# Observer-Based Finite-TimeH∞ Control of Singular Markovian Jump Systems

Authors: Yingqi Zhang; Wei Cheng; Xiaowu Mu; Xiulan Guo
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper addresses the problem of finite-timeH∞ control via observer-based state feedback for a family of
singular Markovian jump systems (SMJSs) with time-varying norm-bounded disturbance. Firstly, the concepts of
singular stochastic finite-time boundedness and singular stochastic finite-time H∞ stabilization via observer-based
state feedback are given. Then an observer-based state feedback controller is designed to ensure singular stochastic
finite-time H∞ stabilization via observer-based state feedback of the resulting closed-loop error dynamic SMJS.
Sufficient criteria are presented for the solvability of the problem, which can be reduced to a feasibility problem
involving linear matrix inequalities with a fixed parameter. As an auxiliary result, we also discuss the problem
of finite-time stabilization via observer-based state feedback of a class of SMJSs and give sufficient conditions of
singular stochastic finite-time stabilization via observer-based state feedback for the class of SMJSs. Finally, illustrative
examples are given to demonstrate the validity of the proposed techniques.


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## Body

## 1. Introduction

In practice, there exist many concerned problems which described that system state does not exceed some bound during some time interval, for instance, large values of the state are not acceptable in the presence of saturations [1–3]. Therefore, we need to check the unacceptable values to see whether the system states remain within the prescribed bound in a fixed finite-time interval. Compared with classical Lyapunov asymptotical stability, in order to deal with these transient performances of control dynamic systems, finite-time stability or short-time stability was introduced in the literatures [4, 5]. Applying Lyapunov function approach, some appealing results were obtained to ensure finite-time stability, finite-time boundedness, and finite-time stabilization of various systems including linear systems, nonlinear systems, and stochastic systems. For instance, Amato et al. [6] investigated the output feedback finite-time stabilization for continuous linear systems. Zhang and An [7] considered finite-time control problems for linear stochastic systems. Recently, Meng and Shen [8] extended the definition of H∞ control to finite-time H∞ control for linear continuous systems, and a state feedback controller was designed to ensure finite-time boundedness of the resulting systems and the effect of the disturbance input on the controlled output satisfying a prescribed level. For more details of the literature related to finite-time stability, finite-time boundedness, and finite-time H∞ control, the reader is referred to [9–20] and the references therein.On the other hand, singular systems referred to as descriptor systems, differential-algebraic systems, generalized state-space systems, or semistate systems have attracted many researchers since the class of systems have been extensively applied to deal with mechanical systems, electric circuits, chemical process, power systems, interconnected systems, and so on; see more practical examples in [21, 22] and the references therein. A great number of results based on the theory of regular systems or state-space systems have been extensively generalized to singular systems with or without time delay, such as stability [23], stabilization [24], H∞ control [25–29], and other issues. Meanwhile, Markovian jump systems are referred to as a special family of hybrid systems and stochastic systems, which are very appropriate to model plants whose parameters are subject to random abrupt changes [30]. Thus, many attracting results and a large variety of control problems have been studied, such as stochastic Lyapunov stability [31–33], sliding mode control [34, 35], robust control [36–40], H∞ filtering [41–45], dissipative control [46], passive control [47], guaranteed cost control [48], tracking control [49], and other issues, the readers are refered to [31] and the references therein. It is pointed out that the problem of state feedback stabilization, just as was mentioned above, requires to assume the complete access to the state vector. Practically this assumption is not realistic for many reasons like the nonexistence of the appropriate sensors to measure some of the states or the limitation in the control strategies. Thus, the observer-based control and output feedback control are probably well suited in such situation for feedback control, such as stability [31], H∞ control [50–54], passive control [55], and finite-time control [6, 20]. However, to date, the problems of observer-based finite-time stabilization of singular stochastic systems have not been investigated. The problems are important and challenging in many practice applications, which motivates the main purpose of our research.In this paper, we consider the problem of finite-timeH∞ control via observer-based state feedback of singular Markovian jump systems (SMJSs) with time-varying norm-bounded disturbance. The results of this paper are totally different from those previous results, although some studies on finite-time control for singular stochastic systems have been conduced, see [18, 56, 57]. The concepts of singular stochastic finite-time boundedness (SSFTB) and singular stochastic finite-time H∞ stabilization via observer-based state feedback of singular stochastic systems are given. The main contribution of the paper is to design an observer-based state feedback controller which ensures singular stochastic finite-time H∞ stabilization via observer-based state feedback of the resulting closed-loop error dynamic SMJS. Sufficient criterions are presented for the solvability of the problem, which can be reduced to a feasibility problem in terms of linear matrix inequalities with a fixed parameter. As an auxiliary result, we also investigate the problem of observer-based finite-time stabilization via state feedback of a class of SMJSs and give sufficient conditions of singular stochastic finite-time stabilization via observer-based state feedback for the class of SMJSs.The rest of this paper is organized as follows. In Section2, the problem formulation and some preliminaries are introduced. The results of singular stochastic finite-time H∞ stabilization via observer-based state feedback are given for a class of SMJSs in Section 3. Section 4 presents numerical examples to show the validity of the proposed methodology. Some conclusions are drawn in Section 5.Notations 1.
Throughout the paper,ℝn and ℝn×m denote the sets of n component real vectors and n×m real matrices, respectively. The superscript T stands for matrix transposition or vector. 𝔼{·} denotes the expectation operator respective to some probability measure ℙ. In addition, the symbol * denotes the transposed elements in the symmetric positions of a matrix, and diag{⋯} stands for a block-diagonal matrix. λmin(P) and λmax(P) denote the smallest and the largest eigenvalues of matrix P, respectively. Notations sup and inf denote the supremum and infimum, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.


## 2. Problem Formulation

Let us consider the following continuous-time singular Markovian jump system (SMJS):(2.1a)E(rt)ẋ(t)=A(rt)x(t)+B(rt)u(t)+G(rt)w(t),(2.1b)z(t)=C(rt)x(t)+D1(rt)u(t)+D2(rt)w(t),(2.1c)y(t)=Cy(rt)x(t),where x(t)∈ℝn is the state variable, z(t)∈ℝl is the controlled output, y(t)∈ℝq is the measured output, w(t)∈ℝp is the controlled input, z(t)∈ℝp is the controlled output, E(rt) is a singular matrix with rank(E(rt))=rrt<n, {rt,t≥0} is continuous-time Markov stochastic process taking values in a finite space 𝕄:={1,2,…,N} with transition matrix Γ=(πij)N×N, and the transition probabilities are described as follows: (2.2)Pr(rt+Δt=j∣rt=i)={πijΔt+o(Δt)ifi≠j,1+πiiΔt+o(Δt)ifi=j,
where limΔ→0o(Δt)/Δt=0, πij satisfies πij≥0(i≠j), and πii=-∑j=1,j≠iNπij for all i,j∈𝕄. Moreover, the disturbance w(t)∈ℝp satisfies the following constraint condition: (2.3)∫0TwT(t)w(t)dt≤d2,d≥0,
and the matrices A(rt),B(rt),G(rt),C(rt),D1(rt), and D2(rt) are coefficient matrices and of appropriate dimension for all rt∈𝕄.For notational simplicity, in the sequel, for each possiblert=i,i∈𝕄, a matrix K(rt) will be denoted by Ki; for instance, A(rt) will be denoted by Ai, B(rt) by Bi, and so on.In this paper, we construct the following state observer and feedback controller:(2.4a)Eix̃̇(t)=Aix̃(t)+Biu(t)+Hi(y(t)-ỹ(t)),(2.4b)ỹ(t)=Cyix̃(t),(2.4c)u(t)=Kix̃(t),where x̃(t) and ỹ(t) are the estimated state and output, x̃(0) is an estimated initial state, Ki is to be a designed state feedback gain, and Hi is an observer gain to be designed. Define the state estimated error e(t)=x(t)-x̃(t) and x̅(t)=[xT(t)eT(t)]T. Then the resulting closed-loop error dynamic SMJS can be written in the form as follows: (2.5a)E̅ix̃̇(t)=A̅ix̅(t)+G̅iw(t),(2.5b)z(t)=C̅ix̅(t)+D2iw(t),where
(2.6)E̅i=[Ei0*Ei],A̅i=[Ai+BiKi-BiKi0Ai-HiCyi],G̅i=[GiGi],C̅i=[Ci+D1iKi-D1iKi].Definition 2.1 (regular and impulse-free, see [21, 22]).
The SMJS (2.1a) with u(t)=0 is said to be regular in time interval [0,T] if the characteristic polynomial det(sEi-Ai) is not identically zero for all t∈[0,T]. The SMJS (2.1a) with u(t)=0 is said to be impulse-free in time interval [0,T] if deg(det(sEi-Ai))=rank(Ei) for all t∈[0,T].Definition 2.2 (singular stochastic finite-time stability (SSFTS)).
The SMJS (2.1a) with w(t)=0 is said to be SSFTS with respect to (c1,c2,T,Ri), with c1<c2 and Ri>0, if the stochastic system is regular and impulse-free in time interval [0,T] and satisfies
(2.7)E{xT(0)EiTRiEix(0)}≤c12⟹E{xT(t)EiTRiEix(t)}<c22,∀t∈[0,T].Definition 2.3 (singular stochastic finite-time boundedness (SSFTB)).
The SMJS (2.1a) which satisfies (2.3) is said to be SSFTB with respect to (c1,c2,T,Ri,d), with c1<c2 and Ri>0, if the stochastic system is regular and impulse-free in time interval [0,T], and condition (2.7) holds.Remark 2.4.
The definition of SSFTB is the generalization of finite-time boundedness [1]. SSFTB implies that the whole mode of the singular stochastic system is finite-time bounded since the static mode is regular and impulse-free.Definition 2.5 (singular stochastic finite-time stabilization via observer-based state feedback).
The error dynamic SMJS (2.5a) and (2.5b) is said to be singular stochastic finite-time stabilization via observer-based state feedback with respect to (c1,c2,T,R̅i,d), with c1<c2 and R̅i>0, if there exists a state feedback control law and a state observer in the form (2.4a)–(2.4c), such that the error dynamic SMJS (2.5a) and (2.5b) is regular and impulse-free in time interval [0,T] and satisfies the following constraint relation:
(2.8)E{x̅T(0)E̅iTR̅iE̅ix̅(0)}≤c12⟹E{x̅T(t)E̅iTR̅iE̅ix̅(t)}<c22,∀t∈[0,T].Definition 2.6 (see [30, 33]).
LetV(x(t),rt=i,t≥0) be the stochastic function, and define its weak infinitesimal operator J of stochastic process {(x(t),rt=i),t≥0} by
(2.9)JV(x(t),rt=i,t)=lim⁡Δt→01Δt{E{V(x(t+Δt),rt+Δt,t+Δt)∣x(t)=x,rt=i}-V(x(t),i,t)}=Vt(x(t),i,t)+Vx(x(t),i,t)ẋ(t,i)+∑j=1NπijV(x(t),j,t).Definition 2.7 (singular stochastic finite-timeH∞ stabilization via observer-based state feedback).
The closed-loop error dynamic SMJS (2.5a) and (2.5b) is said to be singular stochastic finite-time H∞ stabilization via observer-based state feedback with respect to (c1,c2,T,R̅i,γ,d), with c1<c2 and R̅i>0, if there exists a state observer and feedback controller in the form (2.4a)–(2.4c), such that the error dynamic SMJS (2.5a) and (2.5b) is SSFTB with respect to (c1,c2,T, R̅i,d), and under the zero-initial condition, the controlled output z satisfies
(2.10)E{∫0TzT(t)z(t)dt}<γ2E{∫0TwT(t)w(t)dt},
for any nonzero w(t) which satisfies (2.3), where γ is a prescribed positive scalar.
The main objective of this paper being to concentrate on designing a state observer and feedback controller of the form (2.4a)–(2.4c) that ensures singular stochastic finite-time H∞ stabilization via observer-based state feedback of the error dynamic SMJS (2.5a) and (2.5b), we require the following lemmas.Lemma 2.8 (Schur complement lemma, see [57, 58]).
The linear matrix inequalityS=[S11S12*S22]<0 is equivalent to S22<0 and S11-S12S22-1S12T<0 with S11=S11T and S22=S22T.Lemma 2.9 (see [57]).
The following items are true.
(i) Assume thatrank(E)=r, then there exist two orthogonal matrices U and V such that E has the decomposition as
(2.11)E=U[Σr0*0]VT=U[Ir0*0]VT,
where Σr=diag{δ1,δ1,…,δr} with δk>0 for all k=1,2,…,r. Partition U=[U1U2], V=[V1V2], and 𝒱=[V1ΣrV2] with EV2=0 and U2TE=0.
(ii)If P satisfies
(2.12)EPT=PET≥0,
then P̃=UTP𝒱 with U and 𝒱 satisfying (2.11) if and only if
(2.13)P̃=[P11P120P22],
with P11≥0∈ℝr×r. In addition, when P is nonsingular, one has P11>0 and det(P22)≠0. Furthermore, P satisfying (2.12) can be parameterized as
(2.14)P=EV-TXV-1+UZV2T,
where X=diag{P11,Ψ}, Z=[P12TP22T]T, and Ψ∈ℝ(n-r)×(n-r) is an arbitrary parameter matrix.
(iii) IfP is a nonsingular matrix, R and Ψ are two symmetric positive definite matrices, P and E satisfy (2.12), X is a diagonal matrix from (2.14), and the following equality holds:
(2.15)P-1E=ETR1/2SR1/2E.
Then the symmetric positive definite matrix S=R-1/2UX-1UTR-1/2 is a solution of (2.15).


## 3. Main Results

In this section, LMI conditions are established to design a state observer and feedback controller that guarantees the error dynamic SMJS of the class we are considering is singular stochastic finite-timeH∞ stabilization via observer-based state feedback.Theorem 3.1.
The error dynamic SMJS (2.5a) and (2.5b) is singular stochastic finite-time stabilization via observer-based state feedback with respect to (c1,c2,T,R̅i,d) if there exist positive scalars α,c2, a set of mode-dependent nonsingular matrices {P̅i,i∈𝕄}, and sets of mode-dependent symmetric positive-definite matrices {S̅i,i∈𝕄}, {Θi,i∈𝕄}, and for all i∈𝕄, such that the following inequalities hold: (3.1a)P̅iE̅iT=E̅iP̅iT≥0,(3.1b)[A̅iP̅iT+P̅iA̅iT+∑j=1NπijP̅iP̅j-1E̅jP̅iT-αE̅iP̅iTG̅i*-Θi]<0,(3.1c)P̅i-1E̅i=E̅iTR̅i1/2S̅iR̅i1/2E̅i,(3.1d)λ̅1c12+λ̅2d2<c22λ̲1e-αT,where λ̅1=supi∈M{λmax(S̅i)}, λ̲1=infi∈M{λmin(S̅i)}, and λ̅2=supi∈𝕄{λmax(Θi)}.Proof.
Firstly, we prove that the error dynamic SMJS (2.5a) and (2.5b) is regular and impulse-free in time interval [0,T]. Applying Lemma 2.8, condition (3.1b) implies
(3.2)A̅iP̅iT+P̅iA̅iT+(πii-α)E̅iP̅iT<-∑j=1,j≠iNπijP̅iP̅j-1E̅jP̅iT≤0.
Now, there exist two orthogonal matrices U̅i and V̅i such that E̅i has the decomposition as
(3.3)E̅i=U̅i[Σ̅r̅i0*0]V̅iT=U̅i[Ir̅i0*0]V̅iT,
where Σ̅r̅i=diag{δ̅i1,δ̅i2,…,δ̅i̅ri} with δ̅k>0 for all k=1,2,…,r̅i. Partition U̅i=[U̅i1U̅i2], V̅i=[V̅i1V̅i2], and 𝒱̅i=[V̅i1Σ̅r̅V̅i2] with E̅iV̅i2=0 and U̅i2TE̅i=0. Denote
(3.4)U̅iTA̅iV̅i-T=[A̅11iA̅12iA̅21iA̅22i],U̅iTP̅iV̅i=[P̅11iP̅12iP̅21iP̅22i].
Noting that condition (3.1a) and P̅i is a nonsingular matrix, by Lemma 2.9, we have P̅21i=0 and det(P̅22i)≠0. Before and after multiplying (3.2) by U̅iT and U̅i, respectively, this results in that the following matrix inequality holds:
(3.5)[⋆⋆⋆A̅22iP̅22iT+P̅22iA̅22iT]<0,
where the star⋆ will not be used in the following discussion. By Lemma 2.8, we have A̅22iP̅22iT+P̅22iA̅22iT<0. Therefore, A̅22i is nonsingular, which implies that the error dynamic SMJS (2.5a) and (2.5b) is regular and impulse-free in time interval [0,T].
For the given mode-dependent nonsingular matrixP̅i, let us consider the following quadratic function as:
(3.6)V(x̅(t),i)=x̅T(t)P̅i-1E̅ix̅(t).
Computing the weak infinitesimal operator J emanating from the point (x̅,i) at time t along the solution of error dynamic SMJS (2.5a) and (2.5b) and noting the condition (3.1a), we obtain
(3.7)JV(x̅(t),i)=ξT(t)[P̅i-1A̅i+A̅iTP̅i-T+∑j=1NπijP̅j-1E̅jP̅i-1G̅i*0]ξ(t),
where ξ(t)=[x̅T(t),wT(t)]T. Before and after multiplying (3.1b) by diag{P̅i-1,I} and diag{P̅i-T,I}, respectively, this results in the following matrix inequality
(3.8)[P̅i-1A̅i+A̅iTP̅i-T+∑j=1NπijP̅j-1E̅j-αP̅i-1E̅iP̅i-1G̅i*-Θi]<0.
From (3.7) and (3.8), we can obtain
(3.9)JV(x̅(t),i)<αV(x̅(t),i)+wT(t)Θiw(t).
Further, (3.9) can be rewritten as
(3.10)E{J[e-αtV(x̅(t),i)]}<E{e-αtwT(t)Θiw(t)}.
Integrating (3.10) from 0 to t, with t∈[0,T], we obtain
(3.11)e-αtE{V(x̅(t),i)}<E{V(x̅(0),i=r0)}+∫0tE{e-ατwT(τ)Θiw(τ)}dτ.
Noting α≥0,t∈[0,T], and condition (3.1c), we have
(3.12)E{x̅T(t)P̅i-1E̅ix̅(t)}=E{V(x̅(t),i)}<eαt{E{V(x̅(0),i=r0)}+∫0tE{e-ατwT(τ)Θiw(τ)}dτ}≤eαt(λ̅1c12+λ̅2d2).
Taking into account that
(3.13)E{x̅T(t)P̅i-1E̅ix(t)}=E{x̅T(t)E̅iTR̅i1/2S̅iR̅i1/2E̅ix̅(t)}≥λ̲1E{x̅T(t)E̅iTR̅iE̅ix̅(t)},
we obtain
(3.14)E{x̅T(t)E̅iTR̅iE̅ix̅(t)}<eαTλ̅1c12+λ̅2d2λ̲1.
Therefore, it follows that condition (3.1d) implies 𝔼{x̅T(t)E̅iTR̅iE̅ix̅(t)}<c22 for all t∈[0,T]. This completes the proof of the theorem.Theorem 3.2.
The error dynamic SMJS (2.5a) and (2.5b) is singular stochastic finite-time H∞ stabilization via observer-based state feedback with respect to (c1,c2,T,R̅i,γ,d) if there exist positive scalars α,c2,γ, a set of mode-dependent nonsingular matrices {P̅i,i∈𝕄}, and a set of mode-dependent symmetric positive-definite matrices {S̅i,i∈𝕄}, and for all i∈𝕄, such that (3.1a), (3.1c), and the following inequalities: (3.15a)[Ξ̅i+P̅iC̅iTC̅iP̅iTP̅iC̅iTD2i+G̅i*-γ2e-αTI+D2iTD2i]<0,(3.15b)λ¯1c12eαT+γ2d2<c22λ̲1 hold, where Ξ̅i=A̅iP̅iT+P̅iA̅iT+∑j=1NπijP̅iP̅j-1E̅jP̅iT-αE̅iP̅iT.Proof.
Note that(3.16)[P̅iC̅iTC̅iP̅iTP̅iC̅iTD2i*D2iTD2i]=[P̅iC̅iTD2iT][C̅iP̅iTD2i]≥0.
Thus, condition (3.15a) implies that
(3.17)[A̅iP̅iT+P̅iA̅iT+∑j=1NπijP̅iP̅j-1E̅jP̅iT-αE̅iP̅iTG̅i*-γ2e-αTI]<0.
Let Θi=-γ2e-αTI for all i∈𝕄, by Theorem 3.1, conditions (3.1a), (3.1c), (3.15b), and (3.17) can guarantee that the error dynamic SMJS (2.5a) and (2.5b) is singular stochastic finite-time stabilization via observer-based state feedback with respect to (c1,c2,T,R̅i,d). Therefore, we only need to prove that the constraint relation (2.10) holds. Let us choose the Lyapunov-Krasovskii function V(x̅(t),i) in the form (3.6) in Theorem 3.1 and noting (3.7) and (3.15a), we obtain
(3.18)JV(x̅(t),i)<αV(x̅(t),i)+γ2e-αTwT(t)w(t)-zT(t)z(t).
Further, (3.18) can be represented as
(3.19)J[e-αtV(x̅(t),i)]<e-αt[γ2e-αTwT(t)w(t)-zT(t)z(t)].
Integrating (3.19) from 0 to T and noting that under-zero initial condition, we have
(3.20)∫0Te-αt[zT(t)z(t)-γ2e-αTwT(t)w(t)]dt<-∫0TJ[e-αtV(x̅(t),i)]dt≤V(x̅(0),r0)=0.
Using the Dynkin formula, it results that
(3.21)E{∫0Te-αt[zT(t)z(t)-γ2e-αTwT(t)w(t)]wt}<0.
Thus, for all t∈[0,T] and under-zero initial condition, we have
(3.22)E{∫0TzT(t)z(t)dt}≤eαTE{∫0Te-αtzT(t)z(t)dt}<eαTE{∫0Tγ2e-α(t+T)wT(t)w(t)dt}≤γ2E{∫0TwT(t)w(t)dt}.
This completes the proof of the theorem.LetP̅i=diag{Pi,Pi}, S̅i=diag{Si,Si}, and R̅i=diag{Ri,Ri}, then the following theorem gives LMI conditions to ensure singular stochastic finite-time H∞ stabilization via observer-based state feedback of the error dynamic SMJS (2.5a) and (2.5b).Theorem 3.3.
There exist a state feedback controlleru(t)=Kix̃(t) with Ki=YiPi-T and a state observer Hi=-PiCyiT such that the error dynamic SMJS (2.5a) and (2.5b) is singular stochastic finite-time H∞ stabilization via observer-based state feedback with respect to (c1,c2,T,R̅i,γ,d) if there exist positive scalars α,c2,γ,σ1, and sets of mode-dependent symmetric positive-definite matrices {Xi,i∈𝕄}, {Φi,i∈𝕄}, sets of mode-dependent matrices {Yi,i∈𝕄}, {Zi,i∈𝕄}, and for all rt=i∈𝕄, such that the following inequalities hold: (3.23a)0≤PiEiT=EiPiT=EiVi-TXiVi-1EiT≤Φi,(3.23b)[Π11iΠ12i*Π22i]<0,(3.23c)σ1Ri-1<UiXiUiT<Ri-1,(3.23d)[e-αT(-c22+γ2d2)c1*-σ1]<0,where
(3.24)Π11i=[L1(Pi,Yi)-BiYiGi*L2(Pi,Yi)Gi**-γ2e-αTI],Π12i=[PiCiT+YiTD1iT0Υi0-YiTD1iTPiCyiT0ΥiD2iT000],Π22i=-diag⁡{I,I2,Wi,Wi},L1(Pi,Yi)=PiAiT+AiPiT+BiYi+YiTBiT+(πii-α)EiPiT,L2(Pi,Yi)=PiAiT+AiPiT+(πii-α)EiPiT,Υi=[πi1Pi,…,πi(i-1)Pi,πi(i+1)Pi,…,πiNPi],Wi=diag⁡{P1T+P1-Φ1,…,Pi-1T+Pi-1-Φi-1,Pi+1T+Pi+1-Φi+1,…,PNT+PN-ΦN}.
In addition, the form of Pi=Ei𝒱i-TXi𝒱i-1+UiZiVi2T is from (3.36).Proof.
We firstly prove that condition (3.23b) implies condition (3.15a). Let P̅i=diag{Pi,Pi}, S̅i=diag{Si,Si}, and R̅i=diag{Ri,Ri}, then conditions (3.1a), (3.1c), and (3.1d) are equivalent to(3.25a)PiEiT=EiPiT≥0,(3.25b)Pi-1Ei=EiTRi1/2SiRi1/2Ei,(3.25c)κ¯1c12eαT+γ2d2<c22κ̲1,where κ̅1=supi∈M{λmax(Si)} and κ̲1=infi∈M{λmin(Si)}. By condition (3.23a), we have
(3.26)Pj-1Ej≤Pj-1ΦjPj-T,∀j∈M.
Thus, the inequality
(3.27)∑j=1,j≠iNπijPiPj-1EjPiT≤∑j=1,j≠iNπijPiPj-1ΦjPj-TPiT≤ΥiVi-1ΥiT
holds, where
(3.28)Υi=[πi,1Pi,…,πi,i-1Pi,πi,i+1Pi,…,πi,NPi],Vi=diag⁡{P1TΦ1-1P1,…,Pi-1TΦi-1-1Pi-1,Pi+1TΦi+1-1Pi+1,…,PNTΦN-1PN}.
Noting that the inequality
(3.29)PjTΦj-1Pj≥PjT+Pj-Φj
holds for each j∈𝕄, thus,
(3.30)∑j=1,j≠iNπijPiPj-1EjPiT≤ΥiWi-1ΥiT,
where Wi=diag{P1T+P1-Φ1,…,Pi-1T+Pi-1-Φi-1,Pi+1T+Pi+1-Φi+1,…,PNT+PN-ΦN}. Let Υ̅i=diag{Υi,Υi} and W̅i=diag{Wi,Wi}, then a sufficient condition for (3.15a) to guarantee is that
(3.31)[Θ̅i+Υ̅iW̅i-1Υ̅iT+P̅iC̅iTC̅iP̅iTP̅iC̅iTD2i+G̅i*-γ2e-αTI+D2iTD2i]<0,
where Θ̅i=A̅iP̅iT+P̅iA̅iT+(πii-α)E̅iP̅iT. Noting the forms of A̅i,B̅i,C̅i,andG̅i, then the inequality (3.31) is equivalent to the following:
(3.32)[Λ11i-BiKiPiTGiPiCiT+PiKiTD1iT*Λ22iGi-PiKiTD1iT**-γ2e-αTID2iT***-I]<0,
where
(3.33)Λ11i=PiAiT+AiPiT+BiKiPiT+PiKiTBiT+ΥiWi-1ΥiT+(πii-α)EiPiT,Λ22i=PiAiT+AiPiT-HiCyiPiT-PiCyiTHiT+ΥiWi-1ΥiT+(πii-α)EiPiT.
Let Hi=-PiCyiT, we obtain
(3.34)Λ22i=PiAiT+AiPiT+2PiCyiTCyiPiT+ΥiWi-1ΥiT+(πii-α)EiPiT.
Letting Yi=KiPiT and applying Lemma 2.8, it follows that (3.32) is equivalent to (3.23b).
Noting thatPi is nonsingular matrix, by Lemma 2.9, there exist two orthogonal matrices Ui and Vi, such that Ei has the decomposition as
(3.35)Ei=Ui[Σri0*0]ViT=Ui[Iri0*0]ViT,
where Σri=diag{δi1,δi1,…,δiri} with δik>0 for all k=1,2,…,ri. Partition Ui=[Ui1Ui2], Vi=[Vi1Vi2], and 𝒱i=[Vi1ΣriVi2] with EiVi2=0 and Ui2TEi=0. Let P̃i=UiTPi𝒱i, from (3.23a), P̃i is of the following form [P11iP12i0P22i], and Pi can be expressed as
(3.36)Pi=EiVi-TXiVi-1+UiZiVi2T,
where Zi=[P12iTP22iT]T and Xi=diag{P11i,Ψi} with a parameter matrix Ψi. If we choose Ψi as a symmetric positive definite matrix, then Xi is a symmetric positive definite matrix. Thus, Si=Ri-1/2UiXi-1UiTRi-1/2 is a solution of (3.25b), and Pi satisfies PiEiT=EiPiT=Ei𝒱i-TXi𝒱i-1EiT.
LetI<Si<σ1-1I, then it is easy to check that conditions (3.23c) and (3.23d) can guarantee that conditions (3.25c) hold. This completes the proof of the theorem.Corollary 3.4.
There exist a state feedback controlleru(t)=Kix̃(t) with Ki=YiPi-T and a state observer Hi=-PiCyiT such that the error dynamic SMJS (2.5a) is singular stochastic finite-time stabilization via observer-based state feedback with respect to (c1,c2,T,R̅i,d) if there exist positive scalars α,c2,σ1,σ2, and sets of mode-dependent symmetric positive-definite matrices {Xi,i∈𝕄}, {Φi,i∈𝕄}, {Θi,i∈𝕄}, sets of mode-dependent matrices {Yi,i∈𝕄}, {Zi,i∈𝕄}, and for all rt=i∈𝕄, such that the following inequalities hold: (3.37a)0≤PiEiT=EiPiT=EiVi-TXiVi-1EiT≤Φi,(3.37b)[L1(Pi,Yi)-BiYiGi0Υi0*L2(Pi,Yi)GiPiCyiT0Υi**-Θi000***-I200****-Wi0*****-Wi]<0,(3.37c)σ1Ri-1<UiXiUiT<Ri-1,0<Θi<σ2I,(3.37d)[-e-αTc22+σ2d2c1*-σ1]<0,where L1(Pi,Yi),L2(Pi,Yi),Υi,Wi, and Pi are the same as Theorem 3.3.Remark 3.5.
The feasibility of conditions stated in Theorem3.3 and Corollary 3.4 can be turned into the following LMIs-based feasibility problem with a parameter α, respectively:
(3.38)min⁡⁡(β+ρ)Xi,Yi,Zi,Φi,σ1s.t.(3.23a)–(3.23d)withβ=c22,ρ=γ2,(3.39)min⁡⁡βXi,Yi,Zi,Φi,Θi,σ1,σ2s.t.(3.37a)–(3.37d)withβ=c22.
Furthermore, we can also find the parameter α by an unconstrained nonlinear optimization approach, in which a locally convergent solution can be obtained by using the program fminsearch in the optimization toolbox of Matlab.Remark 3.6.
If we can find feasible solution with parameterα=0, by the above discussion, we can obtain the designed state observer, and state feedback controller cannot only ensure SSFTB and stochastic stabilization of the error dynamic SMJS (2.5a) and (2.5b) but also the effect of the disturbance input of the disturbance input on the controlled output satisfying ∥Twz∥<γ for the error dynamic SMJS.


## 4. Numerical Examples

In this section, we present numerical examples to illustrate the proposed methods.Example 4.1.
To show the results of singular stochastic finite-timeH∞ stabilization via observer-based state feedback of the error dynamic SMJS (2.5a) and (2.5b), consider a two-mode SMJS (2.1a)–(2.1c) with parameters as follows:
(i) modeno.1:
(4.1)E1=[1000],A1=[-11.52.2-3],B1=[-10.20.5-0.1],G1=[0.10.1],C1=[1-0.3],D11=[0.5-0.6],D21=[-0.3],Cy1=[0.6-0.6],
mode no.2:
(4.2)E2=[1000],A2=[-0.51.21.6-1.5],B2=[-120.5-1],G2=[0.2-0.1],C2=[1-0.2],D12=[-0.80.6],D22=[-0.4],Cy2=[0.20.4].
In addition, the transition rate matrix is described by Γ=[-0.50.50.6-0.6].
Then, we chooseR1=R2=I2,T=2,d=2,c1=1,andα=2, and Theorem 3.3 yields γ=3.0073, c2=10.7667, and
(4.3)X1=[0.99820.00000.00000.8569],Y1=[0.2129-0.19270.6038-0.2995],Z1=[0.91661.0377],X2=[0.6857-0.0000-0.00000.8569],Y2=[0.50510.03440.11190.2116],Z2=[-0.18950.8802],Φ1=[1.0041-0.0333-0.03330.2897],Φ2=[0.8400-0.2934-0.29340.6147],σ1=0.6852.
This also gives the following gains:
(4.4)K1=[0.3838-0.18570.8700-0.2886],H1=[-0.04900.6226],K2=[0.73660.04010.16320.2470],H2=[-0.5663-0.4151].
Moreover, by Theorem3.3, the optimal bound with minimum value of c22+γ2 relies on parameter α. Let R1=R2=I2,T=2,d=2,andc1=1, then we can find feasible solution when 1.27≤α≤10.11. Figure 1 shows the optimal value with a different value of α. Furthermore, by using the program fminsearch in the optimization toolbox of Matlab starting at α=2, the locally convergent solution can be derived as
(4.5)K1=[0.3251-0.11990.5985-0.1072],H1=[-0.15640.4356],K2=[1.1810-0.11370.6294-0.0173],H2=[-0.4956-0.2904],
with α=1.7043 and the optimal values γ=2.5572 and c2=9.7515.Figure 1
The local optimal bound ofγ and c2.Remark 4.2.
To show the results of singular stochastic finite-time stabilization via observer-based state feedback of the error dynamic SMJS (2.5a), consider a two-mode SMJS (2.1a) and (2.1c) with parameters that the matrical variables and the transition rate matrix are the same as the above example. Let R(1)=R(2)=I2,T=2,d=2,andc1=1, by Corollary 3.4, the optimal bound with minimum value of c22 relies on the parameter α. We can find feasible solution when 1.26≤α≤10.35. Figure 2 shows the optimal value with different value of α. Then using the program fminsearch in the optimization toolbox of Matlab starting at α=2, we can obtain the locally convergent solution c2=9.7037 with α=1.7001.Figure 2
The local optimal bound ofc2.Example 4.3.
To show SSFTB and stochastic stabilization of the error dynamic SMJS (2.5a) and (2.5b) and the effect of the disturbance input of the disturbance input on the controlled output satisfying ∥Twz∥<γ for the error dynamic SMJS, let (4.6)A1=[-41.45-2.5-3.2],A2=[-2.2-1.45-1-1.5],
and other matrical variables and the transition rate matrix are the same as Example 4.1.
Then, letR1=R2=I2,d=2,andc1=1. By Theorem 3.3, we can find the feasible solution when α=0. Noting that when α=0, Theorem 3.3 yields the optimal value γ=0.4001 and c2=1.2817. Thus, the above error dynamic SMJS is stochastically stabilizable, and the effect of the disturbance input of the disturbance input on the controlled output satisfies ∥Twz∥<0.4001.


## 5. Conclusions

This paper investigates the problem of finite-timeH∞ control via observer-based state feedback for a family of SMJSs with time-varying norm-bounded disturbance. An observer-based state feedback controller is designed which ensures singular stochastic finite-time H∞ stabilization via observer-based state feedback of the resulting closed-loop error dynamic SMJS. Sufficient criterions are presented for the solvability of the problem, which can be reduced to a feasibility problem in the form of linear matrix inequalities with a fixed parameter. In addition, we also give the problem of finite-time stabilization via observer-based state feedback of a class of SMJSs and present sufficient conditions of singular stochastic finite-time stabilization via observer-based state feedback for the class of SMJSs. Numerical examples are also given to show the validity of the proposed methodology.



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*Source: 205727-2012-04-04.xml*