𝐻_∞ Control of Singular Markovian Jump Systems with Bounded Transition Probabilities

Abstract

This paper discusses 𝐻_∞ 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.

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. ℝ^𝑚×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+𝑀^𝑇.

2. Preliminaries

2.1. Continuous-Time Markovian Jump Linear Systems

Consider the following continuous-time SMJS described by 𝐸𝑥̇(𝑡)𝜂(𝑡)𝑥(𝑡)𝜂(𝑡)𝑢(𝑡)𝜂(𝑡)𝑤(𝑡)𝑦(𝑡)𝜂(𝑡)𝑥(𝑡)𝜂(𝑡)𝑤(𝑡),=𝐴+𝐵+𝐹=𝐶+𝐷 (1) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector, 𝑢(𝑡)∈ℝ^𝑚 is the control input, 𝑤(𝑡)∈ℝ^𝑝 is the disturbance input, and 𝑦(𝑡)∈ℝ^𝑞 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: Pr𝜂()(𝑡)=𝜆𝑡+Δ𝑡=𝑗∣𝜂=𝑖^𝑖𝑗Δ𝑡+𝑜(Δ𝑡)𝑖≠𝑗1+𝜆_𝑖𝑖Δ𝑡+𝑜(Δ𝑡)𝑖=𝑗, (2) where Δ𝑡>0, lim_Δ𝑡→0(𝑜(Δ𝑡)/Δ𝑡)=0, and the transition probability rate satisfies 𝜆_𝑖𝑗≥0, for 𝑖,𝑗∈𝕊,𝑖≠𝑗, and 𝜆_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗. (3)

For notational simplicity, in the sequel, for each possible 𝜂(𝑡)=𝑖, 𝑖∈𝕊, the matrix 𝐴_𝜂(𝑡) will be denoted by 𝐴_𝑖, 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: 0≤𝜆_𝑖𝑗≤𝜆_𝑖𝑗≤𝜆_𝑖𝑗∀𝑖,𝑗∈𝕊, (4) where 𝑖≠𝑗.

From Assumption 1, when we only know that 𝜆_𝑖=min_{𝑗≠𝑖∈𝕊}𝜆_𝑖𝑗 and 𝜆_𝑖=min_{𝑗≠𝑖∈𝕊}𝜆_𝑖𝑗, we have 𝜆_𝑖≤𝜆_𝑖𝑗≤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−𝐴_𝑖) is not identically zero for every 𝑖∈𝕊.
(2) The nominal system (1) is said to be impulse free if deg(det(𝑠𝐸−𝐴_𝑖))=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 ℰ^∞₀‖𝑥(𝑡)‖²𝑑𝑡∣𝑥₀,𝜂₀≤𝑀𝑥₀,𝜂₀. (5)
(4) The nominal system in (1) is said to be stochastically admissible if it is regular, impulse free, and stochastically stable.

Definition 3. Given 𝛾>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 ℰ^∞₀𝑦(𝑡)²𝑑𝑡<𝛾²𝔼^∞₀‖𝑤(𝑡)‖²𝑑𝑡 (6) holds for zero-initial condition and any nonzero 𝑤(𝑡)∈ℒ₂[0,∞).

In this paper, the 𝐻_∞ controller such that the resulting closed-loop system is stochastically admissible with (6) is 𝑢(𝑡)=𝐾_𝑖𝑥(𝑡), (7) where 𝐾_𝑖 is to be determined.

Lemma 4. Given a scalar 𝛾>0, the unforced system (1) is stochastically admissible with 𝐻_∞ performance if there exist matrices 𝑃_𝑖 such that 𝐸^𝑇𝑃_𝑖=𝑃^𝑇_𝑖𝐴𝐸≥0,_𝑇𝑖𝑃_𝑖+𝑃^𝑇_𝑖𝐴_𝑖+^𝑁_𝑗=1𝜆_𝑖𝑗𝐸^𝑇𝑃_𝑗𝑃^𝑇_𝑖𝐹_𝑖𝐶^𝑇_𝑖∗−𝛾²𝐼𝐷^𝑇_𝑖∗∗−𝐼<0. (8)

2.2. Discrete-Time Markovian Jump Linear Systems

Consider the following discrete-time SMJS described by ()(𝜃(𝑘𝑥(𝑡)(𝜃(𝑘𝑢(𝑘)(𝜃(𝑘𝑤(𝑘)𝑦(𝑘)(𝜃(𝑘𝑥(𝑘)(𝜃(𝑘𝑤(𝑘),𝐸𝑥𝑘+1=𝐴))+𝐵))+𝐹))=𝐶))+𝐷)) (9) where 𝑥(𝑘)∈ℝ^𝑛 is the state vector, 𝑢(𝑘)∈ℝ^𝑚 is the control input, 𝑤(𝑘)∈ℝ^𝑝 is the disturbance input, and 𝑦(𝑘)∈ℝ^𝑞 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: Pr𝜃(𝑘+1)=𝑗∣𝜃(𝑘)=𝑖=𝜋_𝑖𝑗, (10) where 𝜋_𝑖𝑗≥0, for 𝑖,𝑗∈𝕊, and ^𝑁_𝑗=1𝜋_𝑖𝑗=1. (11)

Assumption 5. The transition probabilities for (10) are assumed to be unknown but vary between two known bounds satisfying the following: 0≤𝜋_𝑖𝑗≤𝜋_𝑖𝑗≤𝜋_𝑖𝑗∀𝑖,𝑗∈𝕊. (12)

From [24], we know that ∑^𝑁_𝑗=1𝜋_𝑖𝑗 and ∑^𝑁_𝑗=1𝜋_𝑖𝑗 may not equal 1. It is not necessary to exactly know the transition probabilities but only the bounds of 𝜋_𝑖𝑗, 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−𝐴_𝑖) is not identically zero for every 𝑖∈𝕊.
(2) The nominal system in (9) is said to be causal if deg(det(𝑠𝐸−𝐴_𝑖))=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 ℰ^𝑁_𝑘=0‖𝑥(𝑘)‖²∣𝑥₀,𝜃₀≤𝑀𝑥₀,𝜃₀. (13)
(4) The nominal system in (9) is said to be stochastically admissible if it is regular, causal, and stochastically stable.

Definition 7. Given 𝛾>0, the nominal system in (9) is said to be stochastically admissible with 𝛾-disturbance attenuation such that ℰ^𝑁_𝑘=0𝑦(𝑘)²<𝛾²𝔼^𝑁_𝑘=0‖𝑤(𝑘)‖² (14) holds for zero-initial condition and any nonzero 𝑤(𝑘)∈ℒ₂[0,∞).

The corresponding 𝐻_∞ controller for system (9) is 𝑢(𝑘)=𝐾_𝑖𝑥(𝑘), (15) where 𝐾_𝑖 is to be determined.

Lemma 8. Given a scalar 𝛾>0, the unforced system (9) is stochastically admissible with 𝐻_∞ performance if there exist matrices 𝑃_𝑖=𝑃^𝑇_𝑖 such that 𝐸^𝑇𝑃_𝑖𝐸≥0𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑖−𝐸^𝑇𝑃_𝑖𝐸𝐴^𝑇_𝑖𝑃_𝑖𝐹_𝑖𝐶^𝑇_𝑖∗−𝛾²𝐼+𝐹^𝑇_𝑖𝑃_𝑖𝐹_𝑖𝐷^𝑇_𝑖∗∗−𝐼<0, (16)(17) where 𝑃_𝑖=∑^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗.

Lemma 9 (see[27]). Given any real matrices 𝑋, 𝑌, and 𝑍 with appropriate dimensions and such that 𝑌>0 and is symmetric, then, one has 𝑋^𝑇𝑌𝑋+𝑋^𝑇𝑍+𝑍^𝑇𝑋+𝑍^𝑇𝑌⁻¹𝑍≥0. (18)

3. Main Results

3.1. Continuous-Time Markovian Jump Linear Systems

Theorem 10. Given a scalar 𝛾>0, the unforced system (1) with constraint (4) is stochastically admissible with 𝐻_∞ performance if there exist matrices 𝑋_𝑖 and 𝑇_𝑖𝑗>0 such that 𝑋^𝑇_𝑖𝐸^𝑇=𝐸𝑋_𝑖𝐴≥0_𝑖𝑋_𝑖^⋆−^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗𝑋^𝑇_𝑖𝐸^𝑇𝐹_𝑖𝑋^𝑇_𝑖𝐶^𝑇_𝑖𝑆^𝑇_𝑖(𝑋)𝑆^𝑇_𝑖(𝑋)∗−𝛾²𝐼𝐷^𝑇_𝑖00∗∗−𝐼00∗∗∗𝑅_𝑖0∗∗∗∗𝑅_𝑖<0, (19)
where𝑆^𝑇_𝑖(𝑋)=𝜆_𝑖1𝑋^𝑇_𝑖⋯𝜆_{𝑖(𝑖−1)}𝑋^𝑇_𝑖𝜆_𝑖(𝑖+1)𝑋^𝑇_𝑖⋯𝜆_𝑖𝑁𝑋^𝑇_𝑖𝑆^𝑇_𝑖(𝑋)=𝜆_𝑖1𝑋^𝑇_𝑖𝐸^𝑇⋯𝜆_{𝑖(𝑖−1)}𝑋^𝑇_𝑖𝐸^𝑇𝜆_𝑖(𝑖+1)𝑋^𝑇_𝑖𝐸^𝑇⋯𝜆_𝑖𝑁𝑋^𝑇_𝑖𝐸^𝑇𝑅_𝑖=−diag4𝑇_𝑖1,…,4𝑇_{𝑖(𝑖−1)},4𝑇_𝑖(𝑖+1),…,4𝑇_𝑖𝑁𝑅_𝑖=−diag𝑋₁^⋆−𝑇_𝑖1,…,𝑋_𝑖−1^⋆−𝑇_{𝑖(𝑖−1)},𝑋_𝑖+1^⋆−𝑇_𝑖(𝑖+1),…,𝑋_𝑁^⋆−𝑇_𝑖𝑁 (20)hold for all 𝑖∈𝑆.

Proof. From (3) and (4), we have ^𝑁_𝑗=1𝜆_𝑖𝑗𝐸^𝑇𝑃_𝑗=^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗𝐸^𝑇𝑃_𝑗−^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗𝐸^𝑇𝑃_𝑖≤^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗𝐸^𝑇𝑃_𝑗−^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗𝐸^𝑇𝑃_𝑖. (21)
Then, by (8) and (21), we conclude that 𝐴^𝑇_𝑖𝑃_𝑖^⋆+^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗𝐸^𝑇𝑃_𝑗−^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗𝐸^𝑇𝑃_𝑖𝑃^𝑇_𝑖𝐹_𝑖𝐶^𝑇_𝑖∗−𝛾²𝐼𝐷^𝑇_𝑖∗∗−𝐼 (22)
Since [(1/2)𝑇^−(1/2)_𝑖𝑗−𝐸^𝑇𝑃_𝑗𝑇^(1/2)_𝑖𝑗][(1/2)𝑇^−(1/2)_𝑖𝑗−𝐸^𝑇𝑃_𝑗𝑇^(1/2)_𝑖𝑗]^𝑇≥0 and 𝑇_𝑖𝑗>0,  ∀𝑖,𝑗∈𝕊 and 𝑖≠𝑗, we have 𝐸^𝑇𝑃_𝑗≤14𝑇⁻¹_𝑖𝑗+𝐸^𝑇𝑃_𝑗𝑇_𝑖𝑗𝑃^𝑇_𝑗𝐸,−𝑃_𝑖^−𝑇𝑇⁻¹_𝑖𝑗𝑃_𝑖⁻¹≤𝑇_𝑖𝑗−𝑃_𝑖⁻¹−𝑃_𝑖^−𝑇. (23)(24)
Via (22) and (23), we have (25) implying (22); that is, Ω_𝑖𝑃^𝑇_𝑖𝐹_𝑖𝐶^𝑇_𝑖∗−𝛾²𝐼𝐷^𝑇_𝑖∗∗−𝐼<0, (25) where Ω_𝑖=𝐴^𝑇_𝑖𝑃_𝑖^⋆−^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗𝐸^𝑇𝑃_𝑖+^𝑁_{𝑗=1,𝑗≠𝑖}14𝜆_𝑖𝑗𝑇⁻¹_𝑖𝑗+^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗𝐸^𝑇𝑃_𝑗𝑇_𝑖𝑗𝑃^𝑇_𝑗𝐸. (26)
Let 𝑋_𝑖=𝑃_𝑖⁻¹; then pre- and postmultiply (25) by diag{𝑋^𝑇_𝑖,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 term 𝐸^𝑇𝑃_𝑗 gives some problems, [25] presented a method to deal with problem; that is, 𝐸^𝑇𝑃_𝑗≤(1/4)𝜖_𝑗⁻¹𝐼+𝜖_𝑗𝐸^𝑇𝑃_𝑗𝑃^𝑇_𝑗𝐸. In this paper, we overcome it by (23); let 𝑇⁻¹_𝑖𝑗=𝜖_𝑗⁻¹; we have 𝐸^𝑇𝑃_𝑗≤(1/4)𝜖_𝑗⁻¹𝐼+𝜖_𝑗𝐸^𝑇𝑃_𝑗𝑃^𝑇_𝑗𝐸. Therefore, we have 𝐸^𝑇𝑃_𝑗 ≤ (1/4)𝑇⁻¹_𝑖𝑗 + 𝐸^𝑇𝑃_𝑗𝑇_𝑖𝑗𝑃^𝑇_𝑗𝐸 ≤ (1/4)𝜖_𝑗⁻¹𝐼+𝜖_𝑗𝐸^𝑇𝑃_𝑗𝑃^𝑇_𝑗𝐸.

Theorem 12. Given a scalar 𝛾>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 𝑋_𝑖, 𝑇_𝑖𝑗>0, and 𝑌_𝑖 such that 𝑋^𝑇_𝑖𝐸^𝑇=𝐸𝑋_𝑖≥0,Ω_𝑖𝐹_𝑖𝑋^𝑇_𝑖𝐶^𝑇_𝑖𝑆^𝑇_𝑖(𝑋)𝑆^𝑇_𝑖(𝑋)∗−𝛾²𝐼𝐷^𝑇_𝑖00∗∗−𝐼00∗∗∗𝑅_𝑖0∗∗∗∗𝑅_𝑖<0, (27)(28) whereΩ_𝑖=𝐴_𝑖𝑋_𝑖+𝐵_𝑖𝑌_𝑖^⋆−^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗𝑋^𝑇_𝑖𝐸^𝑇𝑆^𝑇_𝑖(𝑋)=𝜆_𝑖1𝑋^𝑇_𝑖⋯𝜆_{𝑖(𝑖−1)}𝑋^𝑇_𝑖𝜆_𝑖(𝑖+1)𝑋^𝑇_𝑖⋯𝜆_𝑖𝑁𝑋^𝑇_𝑖𝑆^𝑇_𝑖(𝑋)=𝜆_𝑖1𝑋^𝑇_𝑖𝐸^𝑇⋯𝜆_{𝑖(𝑖−1)}𝑋^𝑇_𝑖𝐸^𝑇𝜆_𝑖(𝑖+1)𝑋^𝑇_𝑖𝐸^𝑇⋯𝜆_𝑖𝑁𝑋^𝑇_𝑖𝐸^𝑇𝑅_𝑖=−diag4𝑇_𝑖1,…,4𝑇_{𝑖(𝑖−1)},4𝑇_𝑖(𝑖+1)…,4𝑇_𝑖𝑁𝑅_𝑖=−diag𝑋₁^⋆−𝑇_𝑖1,…,𝑋_𝑖−1^⋆−𝑇_{𝑖(𝑖−1)},𝑋_𝑖+1^⋆−𝑇_𝑖(𝑖+1),…,𝑋_𝑁^⋆−𝑇_𝑖𝑁 (29)hold for all 𝑖∈𝕊 with constraint (4). The controller gain 𝐾_𝑖 can be constructed as 𝐾_𝑖=𝑌_𝑖𝑋_𝑖⁻¹. (30)

Proof. Substituting 𝐴_𝑖 with 𝐴_𝑖+𝐵_𝑖𝐾_𝑖, 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 scalar 𝛾>0, the unforced system (9) with constraint (12) is stochastically admissible with 𝐻_∞ performance if there exist matrices 𝑃_𝑖, 𝐺_𝑖, and 𝑇_𝑖𝑗>0 such that𝐸^𝑇𝑃_𝑖𝐸≥0,Φ_𝑖11Φ_𝑖1200𝐺^𝑇_𝑖𝐶^𝑇_𝑖𝐺^𝑇_𝑖0∗Φ_𝑖22𝐺^𝑇_𝑖𝑇0000∗∗−2𝐼+_𝑖𝑗𝑇_𝑖𝑗𝐹_𝑖00𝑆_𝑖1(𝑃)∗∗∗−𝛾²𝐼+𝐹^𝑇_𝑖𝑇_𝑖𝑗𝐹_𝑖𝐷^𝑇_𝑖0𝑆_𝑖2(𝑃)∗∗∗∗−𝐼00∗∗∗∗∗−𝜋_𝑖𝐸^𝑇𝑃_𝑖𝑅𝐸−2𝐼0∗∗∗∗∗∗_𝑖<0, (31)(32)
where Φ_𝑖11=𝐴_𝑖𝐺_𝑖−𝐺_𝑖^⋆,Φ_𝑖12=𝐺^𝑇_𝑖𝐴^𝑇_𝑖−𝐺_𝑖,Φ_𝑖22=−2𝐺_𝑖−2𝐺^𝑇_𝑖𝑇_𝑖𝑗=14^𝑁_𝑗=1𝜋_𝑖𝑗𝑇_𝑖𝑗,𝐺_𝑖=𝐺_𝑖𝜋+𝐼,_𝑖=^𝑁_𝑗=1𝜋_𝑖𝑗,𝑅_𝑖=−diag𝑇_𝑖1,…,𝑇_𝑖𝑁𝑆^𝑇_𝑖1(𝑃)=𝜋_𝑖1𝑃₁⋯𝜋_𝑖𝑁𝑃_𝑁,𝑆^𝑇_𝑖2(𝑃)=𝜋_𝑖1𝐹^𝑇_𝑖𝑃₁⋯𝜋_𝑖𝑁𝐹^𝑇_𝑖𝑃_𝑁 (33) hold for all 𝑖∈𝑆.

Proof. From (16) and (17), similar to [23], we have (18)=𝑍^𝑇₁𝑍^𝑇₂𝑍^𝑇₃Ψ_𝑖+Π^𝑇_𝑖𝑃_𝑖Π_𝑖𝑍₃𝑍₂𝑍₁, (34) where 𝑍₁=𝐺_𝑖⁻¹𝐺00_𝑖−1𝐴_𝑖000𝐼000𝐼,𝑍₂=𝐼0000𝐼000𝐺_𝑖,𝑍0000𝐼0000𝐼₃=𝐺𝐼00000𝐼00000𝐼00000𝐼00000𝐼_𝑖Π0000^𝑇𝑖=00𝐼𝐹^𝑇_𝑖00,Ψ_𝑖=Φ_𝑖11Φ_𝑖1200𝐺^𝑇_𝑖𝐶^𝑇_𝑖𝐺^𝑇_𝑖∗Φ_𝑖22𝐺^𝑇_𝑖000∗∗−2𝐼000∗∗∗−𝛾²𝐼𝐷^𝑇_𝑖0∗∗∗∗−𝐼0∗∗∗∗∗−𝐸^𝑇𝑃_𝑖.𝐸−2𝐼 (35)
Since ∑^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗≤∑^𝑁_𝑗=1𝜋_𝑖𝑗((1/4)𝑇_𝑖𝑗+𝑃_𝑗𝑇⁻¹_𝑖𝑗𝑃_𝑗) with 𝑇_𝑖𝑗>0, ∀𝑖,𝑗∈𝕊 and (12), we have Π^𝑇_𝑖𝑃_𝑖Π_𝑖≤Π^𝑇_𝑖^𝑁_𝑗=1𝜋_𝑖𝑗14𝑇_𝑖𝑗+𝑃_𝑗𝑇⁻¹_𝑖𝑗𝑃_𝑗Π_𝑖. (36)
From (12), we obtain that 𝜋_𝑖𝐸^𝑇𝑃_𝑖𝐸=^𝑁_𝑗=1𝜋_𝑖𝑗𝐸^𝑇𝑃_𝑖𝐸≤^𝑁_𝑗=1𝜋_𝑖𝑗𝐸^𝑇𝑃_𝑖𝐸. (37)
Taking into account (34), (36), and (37), we have Ψ_𝑖+Π^𝑇_𝑖𝑃_𝑖Π_𝑖≤Ψ_𝑖+Π^𝑇_𝑖^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗𝑇⁻¹_𝑖𝑗𝑃_𝑗Π_𝑖<0, (38) whereΨ_𝑖=Φ_𝑖11Φ_𝑖1200𝐺^𝑇_𝑖𝐶^𝑇_𝑖𝐺^𝑇_𝑖∗Φ_𝑖22𝐺^𝑇_𝑖𝑇000∗∗−2𝐼+_𝑖𝑗𝑇_𝑖𝑗𝐹_𝑖00∗∗∗−𝛾²𝐼+𝐹^𝑇_𝑖𝑇_𝑖𝑗𝐹_𝑖𝐷^𝑇_𝑖0∗∗∗∗−𝐼0∗∗∗∗∗−𝜋_𝑖𝐸^𝑇𝑃_𝑖.𝐸−2𝐼 (39)
Then, by the Schur complement, we see that (32) is equivalent to (38). This completes the proof.

Remark 15. It should be remarked that when 𝐸=𝐼 in (9), [24] gave important criteria under the constraint (12) via ∑^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗≤∑^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗 and ∑^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑖≤∑^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑖, where 𝑃_𝑗 must be positive-definite matrix which is not applicable in SMJSs. In SMJS (9), 𝑃_𝑖 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≤∑^𝑁_𝑗=1𝜋_𝑖𝑗((1/4)𝑇_𝑖𝑗+𝑃_𝑗𝑇⁻¹_𝑖𝑗𝑃_𝑗). Let 𝑇_𝑖𝑗=2𝑃_𝑗; we have (1/4)𝑇_𝑖𝑗+𝑃_𝑗𝑇⁻¹_𝑖𝑗𝑃_𝑗=𝑃_𝑗; thus, we have 𝑃_𝑖≤∑^𝑁_𝑗=1𝜋_𝑖𝑗((1/4)𝑇_𝑖𝑗+𝑃_𝑗𝑇⁻¹_𝑖𝑗𝑃_𝑗)≤∑^𝑁_𝑗=1𝜋_𝑖𝑗𝑃_𝑗.

Theorem 16. Given a scalar 𝛾>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 𝐺_𝑖, 𝑃_𝑖, 𝑌_𝑖, and 𝑇_𝑖𝑗>0 such that𝐸^𝑇𝑃_𝑖𝐸≥0Φ_𝑖11Φ_𝑖1200𝐺^𝑇_𝑖𝐶^𝑇_𝑖𝐺^𝑇_𝑖0∗Φ_𝑖22𝐺^𝑇_𝑖𝑇0000∗∗−2𝐼+_𝑖𝑗𝑇_𝑖𝑗𝐹_𝑖00𝑆_𝑖1(𝑃)∗∗∗−𝛾²𝐼+𝐹^𝑇_𝑖𝑇_𝑖𝑗𝐹_𝑖𝐷^𝑇_𝑖0𝑆_𝑖2(𝑃)∗∗∗∗−𝐼00∗∗∗∗∗−𝜋_𝑖𝐸^𝑇𝑃_𝑖𝑅𝐸−2𝐼0∗∗∗∗∗∗_𝑖<0, (40)(41)where Φ_𝑖11=𝐴_𝑖𝐺_𝑖+𝐵_𝑖𝑌_𝑖−𝐺_𝑖^⋆,Φ_𝑖12=𝐺^𝑇_𝑖𝐴^𝑇_𝑖+𝑌^𝑇_𝑖𝐵^𝑇_𝑖−𝐺_𝑖 (42) hold for all 𝑖∈𝕊 with constraint (12). The controller gain 𝐾_𝑖 can be constructed as 𝐾_𝑖=𝑌_𝑖𝐺_𝑖⁻¹. (43)

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 𝐸=100010000,𝐴₁=,𝐴0.1−0.200.5−1−0.50.100.4₂=0.1−10−0.3−10.400.20.1,𝐴₃=,𝐵0.600.4−0.400.7−0.30.1−0.4₁=10−1,𝐵₂=01−1,𝐵₃=00,𝐹−1₁=0.10.2−0.1,𝐹₂=0.1−0.20.1,𝐹₃=,𝐶−0.10.20.1₁=100,𝐶₂=011,𝐶₃=,𝐷1−11₁=0.2,𝐷₂=0.1,𝐷₃=−0.3. (44)
Assume the bound transition probability matrices are Λ=,−2.51.31.20.8−1.91.10.80.7−1.5Λ=.−2.91.51.41−2.21.210.9−1.9 (45)
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 𝐾₁=,𝐾−6.331510.066996.5153₂=,𝐾2.9966−9.0875−59.5261₃=24.7500−22.4333−17.1457. (46)

Example 2. Consider the following discrete-time SMJS as (9) described as 𝐸=0.50.611.2,𝐴₁=,𝐴0.10.80.151₂=1.10.60.5−1,𝐴₃=,𝐵0.110.5−0.9₁=0−1,𝐵₂=11,𝐵₃=1,𝐹−1₁=0.10.2,𝐹₂=0.1−0.2,𝐹₃=,𝐶−0.10.2₁=10,𝐶₂=01,𝐶₃=,𝐷1−1₁=𝐷₂=0.1,𝐷₃=−0.1. (47)
Assume the bound transition probability matrices are Λ=,0.20.400.300.30.10.40.1Λ=.0.60.800.700.50.30.70.3 (48)
Let 𝛾=1; then via Theorem 16, we obtain the controller gain 𝐾₁=𝐾12.30409.9997₂=𝐾−13.0975−4.5714₃=10.6808−7.6430. (49)

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 𝐴₁=0.8−0.10.20.5,𝐴₂=,𝐴0.60−0.10.2₃=𝐵−0.6100.7¹=00.1,𝐵₂=0,𝐵0.1₃=0𝐶0.2¹=−0.20,𝐶₂=,𝐶0.60₃=0.30𝐷₁=−0.4,𝐷₂=0.2,𝐷₃=0.3. (50)
Assume the bound transition probability matrices are Λ=,0.20.400.300.30.10.40.1Λ=.0.60.800.700.50.30.70.3 (51)
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 of 𝐻_∞ 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.

Acknowledgments

This work was supported by Scientific Research Fund of Liaoning Provincial Education Department under Grant no. L2012464, Young Scientific Research Fund of Shenyang Institute of Engineering under Grant no. LGQN-1308, the National Natural Science Foundation of China under Grants nos. 61104066 and 61374043, and the Program for Liaoning Excellent Talents in University under Grant no. LJQ2013040.