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

Authors: Yan-Mei Xue; Jianwei Yang; Xiao-Mei Liu
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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,PiRi-sliding sector, simplified PiRi-sliding sector, and inner sector and outer sector are given. Second, sufficient conditions for the design of the simplified PiRi-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 PiRi-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 Section2. 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.ℝn denotes the n-dimensional Euclidean space; AT 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+XT. 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:(1)x˙=(A(rt)+ΔA(rt))x+(B(rt)+ΔB(rt))u,
where x(t)∈ℝn and u(t)∈ℝ are state and control input, respectively. ΔArt and ΔB(rt) are the system uncertainties. {rt,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:
(2)Prob{rt+m=j∣rt=i}={λijm+o(m),ifj≠i,1+λiim+o(m),ifj=i,
where m>0, limm→0(o(m)/m)=0 and λij≥0(i,j∈𝒮,j≠i) denote the switching rate from mode i at time t to mode j at time t+m, and λii=-∑j=1,j≠iℕ‍λij for each i∈𝒮.In general, the Markov process transition rate matrixΛ is defined by
(3)Λ=[λij]ℕ×ℕ.In this paper, the transition rates of the jumping process are assumed to be partly available; that is, some elements inΛ 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
(4)Λ=[?λ12λ13?λ21?λ23??a?λ34b?λ43λ44],
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; λij denotes the precisely known element.For clarity, denote𝒮=𝒮k1i∪𝒮k2i∪𝒮uki, i=1,2,…,ℕ by
(5)𝒮k1i={j:λijis  exactly  known},𝒮k2i={j:the  bounds  ofλijis  known},𝒮uki={j:there  is  no  information  available  forλij}.
Furthermore, let 𝒮ki={j:λ_ij≤λij≤λ¯ij}; then 𝒮ki=𝒮k1i∪𝒮k2i. Let 𝒮ki={𝒦1i,𝒦2i,…,𝒦mkii}.   𝒮uki={𝕂1i,𝕂2i,…,𝕂mukii}, where 𝒦l1i denotes the l1th element in 𝒮ki with the index 𝒦l1i in the ith row of the matrix Λ. 𝕂l2i denotes the l2th element in 𝒮uki with the index 𝕂l2i in the ith row of the matrix Λ. mki and muki represent the number of elements in 𝒮ki and 𝒮uki, respectively.Remark 1.
Whenλ_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, letAi=A(rt), ΔAi=ΔA(rt), Bi=B(rt), and ΔBi=ΔB(rt) for each possible value rt=i, i∈𝒮, where Ai and Bi are known constant matrices with appropriate dimensions; then system (1) can be described by
(6)x˙=(Ai+ΔAi)x+(Bi+ΔBi)u.Throughout this paper, for eachi, suppose that the following assumptions are valid.Assumption 2.
The pair(Ai,Bi) is controllable.Assumption 3.
The uncertainties satisfyΔAi=DiEi(t,x)Fi, ΔBi=BiΘi(t), where Di and Fi are known constant matrices with appropriate dimensions, Ei(t,x) is time-varying uncertain matrix satisfying EiEiT≤I, the parameter Θi(t) satisfies |Θi(t)|≤d¯i<1, d¯i is a known positive scalar.For continuous-time Markov jump linear nominal systems(7)x˙=Aix+Biu,
some related notions of sliding sector are firstly presented.Definition 4.
ThePiRi-sliding sector is a subset of ℝn defined as
(8)𝒮i={x∣xT(AiTPi+PiAi+∑j=1ℕ‍λijPj)x=≤-xTRix,x∈ℝn(AiTPi+PiAi+∑j=1ℕ‍λijPj)},
where Pi∈ℝn×n is a positive definite symmetric matrix, Ri∈ℝn×n is a positive semidefinite symmetric matrix, Ri=CiTCi, Ci∈ℝli×n, li≥1, and (Ci,Ai) is an observable pair.The set of thePiRi-sliding sector is nonempty since at least zero state stays inside it.Obviously, while the state of the plant evolves in thePiRi-sliding sector 𝒮i, the Pi-norm |x|Pi=xTPix of the plant (7) decreases with zero control input since
(9)L˙i=xT(AiTPi+PiAi+∑j=1ℕ‍λijPj)x≤-xTRix≤0,∀x∈𝒮i,x≠0,
where Li=|x|Pi2.Remark 5.
WhenAi=A, Pi=P, Ri=R, the PiRi-sliding sector in (8) becomes
(10)𝒮={x∣xT(ATP+PA)x≤-xTRx,x∈ℝn}.
The definition of PR-sliding sector for linear deterministic systems is introduced in [26].Definition 6.
A simplifiedPiRi-sliding sector is a subset of ℝn defined as
(11)𝒮i={x∣|si(x)|≤δi(x),x∈ℝn},
where the linear function si(x) and the square root δi(x) of the quadratic function δi2(x) are determined by
(12)si(x)=Six,Si∈ℝ1×n,δi(x)=xTΩix,Ωi∈ℝn×n,Ωi≥0(Ωi≠0).Remark 7.
Suppose that the following inequalities are satisfied:(13)AiTPi+PiAi+∑j=1ℕ‍λijPj-PiBiBiTPi+Qi<0,i=1,2,…,ℕ.
Then the parameters in the simplified PiRi-sliding sector Pi, Ri, Si, and Ωi can be selected as follows:
(14)Ωi=riQi,Ri=(1-ri)Qi,Si=BiTPi,
where 0<ri<1.Obviously, while the state of the plant evolves in the simplifiedPiRi-sliding sector 𝒮i, the Pi-norm |x|Pi of the plant (7) decreases with zero control input since
(15)L˙i=xT(AiTPi+PiAi+∑j=1ℕ‍λijPj)x=xT(PiBiBiTPi-Qi)x=si2(x)-δi2(x)-xTRix≤-xTRix,∀x∈𝒮i,x≠0.In addition, in order to avoid the chattering phenomenon on the boundary of thePiRi-sliding sector, an inner sector 𝒮iin and an outer sector 𝒮iout are introduced as subsets of the PiRi-sliding sector (11):
(16)𝒮iin={x∣|si(x)|≤αiδi(x),x∈ℝn},(17)𝒮iout={x∣αiδi(x)<|si(x)|≤δi(x),x∈ℝn},
where the positive scalar αi satisfies 0<αi<1. It is not difficult to see that the sets 𝒮iin and 𝒮iout constitute a partition of the simplified PR-sliding sector 𝒮i. That is, 𝒮i=𝒮iin∪𝒮iout and 𝒮iin∩𝒮iout=∅.A hysteresis dead-zone functionσi(si(x),δi(x)) is further introduced for avoiding the chattering. It is defined as
(18)σi(si(x),δi(x))={0,x∈𝒮iinunchanged,x∈𝒮iout1,x∈¯𝒮i.Lemma 8 (see [20]).
For any givenλj≥0 and matrices Pj>0(Pj∈Rn×n,1≤j≤ℕ), if there exists Zi≥Pj, then the following inequality holds:
(19)∑j=1ℕ‍λjPj≤∑j=1ℕ‍λjZi.Lemma 9 (see [27]).
Given a symmetric matrixΠ and matrices M, N with appropriate dimensions, then Π+MF(t)N+NTFT(t)MT<0 for all F(t) satisfying FT(t)F(t)≤I, if and only if there exists a scalar ɛ>0 such that the following inequality holds:
(20)Π+ɛMMT+ɛ-1NTN<0.Lemma 10 (see [28]).
For symmetric and positive definite matricesX and Y, if X≥Y>0, then Y-1≥X-1>0.


## 3. Main Results

First, we present the design of the simplifiedPiRi-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 SimplifiedPiRi-Sliding Sector

Theorem 11.
For continuous-time Markov jump linear uncertain system (6), there exist symmetric and positive definite matrices Xi, Zi, and Yi satisfying the following LMIs:
(21)i∈𝒮ki,{[Γ11iΓ12iΓ13iXiXiFi*-Γ22i000**-Zi00***-Yi0****-ɛiI]<0,Zi≤Xj,j∈𝒮uki,j≠i(22)i∈𝒮uki,{[Φ11iΦ12iXiXiFi*-Xj00**-Yi0***-ɛiI]<0,Xi≤Xj,j∈𝒮uki,j≠i,
where
(23)Γ11i=He(AiXi)+λ¯iiXi-BiBiT+ɛiDiDiT,Γ12i=[λ¯i𝒦1iXi,λ¯i𝒦2iXi,…,λ¯i𝒦mkiiXi],Γ22i=diag{-X𝒦1i,-X𝒦2i,…,-X𝒦mkii},Γ13i=-(∑j∈𝒮ki,j≠i‍λ_ij+λ_ii)Xi,Φ11i=He(AiXi)-∑j∈𝒮ki,j≠i‍λ_ijXi-BiBiT+ɛiDiDiT,Φ12i=[λ¯i𝒦1iXi,λ¯i𝒦2iXi,…,λ¯i𝒦mkiiXi],Φ22i=diag{-X𝒦1i,-X𝒦2i,…,-X𝒦mkii};
then the parameters of PiRi-sliding sector in (14) can be designed as Pi=Xi-1, Qi=Yi-1, Ωi=riYi-1, Ri=(1-ri)Yi-1, and Si=BiTXi-1.Proof.
For system (6), for all ΔAi, i=1,2,…,ℕ, the conditions in (13) are equal to
(24)(Ai+ΔAi)TPi+Pi(Ai+ΔAi)+∑j∈𝒮ki,j≠i‍λijPj+∑j∈𝒮uki,j≠i‍λijPj+λiiPi-PiBiBiTPi+Qi<0.
In the following, we consider two cases: i∈𝒮ki and i∈𝒮uki in the proof.Case I (i∈𝒮ki).
In this case, the conditions in (24) can be guaranteed when for i=1,2,…,ℕ, the following inequalities:
(25)He((Ai+ΔAi)TPi)+∑j∈𝒮ki,j≠i‍λijPj+∑j∈𝒮uki,j≠i‍λijWi+λiiPi-PiBiBiTPi+Qi<0,hhPj≤Wi,j∈𝒮uki,
are satisfied. Substituting ∑j∈𝒮uki,j≠i‍λij=-∑j∈𝒮ki,j≠i‍λij-λii into the first inequality above and utilizing the boundary information of λij, one can see that the above inequalities can be guaranteed by
(26)He((Ai+ΔAi)TPi)+∑j∈𝒮ki,j≠i‍λ¯ijPj-(∑j∈𝒮ki,j≠i‍λ_ij+λ_ii)Wi+λ¯iiPi-PiBiBiTPi+Qi<0,hhPj≤Wi,j∈𝒮uki.
Pre- and postmultiplying Pi-1 on both sides of the first inequality in above inequalities and letting Xi=Pi-1, one has
(27)He((Ai+ΔAi)Xi)+∑j∈𝒮ki,j≠i‍λ¯ijXiXj-1Xi-(∑j∈𝒮ki,j≠i‍λ_ij+λ_ii)XiWiXi+λ¯iiXi-BiBiT+XiQiXi<0.
Let Yi=Qi-1, Zi=Wi-1, and applying Schur-complement technique and Lemma 9, one can obtain the first LMI condition in (21).
In addition, by using Lemma10, one can see that the second inequality in (26) can be converted into the second LMI condition in (21).Case II (i∈𝒮uki).
In this case, substitutingλii=-∑j∈𝒮ki,j≠i‍λij-∑j∈𝒮uki,j≠i‍λij into (24), one has
(28)He((Ai+ΔAi)TPi)+∑j∈𝒮ki,j≠i‍λijPj+∑j∈𝒮uki,j≠i‍λijPj-(∑j∈𝒮ki,j≠i‍λij+∑j∈𝒮uki,j≠i‍λij)Pi-PiBiBiTPi+Qi<0,hi=1,2,…,ℕ.
The conditions in (28) can be guaranteed provided that for i=1,2,…,ℕ, one has
(29)He((Ai+ΔAi)TPi)+∑j∈𝒮ki,j≠i‍λijPj-∑j∈𝒮ki,j≠i‍λijPi-PiBiBiTPi+Qi<0,hhhhPj≤Pi,j∈𝒮uki,j≠i.
Taking use of the boundary information of λij, one can see that the above inequalities can be guaranteed by
(30)He((Ai+ΔAi)TPi)+∑j∈𝒮ki,j≠i‍λ¯ijPj-∑j∈𝒮ki,j≠i‍λ_ijPi-PiBiBiTPi+Qi<0,hhhhPj≤Pi,j∈𝒮uki,j≠i.
Pre- and postmultiplying Pi-1 on both sides of the first inequality in above inequalities and letting Xi=Pi-1 and Yi=Qi-1, one has
(31)He((Ai+ΔAi)Xi)+∑j∈𝒮ki,j≠i‍λ¯ijXiXj-1Xi-∑j∈𝒮ki,j≠i‍λ_ijXi-BiBiT+XiQ¯i-1Xi<0.
Applying Schur-complement technique and Lemma 9 again, the above inequality (31) can be converted into the first LMI condition in (22).
Applying Lemma10, 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 thesimplifiedPiRi-sliding sector𝒮i (11) with the inner sector 𝒮iin (16) and the outer sector 𝒮iout (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 PiRi-sliding sector and obtain mean-square quadratic stability of the closed-loop system. Consider
(32)u=ui1+ui2,
where
(33)ui1=-σi(si(x),δi(x))(SiBi)-1(SiAix+Kisi(x)),(34)ui2=-11-d¯iσi(si(x),δi(x))=×(SiBi)-1sign(si(x))ρi,ρi=|Si||Di||Fi||x|+d¯i|SiAi||x|+d¯iKi|si(x)| and the parameter Ki>max{SiBi/2,Hi} satisfies the following inequality:
(35)2Hiαi2riQi+SiTSiAi+AiTSiTSi>0.Proof.
It is assumed that the initial state of the plant lies outside thePiRi-sliding sector. In this case, we have σi(si(x),δi(x))=1 and ui1=-(SiBi)-1(SiAix+Kisi(x)). It can be easily checked that
(36)si(x)s˙i(x)=-Kisi2(x)+si(x)SiΔAix=+si(x)SiBiui2+si(x)SiΔBiu.
Noting that ΔBi=BiΘi and |Θ|≤d¯i<1, we have
(37)si(x)s˙i(x)≤-Kisi2(x)+si(x)SiBiui2≤+|si(x)||Si||ΔAi||x|+d¯i|si(x)||SiBi||u|.
In view of u=ui1+ui2, it is easy to check that
(38)si(x)s˙i(x)≤-Kisi2(x)+si(x)SiBiui2≤+|si(x)||Si||ΔAi||x|+d¯i|si(x)|≤×|SiBi||ui1|+d¯i|si(x)||SiBi||ui2|.
Substituting ui1=-(SiBi)-1(SiAix+Kisi(x)) into the above inequality and noting that SiBi=BiTPiBi>0, one can see that
(39)si(x)s˙i(x)≤-(1-d¯i)Kisi2(x)+(1-d¯i)si(x)SiBiui2+|si(x)||Si||ΔAi||x|+d¯i|si(x)||SiAi|×|x|+d¯isi(x)SiBiui2+d¯i|si(x)||SiBi||ui2|.
Taking use of ui2 in (34) where ρi=|Si||Di||Fi||x|+d¯i|SiAi||x|+d¯iKi|si(x)|>d¯i|SiAi||x|+d¯iKi|si(x)|, one can see that
(40)si(x)s˙i(x)≤-Kisi2(x)+d¯isi(x)SiBiui2≤+d¯i|si(x)||SiBi||ui2|.
According to the design of ui2 again, one can easily observe that
(41)si(x)SiBiui2+|si(x)||SiBi||ui2|≤-11-d¯i|si(x)|ρi+11-d¯i|si(x)|×|SiBi||(SiBi)-1||sign(si(x))|ρi=0.
It follows from (40) and (41) that
(42)si(x)s˙i(x)≤-Kisi2(x).
Thus the state trajectory of the closed-loop system will be driven into the inner sector of the PiRi-sliding sector.
In the following, we will prove that thePi-norm keeps decreasing. Consider
(43)L˙i(t)=xT{(Ai+ΔAi)TP+P(Ai+ΔAi)+∑j=1ℕ‍λijPj}x+2xTPi(Bi+ΔBi)u≤si2(x)-δi2(x)-xTRix+2xTPi(Bi+ΔBi)u.
Noting that ΔBi=BiΘi and Si=BiTPi, one can see that
(44)L˙i(t)≤si2(x)-δi2(x)-xTRix+2si(x)ui1≤+2si(x)ui2+2si(x)Θi(ui1+ui2).
In the following, we first prove that(45)si(x)ui2+d¯i|si(x)||ui1+ui2|≤0.
Substituting (33) and (34) into the left side of above inequality, one has
(46)si(x)ui2+d¯i|si(x)||ui1+ui2|=-11-d¯i(SiBi)-1|si(x)|ρi+d¯i|si(x)|×|11-d¯i(SiBi)-1SiAix+Ki(SiBi)-1si(x)+11-d¯i(SiBi)-1sign(si(x))ρi|.
Noting that SiBi=BiTPiBi>0 and d¯i<1, we have
(47)si(x)ui2+d¯i|si(x)||ui1+ui2|≤-(SiBi)-1|si(x)|ρi+d¯i(SiBi)-1|si(x)|×|SiAix+Kisi(x)|.
Since ρi=|Si||Di||Fi||x|+d¯i|SiAi||x|+d¯iKi|si(x)|, it is easy to check that
(48)ρi≥d¯i|SiAix+Kisi(x)|.
It follows from (47) and (48) that (45) is established. Thus we have
(49)L˙i(t)≤si2(x)-δi2(x)-xTRix+2si(x)ui1.
Substituting (33) into the above inequality (49) and noting that Ki>(SiBi/2), 2Kiαi2riQi+SiTSiAi+AiTSiTSi>0 and si2(x)≥αiδi2(x), one can see that
(50)L˙i(t)≤si2(x)-δi2(x)-xTRix+2si(x)×(-(SiBi)-1SiAix-Ki(SiBi)-1si(x))=-(SiBi)-1xT(2Kiαi2riQi+SiTSiAi+AiTSiTSi)x-xTRix<-xTRix.
After being forced into the inner sector of the PiRi-sliding sector and before moving out of the PiRi-sliding sector, the control input u(t) is set equal to zero with σi(si(x),δi(x))=0. In this case, |si(x)|≤δi(x) and
(51)L˙i(t)=xT{(Ai+ΔAi)TP+P(Ai+ΔAi)+∑j=1ℕ‍λijPj}x≤si2(x)-δi2(x)-xTRix≤-xTRix.∀x∈𝒮i.
Once the state trajectory of the plant runs out of thePiRi-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 Pi-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 thePiRi-sliding sector into the inside of the inner sector and the Lyapunov function Li(x)=xTPix keeps decreasing in the whole state space; then mean-square quadratic stability of the closed-loop system is established.


## 3.1. The Design of the SimplifiedPiRi-Sliding Sector

Theorem 11.
For continuous-time Markov jump linear uncertain system (6), there exist symmetric and positive definite matrices Xi, Zi, and Yi satisfying the following LMIs:
(21)i∈𝒮ki,{[Γ11iΓ12iΓ13iXiXiFi*-Γ22i000**-Zi00***-Yi0****-ɛiI]<0,Zi≤Xj,j∈𝒮uki,j≠i(22)i∈𝒮uki,{[Φ11iΦ12iXiXiFi*-Xj00**-Yi0***-ɛiI]<0,Xi≤Xj,j∈𝒮uki,j≠i,
where
(23)Γ11i=He(AiXi)+λ¯iiXi-BiBiT+ɛiDiDiT,Γ12i=[λ¯i𝒦1iXi,λ¯i𝒦2iXi,…,λ¯i𝒦mkiiXi],Γ22i=diag{-X𝒦1i,-X𝒦2i,…,-X𝒦mkii},Γ13i=-(∑j∈𝒮ki,j≠i‍λ_ij+λ_ii)Xi,Φ11i=He(AiXi)-∑j∈𝒮ki,j≠i‍λ_ijXi-BiBiT+ɛiDiDiT,Φ12i=[λ¯i𝒦1iXi,λ¯i𝒦2iXi,…,λ¯i𝒦mkiiXi],Φ22i=diag{-X𝒦1i,-X𝒦2i,…,-X𝒦mkii};
then the parameters of PiRi-sliding sector in (14) can be designed as Pi=Xi-1, Qi=Yi-1, Ωi=riYi-1, Ri=(1-ri)Yi-1, and Si=BiTXi-1.Proof.
For system (6), for all ΔAi, i=1,2,…,ℕ, the conditions in (13) are equal to
(24)(Ai+ΔAi)TPi+Pi(Ai+ΔAi)+∑j∈𝒮ki,j≠i‍λijPj+∑j∈𝒮uki,j≠i‍λijPj+λiiPi-PiBiBiTPi+Qi<0.
In the following, we consider two cases: i∈𝒮ki and i∈𝒮uki in the proof.Case I (i∈𝒮ki).
In this case, the conditions in (24) can be guaranteed when for i=1,2,…,ℕ, the following inequalities:
(25)He((Ai+ΔAi)TPi)+∑j∈𝒮ki,j≠i‍λijPj+∑j∈𝒮uki,j≠i‍λijWi+λiiPi-PiBiBiTPi+Qi<0,hhPj≤Wi,j∈𝒮uki,
are satisfied. Substituting ∑j∈𝒮uki,j≠i‍λij=-∑j∈𝒮ki,j≠i‍λij-λii into the first inequality above and utilizing the boundary information of λij, one can see that the above inequalities can be guaranteed by
(26)He((Ai+ΔAi)TPi)+∑j∈𝒮ki,j≠i‍λ¯ijPj-(∑j∈𝒮ki,j≠i‍λ_ij+λ_ii)Wi+λ¯iiPi-PiBiBiTPi+Qi<0,hhPj≤Wi,j∈𝒮uki.
Pre- and postmultiplying Pi-1 on both sides of the first inequality in above inequalities and letting Xi=Pi-1, one has
(27)He((Ai+ΔAi)Xi)+∑j∈𝒮ki,j≠i‍λ¯ijXiXj-1Xi-(∑j∈𝒮ki,j≠i‍λ_ij+λ_ii)XiWiXi+λ¯iiXi-BiBiT+XiQiXi<0.
Let Yi=Qi-1, Zi=Wi-1, and applying Schur-complement technique and Lemma 9, one can obtain the first LMI condition in (21).
In addition, by using Lemma10, one can see that the second inequality in (26) can be converted into the second LMI condition in (21).Case II (i∈𝒮uki).
In this case, substitutingλii=-∑j∈𝒮ki,j≠i‍λij-∑j∈𝒮uki,j≠i‍λij into (24), one has
(28)He((Ai+ΔAi)TPi)+∑j∈𝒮ki,j≠i‍λijPj+∑j∈𝒮uki,j≠i‍λijPj-(∑j∈𝒮ki,j≠i‍λij+∑j∈𝒮uki,j≠i‍λij)Pi-PiBiBiTPi+Qi<0,hi=1,2,…,ℕ.
The conditions in (28) can be guaranteed provided that for i=1,2,…,ℕ, one has
(29)He((Ai+ΔAi)TPi)+∑j∈𝒮ki,j≠i‍λijPj-∑j∈𝒮ki,j≠i‍λijPi-PiBiBiTPi+Qi<0,hhhhPj≤Pi,j∈𝒮uki,j≠i.
Taking use of the boundary information of λij, one can see that the above inequalities can be guaranteed by
(30)He((Ai+ΔAi)TPi)+∑j∈𝒮ki,j≠i‍λ¯ijPj-∑j∈𝒮ki,j≠i‍λ_ijPi-PiBiBiTPi+Qi<0,hhhhPj≤Pi,j∈𝒮uki,j≠i.
Pre- and postmultiplying Pi-1 on both sides of the first inequality in above inequalities and letting Xi=Pi-1 and Yi=Qi-1, one has
(31)He((Ai+ΔAi)Xi)+∑j∈𝒮ki,j≠i‍λ¯ijXiXj-1Xi-∑j∈𝒮ki,j≠i‍λ_ijXi-BiBiT+XiQ¯i-1Xi<0.
Applying Schur-complement technique and Lemma 9 again, the above inequality (31) can be converted into the first LMI condition in (22).
Applying Lemma10, 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 thesimplifiedPiRi-sliding sector𝒮i (11) with the inner sector 𝒮iin (16) and the outer sector 𝒮iout (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 PiRi-sliding sector and obtain mean-square quadratic stability of the closed-loop system. Consider
(32)u=ui1+ui2,
where
(33)ui1=-σi(si(x),δi(x))(SiBi)-1(SiAix+Kisi(x)),(34)ui2=-11-d¯iσi(si(x),δi(x))=×(SiBi)-1sign(si(x))ρi,ρi=|Si||Di||Fi||x|+d¯i|SiAi||x|+d¯iKi|si(x)| and the parameter Ki>max{SiBi/2,Hi} satisfies the following inequality:
(35)2Hiαi2riQi+SiTSiAi+AiTSiTSi>0.Proof.
It is assumed that the initial state of the plant lies outside thePiRi-sliding sector. In this case, we have σi(si(x),δi(x))=1 and ui1=-(SiBi)-1(SiAix+Kisi(x)). It can be easily checked that
(36)si(x)s˙i(x)=-Kisi2(x)+si(x)SiΔAix=+si(x)SiBiui2+si(x)SiΔBiu.
Noting that ΔBi=BiΘi and |Θ|≤d¯i<1, we have
(37)si(x)s˙i(x)≤-Kisi2(x)+si(x)SiBiui2≤+|si(x)||Si||ΔAi||x|+d¯i|si(x)||SiBi||u|.
In view of u=ui1+ui2, it is easy to check that
(38)si(x)s˙i(x)≤-Kisi2(x)+si(x)SiBiui2≤+|si(x)||Si||ΔAi||x|+d¯i|si(x)|≤×|SiBi||ui1|+d¯i|si(x)||SiBi||ui2|.
Substituting ui1=-(SiBi)-1(SiAix+Kisi(x)) into the above inequality and noting that SiBi=BiTPiBi>0, one can see that
(39)si(x)s˙i(x)≤-(1-d¯i)Kisi2(x)+(1-d¯i)si(x)SiBiui2+|si(x)||Si||ΔAi||x|+d¯i|si(x)||SiAi|×|x|+d¯isi(x)SiBiui2+d¯i|si(x)||SiBi||ui2|.
Taking use of ui2 in (34) where ρi=|Si||Di||Fi||x|+d¯i|SiAi||x|+d¯iKi|si(x)|>d¯i|SiAi||x|+d¯iKi|si(x)|, one can see that
(40)si(x)s˙i(x)≤-Kisi2(x)+d¯isi(x)SiBiui2≤+d¯i|si(x)||SiBi||ui2|.
According to the design of ui2 again, one can easily observe that
(41)si(x)SiBiui2+|si(x)||SiBi||ui2|≤-11-d¯i|si(x)|ρi+11-d¯i|si(x)|×|SiBi||(SiBi)-1||sign(si(x))|ρi=0.
It follows from (40) and (41) that
(42)si(x)s˙i(x)≤-Kisi2(x).
Thus the state trajectory of the closed-loop system will be driven into the inner sector of the PiRi-sliding sector.
In the following, we will prove that thePi-norm keeps decreasing. Consider
(43)L˙i(t)=xT{(Ai+ΔAi)TP+P(Ai+ΔAi)+∑j=1ℕ‍λijPj}x+2xTPi(Bi+ΔBi)u≤si2(x)-δi2(x)-xTRix+2xTPi(Bi+ΔBi)u.
Noting that ΔBi=BiΘi and Si=BiTPi, one can see that
(44)L˙i(t)≤si2(x)-δi2(x)-xTRix+2si(x)ui1≤+2si(x)ui2+2si(x)Θi(ui1+ui2).
In the following, we first prove that(45)si(x)ui2+d¯i|si(x)||ui1+ui2|≤0.
Substituting (33) and (34) into the left side of above inequality, one has
(46)si(x)ui2+d¯i|si(x)||ui1+ui2|=-11-d¯i(SiBi)-1|si(x)|ρi+d¯i|si(x)|×|11-d¯i(SiBi)-1SiAix+Ki(SiBi)-1si(x)+11-d¯i(SiBi)-1sign(si(x))ρi|.
Noting that SiBi=BiTPiBi>0 and d¯i<1, we have
(47)si(x)ui2+d¯i|si(x)||ui1+ui2|≤-(SiBi)-1|si(x)|ρi+d¯i(SiBi)-1|si(x)|×|SiAix+Kisi(x)|.
Since ρi=|Si||Di||Fi||x|+d¯i|SiAi||x|+d¯iKi|si(x)|, it is easy to check that
(48)ρi≥d¯i|SiAix+Kisi(x)|.
It follows from (47) and (48) that (45) is established. Thus we have
(49)L˙i(t)≤si2(x)-δi2(x)-xTRix+2si(x)ui1.
Substituting (33) into the above inequality (49) and noting that Ki>(SiBi/2), 2Kiαi2riQi+SiTSiAi+AiTSiTSi>0 and si2(x)≥αiδi2(x), one can see that
(50)L˙i(t)≤si2(x)-δi2(x)-xTRix+2si(x)×(-(SiBi)-1SiAix-Ki(SiBi)-1si(x))=-(SiBi)-1xT(2Kiαi2riQi+SiTSiAi+AiTSiTSi)x-xTRix<-xTRix.
After being forced into the inner sector of the PiRi-sliding sector and before moving out of the PiRi-sliding sector, the control input u(t) is set equal to zero with σi(si(x),δi(x))=0. In this case, |si(x)|≤δi(x) and
(51)L˙i(t)=xT{(Ai+ΔAi)TP+P(Ai+ΔAi)+∑j=1ℕ‍λijPj}x≤si2(x)-δi2(x)-xTRix≤-xTRix.∀x∈𝒮i.
Once the state trajectory of the plant runs out of thePiRi-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 Pi-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 thePiRi-sliding sector into the inside of the inner sector and the Lyapunov function Li(x)=xTPix 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:(52)A1=[1023],A2=[011-2],A3=[200-1],A4=[31-12],B1=[11],B2=[11],B3=[21],B4=[23],D1=[0.10],D2=[0.20.5]D3=[0.20.6],D4=[0.10.5],F1=[0-0.1],F2=[10.3],F3=[0.20.5],F4=[0.10.1].For simulation, we takeE1(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(53)Λ=[-1.50.2????0.50.91?-2?0.1α?β],
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:
(54)P1=[6.2534-6.4840-6.484018.4380],P2=[6.8785-7.1343-7.134320.0458],P3=[8.5811-9.6120-9.612015.6407],P4=[6.1859-1.5374-1.53741.1026],Q1=[6.9519-7.5088-7.508822.4304],Q2=[13.4209-14.1627-14.162740.0263]Q3=[11.4377-11.6823-11.682318.2712],Q4=[10.3808-2.6205-2.62052.0926].
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 d¯1=0.2, d¯2=0.3, d¯3=0.2, and d¯4=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.



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