# Delay-DependentH∞ Control for Descriptor Markovian Jump Systems with Time-Varying Delay

Authors: Jinghao Li; Qingling Zhang; Ding Zhai; Yi Zhang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper is concerned with the delay-dependentH∞ control problem for continuous-time descriptor
Markovian jump systems with time-varying delay. By constructing various Lyapunov-Krasovskii
functionals for different subsystems, together with delay decomposition method, a new delay-dependent
bounded real lemma (BRL) is derived, under which descriptor Markovian jump time-delay systems are
regular, impulse-free, and stochastically stable and satisfy a prescribed H∞ performance level. Since the
reciprocally convex combination approach is adopted to estimate the upper bound of the integral terms,
the BRL obtained in this paper is less conservative than some existing ones. Based on the proposed
BRL, a sufficient condition for the existence of state feedback controller is provided. Finally, three
numerical examples are provided to demonstrate the validity of the proposed methods.


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## Body

## 1. Introduction

Over the past decades, stability problems for time-delay systems have received considerable attention. Many methods have been presented to determine the stability. In the published results, two types of criteria have been widely investigated (see [1]), that is, delay-independent one and delay-dependent one. As we know, the delay-dependent criteria are less conservative than delay-independent ones, especially when the time delay is fairly small. Therefore, in order to reduce the conservatism of stability conditions, a great number of delay-dependent results have been reported, including, descriptor modeled transformation method (e.g., [2]), parameterized modeled transformation one (e.g., [3]), free-weighting matrix one (e.g., [4]), discretized Lyapunov functional one (e.g., [5]), delay decomposition one (e.g., [6]) and so on (e.g., [7, 8]). Among the aforementioned methods, since discretized Lyapunov functional method is complicated and it is difficult to apply this method to design controller, the delay decomposition method is less conservative than others except the discretized Lyapunov functional method.Markovian jump systems are modeled by many stochastic systems, which arise abrupt random changes in their structure, such as component failures or repairs and unexpected environmental changes. A great deal of attention has been devoted to Markovian jump systems (see [9, 10]). Recently, many efforts have been made to study the stability analysis and controller synthesis problems for Markovian jump systems with time-varying and time-invariant delay. Reference [11] divides the time-invariant delay into m segments and uses the delay division method to obtain a less conservative criterion than [12]. In [13], stability and stabilization are discussed for Markovian jump systems with time-varying delay by selecting distinct Lyapunov matrices for different subsystems and introducing a triple-integral term. Reference [14] studies the H∞ analysis for Markovian jump stochastic systems with time-varying delay. By dividing the delay interval into two parts, two improved delay-dependent conditions have been presented to guarantee the stochastic stability.Descriptor systems are also referred to as singular systems, semistate systems, which are a natural representation of dynamical systems and describe a much wider range of systems than standard state-space systems (see, e.g., [10, 15]). When descriptor systems are subject to time delay and stochastic disturbance in their structure, it is very natural to cast this model into descriptor Markovian jump time-delay systems. Very recently, a large number of researches have been published to study descriptor Markovian jump time-delay systems. By constructing more complicated Lyapunov-Krasovskii functionals, [16] studies the passivity and stochastic stability of descriptor Markovian jump systems with time-varying delay. Based on the delay decomposition method, [17] investigates the H∞ control problem for descriptor Markovian jump time-delay systems. To the best of our knowledge, the time-varying delay in most published results has not been tackled adequately, which leaves us much room to improve. All of these motivate the current study.In this paper, we investigateH∞ control problem for continuous-time descriptor Markovian jump systems with time-varying delay. Motivated by delay decomposition method, through designing various Lyapunov-Krasovskii functionals for different subsystems, a delay-dependent BRL is established, which guarantees that the considered system is stochastically admissible and satisfies a prescribed H∞ performance level. The reciprocally convex combination method is used in the proof of the proposed results, which is one of the contributions to this paper. The other is the employed Lyapunov-Krasovskii functionals. Thanks to all of these, the obtained results are less conservative than some existing results.The rest of this paper is organized as follows. Section2 gives problem description and some essential lemmas. Section 3 focuses on H∞ control problem for continuous-time descriptor Markovian jump time-delay systems. A new delay-dependent BRL is proposed, and the design method for the state feedback controller, which guarantees that the resultant closed-loop system is stochastically admissible with a prescribed H∞ performance level, is provided. Section 4 provides numerical examples to illustrate the effectiveness of our methods, and Section 5 concludes the paper.Notation. The notation used throughout this paper is quite standard. ℝn stands for the n-dimensional Euclidean space, and ℝm×n represents the set of all m×n real matrices. The superscripts T and -1 stand for matrix transposition and matrix inverse, respectively. ℒ2[0,∞) represents the space of square integrable vector functions over the interval [0,∞). (Ω,ℱ,𝒫) is the probability space, Ω is the sample space, ℱ is the σ-algebra of subsets of the sample space, and 𝒫 is the probability measure on ℱ. ℰ{·} is the expectation operator with respect to some probability measure 𝒫. ℕ+ represents the set of positive integers. The notation P>0 (P≥0) implies that P is a real symmetric and positive definite (semipositive definite) matrix. For simplicity, Mi is often adopted to denote M(i). The asterisk * in matrix block implies that it can be induced by symmetric position. A block diagonal matrix with diagonal blocks N1,N2,…,Nm will be denoted by diag{N1,N2,…,Nm}. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.


## 2. Problem Formulation and Preliminaries

Fix the probability space(Ω,ℱ,𝒫) and consider the following continuous-time descriptor Markovian jump systems with time-varying delay:
(1)Ex˙(t)=A(rt)x(t)+Ad(rt)x(t-h(t))+B(rt)u(t)+Bw(rt)w(t),z(t)=C(rt)x(t)+Cd(rt)x(t-h(t))+D(rt)u(t)+Dw(rt)w(t),x(θ)=ϕ(θ),∀θ∈[-h,0],
where x(t)∈ℝn is the state vector, u(t)∈ℝm is the input vector, z(t)∈ℝq is the controlled output vector, and w(t)∈ℝp is the disturbance input vector which belongs to ℒ2[0,∞). E is a known real matrix which may be singular, and it is generally assumed that rank (E)=r≤n. The jumping mode {rt,t≥0}, which takes value in a finite set ℓ={1,2,…,N}, is the continuous-time Markov process with the following mode transition probabilities:
(2)𝒫(rt+Δ=j∣rt=i)={γijΔ+o(Δ),ifj≠i1+γiiΔ+o(Δ),ifj=i,
where Δ>0, limt→0(o(Δ)/Δ)=0, and γij≥0(i,j∈ℓ,j≠i), satisfying γii=-∑j∈ℓ,j≠i‍γij, represent the switching rate from mode i at time t to mode j at time t+Δ for all i∈ℓ. A(rt), Ad(rt), B(rt), Bw(rt), C(rt), Cd(rt), D(rt), and Dw(rt) are known real constant matrices representing the deterministic systems for each rt∈ℓ. The considered time-varying delay h(t) satisfies
(3)0≤h(t)≤h,h˙(t)≤hd,
where h and hd are known constants. Throughout this paper, for each rt=i∈ℓ, we use Mi(t) and Ni to denote matrices M(rt,t) and N(rt).To facilitate exposition, the unforced descriptor Markovian jump time-delay systems are given as follows:(4)Ex˙(t)=A(rt)x(t)+Ad(rt)x(t-h(t))+Bw(rt)w(t)z(t)=C(rt)x(t)+Cd(rt)x(t-h(t))+Dw(rt)w(t)x(θ)=ϕ(θ),∀θ∈[-h,0].In the sequel, we recall some indispensable definitions.Definition 1 (see [10]).
When the disturbance signalw(t)≡0, for any given scalars h>0 and hd, consider the following statements.(1)
The continuous-time descriptor Markovian jump time-delay systems (4) are said to be regular and impulse-free for any time-varying delay satisfying (3) if the pairs (E,Ai) are regular and impulse-free for each rt∈ℓ.(2)
The continuous-time descriptor Markovian jump time-delay systems (4) are said to be stochastically stable if there exists a scalar M(ϕ(·),r0)>0 such that
(5)limt→∞ℰ{∫0t‍xT(s,xt,r0)x(s,xt,r0)ds∣xt=x(s)=ϕ(s),s∈[-h,0],r0∈ℓ∫0t‍}≤M(ϕ(·),r0),wherex(t,xt,r0) denotes the solution to systems (4) at time t under the given initial conditions.(3)
The continuous-time descriptor Markovian jump time-delay systems (4) are said to be stochastically admissible if they are regular, impulse-free, and stochastically stable.Definition 2 (see [17]).
For a given scalarγ>0, the descriptor Markovian jump time-delay systems (4) are said to be stochastically admissible with H∞ performance level γ; if the system with w(t)≡0 is stochastically admissible and under zero initial condition, it satisfies
(6)ℰ{∫0∞‍zT(t)z(t)dt}≤γ2∫0∞‍wT(t)w(t)dt
for any nonzero w(t)∈ℒ[0,∞).In this paper, the purpose is to design a mode-dependent state feedback controller for the continuous-time descriptor Markovian jump time-delay systems (1) with a prescribed scalar γ>0 such that the resultant closed-loop systems are stochastically admissible with H∞ performance level γ. Specifically, the desired state feedback controller can be expressed as
(7)u(t)=K(rt)x(t)+Kd(rt)x(t-h(t)).To this end, some essential lemmas are introduced to expedite the following discussion.Lemma 3 (see [18]).
For any constant matrixW∈ℝn×n, W>0, scalar γ>0, and vector function x:[-γ,0]→ℝn such that the following integration is well defined, then
(8)-γ∫t-γt‍xT(s)Wx(s)ds≤-∫t-γt‍xT(s)dsW∫t-γt‍x(s)ds.Lemma 4 (see [8]).
Letf1,f2,…,fN:ℝm↦ℝ have positive values in an open subset 𝔻 of ℝm. Then the reciprocally convex combination of fi over 𝔻 satisfies
(9)min{αi|αi>0,∑iαi=1}∑i‍1αifi(t)=∑i‍fi(t)+maxgi,j(t)∑i≠j‍gi,j(t),
subject to
(10){gi,j:ℝm↦ℝ,gj,i(t)=gi,j(t),[fi(t)gi,j(t)gi,j(t)fi(t)]≥0}.Based on Lemma4, we present the following lemma, which can be regarded as an alternative form of the reciprocally convex combination.Lemma 5.
For any constant matrixW∈ℝn×n, W≥0, and a variable β satisfying a≤β≤b, if there exists a matrix W^ satisfying [WW^W^TW]≥0, then
(11)-1β-ae1(t)TWe1(t)-1b-βe2T(t)We2(t)≤-1b-a[e1(t)e2(t)]T[WW^W^TW][e1(t)e2(t)],
where e1(t) and e2(t) are vector functions with compatible dimensions.Proof.
Taking(12)f1(t)=e1T(t)We1(t),f2(t)=e2T(t)We2(t),g1,2(t)=e1T(t)W^e2(t),g2,1(t)=e2T(t)W^Te1(t),α1=β-ab-a,α2=b-βb-a,
it can be easily verified that the above choice satisfies constraint (10) in Lemma 4. Note that with the equality (9), we have
(13)min{αi|αi>0,∑iαi=1}∑i‍1αifi(t)≤b-aβ-ae1T(t)We1(t)+b-ab-βe2T(t)We2(t),(14)∑i‍fi(t)+maxgi,j(t)∑i≠j‍gi,j(t)≥[e1(t)e2(t)]T[WW^W^TW][e1(t)e2(t)].
By Lemma 4, (13) and (14) yield that
(15)-1β-ae1T(t)We1(t)-1b-βe2T(t)We2(t)≤-1b-a[e1(t)e2(t)]T[WW^W^TW][e1(t)e2(t)].
This completes the proof.Lemma 6 (see [10]).
Given any real matricesS1, S2, and S3 with S1T=S1 and S3>0, then one has
(16)S1+S2S3-1S2T<0
if and only if
(17)[S1S2S2T-S3]<0,
or equivalently
(18)[-S3S2TS2S1]<0.Lemma 7 (see [10]).
Given matricesΦ=ΦT<0, D and E with appropriate dimensions
(19)Φ+DF(t)E+ETF(t)TDT<0
for all F(t) satisfying FT(t)F(t)≤I if and only if there exists scalar ϵ>0 such that
(20)Φ+εDDT+ε-1ETE<0.Lemma 8 (see [19]).
LetP be symmetric such that ERTPER>0, and let Φ be nonsingular. Then, PE+UTΦVT is nonsingular and its inverse is expressed as
(21)(PE+UTΦVT)-1=P-ET+VΦ-U,
where EL and ER are full rank with E=ELERT and UT and V are of full column rank and composed of bases of null space of ET and E, respectively. P- is symmetric and Φ- is nonsingular such that ERTP-ER=(ELTPEL)-1, Φ-=(VTV)-1Φ-1(UUT)-1.


## 3. Main Results

In this section, the delay-dependentH∞ control problem will be studied for the descriptor Markovian jump time-delay system (1) by virtue of the delay decomposition method. To this end, by dividing the delay interval (0,h] into two disjoint subintervals (0,αh] and (αh,h], (0<α<1), the following delay-dependent bounded real lemma (BRL) for the system (4) is firstly proposed, which will play an indispensable role in the derivation of our main results.Theorem 9.
For given scalarsγ>0, h>0, and 0≤hd<1, the descriptor Markovian jump time-delay system (4) is stochastically admissible with H∞ performance level γ for any h(t) satisfying (3), if there exist a set of positive definite matrices Pi, Q1i, Q2i, Q3i, R2i, R3i, U1, U2, W, and matrices Φi, Si, Ti, ∀i∈ℓ, satisfying the following LMIs for each i∈ℓ:(22)[Θi11Θi12Θi130Θi15AiTR2iAiTR3i12h2AiTWCiT*Θi22Θi2300AdiTR2iAdiTR3i12h2AdiTWCdiT**Θi33Θi3400000***Θi4400000****-γ2IBwiTR2iBwiTR3i12h2BwiTWDwiT*****-α1R2i000******-α2R3i00*******-12h2W0********-I]<0,h(t)∈(0,αh](23)[Θi11Θ^i12Θ^i130Θi15AiTR2iAiTR3i12h2AiTWCiT*Θ^i22Θ^i23Θ^i240AdiTR2iAdiTR3i12h2AdiTWCdiT**Θi33Θ^i3400000***Θi4400000****-γ2IBwiTR2iBwiTR3i12h2BwiTWDwiT*****-α1R2i000******-α2R3i00*******-12h2W0********-I]<0,h(t)∈(αh,h](24)∑j=1NγijR2j-W<0,∑j=1NγijR3j-W<0(25)∑j=1NγijQ1j+∑j=1NγijQ2j-U1<0,∑j=1NγijQ2j-U1<0(26)∑j=1NγijQ1j+∑j=1NγijQ3j-U2<0,∑j=1NγijQ3j-U2<0(27)Z1i=[R2iSiSiTR2i]>0,Z2i=[R3iTiTiTR3i]>0,where α1=1/αh, α2=1/(h-αh), and
(28)Θi11=AiT(PiE+UTΦiVT)+(PiE+UTΦiVT)TAiΘi11=+∑j=1NγijETPjE+Q1i+Q2iΘi11=+αhU1+(1-α)hU2-α1ETR2iE,Θi12=(PiE+UTΦiVT)TAdi+α1ETR2iE-α1ETSiE,Θi13=α1ETSiE,Θi15=(PiE+UTΦiVT)TBwi,Θi22=-(1-hd)Q1i-2α1ETR2iE+α1ETSiE+α1ETSiTE,Θi23=α1ETR2iE-α1ETSiE,Θi33=Q3i-Q2i-α1ETR2iE-α2ETR3iE,Θi34=α2ETR3iE,Θi44=-Q3i-α2ETR3iE,Θ^i12=(PiE+UTΦiVT)TAdi,Θ^i13=α1ETR2iE,Θ^i22=-(1-hd)Q1i-2α2ETR3iE+α2ETTiE+α2ETTiTE,Θ^i23=α2ETR3iE-α2ETTiTE,Θ^i24=α2ETR3iE-α2ETTiE,Θ^i34=α2ETTiE.Proof.
Assume that there exist matrices such that LMIs (22)–(27) hold. With these conditions in hand, we firstly show the regularity and absence of impulse of the system (4). To this end, it is from [10] that there exist a couple of nonsingular matrices M and N such that for each i∈ℓ(29)MEN=[Ir000],MAiN=[Ai1Ai2Ai3Ai4],MAdiN=[Adi1Adi2Adi3Adi4].
Decomposing the associated matrices with a compatible dimension, we have(30)M-TPiM-1=[Pi1Pi2Pi3Pi4],M-TUT=[0I]Z1,VTN=Z2[0I],
where Z1∈ℝ(n-r)×(n-r) and Z2∈ℝ(n-r)×(n-r) are arbitrarily nonsingular matrices.
From the(1,1) block in the left-hand matrices of (22) or (23), and observing that Q1i, Q2i, U1, and U2 are positive definite, it can be easily seen that
(31)AiT(PiE+UTΦiVT)+(PiE+UTΦiVT)TAi+ETf(Pj,R2i)E<0,
where f(Pj,R2i) is the function of Pj and R2i and makes no difference to the subsequent proof.
Pre- and postmultiplying (31) by NT and N, respectively, together with (29) and (30), we get
(32)[***Ai4TZ1ΦiZ2+Z2TΦiTZ1TAi4]<0.
According to [10], we can know that the pairs (E,Ai) are regular and impulse-free for each i∈ℓ. Therefore, by Definition 1, the system (4) is regular and impulse-free.
Next, we will show the stochastic stability of the system (4). To this end, we define a new process {(xt,rt),t≥0} by {xt(s)=x(t+s),-2h≤s≤0}; then from [12], we can know {(xt,rt),t≥h} is a Markov process with initial state (ϕ(·),r0). Now, construct the stochastic Lyapunov functional candidate to be
(33)V(xt,rt,t)=V1(xt,rt,t)+V2(xt,rt,t)+V3(xt,rt,t),
where
(34)V1(xt,rt,t)=xT(t)ETP(rt)Ex(t),V2(xt,rt,t)=∫t-h(t)t‍xT(s)Q1(rt)x(s)ds+∫t-αht‍xT(s)Q2(rt)x(s)ds+∫t-ht-αh‍xT(s)Q3(rt)x(s)ds+∫-αh0‍∫t+θt‍xT(s)U1x(s)dsdθ+∫-h-αh‍∫t+θt‍xT(s)U2x(s)dsdθ,V3(xt,rt,t)=∫-αh0‍∫t+θt‍x˙T(s)ETR2(rt)Ex˙(s)dsdθ+∫-h-αh‍∫t+θt‍x˙T(s)ETR3(rt)Ex˙(s)dsdθ+∫-h0‍∫β0‍∫t+θt‍x˙T(s)ETWEx˙(s)dsdθdβ.
Letℒ be the weak infinitesimal generator of the random process {xt,t} acting on V(xt,rt,t) [12]. Then for each rt=i∈ℓ, it can be shown that
(35)ℒV1(xt,i,t)=x˙T(t)ETPiEx(t)+xT(t)ETPiEx˙(t)+xT(t)(∑j=1N‍γijETPjE)x(t)=x˙T(t)ET(PiE+UTΦiVT)x(t)+xT(t)(PiE+UTΦiVT)TEx˙(t)+xT(t)(∑j=1N‍γijETPjE)x(t)=xT(t)AiT(PiE+UTΦiVT)x(t)+xT(t-h(t))AdiT(PiE+UTΦiVT)x(t)+xT(t)(PiE+UTΦiVT)TAix(t)+xT(t)(PiE+UTΦiVT)TAdix(t-h(t))+xT(t)(∑j=1N‍γijETPjE)x(t)=xT(t)(∑j=1N‍AiT(PiE+UTΦiVT)+(PiE+UTΦiVT)TAi+∑j=1N‍γijETPjE)x(t)+xT(t-h(t))AdiT(PiE+UTΦiVT)x(t)+xT(t)(PiE+UTΦiVT)TAdix(t-h(t)),(36)ℒV2(xt,i,t)=xT(t)(Q1i+Q2i+αhU1+(1-α)hU2)x(t)-(1-h˙(t))xT(t-h(t))Q1ix(t-h(t))+xT(t-αh)(Q3i-Q2i)x(t-αh)-xT(t-h)Q3ix(t-h)+∫t-h(t)t‍xT(s)(∑j=1N‍γijQ1j)x(s)ds+∫t-αht‍xT(s)(∑j=1N‍γijQ2j)x(s)ds+∫t-ht-αh‍xT(s)(∑j=1N‍γijQ3j)x(s)ds-∫t-αht‍xT(s)U1x(s)ds-∫t-ht-αh‍xT(s)U2x(s)ds≤xT(t)(Q1i+Q2i+αhU1+(1-α)hU2)x(t)-(1-hd)xT(t-h(t))Q1ix(t-h(t))+xT(t-αh)(Q3i-Q2i)x(t-αh)-xT(t-h)Q3ix(t-h)+∫t-h(t)t‍xT(s)(∑j=1N‍γijQ1j)x(s)ds+∫t-αht‍xT(s)(∑j=1N‍γijQ2j)x(s)ds+∫t-ht-αh‍xT(s)(∑j=1N‍γijQ3j)x(s)ds-∫t-αht‍xT(s)U1x(s)ds-∫t-ht-αh‍xT(s)U2x(s)ds,(37)ℒV3(xt,i,t)=x˙T(t)ET(αhR2i+(1-α)hR3i+12h2W)Ex˙(t)-∫t-αht‍x˙T(s)ETR2iEx˙(s)ds-∫t-ht-αh‍x˙T(s)ETR3iEx˙(s)ds+∫-αh0‍∫t+θt‍x˙T(s)(∑j=1N‍γijETR2jE)x˙(s)dsdθ-∫-αh0‍∫t+θt‍x˙T(s)ETWEx˙(s)dsdθ+∫-h-αh‍∫t+θt‍x˙T(s)(∑j=1N‍γijETR3jE)x˙(s)dsdθ-∫-h-αh‍∫t+θt‍x˙T(s)ETWEx˙(s)dsdθ=x˙T(t)ET(αhR2i+(1-α)hR3i+12h2W)Ex˙(t)-∫t-αht‍x˙T(s)ETR2iEx˙(s)ds+∫-αh0‍∫t+θt‍x˙T(s)ET(∑j=1N‍γijR2j-W)Ex˙(s)dsdθ-∫t-ht-αh‍x˙T(s)ETR3iEx˙(s)ds+∫-h-αh‍∫t+θt‍x˙T(s)ET(∑j=1N‍γijR3j-W)Ex˙(s)dsdθ.
Next, we will estimate the upper bound of the integral terms inℒV2(xt,i,t) and ℒV3(xt,i,t).
Case
1 (0<h(t)≤αh). The upper bound can be evaluated for the last five integral terms in ℒV2(xt,i,t), which can be shown as
(38)∫t-h(t)t‍xT(s)(∑j=1N‍γijQ1j)x(s)ds+∫t-αht‍xT(s)(∑j=1N‍γijQ2j)x(s)ds+∫t-ht-αh‍xT(s)(∑j=1N‍γijQ3j)x(s)ds-∫t-αht‍xT(s)U1x(s)ds-∫t-ht-αh‍xT(s)U2x(s)ds=∫t-h(t)t‍xT(s)(∑j=1N‍γijQ1j+∑j=1N‍γijQ2j-U1)x(s)ds+∫t-αht-h(t)‍xT(s)(∑j=1N‍γijQ2j-U1)x(s)ds+∫t-ht-αh‍xT(s)(∑j=1N‍γijQ3j-U2)x(s)ds.
Likewise, by applying Jensen inequality of Lemma3 and reciprocally convex combination method of Lemma 5, the upper bound for the two definite integrals in ℒV3(xt,i,t) can be shown as
(39)-∫t-αht‍x˙T(s)ETR2iEx˙(s)ds-∫t-ht-αh‍x˙T(s)ETR3iEx˙(s)ds=-∫t-h(t)t‍x˙T(s)ETR2iEx˙(s)ds-∫t-αht-h(t)‍x˙T(s)ETR2iEx˙(s)ds-∫t-ht-αh‍x˙T(s)ETR3iEx˙(s)ds≤-1h(t)(x(t)-x(t-h(t)))T×ETR2iE(x(t)-x(t-h(t)))-1αh-h(t)(x(t-h(t))-x(t-αh))T×ETR2iE(x(t-h(t))-x(t-αh))-1h-αh(x(t-αh)-x(t-h))T×ETR3iE(x(t-αh)-x(t-h))≤-1αh[x(t)-x(t-h(t))x(t-h(t))-x(t-αh)]T×[ETR2iEETSiEETSiTEETR2iE][x(t)-x(t-h(t))x(t-h(t))-x(t-αh)]-1h-αh(x(t-αh)-x(t-h))T×ETR3iE(x(t-αh)-x(t-h))=-[x(t)-x(t-h(t))x(t-h(t))-x(t-αh)x(t-αh)-x(t-h)]T×[α1E-TZ1iE-00α2ETR3iE][x(t)-x(t-h(t))x(t-h(t))-x(t-αh)x(t-αh)-x(t-h)].
Combining (35)–(39), it yields that
(40)ℒV(xt,i,t)≤ηT(t)Ψ1iη(t)+∫t-h(t)t‍xT(s)(∑j=1N‍γijQ1j+∑j=1N‍γijQ2j-U1)x(s)ds+∫-αh0‍∫t+θt‍x˙T(s)ET(∑j=1N‍γijR2j-W)Ex˙(s)dsdθ+∫t-αht-h(t)‍xT(s)(∑j=1N‍γijQ2j-U1)x(s)ds+∫-h-αh‍∫t+θt‍x˙T(s)ET(∑j=1N‍γijR3j-W)Ex˙(s)dsdθ+∫t-ht-αh‍xT(s)(∑j=1N‍γijQ3j-U2)x(s)ds,
where ηT(t)=[xT(t)xT(t-h(t))xT(t-αh)xT(t-h)], and
(41)Ψ1i=Ψi-[I-I000I-I000I-I]T[α1E-TZ1iE-00α2ETR3iE]×[I-I000I-I000I-I],
where E-=diag{E,E} and
(42)Ψi=[Ψi11Ψi1200*Ψi2200**Ψi330***Ψi44],Ψi11=AiT(PiE+UTΦiVT)+(PiE+UTΦiVT)TAiΨi11=+∑j=1N‍γijETPjE+Q1i+Q2i+αhU1+(1-α)hU2Ψi11=+AiT(αhR2i+(1-α)hR3i+12h2W)Ai,Ψi12=(PiE+UTΦiVT)TAdiΨi12=+AiT(αhR2i+(1-α)hR3i+12h2W)Adi,Ψi22=-(1-hd)Q1iΨi22=+AdiT(αhR2i+(1-α)hR3i+12h2W)Adi,Ψi33=Q3i-Q2i,Ψi44=-Q3i.
From the Schur complement lemma of Lemma5, it can be easily verified that Ψ1i<0 holds if (22) holds for 0<h(t)≤αh.
Case
2 (αh<h(t)≤h). The upper bound can be evaluated for the last five integral terms in ℒV2(xt,i,t), which can be shown as
(43)∫t-h(t)t‍xT(s)(∑j=1N‍γijQ1j)x(s)ds+∫t-αht‍xT(s)(∑j=1N‍γijQ2j)x(s)ds+∫t-ht-αh‍xT(s)(∑j=1N‍γijQ3j)x(s)ds-∫t-αht‍xT(s)U1x(s)ds-∫t-ht-αh‍xT(s)U2x(s)ds=∫t-αht‍xT(s)(∑j=1N‍γijQ1j+∑j=1N‍γijQ2j-U1)x(s)ds+∫t-ht-h(t)‍xT(s)(∑j=1N‍γijQ3j-U2)x(s)ds+∫t-h(t)t-αh‍xT(s)(∑j=1N‍γijQ1j+∑j=1N‍γijQ3j-U2)x(s)ds.
Likewise, by applying Jensen inequality of Lemma3 and reciprocally convex combination method of Lemma 5, the upper bound for the two definite integrals in ℒV3(xt,i,t) can be shown as
(44)-∫t-αht‍x˙T(s)ETR2iEx˙(s)ds-∫t-ht-αh‍x˙T(s)ETR3iEx˙(s)ds=-∫t-αht‍x˙T(s)ETR2iEx˙(s)ds-∫t-h(t)t-αh‍x˙T(s)ETR3iEx˙(s)ds-∫t-ht-h(t)‍x˙T(s)ETR3iEx˙(s)ds≤-1αh(x(t)-x(t-αh))T×ETR2iE(x(t)-x(t-αh))-1h(t)-αh(x(t-αh)-x(t-h(t)))T×ETR3iE(x(t-αh)-x(t-h(t)))-1h-h(t)(x(t-h(t))-x(t-h))T×ETR3iE(x(t-h(t))-x(t-h))≤-1αh(x(t)-x(t-αh))T×ETR2iE(x(t)-x(t-αh))-1h-αh[x(t-αh)-x(t-h(t))x(t-h(t))-x(t-h)]T×[ETR3iEETTiEETTiTEETR3iE][x(t-αh)-x(t-h(t))x(t-h(t))-x(t-h)]=-[x(t)-x(t-αh)x(t-αh)-x(t-h(t))x(t-h(t))-x(t-h)]T[α1ETR2iE00α2E-TZ2iE-]×[x(t)-x(t-αh)x(t-αh)-x(t-h(t))x(t-h(t))-x(t-h)].
Combining (35)–(37) and (43)-(44), it yields that
(45)ℒV(xt,i,t)≤ηT(t)Ψ2iη(t)+∫t-αht‍xT(s)(∑j=1N‍γijQ1j+∑j=1N‍γijQ2j-U1)x(s)ds+∫t-h(t)t-αh‍xT(s)(∑j=1N‍γijQ1j+∑j=1N‍γijQ3j-U2)x(s)ds+∫t-ht-h(t)‍xT(s)(∑j=1N‍γijQ3j-U2)x(s)ds+∫-αh0‍∫t+θt‍x˙T(s)ET(∑j=1N‍γijR2j-W)Ex˙(s)dsdθ+∫-h-αh‍∫t+θt‍x˙T(s)ET(∑j=1N‍γijR3j-W)Ex˙(s)dsdθ,
where ηT(t)=[xT(t)xT(t-h(t))xT(t-αh)xT(t-h)], and
(46)Ψ2i=Ψi-[I0-I00-II00I0-I]T[α1ETR2iE00α2E-TZ2iE-]×[I0-I00-II00I0-I].
From the Schur complement lemma of Lemma5, it can be easily verified that Ψ2i<0 holds if (23) holds for αh<h(t)≤h.
Based on the above analysis, we can know that there exists a scalarλ>0 such that for each i∈ℓ(47)ℒV(xt,i,t)≤-λ∥x(t)∥2.
Then, by Dynkin’s formula, we can have for each i∈ℓ and t≥-h(48)ℰV(xt,i,t)≤ℰV(x0,i,0)-λℰ∥x(t)∥2,
which implies that
(49)ℰ∥x(t)∥2≤λ-1ℰV(x0,i,0).
Following a similar line in [12], we can find that the system (4) is stochastically stable under the condition that w(t)=0. By Definition 1, the system (4) is stochastically admissible.
In the following, we will establish theH∞ performance index of the system (4). Using the stochastic Lyapunov functional (33), under the zero initial condition, it is from (22)–(27) that
(50)Jzw(t)=ℰ{∫0t‍[zT(s)z(s)-γ2wT(s)w(s)]ds}=ℰ{∫0t‍[zT(s)z(s)-γ2wT(s)w(s)+ℒV(xs,i,s)]ds}-ℰV(xt,i,t)≤ℰ{∫0t‍[zT(s)z(s)-γ2wT(s)w(s)+ℒV(xs,i,s)]ds}≤ℰ{∫0t‍η^T(s)Θiη^(s)ds},
where η^T(t)=[ηT(t)wT(t)]; for h(t)∈(0,αh],
(51)Θi=[Θi11+CiTCiΘi12+CiTCdiΘi130Θi15+CiTDwi*Θi22+CdiTCdiΘi230CdiTDwi**Θi33Θi340***Θi440****DwiTDwi-γ2I]+[AiTAdiT00BwiT](αhR2i+(1-α)hR3i+12h2W)[AiTAdiT00BwiT]T,
and for h(t)∈(αh,h],
(52)Θi=[Θi11+CiTCiΘ^i12+CiTCdiΘ^i130Θi15+CiTDwi*Θ^i22+CdiTCdiΘ^i23Θ^i24CdiTDwi**Θi33Θ^i340***Θi440****DwiTDwi-γ2I]+[AiTAdiT00BwiT](αhR2i+(1-α)hR3i+12h2W)[AiTAdiT00BwiT]T.
By Schur complement lemma of Lemma6, together with (22)–(27), we have that Jzw(t)<0 for all t>0. Consequently, under the zero initial condition, we have that (6) holds for any nonzero w(t)∈ℒ2[0,∞). Therefore, by Definition 2, the descriptor Markovian jump time-delay system (4) is stochastically admissible with H∞ performance level γ for any time delay h(t) satisfying (3). This completes the proof.Remark 10.
By virtue of the novel Lyapunov-Krasovskii functional, Theorem9 provides a delay-dependent and delay-derivative-dependent H∞ criterion to guarantee the system (4) to be stochastically admissible with H∞ performance level. The novelty of Theorem 9 lies in the new Lyapunov-Krasovskii functional and the reciprocally convex combination method. By dividing the delay interval (0,h] into two subintervals (0,αh] and (αh,h] the following mode-dependent definite integrals
(53)∫t-αht‍xT(s)Q2(rt)x(s)ds+∫t-ht-αh‍xT(s)Q3(rt)x(s)ds
and mode-dependent double integrals
(54)∫-αh0‍∫t+θt‍x˙T(s)ETR2(rt)Ex˙(s)dsdθ+∫-h-αh‍∫t+θt‍x˙T(s)ETR3(rt)Ex˙(s)dsdθ
are introduced. Obviously, the above integral terms take the mode rt into full consideration and contain the mode-independent one [12] as a special case. Since h(t) is restricted to the subinterval (0,αh] or (αh,h] and various Lyapunov-Krasovskii functionals are selected for different subsystems, our method can lead to reduction of conservatism.For the unforced system (4), if disturbance signal w(t)≡0, then we have the following time-delay systems:
(55)Ex˙(t)=A(rt)x(t)+Ad(rt)x(t-h(t))x(θ)=ϕ(θ),∀θ∈[-h,0].Based on Theorem9, the following stochastic admissibility result can be given.Corollary 11.
For given scalarsh>0, 0≤hd<1, the system (55) is stochastically admissible if there exist a set of positive definite matrices Pi, Q1i, Q2i, Q3i, R2i, R3i, U1, U2, W and matrices Φi, Si, Ti, ∀i∈ℓ, satisfying the following LMIs for each i∈ℓ:
(56)Θ1i=Θi-[I-I0000I-I0000I-I00000I]T×[α1E-TZ1iE-000α2ETR3iE0000][I-I0000I-I0000I-I00000I]<0,Θ2i=Θi-[I0-I000-II000I0-I00000I]T×[α1ETR2iE000α2E-TZ2iE-0000][I0-I000-II000I0-I00000I]<0,∑j=1NγijR2j-W<0,∑j=1NγijR3j-W<0,∑j=1NγijQ1j+∑j=1NγijQ2j-U1<0,∑j=1NγijQ2j-U1<0,∑j=1NγijQ1j+∑j=1NγijQ3j-U2<0,∑j=1NγijQ3j-U2<0,(57)Z1i=[R2iSiSiTR2i]>0,Z2i=[R3iTiTiTR3i]>0,
where α1=1/αh, α2=1/(h-αh), E-=diag{E,E}, and
(58)Θi=[Θi11Θi1200Θi15*Θi2200Θi25**Θi3300***Θi440****Θi55],Θi11=AiT(PiE+UTΦiVT)+(PiE+UTΦiVT)TAiΘi11=+∑j=1N‍γijETPjE+Q1i+Q2i+αhU1Θi11=+(1-α)hU2,Θi12=(PiE+UTΦiVT)TAdi,Θi15=AiT(αhR2i+(1-α)hR3i+12h2W),Θi22=-(1-hd)Q1i,Θi25=AdiT(αhR2i+(1-α)hR3i+12h2W),Θi33=Q3i-Q2i,Θi44=-Q3i,Θi55=-(αhR2i+(1-α)hR3i+12h2W).When the time delay is time-invariant, the system (55) reduces to the one
(59)Ex˙(t)=A(rt)x(t)+Ad(rt)x(t-h)x(θ)=ϕ(θ),∀θ∈[-h,0].
Using the similar method of Corollary 11, we have the stochastic admissibility condition for (59).Corollary 12.
For given scalarsh>0, the system (59) is stochastically admissible if there exist a set of positive definite matrices Pi, Q2i, Q3i, R2i, R3i, U1, U2, W and matrices Φi, ∀i∈ℓ, satisfying the following LMIs for each i∈ℓ:
(60)Θ-i=Θi-[I-I000I-I0000I]T[α1ETR2iE000α2ETR3iE0000]×[I-I000I-I0000I]<0,∑j=1NγijR2j-W<0,∑j=1NγijR3j-W<0,∑j=1NγijQ2j-U1<0,∑j=1NγijQ3j-U2<0,
where α1=1/αh, α2=1/(h-αh); and
(61)Θi=[Θi110Θi13Θi14*Θi2200**Θi33Θi34***Θi44],Θi11=AiT(PiE+UTΦiVT)+(PiE+UTΦiVT)TAiΘi11=+∑j=1N‍γijETPjE+Q2i+αhU1+(1-α)hU2,Θi13=(PiE+UTΦiVT)TAdi,Θi14=AiT(αhR2i+(1-α)hR3i+12h2W),Θi22=Q3i-Q2i,Θi33=-Q3i,Θi34=AdiT(αhR2i+(1-α)hR3i+12h2W),Θi44=-(αhR2i+(1-α)hR3i+12h2W).Remark 13.
By decomposing the delay interval(0,h] into two segments (0,αh] and (αh,h], Theorem 9, Corollary 11, and Corollary 12 give delay-dependent conditions for the underlying systems with time-varying and time-invariant delay, respectively. The algorithm about how to select tuning parameter α optimally can be referred to the algorithm in the end of this section.Now, we are ready to deal with the delay-dependentH∞ control problem for the descriptor Markovian jump systems (1). For this purpose, applying the controller in (7) to the system (1), then we can have the following closed-loop systems:
(62)Ex˙(t)=(A(rt)+B(rt)K(rt))x(t)+(Ad(rt)+B(rt)Kd(rt))x(t-h(t))+Bw(rt)w(t),z(t)=(C(rt)+D(rt)K(rt))x(t)+(Cd(rt)+D(rt)Kd(rt))x(t-h(t))+Dw(rt)w(t),x(θ)=ϕ(θ),∀θ∈[-h,0].On the basis of Theorem9, the following result, which provides the design method of state feedback controller and ensures the closed-loop systems (62) to be stochastically admissible with H∞ performance level γ, can be obtained.Theorem 14.
For given scalarsγ>0, εi>0, h>0, and 0≤hd<1, the descriptor Markovian jump time-delay system (1) is stochastically admissible with H∞ performance level γ for any h(t) satisfying (3), if there exist a set of positive definite matrices P-i, Q-1ij, Q-2ij, Q-3ij, R-2i, R-3i, U-1i, U-2i, W- and matrices Φ-i, Hi, Li, ∀i∈ℓ,j∈ℓ, satisfying the following LMIs for each i∈ℓ:(63)[Υi11Υ˘i1200BwiΥi1612h2(AiΛi+BiHi)T(CiΛi+DiHi)TΥi19*Υ˘i22Υ˘i2300Υi2612h2(AdiΛi+BiLi)T(CdiΛi+DiLi)T0**Υi33Υ˘i3400000***Υi4400000****-γ2IΥi5612h2BwiTDwiT0*****Υi66000******-12h2W-00*******-I0********Υi99]<0,h(t)∈(0,αh],(64)[Υi11Υ^i12Υ^i130BwiΥi1612h2(AiΛi+BiHi)T(CiΛi+DiHi)TΥi19*Υ^i22Υ^i23Υ^i240Υi2612h2(AdiΛi+BiLi)T(CdiΛi+DiLi)T0**Υi33000000***Υi4400000****-γ2IΥi5612h2BwiTDwiT0*****Υi66000******-12h2W-00*******-I0********Υi99]<0,h(t)∈(αh,h],(65)γij(N-1)W-<R-2j,γij(N-1)W-<R-3j,∀j≠i,γij≠0,(66)∑j=1NγijQ-1ij+∑j=1NγijQ-2ij-U-1i<0,∑j=1NγijQ-2ij-U-1i<0,(67)∑j=1NγijQ-1ij+∑j=1NγijQ-3ij-U-2i<0,∑j=1NγijQ-3ij-U-2i<0,where α1=1/αh, α2=1/(h-αh), Λi=P-iET+VΦ-iU, and
(68)Υi11=ΛiTAiT+HiTBiT+AiΛi+BiHiΥi11=+γii(ϵiΛiTET+ϵiEΛi-ϵi2EP-iET)Υi11=+Q-1ii+Q-2ii+αhU-1i+(1-α)hU-2iΥi11=-ΛiTET-EΛi+αhR-2i,Υ˘i12=AdiΛi+BiLi+ΛiTET+EΛi-αhR-2i,Υi16=[(AiΛi+BiHi)T(AiΛi+BiHi)T],Υi19=[γi1ΛiTER⋯γi(i-1)ΛiTERγi(i+1)ΛiTER⋯γiNΛiTER],Υ˘i22=-(1-hd)Q-1ii-2ΛiTET-2EΛi+2αhR-2i,Υ˘i23=ΛiTET+EΛi-αhR-2i,Υi26=[(AdiΛi+BiLi)T(AdiΛi+BiLi)T],Υi33=Q-3ii-Q-2ii-2ΛiTET-2EΛi+αhR-2i+(h-αh)R-3i,Υ˘i34=ΛiTET+EΛi-(h-αh)R-3i,Υi44=-Q3ii-ΛiTET-EΛi+(h-αh)R-3i,Υi56=[BwiTBwiT],Υi66=diag{-α1R-2i,-α2R-3i},Υi99=-diag{ERTP-1ER,…,ERTP-i-1ER,ERTP-i+1ER,…,ERTP-NER},Υ^i12=AdiΛi+BiLi,Υ^i13=ΛiTET+EΛi-αhR-2i,Υ^i22=-(1-hd)Q-1ii-2ΛiTET-2EΛi+2(h-αh)R-3i,Υ^i23=ΛiTET+EΛi-(h-αh)R-3i,Υ^i24=Υ^i23.
Furthermore, if the above LMIs have feasible solutions, then the desired state feedback controller can be solved as Ki=HiΛi-1, Kdi=LiΛi-1.Proof.
It follows from Theorem9 that the closed-loop system (62) is stochastically admissible with H∞ performance level γ if there exist a set of matrices such that (22)–(27) hold, where Ai, Adi, Ci, and Cdi are replaced by Ai+BiKi, Adi+BiKdi, Ci+DiKi, and Cdi+DiKdi, respectively.
According to Lemma8, we can set Λi=(PiE+UTΦiVT)-1=P-iET+VΦ-iU; then pre- and postmultiplying (22) and (23) by diag{ΛiT,ΛiT,ΛiT,ΛiT,I,R2i-T,R3i-T,W-T,I} and its transpose and leting Hi=Ki(P-iET+VΦ-iU), Li=Kdi(P-iET+VΦ-iU), we have (69)[Θ˘i11Θ˘i12Θ˘i130BwiΘ˘i1612h2(AiΛi+BiHi)T(CiΛi+DiHi)T*Θ˘i22Θ˘i2300Θ˘i2612h2(AdiΛi+BiLi)T(CdiΛi+DiLi)T**Θ˘i33Θ˘i340000***Θ˘i440000****-γ2IΘ˘i5612h2BwiTDwiT*****Θ˘i6600******-12h2W-10*******-I]<0,h(t)∈(0,αh],(70)[Θ˘i11Θ^i12Θ^i130BwiΘ˘i1612h2(AiΛi+BiHi)T(CiΛi+DiHi)T*Θ^i22Θ^i23Θ^i240Θ˘i2612h2(AdiΛi+BiLi)T(CdiΛi+DiLi)T**Θ˘i33Θ^i340000***Θ˘i440000****-γ2IΘ˘i5612h2BwiTDwiT*****Θ˘i6600******-12h2W-10*******-I]<0,h(t)∈(αh,h],where
(71)Θ˘i11=ΛiTAiT+HiTBiT+AiΛi+BiHiΘ˘i22=+ΛiT(∑j=1N‍γijER(ERTP-jER)-1ERT)ΛiΘ˘i22=+ΛiTQ1iΛi+ΛiTQ2iΛi+αhΛiTU1ΛiΘ˘i22=+(1-α)hΛiTU2Λi-α1ΛiTETR2iEΛi,Θ˘i12=AdiΛi+BiLi+α1ΛiTETR2iEΛi-α1ΛiTETSiEΛi,Θ˘i13=α1ΛiTETSiEΛi,Θ˘i16=[(AiΛi+BiHi)T(AiΛi+BiHi)T],Θ˘i22=-(1-hd)ΛiTQ1iΛi-2α1ΛiTETR2iEΛi,Θ˘i22=+α1ΛiTETSiEΛi+α1ΛiTETSiTEΛi,Θ˘i23=α1ΛiTETR2iEΛiT-α1ΛiTETSiEΛiT,Θ˘i26=[(AdiΛi+BiLi)T(AdiΛi+BiLi)T],Θ˘i33=ΛiTQ3iΛi-ΛiTQ2iΛiΘ˘i33=-α1ΛiTETR2iEΛi-ΛiTα2ETR3iEΛi,Θ˘i34=α2ΛiTETR3iEΛi,Θ˘i44=-ΛiTQ3iΛi-α2ΛiTETR3iEΛi,Θ˘i56=[BwiTBwiT],Θ˘i66=diag{-α1R2i-1,-α2R3i-1},Θ^i12=AdiΛi+BiLi,Θ^i13=α1ΛiTETR2iEΛi,Θ^i22=-(1-hd)ΛiTQ1iΛi-2α2ΛiTETR3iEΛiΘ˘i33=+α2ΛiTETTiEΛi+α1ΛiTETTiTEΛi,Θ^i23=α2ΛiTETR3iEΛi-α2ΛiTETTiTEΛi,Θ^i24=α2ΛiTETR3iEΛi-α2ΛiTETTiEΛi,Θ^i34=α2ΛiTETT3iEΛi.
Note thatγii≤0, we have
(72)γiiΛiTER(ERTP-iER)-1ERTΛi≤γii(εiΛiTET+εiEΛi-εi2EP-iET).
If we choose Si=0 and Ti=0, we have
(73)-[α1ΛiTETR2iEΛi0-α1ΛiTETR2iEΛi002α2ΛiTETR3iEΛi-α2ΛiTETR3iEΛi-α2ΛiTETR3iEΛi-α1ΛiTETR2iEΛi-α2ΛiTETR3iEΛiα1ΛiTETR2iEΛi+α2ΛiTETR3iEΛi00-α2ΛiTETR3iEΛi0α2ΛiTETR3iEΛi]=[I0-I0](-α1ΛiTETR2iEΛi)[I0-I0]+[0I-I0](-α2ΛiTETR3iEΛi)[0I-I0]+[0I0-I](-α2ΛiTETR3iEΛi)[0I0-I]≤[I0-I0](-ΛiTET-EΛi+αhR2i-1)[I0-I0]+[0I-I0](-ΛiTET-EΛi+(h-αh)R3i-1)[0I-I0]+[0I0-I](-ΛiTET-EΛi+(h-αh)R3i-1)[0I0-I],-[α1ΛiTETR2iEΛi-α1ΛiTETR2iEΛi00-α1ΛiTETR2iEΛi2α1ΛiTETR2iEΛi-α1ΛiTETR2iEΛi00-α1ΛiTETR2iEΛiα1ΛiTETR2iEΛi+α2ΛiTETR3iEΛi-α2ΛiTETR3iEΛi00-α2ΛiTETR3iEΛiα2ΛiTETR3iEΛi]=[I-I00](-α1ΛiTETR2iEΛi)[I-I00]+[0I-I0](-α1ΛiTETR2iEΛi)[0I-I0]+[00I-I](-α2ΛiTETR3iEΛi)[00I-I]≤[I-I00](-ΛiTET-EΛi+αhR2i-1)[I-I00]+[0I-I0](-ΛiTET-EΛi+αhR2i-1)[0I-I0]+[00I-I](-ΛiTET-EΛi+(h-αh)R3i-1)[00I-I].
Let Q-1ij=ΛiTQ1jΛi, Q-2ij=ΛiTQ2jΛi, Q-3ij=ΛiTQ3iΛi, R-2i=R2i-1, R-3i=R3i-1, U-1i=ΛiTU1ΛiT, U-2i=ΛiTU2ΛiT, and W-=W-1; then by using Schur complement of Lemma 6, we have that (69) and (70) hold if (63) and (64) hold. Intuitively, we can verify that (66) and (67) are equivalent to (25) and (26).
On the other hand, it is from (65) that for ∀j≠i,γij≠0,
(74)WN-1>γijR2j.
It should also be noted that if γij=0, (74) always holds. Thus, summing (74) for all j∈ℓ, j≠i, implies that
(75)∑j=1,j≠iN‍γijR2j-W<0.
Similarly, we can also obtain
(76)∑j=1,j≠iN‍γijR3j-W<0.
Since γii<0, R2i>0 and R3i>0, it can be verified that (65) guarantees (24). This completes the proof.To show how to decide the parameterα optimally, we construct the following iterative algorithm for Theorem 9. Following a similar way, we can obtain the iterative algorithms for other theorems and corollaries.Algorithm 15.
Select optimal parameterα for maximum delay h.Step1.For a prescribed scalar hd, choose appropriate step lengths Δh and Δα for h and α, such that (22)–(27) in Theorem 9 hold. Set k as the iterative times, and select the initial values k=1, h0=Δh, and α0=Δα. Let the upper bound h=h0+Δh.Step2. Let α=kΔα; if LMIs (22)–(27) are feasible, go to Step 3; otherwise, go to Step 4.Step3. Update h0, α0, k and h as h0=h, α0=α, k=1, and h=h0+Δh, go to Step 2.Step4. Set k=k+1. If kΔα<1; then go to Step 2; otherwise, stop.From the above algorithm, the outputα0 is the optimal α to ensure the delay upper bound h to achieve its maximum value h0.


## 4. Numerical Examples

In this section, we provide three examples to clarify the applicability and less conservatism of the results proposed in this paper.Example 1.
Consider the nominal Markovian jump systems (55) with two modes and the following parameters:
(77)A1=[-3.48880.8057-0.6451-3.2684],A2=[-2.48980.28951.3396-0.0211],Ad1=[-0.8620-1.2919-0.6841-2.0729],Ad2[-2.83060.4978-0.8436-1.0115].
The transition rate matrix is given as
(78)Γ=[γ11γ120.8-0.8].
It is noted that ifE=I, descriptor Markovian jump time-delay systems (55) reduce to the standard state-space Markovian jump time-delay systems. For the time-invariant delay, the stochastic stability problem can be solved by Corollary 12. Table 1 shows the compared results on the maximum allowable upper bound for different γ11 with the recent results [12, 14]. For hd=0.5 and hd=1.5, the maximum upper bound on allowable time delay is shown with given γ11 in Tables 2 and 3, respectively.Table 1
The maximum allowable upper bound for differentγ11 and time-invariant delay.
Method
γ
11
−0.8
−1
−1.8
[12]
h
max
0.5562
0.5463
0.5349
[14]
h
max
0.5812
0.5760
0.5663
Corollary12
h
max
0.7115
0.6712
0.5886Table 2
The maximum allowable upper bound for differentγ11 and hd=0.5.
Method
γ
11
−0.1
−0.5
−0.8
−1
[12]
h
max
0.5420
0.4897
0.4760
0.4704
[14]
h
max
0.5952
0.5757
0.5656
0.5602
Corollary11
h
max
0.8544
0.6535
0.6034
0.5826Table 3
The maximum allowable upper bound for differentγ11 and hd=1.5.
Method
γ
11
−0.1
−0.5
−0.8
−1
[12]
h
max
0.3860
0.3860
0.3860
0.3860
[14]
h
max
0.5828
0.5711
0.5631
0.5582
Corollary11
h
max
0.7445
0.6333
0.5943
0.5772As shown in the above tables, the methods in this paper have advantages over some previous ones in that the obtained maximum upper bound for allowable time delay is much larger. Therefore, the developed results have less conservatism than references [12, 14].Example 2.
Consider the continuous-time descriptor Markovian jump time-delay systems (59) with two modes and the following parameters:
(79)E=[1000],A1=[0.497200-0.9541],A2=[0.512100-0.7215],Ad1=[-1.0101.541500.5449],Ad2=[-0.85211.972100.4321].
The transition rate matrix is given as
(80)Γ=[γ11γ120.3-0.3].Compared with the existing results [17, 20], we find that Theorem 14 in this paper reaches the much larger bounds than the aforementioned results. Therefore, we can conclude that Theorem 14 in this paper is less conservative than the results in references [17, 20]. The compared results can be seen in Table 4.Table 4
The maximum allowable upper bound for differentγ11 and time-invariant delay.
Method
γ
11
−0.6
−0.55
−0.5
−0.45
−0.4
−0.35
[20]
h
max
1.0808
1.0842
1.0878
1.0918
1.0960
1.1006
[17] (k=0.5)
h
max
1.1588
1.1587
1.1587
1.1587
1.1587
1.1587
Corollary12
h
max
1.1769
1.1758
1.1748
1.1737
1.1727
1.1716Example 3.
Consider a Markovian jump system (1) with two modes and the following parameters:
(81)A1=[-2.2460-1.4410-1.5937-2.9289],A2=[-1.89990.81560.6900-0.7881],Ad1=[-0.70981.19080.6686-3.2025],Ad2=[-1.5198-1.6041-0.1567-1.2427],C1=[-0.3775-0.2959],C2=[-1.4751-0.2340],Cd1=[00],Cd2=[00],Bw1=[0.04030.6771],Bw2=[0.5689-0.2556],Dw1=0.1184,Dw2=0.3148.
The transition matrix is given as
(82)Γ=[-330.6-0.6].
WhenE=I, we assume that B1=B2=0, D1=D2=0, and hd=0; the compared results for different γ>0 are given in Table 5. Obviously, the BRL obtained in this paper is less conservative than that in [12].
When we takeE=[1000], then EL=ER=[10] satisfies E=ELERT. The coefficient matrices of the input vector can be given as
(83)B1=[1.50.99],B2=[1.20.99]D1=0.5,D2=-0.3.
In this case, we take V=[01], U=[01], ϵ1=1.5, and ϵ2=1.8.Table 5
The maximum allowable upper bound for differentγ with hd=0.
Method
γ
0.5
1
1.5
2
[12]
h
max
0.4165
0.4739
0.4926
0.5022
Theorem9
h
max
0.4712
0.5364
0.5581
0.5690By applying the LMIs conditions in Theorem14, the minimum attenuation level is γmin=1.15 under the assumption h=0.35 and hd=0.3, and the mode-dependent state feedback controller gain can be obtained. Consider
(84)K1=[2.81740],K2=[-63.9993-0.0246],Kd1=[-0.69960],Kd2=[-50.39730.0141].Therefore, it is easy to see that our results are effective.


## 5. Conclusions

This paper has studied theH∞ control problem for continuous-time descriptor Markovian jump systems with time-varying delay. By constructing novel Lyapunov-Krasovskii functionals, delay-dependent stability criteria and BRL have been developed under which the underlying systems are regular, impulse-free, and stochastically stable with a prescribed H∞ performance level. Compared with other time-varying delay results based on Jensen inequality, the obtained results in this paper have advantages over some previous results, which is partially attributed to the reciprocally convex combination method. The illustrated examples show that the maximum allowable upper bound of time delay is much larger than some latest results.



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*Source: 545636-2013-12-07.xml*