x(0)∥2≤∥P∥∥a∥2etln(Θ/ρ),
where
(3.42)Θ=1-c(μ)λM(P)<1,ρ=mink(Tk-Tk-1).
Noticing that E[V(x(t))]≥λm(P)E∥x(t)∥2, one obtains that
(3.43)E[∥x(t)∥2]≤E[V(x(t))]λm(P)<∥a∥2[∥P∥λm(P)]etln(Θ/ρ)=ψ∥a∥2e-βt,
where
(3.44)ψ=∥P∥λm(P)>1,β=-lnΘρ>0.
Thereforce, by virtue of Definition2.5 with p=2, we know that the equilibrium point of system (3.3) is uniformly exponentially stable in mean square.
Now we will proof thepth moment uniformly exponentially stable in mean of system (3.3).
Forp=2q, q≥1, we have
(3.45)E[V(X(kT+T))q]-E[V(X(kT))q]=E[V(X(kT+T))-V(X(kT))][V(X(kT+T))q-1+⋯+V(X(kT))q-1]≤E[-c(μ)λM(P)V(X(kT))]×[V(X(kT+T))q-1+⋯+V(X(kT))q-1]≤-c(μ)λM(P){E[V(X(kT))q+⋯+V(X(kT))q×V(X(kT+T))]}≤-c(μ)λM(P)E[V(X(kT))q]<0.
It is obvious that
(3.46)E[V(X(kT+T))q]≤[1-c(μ)λM(P)]E[V(X(kT))q].
For t∈[kT,kT+T], we have
(3.47)E[V(x(t))q]≤E[V(X(kT))q]≤[1-c(μ)λM(P)]E[V(X(kT-T))q]≤⋯≤[1-c(μ)λM(P)]kE[V(x(0))q]≤(1-c(μ)λM(P))k∥P∥q∥x(0)∥2q<etln(Θ/ρ)∥P∥q∥x(0)∥2q=∥P∥q∥a∥petln(Θ/ρ).
Similarly, we have
(3.48)E[∥x(t)∥p]≤E[V(x(t))q][λm(P)]q<∥a∥p[∥P∥λm(P)]qetln(Θ/ρ)=ψ~∥a∥pe-βt,
where ψ~=ψp/2>1 is constant.
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 Section4.Remark 3.5.
Thepth 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 thatA is Hurwitz stable, C1≠0, C2=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
(3.49)Mβ<λm(Γ1)+λm2[Γ1]+4λM(Γ~2)2λM(Γ~2)
whenever ∥G∥ is sufficiently small (an upper bound of ∥G∥ has been given in the proof of Theorem 3.3), where Γ1=U1(Tk)C1+(U1(Tk)C1)T, Γ~2=C1T(G1(τk)+G~2(τk)+U~2T(Tk)U~2(Tk))C1 with G~2(τk)=WT(τk)(e-Ad)T(P2-I)e-AdW(τk), U~2(τk)=(P2-I)e-AdW(τk) by choosing P satisfied (eAT)TPeAT-P=-I.


## 4. Examples

Example 4.1.
Consider the system (3.3) with one order Hurwitz stable plant described by transfer functionT(s)=1/(s+1). The state space representation of this system is given by A=-1, Ad=0.1, B=1, C1=1, C2=-1, assuming the period T=1, the time delay d=0.1. Hence W(Tk),Pi,Gi(Tk),Ui(Tk),i=1,2, in Theorem 3.3 are calculated in this case as
(4.1)P1=11-e-2,P2=11-e-1.8,W(Tk)=e2Tk-1Tk,G1(Tk)=(e2Tk-1Tk)21e2-1,G2(Tk)=(e2Tk-1Tk)21e1.8-1,U1(Tk)=e2Tk-1Tk1e2-1,U2(Tk)=e2Tk-1Tk1e1.8-1,
where the pulse width Tk satisfies 0<Tk≤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 μmax=0.7 such that for all μ<μmax. c(μ)=1-2∥Φ∥μ-4∥P∥μ>0 is true. Next, we compute δmax=0.22 such that K·f(∥Ad∥,∥G∥)<μmax is true for all δ<δmax, where K=5K1(1+M2β2∥B∥2∥C∥2)e∥A∥2T and K1=maxs∈[0,T]∥eATseAs∥. In Figure 2, we depict the estimates of the upper bound δmax of ∥G∥ versus Mβ.
We observe thatδmax decreases as Mβ increases. When the states are sufficiently far away from the origin so that Tk=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 δmax), 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.Figure 2
Upper bounds for∥G∥ when Mβ∈(0,0.9398).Figure 3
Sample response of PWM feedback system (3.3) in Example 4.2.Example 4.2.
Consider PWM feedback system (3.3) with transfer function T(s)=1/(s+2)(s+1).
The state space representation of this system is given by(4.2)A=(-200-1),Ad=(0.2000.1),G=(0.02000.006),B=(11)T,C1=[11],C2=[00].
In Figure 3, we plot the sample response of x=(x1x2)T with M=0.2,T=β=1,d=0.1, and x(0)=(20.5)T. We observe the system (3.3) tending to the equilibrium point speed quickly.


## 5. Conclusions

We studied the stochastic PWM feedback systems with time-varying delays and established several Lyapunov and Lagrange criteria for thepth 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.



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*Source: 796104-2012-12-23.xml*