# H

Authors: Hongsheng Lin; Ying Li; Guoliang Wang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper discussesH∞ control problems of continuous-time and discrete-time singular Markovian jump systems (SMJSs) with bounded transition probabilities. Improved sufficient conditions for continuous-time SMJSs to be regular, impulse free, and stochastically stable with γ-disturbance attenuation are established via less conservative inequality to estimate the transition jump rates, so are the discrete-time SMJSs. With the obtained conditions, the design of a state feedback controller which ensures the resulting closed-loop system to be stochastically admissible and with H∞ performance is given in terms of linear matrix inequalities (LMIs). Finally, illustrative examples are presented to show the effectiveness and the benefits of the proposed approaches.


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## Body

## 1. Introduction

Singular systems, also referred to as generalized systems, descriptor systems, implicit systems, differential-algebraic systems, or semistate systems [1], have been extensively studied in the past years due to the fact that singular systems better describe physical systems than regular ones. Many practical problems, such as those in Leontief’s dynamic system [2] and electrical and mechanical systems [3, 4], are modeled as singular systems. Considerable attention has been paid to investigate such continuous-time singular systems [5–8] and discrete-time singular systems [9–11], respectively. On the other hand, Markovian jump systems (MJSs) can be regarded as a special class of hybrid systems with finite operation modes whose structures are subject to random abrupt changes. The studies of MJSs are important in practical applications such as manufacturing systems, aircraft control, target tracking, robotics, solar receiver control, and power systems. Therefore, a great deal of attention has been devoted to the study of this class of systems in recent years such as [12–19].When singular systems experience abrupt changes in their structure and parameters, it is natural to model them as SMJSs. Reference [20] considered the stochastic stability and robust stochastic stability conditions for continuous-time SMJSs. The delay-dependent H∞ control problem of continuous-time SMJSs was studied in [21] via Jensen’s integral inequality approach after introducing free weighting matrices, while [22] gave sufficient conditions for uncertain discrete-time SMJSs with mode-dependent time delays through transforming them into a standard linear system. Particularly, [23] proposed more general condition of H∞  control for discrete-time SMJSs, which does not require any decomposition of the original system, and the construction of H∞ controller gain does not need system decomposition. However, all the above-mentioned criteria on MJSs require the values of transition probabilities to be exactly know. Recently, Boukas discussed H∞ control of discrete-time normal state-space MJSs with bounded transition probabilities in [24] and stabilization of continuous-time SMJSs with full or partial knowledge of transition rates in [25].Following the work of [24, 25], in this paper, the H∞ control problem is considered for both continuous-time and discrete-time SMJSs with bounded transition probabilities. Sufficient criteria guaranteeing SMJSs being stochastically admissible with an H∞ performance are presented in terms of LMIs, which are obtained by using an improved inequality to obtain less conservative estimation on transition probabilities. Based on these, a state feedback controller such that the resulting closed-loop system is stochastically admissible and possesses a prescribed H∞ performance is given. Numerical examples are provided to demonstrate the validity of developed methods.Notation.
ℝ
n denotes the n dimensional Euclidean space. ℝm×n is the set of all m×n real matrices. |·| denotes the Euclidean norm. ε{·} is the expectation operator with respect to some probability measure. In symmetric block matrices, we use “*” as an ellipsis for the terms induced by symmetry, diag{⋯} for a block-diagonal matrix, and (M)⋆≜M+MT.


## 2. Preliminaries

### 2.1. Continuous-Time Markovian Jump Linear Systems

Consider the following continuous-time SMJS described by(1)Ex˙(t)=A(η(t))x(t)+B(η(t))u(t)+F(η(t))w(t)y(t)=C(η(t))x(t)+D(η(t))w(t),
where x(t)∈ℝn is the state vector, u(t)∈ℝm is the control input, w(t)∈ℝp is the disturbance input, and y(t)∈ℝq is the control output. Matrix E may be singular assumed rank(E)=r≤n. Parameter η(t) is the continuous-time Markov processes with right continuous trajectories taking values in a finite set 𝕊={1,2,…,N} with transition probabilities:
(2)Pr(η(t+Δt)=j∣η(t)=i)={λijΔt+o(Δt)i≠j1+λiiΔt+o(Δt)i=j,
where Δt>0, limΔt→0(o(Δt)/Δt)=0, and the transition probability rate satisfies λij≥0, for i,j∈𝕊,i≠j, and
(3)λii=-∑j=1,j≠iN‍λij.For notational simplicity, in the sequel, for each possibleη(t)=i, i∈𝕊, the matrix Aη(t) will be denoted by Ai, and so on.Assumption 1.
The transition probabilities for system (1) are assumed to be unknown but vary between two known bounds satisfying the following:
(4)0≤λ_ij≤λij≤λ¯ij∀i,j∈𝕊,
where i≠j.From Assumption1, when we only know that λ_i=minj≠i∈𝕊λ_ij and λ¯i=minj≠i∈𝕊λ¯ij, we have λ_i≤λij≤λ¯i, which is the same as in [25]. Thus, we may say that Assumption 1 is more natural.Definition 2 (see [26]).
(
1
) The nominal system (1) is said to be regular if det(sE-Ai) is not identically zero for every i∈𝕊.
(
2
) The nominal system (1) is said to be impulse free if deg(det(sE-Ai))=rank(E) for every i∈𝕊.
(
3
) The nominal system in (1) is said to be stochastically stable if, when u(t)=0 and w(t)=0, there exists a constant M(x0,η0) such that
(5)ℰ{∫0∞‍∥x(t)∥2dt∣x0,η0}≤M(x0,η0).
(
4
) The nominal system in (1) is said to be stochastically admissible if it is regular, impulse free, and stochastically stable.Definition 3.
Givenγ>0, the nominal system in (1) is said to be stochastically admissible with γ-disturbance attenuation if it is stochastically admissible and satisfying the following such that
(6)ℰ{∫0∞‍∥y(t)∥2dt}<γ2𝔼{∫0∞‍∥w(t)∥2dt}
holds for zero-initial condition and any nonzero w(t)∈ℒ2[0,∞).In this paper, theH∞ controller such that the resulting closed-loop system is stochastically admissible with (6) is
(7)u(t)=Kix(t),
where Ki is to be determined.Lemma 4.
Given a scalarγ>0, the unforced system (1) is stochastically admissible with H∞ performance if there exist matrices Pi such that
(8)ETPi=PiTE≥0,[AiTPi+PiTAi+∑j=1N‍λijETPjPiTFiCiT*-γ2IDiT**-I]<0.


### 2.2. Discrete-Time Markovian Jump Linear Systems

Consider the following discrete-time SMJS described by(9)Ex(k+1)=A(θ(k))x(t)+B(θ(k))u(k)+F(θ(k))w(k)y(k)=C(θ(k))x(k)+D(θ(k))w(k),
where x(k)∈ℝn is the state vector, u(k)∈ℝm is the control input, w(k)∈ℝp is the disturbance input, and y(k)∈ℝq is the control output. Matrix E may be singular assumed rank(E)=r≤n. Parameter θ(k) is a discrete-time, discrete-state Markovian chain taking values in a finite set 𝕊 with transition probabilities:
(10)Pr(θ(k+1)=j∣θ(k)=i)=πij,
where πij≥0, for i,j∈𝕊, and
(11)∑j=1N‍πij=1.Assumption 5.
The transition probabilities for (10) are assumed to be unknown but vary between two known bounds satisfying the following:
(12)0≤π_ij≤πij≤π¯ij∀i,j∈𝕊.From [24], we know that ∑j=1N‍π_ij and ∑j=1N‍π¯ij may not equal 1. It is not necessary to exactly know the transition probabilities but only the bounds of πij, which are π_ij and π¯ij.Definition 6 (see [26]).
(
1
) The nominal system in (9) is said to be regular if det(sE-Ai) is not identically zero for every i∈𝕊.
(
2
) The nominal system in (9) is said to be causal if deg(det(sE-Ai))=rank(E) for every i∈𝕊.
(
3
) The nominal system in (9) is said to be stochastically stable if, when u(k)=0 and w(k)=0, there exists a constant M(x0,θ0) such that
(13)ℰ{∑k=0N‍∥x(k)∥2∣x0,θ0}≤M(x0,θ0).
(
4
) The nominal system in (9) is said to be stochastically admissible if it is regular, causal, and stochastically stable.Definition 7.
Givenγ>0, the nominal system in (9) is said to be stochastically admissible with γ-disturbance attenuation such that
(14)ℰ{∑k=0N‍∥y(k)∥2}<γ2𝔼{∑k=0N‍∥w(k)∥2}
holds for zero-initial condition and any nonzero w(k)∈ℒ2[0,∞).The correspondingH∞ controller for system (9) is
(15)u(k)=Kix(k),
where Ki is to be determined.Lemma 8.
Given a scalarγ>0, the unforced system (9) is stochastically admissible with H∞ performance if there exist matrices Pi=PiT such that
(16)ETPiE≥0(17)[AiTP^iAi-ETPiEAiTP^iFiCiT*-γ2I+FiTP^iFiDiT**-I]<0,
where P^i=∑j=1N‍πijPj.Lemma 9 (see[27]).
Given any real matricesX, Y, and Z with appropriate dimensions and such that Y>0 and is symmetric, then, one has
(18)XTYX+XTZ+ZTX+ZTY-1Z≥0.


## 2.1. Continuous-Time Markovian Jump Linear Systems

Consider the following continuous-time SMJS described by(1)Ex˙(t)=A(η(t))x(t)+B(η(t))u(t)+F(η(t))w(t)y(t)=C(η(t))x(t)+D(η(t))w(t),
where x(t)∈ℝn is the state vector, u(t)∈ℝm is the control input, w(t)∈ℝp is the disturbance input, and y(t)∈ℝq is the control output. Matrix E may be singular assumed rank(E)=r≤n. Parameter η(t) is the continuous-time Markov processes with right continuous trajectories taking values in a finite set 𝕊={1,2,…,N} with transition probabilities:
(2)Pr(η(t+Δt)=j∣η(t)=i)={λijΔt+o(Δt)i≠j1+λiiΔt+o(Δt)i=j,
where Δt>0, limΔt→0(o(Δt)/Δt)=0, and the transition probability rate satisfies λij≥0, for i,j∈𝕊,i≠j, and
(3)λii=-∑j=1,j≠iN‍λij.For notational simplicity, in the sequel, for each possibleη(t)=i, i∈𝕊, the matrix Aη(t) will be denoted by Ai, and so on.Assumption 1.
The transition probabilities for system (1) are assumed to be unknown but vary between two known bounds satisfying the following:
(4)0≤λ_ij≤λij≤λ¯ij∀i,j∈𝕊,
where i≠j.From Assumption1, when we only know that λ_i=minj≠i∈𝕊λ_ij and λ¯i=minj≠i∈𝕊λ¯ij, we have λ_i≤λij≤λ¯i, which is the same as in [25]. Thus, we may say that Assumption 1 is more natural.Definition 2 (see [26]).
(
1
) The nominal system (1) is said to be regular if det(sE-Ai) is not identically zero for every i∈𝕊.
(
2
) The nominal system (1) is said to be impulse free if deg(det(sE-Ai))=rank(E) for every i∈𝕊.
(
3
) The nominal system in (1) is said to be stochastically stable if, when u(t)=0 and w(t)=0, there exists a constant M(x0,η0) such that
(5)ℰ{∫0∞‍∥x(t)∥2dt∣x0,η0}≤M(x0,η0).
(
4
) The nominal system in (1) is said to be stochastically admissible if it is regular, impulse free, and stochastically stable.Definition 3.
Givenγ>0, the nominal system in (1) is said to be stochastically admissible with γ-disturbance attenuation if it is stochastically admissible and satisfying the following such that
(6)ℰ{∫0∞‍∥y(t)∥2dt}<γ2𝔼{∫0∞‍∥w(t)∥2dt}
holds for zero-initial condition and any nonzero w(t)∈ℒ2[0,∞).In this paper, theH∞ controller such that the resulting closed-loop system is stochastically admissible with (6) is
(7)u(t)=Kix(t),
where Ki is to be determined.Lemma 4.
Given a scalarγ>0, the unforced system (1) is stochastically admissible with H∞ performance if there exist matrices Pi such that
(8)ETPi=PiTE≥0,[AiTPi+PiTAi+∑j=1N‍λijETPjPiTFiCiT*-γ2IDiT**-I]<0.


## 2.2. Discrete-Time Markovian Jump Linear Systems

Consider the following discrete-time SMJS described by(9)Ex(k+1)=A(θ(k))x(t)+B(θ(k))u(k)+F(θ(k))w(k)y(k)=C(θ(k))x(k)+D(θ(k))w(k),
where x(k)∈ℝn is the state vector, u(k)∈ℝm is the control input, w(k)∈ℝp is the disturbance input, and y(k)∈ℝq is the control output. Matrix E may be singular assumed rank(E)=r≤n. Parameter θ(k) is a discrete-time, discrete-state Markovian chain taking values in a finite set 𝕊 with transition probabilities:
(10)Pr(θ(k+1)=j∣θ(k)=i)=πij,
where πij≥0, for i,j∈𝕊, and
(11)∑j=1N‍πij=1.Assumption 5.
The transition probabilities for (10) are assumed to be unknown but vary between two known bounds satisfying the following:
(12)0≤π_ij≤πij≤π¯ij∀i,j∈𝕊.From [24], we know that ∑j=1N‍π_ij and ∑j=1N‍π¯ij may not equal 1. It is not necessary to exactly know the transition probabilities but only the bounds of πij, which are π_ij and π¯ij.Definition 6 (see [26]).
(
1
) The nominal system in (9) is said to be regular if det(sE-Ai) is not identically zero for every i∈𝕊.
(
2
) The nominal system in (9) is said to be causal if deg(det(sE-Ai))=rank(E) for every i∈𝕊.
(
3
) The nominal system in (9) is said to be stochastically stable if, when u(k)=0 and w(k)=0, there exists a constant M(x0,θ0) such that
(13)ℰ{∑k=0N‍∥x(k)∥2∣x0,θ0}≤M(x0,θ0).
(
4
) The nominal system in (9) is said to be stochastically admissible if it is regular, causal, and stochastically stable.Definition 7.
Givenγ>0, the nominal system in (9) is said to be stochastically admissible with γ-disturbance attenuation such that
(14)ℰ{∑k=0N‍∥y(k)∥2}<γ2𝔼{∑k=0N‍∥w(k)∥2}
holds for zero-initial condition and any nonzero w(k)∈ℒ2[0,∞).The correspondingH∞ controller for system (9) is
(15)u(k)=Kix(k),
where Ki is to be determined.Lemma 8.
Given a scalarγ>0, the unforced system (9) is stochastically admissible with H∞ performance if there exist matrices Pi=PiT such that
(16)ETPiE≥0(17)[AiTP^iAi-ETPiEAiTP^iFiCiT*-γ2I+FiTP^iFiDiT**-I]<0,
where P^i=∑j=1N‍πijPj.Lemma 9 (see[27]).
Given any real matricesX, Y, and Z with appropriate dimensions and such that Y>0 and is symmetric, then, one has
(18)XTYX+XTZ+ZTX+ZTY-1Z≥0.


## 3. Main Results

### 3.1. Continuous-Time Markovian Jump Linear Systems

Theorem 10.
Given a scalarγ>0, the unforced system (1) with constraint (4) is stochastically admissible with H∞ performance if there exist matrices Xi and Tij>0 such that
(19)XiTET=EXi≥0[(AiXi)⋆-∑j=1,j≠iN‍λ_ijXiTETFiXiTCiTS-iT(X)S~iT(X)*-γ2IDiT00**-I00***R-i0****R~i]<0,
where(20)S-iT(X)=[λ¯i1XiT⋯λ¯i(i-1)XiTλ¯i(i+1)XiT⋯λ¯iNXiT]S~iT(X)=[λ¯i1XiTET⋯λ¯i(i-1)XiTETλ¯i(i+1)XiTET⋯λ¯iNXiTET]R-i=-diag{4Ti1,…,4Ti(i-1),4Ti(i+1),…,4TiN}R~i=-diag{(X1)⋆-Ti1,…,(Xi-1)⋆-Ti(i-1),(Xi+1)⋆-Ti(i+1),…,(XN)⋆-TiN}hold for all i∈S.Proof.
From (3) and (4), we have
(21)∑j=1N‍λijETPj=∑j=1,j≠iN‍λijETPj-∑j=1,j≠iN‍λijETPi≤∑j=1,j≠iN‍λ¯ijETPj-∑j=1,j≠iN‍λ_ijETPi.
Then, by (8) and (21), we conclude that
(22)[(AiTPi)⋆+∑j=1,j≠iN‍λ¯ijETPj-∑j=1,j≠iN‍λ_ijETPiPiTFiCiT*-γ2IDiT**-I]
Since[(1/2)Tij-(1/2)-ETPjTij(1/2)][(1/2)Tij-(1/2)-ETPjTij(1/2)]T≥0 and Tij>0,  ∀i,j∈𝕊 and i≠j, we have
(23)ETPj≤14Tij-1+ETPjTijPjTE,(24)-Pi-TTij-1Pi-1≤Tij-Pi-1-Pi-T.
Via (22) and (23), we have (25) implying (22); that is,
(25)[Ω-iPiTFiCiT*-γ2IDiT**-I]<0,
where
(26)Ω-i=(AiTPi)⋆-∑j=1,j≠iN‍λ_ijETPi+∑j=1,j≠iN‍14λ¯ijTij-1+∑j=1,j≠iN‍λ¯ijETPjTijPjTE.
LetXi=Pi-1; then pre- and postmultiply (25) by diag{XiT,I,I} and its transpose. Taking into account (24) and via Schur complement, we obtain that (19) and (21) imply (8). This completes the proof.Remark 11.
Since to the termETPj gives some problems, [25] presented a method to deal with problem; that is, ETPj≤(1/4)ϵj-1I+ϵjETPjPjTE. In this paper, we overcome it by (23); let Tij-1=ϵj-1; we have ETPj≤(1/4)ϵj-1I+ϵjETPjPjTE. Therefore, we have ETPj ≤ (1/4)Tij-1 + ETPjTijPjTE ≤ (1/4)ϵj-1I+ϵjETPjPjTE.Theorem 12.
Given a scalarγ>0, there exists a state feedback controller in the form of (7) such that the resulting closed-loop system is stochastically admissible with (6), if there exist matrices Xi, Tij>0, and Yi such that
(27)XiTET=EXi≥0,(28)[ΩiFiXiTCiTS-iT(X)S~iT(X)*-γ2IDiT00**-I00***R-i0****R~i]<0,
where(29)Ωi=(AiXi+BiYi)⋆-∑j=1,j≠iN‍λ_ijXiTETS~iT(X)=[λ¯i1XiT⋯λ¯i(i-1)XiTλ¯i(i+1)XiT⋯λ¯iNXiT]S-iT(X)=[λ¯i1XiTET⋯λ¯i(i-1)XiTETλ¯i(i+1)XiTET⋯λ¯iNXiTET]R-i=-diag{4Ti1,…,4Ti(i-1),4Ti(i+1)…,4TiN}R~i=-diag{(X1)⋆-Ti1,…,(Xi-1)⋆-Ti(i-1),(Xi+1)⋆-Ti(i+1),…,(XN)⋆-TiN}hold for all i∈𝕊 with constraint (4). The controller gain Ki can be constructed as
(30)Ki=YiXi-1.Proof.
SubstitutingAi with Ai+BiKi, then by similar method in Theorem 10 with (28), we can obtain Theorem 12. Thus, it is omitted here.Remark 13.
It is seen that such conditions have equation constraint (28), which are not strict LMIs. In addition, the desired controller (7) is mode dependent, which requires its system mode to be available online. This makes the scope of application limited. In order to deal with this practical condition, mode-independent control is usually used. Based on the methods in [28, 29], such problems can be solved easily.


### 3.2. Discrete-Time Markovian Jump Linear Systems

Theorem 14.
Given a scalarγ>0, the unforced system (9) with constraint (12) is stochastically admissible with H∞ performance if there exist matrices Pi, Gi, and Tij>0 such that(31)ETPiE≥0,(32)[Φi11Φi1200GiTCiTG^iT0*Φi22G^iT0000**-2I+T^ijT^ijFi00S~i1(P)***-γ2I+FiTT^ijFiDiT0S~i2(P)****-I00*****-π_iETPiE-2I0******R~i]<0,
where(33)Φi11=(AiGi-Gi)⋆,Φi12=GiTAiT-Gi,Φi22=-2Gi-2GiTT^ij=14∑j=1N‍π¯ijTij,G^i=Gi+I,π_i=∑j=1N‍π_ij,R~i=-diag{Ti1,…,TiN}S~i1T(P)=[π¯i1P1⋯π¯iNPN],S~i2T(P)=[π¯i1FiTP1⋯π¯iNFiTPN]
hold for all i∈S.Proof.
From (16) and (17), similar to [23], we have
(34)(18)=Z1TZ2TZ3T(Ψi+ΠiTP^iΠi)Z3Z2Z1,
where
(35)Z1=[Gi-100Gi-1Ai000I000I],Z2=[I0000I000Gi0000I0000I],Z3=[I00000I00000I00000I00000IGi0000]ΠiT=[00IFiT00],Ψi=[Φi11Φi1200GiTCiTG^iT*Φi22G^iT000**-2I000***-γ2IDiT0****-I0*****-ETPiE-2I].
Since∑j=1N‍πijPj≤∑j=1N‍πij((1/4)Tij+PjTij-1Pj) with Tij>0, ∀i,j∈𝕊 and (12), we have
(36)ΠiTP^iΠi≤ΠiT(∑j=1N‍π¯ij(14Tij+PjTij-1Pj))Πi.
From (12), we obtain that
(37)π_iETPiE=∑j=1N‍π_ijETPiE≤∑j=1N‍πijETPiE.
Taking into account (34), (36), and (37), we have
(38)Ψi+ΠiTP^iΠi≤Ψ-i+ΠiT(∑j=1N‍π¯ijPjTij-1Pj)Πi<0,
where(39)Ψ-i=[Φi11Φi1200GiTCiTG^iT*Φi22G^iT000**-2I+T^ijT^ijFi00***-γ2I+FiTT^ijFiDiT0****-I0*****-π_iETPiE-2I].
Then, by the Schur complement, we see that (32) is equivalent to (38). This completes the proof.Remark 15.
It should be remarked that whenE=I in (9), [24] gave important criteria under the constraint (12) via ∑j=1N‍πijPj≤∑j=1N‍π¯ijPj and ∑j=1N‍π_ijPi≤∑j=1N‍πijPi, where Pj must be positive-definite matrix which is not applicable in SMJSs. In SMJS (9), Pi is only required to be nonsingular matrix. In the above proof, we remove this constraint via (36). In particular, if E=I, then P^i≤∑j=1N‍π¯ij((1/4)Tij+PjTij-1Pj). Let Tij=2Pj; we have (1/4)Tij+PjTij-1Pj=Pj; thus, we have P^i≤∑j=1N‍π¯ij((1/4)Tij+PjTij-1Pj)≤∑j=1N‍π¯ijPj.Theorem 16.
Given a scalarγ>0, there exists a state feedback controller in the form of (15) such that the resulting closed-loop system (9) is stochastically admissible with (14), if there exist matrices Gi, Pi, Yi, and Tij>0 such that(40)ETPiE≥0(41)[Φi11Φi1200GiTCiTG^iT0*Φi22G^iT0000**-2I+T^ijT^ijFi00S~i1(P)***-γ2I+FiTT^ijFiDiT0S~i2(P)****-I00*****-π_iETPiE-2I0******R~i]<0,where
(42)Φ^i11=(AiGi+BiYi-Gi)⋆,Φ^i12=GiTAiT+YiTBiT-Gi
hold for all i∈𝕊 with constraint (12). The controller gain Ki can be constructed as
(43)Ki=YiGi-1.


## 3.1. Continuous-Time Markovian Jump Linear Systems

Theorem 10.
Given a scalarγ>0, the unforced system (1) with constraint (4) is stochastically admissible with H∞ performance if there exist matrices Xi and Tij>0 such that
(19)XiTET=EXi≥0[(AiXi)⋆-∑j=1,j≠iN‍λ_ijXiTETFiXiTCiTS-iT(X)S~iT(X)*-γ2IDiT00**-I00***R-i0****R~i]<0,
where(20)S-iT(X)=[λ¯i1XiT⋯λ¯i(i-1)XiTλ¯i(i+1)XiT⋯λ¯iNXiT]S~iT(X)=[λ¯i1XiTET⋯λ¯i(i-1)XiTETλ¯i(i+1)XiTET⋯λ¯iNXiTET]R-i=-diag{4Ti1,…,4Ti(i-1),4Ti(i+1),…,4TiN}R~i=-diag{(X1)⋆-Ti1,…,(Xi-1)⋆-Ti(i-1),(Xi+1)⋆-Ti(i+1),…,(XN)⋆-TiN}hold for all i∈S.Proof.
From (3) and (4), we have
(21)∑j=1N‍λijETPj=∑j=1,j≠iN‍λijETPj-∑j=1,j≠iN‍λijETPi≤∑j=1,j≠iN‍λ¯ijETPj-∑j=1,j≠iN‍λ_ijETPi.
Then, by (8) and (21), we conclude that
(22)[(AiTPi)⋆+∑j=1,j≠iN‍λ¯ijETPj-∑j=1,j≠iN‍λ_ijETPiPiTFiCiT*-γ2IDiT**-I]
Since[(1/2)Tij-(1/2)-ETPjTij(1/2)][(1/2)Tij-(1/2)-ETPjTij(1/2)]T≥0 and Tij>0,  ∀i,j∈𝕊 and i≠j, we have
(23)ETPj≤14Tij-1+ETPjTijPjTE,(24)-Pi-TTij-1Pi-1≤Tij-Pi-1-Pi-T.
Via (22) and (23), we have (25) implying (22); that is,
(25)[Ω-iPiTFiCiT*-γ2IDiT**-I]<0,
where
(26)Ω-i=(AiTPi)⋆-∑j=1,j≠iN‍λ_ijETPi+∑j=1,j≠iN‍14λ¯ijTij-1+∑j=1,j≠iN‍λ¯ijETPjTijPjTE.
LetXi=Pi-1; then pre- and postmultiply (25) by diag{XiT,I,I} and its transpose. Taking into account (24) and via Schur complement, we obtain that (19) and (21) imply (8). This completes the proof.Remark 11.
Since to the termETPj gives some problems, [25] presented a method to deal with problem; that is, ETPj≤(1/4)ϵj-1I+ϵjETPjPjTE. In this paper, we overcome it by (23); let Tij-1=ϵj-1; we have ETPj≤(1/4)ϵj-1I+ϵjETPjPjTE. Therefore, we have ETPj ≤ (1/4)Tij-1 + ETPjTijPjTE ≤ (1/4)ϵj-1I+ϵjETPjPjTE.Theorem 12.
Given a scalarγ>0, there exists a state feedback controller in the form of (7) such that the resulting closed-loop system is stochastically admissible with (6), if there exist matrices Xi, Tij>0, and Yi such that
(27)XiTET=EXi≥0,(28)[ΩiFiXiTCiTS-iT(X)S~iT(X)*-γ2IDiT00**-I00***R-i0****R~i]<0,
where(29)Ωi=(AiXi+BiYi)⋆-∑j=1,j≠iN‍λ_ijXiTETS~iT(X)=[λ¯i1XiT⋯λ¯i(i-1)XiTλ¯i(i+1)XiT⋯λ¯iNXiT]S-iT(X)=[λ¯i1XiTET⋯λ¯i(i-1)XiTETλ¯i(i+1)XiTET⋯λ¯iNXiTET]R-i=-diag{4Ti1,…,4Ti(i-1),4Ti(i+1)…,4TiN}R~i=-diag{(X1)⋆-Ti1,…,(Xi-1)⋆-Ti(i-1),(Xi+1)⋆-Ti(i+1),…,(XN)⋆-TiN}hold for all i∈𝕊 with constraint (4). The controller gain Ki can be constructed as
(30)Ki=YiXi-1.Proof.
SubstitutingAi with Ai+BiKi, then by similar method in Theorem 10 with (28), we can obtain Theorem 12. Thus, it is omitted here.Remark 13.
It is seen that such conditions have equation constraint (28), which are not strict LMIs. In addition, the desired controller (7) is mode dependent, which requires its system mode to be available online. This makes the scope of application limited. In order to deal with this practical condition, mode-independent control is usually used. Based on the methods in [28, 29], such problems can be solved easily.


## 3.2. Discrete-Time Markovian Jump Linear Systems

Theorem 14.
Given a scalarγ>0, the unforced system (9) with constraint (12) is stochastically admissible with H∞ performance if there exist matrices Pi, Gi, and Tij>0 such that(31)ETPiE≥0,(32)[Φi11Φi1200GiTCiTG^iT0*Φi22G^iT0000**-2I+T^ijT^ijFi00S~i1(P)***-γ2I+FiTT^ijFiDiT0S~i2(P)****-I00*****-π_iETPiE-2I0******R~i]<0,
where(33)Φi11=(AiGi-Gi)⋆,Φi12=GiTAiT-Gi,Φi22=-2Gi-2GiTT^ij=14∑j=1N‍π¯ijTij,G^i=Gi+I,π_i=∑j=1N‍π_ij,R~i=-diag{Ti1,…,TiN}S~i1T(P)=[π¯i1P1⋯π¯iNPN],S~i2T(P)=[π¯i1FiTP1⋯π¯iNFiTPN]
hold for all i∈S.Proof.
From (16) and (17), similar to [23], we have
(34)(18)=Z1TZ2TZ3T(Ψi+ΠiTP^iΠi)Z3Z2Z1,
where
(35)Z1=[Gi-100Gi-1Ai000I000I],Z2=[I0000I000Gi0000I0000I],Z3=[I00000I00000I00000I00000IGi0000]ΠiT=[00IFiT00],Ψi=[Φi11Φi1200GiTCiTG^iT*Φi22G^iT000**-2I000***-γ2IDiT0****-I0*****-ETPiE-2I].
Since∑j=1N‍πijPj≤∑j=1N‍πij((1/4)Tij+PjTij-1Pj) with Tij>0, ∀i,j∈𝕊 and (12), we have
(36)ΠiTP^iΠi≤ΠiT(∑j=1N‍π¯ij(14Tij+PjTij-1Pj))Πi.
From (12), we obtain that
(37)π_iETPiE=∑j=1N‍π_ijETPiE≤∑j=1N‍πijETPiE.
Taking into account (34), (36), and (37), we have
(38)Ψi+ΠiTP^iΠi≤Ψ-i+ΠiT(∑j=1N‍π¯ijPjTij-1Pj)Πi<0,
where(39)Ψ-i=[Φi11Φi1200GiTCiTG^iT*Φi22G^iT000**-2I+T^ijT^ijFi00***-γ2I+FiTT^ijFiDiT0****-I0*****-π_iETPiE-2I].
Then, by the Schur complement, we see that (32) is equivalent to (38). This completes the proof.Remark 15.
It should be remarked that whenE=I in (9), [24] gave important criteria under the constraint (12) via ∑j=1N‍πijPj≤∑j=1N‍π¯ijPj and ∑j=1N‍π_ijPi≤∑j=1N‍πijPi, where Pj must be positive-definite matrix which is not applicable in SMJSs. In SMJS (9), Pi is only required to be nonsingular matrix. In the above proof, we remove this constraint via (36). In particular, if E=I, then P^i≤∑j=1N‍π¯ij((1/4)Tij+PjTij-1Pj). Let Tij=2Pj; we have (1/4)Tij+PjTij-1Pj=Pj; thus, we have P^i≤∑j=1N‍π¯ij((1/4)Tij+PjTij-1Pj)≤∑j=1N‍π¯ijPj.Theorem 16.
Given a scalarγ>0, there exists a state feedback controller in the form of (15) such that the resulting closed-loop system (9) is stochastically admissible with (14), if there exist matrices Gi, Pi, Yi, and Tij>0 such that(40)ETPiE≥0(41)[Φi11Φi1200GiTCiTG^iT0*Φi22G^iT0000**-2I+T^ijT^ijFi00S~i1(P)***-γ2I+FiTT^ijFiDiT0S~i2(P)****-I00*****-π_iETPiE-2I0******R~i]<0,where
(42)Φ^i11=(AiGi+BiYi-Gi)⋆,Φ^i12=GiTAiT+YiTBiT-Gi
hold for all i∈𝕊 with constraint (12). The controller gain Ki can be constructed as
(43)Ki=YiGi-1.


## 4. Numerical Examples

In this section, numerical examples are given to demonstrate the effectiveness of proposed theory.Example 1.
Consider the following continuous-time SMJS as (1) described as
(44)E=[100010000],A1=[0.1-0.200.5-1-0.50.100.4],A2=[0.1-10-0.3-10.400.20.1],A3=[0.600.4-0.400.7-0.30.1-0.4],B1=[10-1],B2=[0-11],B3=[-100],F1=[0.10.2-0.1],F2=[0.1-0.20.1],F3=[-0.10.20.1],C1=[100],C2=[011],C3=[1-11],D1=0.2,D2=0.1,D3=-0.3.
Assume the bound transition probability matrices are(45)Λ_=[-2.51.31.20.8-1.91.10.80.7-1.5],Λ¯=[-2.91.51.41-2.21.210.9-1.9].
By the method in [25], we find that it is infeasible for this system no matter γ takes any values. However, for γ=0.8, via Theorem 12, we obtain the controller gain
(46)K1=[-6.331510.066996.5153],K2=[2.9966-9.0875-59.5261],K3=[24.7500-22.4333-17.1457].Example 2.
Consider the following discrete-time SMJS as (9) described as
(47)E=[0.50.611.2],A1=[0.10.80.151],A2=[1.10.60.5-1],A3=[0.110.5-0.9],B1=[0-1],B2=[11],B3=[-11],F1=[0.10.2],F2=[0.1-0.2],F3=[-0.10.2],C1=[10],C2=[01],C3=[1-1],D1=D2=0.1,D3=-0.1.
Assume the bound transition probability matrices are(48)Λ_=[0.20.400.300.30.10.40.1],Λ¯=[0.60.800.700.50.30.70.3].
Letγ=1; then via Theorem 16, we obtain the controller gain
(49)K1=[12.30409.9997]K2=[-13.0975-4.5714]K3=[10.6808-7.6430].Example 3.
In order to do some comparison, consider the following discrete-time normal state-space MJS which can be obtained by (9) with E=I and is described as
(50)A1=[0.8-0.10.20.5],A2=[0.60-0.10.2],A3=[-0.6100.7]B1=[00.1],B2=[0.10],B3=[00.2]C1=[-0.20],C2=[0.60],C3=[0.30]D1=-0.4,D2=0.2,D3=0.3.
Assume the bound transition probability matrices are(51)Λ_=[0.20.400.300.30.10.40.1],Λ¯=[0.60.800.700.50.30.70.3].
By the method in [24], we have the minimum γ*=2.45, while via Theorem 16, we obtain γ*=0.43. By this example, it is seen that our method is less conservative.


## 5. Conclusions

In this paper, the problems ofH∞ control for both continuous-time and discrete-time SMJSs with bounded transition probabilities have been investigated. Less conservative sufficient criteria for SMJSs to be stochastically admissible with H∞ performance are given by the LMI approach. Based on the obtained conditions, a kind of state feedback controller, such that not only stochastic admissibility but also a prescribed H∞ performance level is guaranteed, can be computed. Finally, numerical examples are provided to illustrate the advantage and effectiveness of the presented results in this paper.



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*Source: 495194-2013-12-07.xml*