This paper investigates the problem of the stability and stabilization of continuous-time Markovian jump singular systems with partial information on transition probabilities. A new stability criterion which is necessary and sufficient is obtained for these systems. Furthermore, sufficient conditions for the state feedback controller design are derived in terms of linear matrix inequalities. Finally, numerical examples are given to illustrate the effectiveness of the proposed methods.
1. Introduction
In practice, many dynamical systems cannot be represented by the class of linear time-invariant model since the dynamics of these systems are random with some features, for example, abrupt changes, breakdowns of components, changes in the interconnections of subsystems, and so forth. Such class of dynamical systems can be adequately described by the class of stochastic hybrid systems. A special class of hybrid systems referred to as Markovian jump systems (MJS), a class of multimodel systems in which the transitions among different modes are governed by a Markov chain, have attracted a lot of researchers and many problems have been solved, such as stability, stabilization, and H∞ control problems; see [1–7].
However, in most of the studies, complete knowledge of the mode transitions is required as a prerequisite for analysis and synthesis of MJS. This means that the transition probabilities of the underlying Markov chain are assumed to be completely known. However, in practice, incomplete transition probabilities are often encountered especially if adequate samples of the transitions are costly or time consuming to obtain. So, it is necessary to further consider more general jump systems with partial information on transition probabilities. The concept for MJS with partially unknown transition probabilities is first proposed in [8] and a series of studies have been carried out [9–12] recently. A new approach for the analysis and synthesis for Markov jump linear systems with incomplete transition descriptions has been proposed in [12], which can be further used for other analysis and synthesis issues, such as the stability of Markovian jump singular systems (MJSS).
A lot of attention has already been focused on robust stability, robust stabilization, and H∞ control problems for MJSS in recent years, such as the works in [13–17]. However, to the best of the authors’ knowledge, the necessary and sufficient conditions for the stochastic stability and stabilization problems of MJSS have not been fully investigated, especially when the transition probabilities are partially known. The authors in [15, 16] have, respectively, studied the problems of stability and stabilization for a class of continuous-time (discrete-time) singular hybrid systems. New sufficient and necessary conditions for these singular hybrid systems to be regular, impulse-free (causal), and stochastically stable have been proposed in terms of a set of coupled strict linear matrix inequalities (LMIs). But the case of systems with partly known transition probabilities still needs to be considered. In addition to this, it is important to mention that the derivation of strict LMIs for MJSS with incomplete transition probabilities renders the synthesis of the state feedback controllers easier. These problems are important and challenging in both theory and practice, which motivates us for this study.
In this paper, the problem of the stability and stabilization of MJSS with partly known transition probabilities is addressed. Inspired by the ideas in [12], which fully unitized the properties of the transition rate matrix (TRM) and the convexity of the uncertain domains, we explore a new sufficient and necessary condition in terms of strict linear matrix inequalities (LMIs) for the MJSS to be regular, impulsive, and stochastically stable. Then, based on the proposed stability criterion, the conditions for state feedback controller are derived. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.
Compared with the existing works about the stability and stabilization of Markovian jump systems, the current paper has the following novel features. First, the current paper deals with the stability and stabilization problems for MJSS with partly known transition probabilities, while most literatures (e.g., [8–12]) focused on those of normal ones that are special cases of MJSS. Second, the conservatism in the conventional studies [15] is eliminated by considering the fact that the unknown elements of each row in TRM exist. Moreover, the difficulty that the unknown elements contain diagonal elements is also overcome by introducing a lower bound of the diagonal element without additional conservatism.
Notation. The notation used in this technical note is standard. The superscript “T” stands for matrix transposition; ℝn denotes the n dimensional Euclidean space; ℤ+ represents the sets of positive integers, respectively. For the notation (Ω,ℱ,𝒫), Ω represents the sample space, ℱ is the σ-algebra of subsets of the sample space, and 𝒫 is the probability measure on ℱ. E[·] stands for the mathematical expectation. In addition, in symmetric block matrices or long matrix expressions, we use * as an ellipsis for the terms that are introduced by symmetry and diag{X1,X2,…,XN} stands for a block-diagonal matrix constituted by X1,X2,…,XN. The notation X>0 means X is real symmetric positive definite, and Xi is adopted to denote X(i) for brevity. I and 0 represent, respectively, identity matrix and zero matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.
2. Preliminaries and Problem Formulation
Consider the following continuous-time MJSS with Markovian jump parameters:
(1)Ex˙(t)=A(rt)x(t)+B(rt)u(t),
where x(t)∈ℝn is the state vector and u(t)∈ℝm is the control input. The matrix E∈ℝn×n is supposed to be singular with rank(E)=r<n. The stochastic process {rt,t≥0} taking values in a finite set S={1,2,…,N} is described by a continuous-time, discrete-state homogeneous Markov process and has the following mode transition probabilities:
(2)Pr{rt+h=j∣rt=i}={λijh+o(h),ifj≠i,1+λiih+o(h),ifj=i,
where h>0, limh→0(o(h)/h)=0, and λij≥0(i,j∈S,j≠i) denotes the switching rate from mode i at time t to mode j at time t+h, and λii=-∑j∈S,j≠iλij for all i∈S. The TRM is given by
(3)Λ=[λ11λ12⋯λ1Nλ21λ22⋯λ2N⋮⋮⋱⋮λN1λN2⋯λNN].
The set S contains N modes of system (1) and for rt=i∈S, the system matrices of the ith mode are denoted by Ai, Bi, which are known real-valued constant matrices of appropriate dimensions that describe the nominal system.
The transition rates described above are considered to be partially available; that is, some elements in matrix Λ are unknown. Take system (1) with 4 operation modes for example; the TRM Λ may be written as
(4)Λ=[λ11λ12λ^13λ^14λ^21λ^22λ23λ24λ^31λ^32λ33λ34λ41λ42λ^43λ44],
where “·^” denotes the unknown element.
For ∀i∈S, we denote
(5)S=S𝒦i+S𝒰𝒦i,S𝒦i≜{j:λijisknown},S𝒰𝒦i≜{j:λijisunknown}.
If S𝒦i≠∅, S𝒦i is further described as
(6)S𝒦i={𝒦1i,𝒦2i,…,𝒦mi},1≤m≤N,
where 𝒦mi∈ℤ+ represents the index of the mth known element in the ith row of matrix Λ. Also, throughout the technical note, we denote
(7)λ𝒦i=∑j∈S𝒦iλij.
When λ^ii is unknown, it is necessary to provide a lower bound λdi for it and λdi≤-λ𝒦i.
Now, we introduce the following definition for the continuous-time MJSS (1) (with u(t)≡0).
Definition 1 (see [<xref ref-type="bibr" rid="B17">17</xref>]).
The continuous-time MJSS in (1) is said to be regular if, for each i∈S, det(sE-Ai) is not identically zero.
The continuous-time MJSS in (1) is said to be impulsive if, for each i∈S, deg(det(sE-Ai))=rank(E).
The continuous-time MJSS in (1) is said to be stochastically stable if, for any x0∈ℝn and r0∈S, there exists a scalar M(x0,r0)>0 such that
(8)E{∫0∞∥x(t)∥2∣x0,r0}≤M(x0,r0),
where E is the mathematical expectation, and x(t,x0,r0) denotes the solution to system (1) at time t under the initial conditions x0 and r0.
The continuous-time MJSS in (1) is said to be stochastically admissible if it is regular, impulsive, and stochastically stable.
The following lemma is recalled, which will be used in what follows.
Lemma 2 (see [<xref ref-type="bibr" rid="B18">18</xref>]).
Let P∈Rn×n be symmetric such that ERTPER>0, Φ∈Rn×n, and S are nonsingular. Then, PE+STΦRT is nonsingular and its inverse is expressed as
(9)(PE+STΦRT)-1=P-ET+RΦ-S,
where EL and ER are full column rank with E=ELERT, R∈R(n-r)×n, and S∈Rn×(n-r) satisfies RE=0 and ES=0, respectively. P- is symmetric and S is nonsingular such that
(10)ELTP-EL=(ERTPER)-1,Φ-=(RRT)-1Φ-1(SST)-1.
3. Main Results
In this section, we will derive the stochastic stability criteria for system (1) when the transition probabilities are partially unknown and design a state-feedback controller and a static output feedback controller such that the closed-loop system is stochastically stabilizable. The mode-dependent controller considered here has the form
(11)u(t)=K(rt)x(t),
where Ki=K(rt)∈Rm×n(∀rt=i∈S) are the controller gains to be determined. The closed-loop systems obtained by applying controllers (11) to system (1) are
(12)Ex˙(t)=(Ai+BiKi)x(t).
First, we provide the following lemma which presents a necessary and sufficient condition for the continuous-time MJSS with completely known transition probabilities matrix to be stochastically admissible.
Lemma 3 (see [<xref ref-type="bibr" rid="B15">15</xref>]).
System (1) with u(t)=0 is stochastically admissible if and only if there exist matrices Pi∈Rn×n>0, i∈S, and Φi∈R(n-r)×(n-r), such that the following coupled LMIs hold for each i∈S:
(13)AiT(PiE+RTΦiST)+(PiE+RTΦiST)TAi+∑j∈SλijETPjE<0.
Let us first give the stability result for the unforced system (1) (with u(t)≡0). The following theorem presents a necessary and sufficient condition on the stochastic admissibility of the considered system with partially unknown transition probabilities.
Theorem 4.
Consider the unforced system (1) with partially unknown transition probabilities. The corresponding system is stochastically admissible if and only if there exist matrices Pi∈ℝn×n>0 and nonsingular symmetric matrices Φi∈ℝ(n-r)(n-r), such that for each i∈S(14)AiT(PiE+RTΦiST)+(PiE+RTΦiST)TAi+ET𝒫𝒦iE-λ𝒦iETPjE<0,∀j∈S𝒰𝒦i,ifi∈S𝒦i,(15)AiT(PiE+RTΦiST)+(PiE+RTΦiST)TAi+ET𝒫𝒦iE+ET(λdiPi-λdiPj-λ𝒦iPj)E<0,∀j∈S𝒰𝒦i,ifi∈S𝒰𝒦i,
where 𝒫𝒦i=∑j∈S𝒦iλijPj and λdi is a given lower bound for the unknown diagonal element.
Proof.
Consider two cases, i∈S𝒦i and i∈S𝒰𝒦i, and note that system (1) is stochastically stable if and only if (13) holds.
Case 1 (i∈S𝒦i). It should be noted that in this case one has λ𝒦i≤0. We only need to consider λ𝒦i<0 since λ𝒦i=0 means the elements in the ith row of the TRM are known, so it is not considered here. Now the left-hand side of (13) in Lemma 3 can be rewritten as
(16)Θi≜AiT(PiE+RTΦiST)+(PiE+RTΦiST)TAi+∑j∈S𝒦iλijETPjE+∑j∈S𝒰𝒦iλ^ijETPjE=AiT(PiE+RTΦiST)+(PiE+RTΦiST)TAi+ET𝒫𝒦iE-λ𝒦i∑j∈S𝒰𝒦iλ^ij-λ𝒦iETPjE,
where the elements λ^ij,j∈S𝒰𝒦i are unknown. Since 0≤λ^ij/(-λ𝒦i)≤1 and ∑j∈S𝒰𝒦iλ^ij/(-λ𝒦i)=1, we know that
(17)Θi=∑j∈S𝒰𝒦iλ^ij-λ𝒦i×[AiT(PiE+RTΦiST)+(PiE+RTΦiST)TAi+ET𝒫𝒦iE-λ𝒦iETPjE].
Therefore, for 0≤λ^ij≤-λ𝒦i, Θi<0 is equivalent to AiT(PiE+RTΦiST)+(PiE+RTΦiST)TAi+ET𝒫𝒦iE-λ𝒦iETPjE<0,∀j∈S𝒰𝒦i, which implies that, in the presence of unknown elements λ^ij, the system stochastic admissibility is ensured if and only if (14) holds.
Case 2 (i∈S𝒰𝒦i). In this case, λ^ii is unknown, λ𝒦i≥0, and λ^ii≤-λ𝒦i. We also only consider λ^ii<-λ𝒦i since λ^ii=-λ𝒦i; then the ith row of the TRM is completely known.
Now the left-hand side of (15) can be rewritten as
(18)Θi≜AiT(PiE+RTΦiST)+(PiE+RTΦiST)TAi+ET𝒫𝒦iE+λ^iiETPiE+∑j∈S𝒰𝒦i,j≠iλ^ijETPjE=AiT(PiE+RTΦiST)+(PiE+RTΦiST)TAi+ET𝒫𝒦iE+ET[λ^iiPi+(-λ^ii-λ𝒦i)∑j∈S𝒰𝒦i,j≠iλ^ij-λ^ii-λ𝒦iPj]E.
Likewise, since we have 0≤λ^ij/(-λ^ii-λ𝒦i)≤1 and ∑j∈S𝒰𝒦i,j≠iλ^ij/(-λ^ii-λ𝒦i)=1, we know that
(19)Θi=∑j∈S𝒰𝒦i,j≠iλ^ij-λ^ii-λ𝒦i[AiT(PiE+RTΦiST)+(PiE+RTΦiST)TAi+ET𝒫𝒦iE+ET(λ^iiPi-λ^iiPj-λ𝒦iPj)E]
which means that Θi<0 is equivalent to ∀j∈S𝒰𝒦,j≠i,
(20)AiT(PiE+RTΦiST)+(PiE+RTΦiST)TAi+ET𝒫𝒦iE+ET(λ^iiPi-λ^iiPj-λ𝒦iPj)E<0.
As λ^ii is lower bounded by λdi, we have
(21)λdi≤λ^ii<-λ𝒦i
which implies that
(22)λdi≤λ^ii<-λ𝒦i+ϵ
for some ϵ<0 arbitrarily small. Then λ^ii can be further written as a convex combination
(23)λ^ii=-αλ𝒦i+αϵ+(1-α)λdi,
where α takes value arbitrarily in [0,1]. Thus, (14) holds if and only if ∀j∈S𝒰𝒦i,i≠j,
(24)AiT(PiE+RTΦiST)+(PiE+RTΦiST)TAi+ET𝒫𝒦iE+ET(-λ𝒦iPi+ϵ(Pi-Pj))E<0,(25)AiT(PiE+RTΦiST)+(PiE+RTΦiST)TAi+ET𝒫𝒦iE+ET(λdiPi-λdiPj-λ𝒦iPj)E<0
simultaneously hold. Since ϵ is arbitrarily small, (24) holds if and only if
(26)AiT(PiE+RTΦiST)+(PiE+RTΦiST)TAi+ET𝒫𝒦iE-λ𝒦iETPiE<0,
which is the case in (25) when j=i,∀j∈S𝒰𝒦i. Hence (20) is equivalent to (15).
Therefore, we can conclude that the unforced system (1) with unknown elements in the TRM is stochastically admissible if and only if (14) and (15) hold for i∈S𝒦i and i∈S𝒰𝒦i, respectively.
Remark 5.
Theorem 4 presents a new necessary and sufficient condition of stochastic admissibility criterion for the MJSS (1). The approach adopted in Theorem 4, which uses the TRM property (the sum of each row is zero), has extended the result of Theorem 1 in [12] to the MJSS. Note that the lower bound, λdi, of λii is allowed to be arbitrarily negative.
Now let us consider the stabilization problem of system (1) in the presence of unknown elements in the TRM. The following theorem presents a condition for the existence of a mode-dependent stabilizing controller of the form in (11).
Theorem 6.
Let εi be given scalars. Consider the closed-loop system (12) with partially unknown transition probabilities. If there exist matrices P-i∈ℝn×n>0 and nonsingular matrices Φ-i∈ℝ(n-r)×(n-r), matrices Li∈ℝn×m and Hi∈ℝm×(n-r) such that, for each i∈S, the following LMIs hold:(27)[AiYi+YiTAiT+Wi+λii(εiEYi+εiYiTET-εi2EP-iET)YiTFiT(E)-λ𝒦iYiTER*-Xi(P-)0**-ERTP-jER]<0,∀j∈S𝒰𝒦i,ifi∈S𝒦i(28)[AiYi+YiTAiT+Wi+λdi(εiEYi+εiYiTET-εi2EP-iET)YiTFiT(E)-λdi-λ𝒦iYiTER*-Xi(P-)0**-ERTP-jER]<0,∀j∈S𝒰𝒦i,ifi∈S𝒰𝒦i,where
(29)Yi=P-iET+RΦ-iSWi=Bi(LiET+HiR)+(LiET+HiR)TBiTFi(E)=[λi𝒦1ER,…,λi𝒦miER]T,𝒦mi≠iXi(P-)=diag{ERTP-𝒦1ER,…,ERTP-𝒦miER},𝒦mi≠i.
Then there exists a mode-dependent stabilizing controller of the form in (11) such that the closed-loop system is stochastically admissible. The gain of the stabilizing state feedback controller is given by
(30)Ki=(LiET+HiR)(P-iET+RΦ-iS)-1.
Proof.
Consider the closed-loop system (12) and replace Ai by Ai+BiKi in (14) and (15), respectively. Then, if i∈S𝒦i, by Schur complement and performing a congruence transformation to (14) by [YiT00I], with Yi=(PiE+STΦiRT)-1=P-iET+RΦ-iS, we can obtain(31)[AiYi+YiTAiT+BiKiYi+YiTKiTBiT+λiiYiTER(ERTP-iER)-1ERTYiYiTFiT(E)-λ𝒦iYiTER*-Xi(P-)0**-ERTP-jER]<0.Let Li=KiP-i and Hi=KiSΦ-i; we have
(32)BiKiYi+YiTKiTBiT=Bi(LiET+HiR)+(LiET+HiR)TBiT=Wi,Ki=(LiET+HiR)Yi-1=(LiET+HiR)(P-iET+RΦ-iS)-1.
So (31) becomes(33)[AiYi+YiTAiT+Wi+λiiYiTER(ERTP-iER)-1ERTYiYiTFiT(E)-λ𝒦iYiTER*-Xi(P-)0**-ERTP-jER]<0.
Considering the nonlinear term in the above inequalities, the following inequalities are introduced. For any scalars εi, i∈S, by Lemma 2, the following inequalities hold:
(34)0≤[YiTER-εiEL(ERTP-iER)](ERTP-iER)-1×[YiTER-εiEL(ERTP-iER)]T=YiTER(ERTP-iER)-1ERTYi-εiEYi-εiYiTET+εi2EP-iET.
Note that λii≤0; we have
(35)λiiYiTER(ERTP-iER)-1ERTYi≤λii(εiEYi+εiYiTET-εi2EP-iET).
So (33) holds if (27) is fulfilled. In a similar way, if i∈S𝒰𝒦i, (28) can be worked out from (15). Therefore, the closed-loop system is stochastically admissible, and the desired controller gain is given by (30).
Remark 7.
It should be pointed out that if the diagonal elements in the TRM contain unknown ones, the system admissibility, the existence of the admissible controller, and the controller gains solution will be dependent on λdi. The conditions of Theorem 6 are strict LMIs; hence they can be easily tractable by Matlab LMI toolbox.
4. ExamplesExample 1.
Consider system (1) with four operation modes and the following system matrices:
(36)E=[40000.80000],EL=[2000.400],ER=[200200],R=[002],S=[001],A1=[2-71-5-2-124-5],A2=[537793245],A3=[2-54-1-334-68],A4=[143241614],B1=[06-7910],B2=[520560],B3=[350420],B4=[047630].
The transition rate matrix is given as shown in Table 1.
Mode
1
2
3
4
1
−1.2
λ^12
λ^13
0.6
2
0.3
−0.8
0.1
0.4
3
λ^31
λ^32
−0.6
0.3
4
λ^41
λ^42
λ^43
−0.9
Let ε1=1.2,ε2=-1,ε3=-0.2,ε4=2, and λ^ij denote the unknown elements. Using Theorem 6 and the LMI control toolbox of Matlab, we obtain the controller gains for the system as follows:
(37)K1=[3.71233.77080.00052.19862.23250.0006]×104,K2=[-0.7952-3.3671-0.00011.12114.74070.0002]×104,K3=[2.52101.2413-0.00000.59450.2927-0.0000]×105,K4=[5.1907-7.2130-0.00131.7600-2.44730.0008]×103.
The closed-loop dynamic responses and the Markovian chain are shown in Figure 1 with the initial condition x(0)=[0.7,0.5,-2.3]T.
System states and Markovian chain.
Example 2.
Consider system (1) with three operation modes and the following system matrices:
(38)E=[2000],EL=[20],ER=[10],R=[01],S=[02],A1=[1.5-1.40.10.2],A2=[-0.5-0.31-1.2],A3=[-0.10.211],B1=[20],B2=[-1-3],B3=[3-2].
The transition rate matrix is given as shown in Table 2.
Mode
1
2
3
1
−1.2
λ^12
λ^13
2
λ^21
λ^22
0.4
3
0.3
0.5
−0.8
Let ε1=1.2,ε2=-1,ε3=-0.2,λd2=-1. In the 2nd row of TRM, the diagonal element λ^22 is unknown; we assign its lower bound λd2 a priori with different values (λd2∈(-∞,-0.4]). Using Theorem 6 and LMI control toolbox in Matlab, the controller gains for the system are given by
(39)K1=[-7.68340.0014]×105,K2=[-114.1162-0.4001],K3=[529.61950.5013].
When λd2=-2, we obtain the controller gains differently for the system as follows:
(40)K1=[-2.98250.0003]×106,K2=[504.0862-0.4000],K3=[3.00480.0005]×103.
It is seen from above that the obtained controller gains are dependent on λd2. The closed-loop dynamic responses and the Markovian chain are shown in Figure 2 with the initial condition x(0)=[0.7,2.89]T and λd2=-1.
System states and Markovian chain.
Remark 8.
Notice that, in Example 1, all the diagonal elements of TRM are known and, in Example 2, there are unknown diagonal elements in the TRM which illustrate that the controller design is dependent on the lower bound λdi of the corresponding unknown diagonal element. So they cannot be solved by the stabilization criterions developed in [15] which lack considering the case of systems with partly known transition probabilities. Moreover, here examples are for MJSS, while the stabilization criterions developed in [12] which focused on those of normal ones that are special cases of MJSS.
5. Conclusion
The problems of stability and state feedback control for continuous-time MJSS with partly known transition probabilities have been studied. A new sufficient and necessary condition for this class of system to be stochastically admissible has been proposed in terms of strict LMIs. Furthermore, sufficient conditions for the state feedback controller are derived, and numerical examples have also been given to illustrate the main results. However, the study of stability and stabilization of continuous-time MJSS with partly known transition probabilities is a basic problem which only serves as a stepping stone to investigate more complicated systems. However, time-delay appears commonly in various practical systems, and researchers have been paying remarkable attention to the problems of analysis and synthesis for time-delay systems [18–24]. The approaches proposed in this paper could be further extended to time-delay systems in our future work. It is expected that the approach can be further used for other analysis and synthesis issues such as H∞ analysis, H∞ synthesis, and other applications such as Markov jumping neural networks with incomplete transition descriptions.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by Postdoctoral Research Sponsorship in Henan Province (Grant no. 2013013).
ShiP.BoukasE.-K.AgarwalR. K.Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delayBoukasE. K.LiuZ. K.Robust H∞ control of discrete-time Markovian jump linear systems with mode-dependent time-delaysCaoY.-Y.LamJ.Robust H∞ control of uncertain Markovian jump systems with time-delayKangY.ZhangJ. F.GeS. S.Robust output feedback H∞ control of uncertain Markovian jump systems with mode-dependent time-delaysXiongJ.LamJ.GaoH.HoD. W. C.On robust stabilization of Markovian jump systems with uncertain switching probabilitiesKaranM.ShiP.KayaC. Y.Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systemsBoukasE.-K.ShiP.Stochastic stability and guaranteed cost control of discrete-time uncertain systems with Markovian jumping parametersZhangL.BoukasE.-K.Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilitiesZhangL.BoukasE.-K.H∞ control for discrete-time Markovian jump linear systems with partly unknown transition probabilitiesZhangL.BoukasE.-K.LamJ.Analysis and synthesis of Markov jump linear systems with time-varying delays and partially known transition probabilitiesZhangY.HeY.WuM.ZhangJ.Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matricesZhangL.LamJ.Necessary and sufficient conditions for analysis and synthesis of Markov jump linear systems with incomplete transition descriptionsMaS.BoukasE.-K.ChinniahY.Stability and stabilization of discrete-time singular Markov jump systems with time-varying delayWuZ.SuH.ChuJ.Output feedback stabilization for discrete singular systems with random abrupt changesXiaY.BoukasE.-K.ShiP.ZhangJ.Stability and stabilization of continuous-time singular hybrid systemsXiaY.ZhangJ.BoukasE.-K.Control of discrete singular hybrid systemsXuS.LamJ.UezatoE.IkedaM.Strict LMI conditions for stability, robust stabilization, and H∞ control of descriptor systemsProceedings of the 38th IEEE Conference on Decision and Control (CDC '99)December 1999409240972-s2.0-0033312273LiuP.Improved delay-dependent robust exponential stabilization criteria for uncertain time-varying delay singular systemsWuL.ZhengW.GaoH.Dissipativity-based sliding mode control of switched stochastic systemsSuX.ShiP.WuL.A novel control design on discrete-time Takagi-Sugeno fuzzy systems with time-varying delaysParkK. S.LimJ. T.Exponential stability of singularly perturbed discrete systems with time-delayWuL.SuX.ShiP.Sliding mode control with bounded L2 gain performance of Markovian jump singular time-delay systemsWuL.SuX.ShiP.QiuJ.A new approach to stability analysis and stabilization of discrete-time T-S fuzzy time-varying delay systems