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

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

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## 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.


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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, 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 iscritically 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 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
(2.2)P{ω:x(t,ω,a,t0)∈M∀t∈Ta,t0}=1,∀t0∈T,andallx(·,·,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.
S is said to be thepth 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∈Ta,t0, where t0 is a fixed point in X.S is said to be thepth 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 thepth moment uniformly stable in mean. (S,M) is said to be thepth 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 thepth 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 Figure1.Figure 1
Block diagram of PWM feedback systems subjected to multiplicative disturbances [20].The pulse-width modulator is described by(3.1)u(t)=m(e(t))={Msgn(e(kT))t∈[kT,kT+Tk),0t∈[kT+Tk,(k+1)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 sign are given, respectively, by
(3.2)Tk={β|e(kT)|,|e(kT)|≤Tβ;T,|e(kT)|>Tβ,(3.3)sgn(σ)={1σ>00σ=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
(3.4)dx(t)=Ax(t)dt+Bu(t)dt+Gx(t)dWt,y(t)=Cx(t),
where x∈Rn,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 matrixA is stable with one and only one eigenvalue equal to zero. Without loss of generality, we assume that
(3.5)A=(A1000),B=(B1b),C=[C11],G=(G100g),
where A1 is Hurwitz stable, x1∈Rn-1 and x2∈R, and system (3.4) can be rewritten as
(3.6)dx1(t)=A1x1(t)dt+B1u(t)dt+G1x1(t)dWt,dx2(t)=bu(t)dt+gx2(t)dWt,y(t)=C1x1(t)+x2(t).Note that(x1T,x2)=(0T,0) is an equilibrium point of PWM feedback system (3.6).Lemma 3.1 (The Itô isometry).
Assume thatW(t) is a scalar wiener process, and If ϕ(t,ω) is bounded and elementary, then
(3.7)E[(∫STϕ(t,ω)dWt)2]=E[∫STϕ(t,ω)2dt].Proof.
PutΔWj=Wtj+1-Wtj, then(3.8)E[eiejΔWiΔWj]={0ifi≠jE[ej2](tj+1-tj)ifi=j.
Using that eiejΔWi and ΔWj are independent if i<j. Thus
(3.9)E[(∫STϕ(t,ω)dWt)2]=∑i,jE[eiejΔWiΔWj]=∑jej2·(ti-tj)=E[∫STϕ(t,ω)2dt].Theorem 3.2.
Assume that the matrixA1 in (3.6) is Hurwitz stable, b>0, and C1≠0. Then the equilibrium xe=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=(P100ξ) such that Mβ<infTk∈[0,T](2bξ/(G2(Tk)+b2ξ+max(0,θ(Tk)))), and there exists a δ>0 (an upper bounded of δ will be given in the proof ) such that ∥G∥<δ, where
(3.10)θ(Tk)=b2ξ2λM(C1TC1)-bξλm(U(Tk)C1+(U(Tk)C1)T)+λM(U(Tk)UT(Tk)),U(Tk)=(P1-I)(I-e-A1TkTkA1-1B),G2(Tk)=(I-e-A1TkTkA1-1B)T(P1-I)(I-e-A1TkTkA1-1B).Proof.
Since(E∥X∥p)1/p is nondecreasing in p, the equilibrium xe=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
(3.11)x1(t)=eA1(t-kT)x1(kT)+∫kTteA1(t-s)B1u(s)ds+∫kTteA1(t-s)G1x1(s)dWs,x2(t)=x2(kT)+∫kTtbu(s)ds+∫kTtgx2(s)dWs.
Therefore, when t=kT+T,
(3.12)x1(kT+T)=eA1T(x1(kT)-∫0Tke-A1sB1Msgn(C1x1(kT)+x2(kT))ds)+∫kTkT+TeA1(kT+T-s)G1x1(s)dWs=eA1T[(I-MβW(τk)C1)x1(kT)-MβW1(τk)x2(kT)]+∫kTkT+TeA1(kT+T-s)G1x1(s)dWs,(3.13)x2(kT+T)=x2(kT)-∫0TkbMsgn(C1x1(kT)+x2(kT))ds+∫kTkT+Tgx2(s)dWs=-MβbTkτkC1x1(kT)+(1-MβbTkτk)x2(kT)+∫kTkT+Tgx2(s)dWs
with
(3.14)τk=β|C1x1(kT)+x2(kT)|={Tk,Tk<T,≥T,Tk=T,W(τk)={0τk=0,I-e-A1TkTkA1-1Bτk<T,I-e-A1TτkA1-1B=TτkW(T)τk≥T.
To simplify our notations, we let(3.15)H=[I-MβW(τk)-MβW(τk)],Z(kT)=∫kTkT+TeA1(kT+T-s)G1x1(s)dWs,Z~(kT)=∫kTkT+Tgx2(s)dWs,ρ(τk)={0,τk=0,Tkτk,τk∈(0,∞),L(τk)=[-MβbTkτkC11-MβbTkτk]=[-Mβbρ(τk)C11-Mβbρ(τk)].
Then (3.12) and (3.13) are reduced to
(3.16)x1(kT+T)=eA1THx(kT)+Z(kT),(3.17)x2(kT+T)=L(τk)x(kT)+Z~(kT).
SinceA1 is Hurwitz stable, then eA1T is Schur stable. Therefore, there exists a positive definite matrix P1=P1T such that
(3.18)(eA1T)TP1eA1T-P1=-I.
Choosing the quadratic Lyapunov function(3.19)V:Rn→R+,V(x)=xTPx,
where
(3.20)P=(P100ξ),P1>0,ξ>0.
Then,
(3.21)E[V(x(kT+T))]-E[V(x(kT))]=E[xT(kT+T)Px(kT+T)]-E[xT(kT)Px(kT)]=E[x1T(kT+T)P1x1(kT+T)]+E[x2T(kT+T)ξx2(kT+T)]-E[xT(kT)Px(kT)]=E[(eA1THx(kT)+Z(kT))TP1(eA1THx(kT)+Z(kT))]+E[(L(τk)x(kT)+Z~(kT))Tξ(L(τk)x(kT)+Z~(kT))]-E[xT(kT)Px(kT)]=E[xT(kT)(HT(P1-I)H+ξLT(τk)L(τk)-P)x(kT)]+2E[ZT(kT)P1eA1THx1(kT)]+2E[Z~T(kT)ξx2(kT)]+E[ZT(kT)P1Z(kT)]+E[Z~T(kT)ξZ~(kT)]=E[xT(kT)(HT(P1-I)H+ξLT(τk)L(τk)-P)x(kT)]+2E[hT(kT)Φx(kT)]+E[hT(kT)Ph(kT)],
where
(3.22)x(kT)=(x1(kT)x2(kT)),h(kT)=(Z(kT)Z~(kT)),Φ=(P1eA1THξL(τk))=(P1eA1T(I-MβW(τk)C1)-MβP1eA1TW(τk)-Mβbρ(τk)ξC1ξ-Mβbρ(τk)ξ)=Φ1+(000ξ),
with
(3.23)Φ1=(P1eA1T(I-MβW1(τk)C1)-MβP1eA1TW1(τk)Mβbρ(τk)ξ-Mβbρ(τk)ξ).
Let
(3.24)Q=HT(P1-I)H+ξLT(τk)L(τk)-P=(-I-Mβ(U(τk)C1+(U(τk)C1)T)+(γ+2Mβbρ(τk)ξ)C1C1TD12D12Tγ),
where
(3.25)γ=M2β2[G2(τk)+b2ρ2(τk)ξ]-2Mβbρ(τk)ξ,D12=-MβU(τk)+[γ+Mβbρ(τk)ξ]C1T,
with
(3.26)U(τk)=(P1-I)W(τk),G2(τk)=WT(τk)(P1-I)W(τk).
Then (3.21) is reduced to
(3.27)E[V(x(kT+T))]-E[V(x(kT))]=E[xT(kT)Qx(kT)]+2E[hT(kT)Φ1x(kT)]+E[hT(kT)Ph(kT)]+2ξE[Z~(kT)x2(kT)].
By elementary transformation of matrix, we have
(3.28)Q=(ID12γ01)D(I0D12Tγ1),
where
(3.29)D=-(D1100-γ)
with
(3.30)D11=1γ[γI+M2β2(U(τk)UT(τk)-M2β2bρ(τk)ξ(U(τk)C1+(U(τk)C1)T))ab+M2β2b2ρ2(τk)ξ2C1TC1].
Let λm(·) and λM(·) 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
(3.31)λm(D11)≥1+(Mβ)2γ-1{λM[U(τk)UT(τk)]asdfghjkujhik-bρ(τk)ξλm[U(τk)C1+(U(τk)C1)T]+b2ρ2(τk)ξ2λM[C1TC1]}.
Therefore, the claim that D11 is positive definite is true if
(3.32)(Mβ)(G2(τk)+b2ρ2(τk)ξ)-2bρ(τk)ξ+(Mβ){λM[U(τk)UT(τk)]-bρ(τk)ξλm[U(τk)C1+(U(τk)C1)T]asdfghiiii+b2ρ2(τk)ξ2λM[C1TC1]}≤0
with
(3.33)Mβ(G2(τk)+b2ρ2(τk)ξ)-2bρ(τk)ξ<0.
That is,
(3.34)Mβ<infτk∈[0,∞]2bρ(τk)ξG2(τk)+b2ξ+max(0,θ(τk))=infTk∈[0,T]2bξG2(Tk)+b2ξ+max(0,θ(Tk)),
where
(3.35)θ(τk)=b2ρ2(τk)ξ2λM(C1TC1)-bρ(τk)ξλm(U(τk)C1+(U(τk)C1)T)+λM(U(τk)UT(τk)),θ(Tk)=b2ξ2λM(C1TC1)-bξλm(U(Tk)C1+(U(Tk)C1)T)+λM(U(Tk)UT(Tk)).
Thus, wheneverMβ 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
(3.36)Φ1=(IP1eA1TW(τk)bρ(τk)ξ01)Φ2(I0C11),
where
(3.37)Φ2=(P1eA1T00-Mβρ(τk)bξ).
So, we have ∥Φ1∥≤∥Φ2∥.
Now we establish an estimate forh(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.
Fort∈[kT,kT+T], we have
(3.38)x(t)=x(kT)+∫kTt(Ax(s)+Bu(s))ds+∫kTtGx(s)dWs.
Note that
(3.39)E(∫kTt∥B∥|u(s)|ds)2≤M2β2∥B∥2∥C∥2E[∥x(kT)∥2].
Then,
(3.40)E[∥x(t)∥2]4≤E[∥x(kT)∥2]+E[∥∫kTtAx(s)ds∥2]+E[(∫kTt∥B∥|u(s)|ds)2]+E[(∫kTtGx(s)dWs)T(∫kTtGx(s)dWs)].
By Lemma 3.1, one observes that
(3.41)E[(∫kTtGx(s)dWs)T(∫kTtGx(s)dWs)]=E[∫kTtxT(s)GTGx(s)ds].
Then (3.40) is reduced to
(3.42)E[∥x(t)∥2]4≤K0E[∥x(kT)∥2]+(∥A∥2+∥G∥2)∫kTtE[∥x(s)∥2]ds,
where K0=1+M2β2∥B∥2∥C∥2.
By Gronwall inequality, we have(3.43)E[∥x(t)∥2]≤4K0e(∥A∥2+∥G∥2)(t-kT)E[∥x(kT)∥2]≤4K0e(∥A∥2+∥G∥2)TE[∥x(kT)∥2].
Thus,
(3.44)E[∥h(kT)∥2]=E[(∫kTkT+TeA1(kT+T-s)G1x1(s)dWs)T(∫kTkT+TeA1(kT+T-s)G1x1(s)dWs)]+E[(∫kTkT+Tgx2(s)dWs)T(∫kTkT+Tgx2(s)dWs)]=E[∫kTkT+Tx1T(s)G1TeA1T(kT+T-s)eA1(kT+T-s)G1x1(s)ds]+E[∫kTkT+Tg2x22(s)ds]=E[∫kTkT+TxT(s)G~TG~x(s)ds]≤K1∥G∥2∫kTkT+TE[∥x(s)∥2]ds≤4K0K1∥G∥2e(∥A∥2+∥G∥2)TE[∥x(kT)∥2],
where
(3.45)G~=(eA1(kT+T-s)00g),K1=maxs∈[0,T]∥eA1TseA1s∥,K=4K0K1e∥A∥2T.
Thus, there exists aδ>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
(3.46)E[V(x(kT+T))]-E[V(x(kT))]≤-c(μ)λM(P)E[V(x(kT))].
Therefore, E[V(x(kT+T))]-E[V(x(kT))] is negative definite.
If we defineM~={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+,Rn,Rn,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
(3.47)E[V(x(t))]≤λM(P)Ke∥G∥2TE∥x(kT)∥2≤λM(P)λm(P)V(x(kT)).
This implies thatE[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∈Ra~,t0+=Ra,t0+, and for all x~(·,a~,t0)∈S~, whenever a~<δ.
Sincex~(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∈Ra,t0+ and for all x(·,a,t0)∈S, whenever EV(a)<δ, since EV(x(t))/λM(P)≤E∥x(t)∥2≤EV(x(t))/λm(P); hence, for every ε>0,t0∈R+, for any process x(·,·,a,t0)∈S, we have, for t∈Ra,t0+,E∥x(t)∥2≤ε/λm(P), whenever ∥a∥<δ1, where δ1=δ/λM(P). This is precisely the definition of uniform stability in the mean square of the trivial solution xe=0 of S(4). Therefore, the uniform stability of (S~,M~) is equivalent to the uniform stability in the mean square of (S(4),{xe}).
We can now conclude that the equilibriumxe=0 of PWM feedback system (3.6) is uniformly asymptotically stable in mean square.
Forp=2q,q≥1, we have
(3.48)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[(-1λM(P)V(x(kT))+2EhT(kT)Φx(kT)+EhT(kT)Ph(kT))×(V(x(kT+T))q-1+⋯+V(x(kT))q-1)]≤-1λM(P)E[V(x(kT))q]+E[(2hT(kT)Φx(kT)+2hT(kT)Φh(kT))×(V(x(kT+T))q-1+⋯+V(x(kT))q-1)].
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 thepth 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 ofMβ 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 thatA1 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.
WhenC1=0, we have
(3.49)D11=1γ[γI+M2β2U(τk)UT(τk)].
If λm(D11)>0,D11 is positive definite matrix. And if -γ>0, then D is positive negative definite matrix. Therefore, whenever λm(D11)>0 and γ<0, it can easily be verified that
(3.50)(Mβb2-2b)ξ+λM(U(τk)UT(τk))+G2(τk)>0.
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 uniformlypth 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 Theorem3.4, 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(3.51)dx1(t)=A1x1(t)dt+B1u(t)dt+ε1x1(t)dWt,dx2(t)=bu(t)dt+ε2x2(t)dWt,y(t)=C1x1(t)+x2(t),
where ε1∈Rn-1,ε2∈R have positive components. We let ε=(ε1ε2)∈Rn 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 thatA1 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 Theorem3.6 is similar as that of [20].To end this section, we present an algorithm for computing stability bound.Stability Bound Algorithm
An upper bound ofMβ that satisfies (3.34) can be computed and optimized in the following manner.(S1)
Determine the matrixP1 by solving (eA1T)TP1eA1T-P1=-I.(S2)
Choose a precision levelδ>0 and a correspondingly dense partition of the interval [0,T], and say the set {t0=0,t1,…,tN=T}, where 0<tj+1-tj<δ,j=0,1,2,…,N-1.(S3)
For eachi,j=0,1,2,…,N-1, calculate
(3.52)W(tj)=I-e-A1tjtjA1-1B,G2(tj)=W(tj)T(P1-I)W(tj),U(tj)=(P1-I)W(tj).(S4)
Search for the largestMβsuch that
(3.53)Mβ<inftj∈[0,∞]2bρ(tj)ξG2(tj)+b2ξ+max(0,θ(tj)),
where
(3.54)θ(tj)=b2ξ2λM(C1TC1)-bξλm(U(tj)C1+(U(tj)C1)T)+λM(U(tj)UT(tj)).(S5)
Repeat steps 1–4, using finer partition of the interval[0,T] until there is no further significant improvement for Mβopt.


## 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
(4.1)A=(-a000),B=(kk),C=[11].
Hence A1=-a,B1=k,C1=1, and P1,W(Tk),U(Tk) in Theorem 3.2 are calculated in this case as
(4.2)P1=11-e-2aT,W(Tk)=(eaTk-1)kaTk,U(Tk)=k(eaTk-1)aTk(e2aT-1),G2(Tk)=k2(eaTk-1)2a2Tk2(e2aT-1),θ(Tk)=k2ξ2-2k2(eaTk-1)aTk(e2aT-1)ξ+k2(eaTk-1)2a2Tk2(e2aT-1)2=k2(ξ-eaTk-1aTk(e2aT-1))2≥0.
Choosing ξ=1/aT(1+e-aT), it can be seen that the stability bound is given by
(4.3)Mβ<1(((eaT-1)/aT(eaT+1))+(1/2))k.
Assume that M=1, and then the condition for uniformly asymptotic stability in pth mean is given by
(4.4)1kβ>1aT-2aT(eaT+1)+12.
In Example 4.1, we obtain the stability results depicted in Figure 3 (kβopt versus aT).Figure 3
Estimates of upper bounds forkβ 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
(4.5)A=(-1000),B=(11),C=[11].
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 μmax such that c(μ)=1-2(∥R2∥+ξ)μ-∥P∥μ>0 is true for all μ<μmax. Next, we compute δmax such that Kδ2eδ2T<μmax is true for all δ<δmax, where K=4K1(1+M2β2∥B∥2∥C∥2)e∥A∥2T and K1=maxs∈[0,T]∥eA1TseA1s∥. In Figure 4, 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, 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 δmax), 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 Figure5, we plot the sample response of x=(x1x2)T with M=0.2,T=β=1,G=(0.02000.006) 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 thepth 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.



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*Source: 736408-2012-11-19.xml*