Adaptive sliding mode control for stochastic switched systems with actuator faults

Abstract

The problem of adaptive sliding mode control is considered for a class of stochastic switched systems with actuator degradation. In this work, the input matrix for each subsystem is unnecessarily the same. Thus, a weighted sum approach of the input matrices is introduced such that a common sliding surface is designed. By online estimating the loss of effectiveness of the actuators, an adaptive sliding mode controller is designed. It can not only compensate the effect of the actuator degradation effectively, but also reduce the conservatism that the bound of the actuator faults should be known in advance. Moreover, it is shown that the reachability of the sliding surface can be guaranteed. Furthermore, sufficient conditions on the mean-square exponential stability of the sliding mode dynamics are obtained via the average dwell time method. Finally, a numerical simulation example is given to demonstrate the effectiveness of the proposed method.

Keywords

Switched systems actuator degradation sliding mode control average dwell time

Introduction

In practical engineering systems, the occurrence of sensor/actuator faults is inevitable. To increase reliability and availability of the controlled systems, the need for maintaining an acceptable performance for the controlled systems against actuator or sensor failures is becoming more apparent in many complex safety critical systems, such as aircraft, spacecraft, and nuclear power plants (Veillette et al., 1992). Over the past few decades, many methods, such as linear quadratic regulator (LQR) (Veillette, 1995), robust control (Dong and Yang, 2015) and adaptive control techniques (Boskovic et al., 2009) have been proposed. Recently, Yang and Ye (2010) proposed an adaptive reliable $H_{\infty}$ controller for linear systems.

As is well known, sliding mode control (SMC) is an effective robust control method due to its strong robustness against parameter uncertainties and external disturbances. In recent years, the design of SMC subject to actuator faults has been receiving more and more attention (Alwi and Edwards, 2008; Liu and Shi, 2013; Niu and Wang, 2009). Alwi and Edwards (2008) proposed an online SMC allocation scheme, in which the effectiveness level of the actuators was used in the re-distribution of the control signals when faults occur. Moreover, SMC for uncertain time-delay systems with partial actuator degradation was investigated in Niu and Wang (2009). Later, the results were further extended into It $\hat{o}$ stochastic systems (Liu and Shi, 2013). More recently, an adaptive sliding mode controller has been presented for Markovian jumping systems (Li et al., 2017), by which the controller parameters can be updated automatically to compensate for the actuator faults.

On the other hand, switched systems play an important role in the modelling of many real systems subject to known or unknown abrupt parameter variations and sudden change of system structures. Therefore, many results on stability analysis and stabilization (Liberzon, 2003; Zhang and Gao, 2010; Zhao et al., 2017), reduced mode control (Su et al., 2015), admissibility analysis (Darouach and Chadli, 2013), and optimal control (Colanerio et al., 2014) have been obtained. Recently, the SMC method has been extended into switched systems and many constructive results have been obtained (Liu et al., 2015; Wu et al., 2011; Wu and Lam, 2008). Among them, Wu and Lam (2008) investigated SMC for a class of uncertain switched systems with state delay, whose result was further extended to stochastic switched systems inWu et al. (2011). Later, Liu et al. (2015) further considered the switched systems with different input matrices and constructed a common sliding surface via a weighted sum approach. In recent years, the reliable control of the switched systems has drawn more and more attention (Lin et al., 2013; Yin et al., 2017a). Yin et al. (2017a) investigated the state and fault estimation problem for linear switched systems with simultaneous disturbances, sensor and actuator faults. Later, the results were further extended into nonlinear Markovian jump systems (Yin et al., 2017b). Recently, Lin et al. (2013) considered the reliable dissipative control of a class of discrete switched singular systems with stochastic actuator faults. More recently, Liu et al. (2017) further considered the robust $H_{\infty}$ control for a class of discrete switched systems with both random sensor and actuator faults. However, to the author"s best knowledge, the research on SMC of stochastic switched systems subject to actuator faults is still open. Moreover, the existing results (Liu et al., 2015;Wu et al., 2011;Wu and Lam, 2008) cannot be simply extended to stochastic switched systems due to their special structure characteristics.

Motivated by the above discussion, this paper considers adaptive SMC for a class of stochastic switched systems with actuator faults. Our main contribution consists of the following three aspects. (i) A weighted sum approach of the input matrices is utilized to construct a common sliding surface since there may be different input channels for each system. (ii) To compensate for the effects of the actuator degradation, an adaptive SMC law is designed. It not only ensures the reachability of the specified sliding surface, but also attenuates the effect of the actuator degradation by online estimation of the bounded actuator degradation. (iii) By employing the average dwell time approach, sufficient conditions are obtained to guarantee the mean-square exponential stability of the sliding motion.

The rest of this paper is organized as follows. Section ‘Problem statement’ formulates the problem under consideration. A common sliding surface is designed in section ‘Design of the sliding surface’. Section ‘Reachability analysis’ presents an adaptive SMC law and discusses the reachability of the sliding surface. The stability of the sliding motion is analysed in section ‘Stability of the sliding mode dynamics’. A simulation example is given in section ‘Numerical simulation’. The final section gives the conclusion.

Notation: $R^{n}$ denotes the real n-dimensional space; $R^{m \times n}$ denotes the real $m \times n$ matrix space. For any vector $x \in R^{n}$ , $∥ x ∥$ denotes the Euclidean norm or its induced matrix norm. For a real symmetric matrix M, $M > 0 (M < 0)$ means positive (negative) definite, and I is used to represent an identity matrix. $λ_{max} {\cdot}$ and $λ_{min} {\cdot}$ denotes the maximum and minimum eigenvalue of a matrix, respectively, $E {\cdot}$ is the mathematical expectation, rank (·) represents the rank of a matrix, and the part in a matrix induced by symmetry is denoted by *. Matrices, if not explicitly stated, are assumed to have compatible dimensions.

Problem statement

Consider the following stochastic switched systems

\begin{matrix} dx (t) = [(A_{σ} + {\tilde{A}}_{σ}) x (t) + B_{σ} (u^{F} (t) + f_{σ} (x (t)))] dt \\ + (C_{σ} + {\tilde{C}}_{σ}) x (t) dw (t) \end{matrix}

(1)

where $x (t) \in R^{n}$ is the state, $u^{F} (t) \in R^{m}$ is the control input subject to actuator faults, $f_{σ} (x (t))$ is the external disturbance, $w (t)$ is a one-dimensional Brownian motion, $A_{σ}$ , $B_{σ}$ and $C_{σ}$ are known matrices, ${\tilde{A}}_{σ}$ and ${\tilde{C}}_{σ}$ are parameter uncertainties, ${A_{σ}$ , $B_{σ}$ , $C_{σ}, σ \in Γ}$ is a family of matrices depending on an index set $Γ = {1, 2, \dots, s}$ , and $σ (t) : R \to Γ$ is a piecewise constant function of time t called as switching signal that is characterized by the switching sequence ${(i_{0}, t_{0}), (i_{1}, t_{1}), \dots, (i_{N}, t_{N}) | i_{k} \in Γ}$ . Moreover, $σ (t) = i_{k}$ means that the $i_{k}$ th subsystem is being activated when $t \in [t_{k}, t_{k + 1})$ .

For each possible value $σ (t) = i, i \in Γ$ , the parameters associated with the ith subsystem are denoted as

A_{σ} = A_{i}, {\tilde{A}}_{σ} = {\tilde{A}}_{i}, B_{σ} = B_{i}, C_{σ} = C_{i}, {\tilde{C}}_{σ} = {\tilde{C}}_{i}, f_{σ} (x (t)) = f_{i} (x (t))

In this work, it is assumed that the admissible uncertainties ${\tilde{A}}_{i}$ and ${\tilde{C}}_{i}$ satisfy

{\tilde{A}}_{i} = E_{i 1} F_{i 1} (t) H_{i 1}, {\tilde{C}}_{i} = E_{i 2} F_{i 2} (t) H_{i 2}

where $E_{i 1}$ , $H_{i 1}$ , $E_{i 2}$ and $H_{i 2}$ are constant matrices, $F_{i 1} (t)$ and $F_{i 2} (t)$ are unknown matrix functions satisfying $F_{i 1}^{T} (t) F_{i 1} (t) \leq I$ and $F_{i 2}^{T} (t) F_{i 2} (t) \leq I$ .

Besides, the disturbance $f_{i} (x (t))$ is assumed to be norm bounded, that is, $∥ f_{i} (x (t)) ∥ < d_{i}$ , with $d_{i} > 0$ a known constant.

In this work, the actuator degradation model is adopted as follows

u^{F} (t) = (I - Δ (t)) u (t)

(2)

with $Δ (t) = diag (δ_{1} (t), \dots, δ_{m} (t))$ satisfying

0 \leq {\underline{δ}}_{j} \leq δ_{j} (t) \leq {\bar{δ}}_{j} < 1, j = 1, 2, \dots, m

(3)

where the unknown parameter $δ_{j} (t)$ denotes the loss of effectiveness of the jth actuator. The lower and upper bounds ${\underline{δ}}_{j} (t)$ , ${\bar{δ}}_{j} (t)$ of $δ_{j} (t)$ are supposed to be known. For convenience of later development, define $\underline{Δ} = diag ({\underline{δ}}_{1}, \dots, {\underline{δ}}_{m})$ , and $\bar{Δ} = diag ({\bar{δ}}_{1}, \dots, {\bar{δ}}_{m})$ .

Then, the system (1) with actuator degradation (2) is described by

\begin{matrix} dx (t) = [(A_{i} + {\tilde{A}}_{i}) x (t) + B_{i} ((I - Δ (t)) u (t) + f_{i} (x (t)))] dt \\ + (C_{i} + {\tilde{C}}_{i}) x (t) dw (t) \end{matrix}

(4)

Remark 1. It is noted that the actuator model (2) covers the non-faulty case $({\underline{δ}}_{j} = {\bar{δ}}_{j} = 0)$ and partial degradation case $(0 < {\underline{δ}}_{j} \leq {\bar{δ}}_{j} < 1)$ . However, it should be mentioned that the outage case $({\underline{δ}}_{j} = {\bar{δ}}_{j} = 1)$ is not considered. Actually, if this case happens, the full column rank condition of the control matrix $B_{i} (I - Δ (t))$ cannot be guaranteed, which is a general assumption in SMC. Hence, this work does not consider this case.

The control objective of this work is to design an adaptive SMC law such that the stability of the switched system (4) can be guaranteed despite the presence of actuator degradation, uncertainties and stochastic disturbances. To this end, some preliminaries are first introduced, which are useful for the development of the main results.

Assumption 1. The matrix $B_{i}$ is full column rank, that is, $rank (B_{i}) = m$ .

Definition 1. (Liberzon, 2003) For any $T_{2} > T_{1} > 0$ , let $N_{σ} (T_{1}, T_{2})$ denote the number of switchings of $σ (t)$ over $(T_{1}, T_{2})$ . If

N_{σ} (T_{1}, T_{2}) \leq N_{0} + \frac{(T_{2} - T_{1})}{T_{σ}}

holds for $T_{σ} > 0$ and $N_{0} \geq 0$ , then $T_{σ}$ is called the average dwell time.

In this work, let $N_{0} = 0$ , as is usually used in the previous works.

Definition 2. (Wu et al., 2011) The equilibrium $x^{*} = 0$ of system (4) is said to be mean-square exponentially stable if the solution $x (t)$ satisfies

E {∥ x (t) ∥^{2}} \leq γ e^{- κ (t - t_{0})} {∥ x (t_{0}) ∥}^{2}, \forall t \geq t_{0}

for scalars $γ \geq 1$ and $κ > 0$ .

Design of the sliding surface

In this section, a common sliding surface will be designed. It is worth noting that, in system (4), the input matrix $B_{i}$ for each subsystem is not necessarily the same. Hence, the proposed method in this work has a much wider application than some existing ones as in Wu and Lam (2008) and Wu et al. (2011). However, it also brings challenge in designing a common sliding function. To overcome this difficulty, the following weighted sum of the input matrices, as in Liu et al. (2015), is introduced

B = \sum_{i = 1}^{s} α_{i} B_{i}

(5)

where $α_{i}$ is a parameter satisfying $\underline{α} \leq α_{i} \leq \bar{α}, i = 1, 2, \dots, s$ , with $\underline{α}$ and $\bar{α}$ being known constant.

Remark 2. It is noted from (5) that the weighted coefficients $α_{i}$ , $i = 1, 2, \dots, s$ , are required to be bounded, which is a relaxed condition in the controller design. Moreover, it is shown from Assumption 1 that, for general choice of scalars $α_{i}, i = 1, \dots, s$ , B is full column rank.

Then, a common sliding function is constructed as

S (t) = Dx (t)

(6)

where $D = (B^{T} B)^{- 1} B^{T}$ . It can be seen from Remark 2 that B is full column rank generally, which implies that the non-singularity of $B^{T} B$ can be guaranteed.

Remark 3. The sliding surface (6) is a common sliding function, that is, it is mode-independent. By designing such a sliding surface, the jumping of the state trajectories from different sliding surfaces can be avoided in the stability analysis.

Reachability analysis

In the sequel, an adaptive SMC law will be designed such that the state trajectories of system (4) can be driven onto the specified sliding surface $S (t) = 0$ .

In this work, the online estimation mechanism Yang and Ye (2010) is adopted. This can not only compensate the effect of the actuator degradation effectively, but also reduce the conservatism that the bound of the actuator failure should be known in advance. An adaptive SMC controller is designed as follows

u (t) = (I - \hat{Δ} (t))^{- 1} K_{i} x (t) + u_{r} (t)

(7)

with

u_{r} (t) = - \frac{1}{1 - {\bar{δ}}_{max}} (ρ_{i} (t) + η_{i}) sgn ({(D B_{i})}^{T} S (t))

where

\begin{matrix} ρ_{i} (t) = ∥ ({DB}_{i})^{- 1} ∥ ∥ D (A_{i} + B_{i} K_{i}) x (t) ∥ + ∥ (D B_{i})^{- 1} ∥ \\ \times ∥ D E_{i 1} ∥ ∥ H_{i 1} x (t) ∥ + d_{i} + \frac{1}{2} {∥ {(D B_{i})}^{- 1} ∥ ∥ D ∥}^{2} \\ \times ∥ D^{+} ∥ (∥ C_{i} ∥ + ∥ E_{i 2} ∥ ∥ H_{i 2} ∥)^{2} ∥ x (t) ∥, \end{matrix}

$η_{i}$ is a positive scalars, $D^{+}$ is the Moore-Penrose inverse of the matrix D, ${\bar{δ}}_{max} = max {{\bar{δ}}_{j}, j = 1, 2, \dots, m}$ , and $\hat{Δ} (t) = diag ({\hat{δ}}_{1} (t), {\hat{δ}}_{2} (t), \dots, {\hat{δ}}_{m} (t))$ is the estimation of $Δ (t)$ . ${\hat{δ}}_{j} (t)$ updates adaptively according to the following rule

\begin{matrix} {\overset{\cdot}{\hat{δ}}}_{j} (t) = Proj {L_{j} (t)} \\ = {\begin{matrix} 0, if {\hat{δ}}_{j} (t) = {\underline{δ}}_{j} and L_{j} (t) \leq 0, or {\hat{δ}}_{j} (t) = {\bar{δ}}_{j} and L_{j} (t) \geq 0 \\ L_{j} (t), otherwise \end{matrix} \end{matrix}

(8)

where $L_{j} (t) = - λ_{j} (1 - {\hat{δ}}_{j} (t))^{- 1} x^{T} (t) D^{T} D B_{i} K_{i} x (t)$ , $λ_{j} > 0$ is the updating gain, and $Proj {\cdot}$ denotes the projection operator (Ioannou and Sun, 2010), whose role is to project the estimation ${\hat{δ}}_{j} (t)$ to the interval $[{\underline{δ}}_{j}, {\bar{δ}}_{j}]$ , and the matrix $K_{i}$ will be designed in Theorem 2. It is shown from (3) that the non-singularity of the matrix $I - \hat{Δ} (t)$ can be guaranteed.

Remark 4. It can be seen from the SMC law (7) that the matrix $D B_{i}$ is required to be non-singular. As discussed in Remark 2, by selecting the parameters $α_{i}$ properly, the above non-singular condition can be guaranteed generally.

In the following theorem, we will analyse the reachability of the sliding surface $S (t) = 0$ .

Theorem 1. Consider the switched system (4) satisfying Assumption 1. The adaptive SMC law is designed as (7) and (8), then the state trajectories can be driven onto the specified sliding surface $S (t) = 0$ in finite time almost surely.

Proof. In view of (4) and (6), we have

\begin{matrix} dS (t) = [D (A_{i} + {\tilde{A}}_{i}) x (t) + D B_{i} ((I - Δ (t)) u (t) + f_{i} (x (t)))] dt \\ + D (C_{i} + {\tilde{C}}_{i}) x (t) dw (t) \end{matrix}

(9)

Choose the Lyapunov function as

V (t) = S^{T} (t) S (t) + \sum_{j = 1}^{m} \frac{{\tilde{δ}}_{j}^{2} (t)}{λ_{j}}

(10)

where ${\tilde{δ}}_{j} (t) = δ_{j} (t) - {\hat{δ}}_{j} (t), j = 1, 2, \dots m$ . It can be derived from (9) and (10) that

\begin{matrix} L V (t) = 2 S^{T} (t) [D (A_{i} + {\tilde{A}}_{i}) x (t) + D B_{i} [(I - Δ (t)) u (t) + f_{i} (x (t))]] \\ + x (t)^{T} (C_{i} + {\tilde{C}}_{i})^{T} D^{T} D (C_{i} + {\tilde{C}}_{i}) x (t) + 2 \sum_{j = 1}^{m} \frac{{\tilde{δ}}_{j} (t) {\overset{\cdot}{\tilde{δ}}}_{j} (t)}{λ_{j}} \\ = 2 S^{T} (t) [D (A_{i} + {\tilde{A}}_{i}) x (t) + D B_{i} ((I - \hat{Δ} (t)) + \tilde{Δ} (t)) u (t) \\ + D B_{i} f_{i} (x (t))] + x (t)^{T} (C_{i} + {\tilde{C}}_{i})^{T} D^{T} D (C_{i} + {\tilde{C}}_{i}) x (t) \\ + 2 \sum_{j = 1}^{m} \frac{{\tilde{δ}}_{j} (t) {\overset{\cdot}{\tilde{δ}}}_{j} (t)}{λ_{j}} \end{matrix}

(11)

Substituting (7) into (11) yields

\begin{matrix} L V (t) = 2 S^{T} (t) [D (A_{i} + {\tilde{A}}_{i}) x (t) + D B_{i} K_{i} x (t) \\ + D B_{i} \tilde{Δ} (t) (I - \hat{Δ} (t {))}^{- 1} K_{i} x (t) \\ + D B_{i} (I - Δ (t)) u_{r} (t) + D B_{i} f_{i} (x (t))] \\ + x {(t)}^{T} {(C_{i} + {\tilde{C}}_{i})}^{T} D^{T} D (C_{i} + {\tilde{C}}_{i}) x (t) \\ + 2 \sum_{j = 1}^{m} \frac{{\tilde{δ}}_{j} (t) {\overset{\cdot}{\tilde{δ}}}_{j} (t)}{λ_{j}} \end{matrix}

(12)

Considering the adaptive law (8), we have

2 S^{T} (t) D B_{i} \tilde{Δ} (t) (I - \hat{Δ} (t))^{- 1} K_{i} x (t) + 2 \sum_{j = 1}^{m} \frac{{\tilde{δ}}_{j} (t) {\overset{\cdot}{\tilde{δ}}}_{j} (t)}{λ_{j}} \leq 0

(13)

Denote $(D B_{i})^{T} S (t) = [s_{i 1} (t), \dots, s_{im} (t)]^{T}$ . Combining (12) and (13), we have

\begin{matrix} L V (t) \leq 2 S^{T} (t) [D (A_{i} + {\tilde{A}}_{i}) x (t) + {DB}_{i} K_{i} x (t) + {DB}_{i} (I - Δ (t)) u_{r} (t) \\ + {DB}_{i} f_{i} (x (t))] + x (t)^{T} (C_{i} + {\tilde{C}}_{i})^{T} D^{T} D (C_{i} + {\tilde{C}}_{i}) x (t) \\ = 2 S^{T} (t) [D (A_{i} + B_{i} K_{i} + {\tilde{A}}_{i}) x (t) + {DB}_{i} f_{i} (x (t))] \\ - \frac{2}{1 - {\bar{δ}}_{max}} (ρ_{i} (t) + η_{i}) \sum_{j = 1}^{m} (1 - δ_{j} (t)) | s_{ij} (t) | \\ + (D (C_{i} + {\tilde{C}}_{i}) D^{+} S (t {))}^{T} D (C_{i} + {\tilde{C}}_{i}) x (t) \\ \leq 2 S^{T} (t) {DB}_{i} [{({DB}_{i})}^{- 1} D (A_{i} + B_{i} K_{i} + {\tilde{A}}_{i}) x (t) + f_{i} (x (t)) \\ + \frac{1}{2} ({DB}_{i})^{- 1} (D (C_{i} + {\tilde{C}}_{i}) D^{+})^{T} D (C_{i} + {\tilde{C}}_{i}) x (t)] \\ - 2 (ρ_{i} (t) + η_{i}) ‖ ({DB}_{i})^{T} S (t) ‖_{1} \\ \leq 2 ‖ ρ_{i} (t) ‖ ‖ ({DB}_{i})^{T} S (t) ‖ - 2 ‖ (ρ_{i} (t) + η_{i}) ‖ ‖ ({DB}_{i})^{T} S (t) ‖ . \end{matrix}

(14)

In view of (14), it can be obtained that

L V (t) \leq - 2 η_{i} λ_{min} (D B_{i} (D B_{i})^{T}) \sqrt{V (t)}

This implies that

L ∥ S (t) ∥ \leq - η_{i} λ_{min} (D B_{i} (D B_{i})^{T})

(15)

Then, considering the switching sequence and integrating (15) from $t_{k}$ to t and $t_{k - 1}$ to $t_{k}$ $(k = 1, 2, \dots),$ it can be verified that there exists a time instant $τ = \frac{S (0)}{η_{i} λ_{min} (D B_{i} {(D B_{i})}^{T})}$ such that $E {∥ S (τ) ∥} = 0$ , and consequently $E {∥ S (t) ∥} = 0$ , for $t > τ$ (Huang and Mao, 2010). This implies that the state trajectories will be driven onto the specified sliding surface $S (t) = 0$ in finite time almost surely, which completes the proof.

Stability of the sliding mode dynamics

In the above section, we have discussed the reachability of the sliding surface. It is shown that the state trajectories of the system (4) can be driven onto the specified sliding surface $S (t) = 0$ . In the following part, we proceed to consider the stability of the sliding mode dynamics.

Substituting the adaptive SMC law (7) and (8) into system (4), the closed-loop system is obtained as follows

\begin{matrix} dx (t) = [(A_{i} + B_{i} K_{i} + {\tilde{A}}_{i}) x (t) + B_{i} \tilde{Δ} (t) (I - \hat{Δ} (t {))}^{- 1} K_{i} x (t) \\ + B_{i} ((I - Δ (t)) u_{r} (t) + f_{i} (x (t)))] dt + (C_{i} + {\tilde{C}}_{i}) x (t) dw (t) \end{matrix}

(16)

Now, the results on mean-square exponential stability of the sliding motion will be presented in the following theorem.

Theorem 2. Consider the system (4) satisfying Assumption 1. Given the scalar $ς > 0$ , if there exist matrices $X_{i} > 0$ , $T_{i} > 0$ , $Y_{i}$ and parameters $ε_{i} > 0$ , $i \in Γ$ , satisfying the following linear matrix inequalities (LMIs) with equality constraints

[\begin{matrix} Θ_{i} & 0 & X_{i} H_{i 1}^{T} & X_{i} H_{i 2}^{T} \\ - X_{i} + ε_{i} E_{i 2} E_{i 2}^{T} & 0 & 0 \\ * & - ε_{i} I & 0 \\ * & * & - ε_{i} I \end{matrix}] < 0

(17)

B_{i} T_{i} = X_{i} D^{T}, i \in Γ

(18)

with

Θ_{i} = A_{i} X_{i} + X_{i} A_{i}^{T} + B_{i} Y_{i} + Y_{i}^{T} B_{i}^{T} + ς X_{i} + ε_{i} E_{i 1} E_{i 1}^{T}

and the average dwell time $T_{σ}$ satisfying

T_{σ} > T_{σ}^{*} = \frac{\ln μ}{ς}

(19)

with the parameter

μ = max_{i, j \in Γ, i \neq j} \frac{λ_{max} (X_{i}^{- 1})}{λ_{min} (X_{j}^{- 1})}

(20)

the sliding mode dynamics of the system (4) is mean-square exponentially stable for arbitrary switching signal $σ (t)$ . Furthermore, the norm of the state obeys

E {{∥ x (t) ∥}^{2}} \leq γ e^{- κ t} {∥ x (t_{0}) ∥}^{2}

where

\begin{matrix} γ = \frac{b}{a} \geq 1, κ = ς - \frac{\ln μ}{T_{σ}} \\ a = min_{i \in Γ} {λ_{min} (X_{i}^{- 1})}, b = max_{i \in Γ} {λ_{max} (X_{i}^{- 1})} \end{matrix}

Moreover, the gain matrix $K_{i}$ in (7) is designed by $Y_{i} X_{i}^{- 1}$ .

Proof. Consider the Lyapunov function of the ith subsystem as

V (i, t) = x^{T} (t) P_{i} x (t)

Thus, along the state trajectory of system (16), we get

\begin{matrix} L V (i, t) = x^{T} (t) [P_{i} (A_{i} + B_{i} K_{i} + {\tilde{A}}_{i}) + {(A_{i} + B_{i} K_{i} + {\tilde{A}}_{i})}^{T} P_{i}] x (t) \\ + 2 x^{T} (t) P_{i} B_{i} \tilde{Δ} (t) (I - \hat{Δ} (t {))}^{- 1} K_{i} x (t) \\ + 2 x^{T} (t) P_{i} B_{i} [(I - Δ (t)) u_{r} (t) + f_{i} (x (t))] \\ + x^{T} (t) (C_{i} + {\tilde{C}}_{i})^{T} P_{i} (C_{i} + {\tilde{C}}_{i}) x (t) \end{matrix}

(22)

It can be obtained from (18) that

x^{T} (t) P_{i} B_{i} = x^{T} D^{T} T_{i}^{- 1} = S^{T} (t) T_{i}^{- 1} = 0

(23)

Thus, when the system states are driven onto the sliding surface $S (t) = 0$ , we have $x^{T} P_{i} B_{i} = 0$ , which yields

\begin{matrix} L V (i, t) = & x^{T} (t) [P_{i} (A_{i} + B_{i} K_{i} + {\tilde{A}}_{i}) + {(A_{i} + B_{i} K_{i} + {\tilde{A}}_{i})}^{T} P_{i} \\ + {(C_{i} + {\tilde{C}}_{i})}^{T} P_{i} (C_{i} + {\tilde{C}}_{i})] x (t) . \end{matrix}

It is noted that $L V (i, t) + ς V (i, t) < 0$ is equivalent to the following LMI

\begin{matrix} P_{i} (A_{i} + B_{i} K_{i} + {\tilde{A}}_{i}) + (A_{i} + B_{i} K_{i} + {\tilde{A}}_{i})^{T} P_{i} \\ + (C_{i} + {\tilde{C}}_{i})^{T} P_{i} (C_{i} + {\tilde{C}}_{i}) + ς P_{i} < 0 . \end{matrix}

(24)

By Schur’s complement, it can be seen that (24) is implied by the following LMI

[\begin{matrix} P_{i} (A_{i} + B_{i} K_{i} + {\tilde{A}}_{i}) + {(A_{i} + B_{i} K_{i} + {\tilde{A}}_{i})}^{T} P_{i} + ς P_{i} & {(C_{i} + {\tilde{C}}_{i})}^{T} P_{i} \\ * & - P_{i} \end{matrix}] < 0

(25)

Let $K_{i} X_{i} = Y_{i}$ and $P_{i} = X_{i}^{- 1}$ . Multiplying $diag (P_{i}^{- 1}, P_{i}^{- 1})$ and its transpose on the left and right side of (25) yields

[\begin{matrix} A_{i} X_{i} + X_{i} A_{i}^{T} + B_{i} Y_{i} + Y_{i}^{T} B_{i}^{T} + {\tilde{A}}_{i} X_{i} + X_{i} {\tilde{A}}_{i}^{T} + ς X_{i} & X_{i} C_{i}^{T} + X_{i} {\tilde{C}}_{i}^{T} \\ * & - X_{i} \end{matrix}] < 0

which is equivalent to

\begin{matrix} [\begin{matrix} A_{i} X_{i} + X_{i} A_{i}^{T} + B_{i} Y_{i} + Y_{i}^{T} B_{i}^{T} + ς X_{i} & X_{i} C_{i}^{T} \\ - X_{i} \end{matrix}] \\ + Σ_{i} {\bar{F}}_{i} (t) Π_{i} + Σ_{i}^{T} {\bar{F}}_{i}^{T} (t) Π_{i}^{T} < 0 \end{matrix}

(26)

where $Σ_{i} = [\begin{matrix} X_{i} H_{i 1}^{T} & X_{i} H_{i 2}^{T} \\ 0 & 0 \end{matrix}]$ , ${\bar{F}}_{i} (t) = [\begin{matrix} F_{i 1}^{T} (t) & 0 \\ 0 & F_{i 2}^{T} (t) \end{matrix}]$ , and $Π_{i} = [\begin{matrix} E_{i 1}^{T} & 0 \\ 0 & E_{i 2}^{T} \end{matrix}]$ .

In view of Schur’s complement, it is shown that (26) is implied by (17), which together with (24) gives

L V (i, t) < - ς V (i, t)

Thus, it can be obtained that

E {V (i, t)} \leq e^{- ς (t - t_{0})} E {V (i, t_{0})}

(27)

which means that each subsystem of the closed-loop system (16) is mean-square exponentially stable.

Suppose that $t_{k}, k \in {1, 2, \dots, N_{σ}}$ , is the switching instant and the system switches from the jth subsystem to the ith subsystem. Hence, we have $σ (t_{k}^{-}) = j$ and $σ (t_{k}^{+}) = i$ . In view of (20) and (27), we get

E {V (i, t_{k})} \leq μ E {V (j, t_{k}^{-})} and E {V (i, t)} \leq e^{- ς (t - t_{k})} E {V (i, t_{k})}

(28)

Considering (28), we have

\begin{matrix} E {V (i, t)} \leq e^{- ς (t - t_{k})} μ E {V (j, t_{k}^{-})} \\ ⋮ \\ \leq e^{- ς (t - t_{0})} μ^{N_{σ} (t_{0}, t)} V (σ (t_{0}), t_{0}) \\ \leq e^{- (ς - \frac{\ln μ}{T_{σ}}) (t - t_{0})} V (σ (t_{0}), t_{0}) . \end{matrix}

(29)

In view of (21), the following holds

a {∥ x (t) ∥}^{2} \leq E {V (i, t)} and V (σ (t_{0}), t_{0}) \leq b {∥ x (t_{0}) ∥}^{2}

(30)

Therefore, it follows from (29) and (30) that

E {{∥ x (t) ∥}^{2}} \leq \frac{1}{a} E {V (i, t)} \leq \frac{b}{a} e^{- (ς - \frac{\ln μ}{T_{σ}}) (t - t_{0})} {∥ x (t_{0}) ∥}^{2}

(31)

According to (19) and (31), it is shown that the sliding mode dynamics of the system (4) is mean-square exponentially stable, which completes the proof.

Remark 5. It can be seen from Theorems 1 and 2 that if the LMIs (17) and the equality constraint (18) are feasible, the adaptive SMC law (7) and (8) can guarantee the mean-square exponential stability of the switched systems (4). Moreover, it is worth noting that the equality constrains (18) is introduced such that the stability of the closed-loop system (16) can be analysed on the sliding surface $S (t) = 0$ . Accordingly, the result on stability of the sliding mode dynamics is obtained.

Next, a simple algorithm is proposed to solve LMIs with equality constrains as (17) and (18). Consider the equality conditions

B_{i} T_{i} = X_{i} D^{T}, i \in Γ

with $X_{i}$ satisfying LMIs (17), which can be equivalently converted to

trace ({(B_{i} T_{i} - X_{i} D^{T})}^{T} (B_{i} T_{i} - X_{i} D^{T})) = 0

For $ε_{i} > 0$ , $i \in Γ$ , consider the following LMIs

(B_{i} T_{i} - X_{i} D^{T})^{T} (B_{i} T_{i} - X_{i} D^{T}) < ε_{i} I

(32)

By using Shur’s complement, (32) is equivalent to

[\begin{matrix} - ε_{i} I & T_{i} B_{i}^{T} - D X_{i} \\ * & - I \end{matrix}] < 0

(33)

Hence, the stochastic stability problem of the sliding mode dynamics is converted into the problem of searching the global solution of the following minimization problem

min ϵ_{i} subject to (17) and (33) for all i \in Γ

(34)

This is a minimization problem involving linear objective and LMI constrains that can be solved by using LMI tools in Matlab.

Numerical simulation

In this section, a numerical example is given to illustrate the effectiveness of the designed method. Consider the switched systems (4) with two modes and parameters as in the following.

Subsystem 1:

\begin{matrix} A_{1} = [\begin{matrix} - 1 & - 1.5 & 3.5 \\ 1.5 & - 1.5 & - 0.5 \\ - 1.5 & - 0.5 & 0.5 \end{matrix}], B_{1} = [\begin{matrix} 1 & 1 \\ - 1 & 0 \\ 2 & 1.5 \end{matrix}], \\ C_{1} = [\begin{matrix} 0.5 & - 0.2 & 0 \\ - 0.5 & - 0.5 & 0 \\ - 0.5 & 0.2 & - 0.5 \end{matrix}], \\ E_{11} = [1 0 - 1]^{T}, F_{11} (t) = \sin (t), H_{11} = [1 - 1 1], \\ E_{12} = [0 0.2 - 0.2]^{T}, F_{12} (t) = \cos (t), \\ H_{12} = [0 1 0.5], f_{1} (x (t)) = \frac{1}{1 + t^{2}} \end{matrix}

Subsystem 2:

\begin{array}{l} A_{2} = [\begin{matrix} - 2.5 & - 0.5 & - 1.5 \\ - 5.5 & 1.5 & - 2.5 \\ - 2.5 & 1.5 & - 2.5 \end{matrix}], B_{2} = [\begin{matrix} 3 & 2 \\ - 2 & - 1 \\ 1 & 1 \end{matrix}], \\ C_{2} = [\begin{matrix} - 0.5 & 0.5 & - 0.5 \\ 0.5 & - 0.5 & 0 \\ 0.5 & 0.5 & - 0.5 \end{matrix}], \\ E_{21} = {[1 - 1 0]}^{T}, F_{21} (t) = \cos (t), H_{21} = [1 - 1 1], \\ E_{22} = {[1 0 - 0.5]}^{T}, F_{22} (t) = \sin (t), \\ H_{22} = [0.2 - 1.2 0.2], f_{2} (x (t)) = \frac{1}{1 + 2 t^{2}} \end{array}

The actuator faults are supposed to happen all the time. Moreover, assume that the actuator degradation model in (2) is as follows

{\bar{δ}}_{1} = {\bar{δ}}_{2} = diag (0.8, 0.7), {\underline{δ}}_{1} = {\underline{δ}}_{2} = diag (0.3, 0.2)

By choosing $α_{1} = α_{2} = \frac{1}{2}$ , it can be easily shown that $D B_{i}$ is non-singular. Thus, the desirable sliding function (6) is designed as

S (t) = [\begin{matrix} - 0.1081 & - 1.1351 & - 0.3243 \\ 0.5189 & 1.4486 & 0.7568 \end{matrix}] x (t)

Moreover, for $ς = 2.7$ , solving the minimization problem (34) yields

\begin{matrix} ε_{1} = 0.8556, ε_{2} = 0.7414, \\ T_{1} & = [\begin{matrix} 0.3750 & - 0.5006 \\ - 0.5006 & 0.9049 \end{matrix}], T_{2} = [\begin{matrix} 0.3830 & - 0.5061 \\ - 0.5061 & 0.8053 \end{matrix}], \\ Y_{1} = [\begin{matrix} - 0.6952 & 0.8519 & - 3.6380 \\ 1.5584 & - 2.3623 & 2.7171 \end{matrix}], \\ Y_{2} = [\begin{matrix} 0.6966 & - 1.2633 & 4.6329 \\ - 1.3848 & 4.0068 & - 5.4958 \end{matrix}], \\ K_{1} = [\begin{matrix} - 7.4464 & 6.0241 & - 8.0994 \\ 10.4809 & - 12.3049 & 8.2667 \end{matrix}], \\ K_{2} = [\begin{matrix} 6.8612 & - 5.5523 & 9.7557 \\ - 9.5467 & 17.0472 & - 12.3416 \end{matrix}], \\ X_{1} = [\begin{matrix} 0.7932 & 0.2333 & - 0.4699 \\ 0.2333 & 0.3517 & - 0.0581 \\ - 0.4699 & - 0.0581 & 0.8380 \end{matrix}], \\ X_{2} = [\begin{matrix} 0.8653 & 0.0246 & - 0.5231 \\ 0.0246 & 0.2424 & - 0.0088 \\ - 0.5231 & - 0.0088 & 0.8378 \end{matrix}] \end{matrix}

$ϵ_{1} = 9.8525 \times 10^{- 14}$ , and $ϵ_{2} = 1.0487 \times 10^{- 12}$ (that is the constrains $B_{i} T_{i} = X_{i} D^{T}$ , $i \in Γ$ are satisfied).

According to Theorem 2, the parameters $μ$ , $λ$ and $T_{σ}$ are designed, respectively, as follows

μ = max_{i, j} \frac{λ_{max} (X_{i}^{- 1})}{λ_{min} (X_{j}^{- 1})}, i, j \in {1, 2}, and T_{σ} > \frac{\ln μ}{ς}

Therefore, it can be obtained that

μ = 6.8858

and the average dwell time can be designed as

T_{σ} = 0.72

In view of (7) and (8), the adaptive SMC law is designed as

\begin{matrix} u (t) = {\begin{matrix} {(I - \hat{Δ} (t))}^{- 1} [\begin{matrix} - 7.4464 & 6.0241 & - 8.0994 \\ 10.4809 & - 12.3049 & 8.2667 \end{matrix}] \\ x (t) - 5 (ρ_{1} (t) + η_{1}) \\ \times sgn ({(D B_{1})}^{T} S (t)), if i = 1, \\ {(I - \hat{Δ} (t))}^{- 1} [\begin{matrix} 6.8612 & - 5.5523 & 9.7557 \\ - 9.5467 & 17.0472 & - 12.3416 \end{matrix}] \\ x (t) - 5 (ρ_{2} (t) + η_{2}) \\ \times sgn ({(D B_{2})}^{T} S (t)), if i = 2 \end{matrix} \\ L_{j} (t) = {\begin{matrix} - λ_{j} {(1 - {\hat{δ}}_{j} (t))}^{- 1} x^{T} (t) [\begin{matrix} 7.7185 & - 9.7740 & 5.5046 \\ 29.0888 & - 35.2823 & 22.0170 \\ 12.7644 & - 15.8531 & 9.3575 \end{matrix}] \\ x (t), if i = 1, \\ - λ_{j} {(1 - {\hat{δ}}_{j} (t))}^{- 1} x^{T} (t) [\begin{matrix} - 4.3815 & 4.6198 & - 6.0879 \\ - 16.7733 & 11.9533 & - 24.0634 \\ - 7.2980 & 6.5485 & - 10.2918 \end{matrix}] \\ x (t), if i = 2 \end{matrix} \end{matrix}

where

\begin{matrix} ρ_{1} (t) = 1.8629 ‖ [\begin{matrix} - 10.157511.6228 - 7.9530 \\ 13.5079 - 20.166010.4156 \end{matrix}] x (t) ‖ + 0.5988 ∥ H_{1} x (t) ∥ + d_{1} + 21.4520 ∥ x (t) ∥, \end{matrix}

\begin{matrix} ρ_{2} (t) = 1.9652 ‖ [\begin{matrix} 12.7742 & - 1.0026 & 12.2926 \\ - 18.4649 & 12.1874 & - 16.2567 \end{matrix}] x (t) ‖ + 2.7224 ∥ H_{2} x (t) ∥ + d_{2} + 90.9009 ∥ x (t) ∥ \end{matrix}

Then, choose the parameters $λ_{1} = 0.1, λ_{2} = 0.3$ , $d_{1} = d_{2} = 1$ , $ϑ_{1} = ϑ_{2} = 1$ , and $η_{1} = η_{2} = 0.01$ . For the initial state $x (0) = [0.8 0.2 - 0.6]^{T}$ , the simulation results with the proposed adaptive sliding mode controller are illustrated in Figures 1 to 4. The switching signal is given in Figure 1. The control signal is depicted in Figure 4. It can be seen from Figures 2 and 3 that the states $x_{1} (t)$ , $x_{2} (t)$ and $x_{3} (t)$ will be driven onto the domain of the sliding mode and exponentially tend to zero. The state trajectories are attracted towards the designed sliding surface, which shows that the proposed method in this paper can effectively cope with the effects of actuator faults.

Figure 1.

Switching signal $σ (t)$ .

Figure 2.

State trajectories $x (t)$ .

Figure 3.

Sliding variables $S (t)$ .

Figure 4.

Control signals $u (t)$ .

Conclusion

In this paper, the problem of adaptive SMC for a class of stochastic switched systems subject to actuator degradation has been studied. By employing the weighted sum approach, a common sliding surface is designed. Moreover, to compensate the loss of effectiveness of the actuators, an adaptive sliding mode controller is designed. It is noted that in this work only the actuator faults have been considered. In future research, simultaneous sensor and actuator fault phenomena, as in Youssef et al. (2017), may be further considered.

Footnotes

Conflict of interest

The authors declare that there is no conflict of interest.

Funding

This work was supported by the basic research project of Shanghai Dianji University (A1-0227-18-016-04).

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