Observer-based finite-time H∞ control for uncertain discrete-time nonhomogeneous Markov jump systems

Available online at www.sciencedirect.com

Journal of the Franklin Institute 356 (2019) 1730–1749 www.elsevier.com/locate/jfranklin

Observer-based finite-time H∞ control for uncertain discrete-time nonhomogeneous Markov jump systems Xiaobin Gao a, Hongru Ren b, Feiqi Deng c, Qi Zhou a,d,∗ a Guangdong

Province Key Laboratory of Intelligent Decision and Cooperative Control, Guangdong University of Technology, Guangzhou, Guangdong, 510006, China b Department of Automation, University of Science and Technology of China, Hefei, Anhui 230026, China c College of Automation Science and Engineering, South China University of Technology, Guangzhou, Guangdong 510640, China d College of Information Science and Technology, Bohai University, Jinzhou, Liaoning 121013, China Received 7 June 2018; received in revised form 26 September 2018; accepted 8 October 2018 Available online 23 January 2019

Abstract The problem of observer-based finite-time H∞ control for discrete-time Markov jump systems with time-varying transition probabilities and uncertainties is studied in this paper, in which time-varying transition probabilities are modelled as convex polyhedron, and the parameter uncertainty satisfies normbounded. First of all, a Luenberger observer is designed to measure the system state. Then, observerbased controller is constructed to ensure the stochastic finite-time boundedness of the resulting closedloop system with an H∞ performance. Furthermore, sufficient conditions are derived in light of linear matrix inequalities. In the end, the flexibility and applicability of the developed methods are demonstrated by two illustrative examples. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Dynamic systems presenting abrupt changes on their parameters or structures have received significant attention in the last decades. As one effective way to characterize such dy∗ Corresponding author at: School of Automation and Guangdong Province Key Laboratory of Intelligent Decision and Cooperative Control, Guangdong University of Technology, Guangzhou 510006, China. E-mail addresses: [email protected] (X. Gao), [email protected] (H. Ren), [email protected] (F. Deng), [email protected] (Q. Zhou).

https://doi.org/10.1016/j.jfranklin.2018.10.031 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

X. Gao, H. Ren and F. Deng et al. / Journal of the Franklin Institute 356 (2019) 1730–1749

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namic systems, Markov jump systems (MJSs) have gained extensive research interests in both practical and theoretical domains. To date, numerous fundamental works have been done to study the control and filtering problems of MJSs [1–12]. For instance, considering MJSs with partly known transition probabilities (TPs), a mode-dependent filter was designed in [8]. In [4], by employing a novel flexible terminal approach, the authors investigated the sampleddata exponentially synchronization problem for a class of Markovian neural networks with time-varying delayed signals. The work in [1] dealt with the H∞ control issue of descriptor systems with Markov jumping parameters. To solve the issue of event-triggered control for fuzzy MJSs with general switching policies, an asynchronous operation method was introduced in [5]. It is crucial to stress that the TPs of MJSs in most of the literature are assumed to be time invariant, which means that the Markov chain is homogeneous. Unfortunately, this hypothesis is not suitable in many practical systems. Among these cases, the TP matrix therein is not a real constant matrix but a time-dependent one. A quintessential example should be cited that the channel delays and packet dropouts in networked control systems can be described by Markov chain [13,14]. That is, they are different in different periods, which indicate that the TPs of Markov process are time-varying in practice. Therefore, it is essential to investigate the MJSs with time-varying TPs. A reasonable assumption to describe this timevarying characteristics is to be considered in a polytopic set [15–19]. On another research front line, considerable attentions have been attracted to the concept of finite-time stability (FTS). Different from the Lyapunov asymptotic stability which focuses on the state convergence characteristics of the systems with an infinite-time interval [20–29], the FTS puts emphasis on the transient performances of the dynamical systems in finite interval of time and plays a critical role in a multitude of industrial applications, such as robot control systems, networked control systems, and biochemistry reaction systems. Therefore, the problem of FTS has been investigated in the literature intensively [17,30–35]. To mention a few, the finite-time asynchronous H∞ filtering issue for discrete-time MJSs over a lossy network was concerned in [30]. In [36], the authors coped with the finite-time stabilization problem for MJSs with time delay via a switching control approach. In [32], the authors made some attempts to the finite-time problem of MJSs with Gaussian TPs. Very recently, by applying Takagi–Sugeno fuzzy approximation approach, the work in [17] considered the finite-time H∞ filtering problem for nonlinear singular MJSs with nonhomogeneous processes. However, it should be mentioned that, to the best of our knowledge, few of attentions have been paid on the finite-time H∞ control for uncertain discrete-time nonhomogeneous MJSs via observer-based state feedback method, which is the motivation behind this work. In this paper, we concern with the observer-based finite-time H∞ control problem for discrete-time nonhomogeneous MJSs with uncertainties. The main contributions of this paper are summarized as follows. Firstly, compared with homogeneous MJSs [1,7,31,33], the considered systems with nonhomogeneous Markov chain in this paper are more general, and the former can be viewed as a special case of the proposed theoretical methodology. Secondly, in order to guarantee the system behaviors stay within a given threshold during a fixed time interval, an observer-based controller is designed and novel sufficient criteria are derived. Finally, a numerical example and a DC motor device are exploited to illustrate the correctness and applicability of the proposed design technique in this paper. The rest of this paper is outlined as follows. Problem formulation is presented in Section 2. Section 3 addresses the problem of finite-time H∞ control analysis and synthesis. Section 4 contains two examples to verify the theoretical results. Finally, we conclude the paper in Section 5.

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Notations: Throughout this paper, Rn and Z+ , respectively, denote the n-dimensional Euclidean space and the set of nonnegative integers. E{W} means the expectation of W. A symmetric positive-definite matrix L is represented by L > 0. LT stands for the transposition of matrix L. λmin (L) and λmax (L) indicate the minimum eigenvalue and the maximum eigenvalue of matrix L, respectively. The superscript “−1” represents the matrix inverse. I and diag{} stand for the identity matrix and block-diagonal matrix, respectively. ∗ indicates symmetric term. 2. Problem formulation Consider the following discrete-time MJSs with parametric uncertainties: ⎧ ⎨x (k + 1 ) = [A(rk ) + A(rk , k )]x (k ) + B (rk )u (k ) + [G(rk ) + G(rk , k )]w (k ), z (k ) = [C (rk ) + C (rk , k )]x (k ) + [D (rk ) + D (rk , k )]w (k ), ⎩ y (k ) = Cy (rk )x(k),

(1)

where x(k) ∈ Rnx is the state vector, u(k) ∈ Rnu stands for the control input, z(k) ∈ Rnz denotes the control output, y(k) ∈ Rny is the measurement output. w(k) ∈ Rnw is the external disturbance input, which satisfies  N   T E w (k)w(k) ≤ d 2 , d ≥ 0. (2) k=0

In addition, A(rk ), B(rk ), C(rk ), D(rk ), G(rk ), and Cy (rk ) are coefficient matrices with appropriate dimensions for every rk . A(rk , k), C(rk , k), D(rk , k), and G(rk , k) are normbounded parameter uncertainties satisfying  A(rk , k ) C (rk , k ) D (rk , k ) G(rk , k )  = M (rk )F (rk , k ) Na (rk ) Nc (rk ) Nd (rk ) Ng (rk ) , where M(rk ), Na (rk ), Nc (rk ), Nd (rk ), and Ng (rk ) are known real constant matrices with compatible dimensions. F(rk , k) is unknown matrix satisfying FT (rk , k)F(rk , k) ≤ I. The process {rk , k ≥ 0} is assumed to be a discrete Markov chain, which takes values in a finite state set  = {1, . . . , Y }. The TP matrix (k) = {πi j (k)}, i, j ∈  is given as πi j (k) =

Pr(rk+1 = j|rk = i) with π ij (k) ≥ 0 and Yj=1 πi j (k) = 1. To gain more insight, we consider the Markov process to be nonhomogeneous, which means that the TP matrix (k) depends on time instant k and has the following structure: (k) =

w 

αs (k)s ,

s=1

where 0 ≤ αs (k) ≤ 1,

w 

αs (k) = 1,

s=1

and s = {πisj }, s ∈ {1, . . . , w} are given TP matrices. More precisely, the time-varying transition matrix (k) evolves in a polytope defined by its vertices s = {πisj }, s ∈ {1, . . . , w}.

Remark 1. Obviously, if we take ws=1 αs (k)s = , the nonhomogeneous Markov process {rk , k ≥ 0} reduces to a homogeneous one. In other words, the homogeneous Markov pro-

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cess can be regarded as a special case of the nonhomogeneous Markov process, and the nonhomogeneous Markov process is more general and complex than the former. Throughout the paper, for all rk = i ∈ , we use Ai , Bi , Ci , Di , Gi , and Cyi to denote A(rk ), B(rk ), C(rk ), D(rk ), G(rk ), and Cy (rk ), respectively. Here, we construct an observer-based controller for system (1) of the following form ⎧ ⎨x˜(k + 1 ) = Ai x˜(k) + Bi u(k) + Hi (y(k) − y˜(k) ), y˜(k ) = Cyi x˜(k), (3) ⎩ u(k) = Ki x˜(k), where x˜(k) denotes the estimated state. y˜(k) represents the estimated output. Hi and Ki are the observer gain and controller gain to be determined, respectively. Augmenting Eq. (3) to system (1), the resultant closed-loop system satisfies:  x¯(k + 1 ) = A¯ i x¯(k ) + G¯ i w (k ), (4) z (k ) = C¯i x¯(k ) + D¯ i w (k ), where

 T x¯(k ) = x T (k) eT (k) , e(k) = x(k) − x˜(k),

−Bi Ki Ai + Ai,k + Bi Ki ¯ Ai = Ai,k Ai − HiCyi

 −Bi Ki Ai + Bi Ki Mi 0 = A˘ i + A˘ i,k , + F N = 0 Ai − HiCyi Mi i,k ai

Gi + Gi,k Gi Mi ¯ = + F N = G˘ i + M˘ gi Fi,k N˘ gi = G˘ i + G˘ i,k , Gi = Gi + Gi,k Gi Mi i,k gi    C¯i = Ci + Ci,k 0 = Ci 0 + Mi Fi,k Nci 0 = C˘i + C˘i,k , D¯ i = Di + Di,k = Di + Mi Fi,k Ndi = D˘ i + D˘ i,k ,

−Bi Ki A + Bi Ki , A˘ i,k = M˘ ai Fi,k N˘ ai , A˘ i = i 0 Ai − HiCyi  Gi ˘ , G˘ i,k = M˘ gi Fi,k N˘ gi , C˘i = Ci Gi = Gi

0 , C˘i,k = Mi Fi,k N˘ ci , Mi ˘ ˘ ˘ ˘ ˘ , Di = Di , Di,k = Mi Fi,k Ndi , Mai = Mgi = Mi   N˘ ai = Nai 0 , N˘ gi = Ngi , N˘ ci = Nci 0 , N˘ di = Ndi . To proceed further, the following definitions and lemma are introduced. Definition 1 [37]. (Stochastic finite-time stability (SFTS) via observer-based state feedback). The resulting closed-loop system (4) with wk = 0 is said to be SFTS via observer-based state feedback subject to (δ, , Ri , N) with 0 < δ < , Ri > 0, and N ∈ Z+ , if E {x¯T (0 )Ri x¯(0 )} ≤ δ 2 ⇒ E {x¯T (k )Ri x¯(k )} < 2 , holds, where k ∈ {1, 2, . . . , N }. Definition 2 [37]. (Stochastic finite-time boundedness (SFTB) via observer-based state feedback). System (4) is said to be SFTB via observer-based state feedback subject to (δ, , Ri , N,

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d) with 0 < δ < , Ri > 0, N ∈ Z+ , and d ≥ 0, if it is SFTS via observer-based state feedback for any nonzero wk satisfying Eq. (2). Definition 3 [37]. (Stochastic H∞ finite-time boundedness via observer-based state feedback). For 0 < δ < , Ri > 0, γ > 0, N ∈ Z+ , and d ≥ 0, the closed-loop system (4) is said to be stochastic H∞ finite-time boundedness via observer-based state feedback subject to (δ, , Ri , N, d, γ ), and following condition should be satisfied:  N   N    T 2 T E z (k )z (k ) < γ E w (k )w (k ) , k=0

k=0

under the zero-initial condition for any nonzero w(k) satisfying (2). Lemma 1 [38]. Let O, M, F, and N be real matrices with appropriate dimensions. Suppose that F satisfies FFT ≤ I. Then, for any θ > 0, the following requirement holds: O + MF N + N T F T M T ≤ O + θ −1 M M T + θ N T N. 3. Main results In this section, sufficient conditions via observer-based state feedback are presented, which guarantee that the resultant closed-loop system (4) is SFTB with an H∞ performance level. Theorem 1. System (4) is SFTB via observer-based state feedback with respect to (δ, , Ri , N, d), if there exist matrices P¯si > 0, Pˆsqi > 0, Qi , and a scalar μ ≥ 1, such that the following inequalities hold for any i, j ∈ , and s, q ∈ {1, . . . , w} T

A¯ Pˆ A¯ − μP¯si A¯ Ti Pˆsqi G¯ i < 0, sqi = i sqi i (5) ∗ G¯ Ti Pˆsqi G¯ i − Qi sup{λmax (Pˇsi )}δ 2 + sup{λmax (Qi )}d 2 < inf {λmin (Pˇsi )}μ−N 2 , i∈

i∈

i∈

(6)

where Pˆsqi =

Y 

πisj P¯q j , Pˇsi = Ri−1/2 P¯si Ri−1/2 .

j=1

Proof. Construct the Lyapunov–Krasovskii functional as V (k) = V (x¯(k), rk = i ) =

w 

αs (k )x¯T (k )P¯s (rk )x¯(k ),

s=1

where 0 ≤ αs (k) ≤ 1,

w 

αs (k) = 1, P¯si > 0.

s=1

Then E {V (k + 1 )} − μV (k ) − wT (k )Qi w (k )

(7)

X. Gao, H. Ren and F. Deng et al. / Journal of the Franklin Institute 356 (2019) 1730–1749

⎛ = x¯T (k + 1)⎝

Y  w  w 

⎞ αs (k)αs (k + 1)πisj P¯s j ⎠x¯(k + 1) − μx¯T (k)

j=1 s=1 s=1

w 

1735

αs (k)P¯si x¯(k)

s=1

− w (k)Qi w(k), T

denoting w 

αs (k + 1)P¯s j =

s=1

w 

βq (k)P¯q j ,

q=1

thus, we can get E {V (k + 1 )} − μV (k ) − wT (k )Qi w (k ) = x¯T (k + 1)P˜sqi x¯(k + 1) − μx¯T (k)P˜si x¯(k) − wT (k)Qi w(k) = ηT (k)ϒsqi η(k),

(8)

where P˜sqi =

Y  w  w 

αs (k)βq (k)πisj P¯q j ,

j=1 s=1 q=1

P˜si =

w 

αs (k)P¯si , ηT (k) = [x¯T (k) wT (k)],

s=1

ϒsqi

T A¯ P˜ A¯ − μP˜si = i sqi i ∗

A¯ Ti P˜sqi G¯ i . G¯ Ti P˜sqi G¯ i − Qi

Condition (5) is established, which implies that w  w 

αs (k)βq (k) sqi < o,

(9)

s=1 q=1



since 0 ≤ αs (k) ≤ 1, ws=1 αs (k) = 1, and 0 ≤ βq (k) ≤ 1, wq=1 βq (k) = 1, it is easy to prove that ϒ sqi < 0, which means E {V (k + 1 )} < μV (k ) + wT (k )Qi w (k ).

(10)

Hence, we can derive E {V (k + 1 )} ≤ μE {V (k )} + sup{λmax (Qi )}E {wT (k )w (k )}. i∈

Since μ ≥ 1, one can obtain that E {V (k )} < μk E {V (0 )} + sup{λmax (Qi )}E i∈

⎧ k−1 ⎨ ⎩

j=0

≤ μ E {V (0 )} + sup{λmax (Qi )}μ d . k

k 2

i∈

(11)

⎫ ⎬ μk− j−1 wT ( j )w ( j ) ⎭ (12)

Letting Pˇsi = Ri−1/2 P¯si Ri−1/2 , and considering the fact that E {x¯T (0)Ri x¯(0)} ≤ δ 2 , we have   w  1/2 ˇ 1/2 T E {V (0)} = E x¯ (0) αs (0)Ri Psi Ri x¯(0) s=1

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  ≤ sup{λmax (Pˇsi )}E x¯T (0)Ri x¯(0) i∈

≤ sup{λmax (Pˇsi )}δ 2 .

(13)

i∈

On the other hand,  E {V (k )} = E x¯T (k)

w 

 αs (k)P¯si x¯(k)

s=1

 = E x¯ (k) T

w 

 αs (k)Ri1/2 Pˇsi Ri1/2 x¯(k)

s=1

≥ inf {λmin (Pˇsi )}E {x¯T (k)Ri x¯(k)}. i∈

(14)

By utilizing Eqs. (12) − (14), one can derive E {x¯T (k)Ri x¯(k)} <

supi∈ {λmax (Pˇsi )}μk δ 2 + supi∈ {λmax (Qi )}μk d 2 . inf i∈ {λmin (Pˇsi )}

(15)

From Eq. (6), we can further get E {x¯T (k )Ri x¯(k )} < 2 . The proof is completed.  Theorem 2. System (4) is stochastically H∞ finite-time bounded via observer-based state feedback subject to (δ, , Ri , N, d, γ ), if there exist matrices P¯si > 0, Pˆsqi > 0, Xi , and scalars μ ≥ 1 and γ > 0, such that for any i, j ∈ , and s, q ∈ {1, . . . , w} ⎡ ⎤ Pˆsqi − Xi − XiT 0 Xi A¯ i Xi G¯ i ⎢ ⎥ C¯i D¯ i ∗ −I ⎢ ⎥ < 0, (16) ⎣ ⎦ ¯ ∗ ∗ −μPsi 0 ∗ ∗ ∗ −γ 2 μ−N I sup{λmax (Pˇsi )}δ 2 + μ−N γ 2 d 2 < inf {λmin (Pˇsi )}μ−N 2 , i∈

i∈

(17)

hold, where Pˆsqi , P˜sqi , P˜si , and Pˇsi are defined in Theorem 1. Proof. Akin to Theorem 1, we choose the Lyapunov–Krasovskii functional and define (x¯(k), w(k), rk = i ) = E {V (k + 1 )} − μV (k ) + zT (k)z(k) − γ 2 μ−N wT (k)w(k).

(18)

Then, using the similar techniques in Theorem 1, we have (x¯(k), w(k), i ) = ξ T (k)sqi ξ (k),

(19)

where ξ T (k) = [x¯T (k ) wT (k )], T A¯ P˜ A¯ + C¯iT C¯i − μP˜si sqi = i sqi i ∗

A¯ Ti P˜sqi G¯ i + C¯iT D¯ i . G¯ Ti P˜sqi G¯ i + D¯ iT D¯ i − γ 2 μ−N I

(20)

−1 ˆ On the other hand, noting the fact that (Pˆsqi − Xi )Pˆsqi (Psqi − Xi )T ≥ 0 for arbitrary matrix −1 T Xi , ∀i ∈ , we can achieve Pˆsqi − Xi − XiT ≥ −Xi Pˆsqi Xi . Then, from Eq. (16), we can get

X. Gao, H. Ren and F. Deng et al. / Journal of the Franklin Institute 356 (2019) 1730–1749

⎡

−1 T −Xi Pˆsqi Xi ⎢ ∗ ⎢ ⎣ ∗ ∗

0 −I ∗ ∗

⎤ Xi G¯ i ⎥ D¯ i ⎥ < 0. ⎦ 0 2 −N −γ μ I

Xi A¯ i C¯i −μP¯si ∗

1737

(21)

After performing a congruence transformation to Eq. (21) with diag{Xi−T Pˆsqi , I , I , I }, we have ⎡ ⎤ −Pˆsqi 0 Pˆsqi A¯ i Pˆsqi G¯ i ⎢ ∗ ⎥ C¯i D¯ i −I ⎢ ⎥ < 0. (22) ⎣ ∗ ⎦ ¯ ∗ −μPsi 0 ∗ ∗ ∗ −γ 2 μ−N I Multiplying Eq. (22) by the adequate coefficients and adding the resulting inequalities, after using Schur complement, we can get that sqi < 0, which implies that E {V (k + 1 )} < μE {V (k )} − E {z (k)T z (k)} + γ 2 μ−N E {wT (k )w(k )}. Then, it follows that E {V (k )} < μk E {V (0 )} −

k−1 

μk− j−1 E {zT ( j)z( j)} + γ 2 μ−N E

j=0

⎧ k−1 ⎨ ⎩

j=0

⎫ ⎬ μk− j−1 wT ( j)w( j) . (24) ⎭

Under zero initial condition and according to V(k) ≥ 0, ∀k ∈ Z+ , one has ⎧ ⎫ k−1 k−1 ⎨ ⎬    μk− j−1 E zT ( j)z( j) < γ 2 μ−N E μk− j−1 wT ( j)w( j) . ⎩ ⎭ j=0

(23)

(25)

j=0

For μ ≥ 1, we have  N   N    T 2 T E z (k)z(k) < γ E w (k)w(k) . k=0

(26)

k=0

The proof is finished.  Remark 2. By using a slack matrix Xi , the cross coupling between system matrices and Lyapunov matrices is eliminated, which provides a convenient way to calculate the observer gain and controller gain. Theorem 3. For given μ ≥ 1 and σ ≥ 1, the closed-loop system (4) is stochastically H∞ finitetime bounded via observer-based state feedback with regard to (δ, , Ri , N, d, γ ), if there exist matrices P¯si > 0, Pˆsqi > 0, and Xi , and positive scalars γ , θ 1i , θ 2i , and θ 3i , such that the following conditions hold for any i, j ∈ , and s, q ∈ {1, . . . , w} 1

sqi 2sqi < 0, sqi = (27) ∗ 4sqi X1i Bi = BiWi ,

(28)

Ri < P¯si < σ Ri ,

(29)

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σ δ 2 + μ−N γ 2 d 2 < μ−N 2 , where ⎡

1sqi

11 sqi ⎢ ∗ ⎢ ⎢ ∗ =⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Pˆ2sqi 22 sqi ∗ ∗ ∗ ∗

0 0 −I ∗ ∗ ∗

(30)

14 0 Ci 44 sqi ∗ ∗

15 25 sqi 0 45 sqi 55 sqi ∗

⎤ ⎡ X1i Gi X1i Mi ⎢X2i Mi X2i Gi ⎥ ⎥ ⎢ ⎢ Di ⎥ ⎥, 2 = ⎢ 0 T sqi ⎥ ⎢ 0 θ3i Nci Ndi ⎥ ⎢ ⎦ ⎣ 0 0 66 0 sqi

X1i Mi X2i Mi 0 0 0 0

⎤ 0 0⎥ ⎥ Mi ⎥ ⎥, 0⎥ ⎥ 0⎦ 0

4sqi = diag{−θ1i I , −θ2i I , −θ3i I }, T 14 15 ˆ 11 sqi = P1sqi − X1i − X1i , sqi = X1i Ai + BiYi , sqi = −BiYi , T 25 ˆ 22 sqi = P4sqi − X2i − X2i , sqi = X2i Ai − ZiCyi , T T 45 55 ¯ ¯ ¯ 44 sqi = −μP1si + θ1i Nai Nai + θ3i Nci Nci , sqi = −μP2si , sqi = −μP4si , 2 −N 66 I + θ2i NgiT Ngi + θ3i NdiT Ndi , Pˆsqi = sqi = −γ μ

N 

πisj P¯q j , s, q = 1, . . . , w.

j=1

Furthermore, if the conditions (27)–(30) have feasible solutions, the desired observer gain Hi and controller gain Ki can be given by Hi = X2i−1 Zi , Ki = Wi−1Yi .

(31)

Proof. Inequality Eq. (16) can be rewritten as T T T T T T T T i + 1i Fi,k 2i + 2i Fi,k 1i + 3i Fi,k 4i + 4i Fi,k 3i + 5i Fi,k 6i + 6Ti Fi,k 5i < 0,

(32)

where ⎡

i 1 i 3 i 5 i

⎤ Pˆsqi − Xi − XiT 0 Xi A˘ i Xi G˘ i ⎢ ⎥ C˘i D˘ i ∗ −I ⎥, =⎢ ⎣ ⎦ ∗ ∗ −μP¯si 0 ∗ ∗ ∗ −γ 2 μ−N I  T  = M˘ aiT XiT 0 0 0 , 2i = 0 0 N˘ ai 0 ,  T  = M˘ giT XiT 0 0 0 , 4i = 0 0 0 N˘ gi ,  T  = 0 MiT 0 0 , 6i = 0 0 N˘ ci N˘ di .

Applying Lemma 1 to Eq. (32), there exist positive scalars θ 1i , θ 2i , and θ 3i , such that T T −1 T T −1 T T i + θ1−1 i 1i 1i + θ1i 2i 2i + θ2i 3i 3i + θ2i 4i 4i + θ3i 5i 5i + θ3i 6i 6i < 0,

(33)

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then, applying Schur complement, we have ⎡ ⎤ 11 0 Xi A˘ i Xi G˘ i Xi M˘ ai Xi M˘ gi 0 ⎢ ∗ C˘i D˘ i −I 0 0 Mi ⎥ ⎢ ⎥ ⎢ ∗ ∗ 33 34 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ 44 0 0 0 ⎥ ⎢ ⎥ < 0, ⎢ ∗ ⎥ ∗ ∗ ∗ −θ I 0 0 1 i ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ ∗ −θ2i I 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ −θ3i I

1739

(34)

where 11 = Pˆsqi − Xi − XiT , 33 = −μP¯si + θ1i N˘ aiT N˘ ai + θ3i N˘ ciT N˘ ci , 34 = θ3i N˜ ciT N˜ di , 44 = −γ 2 μ−N I + θ2i N˘ giT N˘ gi + θ3i N˘ diT N˘ di . Furthermore, the matrices P¯si , Pˆsqi ,

ˆ P¯ P¯2si ˆsqi = P1sqi , P¯si = 1si P ¯ ∗ P4si ∗

and Xi in Eq. (34) have the following forms



Pˆ2sqi 0 X1i , , X = i 0 X2i Pˆ4sqi

(35)

where Xi is assumed to be non-singular. Substituting Eq. (35) into Eq. (34), and denoting Yi = Wi Ki and Zi = X2i Hi , it is easy to obtain Eq. (27). On the other hand, noticing that Pˇsi = Ri−1/2 P¯si Ri−1/2 in Theorem 1, it follows from Eq. (29) that 1 < inf {λmin (Pˇsi )}, sup{λmax (Pˇsi )} < σ. i∈

i∈

Then, a sufficient condition to guarantee Eq. (17) is obtained as σ δ 2 + μ−N γ 2 d 2 < μ−N 2 . The proof is finished.

(36) 

Remark 3. It is arduous to deal with the condition (28) by applying the linear matrix inequality (LMI) toolbox. To overcome this, condition (28) can be replaced with following constraint [X1i Bi − BiWi ]T [X1i Bi − BiWi ] <  I ,

(37)

with ϖ being a sufficiently small positive scalar. After using Schur complement, the above constraint Eq. (37) is equivalent to the following LMI:

− I [X1i Bi − BiWi ]T < 0. (38) ∗ −I

Remark 4. If ws=1 αs (k)s = , the problem considered in this paper degenerates into the finite-time H∞ control for discrete-time MJSs via observer-based state feedback, which was analysed in [33]. Hence, the problem discussed in this paper is more general. Remark 5. Ref. [19] only investigated the observer-based H∞ control for nonhomogeneous MJSs. However, in many practical applications, the trajectories of state variables are required to stay within a given bound over a fixed finite-time interval, such that the transient behaviour of the system needs to be considered. Moreover, parameter uncertainties are also unavoidable in a multitude of systems and have a negative influence on the system performance. Hence,

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in this paper, we study the problem of observer-based finite-time H∞ control for uncertain discrete-time nonhomogeneous MJSs. It must also be mentioned that, the concept of finitetime stability is entirely different from the Lyapunov asymptotic stability. Hence, our work is more different from the Ref. [19] and has a wider application in practical engineering. 4. Illustrative examples Simulations of two examples are presented in this section to verify the effectiveness and applicability of the observer-based controller in this paper. Example 1. Consider the two-mode discrete-time nonhomogeneous MJS Eq. (1) described by the following coefficient matrices: Mode 1:







0 0 1.2 −0.8 1.2 0.4 0.1 0.1 , B1 = , G1 = , C1 = , A1 = 0.3 1.1 0.1 1.1 0 0.1 0 0.1





 0 1.1 0.1 6.5 0.12 D1 = , Cy1 = , M1 = , Na1 = 0.02 0.1 , 0 0.1 −1.1 4.5 0.1    Ng1 = 0.1 0.14 , Nc1 = 0.13 0.1 , Nd1 = 0.15 0.1 . Mode 2:







0.5 0 0 1.3 1.2 −0.7 0.1 0.1 A2 = , B2 = , G2 = , C2 = , −0.1 1.2 0.2 0.8 0 0.1 0 0.1





 0 2.1 0.1 6 0.1 , Cy2 = M1 = , Na2 = 0.1 0.01 , D2 = 0 0.1 1.3 5 0.11    Ng2 = 0.01 0.05 , Nc2 = 0.1 0.12 , Nd2 = 0.1 0.07 . Moreover, the vertices of the time-varying TP matrix are considered as





0.65 0.35 0.85 0.15 0.2 0.8 0.55 , 2 = , 3 = , 4 = 1 = 0.2 0.8 0.15 0.85 0.3 0.7 0.9

0.45 . 0.1

The related parameters are taken as R1 = R2 = I , μ = 1.1, N = 10, d = 1, σ = 1.1, and δ = 0.1. Here, our main purpose is to design an observer-based controller, which ensures the closed-loop system Eq. (4) to be SFTB with an H∞ performance. For this objective, with the help of the LMI toolbox of MATLAB, we can solve LMIs Eqs. (27)–(30) in Theorem 3, then we get the values min = 2.5577 and γmin = 2.5744, and the observer and state feedback controller gains are calculated as follows:



−0.0237 0.1632 −0.2150 0.2355 , H2 = , H1 = 0.0961 0.2358 −0.0730 0.2014



0.8243 −1.0759 −1.1803 −2.0929 , K2 = . K1 = −0.4922 −0.7833 −0.2756 −2.4655 In order to evaluate the effectiveness of the designed observer-based controller more T clearly, simulations have been carried out. Assume

10the Texternal noise signal as2 w (k) = −0.4k −0.3k [e sin (2k) e cos(2k)], it is easy to check k=0 w (k)w(k) = 0.9905 < d = 1. The initial conditions are set as x T (0) = [0 0], and x˜T (0) = [0 0], which satisfy the condition  T x¯(0) Ri x¯(0) < δ = 0.1. With these conditions, Fig. 1 shows the jumping modes. Fig. 2

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Jumping modes

3

2

1

0

2

4

Time (k)

6

8

10

Fig. 1. The operation modes.

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

2

4

6

8

10

Time (k) Fig. 2. The trajectory of

 x¯(k)T Ri x¯(k).

 (k). From Fig. 2 we can see that in the given finite-time gives the trajectory of x¯(k)T Ri x¯ interval k ∈ [0, 10], the value of x¯(k)T Ri x¯(k) does not exceed the bound min = 2.5577, which means that the closed-loop system (4) is SFTB. The state responses of the original

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0.15

State responses

0.1

0.05

0

-0.05

-0.1

-0.15 0

2

4

Time (k)

6

8

10

8

10

Fig. 3. The trajectories of x1 (k), x˜1 (k) and e1 (k).

0.3

State responses

0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0

2

4

Time (k)

6

Fig. 4. The trajectories of x2 (k), x˜2 (k) and e2 (k).

system and observer are shown in Figs. 3 and 4, from which we can observe that the designed observer-based controller is effective. It can be observed from the simulations that the system (4) with time-varying TPs and parameter uncertainties is SFTB with respect to (0.1, 2.5577, I, 10, 1, 2.5744), and has an H∞ performance index via observer-based state feedback, which indicates the feasibility and effectiveness of the proposed theoretical methodology.

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Fig. 5. DC motor. Table 1 Parameters of a real DC motor device denoting by the nonhomogeneous MJSs. Parameters

(i) a11

(i) a12

(i) a21

(i) a22

b(i) 1

b(i) 2

i=1 i=2 i=3

−0.479908 −1.60262 0.634617

5.1546 9.1632 0.917836

−3.81625 −0.5918697 −0.50569

14.4723 3.0317 2.48116

5.87058212 10.285129 0.7874647

15.50107 2.2282663 1.5302844

Example 2. To demonstrate the applicability of the proposed approach, in this example, a DC motor device driving an inertial load [18,39] in Fig. 5 is considered, which is described by ⎧ ⎨Tk = Km ik , Vmk = Kb vk , ⎩ J vk+1 = −K f vk + Km ik , where Tk is the torque at the DC motor shaft, Km denotes the armature constant, ik and Vmk , respectively, represent the electrical current and the voltage. vk means the angular velocity of the load, Kb is a constant related to certain physical properties of the motor. J stands for the inertial load and K f vk indicates the linear approximation of viscous friction. We define x(k) ≡ [vkT , ikT ]T , then the following model can be used to describe the DC motor: ⎧ ! ! ⎨x (k + 1 ) = Ai + A ! i,k x (k ) + Bi u (k ) + ! Gi + Gi,k w (k ), z (k ) = Ci + Ci,k x (k ) + Di + Di,k w (k ), ⎩ y (k ) = Cyi x(k), where " Ai =

(i) a11

(i) a12

(i) a21

(i) a22

#

(i) b , Bi = 1 0

0 . b(i) 2

The abrupt failures, which generate on the power delivered to the shaft, are assumed to be subject to a nonhomogeneous Markov chain with three operation modes. That is, the normal (i) (i) (i) (i) (rk = i = 1), low (rk = i = 2), and medium (rk = i = 3). The parameters a11 , a12 , a21 , a22 , (i) (i) b1 and b2 are given in Table 1 (for details concerning the modeling, please see [39]).

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The rest of system parameters are given as



0 0.13 0.1 0.21 0.01 , C1 = , D1 = G1 = 0 0.1 0 0.11 0  0.1 , Na1 = Ng1 = Nc1 = Nd1 = 0.1 0.1 , M1 = 0.1



0 0 0.2 0.31 0.21 , C2 = , D2 = G2 = 0 0.14 0 0.01 0.1  0.1 M2 = , Na2 = Ng2 = Nc2 = Nd2 = 0.1 0.1 , 0.1



0 0 0.22 0.17 0.13 , C3 = , D3 = G3 = 0 0.1 0 0.12 0  0.1 M3 = , Na3 = Ng3 = Nc3 = Nd3 = 0.1 0.1 . 0.1 Furthermore, the ⎡ 0.65 0.1 1 = ⎣ 0.4 0.3 0.2 0.7 ⎡ 0.2 0.2 3 = ⎣0.8 0.1 0.3 0.45

vertices of the time-varying TP ⎤ ⎡ 0.25 0.35 0.15 0.3 ⎦, 2 = ⎣ 0.1 0.4 0.1 0.15 0.25 ⎤ ⎡ 0.6 0.15 0.45 0.1 ⎦, 4 = ⎣ 0.7 0.15 0.25 0.3 0.4

0.1 4.7 , Cy1 = 0.15 1.3

1.5 , 14.2

0 4.5 , Cy2 = 0.12 1.9

2.7 , 13.9

0 5.3 , Cy3 = 0.01 1.2

1.5 , 13.6

matrix are set as ⎤ 0.5 0.5⎦, 0.6 ⎤ 0.4 0.15⎦. 0.3

When R1 = R2 = I , μ = 1.03, N = 10, σ = 1.06, d = 1, and δ = 0.2, by Theorem 3, we can obtain that min = 14.7598, γmin = 14.7571, and the observer and state feedback controller gains can be derived as





−0.21020.3871 −0.68990.7921 0.10650.0569 , H2 = , H3 = , −1.13001.1416 −0.24420.2651 −0.14100.1990





0.1067 − 0.9647 0.1595 − 0.9144 −0.8501 − 2.4703 , K2 = , K3 = . K1 = 0.2439 − 0.9401 0.2753 − 1.4196 0.2209 − 1.9366

H1 =

For simulation, the external disturbance, the initial state, and the initial estimated state T are, respectively, taken as wT (k) = [0.55e−0.2k sin (2k) 0.55e−0.1k cos  (2k)], x (0) = [0 0], T T x˜ (0) = [0 0]. Fig. 6 gives the operation modes. Time history of x¯ (k) Ri x¯(k) is depicted in Fig. 7 . It is clear to observe from Fig. 7 that the trajectory of x¯(k)T Ri x¯(k) satisfies   x¯(k)T Ri x¯(k) < 14.7598 under the condition x¯(0)T Ri x¯(0) < 0.2, which indicates that the resultant closed-loop system (4) is SFTB. The state response curves of the real system and the estimated one are plotted in Figs. 8 and 9. Hence, it can be concluded that under the designed observer-based controller, system (4) is SFTB with respect to (0.2, 14.7598, I, 10, 1, 14.7571) and satisfies an H∞ performance index, which demonstrates the applicability of the proposed results. Remark 6. In Example 2, a DC motor device from Yin et al. [18] is considered to illustrate the applicability of our proposed results. Different from Ref. [18] which pays more attention to the asymptotic stability of the DC motor, in this paper we focus on the transient performances

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4

Jumping modes

3

2

1

0

2

4

6

8

10

Time (k) Fig. 6. The operation modes.

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

2

4

6

8

10

Time (k) Fig. 7. The trajectory of

 x¯(k)T Ri x¯(k).

of the DC motor device over a fixed finite-time interval. The behaviour of dynamics during a given time interval is more appropriate in a host of practice applications, a typical case is the robot control systems. Hence, our results are more applicable than Ref. [18] from the practical point of view.

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0.4

Angular velocity (rad/s)

0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0

2

4

Time (k)

6

8

10

8

10

Fig. 8. The trajectories of x1 (k), x˜1 (k) and e1 (k).

0.8

Electrical current (A)

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 0

2

4

6

Time (k) Fig. 9. The trajectories of x2 (k), x˜2 (k) and e2 (k).

5. Conclusions In this paper, we have studied the observer-based finite-time H∞ control problem for discrete-time nonhomogeneous MJSs. A polytope set has been applied to describe time-vary

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TPs. By using LMI techniques, sufficient criteria have been obtained to ensure that the closedloop system is SFTB and achieves a prescribed H∞ performance index via observer-based state feedback. The simulation results have demonstrated the potential of the proposed techniques. Future research will focus on the finite-time control problem for nonhomogeneous MJSs with adaptive control [40,41], packet dropouts [42–44], and asynchronous control strategy [45,46]. Acknowledgement This work was partially supported by the National Natural Science Foundation of China (61673072, 61773072), the Guangdong Natural Science Funds for Distinguished Young Scholar (2017A030306014), the Innovative Research Team Program of Guangdong Province Science Foundation (2018B030312006), the Fundamental Research Funds for the Central Universities (2017FZA5010), the Science and Technology Planning Project of Guangdong Province (2017B010116006), and the Department of Education of Guangdong Province (2016KTSCX030, 2017KZDXM027). References [1] J. Lam, Z. Shu, S. Xu, E.K. Boukas, Robust H∞ control of descriptor discrete-time Markovian jump systems, Int. J. Control 80 (3) (2007) 374–385. [2] Z.-G. Wu, J.H. Park, H. Su, J. Chu, Stochastic stability analysis for discrete-time singular Markov jump systems with time-varying delay and piecewise-constant transition probabilities, J. Frankl. Inst. 349 (9) (2012) 2889–2902. [3] Y. Chen, R. Lu, H. Zou, Stability analysis for stochastic jump systems with time-varying delay, Nonlinear Anal. Hybrid Syst. 14 (2) (2014) 114–125. [4] J. Cheng, J.H. Park, H.R. Karimi, H. Shen, A flexible terminal approach to sampled-data exponentially synchronization of Markovian neural networks with time-varying delayed signals., IEEE Trans. Cybern., to be published, doi:10.1109/TCYB.2017.2729581. [5] J. Cheng, J.H. Park, L. Zhang, Y. Zhu, An asynchronous operation approach to event-triggered control for fuzzy Markovian jump systems with general switching policies, IEEE Trans. Fuzzy Syst. 26 (2018) 6–18. [6] J. Tao, Z.-G. Wu, H. Su, Y. Wu, D. Zhang, Asynchronous and resilient filtering for Markovian jump neural networks subject to extended dissipativity, IEEE Trans. Cybern., to be published, doi:10.1109/TCYB.2018. 2824853. [7] C.E.D. Souza, M.D. Fragoso, H∞ filtering for discrete-time linear systems with Markovian jumping parameters, Int. J. Robust Nonlinear Control 13 (14) (2003) 1299–1316. [8] L. Zhang, E.K. Boukas, Mode-dependent H∞ filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities, Automatica 45 (6) (2009) 1462–1467. [9] H. Li, Y. Wang, D. Yao, R. Lu, A sliding mode approach to stabilization of nonlinear Markovian jump singularly perturbed systems, Automatica 97 (2018) 404–413. [10] D. Yao, B. Zhang, P. Li, H. Li, Event-triggered sliding mode control of discrete-time Markov jump systems, IEEE Trans. Syst. Man Cybern. Syst., to be published, doi:10.1109/TSMC.2018.2836390. [11] R. Lu, J. Tao, P. Shi, H. Su, Z.-G. Wu, Y. Xu, Dissipativity-based resilient filtering of periodic Markovian jump neural networks with quantized measurements, IEEE Trans. Neural Netw. Learn. Syst. 29 (5) (2018) 1888–1899. [12] H. Liang, L. Zhang, H.R. Karimi, Q. Zhou, Fault estimation for a class of nonlinear semi-Markovian jump systems with partly unknown transition rates and output quantization, Int. J. Robust Nonlinear Control 28 (18) (2018) 5380–5962. [13] H. Chen, J. Gao, T. Shi, R. Lu, H∞ control for networked control systems with time delay, data packet dropout and disorder, Neurocomputing 179 (2016) 211–218. [14] H. Zhang, J. Wang, Robust two-mode-dependent controller design for networked control systems with random delays modelled by Markov chains, Int. J. Control 88 (12) (2015) 2499–2509. [15] S. Aberkane, Bounded real lemma for nonhomogeneous Markovian jump linear systems, IEEE Trans. Autom. Control 58 (3) (2013) 797–801.

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