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

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-timeMarkov 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.


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][4][5][6][7][8][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][13][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 wellknown robust control technique against external disturbances, parameter uncertainties, and unmodeled dynamics [15][16][17], and thus there are also some remarkable results published on variable structure control of continuous-time MJLS recently [18][19][20][21][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][24][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 controlbased 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 controlbased sliding sector technique is investigated.The main contribution of this paper is summarized as follows.First, related notions of the sliding sector for MJLS,     -sliding sector, simplified     -sliding sector, and inner sector and outer sector are given.Second, sufficient conditions for the design of the simplified     -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     -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  denotes the -dimensional Euclidean space;   stands for the transpose of matrix ;  and 0 represent the identity matrix and zero matrix in appropriate dimension, respectively.In addition, the symbol He() is used to represent  +   . > 0 ( ≥ 0 ) means that  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.

Problem Statement and Preliminaries
Consider the following time-invariant continuous-time linear uncertain system: where () ∈ R  and () ∈ R are state and control input, respectively.Δ  and Δ(  ) are the system uncertainties.{  ,  ≥ 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: where  > 0, lim  → 0 (()/) = 0 and   ≥ 0 (,  ∈ S,  ̸ = ) denote the switching rate from mode  at time  to mode  at time  + , and   = − ∑ N =1, ̸ =    for each  ∈ S. In general, the Markov process transition rate matrix Λ is defined by 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)  Remark 1.When   =   , 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   = (  ), Δ  = Δ(  ),   = (  ), and Δ  = Δ(  ) for each possible value   = ,  ∈ S, Mathematical Problems in Engineering 3 where   and   are known constant matrices with appropriate dimensions; then system (1) can be described by Throughout this paper, for each , suppose that the following assumptions are valid.For continuous-time Markov jump linear nominal systems some related notions of sliding sector are firstly presented.
Definition 4. The     -sliding sector is a subset of R  defined as where   ∈ R × is a positive definite symmetric matrix,   ∈ R × is a positive semidefinite symmetric matrix,   =      ,   ∈ R   × ,   ≥ 1, and (  ,   ) is an observable pair.
The set of the     -sliding sector is nonempty since at least zero state stays inside it.
Obviously, while the state of the plant evolves in the    sliding sector S  , the   -norm ||   =      of the plant (7) decreases with zero control input since where Remark 5. When   = ,   = ,   = , the     -sliding sector in (8) becomes The definition of -sliding sector for linear deterministic systems is introduced in [26].Definition 6.A simplified     -sliding sector is a subset of R  defined as where the linear function   () and the square root   () of the quadratic function  2  () are determined by Remark 7. Suppose that the following inequalities are satisfied: Then the parameters in the simplified     -sliding sector   ,   ,   , and Ω  can be selected as follows: where 0 <   < 1.
Obviously, while the state of the plant evolves in the simplified     -sliding sector S  , the   -norm ||   of the plant (7) decreases with zero control input since In addition, in order to avoid the chattering phenomenon on the boundary of the     -sliding sector, an inner sector S in  and an outer sector S out  are introduced as subsets of the    sliding sector (11): where the positive scalar   satisfies 0 <   < 1.It is not difficult to see that the sets S in  and S out  constitute a partition of the simplified -sliding sector S  .That is, S  = S in  ∪ S out  and S in  ∩ S out  = 0.A hysteresis dead-zone function   (  (),   ()) is further introduced for avoiding the chattering.It is defined as Lemma 8 (see [20]).For any given   ≥ 0 and matrices   > 0 (  ∈  × , 1 ≤  ≤ N), if there exists   ≥   , then the following inequality holds: Lemma 9 (see [27]).Given a symmetric matrix Π and matrices ,  with appropriate dimensions, then Π + () +     ()  < 0 for all () satisfying   ()() ≤ , if and only if there exists a scalar  > 0 such that the following inequality holds: Lemma 10 (see [28]).For symmetric and positive definite matrices  and , if  ≥  > 0, then  −1 ≥  −1 > 0.

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

The Design of the Simplified 𝑃 𝑖 𝑅 𝑖 -Sliding Sector
Theorem 11.For continuous-time Markov jump linear uncertain system (6), there exist symmetric and positive definite matrices   ,   , and   satisfying the following LMIs: where then the parameters of     -sliding sector in (14) can be designed as Proof.For system (6), for all Δ  ,  = 1, 2, . . ., N, the conditions in ( 13) are equal to In the following, we consider two cases:  ∈ S   and  ∈ S   in the proof.
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 ( ∈ S   ).In this case, substituting 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.

Variable Structure Control Design
Theorem 12. Considering the simplified     -sliding sector S  (11) with the inner sector S in  (16) and the outer sector S 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     -sliding sector and obtain meansquare quadratic stability of the closed-loop system.Consider where It follows from ( 40) and (41) that Thus the state trajectory of the closed-loop system will be driven into the inner sector of the     -sliding sector.
In the following, we will prove that the   -norm keeps decreasing.Consider

L𝑖 (𝑡) = 𝑥
Once the state trajectory of the plant runs out of the    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   -norm ||   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     -sliding sector into the inside of the inner sector and the Lyapunov function   () =      keeps decreasing in the whole state space; then mean-square quadratic stability of the closed-loop system is established.
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.

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.

Figure 1 :Figure 2 :
Figure 1: The response curves of the system states.

Figure 3 :
Figure 3: The response curve of the transition modes.
1 in the th row of the matrix Λ.K   2 denotes the  2 th element in S   with the index K   2 in the th row of the matrix Λ.    and    represent the number of elements in S and S   , respectively.
into the first inequality above and utilizing the boundary information of   , one can see that the above inequalities can be guaranteed by He ((  + Δ  )   +   )   +     −          +   < 0,   ≤   ,  ∈ S   .