# Stochastic Stability for Time-Delay Markovian Jump Systems with Sector-Bounded Nonlinearities and More General Transition Probabilities

Authors: Dan Ye; Quan-Yong Fan; Xin-Gang Zhao; Guang-Hong Yang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper is concerned with delay-dependent stochastic stability for time-delay Markovian jump systems (MJSs) with sector-bounded nonlinearities and more general transition probabilities. Different from the previous results where the transition probability matrix is completely known, a more general transition probability matrix is considered which includes completely known elements, boundary known elements, and completely unknown ones. In order to get less conservative criterion, the state and transition probability information is used as much as possible  to construct the Lyapunov-Krasovskii functional and deal with stability analysis. The delay-dependent sufficient conditions are derived in terms of linear matrix inequalities to guarantee the stability of systems. Finally, numerical examples are exploited to demonstrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

During the past decades, much attention has been devoted to the stochastic systems since stochastic modeling has an important science and engineering application [1, 2]. As an important class of stochastic systems, MJSs have attracted a lot of interest, since they can be used to model many practical dynamical systems, such as power systems, manufacturing systems, and economic systems in which they may experience failure or repairs, abrupt environmental disturbances, and abrupt changes in the operating point of a nonlinear plant (see [3, 4]). Theoretical results of this class of systems have been applied in many processes, such as target tracking, manufactory processes, solar thermal receivers, and fault-detection and fault-tolerant systems. Some examples can be found in [5–7].For MJSs, the transition probabilities of the jumping process are important, but most of the previous issues on this kind of systems usually assumed that the elements of the transition probability matrix are completely known. Based on this condition, a lot of research results have been worked out in the literature, such as [8–13]. However, in many practical engineering applications, we cannot get all of the transition probabilities elements, and some elements might be time variable in some cases. Therefore, the study of Markov jump systems with partly known transition probabilities becomes necessary and some well-known results have been published, see [11–13]. In some cases, the exact value of some elements of the transition probability matrix can not be obtained, but their lower and upper bounds can be determined. The case has been addressed by [8], but its method cannot be used to deal with the case where there is no information available for transition probabilities. In [9], the more general partly known transition probabilities are investigated, which covers the cases that the transition probabilities are exactly known, unknown, and unknown but with known bounds. However, a conservative condition to relax inequality is used to deal with this kind of transition probabilities, and the time delay is not considered.It is well known that time delay is very common in variously practical systems such as manufacturing systems, chemical processes, network control systems (NCSs), and telecommunication and economic systems. They are often the causes of oscillation, instability, and poor performance in many control systems. In the past years, time-delay system has been widely studied, and many analysis results have been reported (see [14, 15]). However, many approaches for time-delay problems only employed partial information of the delay-related terms. The quadratic integral terms of the form ∫t-h(t)t‍f(α)d(α) have been the major choices to construct the Lyapunov-Krasovskii functional, ignoring the employment of the delay upper bound h¯ (see [16]). Recently, Markovian systems with time delay have been widely studied (see [17–19]). But the time delays in [17, 19] are invariant, which is limited in the practical application. Although [18] considers the systems with time-varying delay, some useful terms are ignored to construct the Lyapunov-Krasovskii functional, which may lead to considerable conservativeness. Unfortunately, in the aforementioned papers about time delay, the elements of the considered transition probability matrix are completely known. To the best of the authors’ knowledge, up to now, there are few papers concerning both time delay and more general transition probability to deal with stochastic stability for nonlinear Markov jump systems.Motivated by the previous points, in this paper, we are concerned with the stochastic stability analysis for a class of nonlinear MJSs with time-varying delay and general transition probabilities. The considered nonlinearity is the so-called sector-bounded nonlinearity [20], which is quite more general than the usual Lipschitz conditions. By exploiting full information of the delay-related terms including h(t) and h¯ and inherent relationship between the elements in transition probabilities, less conservative stability criterions are obtained in the framework of LMIs. Numerical examples are given to demonstrate the superiorities and effectiveness of the proposed method.Notations. The superscript T stands for matrix transposition, and Rn denotes the n dimensional Euclidean space. E[·] stands for the n mathematical expectation; He(X) represents the sum of X and XT. In addition, in symmetric block matrices or long matrix expressions, * is used as an ellipsis for the terms that are introduced by symmetry. The notation P>0 (≥0) means that P is real symmetric positive (semipositive) definite. ε[·] stands for the expectation operator with respect to the given probability measure P. Lℱ02([-h-,0];ℝn) denotes the family of ℝn-valued stochastic process x(t),-h-≤t≤0, such that x(t) is ℱ0-measurable and ∫-h-0‍E{||x(t)||2}dt<∞. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.


## 2. Problem Statement

Consider a class of stochastic nonlinear MJSs with sector-bounded nonlinearity and time-varying delay as follows:(1)x˙(t)=A(r(t))x(t)+Ad(r(t))x(t-h(t))+f(t,x(t),r(t))+fd(t,x(t-h(t)),r(t)),x(t)=ψ(t),-h-≤t≤0,
where x(t)∈Rn is the state variable, and A(r(t))∈Rn×n and Ad(r(t))∈Rn×n are known matrix functions of the Markov jump process r(t). f(*) and fd(*) are the vector-valued nonlinear functions. Also h(t)∈[0,h-], h˙(t)≤μ-, and ψ(t)∈Lℱ02([-h-,0];ℝn). r(t) is a time-homogeneous Markov process with right continuous trajectories and takes values on the finite set 𝕃={1,2,…,ℕ} with the following mode transition probabilities:
(2)Pr{r(t+dt)=j∣r(t)=i}={πijdt+o(dt),i≠j,1+πiidt+o(dt),i=j,
where dt>0, where limdt→0(o(dt)/dt)=0. πij≥0, for i≠j, is the transition rate from mode i to mode j, and
(3)∑j=1,i≠jℕ‍πij=-πii,i=(1,…,ℕ).In this paper, more general transition probabilities of the jumping process in [9] are considered. Namely, some elements in matrix have been exactly known, some ones have been merely known with lower and upper bounds, and others may have no information to use. For instance, the transition probability matrix might be described by matrix
(4)[π11?π13⋯π1ℕ??α⋯π2ℕ⋮⋯⋯⋱⋮πℕ1β⋯⋯πℕℕ],
where "?" denotes the inaccessible elements, α and β have known lower and upper bounds (α_≤α≤α- and β_≤β≤β-) and πij is exactly known.For notation clarity, for alli,j∈𝕃, we further rewrite 𝕃=𝕃ki∪𝕃uki with
(5)𝕃ki≜{j:π_ij≤πij≤π-ij},𝕃uki≜{j:j∉𝕃ki},ℒki≜{m∣m∈𝕃ki,m≠i},ℒuki≜{m∣m∈𝕃uki,m≠i},
where π_ij and π-ij are known lower and upper bounds of πij, respectively.Remark 1.
In order to get more general result, this paper considers that𝕃ki includes two cases. One is that πij is exactly known, and the other is that it is unknown but with upper and lower known bounds, which has been considered in [8]. If πij is exactly known, it also can be seen as a rate with equal bounds.
The set𝕃 comprises the various operation modes of system (1), and for each possible value of r(t)=i(i∈𝕃), the matrices and the vector-valued nonlinear functions associated with the ith mode will be denoted by
(6)Ai=A(r(t)=i),Adi=Ad(r(t)=i),fi(t)=f(t,x(t),r(t)=i),fdi(t)=fd(t,x(t-h(t)),r(t)=i).
In system (1), the vector-valued nonlinear functions f(*) and fd(*) are assumed to satisfy the following sector-bounded conditions:
(7)[f(x)-f(y)-M1(x-y)]T×[f(x)-f(y)-M2(x-y)]≤0,[fd(x)-fd(y)-N1(x-y)]T×[fd(x)-fd(y)-N2(x-y)]≤0,
where for all x,y∈Rn, M1,M2,N1,N2∈Rn×n are known constant matrices. Without loss of generality, the following equations are always assumed:
(8)f(0)=0,fd(0)=0.Remark 2.
As in [20], the nonlinear functions f(*), fd(*) are said to belong to sectors. In other words, the nonlinearities are bounded by sectors. Further, both the filter design problems and control analysis have been investigated; see [21, 22]. The nonlinear descriptions in (7) are more general than the usual sigmoid functions and the recently commonly used Lipschitz conditions [21], and M1, M2, N1, N2 are lower and upper slope bounds, respectively.Lemma 3.
Assume thatf(*) is a vector-valued nonlinear function and M1, M2 are known constant matrices. If there exists
(9)[f(x)-M1x]T[f(x)-M2x]≤0,
it can be obtained that
(10)[xf(x)]T[M-1M-2*I][xf(x)]≤0,
with M-1=(M1TM2+M2TM1)/2, M-2=-(M1T+M2T)/2.Proof.
Notice the fact that(11)[f(x)-M1x]T[f(x)-M2x]≤0,[f(x)-M2x]T[f(x)-M1x]≤0;
so,
(12)0.5{[f(x)-M1x]T[f(x)-M2x]+[f(x)-M2x]T[f(x)-M1x]}≤0.
Then, (10) can be obtained. Thus, the proof is complete.For the sake of simplicity, the solutionx(t,x0,r0) of system (1) with r0∈𝕃 is denoted by x(t). It is known that {x(t),r(t)} is a Markov process with an initial state (x0,r0), and its weak infinitesimal generator acting on function V is defined as follows:
(13)ξV(x(t),r(t),t)=limΔ→0+1Δ[ε(V(x(t+Δ),t+Δ,r(t+Δ))∣x(t),r(t)=i)-V(x(t),r(t),t)].Throughout this paper, the following definition is used.Definition 4.
System (1) is said to be stochastically stable if for finite ψ(t)∈Rn defined on [-h-,0] and r(0)∈𝕃, the following is satisfied:
(14)limt→∞ε{∫0t‍xT(s,ψ,r(0))x(s,ψ,r(0))ds}<∞,
where x(s,ψ,r(0)) denotes the solution to system at time t under the initial conditions ψ(t) and r(0).


## 3. Stochastic Stability Analysis

In this section, new LMI-based sufficient conditions for system (1) will be presented based on a special Lyapunov-Krasovskii functional and Lemma 3.For presentation convenience, it is necessary to define(15)χi(t)=[xT(t),xT(t-h(t)),xT(t-h-),x˙T(t),fiT(t),fdiT(t)]T=e1x(t)+e2x(t-h(t))+e3x(t-h-)+e4x˙(t)+e5fi(t)+e6fdi(t),
and the corresponding block entry matrices are ej, j∈{1,…,6}. In addition, there are some other definitions. Consider
(16)λki=∑m∈ℒki‍πim,λ_ki=∑m∈ℒki‍π_im,λuki=∑m∈ℒuki‍πim,𝒫ki=∑m∈ℒki‍πijPm,𝒫-ki=∑m∈ℒki‍π-ijPm,Theorem 5.
For given scalarsh->0 and μ-, the delayed system (1) with h˙(t)≤μ- is stochastically stable if there exist scalars η1i>0 and η2i>0 and matrices Pi>0, Q0i>0, Q1i>0, S0>0, S1>0, R>0, Y11i,Y12i,Y22i,Z11i,Z12i,Z22i, Φi, H1i, and H2i such that the following conditions hold:
(17)He((e1-e2)TΦi[Aie1T+Adie2T-e4T+e5T+e6T]+Z12i(e2-e3)T+Y12i(e1-e2)T)+He(e4Pie1T)-e3Q0ie3T+h-e4(S0+S1)e4T-(1-μ-)e2Q1ie2T+h-Y11i+e1(h-R+Q0i+Q1i+Ψi)e1T-η1i(e1M-1ie1T+e5M-2iTe1T+e1M-2ie5T+e5e5T)-η2i(e2N-1ie2T+e6N-2iTe2T+e2N-2ie6T+e6e6T)<0,(18)He((e1-e2)TΦi[Aie1T+Adie2T-e4T+e5T+e6T]+Z12i(e2-e3)T+Y12i(e1-e2)T)+He(e4Pie1T)-e3Q0ie3T+h-e4S0e4T-(1-μ-)e2Q1ie2T+h-Z11i+e1(h-R+Q0i+Q1i+Ψi)e1T-η1i(e1M-1ie1T+e5M-2iTe1T+e1M-2ie5T+e5e5T)-η2i(e2N-1ie2T+e6N-2iTe2T+e2N-2ie6T+e6e6T)<0,(19)[Y11iY12i*Y22i]≥0,[Z11iZ12i*Z22i]≥0,(20)S0≥Z22i,S0+(1-μ-S1)≥Y22i,(21)i∈𝕃ki{∑m∈ℒki‍π-im(Q0m+Q1m)+π-ii(Q0i+Q1i)-(λ_ki+π_ii)(Q0j+Q1j)-R<0,j∈𝕃uki,∑m∈ℒki‍π-imQ0m+π-iiQ0i-(λ_ki+π_ii)Q0j-R<0,j∈𝕃uki,(22)i∈𝕃uki{∑m∈𝕃ki‍π-im(Q0m+Q1m)-∑m∈𝕃ki‍π_imH2i-R<0,∑m∈𝕃ki‍π-imQ0m-∑m∈𝕃ki‍π_imH2i-R<0,Q0j+Q1j-H2i≤0,j∈𝕃uki,j≠i,Q0j-H2i≥0,j∈𝕃uki,j=i,Pj-H1i≤0,j∈𝕃uki,j≠i,Pj-H1i≥0,j∈𝕃uki,j=i,
where
(23)Ψi={𝒫-ki+π-iiPi-(π_ii+λ_ki)Pl,i∈𝕃ki,l∈𝕃uki,𝒫-ki-∑m∈𝕃ki‍π_imH1i,i∈𝕃uki,M-1=(M1TM2+M2TM1)2,M-2=-(M1T+M2T)2,N-1=(N1TN2+N2TN1)2,N-2=-(N1T+N2T)2,Proof.
First, in order to cast the model involved into the framework of the Markov processes, a new process is defined as{(xt,r(t)),t≥0} by
(24)xt(s)=x(t+s),-2h-≤s≤0;
so, {(xt,r(t)),t≥h-} is a Markov process with initial state (ψ(·),r(0)).
Then, the stochastic Lyapunov-Krasovskii functional is chosen as follows:(25)V(xt,r(t),t)=∑j=16‍Vj(xt,r(t),t),
where
(26)V1(xt,r(t),t)=xT(t)P(r(t))x(t),V2(xt,r(t),t)=∫t-h-t‍xT(α)Q0(r(t))x(α)dα,V3(xt,r(t),t)=∫t-h(t)t‍xT(α)Q1(r(t))x(α)dα,V4(xt,r(t),t)=∫-h-0‍∫t+βt‍x˙T(α)S0x˙(α)dαdβ,V5(xt,r(t),t)=∫-h(t)0‍∫t+βt‍x˙T(α)S1x˙(α)dαdβ,V6(xt,r(t),t)=∫-h-0‍∫t+βt‍xT(α)Rx(α)dαdβ,P(r(t))>0,Q0(r(t))>0,Q1(r(t))>0,S0>0,S1>0,R>0.
The infinitesimal generator of the Markov process acting onV(·) and emanating from the point r(t)=i(i∈𝕃) is given as follows:
(27)ξV(xt,r(t),t)=2χiT(t)e4Pie1Tχi(t)+χiT(t)[e1∑j=1ℕ‍πijPje1T+e1Q0ie1T-e3Q0ie3T]χi(t)+∫t-h-t‍xT(α)∑j=1ℕ‍πijQ0jx(α)dα+χiT(t)[e1Q1ie1T-(1-h˙(t))e2Q1ie2T]χi(t)+∫t-h(t)t‍xT(α)∑j=1ℕ‍πijQ1jx(α)dα+h-χiT(t)e4S0e4Tχi(t)-∫t-h-t‍x˙T(α)S0x˙(α)dα+h(t)χiT(t)e4S1e4Tχi(t)-(1-h˙(t))∫t-h(t)t‍x˙T(α)S1x˙(α)dα+h-χiT(t)e1Re1Tχi(t)-∫t-h-t‍xT(α)Rx(α)dα.
Then, choose
(28)[Y11iY12i*Y22i]≥0,[Z11iZ12i*Z22i]≥0;
it follows that
(29)∫t-h(t)t‍[χi(t)x˙(α)]T[Y11iY12i*Y22i][χi(t)x˙(α)]dα=χiT(t)[He(Y12i(e1-e2)T)+h(t)Y11i]χi(t)+∫t-h(t)t‍x˙T(α)Y22x˙(α)dα≥0,∫t-h-t-h(t)‍[χi(t)x˙(α)]T[Z11iZ12i*Z22i][χi(t)x˙(α)]dα=χiT(t)[He(Z12i(e2-e3)T)+(h--h(t))Z11i]χi(t)+∫t-h-t-h(t)‍x˙T(α)Z22x˙(α)dα≥0.
And the constraints of the model dynamics can be written as χiT(t)Φi[Aie1T+Adie2T-e4T+e5T+e6T]χi(t)≡0 by introducing free variables Φi. In addition, according to Lemma 3 and choosing scalar η1i>0, η2i>0, the following inequalities hold:
(30)-η1iχiT(t)(e1M-1ie1T+e5M-2iTe1T+e1M-2ie5T+e5e5T)χi(t)≥0,-η2iχiT(t)(e2N-1ie2T+e6N-2iTe2T+e2N-2ie6T+e6e6T)χi(t)≥0.
So,ξV(x(t),r(t),t) can be upper bounded as
(31)ξV(xt,r(t),t)≤χiT(t){Ω1(h(t))+e1∑j=1ℕ‍πijPje1T}χi(t)+Ω2+Ω3,
where
(32)Ω1(h(t))=h(t)(Y11i+e4S1e4T)+(h--h(t))Z11i+He[Φi(Aie1T+Adie2T-e4T+e5T+e6T)+e4Pie1T+Y12i(e1-e2)T+Z12i(e2-e3)T]+h-e4S0e4T-e3Q0ie3T-(1-μ-)e2Q1ie2T+e1(Q0i+Q1i+h-R)e1T-η1i(e1M-1ie1T+e5M-2iTe1T+e1M-2ie5T+e5e5T)-η2i(e2N-1ie2T+e6N-2iTe2T+e2N-2ie6T+e6e6T),Ω2=-∫t-h-t-h(t)‍x˙T(α)(S0-Z22i)x˙(α)dα-∫t-h(t)t‍x˙T(α)(S0-Y22i+(1-μ-)S1)x˙(α)dα,Ω3=∫t-h(t)t‍xT(α)[∑j=1ℕ‍πij(Q0j+Q1j)-R]x(α)dα+∫t-h-t-h(t)‍xT(α)[∑j=1ℕ‍πijQ0j-R]x(α)dα.
SinceΩ1(h(t)) is a convex combination of the matrices Y11i+e4S1e4T and Z11i on h(t), it can be handled nonconservatively by two corresponding boundary LMIs: one for h(t)=h- and the other for h(t)=0.
So,Ω1(h(t))<0 is equivalent to
(33)Ω1(0)<0,Ω1(h-)<0.
Then, the proof is separated into two cases, i∈𝕃ki and i∈𝕃uki.
Case
1
(
i
∈
𝕃
k
i
). In this case, πii is bounded or exactly known. Based on ∑j=1ℕπij=λki+πii+λuki=0 and -∑l∈ℒukiπil/(πii+λki)=1, it is easy to obtain that
(34)Ω1(h(t))+e1∑j=1ℕ‍πijPje1T=Ω1(h(t))+e1(∑j∈ℒki‍πijPj+πiiPi+∑l∈ℒuki‍πilPl)e1T=-∑l∈ℒukiπilπii+λki{{[∑j∈Lki‍πijPj+πiiPi-(πii+λki)Pl]}Ω1(h(t))+e1[∑j∈ℒki‍πijPj+πiiPi-(πii+λki)Pl]e1T}.
Considering the fact that0≤-πil/(πii+λki)≤1, it is easy to obtain that Ω1(h(t)) + e1[∑j∈ℒkiπijPj + πiiPi-(πii + λki)Pl]e1T<0, l∈𝕃uki can guarantee Ω1(h(t))+e1∑j=1ℕπijPje1T<0.
Sinceπ_ij≤πij≤π-ij, the following inequality can be obtained:
(35)Ω1(h(t))+e1[∑j∈ℒki‍πijPj+πiiPi-(πii+λki)Pl]e1T≤Ω1(h(t))+e1[𝒫-ki+π-iiPi-(π_ii+λ_ki)Pl]e1T.
So, if (17) and (18) hold, Ω1(h(t))+e1∑j=1ℕπijPje1T<0 can be obtained.
Similarly, (21) ⇒Ω3<0.
And (20) ⇒Ω2<0.
Therefore, when (17)–(21) are satisfied, ξV(xt,r(t),t)<0.
Case
2
(
i
∈
𝕃
u
k
i
). Because ∑j=1ℕπij=0, there exist
(36)-xT(t)∑j=1ℕ‍πijH1ix(t)=0,-∫t-ht‍xT(α)∑j=1ℕ‍πijH2ix(α)dα=0.
Adding the left sides of (36) into (31), the inequality can be written as
(37)ξV(xt,r(t),t)≤χT(t)Γ1(h(t))χ(t)+Γ2+Γ3,
where
(38)Γ1(h(t))=h(t)(Y11i+e4S1e4T)+(h--h(t))Z11i+He[(e1-e2)TΦi(Aie1T+Adie2T-e4T+e5T+e6T)+e4Pie1T+Y12i(e1-e2)T+Z12i(e2-e3)T]+h-e4S0e4T-e3Q0ie3T-(1-μ-)e2Q1ie2T+e1[Q0i+Q1i+h-R+∑j∈𝕃ki‍πij(Pj-H1i)]e1T+e1∑j∈𝕃uki‍πij(Pj-H1i)e1T-η1i(e1M-1ie1T+e5M-2iTe1T+e1M-2ie5T+e5e5T)-η2i(e2N-1ie2T+e6N-2iTe2T+e2N-2ie6T+e6e6T),Γ2=Ω2,Γ3=∫t-h(t)t‍xT(α)[∑j∈𝕃ki‍πij(Q0j+Q1j-H2i)-R+∑j∈𝕃uki‍πij(Q0j+Q1j-H2i)]x(α)dα+∫t-h-t-h(t)‍xT(α)[∑j∈𝕃kiℕ‍πij(Q0j-H2i)-R+∑j∈𝕃ukiℕ‍πij(Q0j-H2i)]x(α)dα.
So,Γ1(h(t)) is also a convex combination of the matrices Y11i+e4S1e4T and Z11i on h(t).
It is known thatπii<0. At the same time, the boundary information π_ij≤πij≤π-ij of the transition probabilities need to be used. It is clear that if (17)–(20) and (22) hold, ξV(xt,r(t),t)<0 can be achieved.
Taking into account (17)–(22), there exists a scalar c>0 for each i∈𝕃 such that
(39)ξV(x(t),r(t),t)≤-c∥x(t)∥2,
which implies that system (1) is stochastically stable.
Therefore, the proof is completed.Iff(t,x(t),r(t))=ΔA(t,r(t))x(t), fd(t,x(t-h(t)), r(t))=ΔAd(t,r(t))x(t-h(t)), system (1) becomes
(40)x˙(t)=(A(r(t))+ΔA(t,r(t)))x(t)+(Ad(r(t))+ΔAd(t,r(t)))x(t-h(t)),x(t)=ψ(t),-h-≤t≤0,
where ΔA(t,r(t)), ΔAd(t,r(t)) are unknown time-varying matrices with appropriate dimensions and are assumed to be of the following form:
(41)[ΔA(t,r(t))ΔAd(t,r(t))]=D(r(t))E(t,r(t))[F(r(t))Fd(r(t))],
where D(r(t)), F(r(t)), Fd(r(t)) are known real constant matrices for all r(t)∈𝕃, and E(t,r(t)), for all r(t)∈𝕃, are the uncertain time-varying matrices satisfying
(42)E(t,r(t))TE(t,r(t))≤I,∀r(t)∈𝕃.It can be easily seen that system (40) is just the MJSs with norm bounded uncertainty (41).Choosingp(t,r(t))=E(t,r(t))q(t,r(t)) and q(t,r(t))=F(r(t))x(t)+Fd(r(t))x(t-h(t)), system (40) is equivalent to the following system:
(43)x˙(t)=A(r(t))x(t)+Ad(r(t))x(t-h(t))+D(r(t))p(t,r(t)),x(t)=ψ(t),-h-≤t≤0.For each possible value ofr(t)=i  (i∈𝕃), p(t,r(t)), D(r(t)), F(r(t)), and Fd(r(t)) can be denoted as pi(t), Di, Fi, and Fdi. Then, it is necessary to choose χi(t)=[xT(t),xT(t-h(t)),xT(t-h-),x˙T(t),piT(t)]T and the corresponding block entry matrices as ej,j∈{1,…,5}.Then, the following corollary is obtained.Corollary 6.
If there exist matricesPi>0, Q0i>0, Q1i>0, S0>0, S1>0, R>0, Y11i, Y12i, Y22i, Z11i, Z12i, Z22i, Φi, H1i, and H2i such that (19)–(22) and the following LMIs are feasible for i=1,2…,ℕ:
(44)He((e1-e2)TΦi[Aie1T+Adie2T-e4T+Die5T]+Z12(e2-e3)T+Y12(e1-e2)T)+He(e4Pie1T)-e3Q0ie3T+h-e4(S0+S1)e4T-(1-μ-)e2Q1ie2T+h-Y11+e1(h-R+Q0i+Q1i+Ψi)e1T+(e1FiT+e2FdiT)(Fie1T+Fdie2T)-e5e5T<0,He((e1-e2)TΦi[Aie1T+Adie2T-e4T+Die5T]+Z12(e2-e3)T+Y12(e1-e2)T)+He(e4Pie1T)-e3Q0ie3T+h-e4S0e4T-(1-μ-)e2Q1ie2T+h-Z11+e1(h-R+Q0i+Q1i+Ψi)e1T+(e1FiT+e2FdiT)(Fie1T+Fdie2T)-e5e5T<0,
then the delayed uncertain system (40) with partly known transition probabilities is stochastically stable.Proof.
Here, the constraints of the model dynamics can be written asχiT(t)Φi[Aie1T+Adie2T-e4T+Die5T]χi(t)≡0. And the additional uncertainty constraint (42) is given as follows:
(45)0≤qT(t)q(t)-pT(t)p(t)=χiT(t){(e1FiT+e2FdiT)(Fie1T+Fdie2T)-e5e5T}χi(t).
Choose the Lyapunov functional as (25); so,
(46)ξV(x(t),r(t),t)≤ξV(x(t),r(t),t)+qT(t)q(t)-pT(t)p(t).
Use the same method as the proof of Theorem5 to make ξV(x(t),r(t),t)+qT(t)q(t)-pT(t)p(t)<0, and then Corollary 6 can be obtained.


## 4. Numerical Examples

In this section, numerical examples are given to demonstrate the benefits and effectiveness of the proposed methods.First, Example 1 in [16] is given to show the superiority of Theorem 5 for MJS (1) with no nonlinear terms and completely known transition probability matrix.Example 7.
Consider MJS (1) with f(t,x(t),r(t))=0 and fd(t,x(t-h(t)),r(t))=0. The two modes are given as
(47)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].
Supposing that μ=0.8 and π22=-0.8, compare the stochastic stability result in Theorem 5 with the stability theorem in [16, 18]. For given π11, the maximum h-, which satisfies the LMIs in (17)–(22), can be calculated.Table1 presents the comparison results, which shows that the stochastic stability result in Theorem 5 is less conservative than those in [16, 18] when no nonlinear terms exist and transition probabilities are completely known.Table 1
Comparison of maximum allowedh-.
π
11
-
0.1
-
0.5
-
1
h
- by [16]
0.4261
0.4206
0.4161
h
- by [18]
0.5050
0.4887
0.4855
h
- by Theorem 5
0.5575
0.5499
0.5355Next numerical example is for uncertain MJS (43) with more general transition probability.Example 8.
Consider the following MJS (43) with four modes, whose system matrices are
(48)A1=[-2.2460-1.4410-1.5937-2.9289],A2=[-1.89990.8156-0.6900-0.7881],A3=[-0.75230.75001.5630-2.3540],A4=[-1.78400.36401.3670-1.5640],Ad1=[-0.5-0.50.050.01],Ad2=[0.0100.05-0.01],Ad3=[-0.01-0.020.050.01],Ad4=[0.030.01-0.2-0.1],F1=F2=F3=F4=[10],Fd1=Fd2=Fd3=Fd4=[-50.5],D1=D2=D3=D4=[0.10.1].
The more general transition probability matrix is chosen as follows, which includes completely known elements, boundary known elements, and completely unknown elements:(49)[-1.30.2????0.30.30.6?-1.5????α].
Choose -1.5≤α≤-0.9. According to Corollary 6, Table 2 can be obtained for different values of μ-. Also, the uncertain MJS (43) is stochastically stable. Moreover, as the upper bound of h˙(t), that is, μ-, increases, the maximum allowed upper bound of h(t) may be reduced.
Furthermore, a nonlinear MJS with sector-bounded nonlinearity is given.Table 2
Maximum allowedh- by Corollary 6.
μ
-
0
0.2
0.6
h
- by Corollary 6
1.1739
1.0161
0.8618Example 9.
Consider MJS (1) with four models, whose Ai, Adi and the transition probability matrix are the same matrices as the ones of Example 8. The nonlinear functions are given as follows:(50)f1(t)=[0.1x1(t)cos2(x1(t))-0.05(x1(t)-x2(t))-0.05x1(t)-0.1x2(t)],f2(t)=2f1(t),f3(t)=[0.15x2(t)cos2(x1(t)+x2(t))+0.1x1(t)+0.25x2(t)0.15x1(t)+0.05x2(t)],f4(t)=2f3(t),fd1(t)=fd2(t)=fd3(t)=fd4(t)=[0.12x1(t-h(t))sin2(x2(t-h(t)))+0.04x1(t-h(t))+0.08x2(t-h(t))0.04x1(t-h(t))+0.04x2(t-h(t))].According to (7), the corresponding matrices, which describe the sector bound, can be chosen as
(51)M11=[0.050.050.050.05],M12=2M11,M13=[0.10.10.1-0.05],M14=2M13,M21=[-0.050.05-0.15-0.25],M22=2M21,M23=[0.10.250.20.15],M24=2M23,N11=N12=N13=N14=[0.040.08-0.040.12],N21=N22=N23=N24=[0.160.080.12-0.04].
Choose -1.5≤α≤-1. Using Matlab LMI toolbox and Theorem 5, the maximum allowed upper bound of h(t) can be obtained for different values of μ-.
From Table3, it can be seen that the stochastic stability can be guaranteed for nonlinear MJS (1) when more general transition probabilities (4) are considered.Table 3
Maximum allowedh- by Theorem 5.
μ
-
0
0.2
0.6
h
- by Theorem 5
1.2999
1.0500
0.9236


## 5. Conclusion

This paper has presented a new method for solving the stochastic stability analysis problem for MJS with time-varying delay and sector-bounded nonlinearity. A general transition probability matrix is adopted, which includes completely known elements, boundary known elements, and completely unknown elements. Based on a new mode-dependent stochastic Lyapunov-Krasovskii functional and some new techniques to deal with transition probabilities, sufficient conditions with less conservativeness are derived in terms of LMIs to obtain stochastic stability of systems. Numerical examples have also demonstrated the effectiveness of the results.



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*Source: 208263-2013-04-18.xml*