# Exponential Stability in Mean Square of Singular Markovian Jump System with Saturating Actuators and Time-Varying Delay

Authors: Fangqing Ding; Xianfa Jiao
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper investigates exponential stability in mean square of singular Markovian jump systems with saturating actuators and time-varying delay. The statistical property of the Markov process is fully used to derive the differential of the function. By using a delay decomposition method, a mode-dependent Lyapunov-Krasovskii function is established. A sufficient condition is proposed for exponential stability in mean square of the system designing the memoryless state feedback. A numerical example shows that the approach proposed is effective.


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## Body

## 1. Introduction

Markovian jump systems are a special class of hybrid systems with both modes and state variables, which is described by a set of time-delayed linear systems with the transitions among models determined by a Markov chain in a finite mode set. In recent years, Markovian jump systems have received much attention, and many problems have been solved, including stability analysis, state feedback, and output feedback controller design [1, 2]. Both time-delay and saturating controls are commonly encountered in various engineering systems, which often lead to poor performance and instability of a control system. Therefore, much attention has been paid to the study on the stability problem of systems with saturating actuator. The problem of robust stabilization of uncertain time-delay systems with a saturating actuator was addressed by Niculescu et al. [3]. Liu has obtained the condition of stability for a class of time-varying delay systems with saturating actuators according to the usage of the linear matrix inequality and Leibniz-Newton formula [4]. For the systems with time-varying delays, the reported results are generally based on the assumption that the derivative of time-varying delays is less than one [3, 4]. Such restriction is very conservative. Furthermore, singular system model is a natural presentation of dynamic systems and can describe a larger class of systems than regular ones, such as large-scale systems, power systems, and constrained control systems, electrical circuits, power systems, and economics  [5]. For a singular time-delay system, it is important to develop conditions which guarantee that the given singular system is not only stable but also regular and impulse free. Wu et al. discussed the stability problem on uncertain singular Markovian jump time-delay systems [6, 7]. Ma et al. investigated the robust stochastic stability problem for discrete-time uncertain singular Markov jump systems with actuator saturation [8]. Boukas et al. established an LMI condition for the singular time-delay systems with Markovian jump to be regular, impulse free, and stochastically stable. However, the result of [9] is delay independent. Generally speaking, delay-independent conditions are more conservative than delay-dependent ones, especially for small time delays. To the best of our knowledge, the robust stability for singular Markov jump systems with actuator saturation and time-varying delay has not been investigated in the literature; this problem is important in both theory and practice.In this paper, we are concerned with stability of singular Markovian jump system with saturating actuators and time-varying delay. A new Lyapunov function can be constructed by using a delay decomposition method and considering the statistical property of Markovian process. In terms of LMI approach, stability conditions are proposed to guarantee the considered systems to be regular, impulse free, and exponentially stable in mean square. Free weighting matrix and reduction method are not introduced in the derivation of the stability criterion. Thus, the results reduce conservation.


## 2. Problem Formulation and Assumption

Consider the singular markovian jump system with saturating actuators and time-varying delay as follows:(1)Ex•=A(rt)x(t)+B(rt)x(t-τ(t))+C(rt)Satu(t-τ(t)),t≥0,x(t)=ϕ(t),t∈[-τ,0],
wherex(t)∈RN andu(t)∈RM are the system states and inputs, respectively, uj(t),j=1,…,M is thejth component ofu(t), and ϕ(t)∈RN is a continuous vector-valued initial function defined on the interval[-τ,0]. The matrixE∈RN×N may be singular, and rankE=r<n. (Ω,F,{Ft}t≥0,P) is the probability space. A(rt), B(rt), and C(rt)are known real constant matrices with appropriate dimension. As rt=i, for the sake of simplicity, Ai, Bi, and Ci denote the matrixes, respectively.τ(t)is the time-varying delay satisfying
(2)0≤τ(t)≤τ,τ•(t)≤μ,
and h, μ are real constant scalars.The saturating function is defined as follows:(3)Sat(u(t))=[Sat(u1(t))Sat(u2(t))⋯Sat(um(t))]T,(4)Sat(uj(t))={Uj,ifuj(t)≤-Uj,uj(t)≤-Uj,uj(t),if-Uj<uj(t)<Uj,Uj,ifuj(t)≥Uj,uj(t)≥Uj,
whereUj>0(j=1,…,m)are known real constants.Assumption 1.
The markovian jumping parameterrtis right-continuous markovian process and takes values in finite set S={1,2,…,N}, and is defined by
(5)P(rt+Δ=j∣rt=i)={ρijΔ+o(Δ),i≠j,1+ρijΔ+o(Δ),i=j,
where Δ>0, limΔ→∞o(Δ)/Δ=0,ρij is the transition rate from i to j and
(6)ρii=-∑j=1,j≠iSρij,∀i,j∈S(ρij≥0,j≠i).Definition 2.
The infinitesimal generator of the solution to system (1) is defined as
(7)LV(xt,rt)=limΔ→0+E[V(xt+Δ,rt+Δ)|xt,rt]-V(xt,rt)Δ.Definition 3.
(1) The singular Markovian jump system with saturating actuators and time-varying delay(8)Ex•(t)=A(rt)x(t)+Ad(rt)x(t-τ(t)),x(t)=ϕ(t),t∈[-τ,0]
is said to be regular and impulse free, if the pairs (E,A(rt)) are singular and impulse free for every rt∈S.
(2) The singular Markovian jump system with saturating actuators and time-varying delay (8) is said to be exponential stable in mean square if there exist scalarsα>0, β>0, such that
(9)E{∥x(t)∥2}≤αe-βt∥ϕ(t)∥τ2,t>0.
(3) The singular Markovian jump system with saturating actuators and time-varying delay (8) is said to be mean-square exponentially admissible, if it is regular, impulse free, and exponentially stable in mean square.Lemma 4 (see [7]).
Suppose that a positive continuous functionf(t) satisfies
(10)E{f(t)}≤ς1E{supt-τ≤s≤tf(s)}+ς2e-εt;
then
(11)E{f(t)}≤e-εt∥f(s)∥τ+ς2e-εt1-ς1eετ,
whereε>0, 0<ς1<1, 0<ς1eετ<1, ς2>0 and d>0.Lemma 5 (see [10]).
D is an M×M diagonal matrix whose entries are either 1 or 0. Note that there are2M elements inD. Suppose that each element in D is labeled as Ds, s∈[1,2M], and letDs-=I-Ds; it is obvious thatDs-∈D, if Ds∈D. Given matrixF,H∈RM×N, one has
(12)Sat(Fx)∈co{DsFx+Ds-Hx:s∈[1,2M]},
whereco denotes convex hull.Lemma 6 (see [11]).
Givenv>0, for a vectorω(s) and a positive-definite matrixΖ with appropriate dimensions, and the integrations concerned are well defined, then
(13)υ∫0υω(s)TΖω(s)ds≥(∫0υω(s)ds)TΖ(∫0υω(s)ds).


## 3. Main Results

Considering the system (1), for each region rt=i∈S, the system (1) becomes
(14)Ex•(t)=Aix(t)+Bix(t-τ(t))+CiSatu(t-τ(t)).Taking the memoryless state feedback control for the systems (14), one has
(15)u(t-τ(t))=Kix(t-τ(t)).Substituting (15) into (14), then the corresponding closed-loop systems become
(16)Ex•(t)=Aix(t)+Bix(t-τ(t))+CiSat(Kix(t-τ(t))).Theorem 7.
For given constantsτ, μ and two positive integers k, l, The system (16) is exponentially stable in mean square. If there exist symmetric positive matricesPi=PiT>0,  Ni=NTi>0,  Rmi=RmiT>0(m=1,…,l),  Mni=MniT>0(n=1,…,k),  T1=T1T>0,  T2=T2T>0,  Qi=QiT>0, where
(17)Qi=QiT=[Q11i⋯Q1ki⋮⋯⋮Q1kiT⋯Qkki],
for everyi∈S, such that(18a)ETPi=PiE,(18b)Σ=[Σ11Q12i+M1i-M2iQ13i+M2i-M3i⋯Q1k-1i+Mk-2i-Mk-1iQ1ki+Mk-1i-MkiMki*Q22i-Q11iQ23i-Q12i⋯Q2k-1i-Q1k-2iQ2ki-Q1k-1i-Q1ki**Q33i-Q22i⋯Q3k-1i-Q2k-2iQ3ki-Q2k-1i-Q2ki⋮⋮⋮⋮⋮⋮⋮***⋯Qk-1k-1i-Qk-2k-2iQk-1ki-Qk-2k-1i-Qk-2k-1i***⋯*Qkki-Qk-1k-1i-Qk-1ki***⋯**-Qkki-1lT2*00⋯000000⋯000000⋯000⋮⋮⋮⋯⋮⋮⋮000⋯000000⋯000*00⋯00*0000002ETT2E00⋯00Σ1k+l+1000⋯000000⋯000⋮⋮⋮⋮⋮⋮⋮000⋯000000⋯000000⋯00-1lETT2E-2ETT2E-(1-μl)Ri1ETT2E0⋯000*-2ETT2E-(1-2μl)Ri2ETT2E⋯0000*-2ETT2E-(1-3μl)Ri3⋯000⋮⋮⋮⋮⋮⋮⋮000⋯-2ETT2E-(1-l-2l)Ril-2ETT2E0000⋯*-2ETT2E-(1-l-1l)Ril-1T2000⋯0*Σk+l+1k+l+1]0<0,where
(18c)Σ11=AiTPi+PiAi+∑j∈SρijETPi+∑m=1lRmi+τ2kAiTT1Ai+τ2lAiTT2Ai+τ2kT1+τ2lT2-M1i-ETT2E+Q11i,Σ1k+l+1=BiTPi+KiTDqTCiTPi+HiTDq-TCiTPi+τ2kBiTT1Ai+τ2kKiTDqTCiTT1Ai+τ2kHiTDq-TCiTT1Ai+τ2lBiTT2Ai+τ2lKiTDqTCiTT2Ai+τ2lHiTDq-TCiTT2Ai,Σk+lk+l=[Bi+CiDqKi+CiDq-Hi]T(τ2kT1+τ2lT2)×[Bi+CiDqKi+CiDq-Hi]-1lETT2E-(1-μ)Ril,[-1hM1i0AiTMni+∑j∈SρijETMnj0-ETT2E[Bi+CiDqKi+CiDq-Hi]TMni**-τT1]≤0,[hΓ10Ni0hΓ2-Ni**∑j∈Sρij(Nj+Qj)]≤0,[∑j∈SρijRmj-τT2]≤0.Proof.
According to the definition of the saturating function and Lemma5, we have
(19)SatKix(t-τ(t))=DqKix(t-τ(t))+Dq-Hix(t-τ(t))q∈[1,2M].
Substituting (19) into (16) gives
(20)Ex•(t)=Aix(t)+[Bi+CiDqKi+CiDq-Hi]x(t-τ(t)).
SincerankE=r<n, there exist nonsingular matrices G and Υ, such that
(21)GEΥ=[Ir000].
Let(22)GAiΥ=[Ai1Ai2Ai3Ai4],G-TPiΥ=[P-i1P-i2P-i3P-i4].
According to (18a), it is easy to obtain that Pi2-=0 for every i∈S. By pre- and postmultiplying Σ11i<0 byΥT and Υ, respectively, we haveAi4TP-i4+P-i4TAi4<0, which implies that Ai4 are nonsingular for every i∈S. Thus, the pairs (E,Ai)are regular and impulse free for everyi∈S. Thus, by Definition 2, the system (8) is regular and impulse free for any time-varying delayτ(t) satisfying (2).
Choose a Lyapunov-Krasovskii function candidate as(23)V(xt,rt,t)=V1(xt,rt,t)+V2(xt,rt,t)+V3(xt,rt,t)+V4(x4,rt,t)+V5(xt,rt,t)+V6(xt,rt,t),
where
(24)V1(xt,rt,t)=x(t)TETP(rt)Ex(t),V2(xt,rt,t)=∫t-htξ(s)TQ(rt)ξ(s)ds,V3(xt,rt,t)=2x(t)T∑n=1kETMn(rt)∫t-nht-(n-1)hx(s)ds,V4(xt,rt,t)=∫t-htds∫t-htξ(s)TN(rt)ξ(θ)dθ,V5(xt,rt,t)=∑m=1l∫t-mσ(t)tx(s)TRm(rt)x(s)ds,V6(xt,rt,t)=h∫-τ0∫t+θtx•(s)TETT1Ex•(s)+x(s)TT1x(s)dsdθ+σ∫-τ0∫t+θtx•(s)TETT2Ex•(s)+x(s)TlT2x(s)dsdθ,ξT(t)=[x(t),x(t-h),…,x(t-(k-1)h)]T,σ(t)=τ(t)l,h=τk,σ=τl.
Moreover, the action of infinitesimal generator (7) on each term function (23) could be expressed as
(25)LV1(xti,t)=2x(t)TETPix•(t)+x(t)T∑j∈SρijETPjx(t),LV2(xt,i,t)=ξ(t)TQiξ(t)-ξ(t-h)TQiξ(t-h)+∫t-htξ(s)T∑j∈SρijQjξ(s)ds,LV3(xt,i,t)=2x•(t)T∑n=1kETMni∫t-nht-(n-1)hx(s)ds-2x(t)T∑n=1kMni[x(t-(n-1)h)000000000000000-x(t-nh)]+2x(t)T∑n=1k∑j∈SρijETMnj∫t-nht-(n-1)hx(s)ds,LV4(xt,i,t)=2[ξ(t)-ξ(t-h)]TNi∫t-htξ(s)ds+∫t-ht∫t-htξ(s)T∑j∈SρijNjξ(θ)dsdθ,LV5(xt,i,t)=x(t)T∑m=1lRmix(t)-∑m=1l(1-mσ•(t))×x(t-mσ(t))TRmix(t-mσ(t))+∑m=1l∫t-mσ(t)tx(s)T∑j∈SρijRmjx(s)ds,LV6(xt,i,t)=hτx•(t)TETT1Ex•(t)+hτx(t)TT1x(t)-h∫t-τtx•(s)TETT1Ex(s)+x(s)TT1x(s)ds+στx•(t)TETT2Ex•(t)+στx(t)TlT2x(t)-σ∫t-τtx•(s)TETT2Ex•(s)+x(s)TlT2x(s)ds.
According to (20), (23), and (25), we have
(26)LV(xt,i,t)≤x(t)T[AiTPi+PiAi+∑j∈SρijETPj+∑m=1lRim000000+hτAiTT1Ai+στAiTT2Ai000000+hτT1+στT2AiTPi+PiAi+∑j∈SρijETPj+∑m=1lRim]×x(t)+2x(t)T[PiBi+PiCiDqKi000000000000000+PiCiDq-Hi000000000000000+hτAiTT1Bi000000000000000+hτAiTT1CiDqKi000000000000000+hτAiTT1CiDq-Hi000000000000000+στAiTT2Bi000000000000000+στAiTT2CiDqKi000000000000000+στAiTT2CiDq-HiPiBi+PiCiDqKi]×x(t-τ(t))+x(t-τ(t))T[Bi+CiDqKi+CiDq-Hi]×(hτT1+στT2)[Bi+CiDqKi+CiDq-Hi]×x(t-τ(t))+ξ(t)TQiξ(t)-ξ(t-h)TQiξ(t-h)+∫t-htξ(s)T∑j∈SρijQjξ(s)ds+∑n=1k∫t-nht-(n-1)h[x(t)x(t-τ(t))x(s)]T×[-1hM1i0AiTMni+∑j∈SρijETMnj00[Bi+CiDqKi+CiDq-Hi]TMni**-hT1]×[x(t)x(t-τ(t))x(s)]-x(t)TM1ix(t)+2x(t)TMi1x(t-h)-2x(t)T∑n=2kMni[x(t-(n-1)h)00000000000000-x(t-nh)]+∫t-ht[ξ(t)-ξ(t-h)]TNiξ(s)+ξ(s)Ni[ξ(t)-ξ(t-h)]+ξ(s)T∑j∈SρijhNjξ(s)ds-∑m=1l(1-mσ•(t))x(t-mσ(t))T×Rmix(t-mσ(t))+∑m=1l∫t-mσ(t)tx(s)T∑j∈SρijRmjx(s)-x(s)TτT2x(s)ds-h∫t-τtx•(s)TETT1Ex•(s)ds×σ∫t-τtx•(s)TETT2Ex•(s)ds,(27)ξ(t)TQiξ(t)-ξ(t-h)TQiξ(t-h)=ζ(t)TΞζ(t),ζ(t)T=[ξ(t)T,x(t-τ)T]T,Ξ=[Q11iQ12iQ13iQ14i···Q1k-1iQ1ki0*Q22i-Q11iQ23i-Q12iQ24i-Q13i···Q2k-1i-Q1k-2iQ2ki-Q1k-1i-Q1ki**Q33i-Q22iQ34i-Q23i···Q3k-1i-Q2k-2iQ3ki-Q2k-1i-Q2ki***Q44i-Q33i···Q4k-1i-Q3k-2iQ4ki-Q3k-1i-Q3ki⋮⋮⋮⋮⋮⋮⋮⋮*****Qk-1k-1i-Qk-2k-2iQk-1ki-Qk-2k-1i-Qk-2ki******Qkki-Qk-1k-1i-Qk-1ki*******-Qkki],(28)-lσ∫t-τtx(s)TT2x(s)ds0000≤-lσ∫t-τ(t)tx(s)TT2x(s)ds0000≤-σ∫t-τ(t)tx(s)TT2x(s)ds000000-σ∫t-(l-1)σ(t)tx(s)TT2x(s)ds-⋯000000-σ∫t-2σ(t)tx(s)TT2x(s)ds000000-σ∫t-σ(t)tx(s)TT2x(s)ds0000=-σ∑m=1l∫t-mσ(t)tx(s)TT2x(s)ds,(29)-h∫t-τtx(s)TT1x(s)ds00=-h∑n=1k∫t-nht-(n-1)hx(s)TT1x(s)ds,(30)-h∫t-τtx•(s)TETT1Ex•(s)ds00=-h∫t-htx•(s)TETT1Ex•(s)ds0000-h∫t-2ht-hx•(s)TETT1Ex•ds-⋯0000-h∫t-(k-2)ht-(k-1)hx•(s)TETT1Ex•(s)ds0000-h∫t-τt-(k-1)hx•(s)TETT1Ex•(s)ds00≤-[∫t-htx•(s)ds]TETT1E[∫t-htx•(s)ds]0000-[∫t-2ht-hx•(s)ds]TETT1E0000×[∫t-2ht-hx•(s)ds]-⋯-[∫t-τt-(k-1)hx•(s)ds]T0000×ETT1E[∫t-τt-(k-1)hx•(s)ds]00=-[x(t)-x(t-h)]TETT1E[x(t)-x(t-h)]0000-[x(t-h)-x(t-2h)]T0000×ETT1E[x(t-h)-x(t-2h)]0000-⋯-[x(t-(k-1)h)-x(t-τ)]T0000×ETT1E[x(t-(k-1)h)-x(t-τ)]00=ξ(t)TΓ1ξ(t)+ξ(t-h)TΓ2ξ(t-h),(31)Γ1=[-ETT1EETT1E0⋯00*-ETT1EETT1E⋯000*-ETT1E⋯00⋮⋮⋮⋮⋮⋮000⋯-ETT1EETT1E000⋯ETT1E-ETT1E],(32)Γ2=[-ETT1E00⋯000-ETT1E0⋯0000-ETT1E⋯00⋮⋮⋮⋮⋮⋮000⋯-ETT1E0000⋯0-ETT1E],(33)-σ∫t-τtx•(s)TETT2Ex•(s)ds=-σ∫t-σ(t)tx•(s)TETT2Ex•(s)ds0000-∫t-2σ(t)t-σ(t)x•(s)TETT2Ex(s)ds0000-⋯-σ∫t-τt-τ(t)x•(s)TETT2Ex(s)ds00≤-σ(t)∫t-σ(t)tx•(s)TETT2Ex•(s)ds0000-σ(t)∫t-2σ(t)t-σ(t)x•(s)TETT2Ex•(s)ds0000-⋯-στ(τ-τ(t))0000×∫t-τt-τ(t)x•(s)TETT2Ex•(s)ds.
Similar to (33), by Lemma 6, it is easy to obtain(34)-σ∫t-τtx•(s)TETT2Ex•(s)ds0000≤ζ2(t)T[-ETT2E0ETT2E0⋯000-1lETT2E00⋯01lETT2E*0-2ETT2EETT2E⋯0000*-2ETT2E⋯00⋮⋮⋮⋮⋯⋮⋮0000⋯-2ETT2EETT2E0*00⋯*-(1+1l)ETT2E]ζ2(t),where(35)ζ2(t)=[x(t)Tx(t-τ)Tx(t-σ(t))Tx(t-2σ(t))T···x(t-(l-1)σ(t))Tx(t-τ(t))T].In conclusion, one has
(36)LV(xt,i,t)00≤∑n=1k∫t-nht-(n-1)h[x(t)Tx(t-τ(t))Tx(s)T]T0000×[-1hMi10AiTMni+∑j∈SρijETMnj0-ETT2E[Bi+CiDqKi+CiDq-Hi]Mni**-τT1]0000×[x(t)x(t-τ(t))x(s)]ds0000+∫t-ht[ξ(t)Tξ(t-h)Tξ(s)T]0000×[hΓ10Ni0hΓ2-Ni**∑j∈Sρij(hNj+Qj)][ξ(t)ξ(t-h)ξ(s)]ds0000+∑m=1l∫t-mσ(t)tx(s)T[∑j∈SρijRjm-τT2]x(s)ds0000+π(t)TΣπ(t),
where (37)π(t)=[ξ(t)T,x(t-τ)T,x(t-σ(t))T,x(t-2σ(t))T,…,x(t-(l-1)σ(t))T,x(t-τ(t))T]T,Σ=[Σ11Q12i+Mi1-Mi2Q13i+Mi2-Mi3⋯Q1k-1i+Mik-2-Mik-1Q1ki+Mik-1-MikMikETT2E*Q22i-Q11iQ23i-Q12i⋯Q2k-1i-Q1k-2iQ2ki-Q1k-1i-Q1ki0**Q33i-Q22i⋯Q3k-1i-Q2k-2iQ3ki-Q2k-1i-Q2ki0⋮⋮⋮⋮⋮⋮⋮⋮***⋯Qk-1k-1i-Qk-2k-2iQk-1ki-Qk-2k-1i-Qk-2k-1i0***⋯*Qkki-Qk-1k-1i-Qk-1ki0***⋯**-Qkki-1lETT2E0*00⋯000-2ETT2E-(1-μl)Ri1000⋯000*000⋯0000⋮⋮⋮⋯⋮⋮⋮⋮000⋯0000000⋯0000*00⋯00*00000⋯00Σ1k+l+100⋯00000⋯000⋮⋮⋮⋮⋮⋮00⋯00000⋯00000⋯00-1lETT2EETT2E0⋯000-2ETT2E-(1-2μl)Ri2ETT2E⋯000*-2ETT2E-(1-3μl)Ri3⋯000⋮⋮⋮⋮⋮⋮00⋯-2ETT2E-(1-(l-2)μl)Ril-2ETT2E000⋯*-2ETT2E-(1-(l-1)μl)Ril-1ETT2E00⋯0*Σk+l+1k+l+1],Σ11=AiTPi+PiAi+∑j∈SρijETPi+∑m=1lRim+τ2kAiTT1Ai+τ2lAiTT2Ai+τ2kT1+τ2lT2-Mi1-ETT2E+Q11i,Σ1k+l-1=BiTPi+KiTDqTCiTPi+HiTDq-TCiTPi+τ2kBiTT1Ai+τ2kKiTDqTCiTT1Ai+τ2kHiTDq-TCiTT1Ai+τ2lBiTT2Ai+τ2lKiTDqTCiTT2Ai+τ2lHiTDq-TCiTT2Ai,Σk+lk+l=[Bi+CiDqKi+CiDq-Hi]T(τ2kT1+τ2lT2)[Bi+CiDqKi+CiDq-Hi]-1lETT2E-(1-μ)Ril;
using (18b) and (18c), it is easy to obtain(38)LV(xt,i,t)≤0.
SinceAi4 are nonsingular for every i∈S, we set
(39)G∧=[Ir-Ai2Ai4-10Ai4-1]G.
It is easy to get(40)G∧EΥ=[Ir000],G∧AiΥ=[A∧i10A∧i3I],
where(41)A∧i1=Ai1-Ai2Ai4-1Ai3,A∧i3=Ai4-1Ai3.
Denote that(42)G∧[Bi+CiDqKi+CiDq-Hi]Υ=[Aid1Aid2Aid3Aid4],G-T∧PiΥ=[Pi1Pi2Pi3Pi4],ΥTRilΥ=[R1ilR2ilR3ilR4il].
Let(43)x-(t)=[x-1(t)x-2(t)]=Υ-1x(t).
Then, for everyi∈S, system (20) is a restricted system equivalent to
(44)x-•1(t)=A∧i1x-1(t)+Aid1x-1(t-τ(t))+Aid2x-2(t-τ(t)),-x-2(t)=A∧i13x-1(t)+Aid3x-1(t-τ(t))+Aid4x-2(t-τ(t)),x-(t)=φ(t)=S-1ϕ(t),t∈[-τ,0].
To prove the exponential stability in mean square, we define a new function as(45)W(xt,i,t)=eεtV(xt,i,t),
where ε>0 and then by Dynkin’s formula, we find that for every i∈S,
(46)EW(xt,rt,t)≤EW(x0,r0,0)+E∫0teεs[εV(xs,rs,s)-λ∥x(s)∥2]ds.
It can be seen from (23), (45), and (46) that ifε is chosen small enough, a constant k>0 can be found such that for t>0,
(47)mini∈S{λmin(Pi1)}E{∥x-1∥2}≤E{xT(t)ETPix(t)}≤ke-εt∥ϕ(t)∥τ2.
Hence, for anyt>0(48)E{∥x-1∥2}≤αe-εt∥ϕ(t)∥τ2,
where α=(mini∈S{λmin(Pi1)})-1k. To study the exponential stability in mean square of x-2(t), we apply the similar analysis method of [12] and define
(49)e(t)=A∧i13x-1(t)+Aid3x-1(t-τ(t)).
Then, according to (48), we have a constant m>0; it can be found that when t>0,
(50)E{∥e(t)∥2}≤me-εt∥ϕ(t)∥τ2.
We construct a function as(51)J(t)=x-2(t)TR4ilx-2(t)-x-2(t-τ(t))TR4ilx-2(t-τ(t)).
By premultiplying the second equation of (44) with x-2(t)TPi4T, we obtain that
(52)0=x-2(t)TPi4Tx-2(t)+x-2(t)TPi4Tx-2(t-τ(t))+x-2(t)TPi4Te(t).
Adding (52) to (51) yields that
(53)J(t)=x-2(t)T(Pi4+Pi4+R4il)x-2(t)+2x-2(t)TPi4TAid4x-2(t-τ(t))-x-2(t-τ(t))TR4ilx-2+2x-2(t)TPi4Te(t)≤[x-2(t)x-2(t-τ(t))]T[Pi4+Pi4T+R4ilPi4TAid4*-R4il]×[x-2(t)x-2(t-τ(t))]+η1x-2(t)Tx-2(t)+η-1e(t)TR4ilR4ilTe(t),
whereη1 is any positive scalar.
By pre- and postmultiplying[Σ11Σ1i+l*Σk+lk+l]<0 by[Υ00Υ]T and[Υ00Υ], respectively, we get a constantη2>0, such that
(54)[Pi4+Pi4T+R4ilPi4TAid4*-Rid4]≤-[η2I000].
On the other hand, sinceη1 can be chosen arbitrarily,η1 is chosen small enough, such that η2-η1>0. Then, we can always find a scalar η3>1, such that
(55)R4il-(η2-η1)I≥η3R4il.
It follows from (51), (53), and (54) that
(56)x-2(t)TR4ilx-2(t)-x-2(t-τ(t))TR4ilx-2(t-τ(t))00≤-η2x-2(t)Tx-2(t)+η1x-2(t)Tx-2(t)0000+η-1e(t)TPi4Pi4Te(t).
It is clear that the above inequalities (55) and (56) imply that
(57)η3x-2(t)TR4ilx-2(t)≤x-2(t)T(R4il-(η1-η2)I)≤x-2(t-τ(t))TR4ilx-2(t-τ(t))+η-1e(t)TPi4Pi4Te(t).
Thus, we get from (57) that
(58)x-2(t)TR4ilx-2(t)≤η3-1x-2(t-τ(t))TR4ilx-2(t-τ(t))0000+(η1η3)-1e(t)TPi4Pi4Te(t).
Letf(t)=x-2(t)TR4ilx-2(t); from the above inequality and the fact (50) and (58), we have
(59)E{f(t)}≤η3-1E{supt-τ≤s≤tf(s)}+γe-εt≤η3-1E{supt-τ≤s≤tf(s)}+γe-δt,
where0<δ<min{ε,d2-1lnη3}, γ=(η1η3)-1maxi∈S∥Pi4∥2∥ϕ(t)∥τ2.
Therefore, applying Lemma4 to the above inequality yields that
(60)E{∥x-2(t)∥2}≤λmin-1(R4il)λmax(R4il)e-δt∥x-2(t)∥τ2+λmin-1(R4il)γe-δt1-η3-1eδd2.
We can find from (50) and the above inequality that the system (16) is exponentially stable in mean square.Remark 8.
It should be pointed out that sometimes too many free variables cannot reduce the conservatism of the obtained results. In the theorem, the system is exponentially stable in mean square without resorting to the free-weighting matrices method, which is in contrast with [6], where the free-weighting matrices method was used.Remark 9.
In the proof of the theorem, the delay interval[0;τ] is divided into N segments of equal length τ/N, such that the information of delayed states x(t-iτ/N), i=1,2,…,N, are all taken into account. It is clear that the Lyapunov function defined in the theorem is more general than the ones in [7].


## 4. Numerical Examples

In this section, we provide a numerical example to verify the effectiveness of the proposed method. Consider systems (1) in R2 with two regines rt∈S={1,2}. The system parameters are described as follows:
(61)E=[1000],A1=[-0.800.6-2.0],A2=[-1.80.50.5-1.8],B1=[1.71.32.52.1],B2=[-0.5-0.30-0.2],C1=[0.2000.2],C2=[0.4000.4],Dq=[1001],Dq-=[0000],φ(s)=[3cos(s)-sin(s)-3cos2(s)],(πij)2×2=[-404010-10],Dq-=[0000],(πij)2×2=[-404010-10],H1=[1001],H2=[1001],τ=1,μ=3.Asl=1,k=1, and (1)
rt=1, by calculating we have
(62)P1=[13.81761.31471.314727.9707],P2=[2.3514-0.2388-0.238827.9707],K1=[-9.9593-6.2795-6.2795-12.1579],Q111=[2.0672-0.1326-0.13262.0672],Q112=[1.9317-0.2833-0.28331.9317],R11=[2.3514-0.2388-0.23882.3514],R21=[0.8620-1.2919-1.29190.8620],M11=[3.5860-0.2001-0.20013.5860],M12=[1.07240.26000.26001.0724],N1=[0.95220.26050.26050.9522],N2=[-2.48980.28950.2895-2.4898],T1=[2.84381.54691.54698.7138],T2=[5.7161005.7161];(2)
rt=2, by calculating we have
(63)P1=[0.95220.26050.26050.9522],P2=[9.04782.22192.221936.3742],K2=[0.1904-0.1109-0.11090.3192],Q111=[2.0672-0.1326-0.13262.0672],Q112=[0.1793-0.7876-0.78760.1793],R11=[2.3514-0.2388-0.23882.3514],R21=[0.8620-1.2919-1.29190.8620],M11=[3.5860-0.2001-0.20013.5860],M12=[1.07240.26000.26001.0724],N1=[0.95220.26050.26050.9522],N2=[-2.48980.28950.2895-2.4898],T1=[2.83011.54991.54998.7230],T2=[5.7161005.7161].Asl=1,k=2, and(1)
rt=1, by calculating we have
(64)P1=[0.95220.26050.26050.9522],P2=[8.48781.29351.293532.5716],K1=[-9.9593-6.2795-6.2795-12.1579],Q111=[2.0672-0.1326-0.13262.0672],Q112=[1.9317-0.2833-0.28331.9317],R11=[2.3514-0.2388-0.23882.3514],Q121=[0.2586-0.3256-0.32560.2586],Q221=[0.1904-0.0521-0.05210.1904],M12=[1.07240.26000.26001.0724],R12=[0.8620-1.2919-1.29190.8620],M11=[3.5860-0.2001-0.20013.5860],M21=[0.1332-0.0926-0.09260.1332],M22=[0.2025-0.2003-0.20030.2025],N1=[0.95220.26050.26050.9522],T1=[2.84381.54691.54698.7138],T2=[5.7161005.7161];(2)
rt=2, by calculating we have
(65)P1=[80.58985.29545.2954229.2114],P2=[2.4898-0.2895-0.28952.4898],K2=[0.1904-0.1109-0.11090.3192],Q111=[2.0672-0.1326-0.13262.0672],Q112=[1.9317-0.2833-0.28331.9317],R11=[2.3514-0.2388-0.23882.3514],R21=[0.8620-1.2919-1.29190.8620],M11=[3.5860-0.2001-0.20013.5860],M12=[1.07240.26000.26001.0724],M21=[0.1332-0.0926-0.09260.1332],M22=[0.2025-0.2003-0.20030.2025],N1=[0.95220.26050.26050.9522],N2=[-2.48980.28950.2895-2.4898],T1=[2.84381.54691.54698.7138],T2=[5.7167005.7161].Asl=2,k=1, and (1)
rt=1, by calculating we have
(66)P1=[2.4898-0.2895-0.28952.1898],P2=[16.53212.82702.827018.5944],K1=[-9.9593-6.2795-6.2795-12.1579],Q111=[2.0672-0.1326-0.13262.0672],Q112=[1.9317-0.2833-0.28331.9317],R11=[2.3514-0.2388-0.23882.3514],R21=[0.8620-1.2919-1.29190.8620],M11=[3.5860-0.2001-0.20013.5860],M12=[1.07240.26000.26001.0724],N1=[0.95220.26050.26050.9522],N2=[-2.48980.28950.2895-2.4898],T1=[2.84381.54691.54698.7138],T2=[5.7161005.7161].(2)
rt=2, by calculating we have
(67)P1=[0.95220.26050.26050.9522],P2=[7.42111.43221.432229.4614],K2=[0.1904-0.1109-0.11090.3192],Q111=[2.0672-0.1326-0.13262.0672],Q112=[0.1793-0.7876-0.78760.1793],R11=[2.3514-0.2388-0.23882.3514],R21=[0.8620-1.2919-1.29190.8620],M11=[3.5860-0.2001-0.20013.5860],M12=[1.07240.26000.26001.0724],N1=[0.95220.26050.26050.9522],N2=[-2.48980.28950.2895-2.4898],T1=[2.84381.54691.54698.7138],T2=[5.7161005.7161].Asl=2,k=2, and (1)
rt=1, by calculating we have
(68)P1=[497.535930.449830.4498728.9455],P2=[2.4898-0.2895-0.28952.4898],Q111=[2.0672-0.1326-0.13262.0672],Q112=[1.9317-0.2833-0.28331.9317],Q121=[0.2586-0.3256-0.32560.2586],R11=[2.3514-0.2388-0.23882.3514],Q221=[0.1904-0.0521-0.05210.1904],R21=[0.8620-1.2919-1.29190.8620],M11=[3.5860-0.2001-0.20013.5860],M12=[1.07240.26000.26001.0724],M21=[0.1332-0.0926-0.09260.1332],M22=[0.2025-0.2003-0.20030.2025],N1=[0.95220.26050.26050.9522],N2=[-2.48980.28950.2895-2.4898],T1=[2.84381.54691.54698.7138],T2=[5.7161005.7161].(2)
rt=2, by calculating we have
(69)P1=[0.95220.26050.26050.9522],P2=[36.1907-7.0282-7.0282101.6875],K2=[0.1904-0.1109-0.11090.3192],Q111=[2.0672-0.1326-0.13262.0672],Q112=[0.1793-0.7876-0.78760.1793],M12=[1.07240.26000.26001.0724],Q121=[0.2586-0.3256-0.32560.2586],Q221=[0.1904-0.0521-0.05210.1904],R11=[2.3514-0.2388-0.23882.3514],R21=[0.8620-1.2919-1.29190.8620],M11=[3.5860-0.2001-0.20013.5860],M12=[1.07240.26000.26001.0724],M21=[0.1332-0.0926-0.09260.1332],N1=[0.95220.26050.26050.9522],N2=[-2.48980.28950.2895-2.4898],T1=[2.84381.54691.54698.7138],T2=[5.7161005.7161].So we can conclude that the condition of the theorem is satisfied; the system (16) is exponentially stable in mean square.


## 5. Conclusions

In this paper, we have investigated the exponential stability in mean square of singular Markovian jump systems with singular Markovian jump systems with saturating actuators and time-varying delay. To guarantee the singular Markovian jump system to be regular, impulse free, and exponentially stable in mean square. Memoryless feedback control has been designed. By exploiting Lyapunov-Krasovskii theory, a new sufficient condition of exponentially stable in mean square is obtained. Finally, the numerical example shows that the approach proposed is effective. To reduce conservatism of the results, our future work focuses on reducing the constraint by an appropriate reduction method.



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*Source: 420420-2013-09-08.xml*