# Stochastic Finite-TimeH∞ Performance Analysis of Continuous-Time Systems

Authors: Bing Wang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The problem ofH∞ control performance analysis of continuous-time systems with random abrupt changes is concerned in this paper. By employing an augmented multiple mode-dependent Lyapunov-Krasovskii functional and using some integral inequalities, new sufficient conditions are obtained relating to finite-time bounded and an H∞ performance index. The finite-time H∞ control performance problem is solved and desired controller is given to ensure the system trajectory stays within a prescribed bound during a given time interval. At last, two numerical examples are provided to show that our results are less conservative than the existing ones.


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## Body

## 1. Introduction

It is well known that Markovian jump systems were introduced when the physical models are always subject to random changes, which can be also regarded as a special class of hybrid systems because of the structures are subject to random abrupt changes [1]. In the recent years, there are a lot of people towards to Markovian jump systems for its widely applications, for example, target tracking, robotics, manufacturing systems, aircraft control, and power systems [2–4]. Markovian jump systems are regarded as a special class of stochastic systems which switches from one to another at different time in the finite operation modes. Many important topics have been studied for Markovian jumping systems such as stability, control synthesis, stabilization, and filter design [5–7].On the other hand, time delay is very common in practical dynamical systems, for example, networked control systems, chemical processes, communication systems, and so on [8–20]. Therefore, during the past two decades, various research topics have been considered for Markovian jump systems with time-varying delays [8–14]. It worth pointing out that when time delay is small enough in linear Markovian jump systems, the delay-dependent criteria are always less conservative than delay-independent ones. Over the past few years, for Markovian jump systems, many important topics related to delay-dependent have been extensively studied [14, 15].Generally speaking, finite-time stability is investigated to address these transient performances of control systems in finite-time interval. Up to now, the concept of finite-time stability has been revisited with different systems, and many important results are obtained for finite-time stability and finite-time boundedness [21–26]. However, to the best of authors' knowledge, the stochastic finite-time H∞ control for Markovian jump systems has not been fully studied. There is some room for next investigation due to the fact that analysis methods in existing references seem still conservative.The major contribution of this paper is that we introduce a newly Lyapunov-Krasovskii functional for Markovian jump system. Some sufficient conditions are obtained to ensure the finite-time stability and bounded of the closed-loop Markovian jump systems. Compared with traditional methods of MJSs, it is shown the less conservative results can be obtained and the desiredH∞ control performance is obtained by employing mode-dependent Lyapunov functional instead of mode-independent Lyapunov functional. The finite-time bounded criterion can be dealt with in the terms of LMIs. Finally, the effectiveness of the developed techniques is also illustrated by two numerical examples.


## 2. Preliminaries

Given the probability space(Ω,F,P), where Ω, F, and P represent the sample space, the algebra of events, and the probability measure defined on F, respectively, the following Markovian jump systems over the probability space (Ω,F,P) are considered:
(1)x˙(t)=Artx(t)+Aτrtx(t-τrt(t))+Brtu(t)+Drtω(t),z(t)=Crtx(t)+Cτrtx(t-τrt(t))+Frtω(t),x(t)=φ(t),t=[-h,0],
where x(t)∈ℛn represents the state vector of Markovian jump system, u(t)∈ℛm is the control input, z(t)∈ℛq denotes the controlled output and φ(t), t=[-h,0], where r0∈𝒩 is initial condition. ω(k)∈ℛq denotes the disturbance input which satisfies
(2)∫0Tω⊺(t)ω(t)dt≤d.Firstly, taking value on the finite set𝒩={1,2,…,N}, let the random form process {rt,t≥0} be the stochastic process with transition rate matrix Ω={πij}, i,j∈𝒩 and let the transition probabilities also be denoted as follows:
(3)Pr(rt+Δ=j∣rt=i)=ρij+πijΔ+o(Δ),
where
(4)ϱij={0,ifi≠j1ifi=j,
and Δ>0, πij≥0, for i≠j, denotes the mode i in time t to time t+Δ with mode j,
(5)-πii=∑j=1,j≠iN‍πij
for each mode i∈𝒩, lim⁡Δ→0+(o(Δ)/Δ)=0. τi(t) denotes the time-varying delay, which satisfies
(6)0<τi(t)≤hi<∞,τ˙i(t)≤μi,
where h=max⁡{hi,i∈𝒩} is the given upper bound of time-varying delays τi(t) and μ=max⁡{μi,i∈𝒩} is the given upper bound of τ˙i(t). All the matrices are known matrices with the appropriate dimension.In this paper, the objective is to design a state feedback controller as follows:(7)u(t)=Kix(t),
where Ki is the controller gains to be designed.Definition 1.
System (1) is said to be finite-time bounded with respect to (c1,c2,T,R,d), if condition (2) and the following inequality hold:
(8)sup⁡-h≤υ≤0𝔼{x⊺(υ)Rx(υ),x˙⊺(υ)Rx˙(υ)}≤c1⟹𝔼{x⊺(t)Rx(t)}<c2,∀t∈[0,T],
where c2>c1≥0 and R>0.Definition 2 (see [8]).
Considering system (1) with the stochastic Lyapunov function V(xt,rt), we get the weak infinitesimal operator as follows:
(9)£V(xt,rt,t)=lim⁡Δt→01Δt[𝔼{V(xt+Δt,rt+Δt,t+Δt)}-V(xt,i,t)]=∂∂tV(xt,i,t)+∂∂xV(xt,i,t)x˙(t,i)+∑j=1N‍πijV(xt,j,t).Definition 3.
Given a constant scalarT>0 and for all admissible ω(t) given in condition (2), if the Markovian jump system (1) is finite-time stochastic bounded and controller outputs satisfy condition (7) with attenuation γ>0,
(10)𝔼{∫0T‍z⊺(t)z(t)dt}≤γ2eηT𝔼{∫0T‍ω⊺(t)ω(t)dt}.
The Markovian jump system (1) is called the finite-time stochastic bounded with a disturbance attenuation γ.Lemma 4 (see [27]).
Letfi: ℛm→ℛ(i=1,2,…,N) have positive values in an open subset 𝒟 of ℛm. Then, the reciprocally convex combination of fi over 𝒟 satisfies
(11)min⁡{βi∣βi>0,∑iβi=1}∑i‍1βifi(t)=∑i‍fi(t)+max⁡gi,j(t)∑i≠j‍gi,j(t)subjectto{[fi(t)gi,j(t)gi,j(t)fj(t)]gi,j:ℛm⟶ℛ,gj,i(t)=gi,j(t),[fi(t)gi,j(t)gi,j(t)fj(t)]≥0}.Lemma 5.
For any constant matrixM∈ℛm×m with M>0, scalars a<b≤0, vector function x:[a,b]→ℛm, such that the integrals in the following are well-defined; then,
(12)-a2-b22∫ab‍∫t+st‍x⊺(s)Mx(s)dsdθ≤-[∫ab‍∫t+st‍x(s)dsdθ]⊺M[∫ab‍∫t+st‍x(s)dsdθ].


## 3. Finite-TimeH∞ Performance Analysis

The issue of stability analysis of Markovian jump system (1) subject to u(t)=0 is given firstly. Therefore, the finite-time stability is obtained in this section.Theorem 6.
System (1) is called the finite-time bounded with respect to (c1,c2,d,R,T), if there exist matrices
(13)Pi>0,Qli>0(l=1,2),Q>0,𝒳i=[X1iX2iX3iX4i]>0,𝒳=[X1X2X3X4]>0,Ys>0(s=1,2),Zi>0,Z>0,H>0,
scalars c1<c2, T>0, λs>0, (s=1,2,…,12), η>0, and Λ>0, such that ∀i,j∈𝒩 and the inequalities hold as follows:
(14)eδh∑j=1N‍πijQ1j+eδh∑j=1,j≠iN‍πijQ2j<Q,(15)∑j=1N‍πij𝒳j-𝒳<0,(16)∑j=1N‍πijZj-Z<0,(17)[𝒳ih𝒮i*𝒳ih]>0,(18)[Y1W1*Y2]>0,(19)[Y1W2*Y2]>0,(20)Ξi=[Ξ11iΞ12iS4iΞ14iΞ15i-S3i+Zi+Zi⊺hPiDi*Ξ22i-S4i+X4ih0X3i2h+X2i⊺2hS3i-S2i⊺-X3i2h-X2i⊺2h0**-Q1i-X4ih-W200S2i⊺+X3i2h+X2i⊺2h0***Ξ44i000****Ξ55i-S1i-Zih2-Zi⊺h20*****-X1ih-Zih2-Zi⊺h20******-δH]<0,(21)c1Λ+dδλ121η(1-e-ηT)<λ1e-ηTc2,
where
(22)Ξ11i=∑j=1N‍πijPj+δPi+PiAi+Ai⊺Pi+eδh(Q1i+Q2i)+hQ+eδh-1δX1i+eδh-δheδh-1δ2X1-X4ih+hY1+W1-Zi-Zi⊺,Ξ12i=PiAτi+X4ih-S4i,Ξ14i=eδh-12δ(X2i+X3i⊺)+eδh-δheδh-12δ2(X2+X3⊺),Ξ15i=-X3i2h-X2i⊺2h+Zih2+Zi⊺h2,Ξ22i=-(1-μi)Q2i-2X4ih+S4i+S4i⊺-W1+W2,Ξ44i=eδh-1δX4i+eδh-δheδh-1δ2X4i+hY2+eδh-δh-1δ2Zi+hδ2eδh+eδh+δh+1δ3Z,Ξ55i=-X4ih-Zih2-Zi⊺h2,Λ=λ2+heδh(λ3+λ4)+h2eδhλ5+h2eδhλ6+12h3eδhλ7+h2eδh(λ8+λ9)+12h3eδhλ10+16h4eδhλ11,λ1=max⁡i∈𝒩λmax⁡(Pi),λ2=max⁡i∈𝒩λmax⁡(P~i),λ3=max⁡i∈𝒩λmax⁡(Q~1i),λ4=max⁡i∈𝒩λmax⁡(Q~2i),λ5=λmax⁡(Q~),λ6=max⁡i∈𝒩λmax⁡(𝒳~i),λ7=max⁡i∈𝒩λmax⁡(𝒳~),λ8=λmax⁡(Y~1),λ9=λmax⁡(Y~2),λ10=max⁡i∈𝒩λmax⁡(Z~i),λ11=λmax⁡(Z~),λ12=λmax⁡(H),P~i=R-(1/2)PiR-(1/2),Q~li=R-(1/2)QliR-(1/2)(l=1,2),Q~=R-(1/2)QR-(1/2),𝒳~i=R-(1/2)𝒳iR-(1/2)(l=1,2),X~=R-(1/2)XR-(1/2),Y~s=R-(1/2)YsR-(1/2)(s=1,2),Z~i=R-(1/2)ZiR-(1/2),Z~=R-(1/2)ZR-(1/2).Proof.
Firstly, a novel process is defined in this paper as follows:(23)xt(s)=x(t+s),s∈[-h,0].
Then, the following Lyapunov-Krasovskii functional is considered:
(24)V(xt,rt,t)=∑l=15‍Vl(xt,rt,t),
where
(25)V1(xt,rt,t)=x(t)⊺eδtPrtx(t),V2(xt,rt,t)=∫t-ht‍eδ(s+h)x⊺(s)Q1rtx(s)ds+∫t-τrt(t)t‍eδ(s+h)x⊺(s)Q2rtx(s)ds+∫-h0‍∫t+θt‍eδ(s+h)x⊺(s)Qx(s)dsdθ,V3(xt,rt,t)=∫-h0‍∫t+θt‍eδ(s-θ)η⊺(s)𝒳rtη(s)dsdθ+∫-h0‍∫θ0‍∫t+υt‍eδ(s-θ)η⊺(s)𝒳η(s)dsdυdθ,V4(xt,rt,t)=∫-h0‍∫t+θt‍eδ(s-θ)x⊺(s)Y1x(s)dsdθ+∫-h0‍∫t+θt‍eδ(s-θ)x˙⊺(s)Y2x˙(s)dsdθ,V5(xt,rt,t)=∫-h0‍∫θ0‍∫t+υt‍eδ(s-θ)x˙⊺(s)Zrtx˙(s)dsdυdθ+∫-h0‍∫ς0‍∫θ0‍∫t+υt‍eδ(s-θ)x˙⊺(s)×Zx˙(s)dsdυdθdς,
where η(t)=[x⊺(t),x˙⊺(t)]⊺.
Lettingi represent the time t, that is, rt=i∈𝒩, one has
(26)£V1(xt,i,t)=δeδtx⊺(t)Pix(t)+2eδtx⊺(t)Pix˙(t)+eδtx⊺(t)(∑j=1NπijPj)x(t)=eδtx⊺(t)(∑j=1NπijPj+δPi)x(t)+2eδtx⊺(t)Pi×(Aix(t)+Aτix(t-τi(t))+Diω(t)).
Noting πij≥0 for j≠i and πii≤0, one has
(27)£V2(xt,i,t)=eδtx⊺(t)(eδhQ1i+eδhQ2i+hQ)x(t)-eδtx⊺(t-h)Q1ix(t-h)-(1-τ˙i(t))eδ(t+h-τi(t))x⊺×(t-τi(t))Q2ix(t-τi(t))+∫t-hteδsx⊺(s)(eδh∑j=1NπijQ1j-Q)x(s)ds+∑j=1Nπij∫t-τj(t)teδ(s+h)x⊺(s)Q2jx(s)ds≤eδtx⊺(t)(eδhQ1i+eδhQ2i+hQ)x(t)-eδtx⊺(t-h)Q1ix(t-h)-(1-τ˙i(t))eδtx⊺(t-τi(t))Q2ix(t-τi(t))+∫t-hteδsx⊺(s)(eδh∑j=1NπijQ1j+eδh×∑j=1,j≠iNπijQ2j-Q)×x(s)ds.
It follows from (15) and (28) that
(28)£V2(xt,i,t)=eδtx⊺(t)(eδhQ1i+eδhQ2i+hQ)x(t)-eδtx⊺(t-h)Q1ix(t-h)-(1-τ˙i(t))eδtx⊺(t-τi(t))Q2ix(t-τi(t)),(29)£V3(xt,i,t)=∫-h0∫t+θteδ(s-θ)η⊺(s)(∑j=1Nπij𝒳ij-𝒳)×η(s)dsdθ+eδtη⊺𝒳iη(t)∫-h0e-δυdυ-eδt∫t-htη⊺(s)𝒳iη(s)ds+eδtη⊺(t)𝒳η(t)∫-h0∫υ0e-δυdθdυ.
By employing Lemma 4, we can obtain that
(30)-∫t-ht‍η⊺(s)𝒳iη(s)ds=-∫t-τi(t)t‍η⊺(s)𝒳iη(s)ds-∫t-ht-τi(t)‍η⊺(s)𝒳iη(s)ds≤-hτi(t)[∫t-τi(t)t‍η(s)ds]⊺𝒳ih[∫t-τi(t)t‍η(s)ds]-hh-τi(t)[∫t-ht-τi(t)η(s)ds]⊺𝒳ih[∫t-ht-τi(t)‍η(s)ds]≤-[∫t-τi(t)t‍η(s)ds∫t-ht-τi(t)‍η(s)ds]⊺[𝒳ih𝒮i*𝒳ih][∫t-τi(t)t‍η(s)ds∫t-ht-τi(t)‍η(s)ds].
It follows from (30) and (31) that
(31)V3(xt,i,t)≤eδtη⊺(t)×(eδh-1δ𝒳i+eδh-δheδh-1δ2𝒳)η(t)-eδt[∫t-τi(t)t‍η(s)ds∫t-ht-τi(t)‍η(s)ds]⊺[𝒳ih𝒮i*𝒳ih]×[∫t-τi(t)t‍η(s)ds∫t-ht-τi(t)‍η(s)ds].
Consider
(32)£V4(xt,i,t)=hx⊺(t)Y1x(t)-∫t-ht‍x⊺(s)Y1x(s)ds+hx˙⊺(t)Y2x˙(t)-∫t-ht‍x˙⊺(s)Y2x˙(s)ds.
Moreover, the following two zero equalities with any symmetric matrices W1 and W2 are considered:
(33)0=x⊺(t)W1x(t)-x⊺(t-τi(t))W1x(t-τi(t))-2∫t-τi(t)t‍x⊺(s)W1x˙(s)ds,(34)0=x⊺(t-τi(t))W2x(t-τi(t))-x⊺(t-h)W2x(t-h)-2∫t-ht-τi(t)‍x⊺(s)W2x˙(s)ds.
With the above two zero equalities (34) and (35), an upper bound of £V4(xt,i,t) is
(35)£4(xt,i,t)=x⊺(t)(hY1+W1)x(t)+hx˙⊺(t)Y2x˙(t)+x⊺(t-τi(t))(W2-W1)x(t-τi(t))-x⊺(t-h)W2x(t-h)-∫t-τi(t)t‍[x(s)x˙(s)]⊺[Y1W1*Y2][x(s)x˙(s)]-∫t-ht-τi(t)‍[x(s)x˙(s)]⊺[Y1W1*Y2][x(s)x˙(s)].
From (19), (20), and (36), one can obtain
(36)£4(xt,i,t)=x⊺(t)(hY1+W1)x(t)+hx˙⊺(t)Y2x˙(t)+x⊺(t-τi(t))(W2-W1)x(t-τi(t))-x⊺(t-h)W2x(t-h).
Now, £V5(xt,i,t) is obtained as follows:
(37)£V5(xt,i,t)=∫-h0‍∫θ0‍∫t+υt‍eδ(s-θ)x˙⊺(s)(∑j=1N‍πijZj-Z)×x˙(s)dsdυdθ+eδtx˙⊺(t)Zix˙(t)∫-h0‍∫θ0‍e-δυdυdθ-eδt∫-h0‍∫t+θt‍x˙⊺(s)Zix˙(s)dsdθ+eδtx˙⊺(t)Zx˙(t)∫-h0‍∫ς0‍∫θ0‍e-δθdυdθdς.
By using Lemma 5, one has
(38)-∫-h0‍∫t+θt‍x˙⊺(s)Zix˙(s)dsdθ≤-2h2∫-h0‍∫t+θt‍x˙⊺(s)dsdθZi∫-h0‍∫t+θt‍x˙(s)dsdθ=-2[x(t)-1h∫t-τi(t)t‍x(s)ds-1h∫t-ht-τi(t)‍x(s)ds]⊺Zi×[x(t)-1h∫t-τi(t)t‍x(s)ds-1h∫t-ht-τi(t)‍x(s)ds].
Together with (38) and (39), it implies that
(39)£V5(xt,i,t)≤eδtx˙⊺(t)eδh-δh-1δ2Zix˙(t)+eδtx˙⊺(t)hδ2eδh+eδh+δh+1δ3Zx˙(t)-2eδt[x(t)-1h∫t-τi(t)t‍x(s)ds-1h∫t-ht-τi(t)‍x(s)ds]⊺Zi×[x(t)-1h∫t-τi(t)t‍x(s)ds-1h∫t-ht-τi(t)‍x(s)ds].
From (26)–(40), we can eventually obtain
(40)£V(xt,rt,t)-δω⊺(t)Hω(t)≤eδtξ⊺(t)Ξiξ(t),
where
(41)ξ⊺(t)=[∫t-ht-τi(t)‍x⊺(t),x⊺(t-τi(t)),x⊺(t-h),∫t-τi(t)t‍x⊺(s)ds,∫t-ht-τi(t)‍x⊺(s)ds,ω⊺(t)].
It follows from (45) that
(42)𝔼{£V(xt,rt,t)}≤𝔼[ηV(xt,rt,t)]+δω⊺(t)Hω(t).
Multiplying the above inequality by e-ηt yields that
(43)𝔼{£[e-ηtV(xt,rt,t)]}≤e-ηtδω⊺(t)Hω(t).
Integrating the inequality from 0 to t, we have
(44)e-ηt𝔼[V(xt,rt,t)]-𝔼[V(x0,r0,0)]≤δ∫0t‍e-ηsω⊺(s)Hω(s)ds.
Denoting P~i=R-(1/2)PiR-(1/2), Q~i=R-(1/2)QiR-(1/2), Q~=R-(1/2)QR-(1/2), 𝒳~i=R-(1/2)𝒳iR-(1/2), 𝒳~=R-(1/2)𝒳R-(1/2), Y~i=R-(1/2)YiR-(1/2), Y~=R-(1/2)YR-(1/2), Z~i=R-(1/2)ZiR-(1/2), and Z~=R-(1/2)ZR-(1/2) yields that
(45)𝔼[V(x0,r0,0)]≤max⁡i∈𝒩λmax⁡(P~i)x⊺(0)Rx(0)+(max⁡i∈𝒩λmax⁡(Q1i)+max⁡i∈𝒩λmax⁡(Q2i))eδh×∫-h0‍eδsx⊺(s)Rx(s)ds+eδhλmax⁡(Q)∫-h0‍∫θ0‍eδsx⊺(s)Rx(s)ds+eδhmax⁡i∈𝒩λmax⁡(𝒳i)∫-h0‍∫θ0‍e-δθη⊺(s)Rη(s)dsdθ+eδhλmax⁡(𝒳)∫-h0‍∫θ0‍∫υ0‍e-δθη⊺(s)Rη(s)dsdθdυ+(λmax⁡(Y1)+λmax⁡(Y2))eδh×∫-h0‍∫θ0‍e-δθx⊺(s)Rx(s)dsdθ+eδhmax⁡i∈𝒩λmax⁡(Zi)∫-h0‍∫θ0‍∫υ0‍e-δθx⊺(s)Rx(s)dsdθ+eδhλmax⁡(Z)∫-h0‍∫ς0‍∫θ0‍∫υ0‍e-δθx⊺(s)Rx(s)dsdθdς≤{max⁡i∈𝒩λmax⁡(P~i)+heδh×(max⁡i∈𝒩λmax⁡(Q1i+max⁡i∈𝒩λmax⁡(Q2i))+h2eδhλmax⁡(Z)+h2eδhmax⁡i∈𝒩λmax⁡(𝒳~i)+12h3eδhλmax⁡(𝒳~)+h2eδh(λmax⁡(Y1)+λmax⁡(Y2))+12h3eδhmax⁡i∈𝒩λmax⁡(Zi)+16h4eδhλmax⁡(Z)}×sup⁡-h≤s≤0{x⊺(s)Rx(s),x˙⊺(s)Rx˙(s)}=c1Λ.
For scalars η>0 and T≥t≥0, (46) turns out to be
(46)𝔼[V(xt,rt,t)]≤𝔼[eηtV(x0,r0,0)]+eηtδ∫0t‍e-ηsω⊺(s)Hω(s)ds≤eηTc1Λ+dδeηTλmax⁡(H)∫0T‍e-ηsds≤eηT{c1Λ+dδλ121η(1-e-ηT)}.
To illustrate the bounded, (26) takes the following form:
(47)𝔼[V(xt,rt,t)]≥𝔼[x⊺(t)eλtPix(t)]≥max⁡i∈𝒩λmin⁡(Pi)𝔼[x⊺(t)Rx(t)]=λ1𝔼[x⊺(t)Rx(t)].
From inequalities (46)–(48), one has
(48)𝔼[x⊺(t)Rx(t)]≤eηTλ1{c1Λ+dδλ121η(1-e-ηT)}.
Finally, inequalities (24) and (49) guarantee that
(49)𝔼[x⊺(t)Rx(t)]<c2.
Therefore, the Markovian jump system (1) is finite-time stochastic bounded with respect to (c1,c2,d,R,T).Remark 7.
It should be noted thatτi(t) and τ˙i(t) may, respectively, get the different upper bound due to the fact that condition (6) holds. However, τi(t) and τ˙i(t) always lead to conservativeness for τi(t)≤h=max⁡{hi,i∈𝒩} and τ˙i(t)≤μ=max⁡{μi,i∈𝒩} in [14–18], and this case can be improved with employing the different Lyapunov-Krasovskii functional (26).Remark 8.
It should be pointed out that, in Theorem6, the novelty of the Lyapunov functional (26) lies in the following: (i) triple-integral terms V3(xt,rt,t) and V5(xt,rt,t) and four-integral term V5(xt,rt,t) are introduced and (ii) the distinct Lyapunov matrices (Pi,Q1i,Q2i,𝒳i,Zi) are chosen for different system modes i(i=1,2,…,N).
For the conditionrt=i, the Markovian jump system given in this paper is followed by
(50)x˙(t)=A¯ix(t)+Aτix(t-τrt(t))+Diω(t),z(t)=Cix(t)+Cτix(t-τi(t))+Fiω(t),
where
(51)A¯i=Ai+BiKi.Theorem 9.
System (53) is finite-time stochastic bounded with respect to (c1,c2,d,R,T) with a disturbance attenuation, if there exist matrices
(52)Pi>0,Qli>0(l=1,2),Q>0,𝒳i=[X1iX2iX3iX4i]>0,𝒳=[X1X2X3X4]>0,Ys>0(s=1,2),Zi>0,Z>0,
scalars c1<c2, T>0, λs>0, (s=1,2,…,12), η>0 and Λ>0, such that for all i,j∈𝒩, (15)–(20) and the following inequalities hold:(53)Σi=[Σ11iΞ12iS4iΞ14iΞ15i-S3i+Zi+Zi⊺hPiDiCi⊺*Ξ22i-S4i+X4ih0X3i2h+X2i⊺2hS3i-S2i⊺-X3i2h-X2i⊺2h0Cτi⊺**-Q1i-X4ih-W200S2i⊺+X3i2h+X2i⊺2h00***Ξ44i0000****Ξ55i-S1i-Zih2-Zi⊺h200*****-X1ih-Zih2-Zi⊺h200******-γ2IFi⊺*******-I]<0,(54)c1Λ+dγ21η(1-e-ηT)<λ1e-ηTc2,
where
(55)Σ11i=∑j=1N‍πijPj+δPi+PiA¯i+A¯i⊺Pi+eδh(Q1i+Q2i)+hQ+eδh-1δX1i+eδh-δheδh-1δ2X1-X4ih+hY1+W1-Zi-Zi⊺.Proof.
Considering the Lyapunov-Krasovskii functional in Theorem6 and from Schur's Lemma, it turns out to be
(56)£V(xt,i,t)+z⊺(t)z(t)-γ2ω⊺(t)ω(t)≤ξ⊺(t)Θi(μpi,hqi)ξ(t).
Thanks to (54), we have
(57)𝔼{£V(xt,i,t)}≤𝔼[ηV(xt,i,t)]+γ2ω⊺(t)ω(t)-𝔼[z⊺(t)z(t)].
Multiplying the (58) by e-ηt, (58) can be written as
(58)𝔼{£[e-ηtV(xt,i,t)]}≤e-ηt[γ2ω⊺(t)ω(t)-z⊺(t)z(t)].
Under the condition of zero initial and 𝔼[V(xt,i,t)]>0, one has
(59)∫0T‍e-ηt[γ2ω⊺(t)ω(t)-z⊺(t)z(t)]dt≤𝔼{∫0T‍£[e-ηtV(xt,i,t)]dt}≤V(x0,r0,0)=0.
Using the Dynkin formula, it results that
(60)𝔼[∫0T‍e-ηυz⊺(υ)z(υ)dυ]≤γ2𝔼[∫0T‍e-ηυω⊺(υ)ω(υ)dυ].
Finally, it is easy to obtains that
(61)𝔼[∫0T‍z⊺(υ)z(υ)dυ]≤γ2eηT𝔼[∫0T‍ω⊺(υ)ω(υ)dυ].
Therefore, the Markovian jump system (53) is finite-time stochastic bounded with an performance γ.


## 4. Finite-TimeH∞ Control

Theorem 10.
System (53) is finite-time stochastic bounded with respects to (c1,c2,d,R,T) with an disturbance attenuation, if there exists matrices
(62)Pi>0,P¯i,Qli>0(l=1,2),K¯i,Q>0,𝒳i=[X1iX2iX3iX4i]>0,𝒳=[X1X2X3X4]>0,Ys>0(s=1,2),Zi>0,Z>0,
scalars c1<c2, T>0, λs>0, (s=1,2,…,12), η>0 and Λ>0, such that for all i,j∈𝒩, (15)–(20) and the following inequalities hold:(63)[Θ11iΞ12iS4iΞ14iΞ15i-S3i+Zi+Zi⊺hPiDiCi⊺*Ξ22i-S4i+X4ih0X3i2h+X2i⊺2hS3i-S2i⊺-X3i2h-X2i⊺2h0Cτi⊺**-Q1i-X4ih-W200S2i⊺+X3i2h+X2i⊺2h00***Ξ44i0000****Ξ55i-S1i-Zih2-Zi⊺h200*****-X1ih-Zih2-Zi⊺h200******-γ2I0*******-I]<0,(64)PiBi=BiP¯i(65)c1Λ+dγ21η(1-e-ηT)<λ1e-ηTc2,
where
(66)Θ11i=∑j=1N‍πijPj+δPi+PiAi+BiK¯i+Ai⊺Pi+K¯i⊺Bi⊺+eδh(Q1i+Q2i)+hQ+eδh-1δX1i+eδh-δheδh-1δ2X1-X4ih+hY1+W1-Zi-Zi⊺,
and the state feedback gain matrices considered in this paper could be designed as follows:
(67)Ki=P¯i-1K¯i,∀i=1,2,…,N.Proof.
This proof can be completed in view of Theorem9 with PiBi=BiP¯i and P¯iKi=K¯i.


## 5. Illustrative Example

Example 1.
Considering the following example with parameters(68)A1=[-0.90.5-0.32-0.8],Aτ1=[-0.5-0.30.3-0.2],B1=[-1.050.8-0.15-1.3],C1=[0.6-0.40.35-0.41],
the transition probabilities matrix is given as follows:
(69)Ω=[-0.20.20.8-0.8].
Given the different upper bounds ofh and δ, the results of the maximum upper bound of decay rates δ and maximum values of h for different time delays are obtained in Tables 1 and 2, respectively. This example indicates fully that the method proposed in the paper plays an important role in reducing conservatism. It can be also seen that our results in this paper show significant improvement over the results obtained in [11, 12]. This clearly shows that our results have less conservatism in the above two cases.Table 1
Comparison of the upper bounds of the decay rate for different delays.
h
=
0.2
h
=
0.5
h
=
0.8
h
=
1
h
=
1.2
[12]
1.2683
1.0064
0.7962
0.6838
0.5900
[11]
1.3618
1.1769
0.9420
0.7694
0.6261
Theorem6
1.3622
1.1771
0.9426
0.7696
0.6263Table 2
Comparison of the allowable values of time delayh for different decay rates.
δ
=
0.6
δ
=
0.8
δ
=
1
δ
=
1.2
δ
=
1.4
[12]
1.1768
0.7938
0.5081
0.2731
0.0657
[11]
1.2435
0.9626
0.7368
0.4651
0.1302
Theorem6
1.2441
0.9630
0.7372
0.4655
0.1304Example 2.
Consider the Markovian jump system (1) where
(70)A1=[-0.81.523],Aτ1=[-0.451-0.52],B1=[-10.20.5-0.1],D1=[0.20.1],C1=[0.2000.1],Cτ1=[0.0300.010.02],E1=[0.0200.010.01],D1=[0.010.001],A2=[-21.214],Aτ2=[-11.20-0.5],B2=[-110.5-2],D2=[0.20.3],C2=[0.10.0200.1],Cτ2=[0.0200.10.02],E2=[0.0400.10.01],F2=[0.040.01],
and corresponding transition rate matrix is
(71)Ω=[-1.21.21-1].
Assuming thatR=I, T=2, c1=1, and d=0.01, by suing LMI toolbox, Theorem 10 provides the following controller gains:
(72)K1=[-11.5351-8.121013.223010.5612],K2=[-21.212315.5613-23.231816.4518].


## 6. Conclusions

We have presented the problems of finite-time stochasticH∞ performance analysis of continuous-time systems with random abrupt changes in this paper. By using a different Lyapunov-Krasovskii functional, several sufficient conditions are provided to ensure the Markovian jump system is finite-time stochastic bounded. The controller gains can be dealt with by LMIs toolbox and optimization techniques. At last, two numerical examples are proposed to illustrate the effective and advantage of the developed theories.



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*Source: 198616-2014-03-03.xml*