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Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems
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Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systemsR Chaoxu Guan a, Zhongyang Fei a,∗, Ting Yang b,c a Research
Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, China b School of Automation, Northwestern Polytechnical University, Xi’an 710072, China c State Key Laboratory of Robotics and System (HIT,SKLRS-2018-KF-13), Harbin Institute of Technology, Harbin 150001, China Received 25 May 2018; received in revised form 17 November 2018; accepted 28 November 2018 Available online xxx
Abstract This paper is concerned with asynchronous stabilization for a class of discrete-time Markovian jump systems. The mode of designed controller is considered to be not perfectly synchronous with the activated mode of the Markovian jump system. In order to achieve the asymptotic stability with asynchronous controller, a conditional probability is introduced to describe the asynchronism of system and controller modes, which is dependent on the active system mode. Besides, due to the difficulty in acquiring all the mode transition probabilities in practice, the transition probabilities of the Markovian jump system and the controllers are supposed to be partially unknown. A necessary and sufficient condition is developed to guarantee the stochastic stability of the resultant closed-loop system and further extended to asynchronous stabilization with partially known transition probabilities. Finally, the effectiveness and advantages of the proposed methods are demonstrated by two illustrative examples. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
R This research is partially supported by the National Science Foundation of China (61873310), Heilongjiang Science Foundation (F2018015), China Postdoctoral Science Foundation (No. 2018M633576), Natural Science Basic Research Plan in Shanxi Province of China (No. 2018JQ6008) and the 111 Project (No. B16014). ∗ Corresponding author. E-mail addresses: [email protected] (C. Guan), [email protected] (Z. Fei), [email protected] (T. Yang).
https://doi.org/10.1016/j.jfranklin.2018.11.025 0016-0032/© 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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1. Introduction In the past few decades, Markovian jump systems have drawn wide attentions from the literature, since its potential applications in communication networks, mechanical engineering, biomedicine and so on. Though the systems are relatively simple, as a special kind of hybrid systems, it is powerful in modeling and describing the stochastic abrupt changes in practical problems. Markovian jump systems have been studied excessively in recent years and a lot of relevant work has been done in the literature [5,9,15,18,22,23,29–31]. To mention a few, the stability and filtering were investigated for a class of Markovian jump linear systems with unknown nonlinearities by employing a Riccati equation method [20]. Wu et al. focused on the Markovian jump singular systems and designed an observer-based sliding-mode controller to constrain state trajectory on an expected sliding-mode surface [26]. An adjoint approach was proposed to study the robust analysis and control for Markovian jump linear systems, which was proved to be more efficient in controller design than small-gain theorem [25]. However, most of the previous results are based on the assumption that mode transition information of the Markov chain is perfectly known, which may be impractical since the complexity and uncertainty in real applications. To overcome this limitation, Zhang and Lam made efforts in the analysis and synthesis of continuous-time and discrete-time Markovian jump systems with partially known transition rates or probabilities [32]. Then various problems were successively considered for Markovian jump systems with partly unknown, uncertain, and generally uncertain transition descriptions [6,8,12,13,16]. As for the synthesis of Markovian jump systems, such as controller design, early researchers usually took advantage of mode-independent design method. Though it is simple for controller design, it will inevitably lead to conservatism since the system dynamic maybe quite different among the subsystems. At present, researchers tend to adopt mode-dependent Lyapunov methods to study the analysis and synthesis of Markovian jump systems. However, this kind of method often requires accurate synchronism between system and controller modes, which is difficult to guarantee, since sometimes the system mode information maybe not accessible due to inevitable device failures or external disturbance in practice. As a result, it is important and necessary to study the asynchronous phenomenon for the synthesis of Markovian jump systems. Recently, a method aimed at asynchronous filtering for discrete-time Markovian jump systems was established by virtue of piecewise homogeneous Markov chain [28]. Furthermore, passivity-based asynchronous control was discussed with the controller mode varying upon the system mode subject to a certain conditional probability [27]. Some related results were also obtained for the nonsynchronous filtering of Markovian jump linear systems, Markovian jump neural networks and Markovian jump Lur’e systems with time-varying delays, missing measurements, redundant channels and so forth [7,33]. However, there is still much room left for improvement on the asynchronous stabilization of Markovian jump systems with partially unknown transition probabilities since the difficulties in dealing with both the asynchronism and unavailable transition probabilities, which inspires the current study. Motivated by the discussions above, this work is devoted to the asynchronous stabilization for discrete-time Markovian jump linear system with partially unknown transition probabilities. Firstly, considering the asynchronism existing in the system control process, we use a conditional probability to represent the relationship of the controller mode in the activated system mode. Secondly, since it is difficult to exactly acquire the system transition probabilities, the system’s Markov chain as well as the controller’s conditional probabilities is assumed to be partially unknown. It should be pointed out that such an assumption has more generality Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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and practicability than previous work [12,17,24], in which only the system’s Markov chain or the controller’s conditional probability is assumed to be partially unknown. Moreover, different from most existing work, where only sufficient conditions are provided for the stability or stabilization of Markovian jump linear system, a necessary and sufficient condition is proposed in this paper to guarantee the stochastic stability of the underlying closed-loop system. Based on this, a state feedback controller is further designed and thus, the asynchronous stabilization is achieved for Markovian jump linear system with partially unknown transition probabilities. In the end, a numerical example, together with a DC motor device, is provided to illustrate the merits of this paper. The rest of this paper is organized as follows. The system description and some fundamental definitions and lemmas are formulated in Section 2. The main results of this work, including necessary and sufficient conditions for stability analysis and synthesis, are elaborated and demonstrated in Section 3. Next, Section 4 provides illustrative examples to verify the effectiveness of the proposed methods in this paper. The conclusions are drawn in Section 5. Notations: The notations throughout the paper are quite standard. Rn represents the n-dimensional Euclidean space. Matrix P is positive or negative definite if it is notated that P > 0 or P < 0. I and 0 denote the identity matrix and zero matrix with proper dimension, respectively. The superscript ‘T’ stands for the transposition of a matrix. For a matrix γ , γ ⊥ denotes a basis for the null-space of γ . The operation E{ · } refers to the mathematical expectation. A matrix is deemed to be compatible for algebraic operations if its dimension is not specially stated. 2. Problem formulation and preliminaries Consider the following discrete-time Markovian jump linear system defined on a probability space (, F, P ): x (k + 1 ) = A(σ (k ) )x (k ) + B (σ (k ) )u (k ),
(1)
where x(k) ∈ Rn is the system state and u(k) ∈ Rm is the control input. {σ (k), k ≥ 0} is a Markov stochastic process taking values in a finite set L = {1, 2, . . . , N }, where N is a positive integer representing the number of the system modes. The transition probability matrix = πi j N×N is given by: Pr {σ (k + 1 ) = j|σ (k ) = i} = πi j , where 0 ≤ πi j ≤ 1, ∀i, j ∈ L and
N
πi j = 1.
j=1
A(σ (k)), B(σ (k)) are known system matrices with appropriate dimensions and abbreviated as Ai , Bi for each σ (k ) = i ∈ L. In this work, the asynchronous controller of system (1) is established in the following form: u(k) = K (η(k ))x(k ),
(2)
where K(η(k)) is the controller gain to be designed. η(k) is the signal governing the switching behavior of controller modes and taking values in another finite set I = {1, 2, . . . , M}, where positive integer M stands for the number of the controller modes. For a certain system mode σ (k) = i ∈ L, the dwell probability of mth (m ∈ I ) controller mode is defined by the Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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conditional probability [3,27]: Pr {η(k ) = m|σ (k ) = i} = vim , (3) M where 0 ≤ vim ≤ 1 and m=1 vim = 1, ∀i ∈ L. The conditional probability matrix is denoted as V = [vim ]N×M . Likewise, when η(k) = m ∈ I, matrix K(η(k)) is written as Km for short. By substituting controller Eq. (2) into system (1), the corresponding closed-loop system is as below x (k + 1 ) = A(σ (k ), η(k ) )x (k ),
(4)
where A(σ (k ), η(k ) ) = A(σ (k ) ) + B (σ (k ) )K (η(k ) ). Remark 1. A signal η(k) is introduced in this paper to describe the mode transition of controllers, which differs from the Markov chain σ (k) of the system, but depends on the system mode. Note that if we choose the finite set I = {1}, which means that only one controller is available for all the subsystems, then the designed controller Eq. (2) degrades into a modeindependent one. If I = L and vim = 1, ∀i = m, then we will obtain the synchronous result. On the other hand, the transition probability matrices and V are supposed to be partially unknown and the two matrices may be expressed as ⎡ ⎤ ⎡ ⎤ π11 πˆ 12 · · · π1N v11 v12 · · · vˆ1M ⎢ πˆ 21 πˆ 22 · · · π2N ⎥ ⎢ v21 vˆ22 · · · v2M ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥, ⎢ ⎥, . . . . ⎣ ⎦ ⎣ ⎦ . . πN1
πN2
···
πˆ NN
vˆN1
vˆN2
···
vNM
where πˆ i j , vˆim (i, j ∈ L, m ∈ I ) denote the unknown matrix elements. In the following, we give some denotations for the latter use. Lik = j : πi j is known , Liuk = j : πi j is unknown , Iik = {m : vim is known}, Iiuk = {m : vim is unknown},
ik = πi j , Vki = vim . j∈Lik
m∈Iik
Definition 1 [32]. System (4) is said to be stochastically stable if for any initial value (x0 , σ 0 , η0 ), the following condition holds ∞
2 x (k ) |x0 , σ0 , η0 < ∞. E k=0
Lemma 1. System (4) is stochastically stable if and only if for any given positive definite matrices Qi , i ∈ L, there exist positive definite matrices Pi , i ∈ L, such that ⎛ ⎞
vim ATi,m ⎝ πi j Pj ⎠Ai,m − Pi = −Qi (5) m∈I
j∈L
holds for any i ∈ L, where Ai,m = Ai + Bi Km . Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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Proof. At first, we prove the sufficiency. The Lyapunov function is chosen as V (x (k ), σ (k ) ) = x T (k )P (σ (k ) )x (k ). Let E {V (k )} = E {V (x (k + 1 ), σ (k + 1 ) )|x (k ), σ (k ) = i} − V (x (k ), i ) and then we have ⎧ ⎫ ⎛ ⎞ ⎨ ⎬
E {V (k )} = E x T (k + 1 )⎝ πi j Pj ⎠x (k + 1 ) − x T (k )Pi x (k ) ⎩ ⎭ j∈L ⎧ ⎫ ⎛ ⎞ ⎨ ⎬
= x T (k ) vim ATi,m ⎝ πi j Pj ⎠Ai,m − Pi x (k ) = −x T (k )Qi x (k ). ⎩ ⎭ m∈I
j∈L
By following the same deduction process as [2], the stochastic stability of the system is derived and the sufficiency is guaranteed. Now, we give the proof of necessity. Suppose the system (4) is stochastically stable, which means that ∞
2 x (k ) |x0 , σ0 , η0 < ∞ E k=0
holds for any initial conditions. For given positive definite matrices Q (σ (n ) ), σ (n ) ∈ L, define N
T T x (k )P (N − k , σ (k ) )x (k ) = E x (n )Q (σ (n ) )x (n )|x (k ), σ (k ) , n=k
where N is a positive integer, P (N − k , σ (k ) ) is a positive definite matrix-valued function of N. From the positive definiteness of Q(σ (n)) and the stochastic stability of system (4), we know that P (N − k , σ (k ) ) is nondecreasing of N and bounded from above [1,2]. Accordingly, there exists one matrix P(σ (k)) such that the following relationship exists x T (k )P (σ (k ) )x (k ) = lim x T (k )P (N − k , σ (k ) )x (k ) N→∞ N
= lim E x T (n )Q (σ (n ) )x (n )|x (k ), σ (k ) , N→∞
n=k
which implies P (σ (k ) ) = lim P (N − k , σ (k ) ). N→∞
Obviously, the matrix P(σ (k)) is positive definite. For σ (k) = i ∈ L, E {x T (k )P (N − k , σ (k ) )x (k ) − x T (k + 1 )P (N − (k + 1 ), σ (k + 1 ) )x (k + 1 )|x (k ), σ (k ) = i} N
T =E { E x (n )Q (σ (n ) )x (n )|x (k ), σ (k ) = i n=k
−E
N
x (n )Q (σ (n ) )x (n )|x (k + 1 ), σ (k + 1 ) |x (k ), σ (k ) = i} = x T (k )Qi x (k ) T
n=k+1
Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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On the other hand, E {x T (k )P (N − k , σ (k ) )x (k ) − x T (k + 1 )P (N − (k + 1 ), σ (k + 1 ) )x (k + 1 )|x (k ), σ (k ) = i} ⎛ ⎞
=x T (k )P (N − k, i )x (k ) − vim x T (k )ATi,m ⎝ πi j P (N − (k + 1 ), j )⎠Ai,m x (k ) m∈I
j∈L
⎛ ⎞
=x T (k ){P (N − k, i ) − vim ATi,m ⎝ πi j P (N − (k + 1 ), j )⎠Ai,m }x (k ). m∈I
j∈L
Thus, we have ⎛ ⎞
P (N − k, i ) − vim ATi,m ⎝ πi j P (N − (k + 1 ), j)⎠Ai,m = Qi . m∈I
j∈L
Let N − k → ∞, ⎛ ⎞
Pi − vim ATi,m ⎝ πi j Pj ⎠Ai,m = Qi , m∈I
j∈L
which finishes the proof of necessity. It is straightforward from Lemma 1 that an LMI-based result could be derived as shown in Lemma 2, which is much easier to check compared with Lemma 1. Lemma 2. System (4) is stochastically stable if and only if there exist positive definite matrices Pi , i ∈ L, such that ⎛ ⎞
vim ATi,m ⎝ πi j Pj ⎠Ai,m − Pi < 0 (6) m∈I
j∈L
holds for any i ∈ L. Lemma 3 (Finsler’s Lemma) [4]. Let ς ∈ Rn , = T ∈ Rn×n , and γ ∈ Rm×n such that rank(γ ) < n. The following statements are equivalent: (i) ς T ς < 0, ∀γ ς = 0, ς = 0; (ii) (γ ⊥ )T γ ⊥ < 0; (iii) ∃X ∈ Rn×m : + X γ + γ T X T < 0. 3. Criteria for asynchronous control In this section, we will investigate the stochastic stability and asynchronous stabilization for the system with partially unknown transition probability matrices. In the first place, we construct a necessary and sufficient condition to guarantee the stability of the switched system based on Lemma 2. Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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Theorem 1. For given controller Km , m ∈ I, system (4) is stochastically stable if and only if there exist matrices Pi > 0, i ∈ L, such that for any i ∈ L,
⎛ vis ATi,s ⎝
s∈Iik
+ 1
⎞
⎛ ⎞
πit Pt ⎠Ai,s + 1 − ik vis ATi,s Pj Ai,s + 1 − Vki ATi,m ⎝ πit Pt ⎠Ai,m
t∈Lik
− Vki
s∈Iik
1−
ik
T Ai,m Pj Ai,m − Pi < 0, ∀ j ∈ Liuk , m ∈ Iiuk .
t∈Lik
(7)
Proof. First, we introduce a notation as ⎛ ⎞ ⎛ ⎞
i = vim ATi,m ⎝ πi j Pj ⎠Ai,m − Pi = vim ATi,m ⎝ πi j Pj ⎠Ai,m m∈I
+
j∈L
⎛
vim ATi,m ⎝
m∈Iik
+
⎞ πˆ i j Pj ⎠Ai,m +
j∈Liuk
⎛ vˆim ATi,m ⎝
m∈Iiuk
⎞
m∈Iik
⎛
vˆim ATi,m ⎝
m∈Iiuk
j∈Lik
⎞
πi j Pj ⎠Ai,m
j∈Lik
πˆ i j Pj ⎠Ai,m − Pi .
j∈Liuk
It is obvious from Lemma 2 that the stochastic stability of system (4) is guaranteed if and only if i < 0 for any i ∈ L. Notice that
⎛ vim ATi,m ⎝
m∈Iik
⎞ πˆ i j Pj ⎠Ai,m
j∈Liuk
⎛ vˆim ATi,m ⎝
m∈Iiuk
⎛
= 1 − ik vim ATi,m ⎝ m∈Iik
⎞
πi j Pj ⎠Ai,m = 1 − Vki
j∈Lik
m∈Iiuk
⎞ πˆ i j P ⎠Ai,m , i j 1 − k j∈Liuk ⎛ ⎞
AT ⎝ πi j Pj ⎠Ai,m , i i,m
vˆim 1 − Vk
j∈Lik
and
⎛ vˆim ATi,m ⎝
m∈Iiuk
⎞ πˆ i j Pj ⎠Ai,m
= 1 − Vki 1 − ik
j∈Liuk
m∈Iiuk
⎛ ⎞
πˆ i j vˆim AT ⎝ Pj ⎠Ai,m , 1 − Vki i,m 1 − ik i j∈Luk
where 0≤
πˆ i j πˆ i j ≤ 1, j ∈ Liuk , = 1, i 1 − k 1 − ik i j∈Luk
and 0≤
vˆim vˆim ≤ 1, m ∈ Iiuk , = 1. i 1 − Vk 1 − Vki i m∈Iuk
Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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Then
⎛ ⎞
i = vim ATi,m ⎝ πi j Pj ⎠Ai,m − Pi =
⎛ ⎞
vˆim πˆ i j vis ATi,s ⎝ πit Pt ⎠ i i 1 − V 1 − k j∈Li k s∈Ii m∈I j∈L m∈Iiuk t∈Lik uk k ⎛ ⎞
× Ai,s + 1 − ik vis ATi,s Pj Ai,s + 1 − Vki ATi,m ⎝ πit Pt ⎠Ai,m s∈Iik
+ 1 − Vki 1 − ik ATi,m Pj Ai,m − Pi
t∈Lik
which yields that the system (4) is stochastically stable if and only if Eq. (7) is satisfied. Theorem 2. The following two propositions are equivalent: (i) There exist matrices Pi > 0, i ∈ L, such that for any i ∈ L, condition (7) holds; (ii) There exist matrices Pi > 0, Ri,m, j > 0, i ∈ L, m ∈ I, j ∈ Liuk , such that for any i ∈ L, the following inequalities hold:
vis Ri,s, j + 1 − Vki Ri,m, j − Pi < 0, ∀ j ∈ Liuk , m ∈ Iiuk , (8) s∈Iik
⎛ ATi,m ⎝
⎞
πit Pt ⎠Ai,m + 1 − ik ATi,m Pj Ai,m − Ri,m, j < 0, ∀ j ∈ Liuk , m ∈ I.
(9)
t∈Lik
Proof. (i)⇒ (ii) According to Eq. (7), we know that there will exist a positive scalar ε such that for any i ∈ L, ⎛ ⎛ ⎞ ⎞
vis ⎝ATi,s ⎝ πit Pt ⎠Ai,s + 1 − ik ATi,s Pj Ai,s + εI ⎠ s∈Iik
t∈Lik
⎛ ⎛ ⎞ ⎞
+ 1 − Vki ⎝ATi,m ⎝ πit Pt ⎠Ai,m + 1 − ik ATi,m Pj Ai,m + εI ⎠ − Pi < 0, ∀ j ∈ Liuk , m ∈ Iiuk , t∈Lik
holds. For any j ∈ Liuk , m ∈ I, we define ⎛ ⎞
Ri,m, j = ATi,m ⎝ πit Pt ⎠Ai,m + 1 − ik ATi,m Pj Ai,m + εI , t∈Lik
and then Eq. (8) is straightforwardly guaranteed. In addition, it is clear to see that the condition (9) is satisfied from the definition of Ri, m, j . (ii)⇒ (i) For any j ∈ Liuk , m ∈ I, Eq. (9) means that ⎛ ⎞
ATi,m ⎝ πit Pt ⎠Ai,m + 1 − ik ATi,m Pj Ai,m < Ri,m, j . t∈Lik
By substituting the inequality above into Eq. (8), we can easily get the result of (i). Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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In the following, we are in the position to design the controller Eq. (2) for Markovian jump system (1) based on Theorem 2. Here we rewrite the denotation Lik = {k1 , . . . , kmi } for the use of Theorem 3. Theorem 3. There exists a state feedback controller Eq. (2) such that the closed-loop system (4) is stochastically stable if and only if there exist matrices Pi > 0, Ri,m, j > 0, Km , Gm , i ∈ L, m ∈ I, j ∈ Liuk , such that for any i ∈ L, inequality (8) and SiT i,m, j Si + Gm Hm + HmT GTm < 0, ∀ j ∈ Liuk , m ∈ I,
(10)
hold, where
i,m, j =diag πik1 Pk1 , . . . , πikmi Pkmi , 1 − ik Pj , −Ri,m, j , ⎡ ⎤ Ai Bi ⎢ .. ⎥ ⎢ ⎥ Si =⎢ . ⎥, Hm = Km −I . ⎣A i B i ⎦ I 0 Proof. Since the equivalence of the two statements in Theorem 2, we know that the system is stochastically stable if and only if there exist matrices Pi > 0, Ri,m, j > 0, Km , Gm , i ∈ L, m ∈ I, j ∈ Liuk , such that for any i ∈ L, inequalities (8) and (9) hold. As a consequence, we only need to prove the equivalence of (9) and (10) and then the result in Theorem 3 is obtained. By substituting Ai + Bi Km for the matrix Ai, m in Eq. (9) and defining ⎡ ⎤ Ai + Bi Km ! ⎢ ⎥ .. I ⎢ ⎥ . , Wi,m = ⎢ , T = ⎥ m Km ⎣Ai + Bi Km ⎦ I we obtain
⎛
i,m, j = − Ri,m, j + ATi,m ⎝
⎞
πit Pt ⎠Ai,m + 1 − ik ATi,m Pj Ai,m
t∈Lik T =Wi,m i,m, j Wi,m
=TmT SiT i,m, j Si Tm .
(11)
Due to Hm Tm = 0, by using Lemma 3, we can derive that i, m, j < 0 holds if and only if there exists one matrix Gm such that Eq. (10) holds. Hence the system (1) can be stabilized by controller Eq. (2) if and only if the conditions of Theorem 3 hold, which completes the proof. If we fix the set I = {1}, then the mode-independent stabilization is straightforwardly derived as shown in Corollary 1 below. Corollary 1. There exists a state feedback controller Eq. (2) such that the closed-loop system (4) is stochastically stable if and only if there exist matrices Pi > 0, Ri, j > 0, K, G, i ∈ L, j ∈ Liuk , such that for any i ∈ L, j ∈ Liuk , the following inequalities hold Ri, j − Pi < 0, ˆ i, j Si + GH + H T GT < 0, SiT Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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where
i, j =diag πik1 Pk1 , . . . , πikmi Pkmi , 1 − ik Pj , −Ri, j , H = K −I . Remark 2. Observing the inequality (9) in Theorem 2, one can find that the controller parameters are coupled with the Lyapunov matrices Pj , which makes it intractable to solve the controller gains. In order to tackle this problem, we separate system mode (Ai , Bi ) and controller mode Km in the proof of Theorem 3 and then make use of Lemma 3 to decouple the system and controller modes. It is worth pointing out that the decoupling method in this paper is different from that used in [27], which applies the inequality −1 −GTφ R¯ iφ Gφ ≤ R¯ iφ − GTφ − Gφ .
(12)
The relaxation in Eq. (12) will lead to a sufficient condition, however, with the decoupling method proposed in this paper, the criterion in Theorem 3 is still a necessary and sufficient one. Although Ref. [32] also presents necessary and sufficient conditions for analysis and synthesis of Markovian jump systems with incomplete transition probabilities, it does not discuss the influence of mode asynchronism and actually, its main result Theorem 3 could be treated as a special case of Theorem 1 in this paper if we set I = L and vim = 1, ∀i = m. Remark 3. This paper is dedicated to the asynchronous control design for discrete-time Markovian jump linear system with partially unknown transition probabilities. By utilizing a conditional probability technique, the asynchronism between system and controller modes is described and novel criteria are obtained for the stability and stabilization even though the transition probability information of both Markovian subsystems and controllers is not perfectly acquired. Note that the proposed method could be extended to some related topics, such as semi-Markovian jump system or hidden Markovian jump system, to exploit their relaxed limitations on the probability distributions for broader applications [10,11], which would be our further research directions.
4. Illustrative examples In this section, we will use two examples to verify the validity and superiority of the proposed results in the above section. The first example is to show the effectiveness of the designed method with asynchronism and partially unknown transition probabilities. Example 1. Consider discrete-time Markovian jump linear system (1) with the following four subsystems: ! ! 1.0 −1.25 0.5 −0.83 , A2 = , A1 = 2.5 2.5 2.5 3.5 ! ! 0.25 −0.25 0.75 −0.57 , A4 = , A3 = 2.5 3.0 2.5 2.75 ! 0.5 , i = 1, . . . 4. Bi = 0.1 Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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Fig. 1. Different conditional probability matrices V. Table 1 Designed controller gains under different cases. K1 = −6.2617 −2.6213 K1 = −6.1113 K2 = −5.7216 −5.3334 K2 = −5.5108 K3 = −4.6721 −4.8625 K3 = −5.0027 K4 = −5.9916 −4.3205 K4 = −5.7056
−2.4961 −5.1908 −4.9535 −4.0413
Case I K1 = −5.9547 K2 = −5.4187 K3 = −5.3747 K4 = −5.4834
−2.2941 −5.1363 −5.7631 −3.5456
Case III
−2.3854 −5.2359 −5.4131 −3.8249
Case II K1 = −5.8499 K2 = −5.2239 K3 = −5.7456 K4 = −5.2089 Case IV
The transition probability matrix = [πi j ]4×4 is assumed as below ⎡ ⎤ 0.3 0.2 0.1 0.4 ⎢πˆ 21 0.2 0.3 πˆ 24 ⎥ ⎥ =⎢ ⎣πˆ 31 πˆ 32 0.5 0.3 ⎦. 0.2 0.2 0.1 0.5 According to the condition in Theorems 1 or 3 in [32], we can easily check that the uncontrolled switched system is not stochastically stable. Now we will utilize Theorem 3 to design state-feedback controllers to stabilize the system. Fig. 1 displays the conditional probability matrix V of different asynchronous cases and Table 1 lists the obtained controller gains based on different cases in Fig. 1. Case I corresponds to the result of synchronous control, while Case II, III, IV reflect the system with different asynchronous degree. The state trajectory of the open-loop system is drawn in Fig. 2 with the initial value x(0) = [0.1, −0.1]T , and Fig. 3 shows the state trajectory of the closed-loop system under asynchronous controller of Case IV. The unknown elements in and V are chosen as random scalars in this simulation. Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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Fig. 2. Open-loop system state trajectory of Example 1.
Fig. 3. Closed-loop system state trajectory of Example 1.
The switching signals of the system and controller modes are displayed in Fig. 4. From these figures, we know that the Markovian jump linear system is stabilized by the designed controllers although there exist asynchronous phenomenon and unknown transition probabilities, which verifies the usefulness and effectiveness of the result in Theorem 3. Example 2. This example focuses on the velocity control issue for a DC motor device, where abrupt failures may happen [14,19,21]. Frequently, abrupt failures make the device behaviour change from normal mode to failure mode, and our purpose is to control the angular velocity of the device under such phenomenon by using the result in Theorem 3. Considering the DC motor testbed with a nonzero constant plus a sinusoid reference, the main idea is to compel Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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Fig. 4. Switching signals of system and controller modes. Table 2 Parameter values of the DC motor device model.
i a11 i a12 i a21 i a22 i a31 i a33 bi1 bi2
i=1
i=2
i=3
−0.4799 5.1546 −3.8162 14.4723 0.1399 −0.9255 5.8705 15.5010
−1.6026 9.1632 −0.5918 3.0317 0.0740 −0.4338 10.2851 2.2282
0.6346 0.9178 −0.5056 2.4811 0.3865 0.0982 0.7874 1.5302
the DC motor to track the sinusoid reference with null steady-state error. By referring to [19], a discrete-time Markovian jump system is established to model the DC motor device, which is in the form of Eq. (1) with the following parameters: ⎡ i ⎤ ⎡ i⎤ i a11 a12 0 b1 i i a22 0 ⎦, Bi = ⎣bi2 ⎦, Ai = ⎣a21 i i a31 0 a33 0 where i ∈ {1, 2, 3} and the numerical values of the symbols above are exhibited in Table 2. The transition probability matrix is presented as ⎡ ⎤ 0.9 πˆ 12 πˆ 13 = ⎣0.36 0.6 0.04⎦. πˆ 31 0.1 πˆ 33 The conditional probability matrix V is given by ⎡ ⎤ 0.7 0.1 0.2 V = ⎣0.1 0.8 0.1⎦. vˆ31 vˆ32 0.6 Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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Table 3 Designed controller gains by Theorem 3. Theorem 3
Controller gains K1 = 0.2258 −0.9182 K2 = 0.1771 −0.9102 K3 = 0.2194 −0.9214
−0.0534 −0.0115 −0.0362
Fig. 5. Closed-loop system state trajectory of Example 2.
Fig. 6. Switching signals of system and controller modes.
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It is easy to check that the uncontrolled switched system is not stable. We would like to design controllers for the model by utilizing Theorem 3 and the corresponding controller gains are shown in Table 3. Figs. 5 and 6 respectively depict the state trajectory of the closed-loop system and the switching signals of both system and controller modes. Each subsystem could be matched with the designed three controllers and the switching of the controller modes is more random. Though there exists great asynchronism between the system and controller modes, the stability is still achieved by the designed controllers, which further illustrates the advantages of the proposed method in this paper. 5. Conclusions In this work, we focus on the stability and asynchronous stabilization for a class of discretetime Markovian jump systems. The asynchronous phenomenon between system and controller modes is considered and investigated by introducing a signal to govern the switching of controller modes, which is closely related to the system mode. The transition probability matrices of both system and controller modes are assumed with partially unknown elements. A necessary and sufficient criterion is developed for the asymptotic stability of the Markovian jump system and a controller is further designed which will tolerate the asynchronism between system and controller modes. Finally, two examples are utilized to verify the validity and superiority of the newly obtained results. References [1] E.K. Boukas, Stochastic Switching Systems: Analysis and Design, Birkhauser, Basel, Berlin, 2005. [2] E.K. Boukas, H. Yang, Stability of discrete-time linear systems with Markovian jumping parameters, Math. Control Sig. Syst. 8 (4) (1995) 390–402. [3] O.L.V. Costa, M.D. Fragoso, M. Todorov, A detector-based approach for the H2 control of Markov jump linear systems with partial information, IEEE Trans. Autom. Control 60 (5) (2015) 1219–1234. [4] M.C. de Oliveira, R.E. Skelton, Stability Tests for Constrained Linear Systems, Springer, Berlin, 2001. 268, 241–257 [5] Z. Feng, P. Shi, Sliding mode control of singular stochastic Markov jump systems, IEEE Trans. Autom. Control 62 (8) (2017) 4266–4273. [6] Z. Feng, W.X. Zheng, On reachable set estimation of delay Markovian jump systems with partially known transition probabilities, J. Frankl. Inst. 353 (15) (2016) 3835–3856. [7] X. Ge, Q.L. Han, Distributed sampled-data asynchronous H∞ filtering of Markovian jump linear systems over sensor networks, Signal Process. 127 (2016) 86–99. [8] I. Ghous, Z. Xiang, H.R. Karimi, H∞ control of 2-D continuous Markovian jump delayed systems with partially unknown transition probabilities, Inf. Sci. 382 (2017) 274–291. [9] C. Guan, Z. Fei, Z. Li, Y. Xu, Improved H∞ filter design for discrete-time Markovian jump systems with time-varying delay, J. Frankl. Inst. 353 (16) (2016) 4156–4175. [10] B. Jiang, Y. Kao, C. Gao, X. Yao, Passification of uncertain singular semi-Markovian jump systems with actuator failures via sliding mode approach, IEEE Trans. Autom. Control 62 (8) (2017) 4138–4143. [11] B. Jiang, Y. Kao, H.R. Karimi, C. Gao, Stability and stabilization for singular switching semi-Markovian jump systems with generally uncertain transition rates, IEEE Trans. Autom. Control 63 (11) (2018) 3919–3926. [12] Y. Kao, J. Xie, C. Wang, Stabilization of singular Markovian jump systems with generally uncertain transition rates, IEEE Trans. Autom. Control 59 (9) (2014) 2604–2610. [13] Y. Kao, T. Yang, J.H. Park, Exponential stability of switched Markovian jumping neutral-type systems with generally incomplete transition rates, Int. J. Robust Nonlinear Control 28 (5) (2018) 1583–1596. [14] S.H. Kim, H2 control of Markovian jump LPV systems with measurement noises: application to a DC-motor device with voltage fluctuations, J. Frankl. Inst. 354 (4) (2017) 1784–1800. [15] F. Li, P. Shi, L. Wu, M.V. Basin, C.C. Lim, Quantized control design for cognitive radio networks modeled as nonlinear semi-Markovian jump systems, IEEE Trans. Ind. Electron. 62 (4) (2015) 2330–2340. Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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[16] X. Li, J. Lam, H. Gao, J. Xiong, H∞ and H2 filtering for linear systems with uncertain Markov transitions, Automatica 67 (2016) 252–266. [17] Z. Li, Z. Fei, H.R. Karimi, New results on stability analysis and stabilization of time-delay continuous Markovian jump systems with partially known rates matrix, Int. J. Robust Nonlinear Control 26 (9) (2016) 1873–1887. [18] Z. Li, Z. Yu, H. Zhao, Further result on H∞ filter design for continuous-time Markovian jump systems with time-varying delay, J. Frankl. Inst. 351 (9) (2014) 4619–4635. [19] R.C.L.F. Oliveira, A.N. Vargas, J.B.R. do Val, P.L.D. Peres, Mode-independent H2 control of a DC motor modeled as a Markov jump linear system, IEEE Trans. Control Syst. Technol. 22 (5) (2014) 1915–1919. [20] P. Shi, M. Karan, C.Y. Kaya, Robust Kalman filter design for Markovian jump linear systems with norm-bounded unknown nonlinearities, Circuits Syst. Signal Process. 24 (2) (2005) 135–150. [21] Y. Shi, J. Huang, B. Yu, Robust tracking control of networked control systems: application to a networked DC motor, IEEE Trans. Industrial Electronics 60 (12) (2013) 5864–5874. [22] J. Song, Y. Niu, J. Lam, Z. Shu, A hybrid design approach for output feedback exponential stabilization of Markovian jump systems, IEEE Trans. Autom. Control 63 (5) (2018) 1404–1417. [23] J. Song, Y. Niu, Y. Zou, Asynchronous output feedback control of time-varying Markovian jump systems within a finite-time interval, J. Frankl. Inst. 354 (15) (2017) 6747–6765. [24] J. Song, Y. Niu, Y. Zou, Asynchronous sliding mode control of Markovian jump systems with time-varying delays and partly accessible mode detection probabilities, Automatica 93 (2018) 33–41. [25] M.G. Todorov, M.D. Fragoso, A new perspective on the robustness of Markov jump linear systems, Automatica 49 (3) (2013) 735–747. [26] L. Wu, P. Shi, H. Gao, State estimation and sliding-mode control of Markovian jump singular systems, IEEE Trans. Autom. Control 55 (5) (2010) 1213–1219. [27] Z.G. Wu, P. Shi, Z. Shu, H. Su, R. Lu, Passivity-based asynchronous control for Markov jump systems, IEEE Trans. Autom. Control 62 (4) (2017) 2020–2025. [28] Z.G. Wu, P. Shi, H. Su, J. Chu, Asynchronous l2 − l∞ filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities, Automatica 50 (1) (2014) 180–186. [29] R. Yang, G.P. Liu, P. Shi, C. Thomas, M.V. Basin, Predictive output feedback control for networked control systems, IEEE Trans. Ind. Electron. 61 (1) (2014) 512–520. [30] T. Yang, L. Zhang, V. Sreeram, A.N. Vargas, T. Hayat, B. Ahmad, Time-varying filter design for semi-Markov jump linear systems with intermittent transmission, Int. J. Robust Nonlinear Control 27 (17) (2017) 4035–4049. [31] T. Yang, L. Zhang, Y. Li, Y. Leng, Stabilisation of Markov jump linear systems subject to both state and mode detection delays, IET Control Theory Appl. 8 (4) (2014) 260–266. [32] L. Zhang, J. Lam, Necessary and sufficient conditions for analysis and synthesis of Markov jump linear systems with incomplete transition descriptions, IEEE Trans. Autom. Control 55 (7) (2010) 1695–1701. [33] Y. Zhu, L. Zhang, W.X. Zheng, Distributed H∞ filtering for a class of discrete-time Markov jump Lur’e systems with redundant channels, IEEE Trans. Ind. Electron. 63 (3) (2016) 1876–1885.
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Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systemsR Chaoxu Guan a, Zhongyang Fei a,∗, Ting Yang b,c a Research
Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, China b School of Automation, Northwestern Polytechnical University, Xi’an 710072, China c State Key Laboratory of Robotics and System (HIT,SKLRS-2018-KF-13), Harbin Institute of Technology, Harbin 150001, China Received 25 May 2018; received in revised form 17 November 2018; accepted 28 November 2018 Available online xxx
Abstract This paper is concerned with asynchronous stabilization for a class of discrete-time Markovian jump systems. The mode of designed controller is considered to be not perfectly synchronous with the activated mode of the Markovian jump system. In order to achieve the asymptotic stability with asynchronous controller, a conditional probability is introduced to describe the asynchronism of system and controller modes, which is dependent on the active system mode. Besides, due to the difficulty in acquiring all the mode transition probabilities in practice, the transition probabilities of the Markovian jump system and the controllers are supposed to be partially unknown. A necessary and sufficient condition is developed to guarantee the stochastic stability of the resultant closed-loop system and further extended to asynchronous stabilization with partially known transition probabilities. Finally, the effectiveness and advantages of the proposed methods are demonstrated by two illustrative examples. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
R This research is partially supported by the National Science Foundation of China (61873310), Heilongjiang Science Foundation (F2018015), China Postdoctoral Science Foundation (No. 2018M633576), Natural Science Basic Research Plan in Shanxi Province of China (No. 2018JQ6008) and the 111 Project (No. B16014). ∗ Corresponding author. E-mail addresses: [email protected] (C. Guan), [email protected] (Z. Fei), [email protected] (T. Yang).
https://doi.org/10.1016/j.jfranklin.2018.11.025 0016-0032/© 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
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1. Introduction In the past few decades, Markovian jump systems have drawn wide attentions from the literature, since its potential applications in communication networks, mechanical engineering, biomedicine and so on. Though the systems are relatively simple, as a special kind of hybrid systems, it is powerful in modeling and describing the stochastic abrupt changes in practical problems. Markovian jump systems have been studied excessively in recent years and a lot of relevant work has been done in the literature [5,9,15,18,22,23,29–31]. To mention a few, the stability and filtering were investigated for a class of Markovian jump linear systems with unknown nonlinearities by employing a Riccati equation method [20]. Wu et al. focused on the Markovian jump singular systems and designed an observer-based sliding-mode controller to constrain state trajectory on an expected sliding-mode surface [26]. An adjoint approach was proposed to study the robust analysis and control for Markovian jump linear systems, which was proved to be more efficient in controller design than small-gain theorem [25]. However, most of the previous results are based on the assumption that mode transition information of the Markov chain is perfectly known, which may be impractical since the complexity and uncertainty in real applications. To overcome this limitation, Zhang and Lam made efforts in the analysis and synthesis of continuous-time and discrete-time Markovian jump systems with partially known transition rates or probabilities [32]. Then various problems were successively considered for Markovian jump systems with partly unknown, uncertain, and generally uncertain transition descriptions [6,8,12,13,16]. As for the synthesis of Markovian jump systems, such as controller design, early researchers usually took advantage of mode-independent design method. Though it is simple for controller design, it will inevitably lead to conservatism since the system dynamic maybe quite different among the subsystems. At present, researchers tend to adopt mode-dependent Lyapunov methods to study the analysis and synthesis of Markovian jump systems. However, this kind of method often requires accurate synchronism between system and controller modes, which is difficult to guarantee, since sometimes the system mode information maybe not accessible due to inevitable device failures or external disturbance in practice. As a result, it is important and necessary to study the asynchronous phenomenon for the synthesis of Markovian jump systems. Recently, a method aimed at asynchronous filtering for discrete-time Markovian jump systems was established by virtue of piecewise homogeneous Markov chain [28]. Furthermore, passivity-based asynchronous control was discussed with the controller mode varying upon the system mode subject to a certain conditional probability [27]. Some related results were also obtained for the nonsynchronous filtering of Markovian jump linear systems, Markovian jump neural networks and Markovian jump Lur’e systems with time-varying delays, missing measurements, redundant channels and so forth [7,33]. However, there is still much room left for improvement on the asynchronous stabilization of Markovian jump systems with partially unknown transition probabilities since the difficulties in dealing with both the asynchronism and unavailable transition probabilities, which inspires the current study. Motivated by the discussions above, this work is devoted to the asynchronous stabilization for discrete-time Markovian jump linear system with partially unknown transition probabilities. Firstly, considering the asynchronism existing in the system control process, we use a conditional probability to represent the relationship of the controller mode in the activated system mode. Secondly, since it is difficult to exactly acquire the system transition probabilities, the system’s Markov chain as well as the controller’s conditional probabilities is assumed to be partially unknown. It should be pointed out that such an assumption has more generality Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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and practicability than previous work [12,17,24], in which only the system’s Markov chain or the controller’s conditional probability is assumed to be partially unknown. Moreover, different from most existing work, where only sufficient conditions are provided for the stability or stabilization of Markovian jump linear system, a necessary and sufficient condition is proposed in this paper to guarantee the stochastic stability of the underlying closed-loop system. Based on this, a state feedback controller is further designed and thus, the asynchronous stabilization is achieved for Markovian jump linear system with partially unknown transition probabilities. In the end, a numerical example, together with a DC motor device, is provided to illustrate the merits of this paper. The rest of this paper is organized as follows. The system description and some fundamental definitions and lemmas are formulated in Section 2. The main results of this work, including necessary and sufficient conditions for stability analysis and synthesis, are elaborated and demonstrated in Section 3. Next, Section 4 provides illustrative examples to verify the effectiveness of the proposed methods in this paper. The conclusions are drawn in Section 5. Notations: The notations throughout the paper are quite standard. Rn represents the n-dimensional Euclidean space. Matrix P is positive or negative definite if it is notated that P > 0 or P < 0. I and 0 denote the identity matrix and zero matrix with proper dimension, respectively. The superscript ‘T’ stands for the transposition of a matrix. For a matrix γ , γ ⊥ denotes a basis for the null-space of γ . The operation E{ · } refers to the mathematical expectation. A matrix is deemed to be compatible for algebraic operations if its dimension is not specially stated. 2. Problem formulation and preliminaries Consider the following discrete-time Markovian jump linear system defined on a probability space (, F, P ): x (k + 1 ) = A(σ (k ) )x (k ) + B (σ (k ) )u (k ),
(1)
where x(k) ∈ Rn is the system state and u(k) ∈ Rm is the control input. {σ (k), k ≥ 0} is a Markov stochastic process taking values in a finite set L = {1, 2, . . . , N }, where N is a positive integer representing the number of the system modes. The transition probability matrix = πi j N×N is given by: Pr {σ (k + 1 ) = j|σ (k ) = i} = πi j , where 0 ≤ πi j ≤ 1, ∀i, j ∈ L and
N
πi j = 1.
j=1
A(σ (k)), B(σ (k)) are known system matrices with appropriate dimensions and abbreviated as Ai , Bi for each σ (k ) = i ∈ L. In this work, the asynchronous controller of system (1) is established in the following form: u(k) = K (η(k ))x(k ),
(2)
where K(η(k)) is the controller gain to be designed. η(k) is the signal governing the switching behavior of controller modes and taking values in another finite set I = {1, 2, . . . , M}, where positive integer M stands for the number of the controller modes. For a certain system mode σ (k) = i ∈ L, the dwell probability of mth (m ∈ I ) controller mode is defined by the Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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conditional probability [3,27]: Pr {η(k ) = m|σ (k ) = i} = vim , (3) M where 0 ≤ vim ≤ 1 and m=1 vim = 1, ∀i ∈ L. The conditional probability matrix is denoted as V = [vim ]N×M . Likewise, when η(k) = m ∈ I, matrix K(η(k)) is written as Km for short. By substituting controller Eq. (2) into system (1), the corresponding closed-loop system is as below x (k + 1 ) = A(σ (k ), η(k ) )x (k ),
(4)
where A(σ (k ), η(k ) ) = A(σ (k ) ) + B (σ (k ) )K (η(k ) ). Remark 1. A signal η(k) is introduced in this paper to describe the mode transition of controllers, which differs from the Markov chain σ (k) of the system, but depends on the system mode. Note that if we choose the finite set I = {1}, which means that only one controller is available for all the subsystems, then the designed controller Eq. (2) degrades into a modeindependent one. If I = L and vim = 1, ∀i = m, then we will obtain the synchronous result. On the other hand, the transition probability matrices and V are supposed to be partially unknown and the two matrices may be expressed as ⎡ ⎤ ⎡ ⎤ π11 πˆ 12 · · · π1N v11 v12 · · · vˆ1M ⎢ πˆ 21 πˆ 22 · · · π2N ⎥ ⎢ v21 vˆ22 · · · v2M ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥, ⎢ ⎥, . . . . ⎣ ⎦ ⎣ ⎦ . . πN1
πN2
···
πˆ NN
vˆN1
vˆN2
···
vNM
where πˆ i j , vˆim (i, j ∈ L, m ∈ I ) denote the unknown matrix elements. In the following, we give some denotations for the latter use. Lik = j : πi j is known , Liuk = j : πi j is unknown , Iik = {m : vim is known}, Iiuk = {m : vim is unknown},
ik = πi j , Vki = vim . j∈Lik
m∈Iik
Definition 1 [32]. System (4) is said to be stochastically stable if for any initial value (x0 , σ 0 , η0 ), the following condition holds ∞
2 x (k ) |x0 , σ0 , η0 < ∞. E k=0
Lemma 1. System (4) is stochastically stable if and only if for any given positive definite matrices Qi , i ∈ L, there exist positive definite matrices Pi , i ∈ L, such that ⎛ ⎞
vim ATi,m ⎝ πi j Pj ⎠Ai,m − Pi = −Qi (5) m∈I
j∈L
holds for any i ∈ L, where Ai,m = Ai + Bi Km . Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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Proof. At first, we prove the sufficiency. The Lyapunov function is chosen as V (x (k ), σ (k ) ) = x T (k )P (σ (k ) )x (k ). Let E {V (k )} = E {V (x (k + 1 ), σ (k + 1 ) )|x (k ), σ (k ) = i} − V (x (k ), i ) and then we have ⎧ ⎫ ⎛ ⎞ ⎨ ⎬
E {V (k )} = E x T (k + 1 )⎝ πi j Pj ⎠x (k + 1 ) − x T (k )Pi x (k ) ⎩ ⎭ j∈L ⎧ ⎫ ⎛ ⎞ ⎨ ⎬
= x T (k ) vim ATi,m ⎝ πi j Pj ⎠Ai,m − Pi x (k ) = −x T (k )Qi x (k ). ⎩ ⎭ m∈I
j∈L
By following the same deduction process as [2], the stochastic stability of the system is derived and the sufficiency is guaranteed. Now, we give the proof of necessity. Suppose the system (4) is stochastically stable, which means that ∞
2 x (k ) |x0 , σ0 , η0 < ∞ E k=0
holds for any initial conditions. For given positive definite matrices Q (σ (n ) ), σ (n ) ∈ L, define N
T T x (k )P (N − k , σ (k ) )x (k ) = E x (n )Q (σ (n ) )x (n )|x (k ), σ (k ) , n=k
where N is a positive integer, P (N − k , σ (k ) ) is a positive definite matrix-valued function of N. From the positive definiteness of Q(σ (n)) and the stochastic stability of system (4), we know that P (N − k , σ (k ) ) is nondecreasing of N and bounded from above [1,2]. Accordingly, there exists one matrix P(σ (k)) such that the following relationship exists x T (k )P (σ (k ) )x (k ) = lim x T (k )P (N − k , σ (k ) )x (k ) N→∞ N
= lim E x T (n )Q (σ (n ) )x (n )|x (k ), σ (k ) , N→∞
n=k
which implies P (σ (k ) ) = lim P (N − k , σ (k ) ). N→∞
Obviously, the matrix P(σ (k)) is positive definite. For σ (k) = i ∈ L, E {x T (k )P (N − k , σ (k ) )x (k ) − x T (k + 1 )P (N − (k + 1 ), σ (k + 1 ) )x (k + 1 )|x (k ), σ (k ) = i} N
T =E { E x (n )Q (σ (n ) )x (n )|x (k ), σ (k ) = i n=k
−E
N
x (n )Q (σ (n ) )x (n )|x (k + 1 ), σ (k + 1 ) |x (k ), σ (k ) = i} = x T (k )Qi x (k ) T
n=k+1
Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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On the other hand, E {x T (k )P (N − k , σ (k ) )x (k ) − x T (k + 1 )P (N − (k + 1 ), σ (k + 1 ) )x (k + 1 )|x (k ), σ (k ) = i} ⎛ ⎞
=x T (k )P (N − k, i )x (k ) − vim x T (k )ATi,m ⎝ πi j P (N − (k + 1 ), j )⎠Ai,m x (k ) m∈I
j∈L
⎛ ⎞
=x T (k ){P (N − k, i ) − vim ATi,m ⎝ πi j P (N − (k + 1 ), j )⎠Ai,m }x (k ). m∈I
j∈L
Thus, we have ⎛ ⎞
P (N − k, i ) − vim ATi,m ⎝ πi j P (N − (k + 1 ), j)⎠Ai,m = Qi . m∈I
j∈L
Let N − k → ∞, ⎛ ⎞
Pi − vim ATi,m ⎝ πi j Pj ⎠Ai,m = Qi , m∈I
j∈L
which finishes the proof of necessity. It is straightforward from Lemma 1 that an LMI-based result could be derived as shown in Lemma 2, which is much easier to check compared with Lemma 1. Lemma 2. System (4) is stochastically stable if and only if there exist positive definite matrices Pi , i ∈ L, such that ⎛ ⎞
vim ATi,m ⎝ πi j Pj ⎠Ai,m − Pi < 0 (6) m∈I
j∈L
holds for any i ∈ L. Lemma 3 (Finsler’s Lemma) [4]. Let ς ∈ Rn , = T ∈ Rn×n , and γ ∈ Rm×n such that rank(γ ) < n. The following statements are equivalent: (i) ς T ς < 0, ∀γ ς = 0, ς = 0; (ii) (γ ⊥ )T γ ⊥ < 0; (iii) ∃X ∈ Rn×m : + X γ + γ T X T < 0. 3. Criteria for asynchronous control In this section, we will investigate the stochastic stability and asynchronous stabilization for the system with partially unknown transition probability matrices. In the first place, we construct a necessary and sufficient condition to guarantee the stability of the switched system based on Lemma 2. Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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Theorem 1. For given controller Km , m ∈ I, system (4) is stochastically stable if and only if there exist matrices Pi > 0, i ∈ L, such that for any i ∈ L,
⎛ vis ATi,s ⎝
s∈Iik
+ 1
⎞
⎛ ⎞
πit Pt ⎠Ai,s + 1 − ik vis ATi,s Pj Ai,s + 1 − Vki ATi,m ⎝ πit Pt ⎠Ai,m
t∈Lik
− Vki
s∈Iik
1−
ik
T Ai,m Pj Ai,m − Pi < 0, ∀ j ∈ Liuk , m ∈ Iiuk .
t∈Lik
(7)
Proof. First, we introduce a notation as ⎛ ⎞ ⎛ ⎞
i = vim ATi,m ⎝ πi j Pj ⎠Ai,m − Pi = vim ATi,m ⎝ πi j Pj ⎠Ai,m m∈I
+
j∈L
⎛
vim ATi,m ⎝
m∈Iik
+
⎞ πˆ i j Pj ⎠Ai,m +
j∈Liuk
⎛ vˆim ATi,m ⎝
m∈Iiuk
⎞
m∈Iik
⎛
vˆim ATi,m ⎝
m∈Iiuk
j∈Lik
⎞
πi j Pj ⎠Ai,m
j∈Lik
πˆ i j Pj ⎠Ai,m − Pi .
j∈Liuk
It is obvious from Lemma 2 that the stochastic stability of system (4) is guaranteed if and only if i < 0 for any i ∈ L. Notice that
⎛ vim ATi,m ⎝
m∈Iik
⎞ πˆ i j Pj ⎠Ai,m
j∈Liuk
⎛ vˆim ATi,m ⎝
m∈Iiuk
⎛
= 1 − ik vim ATi,m ⎝ m∈Iik
⎞
πi j Pj ⎠Ai,m = 1 − Vki
j∈Lik
m∈Iiuk
⎞ πˆ i j P ⎠Ai,m , i j 1 − k j∈Liuk ⎛ ⎞
AT ⎝ πi j Pj ⎠Ai,m , i i,m
vˆim 1 − Vk
j∈Lik
and
⎛ vˆim ATi,m ⎝
m∈Iiuk
⎞ πˆ i j Pj ⎠Ai,m
= 1 − Vki 1 − ik
j∈Liuk
m∈Iiuk
⎛ ⎞
πˆ i j vˆim AT ⎝ Pj ⎠Ai,m , 1 − Vki i,m 1 − ik i j∈Luk
where 0≤
πˆ i j πˆ i j ≤ 1, j ∈ Liuk , = 1, i 1 − k 1 − ik i j∈Luk
and 0≤
vˆim vˆim ≤ 1, m ∈ Iiuk , = 1. i 1 − Vk 1 − Vki i m∈Iuk
Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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Then
⎛ ⎞
i = vim ATi,m ⎝ πi j Pj ⎠Ai,m − Pi =
⎛ ⎞
vˆim πˆ i j vis ATi,s ⎝ πit Pt ⎠ i i 1 − V 1 − k j∈Li k s∈Ii m∈I j∈L m∈Iiuk t∈Lik uk k ⎛ ⎞
× Ai,s + 1 − ik vis ATi,s Pj Ai,s + 1 − Vki ATi,m ⎝ πit Pt ⎠Ai,m s∈Iik
+ 1 − Vki 1 − ik ATi,m Pj Ai,m − Pi
t∈Lik
which yields that the system (4) is stochastically stable if and only if Eq. (7) is satisfied. Theorem 2. The following two propositions are equivalent: (i) There exist matrices Pi > 0, i ∈ L, such that for any i ∈ L, condition (7) holds; (ii) There exist matrices Pi > 0, Ri,m, j > 0, i ∈ L, m ∈ I, j ∈ Liuk , such that for any i ∈ L, the following inequalities hold:
vis Ri,s, j + 1 − Vki Ri,m, j − Pi < 0, ∀ j ∈ Liuk , m ∈ Iiuk , (8) s∈Iik
⎛ ATi,m ⎝
⎞
πit Pt ⎠Ai,m + 1 − ik ATi,m Pj Ai,m − Ri,m, j < 0, ∀ j ∈ Liuk , m ∈ I.
(9)
t∈Lik
Proof. (i)⇒ (ii) According to Eq. (7), we know that there will exist a positive scalar ε such that for any i ∈ L, ⎛ ⎛ ⎞ ⎞
vis ⎝ATi,s ⎝ πit Pt ⎠Ai,s + 1 − ik ATi,s Pj Ai,s + εI ⎠ s∈Iik
t∈Lik
⎛ ⎛ ⎞ ⎞
+ 1 − Vki ⎝ATi,m ⎝ πit Pt ⎠Ai,m + 1 − ik ATi,m Pj Ai,m + εI ⎠ − Pi < 0, ∀ j ∈ Liuk , m ∈ Iiuk , t∈Lik
holds. For any j ∈ Liuk , m ∈ I, we define ⎛ ⎞
Ri,m, j = ATi,m ⎝ πit Pt ⎠Ai,m + 1 − ik ATi,m Pj Ai,m + εI , t∈Lik
and then Eq. (8) is straightforwardly guaranteed. In addition, it is clear to see that the condition (9) is satisfied from the definition of Ri, m, j . (ii)⇒ (i) For any j ∈ Liuk , m ∈ I, Eq. (9) means that ⎛ ⎞
ATi,m ⎝ πit Pt ⎠Ai,m + 1 − ik ATi,m Pj Ai,m < Ri,m, j . t∈Lik
By substituting the inequality above into Eq. (8), we can easily get the result of (i). Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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In the following, we are in the position to design the controller Eq. (2) for Markovian jump system (1) based on Theorem 2. Here we rewrite the denotation Lik = {k1 , . . . , kmi } for the use of Theorem 3. Theorem 3. There exists a state feedback controller Eq. (2) such that the closed-loop system (4) is stochastically stable if and only if there exist matrices Pi > 0, Ri,m, j > 0, Km , Gm , i ∈ L, m ∈ I, j ∈ Liuk , such that for any i ∈ L, inequality (8) and SiT i,m, j Si + Gm Hm + HmT GTm < 0, ∀ j ∈ Liuk , m ∈ I,
(10)
hold, where
i,m, j =diag πik1 Pk1 , . . . , πikmi Pkmi , 1 − ik Pj , −Ri,m, j , ⎡ ⎤ Ai Bi ⎢ .. ⎥ ⎢ ⎥ Si =⎢ . ⎥, Hm = Km −I . ⎣A i B i ⎦ I 0 Proof. Since the equivalence of the two statements in Theorem 2, we know that the system is stochastically stable if and only if there exist matrices Pi > 0, Ri,m, j > 0, Km , Gm , i ∈ L, m ∈ I, j ∈ Liuk , such that for any i ∈ L, inequalities (8) and (9) hold. As a consequence, we only need to prove the equivalence of (9) and (10) and then the result in Theorem 3 is obtained. By substituting Ai + Bi Km for the matrix Ai, m in Eq. (9) and defining ⎡ ⎤ Ai + Bi Km ! ⎢ ⎥ .. I ⎢ ⎥ . , Wi,m = ⎢ , T = ⎥ m Km ⎣Ai + Bi Km ⎦ I we obtain
⎛
i,m, j = − Ri,m, j + ATi,m ⎝
⎞
πit Pt ⎠Ai,m + 1 − ik ATi,m Pj Ai,m
t∈Lik T =Wi,m i,m, j Wi,m
=TmT SiT i,m, j Si Tm .
(11)
Due to Hm Tm = 0, by using Lemma 3, we can derive that i, m, j < 0 holds if and only if there exists one matrix Gm such that Eq. (10) holds. Hence the system (1) can be stabilized by controller Eq. (2) if and only if the conditions of Theorem 3 hold, which completes the proof. If we fix the set I = {1}, then the mode-independent stabilization is straightforwardly derived as shown in Corollary 1 below. Corollary 1. There exists a state feedback controller Eq. (2) such that the closed-loop system (4) is stochastically stable if and only if there exist matrices Pi > 0, Ri, j > 0, K, G, i ∈ L, j ∈ Liuk , such that for any i ∈ L, j ∈ Liuk , the following inequalities hold Ri, j − Pi < 0, ˆ i, j Si + GH + H T GT < 0, SiT Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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where
i, j =diag πik1 Pk1 , . . . , πikmi Pkmi , 1 − ik Pj , −Ri, j , H = K −I . Remark 2. Observing the inequality (9) in Theorem 2, one can find that the controller parameters are coupled with the Lyapunov matrices Pj , which makes it intractable to solve the controller gains. In order to tackle this problem, we separate system mode (Ai , Bi ) and controller mode Km in the proof of Theorem 3 and then make use of Lemma 3 to decouple the system and controller modes. It is worth pointing out that the decoupling method in this paper is different from that used in [27], which applies the inequality −1 −GTφ R¯ iφ Gφ ≤ R¯ iφ − GTφ − Gφ .
(12)
The relaxation in Eq. (12) will lead to a sufficient condition, however, with the decoupling method proposed in this paper, the criterion in Theorem 3 is still a necessary and sufficient one. Although Ref. [32] also presents necessary and sufficient conditions for analysis and synthesis of Markovian jump systems with incomplete transition probabilities, it does not discuss the influence of mode asynchronism and actually, its main result Theorem 3 could be treated as a special case of Theorem 1 in this paper if we set I = L and vim = 1, ∀i = m. Remark 3. This paper is dedicated to the asynchronous control design for discrete-time Markovian jump linear system with partially unknown transition probabilities. By utilizing a conditional probability technique, the asynchronism between system and controller modes is described and novel criteria are obtained for the stability and stabilization even though the transition probability information of both Markovian subsystems and controllers is not perfectly acquired. Note that the proposed method could be extended to some related topics, such as semi-Markovian jump system or hidden Markovian jump system, to exploit their relaxed limitations on the probability distributions for broader applications [10,11], which would be our further research directions.
4. Illustrative examples In this section, we will use two examples to verify the validity and superiority of the proposed results in the above section. The first example is to show the effectiveness of the designed method with asynchronism and partially unknown transition probabilities. Example 1. Consider discrete-time Markovian jump linear system (1) with the following four subsystems: ! ! 1.0 −1.25 0.5 −0.83 , A2 = , A1 = 2.5 2.5 2.5 3.5 ! ! 0.25 −0.25 0.75 −0.57 , A4 = , A3 = 2.5 3.0 2.5 2.75 ! 0.5 , i = 1, . . . 4. Bi = 0.1 Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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Fig. 1. Different conditional probability matrices V. Table 1 Designed controller gains under different cases. K1 = −6.2617 −2.6213 K1 = −6.1113 K2 = −5.7216 −5.3334 K2 = −5.5108 K3 = −4.6721 −4.8625 K3 = −5.0027 K4 = −5.9916 −4.3205 K4 = −5.7056
−2.4961 −5.1908 −4.9535 −4.0413
Case I K1 = −5.9547 K2 = −5.4187 K3 = −5.3747 K4 = −5.4834
−2.2941 −5.1363 −5.7631 −3.5456
Case III
−2.3854 −5.2359 −5.4131 −3.8249
Case II K1 = −5.8499 K2 = −5.2239 K3 = −5.7456 K4 = −5.2089 Case IV
The transition probability matrix = [πi j ]4×4 is assumed as below ⎡ ⎤ 0.3 0.2 0.1 0.4 ⎢πˆ 21 0.2 0.3 πˆ 24 ⎥ ⎥ =⎢ ⎣πˆ 31 πˆ 32 0.5 0.3 ⎦. 0.2 0.2 0.1 0.5 According to the condition in Theorems 1 or 3 in [32], we can easily check that the uncontrolled switched system is not stochastically stable. Now we will utilize Theorem 3 to design state-feedback controllers to stabilize the system. Fig. 1 displays the conditional probability matrix V of different asynchronous cases and Table 1 lists the obtained controller gains based on different cases in Fig. 1. Case I corresponds to the result of synchronous control, while Case II, III, IV reflect the system with different asynchronous degree. The state trajectory of the open-loop system is drawn in Fig. 2 with the initial value x(0) = [0.1, −0.1]T , and Fig. 3 shows the state trajectory of the closed-loop system under asynchronous controller of Case IV. The unknown elements in and V are chosen as random scalars in this simulation. Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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Fig. 2. Open-loop system state trajectory of Example 1.
Fig. 3. Closed-loop system state trajectory of Example 1.
The switching signals of the system and controller modes are displayed in Fig. 4. From these figures, we know that the Markovian jump linear system is stabilized by the designed controllers although there exist asynchronous phenomenon and unknown transition probabilities, which verifies the usefulness and effectiveness of the result in Theorem 3. Example 2. This example focuses on the velocity control issue for a DC motor device, where abrupt failures may happen [14,19,21]. Frequently, abrupt failures make the device behaviour change from normal mode to failure mode, and our purpose is to control the angular velocity of the device under such phenomenon by using the result in Theorem 3. Considering the DC motor testbed with a nonzero constant plus a sinusoid reference, the main idea is to compel Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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Fig. 4. Switching signals of system and controller modes. Table 2 Parameter values of the DC motor device model.
i a11 i a12 i a21 i a22 i a31 i a33 bi1 bi2
i=1
i=2
i=3
−0.4799 5.1546 −3.8162 14.4723 0.1399 −0.9255 5.8705 15.5010
−1.6026 9.1632 −0.5918 3.0317 0.0740 −0.4338 10.2851 2.2282
0.6346 0.9178 −0.5056 2.4811 0.3865 0.0982 0.7874 1.5302
the DC motor to track the sinusoid reference with null steady-state error. By referring to [19], a discrete-time Markovian jump system is established to model the DC motor device, which is in the form of Eq. (1) with the following parameters: ⎡ i ⎤ ⎡ i⎤ i a11 a12 0 b1 i i a22 0 ⎦, Bi = ⎣bi2 ⎦, Ai = ⎣a21 i i a31 0 a33 0 where i ∈ {1, 2, 3} and the numerical values of the symbols above are exhibited in Table 2. The transition probability matrix is presented as ⎡ ⎤ 0.9 πˆ 12 πˆ 13 = ⎣0.36 0.6 0.04⎦. πˆ 31 0.1 πˆ 33 The conditional probability matrix V is given by ⎡ ⎤ 0.7 0.1 0.2 V = ⎣0.1 0.8 0.1⎦. vˆ31 vˆ32 0.6 Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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Table 3 Designed controller gains by Theorem 3. Theorem 3
Controller gains K1 = 0.2258 −0.9182 K2 = 0.1771 −0.9102 K3 = 0.2194 −0.9214
−0.0534 −0.0115 −0.0362
Fig. 5. Closed-loop system state trajectory of Example 2.
Fig. 6. Switching signals of system and controller modes.
Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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It is easy to check that the uncontrolled switched system is not stable. We would like to design controllers for the model by utilizing Theorem 3 and the corresponding controller gains are shown in Table 3. Figs. 5 and 6 respectively depict the state trajectory of the closed-loop system and the switching signals of both system and controller modes. Each subsystem could be matched with the designed three controllers and the switching of the controller modes is more random. Though there exists great asynchronism between the system and controller modes, the stability is still achieved by the designed controllers, which further illustrates the advantages of the proposed method in this paper. 5. Conclusions In this work, we focus on the stability and asynchronous stabilization for a class of discretetime Markovian jump systems. The asynchronous phenomenon between system and controller modes is considered and investigated by introducing a signal to govern the switching of controller modes, which is closely related to the system mode. The transition probability matrices of both system and controller modes are assumed with partially unknown elements. A necessary and sufficient criterion is developed for the asymptotic stability of the Markovian jump system and a controller is further designed which will tolerate the asynchronism between system and controller modes. Finally, two examples are utilized to verify the validity and superiority of the newly obtained results. References [1] E.K. Boukas, Stochastic Switching Systems: Analysis and Design, Birkhauser, Basel, Berlin, 2005. [2] E.K. Boukas, H. Yang, Stability of discrete-time linear systems with Markovian jumping parameters, Math. Control Sig. Syst. 8 (4) (1995) 390–402. [3] O.L.V. Costa, M.D. Fragoso, M. Todorov, A detector-based approach for the H2 control of Markov jump linear systems with partial information, IEEE Trans. Autom. Control 60 (5) (2015) 1219–1234. [4] M.C. de Oliveira, R.E. Skelton, Stability Tests for Constrained Linear Systems, Springer, Berlin, 2001. 268, 241–257 [5] Z. Feng, P. Shi, Sliding mode control of singular stochastic Markov jump systems, IEEE Trans. Autom. Control 62 (8) (2017) 4266–4273. [6] Z. Feng, W.X. Zheng, On reachable set estimation of delay Markovian jump systems with partially known transition probabilities, J. Frankl. Inst. 353 (15) (2016) 3835–3856. [7] X. Ge, Q.L. Han, Distributed sampled-data asynchronous H∞ filtering of Markovian jump linear systems over sensor networks, Signal Process. 127 (2016) 86–99. [8] I. Ghous, Z. Xiang, H.R. Karimi, H∞ control of 2-D continuous Markovian jump delayed systems with partially unknown transition probabilities, Inf. Sci. 382 (2017) 274–291. [9] C. Guan, Z. Fei, Z. Li, Y. Xu, Improved H∞ filter design for discrete-time Markovian jump systems with time-varying delay, J. Frankl. Inst. 353 (16) (2016) 4156–4175. [10] B. Jiang, Y. Kao, C. Gao, X. Yao, Passification of uncertain singular semi-Markovian jump systems with actuator failures via sliding mode approach, IEEE Trans. Autom. Control 62 (8) (2017) 4138–4143. [11] B. Jiang, Y. Kao, H.R. Karimi, C. Gao, Stability and stabilization for singular switching semi-Markovian jump systems with generally uncertain transition rates, IEEE Trans. Autom. Control 63 (11) (2018) 3919–3926. [12] Y. Kao, J. Xie, C. Wang, Stabilization of singular Markovian jump systems with generally uncertain transition rates, IEEE Trans. Autom. Control 59 (9) (2014) 2604–2610. [13] Y. Kao, T. Yang, J.H. Park, Exponential stability of switched Markovian jumping neutral-type systems with generally incomplete transition rates, Int. J. Robust Nonlinear Control 28 (5) (2018) 1583–1596. [14] S.H. Kim, H2 control of Markovian jump LPV systems with measurement noises: application to a DC-motor device with voltage fluctuations, J. Frankl. Inst. 354 (4) (2017) 1784–1800. [15] F. Li, P. Shi, L. Wu, M.V. Basin, C.C. Lim, Quantized control design for cognitive radio networks modeled as nonlinear semi-Markovian jump systems, IEEE Trans. Ind. Electron. 62 (4) (2015) 2330–2340. Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025
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C. Guan, Z. Fei and T. Yang / Journal of the Franklin Institute xxx (xxxx) xxx
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Please cite this article as: C. Guan, Z. Fei and T. Yang, Necessary and sufficient criteria for asynchronous stabilization of Markovian jump systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2018.11.025