𝐻₂ Control for the Continuous-Time Markovian Jump Linear Uncertain Systems with Partly Known Transition Rates and Input Quantization

Abstract

For a class of continuous-time Markovian jump linear uncertain systems with partly known transition rates and input quantization, the 𝐻₂ state-feedback control design is considered. The elements in the transition rates matrix include completely known, boundary known, and completely unknown ones. First, an 𝐻₂ cost index for Markovian jump linear uncertain systems is introduced; then by introducing a new matrix inequality condition, sufficient conditions are formulated in terms of linear matrix inequalities (LMIs) for the 𝐻₂ control of the Markovian jump linear uncertain systems. Less conservativeness is achieved than the result obtained with the existing technique. Finally, a numerical example is given to verify the validity of the theoretical results.

1. Introduction

Recently, much attention has been devoted to the study of the stochastic stability for the Markovian jump systems, and many important results have been published [1–4]. This is because the Markovian jump systems have been widely employed to model many practical systems, such as manufacturing systems, the power systems, and the economic systems in which they may experience abrupt changes in their structures and parameters [5, 6]. It is worth noticing that these results require that the transition probabilities/rates must be known a priori. However, in many practical engineering applications, the likelihood for obtaining the perfect information on all transition probabilities/rates elements is questionable, and the cost might be expensive in some cases. Therefore, the study of the stabilization of the Markovian jump systems with partly known transition probabilities/rates becomes interesting, and some well-known results have been published. The idea for the stochastic stability of the Markovian jump linear uncertain systems with partly known transition probabilities/rates is developed in Zhang et al. [7]. It is then applied to the 𝐻_∞ control design in Zhang and Boukas [8]. In those papers, the feature of the information about the transition probabilities/rates matrix considered includes two kinds of elements: completely known and completely unknown ones. As a matter of fact, the transition probabilities matrix might involve completely known, completely unknown, and boundary known elements. In Shen and Yang [9], an 𝐻₂ state-feedback controller design method is proposed for the continuous-time Markovian jump linear uncertain systems with the three kinds of transition rates matrix elements. In order to yield the design condition for analysis and synthesis, a matrix inequality is introduced to present LMIs conditions. In this paper, less conservative design conditions will be formulated by introducing a new relaxed matrix inequality condition.

On the other hand, in many modern engineering practices, all kinds of information processing devices, such as analog-to-digital and digital-to-analog converters, have been widely used. By the utilization of such information processing devices, some advantages have been brought, for example, lower cost, reduced weight and power, simple installation, and maintenance. However, some new phenomena have also been induced, which might cause server deterioration of system performance or even lead to system instability. Signal quantization is one of the important aspects that should be fully considered in such cases, which always exists in computer-based control systems. Nowadays, many well-known results have been published on quantized feedback control. For example, the feedback stabilization problem is considered by utilizing dynamic quantizers [10–13] and static quantizers [14–17]. In addition, the filter design [18] and the 𝐻_∞ control design [19] are also investigated. Specially, quantization errors have adverse effects on the network control systems which can often be modeled as Markovian jump systems. In Xiao et al. [20], the stabilization problem for single-input discrete Markovian jump linear uncertain systems via mode-dependent quantized state-feedback is addressed, but the transition rates are assumed to be completely known.

To the best of our knowledge, no result has been presented for the control design of the continuous-time Markovian jump linear uncertain systems with partly known transition rates and input signal quantization. In this paper, the 𝐻₂ control for a class of continuous-time Markovian jump linear uncertain systems with respect to partly known transition rates and input signal quantization is addressed. The structure of the controller consists of two parts: the nonlinear part is provided to eliminate the effect of input quantization, and the linear part is obtained by solving LMIs for achieving the 𝐻₂ performance against unknown transition rates and model uncertainties. In comparison with the design utilizing the LMIs technique in Shen and Yang [9], the design method has less conservativeness by introducing a relaxed inequality condition.

The rest of this paper is organized as follows. The problem statement and preliminaries are presented in Section 2. The main results are given in Section 3. In Section 4, a numerical example is presented to illustrate the effectiveness of the results, and the conclusions are drawn in Section 5.

Notations. Throughout this paper, the following notations are used. 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space; 𝐴^𝑇 denotes the transpose of matrix 𝐴; 𝐼 and 0 represent the identity matrix and a zero matrix in appropriate dimensions, respectively; 𝔼{⋅} denotes the mathematical expectation operator; 𝑋>𝑌(𝑋≥𝑌), where 𝑋 and 𝑌 are symmetric matrices, means that 𝑋−𝑌 is positive definite (positive semi-definite); |𝑥|_𝑝 denotes the 𝑝-norm of the vector 𝑥; that is, |𝑥|_𝑝=(|𝑥₁|^𝑝+|𝑥₂|^𝑝+⋯+|𝑥_𝑛|^𝑝)^1/𝑝, where 𝑝≥1. When 𝑝=∞, |𝑥|_∞=max_{1≤𝑖≤𝑛}|𝑥_𝑖|. For matrix 𝑋∈𝑅^𝑚×𝑛, |𝑋|_𝑝 is used to present the matrix 𝑝-norm: |𝑋|_𝑝=sup_𝑦≠0(|𝑋𝑦|_𝑝/|𝑦|_𝑝). The notation |⋅|, in particular denotes the absolute value of a scalar, the standard Euclidean norm of a vector, and the induced norm of a matrix, respectively. In symmetric block matrices, an ∗ is used to represent a term that is induced by symmetry. Finally, the symbol 𝐻𝑒(𝑋) is used to represent 𝑋+𝑋^𝑇.

2. Problem Statement and Preliminaries

Consider a class of the continuous-time Markovian jump linear uncertain systems in the following probability space (Ω,ℱ,𝒫): 𝑥̇(𝑡)=𝐴𝑟_𝑡+Δ𝐴𝑟_𝑡𝑥(𝑡)+𝐵𝑟_𝑡+Δ𝐵𝑟_𝑡𝑞(𝑢(𝑡,𝑥))₀=𝑥(0),𝑟₀=𝑟(0), (1) where 𝑥(𝑡)∈𝑅^𝑛 is the system state and 𝑢(𝑡)∈𝑅^𝑚 is the control input. {𝑟_𝑡,𝑡≥0} is a continuous-time Markovian process with right continuous trajectories taking values in the finite set 𝒮={1,2,…,ℕ}. It governs the switching among the different system modes with the following mode transition probabilities: Pr𝑟_𝑡+ℎ=𝑗∣𝑟_𝑡=𝜆=𝑖^𝑖𝑗ℎ+𝑜(ℎ),if𝑗≠𝑖,1+𝜆_𝑖𝑖ℎ+𝑜(ℎ),if𝑗=𝑖, (2)

where ℎ>0, lim_ℎ→0(𝑜(ℎ)/ℎ)=0, and 𝜆_𝑖𝑗≥0(𝑖,𝑗∈𝒮;𝑗≠𝑖) denote the switching rate from mode 𝑖 at time 𝑡 to mode 𝑗 at time 𝑡+ℎ and that 𝜆_𝑖𝑖=−∑^ℕ_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗 for each 𝑖∈𝒮.

In general, the Markovian process transition rates matrix Λ is defined by: 𝜆Λ=¹¹𝜆₁₂⋯𝜆_1ℕ𝜆₂₁𝜆₂₂⋯𝜆_2ℕ𝜆⋮⋯⋱⋮_ℕ1⋯⋯𝜆_ℕℕ. (3) In this paper, the transition rates of the jumping process are assumed to be partly available; that is, some elements in matrix Λ have been exactly known, some ones have been merely known with lower and upper bounds, and others may have no information to use. For instance, for the system (1) with four operation modes, the transition rates matrix might be described by 𝜆Λ=¹¹??𝜆₁₄??𝜆₂₃???𝛽?𝛼?𝜆₄₃𝜆₄₄, (4)

where ? represents the completely unknown element of the transition rates matrix and the parameters 𝛼 and 𝛽 represent the elements with known lower and upper bounds. That is, 𝛼≤𝛼≤𝛼, and 𝛽≤𝛽≤𝛽, where 𝛼, 𝛼, 𝛽, and 𝛽 are known parameters; 𝜆_𝑖𝑗 denotes the precisely known element.

For clarity, we denote that 𝒮=𝒮^𝑖_𝑘1∪𝒮^𝑖_𝑘2∪𝒮^𝑖_𝑢𝑘,  𝑖=1,2,…,ℕ, with 𝒮^𝑖_𝑘1=𝑗∶𝜆_𝑖𝑗isexactlyknown,𝒮^𝑖_𝑘2=𝑗∶theboundsof𝜆_𝑖𝑗areknown,𝒮^𝑖_𝑢𝑘=𝑗∶thereisnoinformationavailablefor𝜆_𝑖𝑗. (5) Furthermore, let 𝒮^𝑖_𝑘={𝑗∶𝜆_𝑖𝑗≤𝜆_𝑖𝑗≤𝜆_𝑖𝑗}; then one can obtain 𝒮^𝑖_𝑘=𝒮^𝑖_𝑘1∪𝒮^𝑖_𝑘2. If 𝒮^𝑖_𝑘≠∅, it can be described as 𝒮^𝑖_𝑘=𝒦^𝑖₁,𝒦^𝑖₂,…,𝒦^𝑖_𝑚𝑖𝑘. (6) Similarly, if 𝒮^𝑖_𝑢𝑘≠∅, let us denote that 𝒮^𝑖_𝑢𝑘=𝕂^𝑖₁,𝕂^𝑖₂,…,𝕂^𝑖_{𝑚𝑖𝑢𝑘}, (7) where 𝒦^𝑖_𝑙1 denotes the 𝑙₁th element in 𝒮^𝑖_𝑘 with the index 𝒦^𝑖_𝑙1 in the 𝑖th row of the matrix Λ. 𝕂^𝑖_𝑙2 denotes the 𝑙₂th completely unknown element with the index 𝕂^𝑖_𝑙2 in the 𝑖th row of the matrix Λ. 𝑚^𝑖_𝑘 and 𝑚^𝑖_𝑢𝑘 represent the number of elements in 𝒮^𝑖_𝑘 and 𝒮^𝑖_𝑢𝑘, respectively. For example, considering the transition rates matrix (4), one can easily check that 𝒮⁴_𝑘={𝒦⁴₁,𝒦⁴₂,𝒦⁴₃} with 𝒦⁴₁=1, 𝒦⁴₂=3, 𝒦⁴₃=4, 𝒮⁴_𝑢𝑘={𝕂⁴₁} with 𝕂⁴₁=2,  𝑚⁴_𝑘=3, and 𝑚⁴_𝑢𝑘=1.

Remark 1. When the lower and upper bounds of the elements in 𝒮^𝑖_𝑘2 are equal, the transition rates matrix is reduced to the considered case in Zhang et al. [7]. It is obvious that the solving method there can only treat 𝑗∈𝒮^𝑖_𝑘2 as the completely unknown case which can result in some conservativeness.

For convenience, the notations 𝐴_𝑖=𝐴(𝑟_𝑡), 𝐵_𝑖=𝐵(𝑟_𝑡), Δ𝐴_𝑖=Δ𝐴(𝑟_𝑡), and Δ𝐵_𝑖=Δ𝐵(𝑟_𝑡) are used for each possible value 𝑟_𝑖=𝑖, 𝑖∈𝒮, where 𝐴_𝑖 and 𝐵_𝑖 are known constant matrices with appropriate dimensions. Then, the system (1) can be described by 𝑥̇(𝑡)=𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑡)+𝐵_𝑖+Δ𝐵_𝑖𝑞(𝑢(𝑡,𝑥))₀=𝑥(0),𝑟₀=𝑟(0). (8)

The following assumptions are assumed to be valid.

Assumption 1. The pair (𝐴_𝑖,𝐵_𝑖) is controllable.

Assumption 2. Consider the following: Δ𝐴_𝑖=𝐷_𝑖𝐸_𝑖(𝑡)𝐹_𝑖, Δ𝐵_𝑖=𝐵_𝑖𝑀_𝑖Ξ_𝑖(𝑡)𝑁_𝑖, and |𝑀_𝑖Ξ_𝑖(𝑡)𝑁_𝑖|_∞≤𝜓_𝑖, where 𝐷_𝑖, 𝐹_𝑖, 𝑀_𝑖, and 𝑁_𝑖 are known constant matrices with appropriate dimensions, 𝐸_𝑖(𝑡) and Ξ_𝑖(𝑡) are time-varying uncertain matrices satisfying 𝐸_𝑖𝐸^𝑇_𝑖≤𝐼 and Ξ_𝑖Ξ^𝑇_𝑖≤𝐼, and the parameter 𝜓_𝑖 satisfies 0≤𝜓_𝑖<1.

In addition, the quantizer 𝑞(⋅) is defined by an operator function round(⋅) that rounds towards the nearest integer; that is, 𝑞(𝑢(𝑡))=𝜇⋅round𝑢(𝑡)𝜇, (9)

where 𝜇(>0) is called a quantizing level of the quantizer. In computer-based control systems, the value of 𝜇 depends on the sampling accuracy and is known a priori. 𝑞(⋅) is the uniform quantizer with the fixed level 𝜇. Define 𝑒_𝜇=𝑞(𝑢(𝑡))−𝑢(𝑡), since each component of 𝑒_𝜇 is bounded by the half of the quantizing level 𝜇; thus, we have |𝑒_𝜇|_∞≤𝜇/2.

The objective of this paper is to design the state-feedback control law 𝑢(𝑡)=𝐾_𝑖𝑥+𝑢_𝑖𝑐,𝐾_𝑖=𝐾𝑟_𝑡,when𝑟_𝑡=𝑖, (10) such that the resulting closed-loop system is stochastically stable and obtains as small value of the 𝐻₂ cost index for the Markovian jump linear uncertain systems given in the following as possible 𝒥(𝑡)=𝔼^∞_𝑡𝑥^𝑇(𝜏)𝑄𝑟_𝜏𝑥(𝜏)+𝑥^𝑇(𝜏)𝐾^𝑇𝑟_𝜏𝑅𝑟_𝜏𝐾𝑟_𝜏𝑥(𝜏)𝑑𝜏∣𝑟₀, (11) where 𝑄(𝑟_𝑡) and 𝑅(𝑟_𝑡) are positive definite matrices. The nonlinear part of the controller 𝑢_𝑖𝑐 is designed against the effect of signal quantization, and the linear part 𝐾_𝑖𝑥 is proposed to deal with model uncertainties and to unknown transition rates and achieve optimal 𝐻₂ performance.

Remark 2. When 𝑄(𝑟_𝑡)=𝑄>0, 𝑅(𝑟_𝑡)=𝑅>0, and 𝐾(𝑟_𝑡)=𝐾, the above 𝐻₂ cost index is reduced to the following, (12) which is given in Yun et al. [17]: 𝒥(𝑡)=^∞_𝑡𝑥^𝑇(𝜏)𝑄𝑥(𝜏)+𝑥^𝑇(𝜏)𝐾^𝑇𝑅𝐾𝑥(𝜏)𝑑𝜏. (12)

Some useful lemmas are firstly presented before formulating the main result.

Lemma 3 (see [21]). Given a symmetric matrix Π and matrices 𝑀 and 𝑁 with appropriate dimensions, then Π+𝑀𝐹(𝑡)𝑁+𝑁^𝑇𝐹^𝑇(𝑡)𝑀^𝑇<0 for all 𝐹(𝑡) satisfying 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼, if and only if there exists a scalar 𝜀>0 such that the following inequality holds: Π+𝜀𝑀𝑀^𝑇+𝜀⁻¹𝑁^𝑇𝑁<0. (13)

Lemma 4. For any given 𝜆_𝑗≥0 and matrices 𝑃_𝑗>0  (𝑃_𝑗∈𝑅^𝑛×𝑛,1≤𝑗≤ℕ), if there exists 𝑍_𝑖≥𝑃_𝑗, then the following inequality holds: ^ℕ_𝑗=1𝜆_𝑗𝑃_𝑗≤^ℕ_𝑗=1𝜆_𝑗𝑍_𝑖. (14)

Proof. According to 𝑍_𝑖≥𝑃_𝑗 and 𝜆_𝑗≥0, one can obtain that 𝜆_𝑗𝑍_𝑖≥𝜆_𝑗𝑃_𝑗, which further imply that inequality (14) holds.

Remark 5. In Shen and Yang [9], the inequality ∑^ℕ_𝑗=1𝜆_𝑗𝑃_𝑗≤(∑^ℕ_𝑗=1𝜆_𝑗)(∑^ℕ_𝑗=1𝑃_𝑗) is introduced to obtain design conditions. It is clear that the utilization of Lemma 4 will result in less conservativeness since max{𝑃₁,𝑃₂,…,𝑃_ℕ}<∑^ℕ_𝑗=1𝑃_𝑗 for any 𝑃_𝑖>0.

Lemma 6. For 𝛼, 𝛽∈𝑅^𝑛, 𝑝≥1, and 𝑞≥1, the following inequality holds: 𝛼^𝑇𝛽≤|𝛼|_𝑝𝛽_𝑞,𝑝⁻¹+𝑞⁻¹=1. (15)

Lemma 7 (see [22]). For the symmetric and positive definite matrices 𝑃 and 𝑄, if 𝑃≥𝑄>0, then 𝑄⁻¹≥𝑃⁻¹>0.

3. Main Results

Theorem 8. For the system (1) subject to Assumptions 1 and 2, suppose that there exist the symmetric positive definite matrices 𝑋_𝑖, general matrices 𝑌_𝑖, and positive scalars 𝜀_𝑖, 𝛿_𝑖, and 𝛾 such that Γ₁₁𝑋_𝑖𝐹^𝑇_𝑖𝑌^𝑇_𝑖𝑁^𝑇_𝑖Γ₁₄Γ₁₅𝑋_𝑖𝑌_𝑖∗−𝜀_𝑖𝐼00000∗∗−𝛿_𝑖𝐼0000∗∗∗Γ₄₄000∗∗∗∗−𝑍_𝑖00∗∗∗∗∗−𝑄_𝑖⁻¹0∗∗∗∗∗∗−𝑅_𝑖⁻¹𝑍<0,_𝑖≤𝑋_𝑗,𝑗∈𝒮^𝑖_𝑢𝑘,Φ₁₁𝑋_𝑖𝐹^𝑇_𝑖𝑌^𝑇_𝑖𝑁^𝑇_𝑖Φ₁₄𝑋_𝑖𝑌^𝑇_𝑖∗−𝜀_𝑖𝐼0000∗∗−𝛿_𝑖𝐼000∗∗∗Φ₄₄00∗∗∗∗−𝑄_𝑖⁻¹0∗∗∗∗∗−𝑅_𝑖⁻¹𝑋<0,_𝑖≤𝑋_𝑗,𝑗∈𝒮^𝑖_𝑢𝑘,𝛾𝑥^𝑇₀𝑥₀𝑋_𝑖>0, (16)(17)(18) where Γ₁₁=𝐻𝑒𝐴_𝑖𝑋_𝑖+𝐻𝑒𝐵_𝑖𝑌_𝑖+𝜀_𝑖𝐷_𝑖𝐷^𝑇_𝑖+𝛿_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖+𝜆_𝑖𝑖𝑋_𝑖,Γ₁₄=𝜆_𝑖𝒦^𝑖₁𝑋_𝑖,𝜆_𝑖𝒦^𝑖₂𝑋_𝑖,…,𝜆_𝑖𝒦^𝑖_𝑚𝑖𝑘𝑋_𝑖,Γ₄₄=diag−𝑋_𝒦^𝑖₁,−𝑋_𝒦^𝑖₂,…,−𝑋_𝒦^𝑖_𝑚𝑖𝑘,Γ₁₅=−_{𝑗∈𝒮𝑖𝑘}𝜆_𝑖𝑗𝑋_𝑖,Φ₁₁=𝐻𝑒𝐴_𝑖𝑋_𝑖+𝐻𝑒𝐵_𝑖𝑌_𝑖+𝜀_𝑖𝐷_𝑖𝐷^𝑇_𝑖+𝛿_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖−_{𝑗∈𝒮𝑖𝑘}𝜆_𝑖𝑗𝑋_𝑖,Φ₁₄=𝜆_𝑖𝒦^𝑖₁𝑋_𝑖,𝜆_𝑖𝒦^𝑖₂𝑋_𝑖,…,𝜆_𝑖𝒦^𝑖_𝑚𝑖𝑘𝑋_𝑖,Φ₄₄=diag−𝑋_𝒦^𝑖₁,−𝑋_𝒦^𝑖₂,…,−𝑋_𝒦^𝑖_𝑚𝑖𝑘. (19) Then, the controller designed as 𝑢(𝑡)=𝐾_𝑖𝑥(𝑡)+𝑢_𝑖𝑐(𝑡) (20) can drive the state trajectory to the origin asymptotically and can obtain the 𝐻₂ cost by the minimum of 𝑟, where 𝐾_𝑖=𝑌_𝑖𝑋_𝑖⁻¹, 𝑃_𝑖=𝑋_𝑖⁻¹, and 𝑢_𝑖𝑐=(−(1+𝜓)𝜇/2(1−𝜓))sign(𝑥^𝑇𝑃_𝑖𝐵_𝑖).

Proof. Take the Lyapunov function candidate 𝑉=𝑥^𝑇𝑃_𝑖𝑥; then, along the system trajectory of plant (1), the weak infinitesimal operator ℑ^𝑥_𝑎[⋅] of the process {𝑥(𝑡),𝑟_𝑡,𝑡≥0} for plant (8) at the point {𝑡,𝑥,𝑖} is given by Kushner [23] as follows: ℑ^𝑥_𝑎[𝑉]=𝑥̇^𝑇𝑃_𝑖𝑥+𝑥^𝑇𝑃_𝑖𝑥̇+𝑥^𝑇ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗𝑥=𝐴_𝑖+Δ𝐴_𝑖𝑥+𝐵_𝑖+Δ𝐵_𝑖𝐾_𝑖𝑥+𝑢_𝑖𝑐+𝑒_𝜇^𝑇𝑃_𝑖𝑥+𝑥^𝑇𝑃_𝑖𝐴_𝑖+Δ𝐴_𝑖𝑥+𝐵_𝑖+Δ𝐵_𝑖𝐾_𝑖𝑥+𝑢_𝑖𝑐+𝑒_𝜇+𝑥^𝑇ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗𝑥=𝑥^𝑇𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖+Δ𝐴_𝑖+𝑃_𝑖𝐵_𝑖+Δ𝐵_𝑖𝐾_𝑖+𝐾^𝑇_𝑖𝐵_𝑖+Δ𝐵_𝑖^𝑇𝑃_𝑖𝑥+2𝑥^𝑇𝑃_𝑖𝐵_𝑖+Δ𝐵_𝑖𝑢_𝑖𝑐+𝑒_𝜇+𝑥^𝑇ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗𝑥. (21) According to Assumption 2, 𝑢_𝑖𝑐 in Theorem 8, and Lemma 7, one can obtain that 2𝑥^𝑇𝑃_𝑖𝐵_𝑖+Δ𝐵_𝑖𝑢_𝑖𝑐+𝑒_𝜇=2𝑥^𝑇𝑃_𝑖𝐵_𝑖𝑢_𝑖𝑐+2𝑥^𝑇𝑃_𝑖Δ𝐵_𝑖𝑢_𝑖𝑐+𝑒_𝜇+2𝑥^𝑇𝑃_𝑖𝐵_𝑖𝑒_𝜇=2𝑥^𝑇𝑃_𝑖𝐵_𝑖𝑢_𝑖𝑐+2𝑥^𝑇𝑃_𝑖𝐵_𝑖𝑀_𝑖Ξ_𝑖(𝑡)𝑁_𝑖𝑢_𝑖𝑐+𝑒_𝜇+2𝑥^𝑇𝑃_𝑖𝐵_𝑖𝑒_𝜇≤2𝑥^𝑇𝑃_𝑖𝐵_𝑖𝑢_𝑖𝑐+2𝑥^𝑇𝑃_𝑖𝐵_𝑖1𝑀_𝑖Ξ_𝑖(𝑡)𝑁_𝑖∞𝑢_𝑖𝑐∞+𝜇2+2𝑥^𝑇𝑃_𝑖𝐵_𝑖1𝜇2=2𝑥^𝑇𝑃_𝑖𝐵_𝑖𝑢_𝑖𝑐+2𝑥^𝑇𝑃_𝑖𝐵_𝑖1𝜓𝑢_𝑖𝑐∞+𝜇2+𝑥^𝑇𝑃_𝑖𝐵_𝑖1𝜇=2𝑥^𝑇𝑃_𝑖𝐵_𝑖𝑢_𝑖𝑐+1+𝜓𝑥^𝑇𝑃_𝑖𝐵_𝑖1𝜇+2𝜓𝑥^𝑇𝑃_𝑖𝐵_𝑖1𝑢_𝑖𝑐∞𝜇=−1+𝜓𝑥1−𝜓^𝑇𝑃_𝑖𝐵_𝑖1+1+𝜓𝑥^𝑇𝑃_𝑖𝐵_𝑖1𝜇+𝜓1+𝜓𝜇𝑥1−𝜓^𝑇𝑃_𝑖𝐵_𝑖1=0. (22) It follows from (21) and (22) that ℑ^𝑥_𝑎[𝑉]≤𝑥^𝑇𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖+Δ𝐴_𝑖+𝑃_𝑖𝐵_𝑖+Δ𝐵_𝑖𝐾_𝑖+𝐾^𝑇_𝑖𝐵_𝑖+Δ𝐵_𝑖^𝑇𝑃_𝑖𝑥+𝑥^𝑇ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗𝑥. (23) Furthermore, consider the following inequality: 𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖+Δ𝐴_𝑖+𝑃_𝑖𝐵_𝑖+Δ𝐵_𝑖𝐾_𝑖+𝐾^𝑇_𝑖𝐵_𝑖+Δ𝐵_𝑖^𝑇𝑃+^ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗+𝑄_𝑖+𝐾^𝑇_𝑖𝑅_𝑖𝐾_𝑖<0. (24) Since Δ𝐵_𝑖=𝐵_𝑖𝑀_𝑖Ξ_𝑖(𝑡)𝑁_𝑖 and Δ𝐴_𝑖=𝐷_𝑖𝐸_𝑖(𝑡)𝐹_𝑖, the above inequality can be rewritten as 𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝑃_𝑖𝐷_𝑖𝐸_𝑖(𝑡)𝐹_𝑖+𝐹^𝑇_𝑖𝐸^𝑇_𝑖(𝑡)𝐷^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐵_𝑖𝑀_𝑖Ξ_𝑖(𝑡)𝑁_𝑖𝐾_𝑖+𝐾^𝑇_𝑖𝑁^𝑇_𝑖Ξ^𝑇_𝑖(𝑡)𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+^ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗+𝑄_𝑖+𝐾^𝑇_𝑖𝑅_𝑖𝐾_𝑖<0. (25) Using Lemma 3, it is equivalent to 𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝜀_𝑖𝑃_𝑖𝐷_𝑖𝐷^𝑇_𝑖𝑃_𝑖+1𝜀_𝑖𝐹^𝑇_𝑖𝐹_𝑖+𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝛿_𝑖𝑃_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+1𝛿_𝑖𝐾^𝑇_𝑖𝑁^𝑇_𝑖𝑁_𝑖𝐾_𝑖+^ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗+𝑄_𝑖+𝐾^𝑇_𝑖𝑅_𝑖𝐾_𝑖<0. (26) That is, 𝐻𝑒𝐴^𝑇_𝑖𝑃_𝑖+𝐻𝑒𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝜀_𝑖𝑃_𝑖𝐷_𝑖𝐷^𝑇_𝑖𝑃_𝑖+1𝜀_𝑖𝐹^𝑇_𝑖𝐹_𝑖+𝛿_𝑖𝑃_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+1𝛿_𝑖𝐾^𝑇_𝑖𝑁^𝑇_𝑖𝑁_𝑖𝐾_𝑖+_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑗+_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑗+𝜆_𝑖𝑖𝑃_𝑖+𝑄_𝑖+𝐾^𝑇_𝑖𝑅_𝑖𝐾_𝑖<0. (27)
Two cases will be considered.
Case 1 (𝜆_𝑖𝑖∈𝒮^𝑖_𝑘). In this case, using Lemma 4, one can see that 𝐻𝑒𝐴^𝑇_𝑖𝑃_𝑖+𝐻𝑒𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝜀_𝑖𝑃_𝑖𝐷_𝑖𝐷^𝑇_𝑖𝑃_𝑖+1𝜀_𝑖𝐹^𝑇_𝑖𝐹_𝑖+𝛿_𝑖𝑃_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+1𝛿_𝑖𝐾^𝑇_𝑖𝑁^𝑇_𝑖𝑁_𝑖𝐾_𝑖+_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑗+_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑊_𝑖+𝜆_𝑖𝑖𝑃_𝑖+𝑄_𝑖+𝐾^𝑇_𝑖𝑅_𝑖𝐾_𝑖𝑃<0,_𝑗≤𝑊_𝑖,𝑗∈𝒮^𝑖_𝑢𝑘. (28) Substituting ∑_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝜆_𝑖𝑗=−∑_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗−𝜆_𝑖𝑖 into the first inequality in (22) and using the boundary information of the elements of the transition rates matrix, one can achieve that 𝐻𝑒𝐴^𝑇_𝑖𝑃_𝑖+𝐻𝑒𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝜀_𝑖𝑃_𝑖𝐷_𝑖𝐷^𝑇_𝑖𝑃_𝑖+1𝜀_𝑖𝐹^𝑇_𝑖𝐹_𝑖+𝛿_𝑖𝑃_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+1𝛿_𝑖𝐾^𝑇_𝑖𝑁^𝑇_𝑖𝑁_𝑖𝐾_𝑖+_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑗−_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗+𝜆_𝑖𝑖𝑊_𝑖+𝜆_𝑖𝑖𝑃_𝑖+𝑄_𝑖+𝐾^𝑇_𝑖𝑅_𝑖𝐾_𝑖𝑃<0,_𝑗≤𝑊_𝑖,𝑗∈𝒮^𝑖_𝑢𝑘. (29)
Case 2 (𝜆_𝑖𝑖∈𝒮^𝑖_𝑢𝑘; namely, it is completely unknown). In such case, let us take 𝜆_𝑖𝑖=−∑_𝑗∈𝒮^𝑖_𝑘𝜆_𝑖𝑗−∑_𝑗∈𝒮^𝑖_{𝑢𝑘,𝑗≠𝑖}𝜆_𝑖𝑗 into (21); then, one can see that 𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝜀_𝑖𝑃_𝑖𝐷_𝑖𝐷^𝑇_𝑖𝑃_𝑖+1𝜀_𝑖𝐹^𝑇_𝑖𝐹_𝑖+𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝛿_𝑖𝑃_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+1𝛿_𝑖𝐾^𝑇_𝑖𝑁^𝑇_𝑖𝑁_𝑖𝐾_𝑖+_{𝑗∈𝒮𝑖𝑘}𝜆_𝑖𝑗𝑃_𝑗+_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑗+−_{𝑗∈𝒮𝑖𝑘}𝜆_𝑖𝑗−_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑖+𝑄_𝑖+𝐾^𝑇_𝑖𝑅_𝑖𝐾_𝑖<0. (30) Then, the stochastic stability can be guaranteed when 𝐻𝑒𝐴^𝑇_𝑖𝑃_𝑖+𝐻𝑒𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝜀_𝑖𝑃_𝑖𝐷_𝑖𝐷^𝑇_𝑖𝑃_𝑖+1𝜀_𝑖𝐹^𝑇_𝑖𝐹_𝑖+𝛿_𝑖𝑃_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+1𝛿_𝑖𝐾^𝑇_𝑖𝑁^𝑇_𝑖𝑁_𝑖𝐾_𝑖+_{𝑗∈𝒮𝑖𝑘}𝜆_𝑖𝑗𝑃_𝑗−_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑖+𝑄_𝑖+𝐾^𝑇_𝑖𝑅_𝑖𝐾_𝑖𝑃<0,_𝑗≤𝑃_𝑖,𝑗∈𝒮^𝑖_𝑢𝑘. (31) For Case 1, pre- and postmultiplying 𝑃_𝑖⁻¹ in the first inequality in (27) and using Lemma 7 in the second inequality in (21), one can get that 𝐻𝑒𝑃_𝑖⁻¹𝐴^𝑇_𝑖+𝐻𝑒𝐵_𝑖𝐾_𝑖𝑃_𝑖⁻¹+𝜀_𝑖𝐷_𝑖𝐷^𝑇_𝑖+1𝜀_𝑖𝑃_𝑖⁻¹𝐹^𝑇_𝑖𝐹_𝑖𝑃_𝑖⁻¹+𝛿_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖+1𝛿_𝑖𝑃_𝑖⁻¹𝐾^𝑇_𝑖𝑁^𝑇_𝑖𝑁_𝑖𝐾_𝑖𝑃_𝑖⁻¹+_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑖⁻¹𝑃_𝑗𝑃_𝑖⁻¹−_{𝑗∈𝒮𝑖𝑘}𝜆_𝑖𝑗𝑃_𝑖⁻¹𝑊_𝑖𝑃_𝑖⁻¹+𝜆_𝑖𝑖𝑃_𝑖⁻¹+𝑃_𝑖⁻¹𝑄_𝑖𝑃_𝑖⁻¹+𝑃_𝑖⁻¹𝐾^𝑇_𝑖𝑅_𝑖𝐾_𝑖𝑃_𝑖⁻¹𝑊<0,_𝑖−1≤𝑃_𝑗⁻¹,𝑗∈𝒮^𝑖_𝑢𝑘. (32) Let 𝑋_𝑖=𝑃_𝑖⁻¹, 𝑌_𝑖=𝐾_𝑖𝑃_𝑖⁻¹, and 𝑍_𝑖=𝑊_𝑖⁻¹; then, we have 𝐻𝑒𝐴_𝑖𝑋_𝑖+𝐻𝑒𝐵_𝑖𝑌_𝑖+𝜀_𝑖𝐷_𝑖𝐷^𝑇_𝑖+1𝜀_𝑖𝑋_𝑖𝐹^𝑇_𝑖𝐹_𝑖𝑋_𝑖+𝛿_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖+1𝛿_𝑖𝑌^𝑇_𝑖𝑁^𝑇_𝑖𝑁_𝑖𝑌_𝑖+_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑋_𝑖𝑋_𝑗⁻¹𝑋_𝑖−_{𝑗∈𝒮𝑖𝑘}𝜆_𝑖𝑗𝑋_𝑖𝑍_𝑖⁻¹𝑋_𝑖+𝜆_𝑖𝑖𝑋_𝑖+𝑋_𝑖𝑄_𝑖𝑋_𝑖+𝑌^𝑇_𝑖𝑅_𝑖𝑌_𝑖𝑍<0,_𝑖≤𝑋_𝑗,𝑗∈𝒮^𝑖_𝑢𝑘. (33)
Applying the Schur complement formula, one can get (16).
For Case 2, pre- and postmultiplying 𝑃_𝑖⁻¹ in the first inequality in (28) and applying Lemma 7 to the second inequality in (28), one can obtain that 𝐻𝑒𝑃_𝑖⁻¹𝐴^𝑇_𝑖+𝐻𝑒𝐵_𝑖𝐾_𝑖𝑃_𝑖⁻¹+𝜀_𝑖𝐷_𝑖𝐷^𝑇_𝑖+1𝜀_𝑖𝑃_𝑖⁻¹𝐹^𝑇_𝑖𝐹_𝑖𝑃_𝑖⁻¹+𝛿_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖+1𝛿_𝑖𝑃_𝑖⁻¹𝐾^𝑇_𝑖𝑁^𝑇_𝑖𝑁_𝑖𝐾_𝑖𝑃_𝑖⁻¹+_{𝑗∈𝒮𝑖𝑘}𝜆_𝑖𝑗𝑃_𝑖⁻¹𝑃_𝑗𝑃_𝑖⁻¹−_{𝑗∈𝒮𝑖𝑘}𝜆_𝑖𝑗𝑃_𝑖⁻¹+𝑃_𝑖⁻¹𝑄_𝑖𝑃_𝑖⁻¹+𝑃_𝑖⁻¹𝐾^𝑇_𝑖𝑅_𝑖𝐾_𝑖𝑃_𝑖⁻¹𝑃<0,_𝑖−1≤𝑃_𝑗⁻¹,𝑗∈𝒮^𝑖_𝑢𝑘. (34) Let 𝑋_𝑖=𝑃_𝑖⁻¹ and 𝑌_𝑖=𝐾_𝑖𝑃_𝑖⁻¹; then, one can see that 𝐻𝑒𝐴_𝑖𝑋_𝑖+𝐻𝑒𝐵_𝑖𝑌_𝑖+𝜀_𝑖𝐷_𝑖𝐷^𝑇_𝑖+1𝜀_𝑖𝑋_𝑖𝐹^𝑇_𝑖𝐹_𝑖𝑋_𝑖+𝛿_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖+1𝛿_𝑖𝑌^𝑇_𝑖𝑁^𝑇_𝑖𝑁_𝑖𝑌_𝑖+_{𝑗∈𝒮𝑖𝑘}𝜆_𝑖𝑗𝑋_𝑖𝑋_𝑗⁻¹𝑋_𝑖−_{𝑗∈𝒮𝑖𝑘}𝜆_𝑖𝑗𝑋_𝑖+𝑋_𝑖𝑄_𝑖𝑋_𝑖+𝑌^𝑇_𝑖𝑅_𝑖𝑌_𝑖𝑋<0,_𝑖≤𝑋_𝑗,𝑗∈𝒮^𝑖_𝑢𝑘. (35)
Thus, the LMIs in (17) are derived by using the Schur complement formula.
From the above proof, one can see that ℑ^𝑥_𝑎[𝑉]<−𝑥^𝑇𝑄_𝑖𝑥−𝑥^𝑇𝐾^𝑇_𝑖𝑅_𝑖𝐾_𝑖𝑥. (36) It follows from Kushner [23] that 𝔼𝑉(𝑡,𝑥,𝑖)∣𝑟₀−𝑉𝑥₀,𝑟₀≤−𝔼^𝑡₀𝑥^𝑇(𝜏)𝑄_𝑖𝑥(𝜏)−𝑥^𝑇(𝜏)𝐾^𝑇_𝑖𝑅_𝑖𝐾_𝑖𝑥(𝜏)𝑑𝜏. (37) Since 𝔼[𝑉(𝑡,𝑥,𝑖)∣𝑟₀]≥0, by some simple calculation, one can achieve that 𝒥(𝑡)≤𝒥(0)≤𝑉𝑥₀,𝑟₀=𝑥^𝑇₀𝑃_𝑖𝑥₀≤𝛾. (38) Therefore, the minimum cost can be obtained by minimizing 𝛾. Thus, the proof is achieved.

Remark 9. The merit of the proposed results lies in that the transition rates of the jumping process are assumed to be more general, which means that some elements in the transition rates matrix have been exactly known, some ones have been merely known with lower and upper bounds, and others may have no information to use. Dealing with the unknown transition rates, a less conservative method is used. At the same time, the impact of the input signal quantization on the system is also considered. Finally, the controller design conditions are presented in the framework of LMIs.

In order to make comparison with the design method using the LMIs technique in Shen and Yang [9], we present the conditions designed by the utilization of Lemma 2 there.

Proposition 10. For the system (1) subject to Assumptions 1 and 2, suppose that there exist symmetric positive definite matrices 𝑋_𝑖, general matrices 𝑌_𝑖, and positive scalars 𝜀_𝑖, 𝛿_𝑖, and 𝛾 such that Γ₁₁𝑋_𝑖𝐹^𝑇_𝑖𝑌^𝑇_𝑖𝑁^𝑇_𝑖Γ₁₄Γ₁₅𝑋_𝑖𝑌_𝑖∗−𝜀_𝑖𝐼00000∗∗−𝛿_𝑖𝐼0000∗∗∗Γ₄₄000∗∗∗∗−Γ₅₅00∗∗∗∗∗−𝑄_𝑖⁻¹0∗∗∗∗∗∗−𝑅_𝑖⁻¹𝑍<0,_𝑖≤𝑋_𝑗,𝑗∈𝒮^𝑖_𝑢𝑘,Φ₁₁𝑋_𝑖𝐹^𝑇_𝑖𝑌^𝑇_𝑖𝑁^𝑇_𝑖Φ₁₄𝑋_𝑖𝑌^𝑇_𝑖∗−𝜀_𝑖𝐼0000∗∗−𝛿_𝑖𝐼000∗∗∗Φ₄₄00∗∗∗∗−𝑄_𝑖⁻¹0∗∗∗∗∗−𝑅_𝑖⁻¹𝑋<0,_𝑖≤𝑋_𝑗,𝑗∈𝒮^𝑖_𝑢𝑘,𝛾𝑥^𝑇₀𝑥₀𝑋_𝑖>0, (39)(40)(41) where Γ₁₁=𝐻𝑒𝐴_𝑖𝑋_𝑖+𝐻𝑒𝐵_𝑖𝑌_𝑖+𝜀_𝑖𝐷_𝑖𝐷^𝑇_𝑖+𝛿_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖+𝜆_𝑖𝑖𝑋_𝑖,Γ₁₄=𝜆_𝑖𝒦^𝑖₁𝑋_𝑖,𝜆_𝑖𝒦^𝑖₂𝑋_𝑖,…,𝜆_𝑖𝒦^𝑖_𝑚𝑖𝑘𝑋_𝑖,Γ₄₄=diag−𝑋_𝒦^𝑖₁,−𝑋_𝒦^𝑖₂,…,−𝑋_𝒦^𝑖_𝑚𝑖𝑘,Γ₁₅=−_{𝑗∈𝒮𝑖𝑘}𝜆_𝑖𝑗𝑋_𝑖,−_{𝑗∈𝒮𝑖𝑘}𝜆_𝑖𝑗𝑋_𝑖,…,−_{𝑗∈𝒮𝑖𝑘}𝜆_𝑖𝑗𝑋_𝑖,Γ₅₅=diag−𝑋_𝕂^𝑖₁,−𝑋_𝕂^𝑖₂,…,−𝑋_𝕂^𝑖_{𝑚𝑖𝑢𝑘},Φ₁₁=𝐻𝑒𝐴_𝑖𝑋_𝑖+𝐻𝑒𝐵_𝑖𝑌_𝑖+𝜀_𝑖𝐷_𝑖𝐷^𝑇_𝑖+𝛿_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖−_{𝑗∈𝒮𝑖𝑘}𝜆_𝑖𝑗𝑋_𝑖,Φ₁₄=𝜆_𝑖𝒦^𝑖₁𝑋_𝑖,𝜆_𝑖𝒦^𝑖₂𝑋_𝑖,…,𝜆_𝑖𝒦^𝑖_𝑚𝑖𝑘𝑋_𝑖,Φ₄₄=diag−𝑋_𝒦^𝑖₁,−𝑋_𝒦^𝑖₂,…,−𝑋_𝒦^𝑖_𝑚𝑖𝑘. (42) Then, the controller designed as 𝑢(𝑡)=𝐾_𝑖𝑥(𝑡)+𝑢_𝑖𝑐(𝑡) (43) can drive the state trajectory to the origin asymptotically and can obtain the 𝐻₂ cost by minimizing 𝑟, where 𝐾_𝑖=𝑌_𝑖𝑋_𝑖⁻¹, 𝑃_𝑖=𝑋_𝑖⁻¹, and 𝑢_𝑖𝑐=(−(1+𝜓)𝜇/2(1−𝜓))sign(𝑥^𝑇𝑃_𝑖𝐵_𝑖).

Proof. The proof process is similar to that in Theorem 8 but with replacing 𝑃_𝑗≤𝑊_𝑖, 𝑗∈𝒮^𝑖_𝑢𝑘, and ∑_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑊_𝑗 with (∑_𝑗∈𝒮^𝑖_{𝑢𝑘,𝑗≠𝑖}𝜆_𝑖𝑗)(∑_𝑗∈𝒮^𝑖_{𝑢𝑘,𝑗≠𝑖}𝑃_𝑗) in (22). The details are omitted here for space limitation.

4. Numerical Example

An example is presented to illustrate the effectiveness of the proposed method.

Consider the MJLSs with four operation modes as follows: 𝐴₁=10.20.4−4,𝐴₂=,𝐴−1.51.21.4−5₃=−132−2,𝐴₄=,𝐵0340₁=0.50.2,𝐵₂=,𝐵0.10.3₃=01,𝐵₄=01,𝐷₁=02,𝐷₂=10,𝐷₃=0.40.8,𝐷₄=,𝐹−0.50.5₁=10,𝐹₂=,𝐹0.51₃=00.3,𝐹₄=,𝑀0.50.8₁=0.4,𝑀₂=0.6,𝑀₃=0.5,𝑀₄𝑁=1,₁=0.8,𝑁₂=0.2,𝑁₃=0.1,𝑁₄=0.5. (44)

The considered transition rates matrix is given as follows: Λ=−1.30.2???𝛼₁0.30.30.6???0.4𝛼₂?𝛼₃, (45) where −1.2≤𝛼₁≤−0.8, 0.6≤𝛼₂≤0.8, and −2≤𝛼₃≤−1.5. Choosing 𝑥₀=[1;−0.5] and solving (16)–(18) in Theorem 8 and (34)–(38) in Proposition 10, one can obtain the optimal 𝐻₂ performance indices shown in Table 1. It can be seen that less conservativeness is obtained by the proposed method in Theorem 8.

Table 1 

Comparison of optimal 𝛾.

For simulation, one can obtain the controller gains by solving the LMIs in Theorem 8 as follows:

𝐾₁=−5.2949−0.5848, 𝐾₂=−2.4332−1.6826, 𝐾₃=−4.0556−4.6580, and 𝐾₄=−14.2188−12.4624 and by considering the positive definite matrices 𝑃₁=1.51530.10520.10520.1670,𝑃₂=,𝑃1.16400.53370.53370.4595₃=0.77880.42690.42690.4903,𝑃₄=.3.67262.84382.84382.4925 (46)

The following parameters are used in the simulation: 𝜇=0.3, 𝐸₁(𝑡)=0.2cos(𝑡), 𝐸₂(𝑡)=0.05sin(𝑡), 𝐸₃(𝑡)=0.8cos(3𝑡), 𝐸₄(𝑡)=0.4sin(𝑡), Ξ₁(𝑡)=0.5cos(3𝑡), Ξ₂(𝑡)=0.6cos(2𝑡), Ξ₃(𝑡)=0.2sin(𝑡), and Ξ₄(𝑡)=0.05cos(2𝑡). In order to reduce the chattering effects induced by 𝑢_𝑖𝑐, sign(𝑥^𝑇𝑃_𝑖𝐵_𝑖) is substituted by sat(𝑥^𝑇𝑃_𝑖𝐵_𝑖/𝛿_𝑖), where 𝛿₁=0.0015, 𝛿₂=0.0004, 𝛿₃=0.0015, and 𝛿₄=0.0004.

The switching mode, the control input, and the response curves of the system states are presented in Figures 1, 2, and 3, respectively. Among them, Figure 1 shows a possible system modes evolution which meets the transition rates given in this example. As shown in Figure 1, the system has 4 modes and is in different modes at a different time. Figure 2 shows the curve of the control input 𝑢(𝑡). With this controller, Figure 3 depicts the state response curves of the closed-loop system. It can be seen that the considered continuous-time Markovian jump linear uncertain system is stochastically stable in spite of mismatched uncertainty, the input signal quantization, and the partly known transition rates covering the completely known, the boundary unknown, and the completely unknown elements in the transition rates matrix.

Figure 1 

Evolution of the system mode.

Figure 2 

The curve of the control input 𝑢(𝑡).

Figure 3 

The response curve of the state 𝑥.

5. Conclusions

𝐻₂ control design for a class of the continuous-time Markovian jump linear uncertain systems with partly unknown transition rates and input quantization has been investigated. The 𝐻₂ cost index for the Markovian jump systems is first introduced, and then sufficient conditions of the 𝐻₂ control for the Markovian jump linear uncertain systems with unknown transition rates have been presented. The controller is constructed by two parts. The nonlinear part is proposed to eliminate the effect of input quantization. The linear part is designed by solving the LMIs conditions for achieving the 𝐻₂ performance against model uncertainties and unknown transition rates. In comparison with the existing result in the literature, less conservativeness has been obtained by introducing new relaxed inequality conditions. Finally, a numerical example is given to show the effectiveness of the proposed design method.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants nos. 61273155, 61273355, 61322312, and 61273356), the Foundation of State Key Laboratory of Robotics (Grant no. Z2013-06), the New Century Excellent Talents in the University (Grant no. NCET-11-0083), a Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant no. 201157), the Fundamental Research Funds for the Central Universities (Grant no. N120504003).