Adaptive sliding mode controller design of Markov jump systems with time-varying actuator faults and partly unknown transition probabilities

Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

Contents lists available at ScienceDirect

Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Adaptive sliding mode controller design of Markov jump systems with time-varying actuator faults and partly unknown transition probabilities Deyin Yao a , Ming Liu b , Renquan Lu a, *, Yong Xu a , Qi Zhou a a b

School of Automation, Guangdong University of Technology, Guangzhou 510006, Guangdong, China School of Astronautics, Harbin Institute of Technology, Harbin 150001, China

article

info

Article history: Received 19 November 2016 Accepted 21 July 2017

Keywords: Markov jump systems Sliding mode control Adaptive sliding mode controller Partially known transition probabilities

a b s t r a c t This paper addresses the stabilization problem of a class of Markov jump systems with time-varying actuator faults and partially known transition probabilities via the sliding mode control method. In this study, the exact information of the time-varying actuators faults, nonlinearities and external disturbances considered in this paper is unknown for the controller design. By Lyapunov stability theory, some sufficient conditions of stochastic stability for Markov jump systems in the existence of actuators faults, unknown nonlinearities and unknown perturbation are derived. Finally, the effectiveness of the proposed control method is illustrated by a simulation example. © 2017 Published by Elsevier Ltd.

1. Introduction Markov jump systems (MJSs), which are composed of stochastic switched systems [1–6] , have received much attention in recent years [7–11] due to its practical ability of modelling. Many representative results on the analysis and synthesis for MJSs have been published in [12–20] and the references therein. To mention a few, the problem of H∞ filtering for continuous-time MJSs was investigated in [14], where the nonlinearities, time-varying actuator faults and external disturbances problems are not studied. In [19], the problem of robust H∞ filtering for uncertain Markovian jump linear systems (MJLSs) with timevarying delays, which are related to the system mode, was studied. But, the authors did not consider the nonlinearities, time-varying actuator faults and external disturbances problems. The inevitable uncertainties [21–24] including modelling error, parameter perturbations and external disturbances, derived from the practical systems, may cause the control systems to be performance degradation. As is well known, sliding mode control (SMC) is a type of variable-structure control schemes, the main features of which are completely adjusted to the inevitable uncertainties. Therefore, the SMC method is recognized as an effective control algorithm to address the systems with faults, perturbations and disturbances in [25–35]. In [32], the SMC problem for mechanical systems in spite of uncertainties and disturbances was discussed. SMC problem for uncertain nonlinear systems was proposed in [29]. More recently, the SMC problems for MJSs and singular MJSs have been extensively investigated [36–39]. For example, the problem of SMC controller design for networked control systems with packet losses was studied in [36]. The design problem based on sliding mode observer for MJSs subject to actuator faults was investigated in [38]. It should be noted that the problems of MJSs or MJLSs mentioned above are based on the assumption that the transition probabilities are totally known. In fact, the exact information of the transition probabilities is hard to obtain in practical

*

Corresponding author. E-mail addresses: [email protected] (D. Yao), [email protected] (M. Liu), [email protected] (R. Lu), [email protected] (Y. Xu), [email protected] (Q. Zhou). https://doi.org/10.1016/j.nahs.2017.07.007 1751-570X/© 2017 Published by Elsevier Ltd.

106

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

applications [40,41]. As a result, many studies on MJSs with unknown transition probabilities were shown up [18,42–47] . For instance, the problems of a class of continuous-time and discrete-time MJLSs with partly unknown transition probabilities were discussed in [44]. The robust control problem of stochastic stability for MJLSs with respect to norm-bounded transition probabilities was considered in [42]. However, the research of MJSs subjected to time-varying actuator faults and partly unknown transition probabilities is still an open problem, which motivates us to do this work. In this paper, the adaptive SMC problem for MJSs with respect to time-varying actuator faults, partly unknown transition probabilities, unknown matched nonlinearity and unknown external disturbance is tackled. The main contributions of this paper are summarized as follows: firstly, this paper investigates the adaptive SMC of MJSs subject to unknown matched perturbations, unknown matched nonlinearities, unknown transition probabilities, and time-varying actuator faults, which are more general in the realistic industrial systems. Secondly, different from the designed mode-dependent sliding mode surface in [48], in this paper, the stochastic stability of the overall closed-loop system is given in Theorem 2 when system state cannot reach the designed sliding mode surface that relates to system mode. Thirdly, compared with the existing control approaches for MJSs [16,36], a novel adaptive sliding mode controller is constructed to guarantee that all state variables of MJSs can be driven onto the designed sliding mode surface in the presence of the nonlinearities, time-varying actuator faults, and external disturbances in this paper. Finally, a numerical example is provided to show the effectiveness of the proposed scheme. The organization of this paper is given as follows. In Section 2, the system description and some preliminaries are given. Sections 3 and 4 present the design of sliding mode surface and adaptive sliding mode controller, respectively. Section 5 provides the stochastic stability of the closed-loop system. Section 6 provides a numerical example to verify the validity of the proposed control method, and Section 7 concludes this paper. Notations: The notation Pi > 0 (Pi ≥ 0) indicates that Pi is real symmetric and positive definite (semi-positive definite). Im represents m-dimensional square matrix. ∥·∥1 and ∥·∥ are respectively the 1-norm and usual Euclidean vector norm. The notation diag (·) represents a diagonal matrix. The notation (Ω , F , P ) stands for the probability space, where Ω , F and P represent the sample space, σ -algebra of subsets of the sample space and probability measure on F , respectively. E {·} and Pr (·) represent the mathematical expectation and probability, respectively. The star ‘‘∗ ’’ indicates a symmetric term. 2. Problem formulation Consider the following continuous-time MJSs: x˙ (t) = A(rt )x (t ) + B(rt ) uF (t ) + f (x, t) + w (t ) ,

)

(

(1)

where x(t) ∈ Rn , uF (t) ∈ Rm , f (x, t) ∈ Rm and w (t) ∈ Rm are respectively the system state, fault control input, nonlinearity and external disturbance. A(rt ) ∈ Rn×n and B(rt ) ∈ Rn×m are constant system matrices with appropriate dimensions. Besides, let A(rt ) = Ai and B(rt ) = Bi for each mode i. {rt , t ≥ 0} is a right-continuous finite-state Markov process, which takes values in set N = {1, 2, . . . , N }. The transition rates matrix Π = (πij )N ×N of the Markov chain is described as Pr (rt +l = j | rt = i) =

{

πij l + o (l) , i ̸= j, 1 + πii l + o (l) , i = j,

(2)

∑ where l > 0 and liml→0 o(l)/l = 0, and for i ̸ = j, πij > 0 denotes the transition rate from i to j. Besides, πii = − j̸=i πij for i, j∈N. Remark 1. It should be pointed out that the unmatched cases for the nonlinearity and the external disturbance have been considered and addressed in many existing work such as [49,50]. In [49], by utilizing the decomposition method, the unmatched perturbation φ can be decomposed into φ = φm + φu , where φm = BB+ φ and φu = B⊥ B⊥+ φ denote the matched perturbation and the unmatched perturbation, respectively. And B+ is the left pseudo-inverse of B, and the columns of B⊥ span the null space of BT . In this setting, the SMC problem for unmatched nonlinearity and disturbance can be well solved, but the H∞ control approach is introduced simultaneously to tackle the unmatched perturbation φu . In our paper, we did not consider the unmatched case for the following reasons. Firstly, the H∞ control approach is utilized to handle the unmatched perturbations such that the asymptotic stability of the studied system cannot be guaranteed. Secondly, the unmatched cases for the nonlinearity and the external disturbance will unavoidably burden the computational complexity. Thirdly, the unmatched case has not been considered since it is not the main focus of this paper. Therefore, based on the aforementioned discussion, the matched case for the nonlinearity and the external disturbance in this paper is considered. Moreover, assume that the transition rates of the jumping process are partly accessed. For instance, there are four operation modes in system (1), and the transition rates matrix Π may be expressed as

⎡

? ⎢ ?

Π =⎣ π31 π41

?

π22

π13 π23

? ?

? ?

⎤

? ? ⎥

π34 ⎦ π44

,

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

107

where ‘‘?’’ denotes the completely unknown elements. For notational convenience, ∀i ∈ N , we represent N = Nκi ∪ Nuiκ with △

△

Nκi = j : πij is known , Nuiκ = j : πij is unknown .

{

}

{

}

Moreover, if Nκi ̸ = ∅, it can be further described as i Nκi = κ1i , . . . , κm , ∀1 ≤ m ≤ N ,

(

)

i where κm ∈ N+ represents the mth known element with the index κmi in the ith row of matrix Π . Finally, let πκi = j∈Nκi πij throughout this paper. It is well known that, systems, the actuator faults unavoidably arise. Suppose the general actuator [ in the realistic industrial ]T fault model uFσ (t) ≜ uF1σ (t), . . . , uFmσ (t) , which is shown as

∑

uFhσ (t) = ρhσ uh (t) + ςh (t),

(3)

where ρhσ is the unknown actuator efficiency factor (h = 1, 2, . . . , m, σ = 1, 2, . . . , ν ), ςh (t) stands for the unknown time-varying fault-deviation in (the hth actuator)and it is bounded, h and σ denote the hth actuator and the σ th fault mode, σ respectively. Denote ρ σ ≜ diag ρ1σ , ρ2σ , . . . , ρm , and the sets of ρhσ are defined by

]} { [ ( ) △ρ σ ≜ ρ σ : ρ σ = diag ρ1σ , ρ2σ , . . . , ρmσ , ρhσ ∈ ρ σh , ρ σh , where 0 < ρ σ ≤ ρ σh ≤ 1 with ρ σ and ρ σh being known constants. h

h

Remark 2. When 0 < ρ σ ≤ ρhσ ≤ ρ σh ≤ 1, it implies that the hth actuator in the σ th fault mode occurs actuator faults. h For the hth actuator in the σ th fault mode, ρ σ = ρ σh = 1 represents that there is no fault. According to Eq. (3), it can be h σ σ seen that when ρ = ρ h = 0 (h = 1, 2, . . . , m, σ = 1, 2, . . . , ν ), the hth actuator in the hth fault mode will not work. h Meanwhile, the control system has no control input such that the stability of the control system cannot be ensured when ρ σh = ρ σh = 0. According to the work of [51], we can obtain that the unknown time-varying fault-deviation ς (t ) denotes a vector function corresponding to the portion of the control action produced by the actuator that is completely out of control, and ς (t ) satisfies norm-bounded condition. Hence, in this paper, the condition ρ σ = ρ σh = 0 is not considered. h

as

Based on the aforementioned discussion, we have uFσ (t) = ρ σ u(t) + ς (t). For simplicity, the fault model can be described uF (t) = ρ u(t) + ς (t),

(4) ν

△

(

where ρ ≜ diag (ρ1 , . . . , ρm ) ∈{ρ , . . . , ρ }∈△ρ σ , ς (t ) = [ς1 (t ) . . . ςm (t )] . Besides, we define ρ = diag ρ , ρ , . . . , ρ 1

T

1

ρ¯ = diag (ρ¯ 1 , ρ¯ 2 , . . . , ρ¯ m ) . Consequently, system (1) can be rewritten as

2

x˙ (t) = Ai x(t) + Bi (ρ u(t) + ς (t) + f (x, t ) + w (t )) .

) m

,

(5)

The nonlinearity f (x, t ) meets ∥f (x, t )∥ ≤ α + β ∥x (t )∥ , where α > 0 and β > 0 are unknown constants. The △ △ external disturbance vector w (t ) = [w1 (t ) . . . wm (t )]T and fault-deviation vector ς (t ) = [ς1 (t ) . . . ςm (t )]T respectively ¯ h and ς ≤ ςh (t ) ≤ ς¯ h , h = 1, 2, . . . , m, where wh , w ¯ h , ς and ς¯ h are unknown constants. satisfy w h ≤ wh (t ) ≤ w h

△

h

]T

△

△

[

]T

Besides, w (t ) = w 1 (t ) . . . w m (t ) , w ¯ (t ) = [w ¯ 1 (t ) . . . w ¯ m (t )]T , ς (t ) = ς (t ) . . . ς (t )

[

Furthermore, we assume rank (Bi ) = m. The following definitions and lemma are adopted in this paper.

1

m

△

, ς¯ (t ) = [ς¯ 1 (t ) . . . ς¯ m (t )]T .

Definition 1 ([52]). Denote V (x(t), rt , t ≥ 0) = V (x(t), i) satisfied twice differentiable on x (t ) be a Lyapunov functional candidate. The infinitesimal operator LV (x(t), i) is defined as follows: LV (x(t), i) = lim

△t →0+

) } ] 1 [ { ( E V x (t + △t ) , rt +△t | x(t), rt = i − V (x(t), i) .

△t

Definition 2 ([53]). If for every initial condition x0 ∈ Rn and original mode r0 , the following requirement is satisfied: ∞

{∫ E

} ∥x(t)∥2 dt | x0 , r0 < +∞,

0

then the system (1) with u(t) = 0 is said to be stochastically stable. Definition 3 ([48]). If for ϕ ≥ 0 and ϖ > 0, we have

{

ϕ,x lim P sup ⏐xt ⏐ > ϖ

x→0

⏐

ϕ
⏐

}

= 0, P

{

}

ϕ, x lim ⏐xt ⏐ = 0

t→+∞

⏐

⏐

= 1,

108

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122 ϕ, x

where xt denotes the solution at time t of MJSs starting from the state x at time ϕ for ϕ ≤ t . Then the equilibrium point, xt = 0, of MJSs (5) subject to u(t) = 0 is said to be globally asymptotically stable (with probability one). Lemma 1 ([13]). The unforced system x˙ (t ) = Ai x (t ) is stochastically stable if and only if, there exists a set of Pi > 0, i ∈ N satisfying the following condition: Pi Ai + ATi Pi + Qj < 0, where Qj =

(6)

πij Pj .

∑ j∈N

The purpose of this paper is to develop the adaptive SMC approach to stabilize MJSs (5) with time-varying actuators faults and partly unknown transition probabilities such that the obtained sliding mode dynamics are stochastically stable. 3. Sliding mode surface design In this section, we synthesize the sliding mode surface as follows: s (x(t), i) = Γi x(t) = BTi Xi x(t),

(7)

where Xi > 0 is to be determined for i ∈ N . In the following, define a nonsingular matrix Wi and an associated state vector υ (t ) as follows: Wi =

[ T] B˜ i Γi

[ =

]

B˜ Ti

] [ υ1 (t ) υ (t ) = υ2 (t )

,

BTi Xi

with B˜ i being an orthogonal complement of Bi for each i ∈ N , and

[ υ (t) =

B˜ Ti

BTi Xi

] x(t),

(8)

where υ1 (t ) ∈ Rn−m , υ2 (t ) = s (x (t ) , i) ∈ Rm . Thus, x (t ) = Wi−1 υ (t ) ,

[

(9)

(

where Wi−1 = Xi−1 B˜ i B˜ Ti Xi−1 B˜ i

[

υ˙ 1 (t) A¯ (i) = ¯ 11 υ˙ 2 (t) A21 (i)

]

[

)−1 ][

A¯ 12 (i) A¯ 22 (i)

(

Bi BTi Xi Bi

)−1

]

. By the state transformation υ (t ) = Wi x (t ) , we have

] υ1 (t) 0 + (ρ u(t) + ς (t) + f (x, t) + w (t )) , υ2 (t) Γi Bi ]

[

(10)

where

(

A¯ 11 (i) = B˜ Ti Ai Xi−1 B˜ i B˜ Ti Xi−1 B˜ i A¯ 12 (i) = B˜ Ti Ai Bi BTi Xi Bi

(

)−1

)−1

,

(

A¯ 21 (i) = BTi Xi Ai Xi−1 B˜ i B˜ Ti Xi−1 B˜ i A¯ 22 (i) = BTi Xi Ai Bi BTi Xi Bi

(

,

)−1

)−1

,

.

Then, when state trajectories are attracted to the predefined sliding mode surface s (x(t), i) = 0, i.e., υ2 (t) = s (x(t), i) = 0, we can achieve the following sliding mode dynamics:

υ˙ 1 (t) = A¯ 11 (i)υ1 (t).

(11)

Theorem 1. The sliding mode dynamics with partly unknown probabilities in (11) are stochastically stable if there exists Xi > 0 ∈ Rn×n such that for each i ∈ N ,

[( ⎡⎛

1 + πκi

)(

Xi Ai + ATi Xi + πii Xi

)

∗ ⎞

∑ ( ) ⎢⎝1 + πij ⎠ Xi Ai + ATi Xi ⎢ ⎣ j∈Nκi , j̸ =i ∗

] Λiκ < 0, Ψκi ⎤

∀i ∈ Nκi

(12)

Λiκ ⎥ ⎥ < 0, ⎦

∀i ̸∈ Nκi

(13)

Ψκi

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

Xi Ai + ATi Xi

∗

109

˜ ( Bi ) ˜BTj Xj − 2I B˜ j ≤ 0,

∀j ∈ Nuiκ , j ̸= i

(14)

Xi Ai + ATi Xi + Xj ≥ 0,

∀j ∈ Nuiκ , j = i

(15)

]

[

where

[ √

] √ √ √ ˜ ˜ ˜ ˜ πi,κ i Bi , . . . , πi,κ i Bi , πi,κ i Bi , . . . , πi,κmi Bi , Λκ = 1 i −1 i+1 ( ( ) ( ) Xκ i − 2I B˜ κ i , Ψκi = diag B˜ Tκ i Xκ i − 2I B˜ κ i , . . . , B˜ Tκ i 1 1 i−1 i−1 1 i−1 ) ( ) ( ) B˜ T i Xκ i − 2I B˜ κ i , . . . , B˜ T i Xκ i − 2I B˜ κ i . κ κ m m i

i +1

i+1

i+1

m

Proof. Based on Lemma 1, if there exists Pi > 0, i ∈ N satisfying Pi A¯ 11 (i) + A¯ T11 (i) Pi +

N ∑

πij Pj < 0,

(16)

j=1

then the sliding mode dynamics (11) are stochastically stable. Pre- and post-multiplying (16) by Pi−1 and Pi−T , respectively, let B˜ Ti Xi−1 B˜ i = Pi−1 , we have N ∑

Ai Xi−1 + Xi−1 ATi + πii Xi−1 + Xi−1 B˜ i

( )−1 ( ) πij B˜ Tj Xj−1 B˜ j B˜ Ti Xi−1 < 0,

(17)

j=1,j̸ =i

pre- and post-multiplying (17) by Xi and XiT , respectively, we can obtain Xi Ai + ATi Xi + πii Xi + B˜ i

N ∑

( )−1 πij B˜ Tj Xj−1 B˜ j B˜ Ti < 0.

(18)

j=1,j̸ =i

Eq. (18) is equivalent to Xi Ai + ATi Xi + πii Xi + B˜ i

( )−1 πij B˜ Tj Xj−1 B˜ j B˜ Ti + B˜ i

∑ j∈Nκi ,j̸ =i

∑

( )−1 πij B˜ Tj Xj−1 B˜ j B˜ Ti < 0.

(19)

j∈Nuiκ ,j̸ =i

Consider two cases i ∈ Nκi and i ∈ Nuiκ , respectively. Case I ∑ : i ∈ Nκi . Since j∈N πij = 0, the left-hand side of (19) can be rewritten as:

( )−1 πij B˜ Tj Xj−1 B˜ j B˜ Ti

∑

△

Υi = Xi Ai + ATi Xi + πii Xi + B˜ i

j∈Nκi ,j̸ =i

(

∑

+ B˜ i

πij B˜ Tj Xj−1 B˜ j

)−1

B˜ Ti +

∑

( ) πij Xi Ai + ATi Xi .

j∈N

j∈Nuiκ ,j̸ =i

Thus, we have

⎛

⎞

Υi = ⎝1 +

∑

( )−1 ∑ ( ) πij ⎠ Xi Ai + ATi Xi + πii Xi + B˜ i πij B˜ Tj Xj−1 B˜ j B˜ Ti

j∈Nκi

+ B˜ i

∑ j∈Nuiκ ,j̸ =i

j∈Nκi ,j̸ =i

(

πij B˜ Tj Xj−1 B˜ j

)−1

∑

B˜ Ti +

( ) πij Xi Ai + ATi Xi .

j∈Nuiκ

Hence, the inequality (19) can be guaranteed by

⎧⎛ ⎞ ⎪ )−1 ∑ ( ∑ ⎪ ( ) ⎪ ⎪ πij ⎠ Xi Ai + ATi Xi + πii Xi + B˜ i πij B˜ Tj Xj−1 B˜ j B˜ Ti < 0, ⎨ ⎝1 + j∈Nκi

⎪ ⎪ ⎪ ⎪ ⎩

(20)

j∈Nκi

(

B˜ i B˜ Tj Xj−1 B˜ j

)−1

Then, according to (12) and (14), we can achieve Υi < 0.

B˜ Ti + Xi Ai + ATi Xi ≤ 0,

∀j ∈ Nuiκ , j ̸= i.

110

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

Case II : i ∈ Nuiκ . Similarly, the left-hand side of (19) can be described as

⎞

⎛ △

Ξi = ⎝1 +

)−1 ( ∑ ) ( B˜ Ti πij B˜ Tj Xj−1 B˜ j πij ⎠ Xi Ai + ATi Xi + B˜ i

∑

+ πii Xi Ai +

(

(21)

i ,j̸ =i

j∈Nκi , j̸ =i

j∈Nκ

ATi Xi

∑

+ Xi + B˜ i

)

(

πij B˜ Tj Xj−1 B˜ j

)−1

B˜ Ti +

∑

) ( πij Xi Ai + ATi Xi .

j∈Nuiκ , j̸ =i

j∈Nuiκ ,j̸ =i

Thus, (21) can be ensured by

⎧⎛ ⎪ ⎪ ⎪ ⎝1 + ⎪ ⎪ ⎪ ⎨

⎞ ∑

)−1 ( ∑ ) ( B˜ Ti < 0, πij B˜ Tj Xj−1 B˜ j πij ⎠ Xi Ai + ATi Xi + B˜ i

j∈Nκi , j̸ =i

j∈Nκi ,j̸ =i

Xi Ai + ATi Xi + Xj ≥ 0,

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(

Xi Ai + ATi Xi + B˜ i B˜ Tj Xj−1 B˜ j

)−1

B˜ Ti ≤ 0,

∀j ∈ Nuiκ , j = i,

(22)

∀j ∈ Nuiκ , j ̸= i.

Obviously, Ξi < 0 can be obtained by (13), (14) and (15) by using Schur complement. Hence, the proof is over. ■ Remark 3. Notice that (12) and (15) in Theorem 1 cannot be checked at the same time due to Nκi ∩ Nuiκ = ∅. 4. Adaptive sliding mode controller design In this section, an adaptive sliding mode controller is synthesized to guarantee the stochastic stability of the closed-loop system in (5) against actuator fault, nonlinearity and disturbance. However, the parameters of actuator fault, nonlinear function f (x, t) and external disturbance w (t) considered in this paper are unmeasurable and unavailable due to the complicated model structure such that the adaptive control scheme is utilized to estimate these unknown parameters as follows. ˆh (t), w Let ρˆ h (t), αˆ (t), βˆ (t), w ˆh (t), ˆ ς h (t) and ˆ ς h (t) estimate ρh , α, β, w ¯ h , wh , ς h and ς h (h = 1, . . . , m), respectively. Some estimated values are defined as

[ ] [ ]T ˆ1 (t), . . . , w ˆm (t) T , w ˆ(t) = w w ˆ1 (t), . . . , w ˆ(t) = w ˆm (t) , [ ]T [ ]T ˆ ς (t) = ˆ ς 1 (t), . . . , ˆ ς m (t) , ˆ ς (t) = ˆ ς 1 (t), . . . , ˆ ς m (t) . ] [ (i) (i) (i) Then, let Bi = b1 , . . . , bm , where bh ∈ Rn denotes the hth column of Bi . Define the following variable δ (t) ≜ diag (δ1 (t), . . . , δm (t))

(23) (24)

(25)

with δh (t) (h = 1, . . . , m) satisfying

δh (t) ≜ min{1 − sgn(xT (t)Xi bh(i) ), 1},

(26)

where sgn(·) represents the sign function. Remark 4. The parameter δh (t) (h = 1, . . . , m) is essential for the following primary results. Notice that δh (t) = 0 when (i) (i) xT (t)Xi bh > 0; δh (t) = 1 when xT (t)Xi bh ≤ 0. Define the following vector-valued functions

ˆ(t) , ˆ(t) + δ (t) w ˆ(t) − w T (t) ≜ w (

)

M(t) ≜ ˆ ς (t) + δ (t) ˆ ς (t) − ˆ ς (t) ,

(

)

and decompose T (t) and M(t) as T (t) ≜ [T1 (t), . . . , Tm (t)]T ,

Th (t) ∈ R, M(t) ≜ [M1 (t), . . . , Mm (t)]T , Mh (t) ∈ R, (h = 1, . . . , m). √[ ] [ ]T By (9), define ω(t , υ ) ≜ υ1T (t) υ2T (t) Wi−T Wi−1 υ1T (t) υ2T (t) = ∥x(t)∥ . Introduce the following functions △

g(t , υ ) = αˆ (t) + βˆ (t)ω(t , υ ) ∈ R,

(27)

G(t , υ ) ≜ [g(t , υ ), . . . , g(t , υ )] ∈ R . T

m

It can be observed that g(t , υ ) is equivalent to the upper bound of the nonlinear function f (x, t).

(28)

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

111

Note that the designed controller u(t , i) can dispel the effects of terms 2xT (t)Xi Bi f (x, t), 2xT (t)Xi Bi w (t) and 2xT (t)Xi Bi ς (t) by altering unknown parameter ρ in the closed-loop system (5). Therefore, we analyse these terms in order to implement the bottom controller design. The term 2xT (t)Xi Bi f (x, t ) can be converted into the following form:

2xT (t)Xi Bi f (x, t ) ≤

⎧ m ∑ ⎪ ⎪ (i) ⎪ 2 xT (t)Xi bh (α + β ∥x(t)∥) ⎪ ⎪ ⎨

(i)

xT (t)Xi bh > 0,

h=1

m ⎪ ∑ ⎪ (i) ⎪ ⎪ − 2 xT (t)Xi bh (α + β ∥x(t)∥) ⎪ ⎩

(29) x

T

(i) (t)Xi bh

≤ 0.

h=1

By (25) and (26), (29) can be further rewritten as 2xT (t)Xi Bi f (x, t) ≤ 2

m ∑

(i)

xT (t)Xi bh (1 − 2δh (t)) (α + β ∥x(t)∥) .

(30)

h=1

Similarly, the terms 2xT (t)Xi Bi w (t) and 2xT (t)Xi Bi ς (t) become 2xT (t)Xi Bi w (t) ≤ 2

m ∑

(i)

)] [ ( ¯h , w ¯ h + δh (t) wh − w

(31)

(i)

[ ( )] ς¯ h + δh (t) ς h − ς¯ h .

(32)

xT (t)Xi bh

h=1 m

2xT (t)Xi Bi ς (t) ≤ 2

∑

xT (t)Xi bh

h=1

From the above discussion, the adaptive sliding mode controller u (t , i) is designed as follows: u (t , i) ≜ u1 (t , i) + u2 (t , i) + u3 (t , i) + u4 (t , i) + u5 (t , i)

(33)

with

⎧ ( ) u1 (t , i) ≜ −Γi ρˆ (t) x(t), ⎪ ⎪ ⎪ ⎪ −1 ⎪ ⎪ ⎨ u2 (t , i) ≜ −ρˆ (t)T (t), u3 (t , i) ≜ −ρˆ −1 (t) (Im − 2δ (t)) G (t , υ) , ⎪ ⎪ ⎪ ⎪ u4 (t , i) ≜ −ρˆ −1 (t)M(t), ⎪ ⎪ ⎩ u5 (t , i) ≜ −γi sgn(s (x, i)), where γi is a positive scalar for any i ∈ N . The adaptive laws of ρˆ h (t) (h = 1, 2, . . . , m) are designed as follows:

ˆ ρ˙ h (t) ≜ Proj[mink {ρ k },maxk {ρ k }] {Lh (t)} h h ⎧ if ρ ˆ (t) = min{ρ kh }, and Lh (t) ≤ 0 ⎪ h ⎨ k 0 = or ρˆ h (t) = max{ρ kh }, and Lh (t) ≥ 0 ⎪ k ⎩ Lh (t) otherwise, [ (i) (i) Lh (t) ≜ − xT (t)Xi Bi Γih x(t) + ρˆ h−1 (t)xT (t)Xi bh Th (t) + ρˆ h−1 (t)xT (t)Xi bh Mh (t) ] + ρˆ h−1 (t)xT (t)Xi b(hi) (1 − 2δh (t)) g (t , υ) ~h ,

(34)

]

[

where ~h is a positive parameter. Suppose ρˆ h (0) ̸ = 0. Proj {·} can project the estimation ρˆ h (t ) to the range ρ , ρ¯ h . Besides, h

ˆh (t), w the adaptive laws of αˆ (t), βˆ (t), w ˆh (t), ˆ ς h (t) and ˆ ς (t) are designed as follows: h

α˙ˆ (t) ≜ cα

m ∑

(i)

xT (t)Xi bh (1 − 2δh (t)) ,

β˙ˆ (t) ≜ cβ

(i)

xT (t)Xi bh (1 − 2δh (t)) ω (t , υ) ,

(35)

h=1

h=1

˙ (t) ≜ s xT (t)X b(i) , ˆ w h h i h

m ∑

˙ h (t) ≜ sh xT (t)Xi b(i) , ˆ w ˆ ς˙ h (t) ≜ qh xT (t)Xi b(i) ˆ ς˙ h (t) ≜ qh xT (t)Xi b(i) h h , h ,

where cα , cβ , sh and qh (h = 1, . . . , m) are given positive scalars. 5. Stability analysis of the closed-loop system

˜(t) ≜ w ˆ(t) − w, w The definitions of error variables are set as ρ˜ (t) ≜ ρˆ (t) − ρ, w ˜(t) ≜ w ˆ(t) − w, ˜ ς (t) ≜ ˆ ς (t) − ς and ˜ ς (t) ≜ ˆ ς (t) − ς . The stochastic stability of system (5) can be ensured by the following theorem.

112

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

Theorem 2. For each i ∈ N , assume that the inequalities (12)– (15) are feasible, then the adaptive sliding mode controller in (33) can ensure MJSs (5) to be stochastically stable, if there exist Xi > 0 ∈ Rn×n , Γih ∈ Rn×n (h = 1, 2, . . . , m) such that

[(

1 + πκi

)(

Xi Ai + ATi Xi + πii Xi

Ωi

)

∗

⎡⎛

] < 0,

𭟋i

⎢⎝1 + πij ⎠ Xi Ai + ATi Xi ⎢ ⎣ i j∈Nκ , j̸ =i ∗ (

Xi Ai + ATi Xi + Xj ≤ 0, Xi Ai +

ATi Xi

(36)

⎤

⎞ ∑

∀i ∈ Nκi

+ Xj ≥ 0,

Ωi ⎥ ⎥ < 0, ⎦

)

∀i ∈ Nuiκ

(37)

𭟋i

∀j ∈ Nuiκ , j ̸= i

(38)

,j = i

(39)

∀j ∈

Nuiκ

where

[ √

πi,κ i Xκ i , . . . , i+1 i+1 ] √ √ √ √ T T mXi Bi ρ, ¯ mXi Bi , 2(Yi1 ρ¯ 1 ) , . . . , 2(Yim ρ¯ m ) , ⎛

Ωi =

πi,κ i Xκ i , . . . , 1

1

√

πi,κ i Xκ i , i −1

i −1

𭟋i = −diag ⎝Xκ i , . . . , Xκ i , Xκ i 1

i−1

√

i+1

√ πi,κmi Xκmi ,

⎞ m    , . . . , Xκmi , Bi Xi Bi , Bi Xi Bi , Xi , . . . , Xi ⎠ .

Proof. The Lyapunov function is selected as V1 (t) = xT (t)Xi x(t) +

m ∑ ρ˜ 2 (t) h

h=1

~h

+

α˜ 2 (t) cα

+

β˜ 2 (t) cβ

+

m ∑ 1 2 ˜h (t) w h=1

sh

m

m m ) ∑ ) ∑ δh ( ∑ 1 2 δh ( 2 2 ˜ ˜2h (t) + + w ˜2h (t) − w ς h (t) + ˜ ς h (t) − ˜ ς h (t) , h=1

sh

h=1

qh

h=1

qh

where ~h , cα , cβ , sh and qh are given in (34) and (35). The weak infinitesimal generator of V1 (t) is derived as LV1 (t )

⎛ = x (t ) ⎝Xi Ai + T

ATi Xi

+

N ∑

⎞ ( ) πij Xj ⎠ x (t ) − 2xT (t ) Xi Bi ρ Γi ρˆ (t ) x (t ) + 2xT (t ) Xi Bi f (x, t )

j=1

+ 2x (t ) Xi Bi w (t ) + 2xT (t ) Xi Bi ς (t ) − 2xT (t ) Xi Bi ρ ρˆ −1 (t ) T (t ) − 2xT (t ) Xi Bi ρ ρˆ −1 (t ) M (t ) T

− 2xT (t ) Xi Bi ρ ρˆ −1 (t ) (Im − 2δ (t )) G (t , υ) − 2xT (t ) Xi Bi ργi sgn (s (x, i)) m m ∑ ∑ (i) (i) + 2α˜ (t ) xT (t ) Xi bh (1 − 2δh (t )) + 2β˜ (t ) xT (t ) Xi bh (1 − 2δh (t )) ω (t , υ) h=1

+2

+2

m ∑

h=1 (i)

˜ h (t ) + 2 xT (t ) Xi bh w

h=1

h=1

m ∑

m ∑

(i) ς h (t ) + 2 xT (t ) Xi bh ˜

∑

−2

h=1

(

)

(i)

(

)

xT (t ) Xi bh δh (t ) ˜ ς (t ) − ˜ ς h (t ) h

(i)

xT (t ) Xi bh Γih ρ˜ h (t ) x (t ) − 2

h=1 m ∑

(i)

˜h (t ) xT (t ) Xi bh δh (t ) w ˜h (t ) − w

h=1

h=1 m

−2

m ∑

m ∑

(i)

xT (t ) Xi bh ρ˜ h (t ) ρˆ h−1 (t ) Th (t )

h=1 (i)

xT (t ) Xi bh ρ˜ h (t ) ρˆ h−1 (t ) (1 − 2δh (t )) g (t , υ) − 2

m ∑ h=1

(i)

xT (t ) Xi bh ρ˜ h (t ) ρˆ h−1 (t ) Mh (t ) .

(40)

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

113

According to ρ˜ (t ) ≜ ρˆ (t ) − ρ, by (28) and (30), one can obtain

− 2xT (t ) Xi Bi ρ ρˆ −1 (t ) (Im − 2δ (t )) G (t , υ) m ∑ (i) −2 xT (t ) Xi bh ρ˜ h (t ) ρˆ h−1 (t ) (1 − 2δh (t)) g (t , υ) h=1 m ∑

+ 2β˜ (t )

(i)

xT (t ) Xi bh (1 − 2δh (t )) ω (t , υ)

h=1 m ∑

+ 2α˜ (t )

(i)

xT (t ) Xi bh (1 − 2δh (t )) + 2xT (t ) Xi Bi f (x, t ) ≤ 0.

(41)

h=1

˜ (t ) ≜ w ˆ (t ) − w, w According to w ˜ (t ) ≜ w ς (t ) ≜ ˆ ς (t ) − ς and ˜ ς (t ) ≜ ˆ ˆ (t ) − w, ˜ ς (t ) − ς , by (31) and (32), we have 2xT (t ) Xi Bi w (t ) − 2xT (t ) Xi Bi ρ ρˆ −1 (t ) T (t ) − 2

m ∑

(i)

xT (t ) Xi bh ρ˜ h (t ) ρˆ h−1 (t ) Th (t )

h=1

+2

m ∑

(i)

˜h (t ) + 2 xT (t ) Xi bh w

m ∑

(i)

˜h (t ) ≤ 0, ˜h (t ) − w xT (t ) Xi bh δh (t ) w (

)

(42)

h=1

h=1

2xT (t ) Xi Bi ς (t ) − 2xT (t ) Xi Bi ρ ρˆ −1 (t ) M (t ) − 2

m ∑

(i)

xT (t ) Xi bh ρ˜ h (t ) ρˆ h−1 (t ) Mh (t )

h=1

+2

m ∑

(i) xT (t ) Xi bh ˜ ς h (t ) + 2

h=1

m ∑

(

(i)

)

xT (t ) Xi bh δh (t ) ˜ ς (t ) − ˜ ς h (t ) ≤ 0. h

(43)

h=1

) ( )−1 ) ( ( ∑m ∑ T ˆ h (t ) , Γih = BTi Xi Bi BTi Yih , the terms −2xT (t ) Xi Bi ρ Γi ρˆ (t ) x (t ) and −2 m ρˆ (t ) = h=1 Γih ρ h=1 x (t ) ρ˜ (t ) x (t ) in (40) can be rewritten as )−1 ) ( ( − 2xT (t ) Xi Bi ρ Γi ρˆ (t ) x (t ) ≤ mxT (t ) Xi Bi ρ¯ BTi Xi Bi ρ¯ BTi Xi x (t ) m ∑ + xT (t ) (44) (Yih ρ¯ h )T Xi−1 (Yih ρ¯ h ) x (t ) ,

Let Γi (i) Xi bh Γih h

h=1

−2

m ∑

(i)

xT (t ) Xi bh Γih ρ˜ h (t ) x (t ) ≤ mxT (t ) Xi Bi BTi Xi Bi

(

)−1

BTi Xi x (t )

h=1

+ xT (t )

m ∑

(Yih ρ¯ h )T Xi−1 (Yih ρ¯ h ) x (t ) ,

(45)

h=1

and −2xT (t ) Xi Bi ργi sgn (s (x, i)) in (40) can also be simplified as

  − 2xT (t ) Xi Bi ργi sgn (s (x, i)) ≤ −γi ρ max 2xT (t ) Xi Bi  ≤ 0, { } where ρ = max ρ h , h = 1, 2, . . ., m . By (41)–(46), (40) can be simplified as max LV1 (t ) ≤ xT (t ) Σi x (t ) ,

(46)

(47)

where

⎡ ⎢X A + Σi = ⎣ i i Φi =

[√

ATi Xi

mXi Bi ρ, ¯

+

N ∑

⎤ πij Xj

j=1

∗ √

mXi Bi ,

√

Φi ⎥ ⎦, Li

2(Yi1 ρ¯ 1 )T , . . . ,

√

]

2(Yim ρ¯ m )T ,

⎞ m    Li = −diag ⎝Bi Xi Bi , Bi Xi Bi , Xi , . . . , Xi ⎠ . ⎛

If Σi < 0, we have LV1 (t ) < 0 for x (t ) ̸ = 0 from (47). Similar to the proof of Theorem 1, there are respectively two conditions i ∈ Nκi and i ∈ Nuiκ .

114

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

Case III : i ∈ Nκi . ∑N ∑ Since j∈N πij = 0 and πij > 0 (i ̸ = j) , let Pi ≜ Xi Ai + ATi Xi + j=1 πij Xj , and it can be rewritten as

∑

Pi = Xi Ai + ATi Xi + πii Xi +

∑

πij Xj +

j∈Nκi ,j̸ =i

πij Xj +

j∈Nuiκ ,j̸ =i

∑

) ( πij Xi Ai + ATi Xi ,

j∈ N

then, we get

⎛

⎞

Pi = ⎝1 +

∑

∑ ) ( πij ⎠ Xi Ai + ATi Xi + πii Xi + πij Xj

j∈Nκi

∑

+

j∈Nκi ,j̸ =i

πij Xj +

j∈Nuiκ ,j̸ =i

∑

πij Xi Ai +

(

ATi Xi

.

)

j∈Nuiκ

Hence, Pi < 0 can be guaranteed by

⎞ ⎧⎛ ⎪ ∑ ∑ ( ) ⎪ ⎨ ⎝1 + πij ⎠ Xi Ai + ATi Xi + πii Xi + πij Xj < 0, i i j∈Nκ j∈Nκ ,j̸ =i ⎪ ⎪ ⎩ Xj + Xi Ai + ATi Xi ≤ 0,

(48)

∀j ∈ Nuiκ , j ̸= i.

Then, Pi < 0 can be achieved by (36) and (38). Case IV : i ∈ Nuiκ . ∑N Similarly, define Qi ≜ Xi Ai + ATi Xi + j=1 πij Xj , and it can be rewritten as

⎞

⎛

∑ ( ) ( ) πij Xj + πii Xi Ai + ATi Xi + Xi πij ⎠ Xi Ai + ATi Xi +

∑

Qi = ⎝1 +

∑

+

j∈Nκ

∑

πij Xj +

j∈Nuiκ ,j̸ =i

(49)

i ,j̸ =i

j∈Nκi , j̸ =i

( ) πij Xi Ai + ATi Xi .

j∈Nuiκ , j̸ =i

Hence, (49) can be ensured when the following inequalities hold:

⎧⎛ ⎪ ⎪ ⎪ ⎝ ⎪ ⎪ ⎨ 1+

⎞ ∑

∑ ( ) πij ⎠ Xi Ai + ATi Xi + πij Xj < 0,

j∈Nκi , j̸ =i

j∈Nκi ,j̸ =i

⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(50)

Xi Ai + ATi Xi + Xj ≥ 0,

∀j ∈ Nuiκ , j = i,

Xi Ai + ATi Xi + Xj ≤ 0,

∀j ∈ Nuiκ , j ̸= i.

It is straightforward that Qi < 0 can be obtained by (37)–(39). Thus, we can conclude that the closed-loop system with partly unknown probabilities and actuator faults is stochastically stable via the above discussion. This completes the proof. ■ Remark 5. If Nκi = ∅, the closed-loop system will become MJSs with completely unknown transition probabilities, which can be viewed as switched systems under arbitrary switching signal. For ∀i ̸ = j ∈ N , −Xi ≤ −Xj holds according to (38) and (39). Then we can conclude that Xi = Xj = X and ATi X + XAi = −X ≤ 0, which implies that there exists a latent quadratic common Lyapunov functional for all modes. When Nuiκ = ∅, the underlying system becomes MJSs with completely known transition probabilities so that the stochastic stability of the closed-loop system can be easily derived.

5.1. Reachability analysis In this subsection, the reachability of the system state trajectories (5) can be guaranteed by the following theorem. Theorem 3. Assume that Theorems 1 and 2 hold, if there exist positive scalars κ¯ and γi such that Wi ∥x (t )∥ − γi ρ

+ κ¯ ≤ 0, max

{ } ρ max = max ρ h , h = 1, 2, . . ., m ,

∀i ∈ N

where

( 

Wi =  BTi Xi Bi

)−1

 

( 

BTi Xi Ai  + ∥ρ∥ ¯  BTi Xi Bi

m m ( )−1 T  )−1  ∑  ∑ ∥Yih ρ¯ h ∥ +  BTi Xi Bi BTi  ∥Yih ρ¯ h ∥ Bi  h=1

h=1

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

115

  1 ∑  (  )−1 )−1 1 ∑  ( T    + πij BTj Xj Bj BTi Xi  , πij Bj Xj Bj BTi Xi  + 2

2

j∈Nκi

j∈Nuiκ

then, the state trajectories of system (5) can be driven onto the sliding mode surface s (x(t), i) = 0 in finite time with probability one. Proof. The Lyapunov function is chosen as V2 (t ) =

1

[ sT (x (t ) , i) BTi Xi Bi

(

2

m

+

∑ δh ( h=1

sh

)−1

s (x (t ) , i) +

m ∑ ρ˜ 2 (t ) h

α˜ 2 (t )

+

+

β˜ 2 (t )

+

m ∑ 1 2 ˜h (t ) w

~h cα cβ sh h=1 ]h=1 m m ( ) ∑ ) ∑ δh 1 2 2 ˜ ˜2h (t ) + w ˜2h (t ) − w ˜ ς 2h (t ) − ˜ ς h (t ) + ς h (t ) . h=1

qh

h=1

(51)

qh

˙ (t ) = w ˙ (t ) , w ˙ h (t ) = w ˙ h (t ) , ˜ ˜ ˆ Recalling that ˜ ρ˙ h (t ) = ρ˙ˆ h (t ) , w ˜ ˆ α˙ (t ) = ˆ α˙ (t ) , ˜ ς˙ h (t ) = ˆ ς˙ h (t ) , β˙ (t ) = ˆ β˙ (t ) , ˜ h h ς˙ (t ) , by (7), (33) and (34), the infinitesimal operator of V2 (t) is derived as follows: ˜ ς˙ (t ) = ˆ h

h

LV2 (t ) = sT (x (t ) , i) BTi Xi Bi

(

)−1

BTi Xi Ai x (t ) − xT (t ) Xi Bi ρ Γi ρˆ (t ) x (t )

(

)

− xT (t ) Xi Bi ργi sgn (s(x (t ) , i)) − xT (t ) Xi Bi ρ ρˆ −1 (t ) T (t ) m ∑

+

(i)

˜h (t ) − xT (t ) Xi Bi ρ ρˆ −1 (t ) (Im − 2δ) G (t , υ) xT (t ) Xi bh w

h=1

)−1 ( ( )−1 + sT (x (t ) , i) BTi Xi Bi BTi Xi Bi f (x, t ) + sT (x (t ) , i) BTi Xi Bi BTi Xi Bi ς (t ) N ∑ )−1 ( ( )−1 1 + sT (x (t ) , i) BTi Xi Bi BTi Xi Bi w (t ) + sT (x (t ) , i) πij BTj Xj Bj s(x(t), i)

2

∑

−

(i)

xT (t ) Xi bh Γih ρ˜ h (t ) x (t ) −

h=1 m ∑

−

∑

−

(i)

xT (t ) Xi bh ρ˜ h (t ) ρˆ h−1 (t ) (1 − 2δh (t)) g (t , υ)

h=1 (i)

xT (t ) Xi bh ρ˜ h (t ) ρˆ h−1 (t ) Th (t ) + α˜ (t )

h=1 m ∑

j=1

m

m

m ∑

(i)

xT (t ) Xi bh (1 − 2δh (t ))

h=1 (i)

xT (t ) Xi bh ρ˜ h (t ) ρˆ h−1 (t ) Mh (t ) − xT (t ) Xi Bi ρ ρˆ −1 (t ) M (t )

h=1

+ β˜ (t )

m ∑

(i)

xT (t ) Xi bh (1 − 2δh (t )) ω (t , υ) +

h=1

m ∑

(i)

˜ h (t ) xT (t ) Xi bh δh w ˜h (t ) − w (

)

h=1

m

m

h=1

h=1

) ( ∑ ∑ (i) (i) + xT (t ) Xi bh ˜ ς h (t ) + xT (t ) Xi bh δh ˜ ς h (t ) − ˜ ς h (t ) .

(52)

Then, the following inequalities can be obtained sT (x (t ) , i) BTi Xi Bi

(

)−1

BTi Xi Bi f (x, t ) + α˜ (t )

m ∑

(i)

xT (t ) Xi bh (1 − 2δh (t ))

h=1

−

m ∑

(i)

xT (t ) Xi bh ρ˜ h (t ) ρˆ h−1 (t ) (1 − 2δh (t )) g (t , υ)

h=1

+ β˜ (t )

m ∑

(i)

xT (t ) Xi bh (1 − 2δh (t )) ω (t , υ)

h=1

− xT (t ) Xi Bi ρ ρˆ −1 (t) (Im − 2δ (t )) G (t , υ) ≤ 0,

(53)

116

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

sT (x (t ) , i) BTi Xi Bi

(

−

m ∑

)−1

BTi Xi Bi w (t ) − xT (t ) Xi Bi ρ ρˆ −1 (t ) T (t )

(i)

xT (t ) Xi bh ρ˜ h (t ) ρˆ h−1 (t ) Th (t ) +

m ∑ h=1

h=1

+

m ∑

(i) ˜ xT (t ) Xi bh w h (t )

) (i) ˜h (t ) ≤ 0, ˜h (t ) − w xT (t ) Xi bh δh w (

(54)

h=1

sT (x (t ) , i) BTi Xi Bi

(

−

m ∑

)−1

BTi Xi Bi ς (t ) − xT (t ) Xi Bi ρ ρˆ −1 (t ) M (t )

(i)

xT (t ) Xi bh ρ˜ h (t ) ρˆ h−1 (t ) Mh (t ) +

+

(i) xT (t ) Xi bh ˜ ς h (t )

h=1

h=1 m ∑

m ∑

) ( (i) ς h (t) ≤ 0, xT (t ) Xi bh δh ˜ ς h (t ) − ˜

h=1   − xT (t ) Xi Bi ργi sgn (s (x (t ) , i)) ≤ −γi ρ max xT (t ) Xi Bi  .

(55)

By (53)–(55), (52) can be simplified as sT (x (t ) , i) BTi Xi Bi

(

)−1

BTi Xi Ai x (t ) − xT (t ) Xi Bi ρ Γi ρˆ (t ) x (t ) −

)

(

m ∑

(i)

xT (t ) Xi bh Γih ρ˜ h (t ) x (t )

h=1

1

+ sT (x (t ) , i) 2

N ∑

πij BTj Xj Bj

(

)−1

s (x (t ) , i) − xT (t ) Xi Bi ργi sgn (s (x (t ) , i))

j=1

[

] ≤ ∥s (x (t ) , i)∥ Wi ∥x (t )∥ − γi ρ max + κ¯ − κ¯ ∥s (x (t ) , i)∥ ,

(56)

where

( 

Wi =  BTi Xi Bi

)−1

 

( 

BTi Xi Ai  +  BTi Xi Bi

m )−1 T  ∑ ∥Yih ρ¯ h ∥ Bi  h=1

m (   )−1 )−1  1 ∑  ( T  ∑  ∥Yih ρ¯ h ∥ + + ∥ρ∥ ¯  BTi Xi Bi BTi  πij Bj Xj Bj BTi Xi 

2

h=1

j∈Nκi

  )−1 1 ∑  ( T  + πij Bj Xj Bj BTi Xi  2

j∈Nuiκ

with κ¯ being a small positive scalar. LV2 (t ) ≤ −κ¯ ∥s (x (t ) , i)∥ for s (x (t ) , i) ̸ = 0 holds as Wi ∥x (t )∥ − γi ρ + κ¯ ≤ 0. max Hence, the state trajectories reachability is obtained via the above proof. This completes the proof. ■ Remark 6. It should be noted that there exist the unknown bounded transition probabilities πij in (56), however, we can properly design the constant κ¯ such that the inequality (56) holds. 6. Simulation results For MJSs (5) with four subsystems, the matrices are given as A (1) =

A (3) =

[ −1.86 0 0

[ −1.86 0 0

1.5 −5.279 2.928

0 .2 −2.3 , −2.082

0.1 −2 , −0.1

0.11 0.2 0.15

0.1 −2 , −0.1

[ B (3) =

0 .2 −2.3 , −2.082

0.11 0.2 0.15

[ B (1) =

1.5 −3.279 2.928

[

]

A (2) =

]

]

B (4) =

0 0

[ A (4) =

−1.86

[

0.1 0.15 0.1

0.1 −2 , −0.1

[

0.1 0.15 0.1

0.1 −2 . −0.1

B (2) =

]

−1.86

]

]

0 0

1 .5 −4.279 2.928

0.2 −2.3 , −2.082

1 .5 −6.279 2.928

0.2 −2.3 , −2.082

]

]

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

117

Fig. 1. Switching signal rt .

Fig. 2. State responses of the closed-loop system.

The transition matrix Π is given as

⎡

? ⎢ ? Π =⎣ 1.2 2.6

2.2 −1.9 ? ?

? 1.7 −1.8 ?

⎤

? ? ⎥ , ? ⎦ −2.8

where ‘‘?’’ represents the unknown element. There are two actuators in this condition. The following two possible faulty modes are considered from (3). Normal mode 1: Two actuators are normal, i.e., ρ11 = ρ21 = ρ12 = ρ22 = 1. Fault mode 2: The first actuator has lost of effectiveness with the bounds ρ 1 = 0.1, ρ¯ 11 = 0.9; the second actuator has 1 lost effectiveness with the bounds ρ 1 = 0.15, ρ¯ 21 = 0.95. 2 The fault mode of (4) in this paper is assumed as follows:

ρ = diag (0.1, 0.15) ,

ρ¯ = diag (0.95, 0.80) .

118

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

Fig. 3. The trajectories of s (x (t ) , 1).

Fig. 4. The trajectories of s (x (t ) , 2).

The nonlinear function f (x, t ) , external disturbance w (t ) and the unknown fault-deviation function ς (t ) of four subsystems are respectively set as follows: f (x, t ) =

ς (t ) =

0.28 + 0.2 sin (200t ) x1 (t ) , 0.28 + 0.2 sin (200t ) x3 (t )

[

]

w (t ) =

0.28 + 0.2 sin (200t ) , 0.28 − 0.2 cos (200t )

[

]

0.28 + 0.2 sin (200t ) . 0.28 + 0.2 cos (200t )

[

]

]T

The initial state of MJSs is set as x (0) = 2.5 −0.5 0.2 . The adaptive laws are determined by (34) and (35), and the ˆ¯ 1 (0) = 0.25, w designed parameters are selected as ρˆ 1 (0) = 0.60, ρˆ 2 (0) = 0.60, αˆ (0) = 0.1, βˆ (0) = 0.2, w ˆ 1 (0) = 0.15, ˆς¯ 1 (0) = 0.28, ςˆ (0) = 0.05, w ˆ¯ 2 (0) = 0.25, w ˆ ˆ 0) = 0.15, ς¯ 2 (0) = 0.28, ςˆ (0) = 0.05, and ~1 = ~2 = 0.1, cα = 0.15, ( 2 1 2 cβ = 0.1, s1 = s2 = 1.5, q1 = q2 = 1. The parameters in (33) are selected as γ1 = 2.5, γ2 = 2.0, γ3 = 1.0, γ4 = 1.5. We s(x(t ),i) replace sgn (s (x (t ) , i)) with ∥s(x(t ),i)∥+0.02 to prevent the chattering. By solving the conditions (12)–(15) and (36)–(39), the

[

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

Fig. 5. The trajectories of s (x (t ) , 3).

Fig. 6. The trajectories of s (x (t ) , 4).

Fig. 7. The estimation α (t).

119

120

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

Fig. 8. The estimation β (t).

parameter matrices X (i) are achieved as 2.4786 −1.8514 1.1210

−1.8514 2.4079 −0.7719 −1.2031 1.4050 −1.1289

[ X (1) =

1.8562 −1.2031 0.8279

[ X (3) =

1.1210 −0.7719 , 3.2400

]

0.8279 −1.1289 , 2.9258

2.9617 −1.7021 1.0776

[ X (2) =

]

6.3483 −4.5678 5.0782

[ X (4) =

−1.7021 1.5477 −0.8377 −4.5678 5.2570 −3.7635

1.0776 −0.8377 , 1.8645

]

5.0782 −3.7635 . 6.4634

]

Then, the designed sliding mode surfaces (7) are calculated as

s (x (t ) , i) =

⎧ ⎪ ⎪ = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪= ⎨ ⎪ ⎪ ⎪ ⎪ ⎪= ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩=

0.0705 3.8386

0.1621 −4.9237

0.4549 x (t ) 1.3320

i = 1,

0.1486 3.5926

−0.0218 −3.1818

0.1686 x (t ) 1.5967

i = 2,

0.0877 2.5091

−0.0207 −2.8174

0.3041 x (t ) 2.0481

i = 3,

0.4575 9.2626

−0.0446 −10.5943

[ [ [ [

]

]

]

0.5896 x (t ) 7.3886

]

i = 4.

The simulation results are provided in Figs. 1–8. Fig. 1 shows the possible switching signal r (t ) and Fig. 2 plots the state responses of the closed-loop system. The responses of four sliding mode surfaces s (x (t ) , i) are respectively depicted in Figs. 3–6. Figs. 7–8 plot the estimated values αˆ (t ) and βˆ (t ), respectively. 7. Conclusion This paper has investigated the problem of adaptive SMC for continuous-time MJSs with time-varying actuator faults and partially known probabilities. By using Lyapunov stability theory, some new sufficient conditions of stochastic stability for MJSs are obtained. The given numerical example has shown the applicability of the proposed control method. Acknowledgements This work is supported by the Funds for China National Funds for Distinguished Young Scientists (61425009), the National Natural Science Foundation of China (U1611262, 61473096), Guangdong Province Higher Vocational Colleges & Schools Pearl River Scholar approved in 2015 and the China National 863 Technology Projects (2015BAF32B03-05), the Department of Education of Guangdong Province (2016KTSCX030).

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

121

References [1] J. Fu, T.-F. Li, T. Chai, C.-Y. Su, Sampled-data-based stabilization of switched linear neutral systems, Automatica 72 (2016) 92–99. [2] J. Fu, R. Ma, T. Chai, Global finite-time stabilization of a class of switched nonlinear systems with the powers of positive odd rational numbers, Automatica 54 (2015) 360–373. [3] L. Liu, Q. Zhou, H. Liang, L. Wang, Stability and stabilization of nonlinear switched systems under average dwell time, Appl. Math. Comput. 298 (2017) 77–94. [4] S. Vazquez, A. Marquez, R. Aguilera, D. Quevedo, J.I. Leon, L.G. Franquelo, Predictive optimal switching sequence direct power control for grid-connected power converters, IEEE Trans. Ind. Electron. 62 (4) (2015) 2010–2020. [5] H. Yang, B. Jiang, H. Zhang, Stabilization of non-minimum phase switched nonlinear systems with application to multi-agent systems, Systems Control Lett. 61 (10) (2012) 1023–1031. [6] L. Zhang, G.-H. Yang, Adaptive fuzzy output constrained decentralized control for switched nonlinear large-scale systems with unknown dead zones, Nonlinear Anal. Hybrid Syst. 23 (2017) 61–75. [7] E.-K. Boukas, Z. Liu, Robust H∞ control of discrete-time Markovian jump linear systems with mode-dependent time-delays, IEEE Trans. Automat. Control 46 (12) (2001) 1918–1924. [8] M. Liu, D.W. Ho, P. Shi, Adaptive fault-tolerant compensation control for Markovian jump systems with mismatched external disturbance, Automatica 58 (2015) 5–14. [9] J. Lam, Z. Shu, S. Xu, E.-K. Boukas, Robust control of descriptor discrete-time Markovian jump systems, Internat. J. Control 80 (3) (2007) 374–385. [10] J. Tao, R. Lu, P. Shi, H. Su, Z.-G. Wu, Dissipativity-based reliable control for fuzzy markov jump systems with actuator faults, IEEE Trans. Cybern. (2016). http://dx.doi.org/10.1109/TCYB.2016.2584087. [11] L. Wu, P. Shi, H. Gao, C. Wang, H∞ filtering for 2D Markovian jump systems, Automatica 44 (7) (2008) 1849–1858. [12] M.S. Ali, S. Saravanan, J. Cao, Finite-time boundedness, L2 -gain analysis and control of Markovian jump switched neural networks with additive timevarying delays, Nonlinear Anal. Hybrid Syst. 23 (2017) 27–43. [13] E.-K. Boukas, Stochastic Switching Systems: Analysis and Design, Springer Science & Business Media, 2007. [14] C.E. de Souza, A. Trofino, K.A. Barbosa, Mode-independent filters for Markovian jump linear systems, IEEE Trans. Automat. Control 51 (11) (2006) 1837–1841. [15] J. Dong, G.H. Yang, Robust H2 control of continuous-time Markov jump linear systems, Automatica 44 (5) (2008) 1431–1436. [16] M. Liu, P. Shi, L. Zhang, X. Zhao, Fault-tolerant control for nonlinear Markovian jump systems via proportional and derivative sliding mode observer technique, IEEE Trans. Circuits Syst. I. Regul. Pap. 58 (11) (2011) 2755–2764. [17] S. Long, S. Zhong, Improved results for stochastic stabilization of a class of discrete-time singular Markovian jump systems with time-varying delay, Nonlinear Anal. Hybrid Syst. 23 (2017) 11–26. [18] J. Xiong, J. Lam, H. Gao, D.W. Ho, On robust stabilization of Markovian jump systems with uncertain switching probabilities, Automatica 41 (5) (2005) 897–903. [19] S. Xu, T. Chen, J. Lam, Robust H∞ filtering for uncertain Markovian jump systems with mode-dependent time delays, IEEE Trans. Automat. Control 48 (5) (2003) 900–907. [20] H. Yang, B. Jiang, V. Cocquempot, H. Zhang, Stabilization of switched nonlinear systems with all unstable modes: application to multi-agent systems, IEEE Trans. Automat. Control 56 (9) (2011) 2230–2235. [21] L. Bai, Q. Zhou, L. Wang, Z. Yu, Observer-based adaptive control for stochastic nonstrict-feedback systems with unknown backlash-like hysteresis, Int. J. Adapt. Control Signal Process. (2017). http://dx.doi.org/10.1002/Acs.2780. [22] Q. Zhou, D. Yao, J. Wang, C. Wu, Robust control of uncertain semi-Markovian jump systems using sliding mode control method, Appl. Math. Comput. 286 (2016) 72–87. [23] H. Yang, V. Cocquempot, B. Jiang, Robust fault tolerant tracking control with application to hybrid nonlinear systems, IET Control Theory Appl. 3 (2) (2009) 211–224. [24] Q. Zhou, C. Wu, P. Shi, Observer-based adaptive fuzzy tracking control of nonlinear systems with time delay and input saturation, Fuzzy Sets and Systems 316 (2017) 49–68. [25] G. Bartolini, A. Ferrara, V.I. Utkin, Adaptive sliding mode control in discrete-time systems, Automatica 31 (5) (1995) 769–773. [26] M.V. Basin, P.C. Rodriguez-Ramirez, Sliding mode controller design for stochastic polynomial systems with unmeasured states, IEEE Trans. Ind. Electron. 61 (1) (2014) 387–396. [27] C. Edwards, S. Spurgeon, Sliding Mode Control: Theory and Applications, CRC Press, 1998. [28] M.T. Hamayun, C. Edwards, H. Alwi, Design and analysis of an integral sliding mode fault-tolerant control scheme, IEEE Trans. Automat. Control 57 (7) (2012) 1783–1789. [29] S. Laghrouche, F. Plestan, A. Glumineau, Higher order sliding mode control based on integral sliding mode, Automatica 43 (3) (2007) 531–537. [30] M. Liu, L. Zhang, P. Shi, Y. Zhao, Sliding mode control of continuous-time Markovian jump systems with digital data transmission, Automatica 80 (2017) 200–209. [31] J. Liu, S. Vazquez, L. Wu, A. Marquez, H. Gao, L.G. Franquelo, Extended state observer-based sliding-mode control for three-phase power converters, IEEE Trans. Ind. Electron. 64 (1) (2017) 22–31. [32] V.I. Utkin, J. Guldner, M. Shijun, Sliding Mode Control in Electro-Mechanical Systems, Vol. 34, CRC press, 1999. [33] V.I. Utkin, Sliding Modes in Control and Optimization, Springer Science & Business Media, 2013. [34] J. Wang, Y. Gao, J. Qiu, C.K. Ahn, Sliding mode control for nonlinear systems by T-S fuzzy model and delta operator approaches, IET Control Theory Appl. (2016). http://dx.doi.org/10.1049/iet-cta.2016.0231. [35] K.D. Young, V.I. Utkin, U. Ozguner, A control engineer’s guide to sliding mode control, IEEE Trans. Control Syst. Technol. 7 (3) (1999) 328–342. [36] Y. Niu, D.W.C. Ho, Design of sliding mode control subject to packet losses, IEEE Trans. Automat. Control 55 (11) (2010) 2623–2628. [37] P. Shi, Y. Xia, G. Liu, D. Rees, On designing of sliding-mode control for stochastic jump systems, IEEE Trans. Automat. Control 51 (1) (2006) 97–103. [38] P. Shi, M. Liu, L. Zhang, Fault-tolerant sliding-mode-observer synthesis of Markovian jump systems using quantized measurements, IEEE Trans. Ind. Electron. 62 (9) (2015) 5910–5918. [39] L. Wu, P. Shi, H. Gao, State estimation and sliding-mode control of Markovian jump singular systems, IEEE Trans. Automat. Control 55 (5) (2010) 1213–1219. [40] H. Zhang, Y. Shi, A.S. Mehr, Robust static output feedback control and remote PID design for networked motor systems, IEEE Trans. Ind. Electron. 58 (12) (2011) 5396–5405. [41] H. Zhang, Y. Shi, A.S. Mehr, Robust H∞ PID control for multivariable networked control systems with disturbance/noise attenuation, Internat. J. Robust Nonlinear Control 22 (2) (2012) 183–204. [42] M. Karan, P. Shi, C.Y. Kaya, Transition probability bounds for the stochastic stability robustness of continuous-and discrete-time Markovian jump linear systems, Automatica 42 (12) (2006) 2159–2168.

122

D. Yao et al. / Nonlinear Analysis: Hybrid Systems 28 (2018) 105–122

[43] H. Yang, B. Jiang, G. Tao, D. Zhou, Robust stability of switched nonlinear systems with switching uncertainties, IEEE Trans. Automat. Control 61 (9) (2016) 2531–2537. [44] L. Zhang, E.-K. Boukas, Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities, Automatica 45 (2) (2009) 463–468. [45] 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. [46] L. Zhang, J. Lam, Necessary and sufficient conditions for analysis and synthesis of Markov jump linear systems with incomplete transition descriptions, IEEE Trans. Automat. Control 55 (7) (2010) 1695–1701. [47] Y. Zhang, Y. He, M. Wu, J. Zhang, Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices, Automatica 47 (1) (2011) 79–84. [48] B. Chen, Y. Niu, Y. Zou, Adaptive sliding mode control for stochastic Markovian jumping systems with actuator degradation, Automatica 49 (6) (2013) 1748–1754. [49] F. Castanos, L. Fridman, Analysis and design of integral sliding manifolds for systems with unmatched perturbations, IEEE Trans. Automat. Control 51 (5) (2006) 853–858. [50] M. Rubagotti, A. Estrada, F. Castanos, A. Ferrara, L. Fridman, Integral sliding mode control for nonlinear systems with matched and unmatched perturbations, IEEE Trans. Automat. Control 56 (11) (2011) 2699–2704. [51] Xiao-Zheng, G.-H. Yang, Robust adaptive fault-tolerant compensation control with actuator failures and bounded disturbances, Acta Automat. Sinica 35 (3) (2009) 305–309. [52] S.P. Meyn, R.L. Tweedie, Markov Chains and Stochastic Stability, Springer Science & Business Media, 2012. [53] X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl. 79 (1) (1999) 45–67.