A New Proof to the Necessity of a Second Moment Stability Condition of Discrete-Time Markov Jump Linear Systems with Real States

Abstract

This paper studies the second moment stability of a discrete-time jump linear system with real states and the system matrix switching in a Markovian fashion. A sufficient stability condition was proposed by Fang and Loparo (2002), which only needs to check the eigenvalues of a deterministic matrix and is much more computationally efficient than other equivalent conditions. The proof to the necessity of that condition, however, is a challenging problem. In the paper by Costa and Fragoso (2004), a proof was given by extending the state domain to the complex space. This paper proposes an alternative necessity proof, which does not need to extend the state domain. The proof in this paper demonstrates well the essential properties of the Markov jump systems and achieves the desired result in the real state space.

1. Introduction

1.1. Background of the Discrete-Time Markov Jump Linear Systems

This paper studies the stability condition of discrete-time jump linear systems in the real state domain. In a jump linear system, the system parameters are subject to abrupt jumps. We are concerned with the stability condition when these jumps are governed by a finite Markov chain. A general model is shown as follows: 𝑥𝑥[[𝑘0]+=1]𝑥=₀𝐴[𝑞[𝑘]]𝑥[𝑘],,𝑞[𝑘]=𝑞₀,(1.1) where 𝑥[𝑘]∈𝑅^𝑛 is the state and {𝑞[𝑘]} is a discrete-time Markov chain with a finite state space {𝑞₁,𝑞₂,…,𝑞_𝑁} and a transition matrix 𝑄=(𝑞_𝑖𝑗)_𝑁×𝑁, where 𝑞_𝑖𝑗=𝑃(𝑞[𝑘+1]=𝑞_𝑗∣𝑞[𝑘]=𝑞_𝑖). 𝑥₀∈𝑅^𝑛 is the initial state. 𝑞₀ is the initial Markov state, whose distribution is denoted as 𝑝=[𝑝₁𝑝₂⋯𝑝_𝑁] with 𝑝_𝑖=𝑃(𝑞₀=𝑞_𝑖). {𝑞[𝑘]} is assumed to be a time-homogeneous aperiodic Markov chain. When 𝑞[𝑘]=𝑞_𝑖, 𝐴[𝑞[𝑘]]=𝐴_𝑖(𝑖=1,…,𝑁), that is, 𝐴[𝑞[𝑘]] switches among {𝐴_𝑖}^𝑁𝑖=1. A compound matrix is constructed from 𝐴_𝑖 as 𝐴_[2]=𝑄^𝑇⊗𝐼_𝑛2diag𝐴_𝑖⊗𝐴_𝑖^𝑁𝑖=1,(1.2) where 𝐼_𝑛2 denotes an identity matrix with the order of 𝑛² and ⊗ denotes the Kronecker product [1]. A brief introduction on the Kronecker product will be given in Section 2.1.

For the jump linear system in (1.1), the first question to be asked is “is the system stable?” There has been plenty of work on this topic, especially in 90s, [2–6]. Recently this topic has caught academic interest again because of the emergence of networked control systems [7]. Networked control systems often suffer from the network delay and dropouts, which may be modelled as Markov chains, so that networked control systems can be classified into discrete-time jump linear systems [8–11]. Therefore, the stability of the networked control systems can be determined through studying the stability of the corresponding jump linear systems. Before proceeding further, we review the related work.

1.2. Related Work

At the beginning, the definitions of stability of jump linear systems are considered. In [6], three types of second moment stability are defined.

Definition 1.1. For the jump linear system in (1.1), the equilibrium point 0 is(1)stochastically stable, if, for every initial condition (𝑥[0]=𝑥₀, 𝑞[0]=𝑞₀),  𝐄^∞_𝑘=0‖𝑥[𝑘]‖²∣𝑥₀,𝑞₀<∞,(1.3) where ‖⋅‖ denotes the 2-norm of a vector;(2)mean square stable (MSS), if, for every initial condition (𝑥₀,𝑞₀),  lim_𝑘→∞𝐄‖𝑥[𝑘]‖²∣𝑥₀,𝑞₀=0;(1.4)(3)exponentially mean square stable, if, for every initial condition (𝑥₀,𝑞₀), there exist constants 0<𝛼<1 and 𝛽>0 such that for all 𝑘≥0, 𝐄‖𝑥[𝑘]‖²∣𝑥₀,𝑞₀<𝛽𝛼^𝑘‖‖𝑥₀‖‖²,(1.5) where 𝛼 and 𝛽 are independent of 𝑥₀ and 𝑞₀.

In [6], the above 3 types of stabilities are proven to be equivalent. So we can study mean square stability without loss of generality. In [6], a necessary and sufficient stability condition is proposed.

Theorem 1.2 (see [6]). The jump linear system in (1.1) is mean square stable, if and only if, for any given set of positive definite matrices {𝑊_𝑖∶𝑖=1,…,𝑁}, the following coupled matrix equations have unique positive definite solutions {𝑀_𝑖∶𝑖=1,…,𝑁}: ^𝑁_𝑗=1𝑞_𝑖𝑗𝐴^𝑇𝑖𝑀_𝑗𝐴_𝑖−𝑀_𝑖=−𝑊_𝑖.(1.6) Although the above condition is necessary and sufficient, it is difficult to verify because it claims validity for any group of positive definite matrices {𝑊_𝑖∶𝑖=1,…,𝑁}. A more computationally efficient testing criterion was, therefore, pursued [3, 4, 12–15]. Theorem 1.3 gives a sufficient mean square stability condition.

Theorem 1.3 (see [4, 12]). The jump linear system in (1.1) is mean square stable, if all eigenvalues of the compound matrix 𝐴_[2] in (1.2) lie within the unit circle.

Remark 1.4. By Theorem 1.3, the mean square stability of a jump linear system can be reduced to the stability of a deterministic system in the form 𝑦_𝑘+1=𝐴_[2]𝑦_𝑘 [13]. Thus the complexity of the stability problem is greatly reduced. Theorem 1.3 only provides a sufficient condition for stability. The condition was conjectured to be necessary as well [2, 15]. In the following, we briefly review the research results related to Theorem 1.3.

In [14], Theorem 1.3 was proven to be necessary and sufficient for a scalar case, that is, 𝐴_𝑖(𝑖=1,…,𝑁) are scalar. In [15], the necessity of Theorem 1.3 was proven for a special case with 𝑁=2 and 𝑛=2. In [4, 12], Theorem 1.3 was asserted to be necessary and sufficient for more general jump linear systems. Specifically, Bhaurucha [12] considered a random sampling system with the sampling intervals governed by a Markov chain while Mariton [4] studied a continuous-time jump linear system. Although their sufficiency proof is convincing, their necessity proof is incomplete.

The work in [3] may shed light on the proof of the necessity of Theorem 1.3. In [3], a jump linear system model being a little different from (1.1) is considered. The difference lies in(i)𝑥[𝑘]∈𝐶^𝑛, where 𝐶 stands for the set of complex numbers,(ii)𝑥₀∈𝑆_𝑐, where 𝑆_𝑐 is the set of complex vectors with finite second-order moments in the complex state space.

The mean square stability in [3] is defined as Lim_𝑘→∞𝐄𝑥[𝑘]𝑥^∗[𝑘]∣𝑥₀,𝑞₀=0,∀𝑥₀∈𝑆_𝑐,∀𝑞₀,(1.7) where * stands for the conjugate transpose. Corresponding to the definition in (1.7), the mean square stability in (1.4) can be eewritten into (because 𝑥[𝑘]∈𝑅^𝑛 in (1.4), there is no difference between 𝑥^𝑇[𝑘]and 𝑥^∗[𝑘]), Lim_𝑘→∞𝐄𝑥[𝑘]𝑥^∗[𝑘]∣𝑥₀,𝑞₀=0,∀𝑥₀∈𝑆_𝑗,∀𝑞₀,(1.8) where 𝑆_𝑗 is the set of all vectors in 𝑅^𝑛. For any vector 𝑥∈𝑅^𝑛, we can treat it as a random vector with a single element in 𝑅^𝑛, and also a random vector in 𝐶^𝑛. Of course, such random vectors have finite second-order moments. Therefore, we know 𝑆_𝑗⊂𝑆_𝑐,𝑆_𝑗∩𝑆_𝑐≠𝑆_𝑐.(1.9) It can be seen that the mean square stability in (1.7) requires stronger condition (𝑥₀∈𝑆_𝑐) than the one in (1.8) (𝑥₀∈𝑆_𝑗). When 𝐴_𝑖(𝑖=1,…,𝑁) are real matrices, a necessary and sufficient stability condition was given in the complex state domain.

Theorem 1.5 (see [3]). The jump linear system in (1.1) (with complex states) is mean square stable in the sense of (1.7) if and only if 𝐴_[2] is Schur stable.
Due to the relationship of 𝑆_𝑗⊂𝑆_𝑐 and Theorem 1.5, we can establish the relationship diagram in Figure 1. As it shows, the Schur stability of 𝐴_[2] is a sufficient condition for mean square stability with 𝑥₀∈𝑆_𝑗 at the first look.

Figure 1 

Relationship between different senses of mean square stability.

We are still wondering “whether the condition in Theorem 1.3 is necessary too?” the answer is definitely “yes.” That necessity was conjectured in [2]. A proof to the necessity of that condition was first given in [16], which extends the state domain to the complex space and establishes the desired necessity in the stability sense of (1.7). As mentioned before, our concerned stability (in the sense of (1.8)) is weaker than that in (1.7). This paper proves that the weaker condition in (1.8) still yields the schur stability of 𝐴_[2], that is, the necessity of theorem 1.3 is confirmed. This paper confines the state to the real space domain and makes the best use of the essential properties of the markov jump linear systems to reach the desired necessity goal. In Section 2, a necessary and sufficient version of Theorem 1.3 is stated and its necessity is strictly proven. In Section 3, final remarks are placed.

2. A Necessary and Sufficient Condition for Mean Square Stability

This section will give a necessary and sufficient version of Theorem 1.3. Throughout this section, we will define mean square stability in the sense of (1.4) (𝑥₀∈𝑆_𝑗). At the beginning, we will give a brief introduction to the Kronecker product and list some of its properties. After then, the main result, a necessary and sufficient condition for the mean square stability, is presented in Theorem 2.1 and its necessity is proven by direct matrix computations.

2.1. Mathematical Preliminaries

Some of the technical proofs in this paper make use of the Kronecker product, ⊗ [1]. The Kronecker product of two matrices 𝐴=(𝑎_𝑖𝑗)_𝑀×𝑁, 𝐵=(𝑏_𝑝𝑞)_𝑃×𝑄 is defined as𝐴⊗𝐵=⎡⎢⎢⎢⎢⎢⎢⎣𝑎¹¹𝑎₂₁𝐵𝑎₁₂𝐵⋯𝑎_1𝑁𝐵𝑎⋮𝐵𝑎⋮₂₂𝐵⋱⋯𝑎⋮_2𝑁𝑀1𝐵𝐵𝑎_𝑀2𝐵⋯𝑎_𝑀𝑁𝐵⎤⎥⎥⎥⎥⎥⎥⎦_{𝑀𝑃×𝑁𝑄}.(2.1) For simplicity, 𝐴⊗𝐴 is denoted as 𝐴^[2] and 𝐴⊗𝐴^[𝑛] is denoted as 𝐴^[𝑛+1](𝑛≥2).

For two vectors 𝑥 and 𝑦, 𝑥⊗𝑦 simply rearranges the columns of 𝑥𝑦^𝑇 into a vector. So for two stochastic processes {𝑥[𝑛]} and {𝑦[𝑛]}, lim_𝑛→∞𝐄[𝑥[𝑛]⊗𝑦[𝑛]]=0 if and only if lim_𝑛→∞𝐄[𝑥[𝑛]𝑦^𝑇[𝑛]]=0. Furthermore, if lim_𝑛→∞𝐄[𝑥^[2][𝑛]]=0 and lim_𝑛→∞𝐄[𝑦^[2][𝑛]]=0, then lim_𝑛→∞𝐄[𝑥[𝑛]⊗𝑦[𝑛]]=0.(2.2)

The following property of the Kronecker product will be frequently used in the technical proofs 𝐴₁𝐴₂⋯𝐴_𝑛⊗𝐵₁𝐵₂⋯𝐵_𝑛=𝐴₁⊗𝐵₁𝐴₂⊗𝐵₂⋯𝐴_𝑛⊗𝐵_𝑛,(2.3) where 𝐴_𝑖, 𝐵_𝑖(𝑖=1,2,…,𝑛) are all matrices with appropriate dimensions.

Our computations need two linear operators, vec and devec. The vec operator transforms a matrix 𝐴=(𝑎_𝑖𝑗)_𝑀×𝑁 into a vector as vec(𝐴)=𝑎₁₁⋯𝑎_𝑀1𝑎₁₂⋯𝑎_𝑀2⋯𝑎_1𝑁⋯𝑎_𝑀𝑁^𝑇.(2.4) The devec operator inverts the vec operator for a square matrix, that is, devec(vec(𝐴))=𝐴,(2.5) where 𝐴 is a square matrix.

2.2. Main Results

Theorem 2.1. The jump linear system in (1.1) is mean square stable if and only if 𝐴_[2] is Schur stable, that is, all eigenvalues of 𝐴_[2] lie within the unit circle.
There are already some complete proofs for sufficiency of Theorem 2.1, [3, 12, 13]. So we will focus on the necessity proof. Throughout this section, the following notational conventions will be followed.

The initial condition of the jump linear system in (1.1) is denoted as 𝑥[0]=𝑥₀, 𝑞[0]=𝑞₀ and the distribution of 𝑞₀ is denoted as 𝑝=[𝑝₁𝑝₂⋯𝑝_𝑁] (𝑃(𝑞[0]=𝑞_𝑖∣𝑞₀)=𝑝_𝑖).

The system transition matrix in (1.1) is defined as Φ(𝑘;𝑚)=⎧⎪⎨⎪⎩^𝑘−1_𝑙=𝑚𝐼𝐴[𝑞[𝑙]],if𝑚<𝑘,_𝑛,if𝑚≥𝑘,(2.6) where 𝐼_𝑛 is an identity matrix with the order of 𝑛. With this matrix, the system’s state at time instant 𝑘 can be expressed as 𝑥[𝑘]=Φ(𝑘;0)𝑥₀.(2.7)

A conditional expectation is defined as Φ_𝑖[𝑘]=𝑃𝑞[𝑘]=𝑞_𝑖∣𝑞₀𝐄(Φ(𝑘;0))^[2]∣𝑞[𝑘]=𝑞_𝑖,𝑞₀,(2.8) where 𝑖=1,2,…,𝑁. Specially Φ_𝑖[0]=𝑝_𝑖𝐼_𝑛2 (𝑖=1,…,𝑁). Based on the definition of Φ_𝑖[𝑘], we obtain 𝐄(Φ(𝑘;0))^[2]∣𝑞₀=^𝑁_𝑖=1Φ_𝑖[𝑘].(2.9) By combining all Φ_𝑖[𝑘](𝑖=1,2,…,𝑁) into a bigger matrix, we define 𝑉_Φ[𝑘]=Φ^𝑇1[𝑘]Φ^𝑇2[𝑘]⋯Φ^𝑇𝑁[𝑘]^𝑇.(2.10) Thus, 𝑉_Φ[0]=𝑝^𝑇⊗𝐼_𝑛2.

The necessity proof of Theorem 2.1 needs the following three preliminary Lemmas.

Lemma 2.2. If the jump linear system in (1.1) is mean square stable, then lim_𝑘→∞𝐄(Φ[𝑘;0])^[2]∣𝑞₀=0,∀𝑞₀.(2.11)

Proof of Lemma 2.2. Because the system is mean square stable, we get lim_𝑘→∞𝐄𝑥^[2][𝑘]∣𝑥₀,𝑞₀=0,∀𝑥₀,𝑞₀.(2.12) The expression of 𝑥[𝑘]=Φ(𝑘;0)𝑥₀ yields lim_𝑘→∞𝐄Φ(𝑘;0)𝑥₀^[2]∣𝑥₀,𝑞₀=0.(2.13)Φ(𝑘;0) is an 𝑛×𝑛 matrix. So we can denote it as Φ(𝑘;0)=[𝑎₁(𝑘),𝑎₂(𝑘),…,𝑎_𝑛(𝑘)], where 𝑎_𝑖(𝑘) is a column vector. By choosing 𝑥₀=𝑒_𝑖 (𝑒_𝑖 is an 𝑅^𝑛×1 vector with the 𝑖th element as 1 and the others as 0), (2.13) yields lim_𝑘→∞𝐄𝑎^𝑖[2][𝑘]∣𝑞₀=0,𝑖=1,2,…,𝑛.(2.14) By the definition of the Kronecker product, we know (Φ(𝑘;0))^[2]=𝑎₁[𝑘]⊗𝑎₁[𝑘],…,𝑎₁[𝑘]⊗𝑎_𝑛[𝑘],…,𝑎_𝑛[𝑘]⊗𝑎₁[𝑘],…,𝑎_𝑛[𝑘]⊗𝑎_𝑛[𝑘].(2.15) So (2.14) yields lim_𝑘→∞𝐄(Φ[𝑘;0])^[2]∣𝑞₀=0,∀𝑞₀.(2.16)

Lemma 2.3. If the jump linear system in (1.1) is mean square stable, then lim_𝑘→∞Φ_𝑖[𝑘]=0,𝑖=1,…,𝑁,∀𝑞₀.(2.17)

Proof of Lemma 2.3. Choose any 𝑧₀, 𝑤₀∈𝑅^𝑛. Lemma 2.2 guarantees lim_𝑘→∞𝐄𝑧^0[2]^𝑇(Φ(𝑘;0))^[2]𝑤^0[2]∣𝑞₀=0.(2.18) By the definition of the Kronecker product, we know 𝐄𝑧^0[2]^𝑇(Φ(𝑘;0))^[2]𝑤^0[2]∣𝑞₀=𝐄𝑧^𝑇0Φ(𝑘;0)𝑤₀²∣𝑞₀.(2.19) By (2.8), (2.9), and (2.19), we get 𝐄𝑧^𝑇0Φ[𝑘;0]𝑤₀²∣𝑞₀=^𝑁_𝑖=1𝑃𝑞[𝑘]=𝑞_𝑖∣𝑞₀𝐄𝑧^𝑇0Φ[𝑘,0]𝑤₀²∣𝑞[𝑘]=𝑞_𝑖,𝑞₀.(2.20) Because 𝑃(𝑞[𝑘]=𝑞_𝑖|𝑞₀)≥0 and 𝐄[(𝑧^𝑇0Φ[𝑘;0]𝑤₀)²|𝑞[𝑘]=𝑞_𝑖,𝑞₀]≥0, the combination of (2.18) and (2.20) yields lim_𝑘→∞𝑃𝑞[𝑘]=𝑞_𝑖∣𝑞₀𝐄𝑧^𝑇0Φ(𝑘;0)𝑤₀²∣𝑞[𝑘]=𝑞_𝑖,𝑞₀=0.(2.21)Φ(𝑘;0) is an 𝑛×𝑛 matrix. So it can be denoted as Φ(𝑘;0)=(𝑎_𝑚𝑗(𝑘))_{𝑚=1,…,𝑛;𝑗=1,…,𝑛}. In (2.21), we choose 𝑧₀=𝑒_𝑚 and 𝑤₀=𝑒_𝑗 and get lim_𝑘→∞𝑃𝑞[𝑘]=𝑞_𝑖∣𝑞₀𝐄𝑎_𝑚𝑗(𝑛)²∣𝑞[𝑘]=𝑞_𝑖,𝑞₀=0,(2.22) where 𝑖=1,2,…,𝑁, 𝑚=1,…,𝑛 and 𝑗=1,…,𝑛. By the definition of Φ_𝑖[𝑘], we know the elements of Φ_𝑖[𝑘] take the form of 𝑃𝑞[𝑘]=𝑞_𝑖∣𝑞₀𝐄𝑎_𝑚1𝑗1(𝑘)𝑎_𝑚2𝑗2(𝑘)∣𝑞[𝑘]=𝑞_𝑖,𝑞₀,(2.23) where 𝑚₁, 𝑚₂, 𝑗₁, 𝑗₂=1,…,𝑛. So (2.22) guarantees lim_𝑘→∞Φ_𝑖[𝑘]=0,∀𝑞₀.(2.24)

Lemma 2.4. 𝑉_Φ[𝑘] is governed by the following dynamic equation 𝑉_Φ[𝑘]=𝐴_[2]𝑉_Φ[𝑘−1],(2.25) with 𝑉_Φ[0]=𝑝^𝑇⊗𝐼_𝑛2.

Proof of Lemma 2.4. By the definition in (2.8), we can recursively compute Φ_𝑖[𝑘] as follows: Φ_𝑖[𝑘]==𝑃𝑃𝑞𝑞[[𝑘𝑘]]==𝑞𝑞_𝑖𝑖∣𝑞₀𝐄(𝐴[𝑞[𝑘−1]]Φ(𝑘−1;0))^[2]∣𝑞[𝑘]=𝑞_𝑖,𝑞₀=𝑃𝑞[𝑘]=𝑞_𝑖∣𝑞₀𝐄(𝐴[𝑞[𝑘−1]])^[2](Φ(𝑘−1;0))^[2]∣𝑞[𝑘]=𝑞_𝑖,𝑞₀∣𝑞₀^𝑁_𝑗=1×𝐄(𝐴𝑃[𝑞𝑞[𝑘[𝑘−−11]]])=_[2]𝑞_𝑗∣𝑞[𝑘]=𝑞_𝑖,𝑞₀=_𝑁(Φ(𝑘−1;0))^[2]∣𝑞[𝑘]=𝑞_𝑖,𝑞[𝑘−1]=𝑞_𝑗,𝑞₀_𝑗=1×𝐴𝐄^𝑗[2]𝑃(Φ𝑞(𝑘[𝑘−]1=;0𝑞)_𝑖)∣𝑞_0[2]𝑃𝑞[𝑘−1]=𝑞_𝑗∣𝑞[𝑘]=𝑞_𝑖,𝑞₀∣𝑞[𝑘]=𝑞_𝑖,𝑞[𝑘−1]=𝑞_𝑗,𝑞₀.(2.26) Because Φ(𝑘−1;0) depends on only {𝑞[𝑘−2],𝑞[𝑘−3],…,𝑞[0]} and the jump sequence {𝑞[𝑘]} is Markovian, we know 𝐄(Φ(𝑘−1;0))^[2]∣𝑞[𝑘]=𝑞_𝑖,𝑞[𝑘−1]=𝑞_𝑗,𝑞₀=𝐄(Φ(𝑘−1;0))^[2]∣𝑞[𝑘−1]=𝑞_𝑗,𝑞₀.(2.27)𝑃(𝑞[𝑘]=𝑞_𝑖∣𝑞₀)𝑃(𝑞[𝑘−1]=𝑞_𝑗∣𝑞[𝑘]=𝑞_𝑖,𝑞₀) can be computed as 𝑃=𝑞𝑃[𝑘]𝑞=[𝑘𝑞_𝑖−∣𝑞1₀]=𝑃𝑞_𝑗𝑞[𝑘−1]=𝑞_𝑗∣𝑞[𝑘]=𝑞_𝑖,𝑞₀=𝑃𝑞[𝑘]=𝑞_𝑖,𝑞[𝑘]=𝑞_𝑖∣𝑞₀=𝑃𝑞[𝑘]=𝑞_𝑖∣𝑞[𝑘−1]=𝑞_𝑗,𝑞₀𝑃𝑞[𝑘−1]=𝑞_𝑗∣𝑞₀=𝑞_𝑗𝑖∣𝑞[𝑘−1]=𝑞_𝑗𝑃𝑞[𝑘−1]=𝑞_𝑗∣𝑞₀𝑃𝑞[𝑘−1]=𝑞_𝑗∣𝑞₀.(2.28) Substituting (2.27) and (2.28) into the expression of Φ_𝑖[𝑘], we get Φ_𝑖[𝑘]=^𝑁_𝑗=1𝑞_𝑗𝑖𝐴^𝑗[2]Φ_𝑗[𝑘−1].(2.29) After combining Φ_𝑖[𝑘](𝑖=1,2,…,𝑁) into 𝑉_Φ[𝑘] as (2.10), we get 𝑉_Φ[𝑘]=𝐴_[2]𝑉_Φ[𝑘−1].(2.30) We can trivially get 𝑉_Φ[0] from Φ_𝑖[0] by (2.10).

Proof of Necessity of Theorem 2.1. By Lemma 2.3, we get lim_𝑘→∞𝑉_Φ[𝑘]=0.(2.31) By Lemma 2.4, we get 𝑉_Φ[𝑘]=𝐴^𝑘[2]𝑉_Φ[0] and 𝑉_Φ[0]=𝑝^𝑇⊗𝐼_𝑛2. Therefore, (2.31) yields lim_𝑘→∞𝐴^𝑘[2]𝑝^𝑇⊗𝐼_𝑛2=0,(2.32) for any 𝑝 (the initial distribution of 𝑞₀).
𝐴^𝑘[2] is an 𝑁𝑛²×𝑁𝑛² matrix. We can write 𝐴^𝑘[2] as 𝐴^𝑘[2]=[𝐴¹(𝑘),𝐴²(𝑘),…,𝐴^𝑁(𝑘)] where 𝐴^𝑖(𝑛)(𝑖=1,…,𝑁) is an 𝑁𝑛²×𝑛² matrix. By taking 𝑝_𝑖=1 and 𝑝_𝑗=0(𝑗=1,…,𝑖−1,𝑖+1,…,𝑁), (2.32) yields lim_𝑘→∞𝐴^𝑖(𝑘)=0.(2.33) Thus we can get lim_𝑘→∞𝐴^𝑘[2]=0.(2.34) So 𝐴_[2] is Schur stable. The proof is completed.

3. Conclusion

This paper presents a necessary and sufficient condition for the second moment stability of a discrete-time Markovian jump linear system. Specifically this paper provides proof for the necessity part. Different from the previous necessity proof, this paper confines the state domain to the real space. It investigates the structures of relevant matrices and make a good use of the essential properties of Markov jump linear systems, which may guide the future research on such systems.

Acknowledgment

This work was supported in part by the National Natural Science Foundation of China (60904012), the Program for New Century Excellent Talents in University (NCET-10-0917) and the Doctoral Fund of Ministry of Education of China (20093402120017).