Quantized State-Feedback Stabilization for Delayed Markovian Jump Linear Systems with Generally Incomplete Transition Rates

Abstract

This paper is concerned with the robust quantized state-feedback controller design problem for a class of continuous-time Markovian jump linear uncertain systems with general uncertain transition rates and input quantization. The uncertainties under consideration emerge in both system parameters and mode transition rates. This new uncertain model is more general than the existing ones and can be applicable to more practical situations because each transition rate can be completely unknown or only its estimate value is known. Based on linear matrix inequalities, the quantized state-feedback controller is formulated to ensure the closed-loop system is stable in mean square. Finally, a numerical example is presented to verify the validity of the developed theoretical results.

1. Introduction

Markovian jump systems have been serving as popular tools for analyzing plants subjected to random abrupt changes, such as random component failures, abrupt environment changes, disturbance, and changes in the interconnections of subsystems; see [1–11] and the references therein. In practice, transition rates are difficult to precisely estimate the transition rates. Therefore, developing the analysis and synthesis method for MJS with uncertain transition rates is of great importance. Bounded uncertain transition rates (BUTRs) are introduced when the precise values of the transition rates are unknown, but their bounds (upper bounds and lower bounds) are known; see, for example, [12–14]. However, in some practical cases, to obtain the bound of every TR is difficult or even impossible. Zhang et al. [15–18] proposed another description for the uncertain TRs, which is partly unknown TRs (PUTRs). In this description, some of the TRs can be unknown. This model also simulates researchers’ interests (see, e.g., [19–22]). In this PUTR model, every transition rate is either exactly known or completely unknown, which may be too restrictive in many practical situations. Therefore, Guo and Wang [23] proposed generally uncertain TRs to deal with a more practical situation where the transition rates can be completely unknown or only its bound is known.

On the other hand, in many modern engineering practices, information processing devices, such as analog-to-digital and digital-to-analog converters, have been widely used and brought about some advantages, such as lower cost, reduced weight and power, and simple installation and maintenance. However, server deterioration of system performance or even system instability may also be induced. Signal quantization should be fully considered in such cases. Nowadays, the feedback stabilization problem is considered by utilizing dynamic quantizers [24–26] and static quantizers [27–36]. The stabilization problem for single-input discrete Markov jump linear systems via mode dependent quantized state feedback is addressed by Xiao et al. in [37], but the transition rates are assumed to be completely known. Ye et al. [38, 39] considered 𝐻₂ control of Markov jump linear systems with unknown transition rates and input quantization. To the best of our knowledge, no result has been presented for control design of delayed continuous-time Markov jump linear uncertain systems with generally unknown transition rates and input signal quantization.

In this paper, robust quantized state-feedback stabilization for delayed Markovian jump linear systems with generally uncertain transition rates is addressed. In Section 2, the considered systems are formulated and the purposes of the paper are stated. In Section 3, the main results are derived via linear matrix inequalities. The structure of the controller consists of two parts. The nonlinear part is provided to eliminate the effect of input quantization. The linear part is obtained by solving LMIs to deal with model uncertainties and unknown transition rates. Section 4 concludes the paper.

Notation. In this paper, ℝ^𝑛 and ℝ^𝑛×𝑚 denote the 𝑛-dimensional Euclidean space and the set of all 𝑛×𝑚 real matrices, respectively. ℕ⁺ represents the set of positive integers. The notation 𝑃>0(𝑃≥0) means that 𝑃 is a real symmetric and positive-definite (semi-positive-definite) matrix. For notation (Ω,ℱ,𝑃),Ω represents the space, ℱ is the 𝜎-algebra of subsets of the sample space, and 𝑃 is the probability measure on ℱ. 𝐸{⋅} stands for the mathematical expectation. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Formulation

Consider the following stochastic system with Markovian jump parameters, defined on a complete probability space (Ω,ℱ,𝑃): 𝑥̇(𝑡)=𝐴𝑟_𝑡𝑥(𝑡)+𝐴_𝑑𝑟_𝑡𝑥()+𝐵𝑟𝑡−𝜏_𝑡+Δ𝐵𝑟_𝑡𝑞(𝑢(𝑡𝑥)),𝑡≥0,₀=𝑥(0),𝑟₀=𝑟(0), (1) where 𝑥(𝑡)∈ℝ^𝑛 is the system state. The mode jumping process {𝑟_𝑡,𝑡≥0} is a right-continuous Markov process on the probability space taking values in a finite state space 𝕊={1,2,…,𝑠} with the mode transition probabilities: 𝑃𝑟𝑟_𝑡+Δ=𝑗∣𝑟_𝑡=𝜋=𝑖^𝑖𝑗Δ+𝑜(Δ),if𝑖≠𝑗,1+𝜋_𝑖𝑗Δ+𝑜(Δ),if𝑖=𝑗, (2) where Δ>0, lim_Δ→0(𝑜(Δ)/Δ)=0, and 𝜋_𝑖𝑗≥0(𝑖,𝑗∈𝕊,𝑗≠𝑖) is the TR from mode 𝑖 at time 𝑡 to mode 𝑗 at time 𝑡+Δ, and 𝜋_𝑖𝑖=−^𝑠_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗. (3) for each 𝑖∈𝕊.

The mode TR matrix Π≜(𝜋_𝑖𝑗) is considered to be generally uncertain. For instance, the TR matrix for system (1) with 𝑠 operation modes may be expressed as 𝜋₁₁+Δ₁₁?𝜋₁₃+Δ₁₃𝜋⋯???₂₃+Δ₂₃⋯𝜋_2𝑠+Δ_2𝑠?𝜋⋮⋮⋮⋱⋮_𝑠2+Δ_𝑠2?⋯𝜋_𝑠𝑠+Δ_𝑠𝑠, (4) where 𝜋_𝑖𝑗 and Δ_𝑖𝑗∈[−𝛿_𝑖𝑗,𝛿_𝑖𝑗](𝛿_𝑖𝑗≥0) represent the estimate value and estimate error of the uncertain TR 𝜋_𝑖𝑗, respectively, where 𝜋_𝑖𝑗 and 𝛿_𝑖𝑗 are known. “?’’ represents the complete unknown TR, which means its estimate value 𝜋_𝑖𝑗 and estimate error bound are unknown. For notational clarity, for all 𝑖∈𝕊, the set 𝑈^𝑖 denotes 𝑈^𝑖=𝑈^𝑖_𝑘∪𝑈^𝑖_𝑢𝑘 with 𝑈^𝑖_𝑘≜{𝑗∶theestimatevalueof𝜆_𝑖𝑗isknownfor𝑗∈𝕊}, 𝑈^𝑖_𝑢𝑘≜{𝑗∶theestimatevalueof𝜆_𝑖𝑗isunknownfor𝑗∈𝕊}. Moreover, if 𝑈^𝑖_𝑘≠∅, it is further described as 𝑈^𝑖_𝑘={𝑘^𝑖₁,𝑘^𝑖₂,…,𝑘^𝑖_𝑚}, where 𝑘^𝑖_𝑚∈ℕ⁺ represent the 𝑚th bound-known element with the index 𝑘^𝑖_𝑚 in the 𝑖th row of matrix Π. According to the properties of the TRs (e.g.,𝜆_𝑖𝑗≥0(∀𝑖,𝑗∉𝕊,𝑖≠𝑗)and𝜆_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗), we assume that the known estimate values of the TRs are well defined. That is, the following assumptions hold.

Assumption 1. If 𝑈^𝑖_𝑘=𝕊, then 𝜆_𝑖𝑗−𝛿_𝑖𝑗≥0(∀𝑗∈𝕊,𝑗≠𝑖),𝜆_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗≤0, and 𝛿_𝑖𝑖=∑^𝑁_{𝑗=1,𝑗≠𝑖}𝛿_𝑖𝑗>0.

Assumption 2. If 𝑈^𝑖_𝑘≠𝕊,and𝑖∈𝑈^𝑖_𝑘, then 𝜆_𝑖𝑗−𝛿_𝑖𝑗≥0(∀𝑗∈𝑈^𝑖_𝑘,𝑗≠𝑖),𝜆_𝑖𝑖+𝛿_𝑖𝑖≤0 and ∑_𝑗∈𝑈^𝑖_𝑘𝜆_𝑖𝑗≤0.

Assumption 3. If 𝑈^𝑖_𝑘≠𝕊 and 𝑖∉𝑈^𝑖_𝑘, then 𝜆_𝑖𝑗−𝛿_𝑖𝑗≥0(∀𝑗∈𝑈^𝑖_𝑘).

Remark 4. The above assumption is reasonable, since it is the direct result from the properties of the TRs e.g.,𝜋_𝑖𝑗≥0∀𝑖,𝑗∉𝕊,𝑖≠𝑗,𝜋_𝑖𝑖=−^𝑠_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗. (5) The above description about uncertain TRs is more general than either the BUTR or PUTR models. To show this, we rewrite the two uncertain models as follows: BUTR model (see [12–14]): 𝜋₁₁+Δ₁₁𝜋₁₂+Δ₁₂⋯𝜋_1𝑠+Δ_1𝑠𝜋₂₁+Δ₂₁𝜋₂₂+Δ₂₂⋯𝜋_2𝑠+Δ_2𝑠𝜋⋮⋮⋱⋮_𝑠1+Δ_𝑠1𝜋_𝑠2+Δ_𝑠2⋯𝜋_𝑠𝑠+Δ_𝑠𝑠 (6) with 𝜋_𝑖𝑗−𝛿_𝑖𝑗≥0(∀𝑗∈𝕊,𝑗≠𝑖), 𝜋_𝑖𝑖=−∑^𝑠_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗,and𝛿_𝑖𝑖=−∑^𝑠_{𝑗=1,𝑗≠𝑖}𝛿_𝑖𝑗. PUTR model (see [15–22]): 𝜋₁₁?𝜋₁₃⋯???𝜋₂₃⋯𝜋_2𝑠⋮⋮⋮⋱⋮?𝜋_𝑠2?⋯𝜋_𝑠𝑠. (7) Obviously, if 𝑈^𝑖_𝑘∅,∀𝑖∈𝕊, the GUTR model (4) will reduce to the BUTR model (6); if 𝛿_𝑖𝑗=0,∀𝑖∈𝕊,∀𝑗∈𝑈^𝑖_𝑘, the GUTR model (4) will reduce to the PUTR model (7). The GUTR model (4) is more general than the other two models; therefore, it is more practicable.

For convenience, the system matrices 𝐴_𝑖≜𝐴(𝑟_𝑡=𝑖), 𝐴_𝑑𝑖≜𝐴_𝑑(𝑟_𝑡=𝑖), and 𝐵_𝑖≜𝐵(𝑟_𝑡=𝑖),Δ𝐵_𝑖≜Δ𝐵(𝑟_𝑡=𝑖),𝑖∈𝕊, are known matrix functions of the Markovian process. Then system (1) can be described by 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐴_𝑑𝑖𝑥()+𝐵𝑡−𝜏_𝑖+Δ𝐵_𝑖𝑞(𝑢(𝑡𝑥)),𝑡≥0,₀=𝑥(0),𝑟₀=𝑟(0). (8)

The following assumptions are assumed to be valid.

Assumption 5. Δ𝐵_𝑖=𝐵_𝑖𝑀_𝑖Ξ_𝑖(𝑡)𝐹_𝑖 and |𝑀_𝑖Ξ_𝑖(𝑡)𝐹_𝑖|_∞≤𝜓_𝑖, where 𝑀_𝑖 and 𝐹_𝑖 are known constant matrices with appropriate dimensions, Ξ_𝑖(𝑡) is time-varying uncertain matrix satisfying Ξ_𝑖(𝑡)Ξ^𝑇_𝑖(𝑡)≤𝐼, and parameter 𝜓_𝑖 satisfies 0≤𝜓_𝑖<1.

In addition, the quantizer 𝑞(⋅) is defined by an operator function round(⋅) which rounds to the nearest integer; that is, 𝑞(𝑢(𝑡𝑢(𝑡)))=𝜇⋅round𝜇, (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, we have |𝑒_𝜇|_∞≤𝜇/2.

The objective of this paper is to design a 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. 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 unknown transition rates.

Lemma 6 (Petersen, 1987). Given a symmetric matrix Π and matrices 𝑀, 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. (11)

Lemma 7 (14). Given any real number 𝜀 and any matrix 𝑄, the matrix inequality 𝜀𝑄+𝑄^𝑇≤𝜀²𝑇+𝑄𝑇⁻¹𝑄^𝑇 (12) holds for any matrix 𝑇>0.

3. Stochastic Stability Analysis

The goal of this section is to develop an analysis result of stability for system (1) with general uncertain TRs.

Theorem 8. Consider that uncertain Markovian jump system (8) with a GUTR matrix (4) is stochastically stable if there exist matrices 𝑃_𝑖>0(𝑖∈𝕊), 𝑇_𝑖𝑗>0(𝑖∉𝑈^𝑖_𝑘,𝑗∈𝑈^𝑖_𝑘), 𝑉_𝑖𝑗𝑙>0(𝑖,𝑗∈𝑈^𝑖_𝑘,𝑈^𝑖_𝑢𝑘≠∅,𝑙∈𝑈^𝑖_𝑢𝑘),𝑊_𝑖𝑗>0(𝑖,𝑗∈𝑈^𝑖_𝑘,𝑈^𝑖_𝑢𝑘=∅), and 𝑄>0 such that the following LMIs are feasible for 𝑖=1,2,…,𝑠.
If 𝑖∉𝑈^𝑖_𝑘, Γ₁₁𝑃_𝑖𝐴_𝑑𝑖𝑃_𝑖𝑘^𝑖₁−𝑃_𝑖⋯𝑃_𝑖𝑘^𝑖_𝑚−𝑃_𝑖∗−𝑄0⋯0∗∗−𝑇_𝑖𝑘^𝑖₁⋯0∗∗∗⋱⋮∗∗∗∗−𝑇_𝑖𝑘^𝑖_𝑚<0𝑃_𝑖−𝑃_𝑗≥0,∀𝑗∈𝑈^𝑖_𝑢𝑘,𝑗≠𝑖. (13)(14)
If 𝑖∈𝑈^𝑖_𝑘 and 𝑈^𝑖_𝑢𝑘≠∅,forone𝑙∈𝑈^𝑖_𝑢𝑘, Θ₁₁𝑃_𝑖𝐴_𝑑𝑖𝑃_𝑘^𝑖₁−𝑃_𝑙⋯𝑃_𝑘^𝑖_𝑚−𝑃_𝑙∗−𝑄0⋯0∗∗−𝑉_𝑖𝑘^𝑖_1𝑙⋯0∗∗∗⋱⋮∗∗∗∗−𝑉_𝑖𝑘^𝑖_𝑚𝑙<0. (15)
If 𝑖∈𝑈^𝑖_𝑘 and 𝑈^𝑖_𝑢𝑘=∅,Ψ₁₁𝑃_𝑖𝐴_𝑑𝑖𝑃₁−𝑃_𝑖⋯𝑃_𝑖−1−𝑃_𝑖𝑃_𝑖+1−𝑃_𝑖⋯𝑃_𝑠−𝑃_𝑖∗−𝑄0⋯00⋯0∗∗−𝑊_𝑖1⋯00⋯0∗∗∗⋱⋮⋮⋮⋮∗∗∗∗−𝑊_{𝑖(𝑖−1)}0⋮⋮∗∗∗∗∗−𝑊_𝑖(𝑖+1)⋮⋮∗∗∗∗∗∗⋱⋮∗∗∗∗∗∗∗−𝑊_𝑖𝑠<0, (16)where Γ₁₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝜎_𝑖𝑃_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+1𝜎_𝑖𝐾^𝑇_𝑖𝐹^𝑇_𝑖𝐹_𝑖𝐾_𝑖+_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑖+_{𝑗∈𝑈𝑖𝑘}𝛿²_𝑖𝑗4𝑇_𝑖𝑗Θ+𝑄₁₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝜎_𝑖𝑃_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+1𝜎_𝑖𝐾^𝑇_𝑖𝐹^𝑇_𝑖𝐹_𝑖𝐾_𝑖+_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑙+_{𝑗∈𝑈𝑖𝑘}𝛿²_𝑖𝑗4𝑉_𝑖𝑗𝑙Ψ+𝑄₁₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝜎_𝑖𝑃_𝑖𝐵_𝑖𝑀_𝑖𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+1𝜎_𝑖𝐾^𝑇_𝑖𝐹^𝑇_𝑖𝐹_𝑖𝐾_𝑖+_{𝑗∈𝕊,𝑗≠𝑖}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑖+_{𝑗∈𝕊,𝑗≠𝑖}𝛿²_𝑖𝑗4𝑊_𝑖𝑗+𝑄. (17) Then the controller designed as 𝑢(𝑡)=𝐾_𝑖𝑥+𝑢_𝑖𝑐 (18) can drive the state trajectory to the origin asymptotically, where 𝐾_𝑖=𝑌_𝑖𝑋_𝑖⁻¹,𝑃_𝑖=𝑋_𝑖⁻¹, and 𝑢_𝑖𝑐=(1+𝜓)𝑢/(2(1−𝜓))sign(𝑥^𝑇𝑃_𝑖𝐵_𝑖).

Proof. Take the Lyapunov function candidate 𝑉=𝑥^𝑇𝑃_𝑖𝑥+∫^𝑡_𝑡−𝜏𝑥^𝑇(𝛼)𝑄𝑥(𝛼)𝑑𝛼; then along the system trajectory of plant (8), the weak infinitesimal operator 𝔍^𝑥_𝑎[⋅] of the process 𝑥(𝑡),𝑟_𝑡,𝑡≥0, for plant (8) at the point 𝑡,𝑥,𝑖 is as follows: 𝔍^𝑥_𝑎[𝑉]=𝑥̇^𝑇𝑃_𝑖𝑥+𝑥^𝑇𝑃_𝑖𝑥̇+𝑥^𝑇ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗𝑥=𝐴_𝑖𝑥+𝐴_𝑑𝑖𝑥()+𝐵𝑡−𝜏_𝑖+Δ𝐵_𝑖𝐾_𝑖𝑥+𝑢_𝑖𝑐+𝑒_𝜇^𝑇𝑃_𝑖𝑥+𝑥^𝑇𝑃_𝑖𝐴_𝑖𝑥+𝐴_𝑑𝑖𝑥()+𝐵𝑡−𝜏_𝑖+Δ𝐵_𝑖𝐾_𝑖𝑥+𝑢_𝑖𝑐+𝑒_𝜇+𝑥^𝑇ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗𝑥=𝑥^𝑇𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝑃_𝑖𝐵_𝑖+Δ𝐵_𝑖𝐾_𝑖+𝐾^𝑇_𝑖𝐵_𝑖+Δ𝐵_𝑖^𝑇𝑃_𝑖+𝑄𝑥+𝑥^𝑇(𝑡−𝜏)𝐴^𝑇_𝑑𝑖𝑃_𝑖𝑥+𝑥^𝑇𝑃_𝑖𝐴_𝑑𝑖𝑥(𝑡−𝜏)−𝑥^𝑇(𝑡−𝜏)𝑄𝑥(𝑡−𝜏)+2𝑥^𝑇𝑃_𝑖𝐵_𝑖+Δ𝐵_𝑖𝑢_𝑖𝑐+𝑒_𝜇+𝑥^𝑇ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗𝑥. (19) According to Assumption 2, 𝑢_𝑖𝑐 in Theorem 8, and Lemma 6, 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)𝐹_𝑖𝑢_𝑖𝑐+𝑒_𝜇+2𝑥^𝑇𝑃_𝑖𝐵_𝑖𝑒_𝜇≤2𝑥^𝑇𝑃_𝑖𝐵_𝑖𝑢_𝑖𝑐+2𝑥^𝑇𝑃_𝑖𝐵_𝑖1𝑀_𝑖Ξ_𝑖(𝑡)𝐹_𝑖∞𝑢_𝑖𝑐∞+𝜇2+2𝑥^𝑇𝑃_𝑖𝐵_𝑖1𝜇2=2𝑥^𝑇𝑃_𝑖𝐵_𝑖𝑢_𝑖𝑐+2𝑥^𝑇𝑃_𝑖𝐵_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. (20) It follows from (19) and (20) that 𝔍^𝑥_𝑎[𝑉]≤𝑥^𝑇𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝑃_𝑖𝐵_𝑖+Δ𝐵_𝑖𝐾_𝑖+𝐾^𝑇_𝑖𝐵_𝑖+Δ𝐵_𝑖^𝑇𝑃_𝑖+𝑄𝑥+𝑥^𝑇(𝑡−𝜏)𝐴^𝑇_𝑑𝑖𝑃_𝑖𝑥+𝑥^𝑇𝑃_𝑖𝐴_𝑑𝑖𝑥(𝑡−𝜏)−𝑥^𝑇(𝑡−𝜏)𝑄𝑥(𝑡−𝜏)+𝑥^𝑇ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗𝑥. (21) Since Δ𝐵_𝑖=𝐵_𝑖𝑀_𝑖Ξ_𝑖(𝑡)𝐹_𝑖, we introduce Φ_𝑖≜𝑥^𝑇𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝑃_𝑖𝐵_𝑖𝑀_𝑖Ξ_𝑖(𝑡)𝐹_𝑖𝐾_𝑖+𝐾^𝑇_𝑖𝐹^𝑇_𝑖Ξ^𝑇_𝑖(𝑡)𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑄𝑥+𝑥^𝑇(𝑡−𝜏)𝐴^𝑇_𝑑𝑖𝑃_𝑖𝑥+𝑥^𝑇𝑃_𝑖𝐴_𝑑𝑖𝑥(𝑡−𝜏)−𝑥^𝑇(𝑡−𝜏)𝑄𝑥(𝑡−𝜏). (22) Then, the above inequality can be rewritten as 𝔍^𝑥_𝑎[𝑉]≤Φ_𝑖+𝑥^𝑇ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗𝑥. (23) Three cases should be considered.
Case I (𝑖∉𝑈^𝑖_𝑘). In this case, note that ∑_𝑗∈𝑈^𝑖_{𝑢𝑘,𝑗≠𝑖}𝜋_𝑖𝑗=−𝜋_𝑖𝑖−∑_𝑗∈𝑈^𝑖_𝑘𝜋_𝑖𝑗 and 𝜋_𝑖𝑗≥0,𝑗∈𝑈^𝑖_𝑢𝑘,𝑗≠𝑖; then from (23), we have 𝔍^𝑥_𝑎[𝑉]≤Φ_𝑖+𝑥^𝑇_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗+𝜋_𝑖𝑖𝑃_𝑖+_{𝑗∈𝑈𝑖𝑢𝑘,𝑗≠𝑖}𝜋_𝑖𝑗𝑃_𝑖𝑥=Φ_𝑖+𝑥^𝑇_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗+𝜋_𝑖𝑖𝑃_𝑖+−𝜋_𝑖𝑖−_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑖𝑥=Φ_𝑖+𝑥^𝑇_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑖𝑥=Φ_𝑖+𝑥^𝑇_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗+Δ_𝑖𝑗𝑃_𝑗−𝑃_𝑖𝑥=Φ_𝑖+𝑥^𝑇_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑖+_{𝑗∈𝑈𝑖𝑘}Δ_𝑖𝑗𝑃_𝑗−𝑃_𝑖𝑥. (24) There, note that 𝜋_𝑖𝑗=𝜋_𝑖𝑗+Δ_𝑖𝑗 for 𝑗∈𝑈^𝑖_𝑘.
On the other hand, in view of Lemma 7, we have _{𝑗∈𝑈𝑖𝑘}Δ_𝑖𝑗𝑃_𝑗−𝑃_𝑖=_{𝑗∈𝑈𝑖𝑘}12Δ_𝑖𝑗𝑃_𝑗−𝑃_𝑖+12Δ_𝑖𝑗𝑃_𝑗−𝑃_𝑖≤_{𝑗∈𝑈𝑖𝑘}12Δ_𝑖𝑗²𝑇_𝑖𝑗+𝑃_𝑗−𝑃_𝑖𝑇⁻¹_𝑖𝑗𝑃_𝑗−𝑃_𝑖≤_{𝑗∈𝑈𝑖𝑘}𝛿²_𝑖𝑗4𝑇_𝑖𝑗+𝑃_𝑗−𝑃_𝑖𝑇⁻¹_𝑖𝑗𝑃_𝑗−𝑃_𝑖. (25) From (24) and (25), we have 𝔍^𝑥_𝑎[𝑉]≤Φ_𝑖+𝑥^𝑇×_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑖+_{𝑗∈𝑈𝑖𝑘}𝛿²_𝑖𝑗4𝑇_𝑖𝑗+𝑃_𝑗−𝑃_𝑖𝑇⁻¹_𝑖𝑗𝑃_𝑗−𝑃_𝑖𝑥. (26) Hence, 𝔍^𝑥_𝑎[𝑉]<0 holds if 𝒜ℋ=¹¹𝑃_𝑖𝐴_𝑑𝑖∗−𝑄<0, (27) where 𝒜₁₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝑃_𝑖𝐵_𝑖𝑀_𝑖Ξ_𝑖(𝑡)𝐹_𝑖𝐾_𝑖+𝐾^𝑇_𝑖𝐹^𝑇_𝑖Ξ^𝑇_𝑖(𝑡)𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑄+_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑖+_{𝑗∈𝑈𝑖𝑘}𝛿²_𝑖𝑗4𝑇_𝑖𝑗+𝑃_𝑗−𝑃_𝑖𝑇⁻¹_𝑖𝑗𝑃_𝑗−𝑃_𝑖. (28) Let 𝒜₁₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝑄+_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑖+_{𝑗∈𝑈𝑖𝑘}𝛿²_𝑖𝑗4𝑇_𝑖𝑗+𝑃_𝑗−𝑃_𝑖𝑇⁻¹_𝑖𝑗𝑃_𝑗−𝑃_𝑖; (29) then we have ℋ=𝒜₁₁𝑃_𝑖𝐴_𝑑𝑖+𝑃∗−𝑄^𝑖𝐵_𝑖𝑀_𝑖0Ξ_𝑖(𝑡)𝐹_𝑖𝐾_𝑖0+𝐹_𝑖𝐾_𝑖0^𝑇Ξ^𝑇_𝑖(𝑡)𝑃_𝑖𝐵_𝑖𝑀_𝑖0^𝑇<0, (30) which is equivalent to (13) by Schur complement.
Case II (𝑖∈𝑈^𝑖_𝑘and𝑈^𝑖_𝑢𝑘≠∅). There must be an 𝑙∈𝑈^𝑖_𝑢𝑘 such that 𝑃(𝑙)−𝑃(𝑗)≥0, ∀𝑗∈𝑈^𝑖_𝑢𝑘. We define 𝔍^𝑥_𝑎[𝑉]≤Φ_𝑖+𝑥^𝑇_{𝑗∈𝑈𝑖𝑘}𝜆_𝑖𝑗𝑃𝑗+_{𝑗∈𝑈𝑖𝑢𝑘}𝜆_𝑖𝑗𝑃(𝑙)𝑥=Φ_𝑖+𝑥^𝑇_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑙𝑥=Φ_𝑖+𝑥^𝑇_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑙𝑥=Φ_𝑖+𝑥^𝑇_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗+Δ_𝑖𝑗𝑃_𝑗−𝑃_𝑙𝑥=Φ_𝑖+𝑥^𝑇_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑙+_{𝑗∈𝑈𝑖𝑘}Δ_𝑖𝑗𝑃_𝑗−𝑃_𝑙𝑥. (31) By using Lemma 7 again, we have _{𝑗∈𝑈𝑖𝑘}Δ_𝑖𝑗𝑃_𝑗−𝑃_𝑙=_{𝑗∈𝑈𝑖𝑘}12Δ_𝑖𝑗𝑃_𝑗−𝑃_𝑙+12Δ_𝑖𝑗𝑃_𝑗−𝑃_𝑙≤_{𝑗∈𝑈𝑖𝑘}12Δ_𝑖𝑗²𝑉_𝑖𝑗𝑙+𝑃_𝑗−𝑃_𝑙𝑉⁻¹_𝑖𝑗𝑙𝑃_𝑗−𝑃_𝑙≤_{𝑗∈𝑈𝑖𝑘}𝛿²_𝑖𝑗4𝑉_𝑖𝑗𝑙+𝑃_𝑗−𝑃_𝑙𝑉⁻¹_𝑖𝑗𝑙𝑃_𝑗−𝑃_𝑙. (32) From (31) and (32), we have 𝔍^𝑥_𝑎[𝑉]≤Φ_𝑖+𝑥^𝑇×_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑙+_{𝑗∈𝑈𝑖𝑘}𝛿²_𝑖𝑗4𝑉_𝑖𝑗𝑙+𝑃_𝑗−𝑃_𝑙𝑉⁻¹_𝑖𝑗𝑙𝑃_𝑗−𝑃_𝑙𝑥. (33) Hence, 𝔍^𝑥_𝑎[𝑉]<0 holds if 𝔄𝔄=¹¹𝑃_𝑖𝐴_𝑑𝑖∗−𝑄<0, (34) where 𝔄₁₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝑃_𝑖𝐵_𝑖𝑀_𝑖Ξ_𝑖(𝑡)𝐹_𝑖𝐾_𝑖+𝐾^𝑇_𝑖𝐹^𝑇_𝑖Ξ^𝑇_𝑖(𝑡)𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑄+_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑙+_{𝑗∈𝑈𝑖𝑘}𝛿²_𝑖𝑗4𝑉_𝑖𝑗𝑙+𝑃_𝑗−𝑃_𝑙𝑉⁻¹_𝑖𝑗𝑙𝑃_𝑗−𝑃_𝑙. (35) Let 𝔄₁₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝑄+_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑙+_{𝑗∈𝑈𝑖𝑘}𝛿²_𝑖𝑗4𝑉_𝑖𝑗𝑙+𝑃_𝑗−𝑃_𝑙𝑉⁻¹_𝑖𝑗𝑙𝑃_𝑗−𝑃_𝑙; (36) then we have 𝔄=𝔄₁₁𝑃_𝑖𝐴_𝑑𝑖+𝑃∗−𝑄^𝑖𝐵_𝑖𝑀_𝑖0Ξ_𝑖(𝑡)𝐹_𝑖𝐾_𝑖0+𝐹_𝑖𝐾_𝑖0^𝑇Ξ^𝑇_𝑖(𝑡)𝑃_𝑖𝐵_𝑖𝑀_𝑖0^𝑇<0, (37) which is equivalent to (14) by Schur complement.
Case III (𝑖∈𝑈^𝑖_𝑘and𝑈^𝑖_𝑢𝑘=∅). In this case, 𝔍^𝑥_𝑎[𝑉]≤Φ_𝑖+𝑥^𝑇𝑁_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗𝑃𝑗+𝜆_𝑖𝑖𝑃(𝑖)𝑥=Φ_𝑖+𝑥^𝑇_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑖𝑥=Φ_𝑖+𝑥^𝑇_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑖𝑥=Φ_𝑖+𝑥^𝑇_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗+Δ_𝑖𝑗𝑃_𝑗−𝑃_𝑖𝑥=Φ_𝑖+𝑥^𝑇_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑖+_{𝑗∈𝑈𝑖𝑘}Δ_𝑖𝑗𝑃_𝑗−𝑃_𝑖𝑥. (38) In view of Lemma 7, we have ^𝑠_{𝑗=1,𝑗≠𝑖}Δ_𝑖𝑗𝑃_𝑗−𝑃_𝑖=^𝑠_{𝑗=1,𝑗≠𝑖}12Δ_𝑖𝑗𝑃_𝑗−𝑃_𝑖+12Δ_𝑖𝑗𝑃_𝑗−𝑃_𝑖≤^𝑠_{𝑗=1,𝑗≠𝑖}12Δ_𝑖𝑗²𝑊_𝑖𝑗+𝑃_𝑗−𝑃_𝑖𝑊⁻¹_𝑖𝑗𝑃_𝑗−𝑃_𝑖≤^𝑠_{𝑗=1,𝑗≠𝑖}𝛿²_𝑖𝑗4𝑊_𝑖𝑗+𝑃_𝑗−𝑃_𝑖𝑊⁻¹_𝑖𝑗𝑃_𝑗−𝑃_𝑖. (39) From (38) and (39), we have 𝔍^𝑥_𝑎[𝑉]≤Φ_𝑖+𝑥^𝑇×^𝑠_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑖+^𝑠_{𝑗=1,𝑗≠𝑖}𝛿²_𝑖𝑗4𝑊_𝑖𝑗+𝑃_𝑗−𝑃_𝑖×𝑊⁻¹_𝑖𝑗𝑃_𝑗−𝑃_𝑖𝑥. (40) Hence, 𝔍^𝑥_𝑎[𝑉]<0 holds if ℛℛ=¹¹𝑃_𝑖𝐴_𝑑𝑖∗−𝑄<0, (41) where ℛ₁₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝑃_𝑖𝐵_𝑖𝑀_𝑖Ξ_𝑖(𝑡)𝐹_𝑖𝐾_𝑖+𝐾^𝑇_𝑖𝐹^𝑇_𝑖Ξ^𝑇_𝑖(𝑡)𝑀^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑄+_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑖+_{𝑗∈𝑈𝑖𝑘}𝛿²_𝑖𝑗4𝑊_𝑖𝑗+𝑃_𝑗−𝑃_𝑖𝑊⁻¹_𝑖𝑗𝑃_𝑗−𝑃_𝑖. (42) Let ℛ₁₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝐾^𝑇_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐵_𝑖𝐾_𝑖+𝑄+_{𝑗∈𝑈𝑖𝑘}𝜋_𝑖𝑗𝑃_𝑗−𝑃_𝑖+_{𝑗∈𝑈𝑖𝑘}𝛿²_𝑖𝑗4𝑊_𝑖𝑗+𝑃_𝑗−𝑃_𝑖𝑊⁻¹_𝑖𝑗𝑃_𝑗−𝑃_𝑖; (43) then we have ℛ=ℛ₁₁𝑃_𝑖𝐴_𝑑𝑖+𝑃∗−𝑄^𝑖𝐵_𝑖𝑀_𝑖0Ξ_𝑖(𝑡)𝐹_𝑖𝐾_𝑖0+𝐹_𝑖𝐾_𝑖0^𝑇Ξ^𝑇_𝑖(𝑡)𝑃_𝑖𝐵_𝑖𝑀_𝑖0^𝑇<0, (44) which is equivalent to (15) by Schur complement. The proof is completed.

Remark 9. As mentioned in Section 2, if, 𝑈^𝑖_𝑢𝑘=∅∀𝑖∈𝕊, then GUTR matrix reduces to BUTR one. Similarly, if, 𝛿_𝑖𝑗=0∀𝑗∈𝑈^𝑖_𝑘,∀𝑖∈𝕊, GUTR matrix reduces to PUTR one. Therefore, Theorem 8 can also be applicable to the MJSs with BUTRs or PUTRs. Because the BUTR methods require the estimate of every rate to be known, such methods cannot be applied to the GUTR model. By replacing the uncertain TRs with unknown ones, the generally uncertain TR matrix can become partly known TR matrix, so that the PUTR methods can be applied to GUTR model. However, such methods are inevitably conservative for GUTRs since the information of the TRs’ estimates cannot be utilized.

4. Conclusions

The stability problems for a class of Markovian jump linear systems with generally uncertain transition rates are investigated in this paper. The considered systems are more general than the systems with bounded uncertain transition rates or partly unknown transition rates, which can be viewed as two special cases of the systems we tackled here. The LMI-based stochastic stability condition for the underlying systems is derived. There are some possible directions to extend the proposed model and method. Tracking control and fault-tolerant control are two important research areas due to their wide application in the practical systems [40–46]. However, for the tracking control and fault-tolerant control, no research has focused on the case of general uncertain transition rates. Therefore, it is worth further extending the proposed method to deal with these problems.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and constructive suggestions. This research is supported by the Natural Science Foundation of Guangxi Autonomous Region (nos. 2012GXNSFBA053003, 2013YB141), the National Natural Science Foundations of China (60973048, 61272077, 60974025, 60673101, and 60939003), National 863 Plan Project (2008AA04Z401, 2009AA043404), the Natural Science Foundation of Shandong Province (no. Y2007G30), the Scientific and Technological Project of Shandong Province (no. 2007GG3WZ04016), the Science Foundation of Harbin Institute of Technology (Weihai) (HIT(WH) 200807), the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT.NSRIF. 2001120), and Shandong Provincial Key Laboratory of Industrialy.