Sliding Sector-Based Variable Structure Control of Continuous-Time Markov Jump Linear Systems Subject to Unknown Transition Rates

Abstract

Based on sliding sector technique, the variable structure control for a class of uncertain continuous-time Markovian jump linear systems (MJLS) is investigated. The elements in the transition rate matrix include completely known, boundary known, and completely unknown ones. First, the related notions about sliding sector for continuous-time Markov jump linear systems are given; then based on linear matrix inequalities (LMIs) technique, sufficient conditions for the design of the sliding sector are provided. Second, a variable structure control law is presented to guarantee the mean-square quadratic stability of the closed-loop system in spite of the effects of the existing uncertainties and unknown/uncertain transition rates. Finally, an example is given to verify the validity of the theoretical results.

1. Introduction

The subject of this paper is variable structure control of continuous-time Markov jump linear uncertain systems. Control design problems of Markov jump systems are motivated by the fact that these systems represent a class of important stochastic systems which is popular in modeling practical engineering systems subject to failures or changes in structures and parameters. Such as in [1, 2], Markovian switching technique has been used to model the switching action of mobile manipulators. In addition, some important theoretical results have also been published so far; see, for example, [3–9] and the references therein. It is worth noticing that these results require that the transition probabilities/rates must be known precisely. In fact, the likelihood of obtaining the perfect information on all transition probabilities/rates is questionable, and the cost might be expensive in some areas. As a result, the study of control synthesis for Markov jump systems with uncertain or unknown transition probabilities/rates becomes interesting, and some important results have been established. For instance, the robust control designs for MJLS subject to the norm-bounded or polytopic uncertainties in the transition rates were considered in [10, 11]. However, they both require to know the structure and “nominal” terms of the uncertain transition rates. Furthermore, control designs of MJLS with partly unknown transition rates were developed in [12–14], in which the transition rate matrix included two kinds of elements: completely known and completely unknown ones. In fact, besides the two kinds of elements, it might also own boundary known elements in practice, and we will consider this more general case for reducing the conservatism of control design in this paper.

On the other hand, variable structure control is a well-known robust control technique against external disturbances, parameter uncertainties, and unmodeled dynamics [15–17], and thus there are also some remarkable results published on variable structure control of continuous-time MJLS recently [18–22]. In [21], the robust stabilization of variable structure control for continuous-time MJLS with matched uncertainties was first investigated, and then in [22], a singular system approach was introduced for systems with parameter uncertainties and unknown nonlinear function. It is worth mentioning that these works require the transition rates to be precisely known a prior, and to the best of our knowledge, no results have been reported to variable structure control of continuous-time MJLS with unknown transition rates. In addition, one of the disadvantages about variable structure control is that chattering phenomenon might occur on the neighbor of the sliding manifold which is harmful to actuator devices, and thus it is a barrier for the use of the variable structure control strategy [23–25]. In order to avoid the occurrence of chattering phenomenon, it is worth noting that for a class of linear deterministic systems, a variable structure control-based sliding sector is first introduced in [26]. The main feature of the sliding sector-based variable structure control scheme is that zero control input can be used in the sliding sector of the state space and then the chattering phenomenon is avoided.

Motivated by the aforementioned discussion, for a class of continuous-time uncertain MJLS subject to unknown transition rates, a novel robust variable structure control-based sliding sector technique is investigated. The main contribution of this paper is summarized as follows. First, related notions of the sliding sector for MJLS, 𝑃_𝑖𝑅_𝑖-sliding sector, simplified 𝑃_𝑖𝑅_𝑖-sliding sector, and inner sector and outer sector are given. Second, sufficient conditions for the design of the simplified 𝑃_𝑖𝑅_𝑖-sliding sector for the considered Markov jump linear uncertain system are presented in terms of LMI technique. Finally, a variable structure control law is established to drive the state trajectory to converge to the inner sector of the 𝑃_𝑖𝑅_𝑖-sliding sector and guarantee mean square quadratic stability of the closed-loop system in spite of the effect of unknown transition rates and system uncertainties.

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.

Throughout this paper, the following notations are used. ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space; 𝐴^𝑇 stands for the transpose of matrix 𝐴; 𝐼 and 0 represent the identity matrix and zero matrix in appropriate dimension, respectively. In addition, the symbol He(𝑋) is used to represent 𝑋+𝑋^𝑇. 𝑃>0 (𝑃≥0 ) means that 𝑃 is real symmetric and (semi-) positive definite. |⋅| denotes the absolute value of a scalar, the standard Euclidean norm of a vector, or the induced norm of a matrix, respectively. In symmetric block matrices, the notation ∗ is used to represent a term that is induced by symmetry.

2. Problem Statement and Preliminaries

Consider the following time-invariant continuous-time linear uncertain system: 𝑥̇=𝐴𝑟_𝑡+Δ𝐴𝑟_𝑡𝑥+𝐵𝑟_𝑡+Δ𝐵𝑟_𝑡𝑢, (1) where 𝑥(𝑡)∈ℝ^𝑛 and 𝑢(𝑡)∈ℝ are state and control input, respectively. Δ𝐴𝑟_𝑡 and Δ𝐵(𝑟_𝑡) are the system uncertainties. {𝑟_𝑡,𝑡≥0} is a continuous-time Markovian process with right continuous trajectories and taking values in a finite set 𝒮={1,2,…,ℕ}. It governs the switching among the different system modes with the following mode transition probabilities: Prob𝑟_𝑡+𝑚=𝑗∣𝑟_𝑡=𝜆=𝑖^𝑖𝑗𝑚+𝑜(𝑚),if𝑗≠𝑖,1+𝜆_𝑖𝑖𝑚+𝑜(𝑚),if𝑗=𝑖, (2) where 𝑚>0, lim_𝑚→0(𝑜(𝑚)/𝑚)=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+𝑚, and 𝜆_𝑖𝑖=−∑^ℕ_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗 for each 𝑖∈𝒮.

In general, the Markov process transition rate matrix Λ is defined by Λ=𝜆_{𝑖𝑗ℕ×ℕ}. (3)

In this paper, the transition rates of the jumping process are assumed to be partly available; that is, some elements in Λ have been exactly known; some ones are uncertain but with known boundaries; others may have no information to use. For instance, for system (1) with four operation modes, the transition rate matrix might be described by Λ=?𝜆₁₂𝜆₁₃?𝜆₂₁?𝜆₂₃??𝑎?𝜆₃₄𝑏?𝜆₄₃𝜆₄₄, (4) where ? represents the completely unknown element in the transition rate matrix; parameters 𝑎 and 𝑏 represent the elements with known lower and upper bounds. That is, 𝑎≤𝑎≤𝑎 and 𝑏≤𝑏≤𝑏, where 𝑎,𝑎,𝑏, and 𝑏 are known parameters; 𝜆_𝑖𝑗 denotes the precisely known element.

For clarity, denote 𝒮=𝒮^𝑖_𝑘1∪𝒮^𝑖_𝑘2∪𝒮^𝑖_𝑢𝑘, 𝑖=1,2,…,ℕ by 𝒮^𝑖_𝑘1=𝑗∶𝜆_𝑖𝑗,𝒮isexactlyknown_𝑖𝑘2=𝑗∶theboundsof𝜆_𝑖𝑗,𝒮isknown_𝑖𝑢𝑘=𝑗∶thereisnoinformationavailablefor𝜆_𝑖𝑗. (5) Furthermore, let 𝒮^𝑖_𝑘={𝑗∶𝜆_𝑖𝑗≤𝜆_𝑖𝑗≤𝜆_𝑖𝑗}; then 𝒮^𝑖_𝑘=𝒮^𝑖_𝑘1∪𝒮^𝑖_𝑘2. Let 𝒮^𝑖_𝑘={𝒦^𝑖₁,𝒦^𝑖₂,…,𝒦^𝑖_𝑚𝑖𝑘}.   𝒮^𝑖_𝑢𝑘={𝕂^𝑖₁,𝕂^𝑖₂,…,𝕂^𝑖_{𝑚𝑖𝑢𝑘}}, 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 element in 𝒮^𝑖_𝑢𝑘 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.

Remark 1. When 𝜆_𝑖𝑗=𝜆_𝑖𝑗, the transition rates are reduced to the situation in [12]. Obviously, the solving method there can only treat the above known bound case as the completely unknown case, and some conservativeness has been brought in.

For convenience, let 𝐴_𝑖=𝐴(𝑟_𝑡), Δ𝐴_𝑖=Δ𝐴(𝑟_𝑡), 𝐵_𝑖=𝐵(𝑟_𝑡), and Δ𝐵_𝑖=Δ𝐵(𝑟_𝑡) for each possible value 𝑟_𝑡=𝑖, 𝑖∈𝒮, where 𝐴_𝑖 and 𝐵_𝑖 are known constant matrices with appropriate dimensions; then system (1) can be described by 𝑥̇=𝐴_𝑖+Δ𝐴_𝑖𝑥+𝐵_𝑖+Δ𝐵_𝑖𝑢. (6)

Throughout this paper, for each 𝑖, suppose that the following assumptions are valid.

Assumption 2. The pair (𝐴_𝑖,𝐵_𝑖) is controllable.

Assumption 3. The uncertainties satisfy Δ𝐴_𝑖=𝐷_𝑖𝐸_𝑖(𝑡,𝑥)𝐹_𝑖, Δ𝐵_𝑖=𝐵_𝑖Θ_𝑖(𝑡), where 𝐷_𝑖 and 𝐹_𝑖 are known constant matrices with appropriate dimensions, 𝐸_𝑖(𝑡,𝑥) is time-varying uncertain matrix satisfying 𝐸_𝑖𝐸^𝑇_𝑖≤𝐼, the parameter Θ_𝑖(𝑡) satisfies |Θ_𝑖(𝑡)|≤𝑑_𝑖<1, 𝑑_𝑖 is a known positive scalar.

For continuous-time Markov jump linear nominal systems 𝑥̇=𝐴_𝑖𝑥+𝐵_𝑖𝑢, (7) some related notions of sliding sector are firstly presented.

Definition 4. The 𝑃_𝑖𝑅_𝑖-sliding sector is a subset of ℝ^𝑛 defined as 𝒮_𝑖=𝑥∣𝑥^𝑇𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+^ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗𝑥≤−𝑥^𝑇𝑅_𝑖𝑥,𝑥∈ℝ^𝑛, (8) where 𝑃_𝑖∈ℝ^𝑛×𝑛 is a positive definite symmetric matrix, 𝑅_𝑖∈ℝ^𝑛×𝑛 is a positive semidefinite symmetric matrix, 𝑅_𝑖=𝐶^𝑇_𝑖𝐶_𝑖, 𝐶_𝑖∈ℝ^{𝑙𝑖×𝑛}, 𝑙_𝑖≥1, and (𝐶_𝑖,𝐴_𝑖) is an observable pair.

The set of the 𝑃_𝑖𝑅_𝑖-sliding sector is nonempty since at least zero state stays inside it.

Obviously, while the state of the plant evolves in the 𝑃_𝑖𝑅_𝑖-sliding sector 𝒮_𝑖, the 𝑃_𝑖-norm |𝑥|_𝑃𝑖=𝑥^𝑇𝑃_𝑖𝑥 of the plant (7) decreases with zero control input since 𝐿̇_𝑖=𝑥^𝑇𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+^ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗𝑥≤−𝑥^𝑇𝑅_𝑖𝑥≤0,∀𝑥∈𝒮_𝑖,𝑥≠0, (9) where 𝐿_𝑖=|𝑥|²_𝑃𝑖.

Remark 5. When 𝐴_𝑖=𝐴, 𝑃_𝑖=𝑃, 𝑅_𝑖=𝑅, the 𝑃_𝑖𝑅_𝑖-sliding sector in (8) becomes 𝒮=𝑥∣𝑥^𝑇𝐴^𝑇𝑃+𝑃𝐴𝑥≤−𝑥^𝑇𝑅𝑥,𝑥∈ℝ^𝑛. (10) The definition of 𝑃𝑅-sliding sector for linear deterministic systems is introduced in [26].

Definition 6. A simplified 𝑃_𝑖𝑅_𝑖-sliding sector is a subset of ℝ^𝑛 defined as 𝒮_𝑖=𝑥∣𝑠_𝑖(𝑥)≤𝛿_𝑖(𝑥),𝑥∈ℝ^𝑛, (11) where the linear function 𝑠_𝑖(𝑥) and the square root 𝛿_𝑖(𝑥) of the quadratic function 𝛿²_𝑖(𝑥) are determined by 𝑠_𝑖(𝑥)=𝑆_𝑖𝑥,𝑆_𝑖∈ℝ^1×𝑛,𝛿_𝑖(𝑥)=𝑥^𝑇Ω_𝑖𝑥,Ω_𝑖∈ℝ^𝑛×𝑛,Ω_𝑖≥0Ω_𝑖≠0. (12)

Remark 7. Suppose that the following inequalities are satisfied: 𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+^ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗−𝑃_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑄_𝑖<0,𝑖=1,2,…,ℕ. (13) Then the parameters in the simplified 𝑃_𝑖𝑅_𝑖-sliding sector 𝑃_𝑖, 𝑅_𝑖, 𝑆_𝑖, and Ω_𝑖 can be selected as follows: Ω_𝑖=𝑟_𝑖𝑄_𝑖,𝑅_𝑖=1−𝑟_𝑖𝑄_𝑖,𝑆_𝑖=𝐵^𝑇_𝑖𝑃_𝑖, (14) where 0<𝑟_𝑖<1.

Obviously, while the state of the plant evolves in the simplified 𝑃_𝑖𝑅_𝑖-sliding sector 𝒮_𝑖, the 𝑃_𝑖-norm |𝑥|_𝑃𝑖 of the plant (7) decreases with zero control input since 𝐿̇_𝑖=𝑥^𝑇𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+^ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗𝑥=𝑥^𝑇𝑃_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑃_𝑖−𝑄_𝑖𝑥=𝑠²_𝑖(𝑥)−𝛿²_𝑖(𝑥)−𝑥^𝑇𝑅_𝑖𝑥≤−𝑥^𝑇𝑅_𝑖𝑥,∀𝑥∈𝒮_𝑖,𝑥≠0. (15)

In addition, in order to avoid the chattering phenomenon on the boundary of the 𝑃_𝑖𝑅_𝑖-sliding sector, an inner sector 𝒮ⁱⁿ_𝑖 and an outer sector 𝒮^out_𝑖 are introduced as subsets of the 𝑃_𝑖𝑅_𝑖-sliding sector (11): 𝒮ⁱⁿ_𝑖=𝑥∣𝑠_𝑖(𝑥)≤𝛼_𝑖𝛿_𝑖(𝑥),𝑥∈ℝ^𝑛,𝒮^out_𝑖=𝑥∣𝛼_𝑖𝛿_𝑖(𝑥)<𝑠_𝑖(𝑥)≤𝛿_𝑖(𝑥),𝑥∈ℝ^𝑛, (16)(17) where the positive scalar 𝛼_𝑖 satisfies 0<𝛼_𝑖<1. It is not difficult to see that the sets 𝒮ⁱⁿ_𝑖 and 𝒮^out_𝑖 constitute a partition of the simplified 𝑃𝑅-sliding sector 𝒮_𝑖. That is, 𝒮_𝑖=𝒮ⁱⁿ_𝑖∪𝒮^out_𝑖 and 𝒮ⁱⁿ_𝑖∩𝒮^out_𝑖=∅.

A hysteresis dead-zone function 𝜎_𝑖(𝑠_𝑖(𝑥),𝛿_𝑖(𝑥)) is further introduced for avoiding the chattering. It is defined as 𝜎_𝑖𝑠_𝑖(𝑥),𝛿_𝑖(𝑥)=0,𝑥∈𝒮ⁱⁿ_𝑖unchanged,𝑥∈𝒮^out_𝑖1,𝑥∈𝒮_𝑖. (18)

Lemma 8 (see [20]). 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. (19)

Lemma 9 (see [27]). 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. (20)

Lemma 10 (see [28]). For symmetric and positive definite matrices 𝑋 and 𝑌, if 𝑋≥𝑌>0, then 𝑌⁻¹≥𝑋⁻¹>0.

3. Main Results

First, we present the design of the simplified 𝑃_𝑖𝑅_𝑖-sliding sector for the considered continuous-time Markov jump linear uncertain system (6). Second, the controller design for mean-square quadratic stability of the continuous-time Markov jump linear uncertain system is developed.

3.1. The Design of the Simplified 𝑃_𝑖𝑅_𝑖-Sliding Sector

Theorem 11. For continuous-time Markov jump linear uncertain system (6), there exist symmetric and positive definite matrices 𝑋_𝑖, 𝑍_𝑖, and 𝑌_𝑖 satisfying the following LMIs: 𝑖∈𝒮^𝑖_𝑘,Γ_11𝑖Γ_12𝑖Γ_13𝑖𝑋_𝑖𝑋_𝑖𝐹_𝑖∗−Γ_22𝑖000∗∗−𝑍_𝑖00∗∗∗−𝑌_𝑖0∗∗∗∗−𝜀_𝑖𝐼𝑍<0,_𝑖≤𝑋_𝑗,𝑗∈𝒮^𝑖_𝑢𝑘,𝑗≠𝑖𝑖∈𝒮^𝑖_𝑢𝑘,Φ_11𝑖Φ_12𝑖𝑋_𝑖𝑋_𝑖𝐹_𝑖∗−𝑋_𝑗00∗∗−𝑌_𝑖0∗∗∗−𝜀_𝑖𝐼𝑋<0,_𝑖≤𝑋_𝑗,𝑗∈𝒮^𝑖_𝑢𝑘,𝑗≠𝑖, (21)(22) where Γ_11𝑖=He𝐴_𝑖𝑋_𝑖+𝜆_𝑖𝑖𝑋_𝑖−𝐵_𝑖𝐵^𝑇_𝑖+𝜀_𝑖𝐷_𝑖𝐷^𝑇_𝑖,Γ_12𝑖=𝜆_𝑖𝒦^𝑖₁𝑋_𝑖,𝜆_𝑖𝒦^𝑖₂𝑋_𝑖,…,𝜆_𝑖𝒦^𝑖_𝑚𝑖𝑘𝑋_𝑖,Γ_22𝑖=diag−𝑋_𝒦^𝑖₁,−𝑋_𝒦^𝑖₂,…,−𝑋_𝒦^𝑖_𝑚𝑖𝑘,Γ_13𝑖=−_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗+𝜆_𝑖𝑖𝑋_𝑖,Φ_11𝑖=He𝐴_𝑖𝑋_𝑖−_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑋_𝑖−𝐵_𝑖𝐵^𝑇_𝑖+𝜀_𝑖𝐷_𝑖𝐷^𝑇_𝑖,Φ_12𝑖=𝜆_𝑖𝒦^𝑖₁𝑋_𝑖,𝜆_𝑖𝒦^𝑖₂𝑋_𝑖,…,𝜆_𝑖𝒦^𝑖_𝑚𝑖𝑘𝑋_𝑖,Φ_22𝑖=diag−𝑋_𝒦^𝑖₁,−𝑋_𝒦^𝑖₂,…,−𝑋_𝒦^𝑖_𝑚𝑖𝑘; (23) then the parameters of 𝑃_𝑖𝑅_𝑖-sliding sector in (14) can be designed as 𝑃_𝑖=𝑋_𝑖⁻¹, 𝑄_𝑖=𝑌_𝑖⁻¹, Ω_𝑖=𝑟_𝑖𝑌_𝑖⁻¹, 𝑅_𝑖=(1−𝑟_𝑖)𝑌_𝑖⁻¹, and 𝑆_𝑖=𝐵^𝑇_𝑖𝑋_𝑖⁻¹.

Proof. For system (6), for all Δ𝐴_𝑖, 𝑖=1,2,…,ℕ, the conditions in (13) are equal to 𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖+Δ𝐴_𝑖+_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑗+_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑗+𝜆_𝑖𝑖𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑄_𝑖<0. (24) In the following, we consider two cases: 𝑖∈𝒮^𝑖_𝑘 and 𝑖∈𝒮^𝑖_𝑢𝑘 in the proof.

Case I (𝑖∈𝒮^𝑖_𝑘). In this case, the conditions in (24) can be guaranteed when for 𝑖=1,2,…,ℕ, the following inequalities: He𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑃_𝑖+_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑗+_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑊_𝑖+𝜆_𝑖𝑖𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑄_𝑖𝑃<0,_𝑗≤𝑊_𝑖,𝑗∈𝒮^𝑖_𝑢𝑘, (25) are satisfied. Substituting ∑_𝑗∈𝒮^𝑖_{𝑢𝑘,𝑗≠𝑖}𝜆_𝑖𝑗=−∑_𝑗∈𝒮^𝑖_{𝑘,𝑗≠𝑖}𝜆_𝑖𝑗−𝜆_𝑖𝑖 into the first inequality above and utilizing the boundary information of 𝜆_𝑖𝑗, one can see that the above inequalities can be guaranteed by He𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑃_𝑖+_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑗−_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗+𝜆_𝑖𝑖𝑊_𝑖+𝜆_𝑖𝑖𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑄_𝑖𝑃<0,_𝑗≤𝑊_𝑖,𝑗∈𝒮^𝑖_𝑢𝑘. (26) Pre- and postmultiplying 𝑃_𝑖⁻¹ on both sides of the first inequality in above inequalities and letting 𝑋_𝑖=𝑃_𝑖⁻¹, one has He𝐴_𝑖+Δ𝐴_𝑖𝑋_𝑖+_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑋_𝑖𝑋_𝑗⁻¹𝑋_𝑖−_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗+𝜆_𝑖𝑖𝑋_𝑖𝑊_𝑖𝑋_𝑖+𝜆_𝑖𝑖𝑋_𝑖−𝐵_𝑖𝐵^𝑇_𝑖+𝑋_𝑖𝑄_𝑖𝑋_𝑖<0. (27) Let 𝑌_𝑖=𝑄_𝑖⁻¹, 𝑍_𝑖=𝑊_𝑖⁻¹, and applying Schur-complement technique and Lemma 9, one can obtain the first LMI condition in (21).
In addition, by using Lemma 10, one can see that the second inequality in (26) can be converted into the second LMI condition in (21).

Case II (𝑖∈𝒮^𝑖_𝑢𝑘). In this case, substituting 𝜆_𝑖𝑖=−∑_𝑗∈𝒮^𝑖_{𝑘,𝑗≠𝑖}𝜆_𝑖𝑗−∑_𝑗∈𝒮^𝑖_{𝑢𝑘,𝑗≠𝑖}𝜆_𝑖𝑗 into (24), one has He𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑃_𝑖+_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑗+_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑗−_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗+_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑄_𝑖<0,𝑖=1,2,…,ℕ. (28)
The conditions in (28) can be guaranteed provided that for 𝑖=1,2,…,ℕ, one has He𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑃_𝑖+_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑗−_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑄_𝑖𝑃<0,_𝑗≤𝑃_𝑖,𝑗∈𝒮^𝑖_𝑢𝑘,𝑗≠𝑖. (29) Taking use of the boundary information of 𝜆_𝑖𝑗, one can see that the above inequalities can be guaranteed by He𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑃_𝑖+_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑗−_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑃_𝑖−𝑃_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝑄_𝑖𝑃<0,_𝑗≤𝑃_𝑖,𝑗∈𝒮^𝑖_𝑢𝑘,𝑗≠𝑖. (30) Pre- and postmultiplying 𝑃_𝑖⁻¹ on both sides of the first inequality in above inequalities and letting 𝑋_𝑖=𝑃_𝑖⁻¹ and 𝑌_𝑖=𝑄_𝑖⁻¹, one has He𝐴_𝑖+Δ𝐴_𝑖𝑋_𝑖+_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑋_𝑖𝑋_𝑗⁻¹𝑋_𝑖−_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝜆_𝑖𝑗𝑋_𝑖−𝐵_𝑖𝐵^𝑇_𝑖+𝑋_𝑖𝑄_𝑖⁻¹𝑋_𝑖<0. (31) Applying Schur-complement technique and Lemma 9 again, the above inequality (31) can be converted into the first LMI condition in (22).
Applying Lemma 10, one can see that the second inequality in (30) can be converted into the second condition in (22). By the design of the parameters of PR-sliding sector in (14), the proof of Theorem 11 is completed.

3.2. Variable Structure Control Design

Theorem 12. Considering the simplified 𝑃_𝑖𝑅_𝑖-sliding sector 𝒮_𝑖 (11) with the inner sector 𝒮_𝑖^𝑖𝑛 (16) and the outer sector 𝒮_𝑖^𝑜𝑢𝑡 (17), the following variable structure controller is designed to ensure that the state trajectories of the plant (6) enter into the inner sector from the outside of 𝑃_𝑖𝑅_𝑖-sliding sector and obtain mean-square quadratic stability of the closed-loop system. Consider 𝑢=𝑢_𝑖1+𝑢_𝑖2, (32) where 𝑢_𝑖1=−𝜎_𝑖𝑠_𝑖(𝑥),𝛿_𝑖(𝑥)𝑆_𝑖𝐵_𝑖⁻¹𝑆_𝑖𝐴_𝑖𝑥+𝐾_𝑖𝑠_𝑖(𝑥),𝑢_𝑖21=−1−𝑑_𝑖𝜎_𝑖𝑠_𝑖(𝑥),𝛿_𝑖(𝑥)×𝑆_𝑖𝐵_𝑖⁻¹sign𝑠_𝑖(𝑥)𝜌_𝑖, (33)(34)𝜌_𝑖=|𝑆_𝑖||𝐷_𝑖||𝐹_𝑖||𝑥|+𝑑_𝑖|𝑆_𝑖𝐴_𝑖||𝑥|+𝑑_𝑖𝐾_𝑖|𝑠_𝑖(𝑥)| and the parameter 𝐾_𝑖>max{𝑆_𝑖𝐵_𝑖/2,𝐻_𝑖} satisfies the following inequality: 2𝐻_𝑖𝛼²_𝑖𝑟_𝑖𝑄_𝑖+𝑆^𝑇_𝑖𝑆_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑆^𝑇_𝑖𝑆_𝑖>0. (35)

Proof. It is assumed that the initial state of the plant lies outside the 𝑃_𝑖𝑅_𝑖-sliding sector. In this case, we have 𝜎_𝑖(𝑠_𝑖(𝑥),𝛿_𝑖(𝑥))=1 and 𝑢_𝑖1=−(𝑆_𝑖𝐵_𝑖)⁻¹(𝑆_𝑖𝐴_𝑖𝑥+𝐾_𝑖𝑠_𝑖(𝑥)). It can be easily checked that 𝑠_𝑖(𝑥)𝑠_𝑖̇(𝑥)=−𝐾_𝑖𝑠²_𝑖(𝑥)+𝑠_𝑖(𝑥)𝑆_𝑖Δ𝐴_𝑖𝑥+𝑠_𝑖(𝑥)𝑆_𝑖𝐵_𝑖𝑢_𝑖2+𝑠_𝑖(𝑥)𝑆_𝑖Δ𝐵_𝑖𝑢. (36) Noting that Δ𝐵_𝑖=𝐵_𝑖Θ_𝑖 and |Θ|≤𝑑_𝑖<1, we have 𝑠_𝑖(𝑥)𝑠_𝑖̇(𝑥)≤−𝐾_𝑖𝑠²_𝑖(𝑥)+𝑠_𝑖(𝑥)𝑆_𝑖𝐵_𝑖𝑢_𝑖2+𝑠_𝑖(𝑥)𝑆_𝑖Δ𝐴_𝑖|𝑥|+𝑑_𝑖𝑠_𝑖(𝑥)𝑆_𝑖𝐵_𝑖|𝑢|. (37) In view of 𝑢=𝑢_𝑖1+𝑢_𝑖2, it is easy to check that 𝑠_𝑖(𝑥)𝑠_𝑖̇(𝑥)≤−𝐾_𝑖𝑠²_𝑖(𝑥)+𝑠_𝑖(𝑥)𝑆_𝑖𝐵_𝑖𝑢_𝑖2+𝑠_𝑖(𝑥)𝑆_𝑖Δ𝐴_𝑖|𝑥|+𝑑_𝑖𝑠_𝑖(𝑥×𝑆_𝑖)𝐵_𝑖𝑢_𝑖1+𝑑_𝑖𝑠_𝑖(𝑥)𝑆_𝑖𝐵_𝑖𝑢_𝑖2. (38) Substituting 𝑢_𝑖1=−(𝑆_𝑖𝐵_𝑖)⁻¹(𝑆_𝑖𝐴_𝑖𝑥+𝐾_𝑖𝑠_𝑖(𝑥)) into the above inequality and noting that 𝑆_𝑖𝐵_𝑖=𝐵^𝑇_𝑖𝑃_𝑖𝐵_𝑖>0, one can see that 𝑠_𝑖(𝑥)𝑠_𝑖̇(𝑥)≤−1−𝑑_𝑖𝐾_𝑖𝑠²_𝑖(𝑥)+1−𝑑_𝑖𝑠_𝑖(𝑥)𝑆_𝑖𝐵_𝑖𝑢_𝑖2+𝑠_𝑖(𝑥)𝑆_𝑖Δ𝐴_𝑖|𝑥|+𝑑_𝑖𝑠_𝑖(𝑥)𝑆_𝑖𝐴_𝑖×|𝑥|+𝑑_𝑖𝑠_𝑖(𝑥)𝑆_𝑖𝐵_𝑖𝑢_𝑖2+𝑑_𝑖𝑠_𝑖(𝑥)𝑆_𝑖𝐵_𝑖𝑢_𝑖2. (39) Taking use of 𝑢_𝑖2 in (34) where 𝜌_𝑖=|𝑆_𝑖||𝐷_𝑖||𝐹_𝑖||𝑥|+𝑑_𝑖|𝑆_𝑖𝐴_𝑖||𝑥|+𝑑_𝑖𝐾_𝑖|𝑠_𝑖(𝑥)|>𝑑_𝑖|𝑆_𝑖𝐴_𝑖||𝑥|+𝑑_𝑖𝐾_𝑖|𝑠_𝑖(𝑥)|, one can see that 𝑠_𝑖(𝑥)𝑠_𝑖̇(𝑥)≤−𝐾_𝑖𝑠²_𝑖(𝑥)+𝑑_𝑖𝑠_𝑖(𝑥)𝑆_𝑖𝐵_𝑖𝑢_𝑖2+𝑑_𝑖𝑠_𝑖(𝑥)𝑆_𝑖𝐵_𝑖𝑢_𝑖2. (40) According to the design of 𝑢_𝑖2 again, one can easily observe that 𝑠_𝑖(𝑥)𝑆_𝑖𝐵_𝑖𝑢_𝑖2+𝑠_𝑖(𝑥)𝑆_𝑖𝐵_𝑖𝑢_𝑖21≤−1−𝑑_𝑖𝑠_𝑖(𝑥)𝜌_𝑖+11−𝑑_𝑖𝑠_𝑖(𝑥×𝑆_𝑖)𝐵_𝑖𝑆_𝑖𝐵_𝑖⁻¹sign𝑠_𝑖(𝑥)𝜌_𝑖=0. (41) It follows from (40) and (41) that 𝑠_𝑖(𝑥)𝑠_𝑖̇(𝑥)≤−𝐾_𝑖𝑠²_𝑖(𝑥). (42) Thus the state trajectory of the closed-loop system will be driven into the inner sector of the 𝑃_𝑖𝑅_𝑖-sliding sector.
In the following, we will prove that the 𝑃_𝑖-norm keeps decreasing. Consider 𝐿̇_𝑖(𝑡)=𝑥^𝑇𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑃+𝑃𝐴_𝑖+Δ𝐴_𝑖+^ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗𝑥+2𝑥^𝑇𝑃_𝑖𝐵_𝑖+Δ𝐵_𝑖𝑢≤𝑠²_𝑖(𝑥)−𝛿²_𝑖(𝑥)−𝑥^𝑇𝑅_𝑖𝑥+2𝑥^𝑇𝑃_𝑖𝐵_𝑖+Δ𝐵_𝑖𝑢. (43) Noting that Δ𝐵_𝑖=𝐵_𝑖Θ_𝑖 and 𝑆_𝑖=𝐵^𝑇_𝑖𝑃_𝑖, one can see that 𝐿̇_𝑖(𝑡)≤𝑠²_𝑖(𝑥)−𝛿²_𝑖(𝑥)−𝑥^𝑇𝑅_𝑖𝑥+2𝑠_𝑖(𝑥)𝑢_𝑖1+2𝑠_𝑖(𝑥)𝑢_𝑖2+2𝑠_𝑖(𝑥)Θ_𝑖𝑢_𝑖1+𝑢_𝑖2. (44)
In the following, we first prove that 𝑠_𝑖(𝑥)𝑢_𝑖2+𝑑_𝑖𝑠_𝑖(𝑥)𝑢_𝑖1+𝑢_𝑖2≤0. (45) Substituting (33) and (34) into the left side of above inequality, one has 𝑠_𝑖(𝑥)𝑢_𝑖2+𝑑_𝑖𝑠_𝑖(𝑥)𝑢_𝑖1+𝑢_𝑖21=−1−𝑑_𝑖𝑆_𝑖𝐵_𝑖⁻¹𝑠_𝑖(𝑥)𝜌_𝑖+𝑑_𝑖𝑠_𝑖(𝑥×𝑆_𝑖)𝐵_𝑖⁻¹𝑆_𝑖𝐴_𝑖𝑥+𝐾_𝑖𝑆_𝑖𝐵_𝑖⁻¹𝑠_𝑖(𝑥)+11−𝑑_𝑖𝑆_𝑖𝐵_𝑖⁻¹sign𝑠_𝑖(𝑥)𝜌_𝑖. (46) Noting that 𝑆_𝑖𝐵_𝑖=𝐵^𝑇_𝑖𝑃_𝑖𝐵_𝑖>0 and 𝑑_𝑖<1, we have 𝑠_𝑖(𝑥)𝑢_𝑖2+𝑑_𝑖𝑠_𝑖(𝑥)𝑢_𝑖1+𝑢_𝑖2≤−𝑆_𝑖𝐵_𝑖⁻¹𝑠_𝑖(𝑥)𝜌_𝑖+𝑑_𝑖𝑆_𝑖𝐵_𝑖⁻¹𝑠_𝑖(𝑥×𝑆_𝑖)𝐴_𝑖𝑥+𝐾_𝑖𝑠_𝑖(𝑥). (47) Since 𝜌_𝑖=|𝑆_𝑖||𝐷_𝑖||𝐹_𝑖||𝑥|+𝑑_𝑖|𝑆_𝑖𝐴_𝑖||𝑥|+𝑑_𝑖𝐾_𝑖|𝑠_𝑖(𝑥)|, it is easy to check that 𝜌_𝑖≥𝑑_𝑖𝑆_𝑖𝐴_𝑖𝑥+𝐾_𝑖𝑠_𝑖(𝑥). (48) It follows from (47) and (48) that (45) is established. Thus we have 𝐿̇_𝑖(𝑡)≤𝑠²_𝑖(𝑥)−𝛿²_𝑖(𝑥)−𝑥^𝑇𝑅_𝑖𝑥+2𝑠_𝑖(𝑥)𝑢_𝑖1. (49) Substituting (33) into the above inequality (49) and noting that 𝐾_𝑖>(𝑆_𝑖𝐵_𝑖/2), 2𝐾_𝑖𝛼²_𝑖𝑟_𝑖𝑄_𝑖+𝑆^𝑇_𝑖𝑆_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑆^𝑇_𝑖𝑆_𝑖>0 and 𝑠²_𝑖(𝑥)≥𝛼_𝑖𝛿²_𝑖(𝑥), one can see that 𝐿̇_𝑖(𝑡)≤𝑠²_𝑖(𝑥)−𝛿²_𝑖(𝑥)−𝑥^𝑇𝑅_𝑖𝑥+2𝑠_𝑖(𝑥)×−𝑆_𝑖𝐵_𝑖⁻¹𝑆_𝑖𝐴_𝑖𝑥−𝐾_𝑖𝑆_𝑖𝐵_𝑖⁻¹𝑠_𝑖(𝑥)=−𝑆_𝑖𝐵_𝑖⁻¹𝑥^𝑇2𝐾_𝑖𝛼²_𝑖𝑟_𝑖𝑄_𝑖+𝑆^𝑇_𝑖𝑆_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑆^𝑇_𝑖𝑆_𝑖𝑥−𝑥^𝑇𝑅_𝑖𝑥<−𝑥^𝑇𝑅_𝑖𝑥. (50) After being forced into the inner sector of the 𝑃_𝑖𝑅_𝑖-sliding sector and before moving out of the 𝑃_𝑖𝑅_𝑖-sliding sector, the control input 𝑢(𝑡) is set equal to zero with 𝜎_𝑖(𝑠_𝑖(𝑥),𝛿_𝑖(𝑥))=0. In this case, |𝑠_𝑖(𝑥)|≤𝛿_𝑖(𝑥) and 𝐿̇_𝑖(𝑡)=𝑥^𝑇𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑃+𝑃𝐴_𝑖+Δ𝐴_𝑖+^ℕ_𝑗=1𝜆_𝑖𝑗𝑃_𝑗𝑥≤𝑠²_𝑖(𝑥)−𝛿²_𝑖(𝑥)−𝑥^𝑇𝑅_𝑖𝑥≤−𝑥^𝑇𝑅_𝑖𝑥.∀𝑥∈𝒮_𝑖. (51)
Once the state trajectory of the plant runs out of the 𝑃_𝑖𝑅_𝑖-sliding sector with zero control input, the control input (32)–(34) will make it move back to the inside of the inner sector again, while the 𝑃_𝑖-norm |𝑥|_𝑃𝑖 keeps decreasing.
Thus, from the above proof, one can see that it guarantees that the state trajectory of the plant moves from the outside of the 𝑃_𝑖𝑅_𝑖-sliding sector into the inside of the inner sector and the Lyapunov function 𝐿_𝑖(𝑥)=𝑥^𝑇𝑃_𝑖𝑥 keeps decreasing in the whole state space; then mean-square quadratic stability of the closed-loop system is established.

4. Illustrative Example

Consider the Markov jump linear system with four operation modes: 𝐴₁=1023,𝐴₂=,𝐴011−2₃=200−1,𝐴₄=,𝐵31−12₁=11,𝐵₂=11,𝐵₃=21,𝐵₄=23,𝐷₁=00.1,𝐷₂=𝐷0.20.5₃=0.20.6,𝐷₄=,𝐹0.10.5₁=0−0.1,𝐹₂=,𝐹10.3₃=0.20.5,𝐹₄=0.10.1. (52)

For simulation, we take 𝐸₁(𝑡,𝑥)=0.1sin(𝑡), 𝐸₂(𝑡,𝑥)=0.1sin(2𝑡), 𝐸₃(𝑡,𝑥)=0.5sin(𝑡), 𝐸₄(𝑡,𝑥)=0.4sin(2𝑡), Θ₁=0.2sin(2), Θ₂=0.3cos(2𝑡), Θ₃=0.2cos(𝑡), and Θ₄=0.2sin(2𝑡).

The transition rates matrix is Λ=−1.50.2????0.50.91?−2?0.1𝛼?𝛽, (53) where 0.5≤𝛼≤2 and −0.8≤𝛽≤−0.1. Since there exist the completely unknown transition rates “?” and boundary known transition rates 𝛼 and 𝛽, the reported methods in [21, 22] cannot work. Solving the LMIs conditions (21)-(22) in Theorem 11, one can obtain that positive definite matrices are as follows: 𝑃₁=6.2534−6.4840−6.484018.4380,𝑃₂=,𝑃6.8785−7.1343−7.134320.0458₃=8.5811−9.6120−9.612015.6407,𝑃₄=,𝑄6.1859−1.5374−1.53741.1026₁=,𝑄6.9519−7.5088−7.508822.4304₂=𝑄13.4209−14.1627−14.162740.0263₃=,𝑄11.4377−11.6823−11.682318.2712₄=.10.3808−2.6205−2.62052.0926 (54) Selecting 𝛼₁=0.8, 𝛼₂=0.9, 𝛼₃=0.6, 𝛼₄=0.6, 𝑟₁=0.8, 𝑟₂=0.9, 𝑟₃=0.8, 𝑟₄=0.8, 𝐾₁=20, 𝐾₂=120, 𝐾₃=40, 𝐾₄=50, then the condition in Theorem 12 is satisfied. For the purpose of simulation, let 𝑑₁=0.2, 𝑑₂=0.3, 𝑑₃=0.2, and 𝑑₄=0.2. Using these data, a simulation program has been done in Matlab. The simulation results are presented in Figures 1–3. It can be seen from Figure 1 that the system state can converge to the equilibrium point in less than 5 seconds. Figure 2 shows the response curve of the control input. One can easily see that no chattering phenomenon happens in the whole control process. The transition modes of system are given in Figure 3. It can be seen that the considered continuous-time Markov jump linear uncertain system is stochastic quadratic stable in spite of mismatched uncertainty and the transition rate matrix which covers completely known, boundary unknown, and completely unknown elements.

Figure 1 

The response curves of the system states.

Figure 2 

The response curve of the control input.

Figure 3 

The response curve of the transition modes.

5. Conclusions

In this paper, the problem of designing sliding sector-based variable structure controllers is considered for Markov jump linear uncertain continuous-time time-invariant systems with unknown transition rates. First, the related notions about sliding sector for continuous-time MJLSs are presented and sufficient conditions for the design of the sliding sector are given by LMIs. Second, the variable structure control law is presented to ensure the mean-square quadratic stability of the closed-loop system, and simulation results finally have verified the effectiveness of the proposed method. In our future research, how to design dynamic output feedback variable structure controller based on sliding sector technique and considering the applying of the sliding sector technique in industrial robotic systems are interesting work. Finally, the authors would like to thank the associate editor and all the reviewers for their constructive comments and suggestions for improving this paper.

Acknowledgments

This work was supported in part by the Funds of National Science of China (nos. 11301276,61104123), the Natural Science Funds of Jiangsu Province (BK20130984,BK20131000), the Scientific Research Foundation of Nanjing University of Information Science & Technology (S8112075001,S8112081001), and Laboratory Open Fund of Nanjing University of Information Science & Technology (N1081005112).