Stochastic Stabilization for Nonhomogeneous Markovian Jump Discrete-Time Singula

CHEN Lixiong()， CUI Wenxia()， LIU Yuhao()

School of Mathematics Physics and Statistics， Shanghai University of Engineering Science， Shanghai 201620， China

Abstract: Stochastic stability analysis and control synthesis problems are studied for a class of nonhomogeneous Markovian jump discrete-time singular systems (MJDSS). The time-varying character is considered to be the model in a polytopic sense. Based on the parameter dependent stochastic Lyapunov functional and the matrix analysis techniques， sufficient criteria are derived to ensure regularity， causality and stochastic stability of the closed-loop singular system in terms of linear matrix inequalities. Finally， one example is provided to illustrate the effectiveness of our results.

Key words: stochastic stability; nonhomogeneous; Lyapunov function; singular system; Markovian switching

Introduction

Singular systems are also referred to as generalized systems， descriptor systems， differential-algebraic systems， or implicit systems， which are a natural representation of dynamic systems and describe a larger family of systems than the normal linear systems[1]. Recently， increasing interest has been shown on singular systems since the class of systems has been found successful applications in broad range of scientific areas such as mechanical systems， electric circuits， chemical process， power systems， and other areas in the past decades[2]. It should be pointed out that the study on singular systems is much more complicated than that research for state-space systems， because it needs to consider not only stability， but also regularity and impulse free (for continuous singular systems) or causality (for discrete singular systems) simultaneously， while the latter two do not occur in the regular ones[3-5].

One popular system model for randomness in the sciences and industries is the Markovian switching model[6]. This is partly because Markovian jump is a suitable mathematical pattern to represent a class of stochastic system subject to random abrupt variations in the structures[7]. One typical example is networked systems， in which packet dropouts and network delays evolve in Markov chains or Markov processes[8]. Moreover， Markovian jump singular systems (MJSS) belong to a class of stochastic switched systems where the switching law is governed by a Markovian process (or chain in the discrete-time case). The theory of admissibility， dissipativity， observer and sliding mode control， as well as important applications of such singular systems， can be found in several references in the current literature， for instance in Refs. [9， 10-12]， for continuous-time case， and Refs. [13-17] for the discrete-time case.

However， almost all the aforementioned works assume that the Markovian process (or chain) is homogeneous， that is the transition probabilities are time-invariant[6-7， 11-12]. However， in many applications， this assumption may not be verified. For example， random component failures are considered in singular systems. Usually， it is often assumed that the component failure rates (or probabilities) are time-independent and independent of system state. In other words， the underlying Markovian chain used to model random failures is homogeneous. However， this assumption is often violated and the failure rate of a component usually depends on many factors in reality such as its age and working time[8]. In most cases， it is reasonable to assume that if a component is more solicited， it is more likely to fail. Another example can be referred to the study of MJSS with uncertain transition probabilities. Most of the developed results in the field of Markovian jump systems (MJS) are based on the critical assumption that transition probabilities are known precisely. However， the estimated values of transition probabilities are available in practice. Estimation errors， also referred to as transition probability uncertainties， may lead to instability or at least degraded system performances[18-19]， and the Markovian chain is homogeneous. If the time-varying uncertainties are considered， it will lead to nonhomogeneous Markovian chains. Abekane dealt with stochastic stabilization of a class of nonhomogeneous Markovian linear systems by using a parameter dependent stochastic Lyapunov function[20]. In Ref.[8]， filter was studied for nonhomogeneous Markovian jump discrete-time singular systems(MJDSS)with uncertainties. There are few results on the stabilization problem of nonhomogeneous MJDSS.

In consequence of the above discussion， in this paper， we consider the problem of stochastic stability analysis and control synthesis of MJSS with time-varying transition probabilities (nonhomogeneous MJSS) in the discrete-time domain. The time-varying character is considered to be in a polytopic sense. The approach followed in this note is based on the use of a parameter dependent stochastic Lyapunov functional.

1 System Description

Fix an underlying probability space (Ω，，P) and consider the following discrete-time MJSS:

Ex(k+1)=A(rk)x(k)+B(rk)u(k)，

(1)

wherex(k)∈nis the system state，u(k)∈mis the system input， andy(k)∈qis the system measured output.E，A(rk)，B(rk) andC(rk)are known as real matrices with appropriate dimensions. The process{rk，k≥0} is described by a discrete-time Markovian chain with finite state-spaceΛ={1， 2，…，σ} and mode transition probabilities.

πij(k)=Prrk+1=jrk=i，

(2)

(3)

where

(4)

andΠl，l=1， 2， …，Nare given transition probability matrices. That is， the time-varying transition probability matrixΠ(ξ(k)) as a polytopic time-varying transition matrix. In Ref.[1]， the indicator function is defined asξ(k)=ξ1(k)ξ2(k) …ξN(k)T， and for ∀l∈{1， 2，…，N}.

Definition1(Regular and Impulse Free[1, 21]).

(1) The matrix pair (E，Ai) is said to be regular if， for eachi∈Λ，the characteristic polynomialdet(sE-Ai) is not identically zero.

(2) The matrix pair (E，Ai)is said to be causal if， for eachi∈Λ，deg(det(sE-Ai))=rank(E).

Definition2[1， 8， 20， 22]

(3) System (1) withu(k)=0 is said to be regular and causal， if the pair(E，Ai) is regular， causal for eachi∈Λ.

In this paper， there are two purposes. The first one is to develop the conditions， which guarantee that system (1) withu(k)=0is regular， causal， and is stochastically stable. The second is to design a state feedback controlleruk≡K(rk)xkfor system (1)， and it’s got that the resulting closed-loop system is regular， causal， and is stochastically stable.

2 Main Results

2.1 Stochastic stability

In this section， we give the condition for stochastic stability of system (1) withu(k)=0 in terms of the linear matrix inequality(LMI) feasibility problem.

(6)

ProofWe firstly verify that singular system (1) withu(k)=0 is regular and causal. Using the Schur complement property[23]and noting that condition (6)， we have

(7)

From formula (7)， it follows that

(8)

(9)

Assume that rank(E)=r

Therefore，A4iis nonsingular， which implies that the system (1) withu(k)=0 is regular and causal. Next， we need to prove the system (1) withu(k)=0 is stochastically stability. Let

which is theσ-algebra generated by (x(t)，rt)， 0≤t≤k.Consider the parameter dependent stochastic Lyapunov functional

(10)

Pj(ξ(k+1)))Ex(k+1)+2xT(k+1)

(11)

then

ΔVk(x(k)，rk，ξ(k))= Ξ[V(k+1)]-V(x(k)，rk，ξ(k))=

Pj(ξ(k+1)))Ex(k+1)-

xT(k)ETPi(ξ(k))Ex(k)+

(12)

where

(13)

(14)

Replacing Eqs. (12) and (13) in Eq. (11)， we get

ΔVk(x(k)，rk=i，ξ(k))=

xT(k)Ωi(k)x(k)，

(15)

where

From formula (7)， we have Ωi(k)<0. Hence，

ΔVk(x(k)，rk=i，ξ(k))≤-ρixT(k)x(k)≤-ρxT(k)x(k)，

(16)

where

(17)

andλmin(-Ωi(k)) denotes the minimal eigenvalue of -Ωi(k). From formula (16)， we have that for anyT≥1，

(18)

which yields the following for anyT≥1，

(19)

It is implied that

(20)

FromDefinition2， it implies that system (1) is stochastically stable. The proof ofTheorem1has been completed.

2.2 Stochastic stabilization

In this section， we consider the stochastic stabilization problem for system (1)， and design the controller. Two different cases are addressed which depend on the information level we have

(a) Only the system mode is available in real time. We design a state feedback control law of the form

u(k)=K(r(k))x(k)，

(21)

which ensure stochastic stability of the closed loop singular system.

(b) The scheduling parameters and the system mode are not available for feedback. In this case the control law is given by

u(k)=Kx(k).

(22)

(23)

Remark1Stochastic switched systems were studied in Refs.[5-6， 13-14， 17]， where the switching law is governed by a Markovian process (or chain in the discrete-time case). And transition probabilities are all homogeneous in the above references. However， time-varying transition probabilities (nonhomogeneous) are considered in MJSS of this paper.

Using the result ofTheorem2is difficult to solve the matrix inequalities. In the following， we will give the solvable condition to Eq.(23).

(24)

ProofIf there exist a set of positive scalarsθi，

then

(25)

From Eq. (24)， we have

(26)

By the Schur complement property Ref.[19]， and from inequality(24)， it is obtained that

(27)

then the inequality (23) is established. According to Theorem 1， the singular system (1) is regular， causal and stochastic stabilized by Eq. (21). The proof of Corollary 1 has been completed.

(28)

Remark2It is pointed out that conditions (24) and (28) are not strict linear matrix inequalities (LMIs). However， once we fix the parametersθi， andθ， the conditions can be turned into LMIs based feasibility problem. Thus， the feasible conditions stated in Corollary 1 and Corollary 2 can be turned into the LMIs based feasibility problem with fixed parameters.

Remark3As a special form of MJSS， some results on interconnected MJSS have been investigated， such as observer[8, 15]， sliding mode control[10]and finite-time control[16]. Based on the descriptor system scheme， the finite-time control problems of nonhomogeneous MJSS could be tackled for continuous or discrete time systems， and the corresponding results will be established in our future work.

3 Numerical Examples

In this section， the simulation results are presented to illustrate the theoretical results derived in this paper.

Example1Consider the MJSS model (1) with two operation modes and the following state-space representation:

The transition probability matrix is assumed to be time-varying in a polytope defined by its vertices:

We consider the control synthesis problem (b) (scheduling parameters and system modes are not available). Lettingθ=2 and applyingCorollary2， we obtained the feasible matrices and the feedback gain as

From Corollary 2， the singular system (1) is regular， causal and stochastic stabilized. And the numerical simulation further verifies this result with the controller (23) (Fig. 1).

Fig.1 State trajectory x(k) in the system (1) for Example 1

4 Conclusions

In this paper， we have presented a theoretical framework to analyze stability and stabilization of the nonhomogeneous Markovian jump discrete-time singular systems. We characterize the time-varying character of Markovian switching by using a polytopic sense. We develop a controller based on the parameter dependent stochastic Lyapunov function and the matrix analysis techniques， and ensure that the closed-loop singular system is regular， causal and stochastic stable important aspects requiring further investigation include: 1) with the nonhomogeneous Markovian jump parameters， dissipativity analysis problem needs to be studied for the singular systems; 2) the extension of nonhomogeneous Markovian jump to the design of optimalH2feedback controllers.