# Stability Analysis for Markovian Jump Neutral Systems with Mixed Delays and Partially Known Transition Rates

Authors: Lianglin Xiong; Xiaobing Zhou; Jie Qiu; Jing Lei
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The delay-dependent stability problem is studied for Markovian jump neutral systems with partial information on transition probabilities, and the considered delays are mixed and model dependent. By constructing the new stochastic Lyapunov-Krasovskii functional, which combined the introduced free matrices with the analysis technique of matrix inequalities, a sufficient condition for the systems with fully known transition rates is
firstly established. Then, making full use of the transition rate matrix, the results are obtained for the other case, and the uncertain neutral Markovian jump system with incomplete transition rates is also considered. Finally, to show the validity of the obtained results, three numerical examples are provided.


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## Body

## 1. Introduction

A switched system is a dynamic system consisted of a number of subsystems and a rule that manages the switching between these subsystems. In the past, a large number of excellent papers and monographs on the stability of switched systems have been published such as [1–7] and the references cited therein. Among the results for switched systems, the stabilization problem of switched neutral systems has also been explored by some researchers [8–22], and mainly two kinds of switching rule are designed in these articles. Some state-dependent switching rules are obtained assuming the convex combination of the systems matrix, see, for example, [8, 10, 20]. To reduce the conservative, the authors in [11] have investigated the stabilization for switched neutral systems without the assumption that the restraint of the convex combination on systems matrices, the dwell time, and state-dependent rules were designed. Similarly, the authors in [12] have also studied the problem of the BIBO stability for switched neutral systems. Using the Razumikhin-like approach [13] and the Leibniz-Newton formula, the global exponential stability for a class of switched neutral systems with interval-time-varying state delay and two classes of perturbations is investigated in [14], with arbitrary switching signal. Moreover, with the constructed state-dependent switching rule, the authors in [15] have investigated the global exponential stability of switched neutral systems. With the dwell time approach, the improved stability conditions for a class of switched neutral systems with mixed time-varying delays have also been obtained in [16], and the upper bound of derivative of the discrete time-varying delay is not restricted to one. In [17], the robust reliable stabilization of uncertain switched neutral systems with delayed switching has been considered. The H∞ fault estimator for switched nonlinear systems of neutral type has been designed in [9]. In [18], the authors have studied the problem of exponential stability for neutral switched systems with interval time-varying mixed delays and nonlinear perturbations, obtaining the less conservative conditions based on the introduced free matrices. More recently, the markovian jump parameters have been considered for the analysis of switched neutral systems in [19].In the past few decades, as a special switched system, markovian jump systems (MJSs) have been widely studied due to the fact that many dynamical systems subject to random abrupt variations can be modeled by MJSs such as manufacturing systems, networked control systems, and fault-tolerant control systems. There are a lot of useful results that have been presented in the literature, such as [23–29], and the references therein. For the analysis of MJSs, the transition probabilities in the jumping process determine the system behavior to a large extent. However, the likelihood of obtaining such available knowledge is actually questionable, and the cost is probably expensive. Rather than having a large complexity to measure or estimate all the transition probabilities, it is significant and necessary, from control perspectives, to further study more general jump systems with partly unknown transition probabilities. Recently, many results on the Markovian jump systems with partly unknown transition probabilities are obtained [30–37]. By introducing the free matrices based on the property of transition rate matrix, [32] gave less conservative conditions than that in [30] for Markovian jump systems with partial information on transition probability. And [33] provided with a new approach to obtain the necessary and sufficient conditions for markovian jump linear systems with incomplete transition probabilities, which may be appropriate to discuss the counterpart of delay systems. Most of these improved results just require some free matrices or the knowledge of the known elements in transition rate matrix, such as the bounds or structures of uncertainties, and some else of the unknown elements need not be considered. It is a great progress on the analysis of markovian jump systems. However, a few of these papers have considered the effect of delay on the stability or stabilization for the corresponding neutral systems, except for [19]. The global exponential stability of the markovian jumping neutral systems with interval time-varying delays has been studied by [19]; however, the transition probabilities are fully known, and the constructed Lyapunov did not fully consider the effect of the transition probabilities on the integrand. To the best of the authors' knowledge, the markovian jump neutral systems have not been fully investigated, and it is very challenging. All of these motivate this paper.In this paper, the delay-dependent stability problem of neutral Markovian jump linear systems with partly unknown transition probabilities is investigated. The obtained results are presented in the form of linear matrix inequalities, which is easily computed by the Matlab toolbox. The considered systems are more general than the systems with completely known or completely unknown transition probabilities, which can be viewed as two special cases of the ones tackled here. Moreover, in contrast with the recent research on uncertain transition probabilities, our proposed concept of the partly unknown transition probabilities does not require any knowledge of the unknown elements, such as the bounds or structures of uncertainties. In addition, the relationship between the stability criteria currently obtained for the usual MJLS and switched linear system under arbitrary switching is exposed by our proposed systems. Furthermore, the number of matrix inequalities conditions obtained in this paper is much more than some existing results due to the introduced free matrices based on the system itself and the information of transition probabilities in this paper, which may increase the complexity of computation. However, it would decrease the conservativeness for the delay-dependent stability conditions. Finally, two numerical examples are provided to illustrate the effectiveness of our results.


## 2. Problem Statement and Preliminaries

Consider the following neutral system with markovian jump parameters:(2.1)x˙(t)-C(rt)x˙(t-τ2(rt))=A(rt)x(t)+B(rt)x(t-τ1(rt)),x(t0+θ)=φ(θ),∀θ∈[-τ,0],
where {rt},t≥0 is a right-continuous Markov process on the probability space taking values in a finite state space, ℘={1,2,…,N} with generator 𝕀=(λij),i,j∈℘ given by
(2.2)Pr{rt+Δ=j∣rt=i}={λijΔ+o(Δ),j≠i,1+λiiΔ+o(Δ),j=i,
where Δ>0,limΔ→0(o(Δ)/Δ)=0, λij≥0, for j≠i, is the transition rate from mode i at time t to mode j at time t+Δ, λii=-∑j=1N‍λij. A(rt), B(rt), and C(rt) are known matrix functions of the markovian process, x(t)∈Rn is the state vector, and φ(θ) is the initial condition function. τ1(rt) and τ2(rt) are mode-dependent delays, when rt=i∈℘, τ1(rt)=τ1i,τ2(rt)=τ2i, and τ=max(τ1i,τ2i).Since the transition probability depends on the transition rates for the continuous-time MJSs, the transition rates of the jumping process are considered to be partly accessible in this paper. For instance, the transition rate matrix𝕀 with ℕ operation modes may be expressed as
(2.3)𝕀=(λ11?λ13⋯?????λ2N⋮⋮⋮⋱⋮?λN2λN3⋯λNN),
where ? represents the unknown transition rate. For notational clarity, foralli∈℘, the set Ui denotes Ui=Uki⋃Uuki with Uki≜{j:λijisknownforj∈℘}, Uuki≜{j:λijisunknownforj∈℘}, and λki≜∑j∈Uki‍λij.Moreover, ifUki≠∅, it is further described as
(2.4)Uki={k1i,k2i,…,kmi,},
where m is a nonnegation integer with 1≤m≤N, and kji∈Z+,1≤kji≤Nj=1,2,…,m represent the jth known element of the set Uki in the ith row of the transition rate matrix 𝕀.Remark 2.1.
It is worthwhile to note that ifUki=∅,Ui=Uuki which means that any information between the ith mode and the other N-1 modes is not accessible, then MJSs with N modes can be regarded as ones with N-1 modes. It is clear that when Uuki=∅,Ui=Uki, the system (2.1) becomes the usual assumption case.For the sake of simplicity, the solutionx(t,φ(θ),r0) with r0∈℘ is denoted by x(t). It is known from [38] that {x(t),t} is a Markov process with an initial state {φ(θ),r0}, and its weak infinitesimal generator, acting on function V, is defined in [39]:
(2.5)LV(x(t),t,i)=limΔ→0+1Δ[ε(V(x(t+Δ),t+Δ,rt+Δ)∣x(t),rt=i)-V(x(t),t,i)].Throughout this paper, the following definition is necessary. More details refer to [23].Definition 2.2 (see, [32]).
The system (2.1) is said to be stochastically stable if the following holds:
(2.6)ε{∫0∞‍∥x(t)∥2dt∣φ,r0}<∞,
for every initial condition φ∈Rn and r0∈℘.


## 3. Stability Analysis for Neutral Markovian Jump Systems

The purpose of this section is to state the stability analysis for neutral markovian jump systems with partly unknown transition rates. Throughout the paper, the matrixC(rt) is assumed to be ρ(C(rt))<1. Before giving the stability result of systems (2.1) with a partly unknown transition rate matrix (2.3), the stability of neutral markovian jump systems (2.1) with all transition probabilities known is firstly investigated. With the introduced free matrices and the novel analysis technique of matrix, the stability conditions are presented in this section.Theorem 3.1.
The system (2.1) with a fully known transition rate matrix is stochastically stable if there exist matrices Pi>0,Q1i>0,Q2i>0,R1>0, and R2>0 and any matrices N1,N2,N3,andN4 with appropriate dimensions satisfying the following linear matrix inequalities:
(3.1)φi=(φi11φi12φi13φi14⋆φi22φi23φi24⋆⋆φi33φi34⋆⋆⋆φi44)<0,(3.2)∑j=1N‍λijQ1j+∑j=1N‍λijτ1jR1-R1≤0,(3.3)∑j=1N‍λijQ2j+∑j=1N‍λijτ2jR2-R2≤0,
with
(3.4)φi11=PiAi+AiTPi+∑j=1N‍λijPj+Q1i+N1Ai+AiTN1T+τ1iR1,φi12=-N1+AiTN2T,φi13=N1Bi+AiTN3T+PiBi,φi14=N1Ci+AiTN4T+PiCi,φi22=-N2-N2T+Q2i+τ2iR2,φi23=N2Bi-N3T,φi24=N2Ci-N4T,φi33=N3Bi+BiTN3T-Q1i+∑j=1N‍λijτ1jQ1i,φi34=N3Ci+BiTN4T,φi44=N4Ci+CiTN4T-Q2i+∑j=1N‍λijτ2jQ2i.Proof.
Construct a stochastic Lyapunov functional candidate as(3.5)V(xt,t,rt)=∑i=15‍Vi(xt,t,rt),
where
(3.6)V1(xt,t,rt)=xT(t)P(rt)x(t),V2(xt,t,rt)=∫t-τ1(rt)t‍xT(s)Q1(rt)x(s)ds,V3(xt,t,rt)=∫t-τ2(rt)t‍x˙T(s)Q2(rt)x˙(s)ds,V4(xt,t,rt)=∫-τ1(rt)0‍∫t+θt‍xT(s)R1x(s)dsdθ,V5(xt,t,rt)=∫-τ2(rt)0‍∫t+θt‍x˙T(s)R2x˙(s)dsdθ,
where P(rt),Q1(rt),Q2(rt),R1,R2,andrt∈℘ are all positive definite matrices with appropriate dimensions to be determined. Then, for given rt=i∈℘,P(rt)=Pi,Q1(rt)=Q1i,Q2(rt)=Q2i, and the weak infinitesimal operator L of the stochastic process x(t) along the evolution of Vk(xt,t,i)(k=1,…,7) are given as
(3.7)LV1(xt,t,i)=2xT(t)Pix˙(t)+xT(t)∑j=1N‍λijPjx(t)=2xT(t)Pi(Aix(t)+Bix(t-τ1i)+Cix˙(t-τ2i))+xT(t)∑j=1N‍λijPjx(t).
According to the definition of the weak infinitesimal operator L and the expression (2.2), it can be shown that
(3.8)LV2(xt,t,i)=limΔ→0+1Δ[ε(∫t+Δ-τ1(rt+Δ)t+Δ‍xT(s)Q1(rt+Δ)x(s)ds)|x(t),rt=i)-V2(x(t),t,i)]=limΔ→0+1Δ[∫t+Δ-τ1(rt+Δ)t+Δ‍xT(s)∑j=1N‍Pr{rt+Δ=j∣rt=i}Q1jx(s)ds-V2(x(t),t,i)]=limΔ→0+1Δ[∫t+Δ-τ1i-∑j=1N(λijΔ+O(Δ))τ1jt+Δ‍xT(s)[Q1i+∑j=1N‍(λijΔ+O(Δ))Q1j]x(s)dslimΔ→0+1Δ1Δ-∫t-τ1it‍xT(s)Q1ix(s)ds[Q1i+∑j=1N‍(λijΔ+O(Δ))Q1j][Q1i+∑j=1N‍(λijΔ+O(Δ))Q1j]]=limΔ→0+1Δ[∫t+Δ-τ1i-∑j=1N(λijΔ+O(Δ))τ1jt+Δ‍xT(s)Q1ix(s)ds-∫t-τ1it‍xT(s)Q1ix(s)ds]+limΔ→0+1Δ[∫t+Δ-τ1i-∑j=1N(λijΔ+O(Δ))τ1jt+Δ‍xT(s)(∑j=1N‍(λijΔ+O(Δ))Q1j)x(s)ds]=limΔ→0+1Δ∫tt+Δ‍xT(s)Q1ix(s)ds+limΔ→0+1Δ∫t+Δ-τ1i-∑j=1N(λijΔ+O(Δ))τ1jt-τ1i‍xT(s)Q1ix(s)ds+limΔ→0+1Δ∫t+Δ-τ1i-∑j=1N(λijΔ+O(Δ))τ1jt-τ1i‍xT(s)∑j=1N‍(λijΔ+O(Δ))Q1jx(s)ds=xT(t)Q1ix(t)-(1-∑j=1N‍λijτ1j)xT(t-τ1i)Q1ix(t-τ1i)+∫t-τ1it‍xT(s)∑j=1N‍λijQ1jx(s)ds,LV4(xt,t,i)=limΔ→0+1Δ[ε(∫-τ1(rt+Δ)0‍∫t+Δ+θt+Δ‍xT(s)R1x(s)ds)|x(t),rt=i)-∫-τ1i0‍∫t+θt‍xT(s)R1x(s)ds]=limΔ→0+1Δ[ε(∫-τ1i-∑j=1N(λijΔ+O(Δ))τ1j0‍∫t+Δ+θt+Δ‍xT(s)R1x(s)ds)|x(t),rt=i)limΔ→0+1Δ1Δ1Δ-∫-τ1i0‍∫t+θt‍xT(s)R1x(s)ds(∫-τ1i-∑j=1N(λijΔ+O(Δ))τ1j0‍∫t+Δ+θt+Δ‍xT(s)R1x(s)ds)(∫-τ1i-∑j=1N(λijΔ+O(Δ))τ1j0‍∫t+Δ+θt+Δ‍xT(s)R1x(s)ds)]=limΔ→0+1Δ[ε(∫-τ1i-∑j=1N(λijΔ+O(Δ))τ1j-τi‍∫t+Δ+θt+Δ‍xT(s)R1x(s)ds)|x(t),rt=i)limΔ→0+1Δ+∫-τ1i0‍∫t+Δ+θt+Δ‍xT(s)R1x(s)ds-∫-τ1i0‍∫t+θt‍xT(s)R1x(s)ds(∫-τ1i-∑j=1N(λijΔ+O(Δ))τ1j-τi‍∫t+Δ+θt+Δ‍xT(s)R1x(s)ds)]=τ1ixT(t)R1x(t)-∫t-τ1it‍xT(s)R1x(s)ds+∫t-τ1it‍xT(s)∑j=1N‍λijτ1jR1x(s)ds.
Similar to the above, we can obtain
(3.9)LV3(xt,t,i)=x˙T(t)Q2ix˙(t)-(1-∑j=1N‍λijτ2j)x˙T(t-τ2i)Q2ix˙(t-τ2i)+∫t-τ2it‍x˙T(s)∑j=1N‍λijQ2jx˙(s)ds,LV5(xt,t,i)=τ2ix˙T(t)R2x˙(t)-∫t-τ2it‍x˙T(s)R2x˙(s)ds+∫t-τ2it‍x˙T(s)∑j=1N‍λijτ2jR2x˙(s)ds.
Moreover, there exist matrices Nk(k=1,…,4) with appropriate dimensions, such that the following equality holds according to (2.1):
(3.10)2ξT(t)N(-x˙(t)+Aix(t)+Bix(t-τ1i)+Cix˙(t-τ2i))=0,
where
(3.11)NT=(N1TN2TN3TN4T),ξT(t)=(xT(t)x˙T(t)xT(t-τ1i)x˙T(t-τ2i)).
From (3.7)–(3.10) and with (3.2)-(3.3), one can obtain that
(3.12)LV(xt,t,i)=∑j=14‍LVj(xt,t,i)=xT(t)φix(t)<0,
where φi are defined in this theorem. Therefore,
(3.13)ε{∫0∞‍∥x(t)∥2dt∣φ,r0}<∞,
which means that systems (2.1) are stochastic stability. The proof is completed.Based on the result of Theorem3.1, the next theorem will relate to the stability condition of systems (2.1) with partially known transition probabilities.Theorem 3.2.
The system (2.1) with a partly unknown transition rate matrix (2.4) is stochastically stable if there exist matrices Pi>0,Q1i>0,Q2i>0(i=1,2,…,N),R1>0, and R2>0 and any matrices Nk(k=1,…,4),Wi=WiT with appropriate dimensions satisfying the following linear matrix inequalities:
(3.14)ϕi=(ϕi11ϕi12ϕi13ϕi14⋆ϕi22ϕi23ϕi24⋆⋆ϕi33ϕi34⋆⋆⋆ϕi44)<0,∀j∈Uukiifi∈Uki,(3.15)ψi=(ψi11ψi12ψi13ψi14⋆ψi22ψi23ψi24⋆⋆ψi33ψi34⋆⋆⋆ψi44)<0,∀j∈Uukiifi∈Uuki,(3.16)∑j∈Uki‍λijQ1j+∑j∈Uki‍λijτ1jR1-R1-λkiQ1j-λkiτ1jR1≤0,∀j∈Uukiifi∈Uki,(3.17)∑j∈Uki‍λijQ1j+∑j∈Uki‍λijτ1jR1-R1+λdiQ1i-λdiQ1j-λkiQ1j+λdiτ1iR1-λdiτ1jR1-λkiτ1jR1≤0,∀j∈Uukiifi∈Uki,(3.18)∑j∈Uki‍λijQ2j+∑j∈Uki‍λijτ2jR2-R2-λkiQ2j-λkiτ2jR2≤0,∀j∈Uukiifi∈Uki,(3.19)∑j∈Uki‍λijQ2j+∑j∈Uki‍λijτ2jR2-R2+λdiQ2i-λdiQ2j-λkiQ2j+λdiτ2iR2-λdiτ2jR2-λkiτ2jR2≤0,∀j∈Uukiifi∈Uki,(3.20)Pj-Wi≤0,j∈Uuki,j≠i,(3.21)Pj-Wi≥0,j∈Uuki,j=i,
with
(3.22)ϕi11=PiAi+AiTPi+∑j∈Uki‍λij(Pj-Wi)+Q1i+N1Ai+AiTN1T+τ1iR1,ϕi12=-N1+AiTN2T,ϕi13=N1Bi+AiTN3T+PiBi,ϕi14=N1Ci+AiTN4T+PiCi,ϕi22=-N2-N2T+Q2i+τ2iR2,ϕi23=N2Bi-N3T,ϕi24=N2Ci-N4T,ϕi33=N3Bi+BiTN3T-Q1i+∑j∈Uki‍λijτ1jQ1i-λkiτ1jQ1i,ϕi34=N3Ci+BiTN4T,ϕi44=N4Ci+CiTN4T-Q2i+∑j∈Uki‍λijτ2jQ2i-λkiτ2jQ2i.ψi33=N3Bi+BiTN3T-Q1i+∑j∈Uki‍λijτ1jQ1i+λdiτ1iQ1i-λdiτ1jQ1i-λkiτ1jQ1i,ψi44=N4Ci+CiTN4T-Q2i+∑j∈Uki‍λijτ2jQ2i+λdiτ2iQ2i-λdiτ2jQ2i-λkiτ2jQ2i,ψi11=ϕi11,ψi12=ϕi12,ψi13=ϕi13,ψi14=ϕi14,ψi22=ϕi22,ψi23=ϕi23,ψi24=ϕi24,ψi34=ϕi34,
and λdi is a given lower bound for the unknown diagonal element.Proof.
For the case of the systems (2.1) with partly unknown transition probabilities, and taking into account the situation that the information of transition probabilities is not accessible completely, due to ∑j=1Nλij=0, the following zero equation holds for arbitrary matrices Wi=WiT is satisfied:
(3.23)-xT(t)(∑j=1N‍λijWi)x(t)=0,∀i∈℘,
and the inequality (3.12) can be rewritten as
(3.24)LV(xt,t,i)=xT(t)∑j∈Uuki‍λij(Pj-Wi)x(t)+ξT(t)φ~iξ(t)<0,
where ξ(t) has already been defined on the above
(3.25)φ~i=(φ~i11φ~i12φ~i13φ~i14⋆φ~i22φ~i23φ~i24⋆⋆φ~i33φ~i34⋆⋆⋆φ~i44),i=1,…,N,
with the elements are the same as those in φi, except for
(3.26)φ~i11=PiAi+AiTPi+∑j∈Uki‍λij(Pj-Wi)+Q1i+N1Ai+AiTN1T,
and note that λii=-∑j∈Ukiλij and λij≥0 for all j≠i, namely, λii<0 for all i∈℘. Therefore, it follows from easy computation that if i∈Uki, inequalities (3.20) and the formula (3.25) less than 0 imply that
(3.27)LV(xt,t,i)<0.
On the other hand, for the same reason, if i∈Uuki, inequalities (3.20)-(3.21) and the formula (3.25) less than 0 also imply that inequality (3.27) holds. Therefore,
(3.28)ε{∫0∞‍∥x(t)∥2dt∣φ,r0}<∞,
which means that systems (2.1) with partly unknown transition probabilities are stochastically stable.
Note that the formula (3.25) less than 0 can be represented as follows:
(3.29)φ~i=Φi+∑j∈Uuki‍λ¯ijdiag(0,0,τ1jQ1i,τ2jQ2i)<0,
where λ¯ij is an unknown element in transition probabilities matrix, and
(3.30)Φi=(Φi11Φi12Φi13Φi14⋆Φi22Φi23Φi24⋆⋆Φi33Φi34⋆⋆⋆Φi44),
with
(3.31)Φi11=φ~i11,Φi12=φ~i12,Φi13=φ~i13,Φi14=φ~i14,Φi22=φ~i22,Φi23=φ~i23,Φi24=φ~i24,Φi34=φ~i34,Φi33=N3Bi+BiTN3T-Q1i+∑j∈Uki‍λijτ1jQ1i,Φi44=N4Ci+CiTN4T-Q2i+∑j∈Uki‍λijτ2jQ2i.
One can note that (3.29) can be separated into two cases, i∈Uki and i∈Uuki.
Case I (i∈Uki): it should be first noted that in this case one has λki<0. In fact, we only need to consider λki<0 because λki=0 means all the elements in the i throw of the transition rate matrix.
Now the formula (3.29) can be rewritten as
(3.32)φ~i=Φi+∑j∈Uuki‍λ¯ijdiag(0,0,τ1jQ1i,τ2jQ2i)=Φi-λki∑j∈Uuki‍λ¯ij-λkidiag(0,0,τ1jQ1i,τ2jQ2i)<0.
It follows from 0≤λ¯ij/-λki≤1 and ∑j∈Uuki‍(λ¯ij/-λki)=1 that
(3.33)φ~i=∑j∈Uuki‍λ¯ij-λki(Φi-λkidiag(0,0,τ1jQ1i,τ2jQ2i))<0.
Similar to the above proof, (3.2) and (3.3) can be rewritten as (3.16) and (3.18), respectively, for this case. Accordingly, for 0≤λ¯ij≤λki, φ~i<0 is equivalent to (3.14) which is satisfied for all j∈Uuki, which also implies that, in the presence of unknown elements λ¯ij, the system stability is ensured if (3.14), (3.16), (3.18), and (3.20) hold.
Case II (i∈Uuki): for the sake of simple expression, let Ψj=diag(0,0,τ1jQ1i,τ2jQ2i).
In this case,λ¯ii is unknown, λki≥0, and λ¯ii≤0, and following the same analysis of the above case, we just consider λ¯ii<-λki. And now the formula (3.29) can be rewritten as
(3.34)φ~i=Φi+λ¯iidiag(0,0,τ1iQ1,τ2iQ2)+∑j∈Uuki,j≠i‍λ¯ijdiag(0,0,τ1jQ1i,τ2jQ2i)=Φi+λ¯iidiag(0,0,τ1iQ1i,τ2iQ2i)+(-λ¯ii-λki)∑j∈Uuki,j≠i‍λ¯ij-λ¯ii-λkiΨj.
Similarly, since we have
(3.35)0≤λ¯ij-λ¯ii-λki≤1,∑j∈Uuki,j≠i‍λ¯ij-λ¯ii-λki=1,
one can know that
(3.36)φ~i=∑j∈Uuki‍λ¯ij-λ¯ii-λki(Φi+λ¯iiΨi+(-λ¯ii-λki)Ψj),
which means that φ~i<0 is equivalent to allj∈Uuki,j≠i,
(3.37)Φi+λ¯iiΨi+(-λ¯ii-λki)Ψj<0,
and from the defined λdi in Theorem 3.2, we have that λdi≤λ¯ii<-λki, which means that λ¯ii may take any value between [λdi,-λki+ε] for some ε<0 arbitrarily small. Then, λ¯ii can be further written as a convex combination
(3.38)λ¯ii=α(-λki+ε)+(1-α)λdi,
where α takes value arbitrarily in [0,1]. So, (3.37) holds if and only if forallj∈Uuki,j≠i,
(3.39)Φi-λkiΨi+ε(Ψi-Ψj)<0,(3.40)Φi+λdiΨi-λdiΨj-λkiΨi<0
simultaneously hold. Since ε is arbitrarily small, (3.39) holds if and only if
(3.41)Φi-λkiΨi<0,
which is the case in (3.40) when j=i for allj∈Uuki. Hence, (3.37) is equivalent to (3.15). Furthermore, following the same line of this proof, (3.2) and (3.3) can be represented as (3.17) and (3.19), respectively, in this case.
Therefore, from the above discussion, in the presence of unknown elements in the transition probabilities matrix, we can readily draw a conclusion that the system (2.1) with partly known transition rates is stable if the inequalities in Theorem 3.2 are satisfied. It completes this proof.Remark 3.3.
In order to obtain the less conservative stability criterion of MJSs with partial information on transition probabilities, similar to [32], the free-connection weighting matrices are introduced by making use of the relationship of the transition rates among various subsystems, that is, ∑j=1Nλij=0 for all i∈℘, which overcomes the conservativeness of using the fixed-connection weighting matrices. However, it is difficult to decrease the conservative using free-connection matrices only based on the above equalities, but not on the systems and the themselves Newton-Leibniz formula. Moreover, this paper is inspired by [30], and the delay-dependent stability results in this theorem are the extension of [30] to delay systems to some extent. Although the large number of introduced free weighting matrices may increase the complexity of computation, using the technique of free weighting matrices would reduce the conservativeness, which would be reflected in the fifth section.Remark 3.4.
It should be noted that the more known elements are there in (2.3), the lower the conservative of the condition will be. In other word, the more unknown elements are there in (2.3), the lower the maximum of time delay will be in Theorem 3.2. Actually, if all transition probabilities are unknown, the corresponding system can be viewed as a switched linear system under arbitrary switching. Thus, the conditions obtained in Theorem 3.2 will thereby cover the results for arbitrary switched linear system with mixed delays. In that case, one can see that the stability condition in Theorem 3.2 becomes seriously conservative, for many constraints. Fortunately, we can use the common Lyapunov functional method to analyze the stability for the system under the assumption that all transition probabilities are not known.For the stability analysis of the neutral markovian jump systems with all transition probabilities is not known, one can construct the following common Lyapunov functional:(3.42)V(xt,t,rt)=xT(t)Px(t)+∫t-τ1(rt)t‍xT(s)Q1x(s)ds+∫t-τ2(rt)t‍x˙T(s)Q2x˙(s)ds+∫-τ1(rt)0‍∫t+θt‍xT(s)R1x(s)dsdθ+∫-τ2(rt)0‍∫t+θt‍x˙T(s)R2x˙(s)dsdθ,
and following a similar line as in the proof of Theorem 3.1, we can obtain the following corollary.Corollary 3.5.
The system (2.1) with all elements unknown in transition rate matrix (2.3) is stochastically stable if there exist positive definite matrices P,Q1,Q2,R1>0,andR2>0 and any matrices Nk(k=1,…,4) with appropriate dimensions satisfying the following linear matrix inequalities:
(3.43)χi=(χi11χi12χi13χi14⋆χi22χi23χi24⋆⋆χi33χi34⋆⋆⋆χi44)<0,
with
(3.44)χi11=PAi+AiTP+Q1+N1Ai+AiTN1T+τ1iR1,χi12=-N1+AiTN2T,χi13=N1Bi+AiTN3T+PBi,χi14=N1Ci+AiTN4T+PCi,χi22=-N2-N2T+Q2+τ2iR2,χi23=N2Bi-N3T,χi24=N2Ci-N4T,χi33=N3Bi+BiTN3T-Q1,χi34=N3Ci+BiTN4T,χi44=N4Ci+CiTN4T-Q2.


## 4. Extension to Uncertain Neutral Markov Jump Systems

In this section, we will consider the uncertain neutral Markov jump systems with partially unknown transition probabilities as follows:(4.1)x˙(t)-C(rt)x˙(t-τ2(rt))=(A(rt)+ΔA(rt))x(t)+(B(rt)+ΔB(rt))x(t-τ1(rt)).A(rt),B(rt) are known mode-dependent constant matrices with appropriate dimensions, while ΔA(rt), ΔB(rt) are the time-varying but norm-bounded uncertainties satisfying
(4.2)(ΔA(rt)ΔB(rt))=LrtFrt(t)(H1(rt)H2(rt)),
where Lrt,H1(rt), and H2(rt) are known mode-dependent matrices with appropriate dimensions, and Frt(t) is the time-varying unknown matrix function with Lebesgue norm measurable elements satisfying FrtT(t)Frt(t)≤I.Theorem 4.1.
The uncertain neutral markovian jump system (4.1) with a partly unknown transition rate matrix (2.3) is stochastically stable if there exist matrices P~i>0,Q~1i>0,Q~2i>0,R~1>0, and R~2>0 and any matrices W~i=W~iT,M~i=M~iT,S~i=S~iT,F~k,G~k, and N~k(k=1,…,6) with appropriate dimensions satisfying the following linear matrix inequalities:
(4.3)ϕ~i=(ϕ~i11ϕ~i12ϕ~i13ϕ~i14H1iTP~iLi+N~1Li⋆ϕ~i22ϕ~i23ϕ~i240N~2Li⋆⋆ϕ~i33ϕ~i34H2iTN~3Li⋆⋆⋆ϕ~i440N~4Li⋆⋆⋆⋆-I0⋆⋆⋆⋆⋆-I)<0,∀j∈Uukiifi∈Uki,(4.4)ψ~i=(ψ~i11ψ~i12ψ~i13ψ~i14H1iTP~iLi+N~1Li⋆ψ~i22ψ~i23ψ~i240N~2Li⋆⋆ψ~i33ψ~i34H2iTN~3Li⋆⋆⋆ψ~i440N~4Li⋆⋆⋆⋆-I0⋆⋆⋆⋆⋆-I)<0,∀j∈Uukiifi∈Uuki,(4.5)∑j∈Uki‍λijQ~1j+∑j∈Uki‍λijτ1jR~1-R~1-λkiQ~1j-λkiτ1jR~1≤0,∀j∈Uukiifi∈Uki,(4.6)∑j∈Uki‍λijQ~1j+∑j∈Uki‍λijτ1jR~1-R~1+λdiQ~1i-λdiQ~1j-λkiQ~1j+λdiτ1iR~1-λdiτ1jR~1-λkiτ1jR~1≤0,∀j∈Uukiifi∈Uki,(4.7)∑j∈Uki‍λijQ~2j+∑j∈Uki‍λijτ2jR~2-R~2-λkiQ~2j-λkiτ2jR~2≤0,∀j∈Uukiifi∈Uki,(4.8)∑j∈Uki‍λijQ~2j+∑j∈Uki‍λijτ2jR~2-R~2+λdiQ~2i-λdiQ~2j-λkiQ~2j+λdiτ2iR~2-λdiτ2jR~2-λkiτ2jR~2≤0,∀j∈Uukiifi∈Uki,(4.9)P~j-W~i≤0,j∈Uuki,j≠i,(4.10)P~j-W~i≥0,j∈Uuki,j=i,
with
(4.11)ϕ~i11=P~iAi+AiTP~i+∑j∈Uki‍λij(P~j-Wi)+Q~1i+N~1Ai+AiTN~1T+τ1iR~1,ϕ~i12=-N~1+AiTN~2T,ϕ~i13=N~1Bi+AiTN~3T+P~iBi,ϕ~i14=N~1Ci+AiTN~4T+P~iCi,ϕ~i22=-N~2-N~2T+Q~2i+τ2iR~2,ϕ~i23=N~2Bi-N~3T,ϕ~i24=N~2Ci-N~4T,ϕ~i33=N~3Bi+BiTN~3T-Q~1i+∑j∈Uki‍λijτ1jQ~1i-λkiτ1jQ~1i,ϕ~i34=N~3Ci+BiTN~4T,ϕ~i44=N~4Ci+CiTN~4T-Q~2i+∑j∈Uki‍λijτ2jQ~2i-λkiτ2jQ~2i,ψ~i33=N~3Bi+BiTN~3T-Q~1i+∑j∈Uki‍λijτ1jQ~1i+λdiτ1iQ~1i-λdiτ1jQ~1i-λkiτ1jQ~1i,ψ~i44=N~4Ci+CiTN~4T-Q~2i+∑j∈Uki‍λijτ2jQ~2i+λdiτ2iQ~2i-λdiτ2jQ~2i-λkiτ2jQ~2i,ψ~i11=ϕ~i11,ψ~i12=ϕ~i12,ψ~i13=ϕ~i13,ψ~i14=ϕ~i14,ψ~i22=ϕ~i22,ψ~i23=ϕ~i23,ψ~i24=ϕ~i24,ψ~i34=ϕ~i34,
and λdi is a given lower bound for the unknown diagonal element.Proof.
ϕ
~
i
<
0 can be written as
(4.12)ϕ~i=ϕi+𝔻iTFi(t)𝔾iT+𝔾iFiT(t)𝔻i<0,
where
(4.13)𝔾iT=(LiTPi+LiTN1TLiTN2TLiTN3TLiTN4T),𝔻i=(H1i0H2i0),
and ϕi are defined in Theorem 3.2. According to the approach in [40] with Lemma 2.4 in [41], there exists a scalar ε such that (4.12) are equivalents to
(4.14)φ~i=εφi+ε2𝔻iT𝔻i+𝔾i𝔾iT<0.
Introducing new variablesP~i=εPi,  Q~1i=εQ1i,Q~2i=εQ2i,R~1=εR1,R~2=εR2, W~i=εWi, and N~k=εNk(k=1,…,4), and with Schur's complement [42], yields inequalities (4.3). Similarly, it concludes (4.4) with the same proof. On the other hand, with the same variables substitution, we note that pre- and postmultiplying, respectively, (3.16)–(3.21) by a scalar ε yield (4.5)–(4.10), which completes this proof.Remark 4.2.
It should be mentioned that Theorem4.1 is an extension of (2.1) to uncertain neutral markovian jump systems (4.1) with incomplete transition descriptions. In fact, this technology is frequently adopted in dealing with the robust stability analysis of uncertain systems.


## 5. Examples

In order to show the effectiveness of the approaches presented in the above sections, two numerical examples are provided.Example 5.1.
Consider the MJLS (2.1) with four operation modes whose state matrices are listed as follows:
(5.1)A1=(-1.1500-1.5),A2=(-3.1500-7.1),A3=(-1.3800-1.8),A4=(-2.9500-3.65),B1=(0.02-0.020.04-0.05),B2=(-0.05-0.060.07-0.07),B3=(0.08-0.08-0.030.06),B4=(0.07-0.050.090.03),C1=(0.29000.36),C2=(0.3500-0.07),(5.2)C3=(0.03000.06),C4=(0.280.010.020.07).
The partly transition rate matrix Π is considered as
(5.3)CaseI:Π=[-1.30.2????0.30.30.6?-1.5?0.4???],CaseII:Π=[-1.30.20.60.50.2-0.80.30.30.60.7-1.50.20.40.10.4-0.9].
Our purpose here is to check the stability of the above system for the two different cases of transition probabilities. For Case I, it is clear to see thatλ22 and λ44 are not valued, one can set λd2=-0.8,λd2=-0.9, and let τ12=1.1000,τ13=0.8000,τ14=0.9900,τ21=τ11,τ22=τ12,τ23=τ13, and τ24=τ14. Solving the inequalities in Theorem 3.2 using LMI toolbox, the maximum of the time delay τ11 can be computed as τ11=1.1210. However, in Case II, the maximum of the time delay τ11 can be computed as τ11=2.0530 by Theorem 3.1. It is easily seen that the more transition probabilities knowledge we have, the larger the maximum of delay can be obtained for ensuring stability. This shows the trade-off between the cost of obtaining transition probabilities and the system performance.
Furthermore, when the transition probabilities are not fully known, as the delay for one of the subsystems decreases, the maximum of other delays may increase. However, when all transition probabilities are fully known, the conclusion may be on the opposite in some interval. In fact, the above observation is in accordance with the actual. Then, we assume thatλd2=-0.8,λd2=-0.9, and let τ12=1.1000,τ14=0.9900,τ21=τ11,τ22=τ12,τ23=τ13, and τ24=τ14, τ13 be different with τ11, and with the same computation in Theorem 3.1, as shown in Table 1. However, just according to the approach of Theorem 3 in [32], not introducing some other free matrices and some other skills, we cannot find the feasible solutions which contain time delay to verify the stability of the system. Therefore, this example shows that the stability criterion in this paper gives much less conservative delay-dependent stability conditions. This example also shows that the approach presented in this paper is effectiveness.Table 1
The maximal allowable delaysτ11 of Example 5.1 for different values of τ12 in a different case.
Case
I
II
τ
13
0.80
0.75
0.70
0.65
0.60
0.75
0.80
0.85
0.90
0.95
τ
11
1.1210
1.1960
1.2710
1.3460
1.4210
1.9280
2.0530
2.1780
2.3030
2.4279Example 5.2.
Consider the above MJLS in Example5.1 with partially unknown transition probabilities of Case I, and the uncertain structure matrices are described by (4.2) where
(5.4)L1=(0.10.2),L2=(0.2-0.4),L3=(0.40.3),L4=(0.250.27),H11=(0.721.14),H12=(1.221.66),H13=(1.41-1.13),H14=(-1.391.63),H21=(-1.111.32),H22=(1.311.14),H23=(1.271.15),H24=(-1.261.37).
In this case, we check the robust stability result provided by Theorem 4.1. One can also set λd2=-0.8,λd2=-0.9, and let τ12=1.1000,τ13=0.7500,τ14=0.9900,τ21=τ11,τ22=τ12,τ23=τ13, and τ24=τ14. Solving the inequalities in Theorem 4.1 by using LMI toolbox, the maximum of the time delay τ11 can be computed as τ11=1.0892. Some of the feasible solutions can be obtained as follows:
(5.5)P1=(6.6762-1.8338-1.83389.5418),P2=(6.6930-1.8283-1.82839.5446),P3=(10.7916-5.0166-5.016612.0252),P4=(10.7923-5.0178-5.017812.0274),W1=(10.7928-270.5249260.488012.0285),W2=(6.6915371.7539-375.41159.5439),W3=(10.794382.9434-92.986112.0338),W4=(10.7920-212.1720202.137412.0265),(5.6)N4=(-0.30690.13420.05480.1402).
In a word, this example shows that the robust stability condition of Theorem 4.1 is feasible. It is also approved that the approach provided in this paper is effectiveness.Example 5.3.
Consider the MJLS (2.1) with two operation modes whose state matrices are listed as follows:
(5.7)A1=(-2.160.05-0.15-1.35),A2=(-3.13-0.190.21-2.21),B1=(0.22-0.020.54-0.05),B2=(0.43-0.160.271.07),C1=(-0.08-0.060.030.04),C2=(0.050.310.23-0.06).
The partly transition rate matrix Π is considered as
(5.8)Π=[????].
The above matrix (5.8) implies that systems (2.1) with all transition probabilities are not known, which viewed the systems as  switched systems with arbitrary switching. Given that τ11=2.4279,τ12=1.1000,τ13=0.95,τ14=0.9900,τ21=τ11,τ22=τ12,τ23=τ13, and τ24=τ14, according to the approach of Theorem 3.2, we cannot find the feasible solutions. However, using the Matlab LMI toolbox, we solve the LMI in Corollary 3.5, and the feasible solutions can be obtained as follows:
(5.9)P=(15.93640.74910.749111.2785),R1=(2.76190.23820.23821.5829),R2=(0.99400.09740.09740.5010),Q1=(6.86450.31730.31737.3064),Q2=(4.31090.17520.17523.8836).
For the case of all transition probabilities that are unknown, this example also shows that Corollary 3.5 is less conservative than Theorem 3.2 on the stability analysis for the neutral markovian jump system.


## 6. Conclusion

The delay-dependent stability for neutral markovian jump systems with partly known transition probabilities has been investigated. Based on a new class of stochastic Lyapunov-Krasovskii functionals constructed, and combined with the technique of analysis for matrix inequalities, some new stability criteria are obtained. The main contribution of this paper contains the following two-fold: one is the extension of delay-dependent stability conditions for markovian jump delay systems to markovian neutral jump systems; the other is the new method presented to decrease the conservative brought by the markovian jump with partly known transition probabilities. The future work is to investigate the systems with mode-dependent interval mixed time delays and the systems with unsynchronised control input. Three examples have shown the effectiveness of the conditions presented in this paper.



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*Source: 450168-2012-10-24.xml*