# Estimation and Synthesis of Reachable Set for Singular Markovian Jump Systems

Authors: Yucai Ding; Hui Liu
Journal: Complexity
 (2018)

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## Abstract

The problems of reachable set estimation and state-feedback controller design are investigated for singular Markovian jump systems with bounded input disturbances. Based on the Lyapunov approach, several new sufficient conditions on state reachable set and output reachable set are derived to ensure the existence of ellipsoids that bound the system states and output, respectively. Moreover, a state-feedback controller is also designed based on the estimated reachable set. The derived sufficient conditions are expressed in terms of linear matrix inequalities. The effectiveness of the proposed results is illustrated by numerical examples.


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## Body

## 1. Introduction

The research on singular systems has attracted significant attention in the past years due to the fact that singular systems can better describe a larger class of physical systems such as robotic systems, electric circuits, and mechanical systems. When singular systems experience abrupt changes in their structures, it is natural to model them as singular Markovian jump systems [1, 2]. The analysis and synthesis of such class of systems have gained considerable attention because of their importance in applications (see, e.g., the literature [3–10] and the references therein).Reachable set is one of the important techniques for parameter estimation or state estimation problems [11]. Reachable set for a dynamic system is the set containing all the system states starting from the origin under bounded input disturbances. However, the exact shape of reachable sets of a dynamic system is very complex and hard to obtain; for this reason a number of researchers began to turn their attention to the reachable set estimation problem. The common strategies for reachable set estimation are ellipsoidal method [12] and polyhedron method [13]. The main idea of these methods is to detect simple convex shapes like ellipsoid or polyhedron, which contains all the system states. Compared with polyhedron method, the primary advantage of ellipsoidal method is that the ellipsoid structure is simple and directly related to quadratic Lyapunov functions. As a result, linear matrix inequalities (LMIs) techniques can be used to determine bounding ellipsoids. In the framework of bounding ellipsoid, the reachable set estimation problem for linear time delay systems has received significant research attention in recent years. In [14] sufficient conditions for the existence of bounding ellipsoids containing the reachable set of continuous-time linear systems with time-varying delays were derived by using the Lyapunov-Razumikhin function. In [15], by using Lyapunov-Krasovskii functional method, the author derived some less conservative conditions than those in [14]. In [16] the reachable set of delayed systems with polytopic uncertainties was investigated by using the maximal Lyapunov-Krasovskii functional approach, and some new conditions bounding the set of reachable states are derived. Interesting results on reachable set of delayed systems with polytopic uncertainties can also be found in [17–20]. In addition, some other strategies without using Lyapunov-Krasovskii functional have been provided to estimate the reachable set of continuous-time linear time-varying systems [21] and nonlinear time delay systems [22, 23]. The authors in [24] extended the ideas of reachable set estimation of continuous-time systems to discrete-time systems, wherein a fundamental result (Lemma 2.1 [24]) for the reachable set estimation of discrete-time systems was proposed. The authors in [25] improved the fundamental result obtained in [24] and provided a basic tool (Lemma 4 [25]) for the reachable set estimation of discrete-time systems. On the basis of the general ideas proposed in [25], the reachable set estimation problem was also extended to some classes of complicated systems, such as singular systems [26], Markovian jump systems [27], switched linear systems [28], and T-S fuzzy systems [29, 30]. For the reachable set estimation of discrete-time systems, the other important contributions can be found in [31, 32]. On the other hand, the problem of controller design for specifications involved with the reachable set of a control system is also a very important issue [33]. The controller design problems concerning reachable set were studied in [34] and [35] by using ellipsoidal method and polyhedron method, respectively. Two issues were raised in [34]: the first one is to design a controller such that the reachable set of the closed-loop system is contained in an ellipsoid, and the admissible ellipsoid should be as small as possible; the second one is to design a controller such that the reachable set of the closed-loop system is contained in a given ellipsoid. By constructing suitable Lyapunov-Krasovskii functional, LMI-based sufficient conditions for the existence of controller guaranteeing the ellipsoid bounds as small as possible have been derived for continuous-time delay systems [34] and discrete-time periodic systems [36]. It is obvious that the LMI-based controller design is quite simple and numerically tractable. However, it should be pointed out that the reachable set estimation and synthesis problems of singular Markovian jump systems are much more difficult and challenging than that for nonsingular Markovian jump systems since the ellipsoid containing the reachable set is not directly related to quadratic Lyapunov functions. To the best of the authors’ knowledge, no related results have been established for reachable set estimation and synthesis of singular Markovian jump systems, which has motivated this paper.In this paper, we consider the problems of reachable set estimation and synthesis of singular Markovian jump systems. By using the Lyapunov approach, the estimation conditions on state reachable set and output reachable set are derived, respectively. Moreover, the desired state-feedback controller is designed based on the estimated reachable set.Notation. Throughout this paper, Rn denotes the n-dimensional Euclidean space; AT represents the transpose of A; Sym(M) stands for M+MT; X>0  (<0) means X is a symmetric positive (negative) definite matrix; E· refers to the expectation; Xl×m denotes the matrix composed of elements of first l rows and m columns of matrix X; · refers to the Euclidean vector norm; the symbol “∗” in LMIs denotes the symmetric term of the matrix; I is the unit matrix with appropriate dimensions.


## 2. Problem Formulation

Consider the following singular Markovian jump system:(1)Ex˙t=Artxt+Brtut+Drtωtyt=Crtxt,where x(t)∈Rn is the state vector, u(t)∈Rm is the control input, y(t)∈Rp is the measured output, and ω(t)∈Rq is the exogenous disturbance which satisfies(2)ωTtωt≤ω¯2.E, A(rt), B(rt), C(rt), and D(rt) are real constant matrices with appropriate dimensions and rank(E)=l<n. {rt,t⩾0} is a continuous-time Markovian process with transition rate matrix Π=[πij]i∈S=1,2,…,N and the evolution of Markovian process is governed by the following transition rate:(3)Pr⁡rt+Δ=j∣rt=i=πijΔ+oΔ,j≠i1+πiiΔ+oΔ,j=i,where Δ>0 and limΔ→0(oΔ/Δ)=0; πij⩾0 for i≠j is the transition rate from mode i to mode j and πii=-∑j=1,j≠iNπij.For notational simplicity, in the sequel, for each possiblert=i, i∈S, matrices A(rt), B(rt), C(rt), and D(rt) will be denoted by Ai, Bi, Ci, and Di, respectively. When u(t)=0, the system Ex˙(t)=Aix(t)+Diω(t) is referred to as a free system.In this paper we are interested in determining ellipsoids that contain, respectively, the state reachable set and output reachable set. In the reachable set analysis, it is required that systems should be asymptotically stable. When this requirement is not met, we will further design a state-feedback controller such that the reachable set of the closed-loop system is contained in the smallest ellipsoid.The state reachable set of the free system in (1) is defined by (4)Rx=xt∈Rn∣xt,t≥0,is  a  solution  of1forx0=0.An ellipsoid E(X) bounding the reachable set can be always represented as follows: (5)EX=x∈Rn∣xTXx≤1,X>0.Particularly, when X=κI for ∀κ>0, the ellipsoid E(X) will become a ball which is denoted by B(X).Sincerank(E)=l<n, there exist two nonsingular matrices M and N such that(6)MEN=I000,MAiN=A11iA12iA21iA22i,MDi=D1iD2i.Let N-1x(t)=x~(t)=[x~1T(t)x~2T(t)]T. Then the free system can be rewritten as the following differential-algebraic form:(7)x~˙1t=A11ix~1t+A12ix~2t+D1iωt(8)0=A21ix~1t+A22ix~2t+D2iωt.The following definition and lemma are also useful in deriving the main results.Definition 1 (see [2]).
(I) The free system is said to be regular ifdet⁡(sE-Ai) is not identically zero for each i∈S.
(II) The free system is said to be impulse-free ifdeg⁡det⁡sE-Ai=rank(E) for each i∈S.Lemma 2 (see [2]).
For any matricesU and V∈Rn×n with V>0, one has UV-1UT≥U+UT-V.Lemma 3 (see [12]).
LetV(x(t)) be a Lyapunov function and V(x(0))=0. If (9)V˙xt+αVxt-αω¯2ωTtωt≤0,α>0,then V(x(t))≤1, ∀t≥0.


## 3. Main Results

### 3.1. State Reachable Set Estimation

In this subsection, we will focus our attention on determining a ball which contains the state reachable set of the free system.Theorem 4.
If there exist nonsingular matricesXi∈Rn×n and a scalar α>0 such that the following LMIs hold for each i∈S,(10)ETXi=XiTE≥0,(11)H=SymAiTXi+∑j=1NπijETXj+αETXiXiTDi∗-αω¯2I<0,then the state reachable set of free system starting from the origin is mean-square bounded within the following set:(12)⋂i=1NEX^i=⋂i=1Nx∈Rn∣xTX^ix≤1,where (13)X^i=1ri2N-TN-1ri=1mini∈S⁡λminX~i+A22i-1A21imini∈S⁡λminX~i+D2iω¯X~i=M-TXiNl×l.Proof.
We first prove the regularity and nonimpulsiveness of the free system. LetM-TXiN=X11iX12iX21iX22i. Then, by (10), we obtain that X12i=0. From (11), it is easy to show that(14)SymAiTXi+∑j=1NπijETXj+αETXi<0.Pre- and postmultiplying (14) by NT and N, respectively, we get(15)⋆⋆∗A22iTX22i+X22iTA22i<0,where ⋆ will be irrelevant to the results of the following discussion; thus the real expressions of these two variables are omitted. It follows from (15) that(16)A22iTX22i+X22iTA22i<0which implies that A22i is nonsingular for each i∈S. Therefore, by Definition 1, we have that the free system is regular and nonimpulsive.
Next, we will show the state reachable set of free system is mean-square bounded within the setE(X^). Consider the following Lyapunov function:(17)Vxt,rt=xTtETXixt.Let L be the weak infinitesimal generator of the random process {x(t),rt}. Calculating the difference of V(x(t),rt) along the trajectories of the free system, we get(18)LVxt,rt=x˙TtETXixt+xTtXiTEx˙t+xTt∑j=1NπijETXjxt.Defining the augmented system variable as ξ(t)=[xT(t)ωT(t)]T and using conditions (10) and (11), we have(19)LVxt,rt+αVxt,rt-αω¯2ωTtωt=ξTtHξt<0.Then we can deduce from Lemma 3 that ExTtETXixt≤1, which infers that(20)Ex~TtNTETMTM-TXiNx~t≤1.Recalling that X12i=0, it follows from (20) that(21)Ex~1TtX~ix~1t≤1.From (21) we have mini∈SλminX~iEx~1t2≤Ex~1TtX~ix~1t≤1, which implies that Ex~1t≤1/mini∈S⁡λmin(X~i).
SinceA22i is nonsingular for each i∈S, (8) can be rewritten as(22)x~2t=-A22i-1A21ix~1t+D2iωt.Then we can deduce that(23)Ex~2t≤A22i-1A21imini∈S⁡λminX~i+D2iω¯.It follows from the fact x~(t)=[x~1T(t)0]T+[0x~2T(t)]T that(24)Ex~t≤Ex~1t+x~2t≤ri,where ri=1/mini∈Sλmin(X~i)+A22i-1A21i/mini∈Sλmin(X~i)+D2iω¯.By (24), it can be seen that(25)1ri2Ex~Ttx~t≤1which implies that the trajectories of (7)-(8) are mean-square bounded within the set ⋂i=1NB(1/ri2)I. Moreover, notice that x~(t)=N-1x(t), and (25) can be rewritten as (1/ri2)ExT(t)N-TN-1x(t)≤1. By denoting (1/ri2)N-TN-1=X^i, the state reachable set of free system is mean-square bounded within the set ⋂i=1NE(X^i).It should be noted that inequality (10) represents a nonstrict LMI. This may lead to numerical problems since equality constraints are usually not satisfied perfectly. Below, we will develop a numerically tractable and nonconservative LMI condition.Theorem 5.
If there exist symmetric positive definite matricesPi∈Rn×n, nonsingular matrices Qi, and a scalar α>0 such that the following LMI holds for each i∈S,(26)ΨiPiE+RTQiSTTDi∗-αω¯2I<0,then the state reachable set of free system is mean-square bounded within the following set:(27)⋂i=1NEP^i=⋂i=1Nx∈Rn∣xTP^ix≤1,where (28)Ψi=SymAiTPiE+RTQiST+∑j=1NπijETPjE+αETPiEP^i=1r~i2N-TN-1r~i=1mini∈S⁡λminP~i+A22i-1A21imini∈S⁡λminP~i+D2iω¯P~i=M-TPiM-1l×l,R∈R(n-l)×n is any matrix with full row rank and satisfies RE=0; S∈Rn×(n-l) is any matrix with full column rank and satisfies ES=0.Proof.
LetXi=PiE+RTQiST in (26); it is easy to obtain (10) and (11). In this case, inequality (20) will be replaced by Ex~TtNTETMTM-TPiM-1MENx~t≤1, which infers that Ex~1TtP~ix~1t≤1 with P~i=M-TPiM-1l×l. Then, following the same lines as (22)–(25) in Theorem 4, we can get ExTtP^ixt≤1 for any i∈S. Therefore, the state reachable set of free system is mean-square bounded within the set ⋂i=1NE(P^i).Remark 6.
In order to make the ellipsoidE(P^i) as small as possible, we require trace(P^i)→max. For this purpose, we can add the additional requirement P~i>ϵI and then maximize a positive scalar ϵ, which is equivalent to the following minimization problem:(29)minϵ¯s.t.26,-ϵ¯III-P~i<0,where ϵ¯=1/ϵ.


### 3.2. Output Reachable Set Estimation

Theorem 7.
If there exist symmetric positive definite matricesPi∈Rn×n and Yi∈Rn×n, nonsingular matrices Qi, and a scalar α>0 such that the following LMIs hold for each i∈S,(30)ΨiPiE+RTQiSTTDi∗-αω¯2I<0(31)CiTYiCi≤P^i,then the output reachable set of free system is mean-square bounded within the following set:(32)⋂i=1NEYi=⋂i=1Ny∈Rp∣yTYiy≤1,where Ψi, R, S, and P^i are defined in Theorem 5.Proof.
By Theorem5, LMI (30) ensures(33)ExTtP^ixt≤1,∀i∈S.With this and (31), we obtain that(34)ExTtCiTYiCixt≤ExTtP^ixt≤1.Due to the fact that y(t)=Cix(t), (34) can be rewritten as EyTtYiyt≤1. Thus, the output reachable set of free system is mean-square bounded within the set ⋂i=1NE(Yi).Remark 8.
The output reachable set is also expected to be as small as possible. To achieve this goal, we first solve LMI (30) and get P^i satisfying trace(P^i)→max, which can be implemented by using (29). Then we add the additional requirement CiTYiCi>δI and maximize a positive scalar δ, which is equivalent to the following minimization problem:(35)minδ¯s.t.31,-δ¯III-CiTYiCi<0,where δ¯=1/δ.


### 3.3. State-Feedback Controller Design

In this section, we turn our attention to the state-feedback control problem. Our goal here is to find a state-feedback controller, which not only stabilizes the closed-loop system, but also makes the ellipsoid bound on the reachable set of closed-loop system as small as possible.Now, consider the state-feedback controlleru(t)=Kix(t), where Ki is a gain matrix to be determined later. By using this controller, the closed-loop system can be obtained as(36)Ex˙t=A~ixt+Diωtyt=Cixt,where A~i=Ai+BiKi.Theorem 9.
Consider singular Markov jump system (1). If there exist nonsingular matrices Pi∈Rn×n, matrices Si, and scalars α>0, δi>0 such that the following LMIs hold for each i∈S,(37)PiTET=EPi≥0(38)PiTET≤δiI(39)ΦiDiWi∗-αω¯2I0∗∗-Ji<0,then the reachable set of system (1) is mean-square bounded within the set ⋂i=1NE(P¯i)=⋂i=1Nx∈Rn∣xTP¯ix≤1, and the desired controller gain matrix is given by Ki=SiPi-1, where (40)Φi=SymPiTAiT+SiTBiT+α+πiiPiTETWi=πi1PiT⋯πii-1PiTπii+1PiT⋯πiNPiTJi=diag⁡P1+P1T-δ1I,…,Pi-1+Pi-1T-δi-1I,Pi+1+Pi+1T-δi+1I,…,PN+PNT-δNIP¯i=1r^i2N-TN-1r^i=maxi∈S⁡λmaxPˇi+A22i-1A21imaxi∈S⁡λmaxPˇi+D2iω¯Pˇi=N-1PiMTl×l.Proof.
DenoteXi=Pi-1 and Si=KiPi for each i∈S. Then, pre- and postmultiplying (39) by diag(XiT,I,I) and its transpose, respectively, we obtain(41)Φ~iXiTDiW~i∗-αω¯2I0∗∗-Ji<0,where Φ~i=SymA~iTXi+(α+πii)ETXi, W~i=πi1I⋯πi(i-1)Iπi(i+1)I⋯πiNI.
Using Lemma2, we have(42)δi-1PjPjT≥Pj+PjT-δiI,j=1,2,…,i-1,i+1,…,N.From (41) and (42), it is easy to obtain that(43)Φ~iXiTDiW~i∗-αω¯2I0∗∗-J~i<0,where J~i=diag⁡(δ1-1P1P1T,…,δi-1-1Pi-1Pi-1T,δi+1-1Pi+1Pi+1T,…, δN-1PNPNT).
By Schur complement, the previous matrix inequality becomes(44)Φ^iXiTDi∗-αω¯2I<0,where Φ^i=SymA~iTXi+(α+πii)ETXi+∑j=1,j≠iN‍δjπijXjTXj.
Sincej∈S, (38) infers that PjTET≤δjI. Pre- and postmultiplying the previous matrix inequality by XjT and Xj, respectively, we have ETXj≤δjXjTXj. This together with (44) implies(45)Φ¯iXiTDi∗-αω¯2I<0,where Φ¯i=SymA~iTXi+∑j=1N‍πijETXj+αETXi.
Pre- and postmultiplying (37) by XiT and Xi, respectively, we obtain(46)ETXi=XiTE≥0.From the above discussion, we show that if (37)–(39) hold, then (45) and (46) hold. Thus, it follows from Theorem 4 that the closed-loop system can be stabilized by the designed state-feedback controller.
Next, we show that the reachable set of the closed-loop system (36) is mean-square bounded within the set ⋂i=1NE(P¯i). From (45) and (46), it is easy to show that there exists a Lyapunov function V(x(t),rt)=xT(t)ETXix(t) such that LV(x(t),rt)+αV(x(t),rt)-(α/ω¯2)ωT(t)ω(t)<0, where LV(x(t),rt) denotes the difference of V(x(t),rt) along the trajectories of (36). It follows from Lemma 3 that ExTtETXixt≤1, which implies(47)Ex~TtNTETMTM-TXiNx~t≤1.
Noting thatXi=Pi-1, then (47) can be rewritten as(48)Ex~TtNTETMTM-TPi-1Nx~t≤1.Recalling that M-TPi-1N is a lower triangular matrix, (48) infers that(49)Ex~1TtPˇi-1x~1t≤1,where Pˇi-1=M-TPi-1Nl×l.
By (49), it can be seen that(50)mini∈SλminPˇi-1Ex~1t2≤Ex~1TtPˇi-1x~1t≤1.Using the fact that mini∈Sλmin(Pˇi-1)=1/maxi∈S⁡λmax(Pˇi), we get(51)Ex~1t≤maxi∈S⁡λmaxPˇi.This together with (24) yields(52)Ex~2t≤A22i-1A21imaxi∈SλmaxPˇi+D2iω¯.From (24), (51), and (52), we have that(53)Ex~t≤r^i,where r^i=maxi∈Sλmax(Pˇi)+A22i-1A21imaxi∈Sλmax(Pˇi)+D2iω¯. This implies that (1/r^i2)Ex~Ttx~t≤1. Recalling that x~(t)=N-1x(t), we have(54)1r^i2ExTtN-TN-1xt≤1.By denoting (1/r^i2)N-TN-1=P¯i, the reachable set of closed-loop system (36) is mean-square bounded within the set ⋂i=1NE(P¯i).Remark 10.
In order to make the ellipsoidE(P¯i) as small as possible, we shall carry out the following minimization problem:(55)minϵs.t.37,38,39,Pˇi≤ϵI.


## 3.1. State Reachable Set Estimation

In this subsection, we will focus our attention on determining a ball which contains the state reachable set of the free system.Theorem 4.
If there exist nonsingular matricesXi∈Rn×n and a scalar α>0 such that the following LMIs hold for each i∈S,(10)ETXi=XiTE≥0,(11)H=SymAiTXi+∑j=1NπijETXj+αETXiXiTDi∗-αω¯2I<0,then the state reachable set of free system starting from the origin is mean-square bounded within the following set:(12)⋂i=1NEX^i=⋂i=1Nx∈Rn∣xTX^ix≤1,where (13)X^i=1ri2N-TN-1ri=1mini∈S⁡λminX~i+A22i-1A21imini∈S⁡λminX~i+D2iω¯X~i=M-TXiNl×l.Proof.
We first prove the regularity and nonimpulsiveness of the free system. LetM-TXiN=X11iX12iX21iX22i. Then, by (10), we obtain that X12i=0. From (11), it is easy to show that(14)SymAiTXi+∑j=1NπijETXj+αETXi<0.Pre- and postmultiplying (14) by NT and N, respectively, we get(15)⋆⋆∗A22iTX22i+X22iTA22i<0,where ⋆ will be irrelevant to the results of the following discussion; thus the real expressions of these two variables are omitted. It follows from (15) that(16)A22iTX22i+X22iTA22i<0which implies that A22i is nonsingular for each i∈S. Therefore, by Definition 1, we have that the free system is regular and nonimpulsive.
Next, we will show the state reachable set of free system is mean-square bounded within the setE(X^). Consider the following Lyapunov function:(17)Vxt,rt=xTtETXixt.Let L be the weak infinitesimal generator of the random process {x(t),rt}. Calculating the difference of V(x(t),rt) along the trajectories of the free system, we get(18)LVxt,rt=x˙TtETXixt+xTtXiTEx˙t+xTt∑j=1NπijETXjxt.Defining the augmented system variable as ξ(t)=[xT(t)ωT(t)]T and using conditions (10) and (11), we have(19)LVxt,rt+αVxt,rt-αω¯2ωTtωt=ξTtHξt<0.Then we can deduce from Lemma 3 that ExTtETXixt≤1, which infers that(20)Ex~TtNTETMTM-TXiNx~t≤1.Recalling that X12i=0, it follows from (20) that(21)Ex~1TtX~ix~1t≤1.From (21) we have mini∈SλminX~iEx~1t2≤Ex~1TtX~ix~1t≤1, which implies that Ex~1t≤1/mini∈S⁡λmin(X~i).
SinceA22i is nonsingular for each i∈S, (8) can be rewritten as(22)x~2t=-A22i-1A21ix~1t+D2iωt.Then we can deduce that(23)Ex~2t≤A22i-1A21imini∈S⁡λminX~i+D2iω¯.It follows from the fact x~(t)=[x~1T(t)0]T+[0x~2T(t)]T that(24)Ex~t≤Ex~1t+x~2t≤ri,where ri=1/mini∈Sλmin(X~i)+A22i-1A21i/mini∈Sλmin(X~i)+D2iω¯.By (24), it can be seen that(25)1ri2Ex~Ttx~t≤1which implies that the trajectories of (7)-(8) are mean-square bounded within the set ⋂i=1NB(1/ri2)I. Moreover, notice that x~(t)=N-1x(t), and (25) can be rewritten as (1/ri2)ExT(t)N-TN-1x(t)≤1. By denoting (1/ri2)N-TN-1=X^i, the state reachable set of free system is mean-square bounded within the set ⋂i=1NE(X^i).It should be noted that inequality (10) represents a nonstrict LMI. This may lead to numerical problems since equality constraints are usually not satisfied perfectly. Below, we will develop a numerically tractable and nonconservative LMI condition.Theorem 5.
If there exist symmetric positive definite matricesPi∈Rn×n, nonsingular matrices Qi, and a scalar α>0 such that the following LMI holds for each i∈S,(26)ΨiPiE+RTQiSTTDi∗-αω¯2I<0,then the state reachable set of free system is mean-square bounded within the following set:(27)⋂i=1NEP^i=⋂i=1Nx∈Rn∣xTP^ix≤1,where (28)Ψi=SymAiTPiE+RTQiST+∑j=1NπijETPjE+αETPiEP^i=1r~i2N-TN-1r~i=1mini∈S⁡λminP~i+A22i-1A21imini∈S⁡λminP~i+D2iω¯P~i=M-TPiM-1l×l,R∈R(n-l)×n is any matrix with full row rank and satisfies RE=0; S∈Rn×(n-l) is any matrix with full column rank and satisfies ES=0.Proof.
LetXi=PiE+RTQiST in (26); it is easy to obtain (10) and (11). In this case, inequality (20) will be replaced by Ex~TtNTETMTM-TPiM-1MENx~t≤1, which infers that Ex~1TtP~ix~1t≤1 with P~i=M-TPiM-1l×l. Then, following the same lines as (22)–(25) in Theorem 4, we can get ExTtP^ixt≤1 for any i∈S. Therefore, the state reachable set of free system is mean-square bounded within the set ⋂i=1NE(P^i).Remark 6.
In order to make the ellipsoidE(P^i) as small as possible, we require trace(P^i)→max. For this purpose, we can add the additional requirement P~i>ϵI and then maximize a positive scalar ϵ, which is equivalent to the following minimization problem:(29)minϵ¯s.t.26,-ϵ¯III-P~i<0,where ϵ¯=1/ϵ.


## 3.2. Output Reachable Set Estimation

Theorem 7.
If there exist symmetric positive definite matricesPi∈Rn×n and Yi∈Rn×n, nonsingular matrices Qi, and a scalar α>0 such that the following LMIs hold for each i∈S,(30)ΨiPiE+RTQiSTTDi∗-αω¯2I<0(31)CiTYiCi≤P^i,then the output reachable set of free system is mean-square bounded within the following set:(32)⋂i=1NEYi=⋂i=1Ny∈Rp∣yTYiy≤1,where Ψi, R, S, and P^i are defined in Theorem 5.Proof.
By Theorem5, LMI (30) ensures(33)ExTtP^ixt≤1,∀i∈S.With this and (31), we obtain that(34)ExTtCiTYiCixt≤ExTtP^ixt≤1.Due to the fact that y(t)=Cix(t), (34) can be rewritten as EyTtYiyt≤1. Thus, the output reachable set of free system is mean-square bounded within the set ⋂i=1NE(Yi).Remark 8.
The output reachable set is also expected to be as small as possible. To achieve this goal, we first solve LMI (30) and get P^i satisfying trace(P^i)→max, which can be implemented by using (29). Then we add the additional requirement CiTYiCi>δI and maximize a positive scalar δ, which is equivalent to the following minimization problem:(35)minδ¯s.t.31,-δ¯III-CiTYiCi<0,where δ¯=1/δ.


## 3.3. State-Feedback Controller Design

In this section, we turn our attention to the state-feedback control problem. Our goal here is to find a state-feedback controller, which not only stabilizes the closed-loop system, but also makes the ellipsoid bound on the reachable set of closed-loop system as small as possible.Now, consider the state-feedback controlleru(t)=Kix(t), where Ki is a gain matrix to be determined later. By using this controller, the closed-loop system can be obtained as(36)Ex˙t=A~ixt+Diωtyt=Cixt,where A~i=Ai+BiKi.Theorem 9.
Consider singular Markov jump system (1). If there exist nonsingular matrices Pi∈Rn×n, matrices Si, and scalars α>0, δi>0 such that the following LMIs hold for each i∈S,(37)PiTET=EPi≥0(38)PiTET≤δiI(39)ΦiDiWi∗-αω¯2I0∗∗-Ji<0,then the reachable set of system (1) is mean-square bounded within the set ⋂i=1NE(P¯i)=⋂i=1Nx∈Rn∣xTP¯ix≤1, and the desired controller gain matrix is given by Ki=SiPi-1, where (40)Φi=SymPiTAiT+SiTBiT+α+πiiPiTETWi=πi1PiT⋯πii-1PiTπii+1PiT⋯πiNPiTJi=diag⁡P1+P1T-δ1I,…,Pi-1+Pi-1T-δi-1I,Pi+1+Pi+1T-δi+1I,…,PN+PNT-δNIP¯i=1r^i2N-TN-1r^i=maxi∈S⁡λmaxPˇi+A22i-1A21imaxi∈S⁡λmaxPˇi+D2iω¯Pˇi=N-1PiMTl×l.Proof.
DenoteXi=Pi-1 and Si=KiPi for each i∈S. Then, pre- and postmultiplying (39) by diag(XiT,I,I) and its transpose, respectively, we obtain(41)Φ~iXiTDiW~i∗-αω¯2I0∗∗-Ji<0,where Φ~i=SymA~iTXi+(α+πii)ETXi, W~i=πi1I⋯πi(i-1)Iπi(i+1)I⋯πiNI.
Using Lemma2, we have(42)δi-1PjPjT≥Pj+PjT-δiI,j=1,2,…,i-1,i+1,…,N.From (41) and (42), it is easy to obtain that(43)Φ~iXiTDiW~i∗-αω¯2I0∗∗-J~i<0,where J~i=diag⁡(δ1-1P1P1T,…,δi-1-1Pi-1Pi-1T,δi+1-1Pi+1Pi+1T,…, δN-1PNPNT).
By Schur complement, the previous matrix inequality becomes(44)Φ^iXiTDi∗-αω¯2I<0,where Φ^i=SymA~iTXi+(α+πii)ETXi+∑j=1,j≠iN‍δjπijXjTXj.
Sincej∈S, (38) infers that PjTET≤δjI. Pre- and postmultiplying the previous matrix inequality by XjT and Xj, respectively, we have ETXj≤δjXjTXj. This together with (44) implies(45)Φ¯iXiTDi∗-αω¯2I<0,where Φ¯i=SymA~iTXi+∑j=1N‍πijETXj+αETXi.
Pre- and postmultiplying (37) by XiT and Xi, respectively, we obtain(46)ETXi=XiTE≥0.From the above discussion, we show that if (37)–(39) hold, then (45) and (46) hold. Thus, it follows from Theorem 4 that the closed-loop system can be stabilized by the designed state-feedback controller.
Next, we show that the reachable set of the closed-loop system (36) is mean-square bounded within the set ⋂i=1NE(P¯i). From (45) and (46), it is easy to show that there exists a Lyapunov function V(x(t),rt)=xT(t)ETXix(t) such that LV(x(t),rt)+αV(x(t),rt)-(α/ω¯2)ωT(t)ω(t)<0, where LV(x(t),rt) denotes the difference of V(x(t),rt) along the trajectories of (36). It follows from Lemma 3 that ExTtETXixt≤1, which implies(47)Ex~TtNTETMTM-TXiNx~t≤1.
Noting thatXi=Pi-1, then (47) can be rewritten as(48)Ex~TtNTETMTM-TPi-1Nx~t≤1.Recalling that M-TPi-1N is a lower triangular matrix, (48) infers that(49)Ex~1TtPˇi-1x~1t≤1,where Pˇi-1=M-TPi-1Nl×l.
By (49), it can be seen that(50)mini∈SλminPˇi-1Ex~1t2≤Ex~1TtPˇi-1x~1t≤1.Using the fact that mini∈Sλmin(Pˇi-1)=1/maxi∈S⁡λmax(Pˇi), we get(51)Ex~1t≤maxi∈S⁡λmaxPˇi.This together with (24) yields(52)Ex~2t≤A22i-1A21imaxi∈SλmaxPˇi+D2iω¯.From (24), (51), and (52), we have that(53)Ex~t≤r^i,where r^i=maxi∈Sλmax(Pˇi)+A22i-1A21imaxi∈Sλmax(Pˇi)+D2iω¯. This implies that (1/r^i2)Ex~Ttx~t≤1. Recalling that x~(t)=N-1x(t), we have(54)1r^i2ExTtN-TN-1xt≤1.By denoting (1/r^i2)N-TN-1=P¯i, the reachable set of closed-loop system (36) is mean-square bounded within the set ⋂i=1NE(P¯i).Remark 10.
In order to make the ellipsoidE(P¯i) as small as possible, we shall carry out the following minimization problem:(55)minϵs.t.37,38,39,Pˇi≤ϵI.


## 4. Numerical Examples

In this section, two numerical simulation examples are given to show the effectiveness of the main results derived above.Example 11.
Consider the free system in (1) with the following parameters: (56)E=1000,A1=-0.8695-1.5760-0.23891.8258,A2=-0.50430.1206-1.1634-1.4435,D1=D2=0.50.5,C1=1001,C2=1101.The switching between two modes is described by the following transition rate matrix: (57)Π=-336-6.In this example, we chooseM=I, N=I. By solving optimization problem (29) with the aid of fminsearch, the minimal ϵ¯ and the corresponding α are 0.1312 and 1.7419, respectively. Using the above parameter values, we can obtain P^1=(1/r~12)I=2.1414I and P^2=(1/r~22)I=0.9992I by solving (26). Owing to Theorem 5, the state reachable set Rx is mean-square bounded within the set B(P^1)∩B(P^2)=B(P^1).By applying Theorem7, we have the following results:(58)Y1=1.6561001.6561,Y2=0.9975-0.9975-0.99751.9950.Therefore, the output reachable set of free system is mean-square bounded within the set B(Y1)∩E(Y2).For simulation we assume thatx0=[0.20.1303]T and the disturbance is chosen as ω(t)=sin⁡(t). A case for stochastic variation with transition rate matrix Π is shown in Figure 1. The state reachable set Rx and the ball B(P^2) are depicted in Figure 2. Figure 2 shows that the trajectory of the system is mean-square bounded within the region B(P^2). The output reachable set Ry, the ball B(Y1), and the ellipsoid E(Y2) are depicted in Figure 3. Figure 3 shows that the output reachable set Ry is mean-square bounded within the region B(Y1)∩E(Y2).Figure 1
The switching between two modes.Figure 2
The state reachable setRx and the bounding ball B(P^1).Figure 3
The output reachable setRy and the bounding ellipsoids.Example 12.
Consider system (1) with the following parameters: (59)E=1000,A1=0.251.970.442.31,A2=1.90-1.721.870.64,B1=0.140.79,B2=0.34-0.50,D1=0.500.50,D2=0.700.70.The switching between two modes is described by the following transition rate matrix: (60)Π=-336-6.
By Theorem9, we get the following results: (61)α=0.1997,δ1=22.1356,δ2=18.0362,P1=18.021103.8045102.2184,P2=14.70290-8.624067.2193.Therefore, the gain matrices of state-feedback controller can be obtained as (62)K1=-12.7027-5.3634,K2=-7.19574.6404.The corresponding parameter values r^12 and r^22 are, respectively, 27.7747 and 314.7993, which imply P¯1=(1/r^12)I=0.0360I and P¯2=(1/r^22)I=0.0032I. Applying this controller makes the state reachable set of closed-loop system (36) mean-square bounded within the region B(P¯1).
For the purpose of the simulation, we assume the initial conditionx0=[0.30.9763]T and the disturbance is chosen as ω(t)=sin⁡(0.2t). Figure 4 shows one possible switching between two modes. Figure 5 depicts the state reachable set of closed-loop system (36).Figure 4
The switching between two modes.Figure 5
The state reachable setRx and the bounding ball B(P¯1).


## 5. Conclusions

This paper has dealt with the problems of reachable set estimation and state-feedback controller design for singular Markovian jump systems. New sufficient conditions for the state reachable set estimation and output reachable set estimation have been, respectively, derived in terms of linear matrix inequalities. Based on the estimated reachable set, the state-feedback controller has also been designed. Numerical examples and simulation results have been provided to demonstrate the effectiveness of the proposed methods.



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