Distributed Adaptive Fault-Tolerant Output Regulation of Heterogeneous Multi-Agent Systems With Coupling Uncertainties and Actuator Faults

I. Introduction

THE application of multi-agent systems (MASs) [1]–[6] is wide, including smart grid systems [1], intelligent transportation systems [2] and multi-robot systems [3]. In the last decade, rapid development has achieved in MASs. The cooperative output regulation (COR), one of the important research topics on MASs, has received considerable attention. The control objective of COR is to design the control protocol of subsystems to achieve the disturbance rejection and trajectory tracking for each of them. Some effective COR methods have been designed for linear heterogeneous MASs to achieve its control objective [7]–[13]. In particular, a distributed control approach is proposed in [7] to solve the robust output regulation problem. In [8], the authors develop a new robust COR method by relaxing the assumption that circles are allowed in the network topology. In [9], the adaptive distributed observer approach is proposed to solve the COR problem assuming that the neighbours of the exosystem know the system matrix of the exosystem. In [10], a novel data-driven solution is given to solve the COR problem under switching network topology.

Practically, actuator faults often occur owing to the long-running operation of the systems. The presence of actuator faults deteriorate the performance or even destabilize the system [14]–[16]. For the sake of reliability and safety, some valuable distributed fault-tolerant control methods have been proposed to compensate for the influence of actuator faults. In [17], an effective distributed fault estimator is designed for MASs under the directed network topology. In [18], the authors concern with the output tracking consensus for a class of pure-feedback nonlinear MASs with actuator failures. In [19], the fixed-time fault-tolerant controller is designed for second-order nonlinear MASs. In [20], a distributed adaptive fuzzy fault-tolerant controller is proposed for the high-order nonlinear MASs with actuator bias faults. A distributed adaptive event-triggered fault-tolerant method is designed in [21] to reduce the communication burden among the communication networks. In [22], the authors design a circuit implementation method for the developed fault-tolerant consensus controller and apply to a multiple coupled nonlinear forced pendulum system. However, it is still difficult to solve the cooperative fault-tolerant output regulation problem directly through the existing methods. The major challenges are: 1) to find an effective way to ensure the COR problem is solvable under the influenced of actuator faults; 2) how to present a fault-tolerant controller to compensate for actuator faults. In [23], a sufficient condition is established to ensure that the regulator equations are solvable and then a distributed fault-tolerant controller is designed. In [24], the authors solve the COR for linear heterogeneous MASs under the influence of denial-of-service attacks and actuator faults. However, only nominal and certain MASs are considered in the aforementioned results.

Another practical issue on MASs controller design is to address the effect of uncertainties, including both system uncertainty and coupling uncertainty. In [25], the COR problems of MASs with coupling uncertainties among subsystems are studied. However, only the matched coupling uncertainties are considered. It is well known that mismatched coupling uncertainties are more general in practical applications. Moreover, the existing cyclic-small-gain theorem method in [25] cannot be directly used to settle this problem. Therefore, to develop a fault-tolerant and robust distributed control method for heterogeneous MASs with system uncertainties and mismatched coupling uncertainties to resist the influence of actuator faults is an open and interesting problem.

In this paper, we will solve the distributed adaptive fault-tolerant output regulation problem for heterogeneous MASs with system uncertainty and mismatched coupling uncertainties under the influence of actuator faults. Compared with existing results, the contributions are threefold.

1) A novel distributed adaptive fault-tolerant control method is proposed to solve the fault-tolerant output regulation problem for heterogeneous MASs with matched system uncertainties and mismatched coupling uncertainties among subsystems. To the best of our knowledge, this paper is the first attempt to integrate cyclic-small-gain techniques, COR theory and distributed fault-tolerant method.

2) Different from the existing distributed fault-tolerant control result [23], a more general directed network topology is considered in this paper. To observe the state of the exosystem, novel distributed finite-time observers are designed. In particular, a new variable is introduced in observers to identify which subsystem accurately estimates the state of the exosystem. Besides, we develop a novel distributed fault-tolerant control method that is able to handle unknown matched and bounded uncertainties. Besides, it is shown that all subsystems can achieve trajectory tracking while rejecting nonvanishing disturbances.

3) Different from our previous work [25], in which only the matched coupling uncertainties are considered, the more general unmatched coupling uncertainties are considered in this paper. To deal with it, a novel sufficient condition with cyclic-small-gain condition is proposed by using the linear matrix inequality technique.

Notations: For vectors $x_i\in {\mathbb R}^{m}$ , the vector $[x_1^T,\ldots,x_N^T]^T$ is denoted by $col\{x_1,\ldots,x_N\}$ . $I$ and $0$ represent respectively the identity matrix and zero matrix (or vector). A matrix with matrices $N_1,\ldots,N_m$ on its principal diagonal is symbolized by $diag\{N_1,\ldots,N_m\}$ . ${\bf{1}} = col\{1,\ldots,1\}\in {\mathbb R}^N$ . ${\rm sign}(x)$ denotes the sign function of a scalar $x$ .

II. Problem Formulation

A Systems Dynamics

Consider the heterogeneous uncertain MASs generalized from [25]

where $x_i (t)\in {\mathbb R}^{n_i }$ , $u_i (t)\in {\mathbb R}^{m_i }$ and $e_i (t)\in {\mathbb R}^{p_i }$ are the state, control input and regulated output vectors, respectively. $v(t)\in {\mathbb R}^q$ is the exogenous signal. $S$ , $A_i$ , $B_i$ , $C_i$ , $E_i$ and $F_i$ are system matrices with proper dimensions. The matched system uncertainty satisfies $\Delta A_i = B_iG_i(t)$ . The function $\Phi_i(e,v)$ denotes unknown interconnections satisfying $|\Phi_i(e,v)|\leq d_i|e|$ , where $e = col\{e_1,\ldots,e_N\}$ and $d_i$ ( $i = 1,2,\ldots,N$ ) are know constants.

We introduce the following assumptions.

Assumption 1: $(A_i,B_i)$ is stabilizable for $i = 1,2,\ldots, N$ .

Assumption 2: There exist solution matrices $X_i$ and $U_i$ such that the following regulator equation are satisfied

Remark 1: As discussed in [9] and [26], Assumptions 1 and 2 are necessary to ensure that the COR problem is solvable.

In this paper, we consider a directed graph ${\cal G} = ({\cal N},{\cal S})$ , which consists of nodes ${\cal N} = \{v_1, v_2 ,\ldots,v_N\}$ and edges ${\cal S} = \{(v_i,v_j)|v_i,v_j\in{\cal N}\}$ . If node $v_i$ can receive information from node $v_j$ , then $(v_i,v_j)\in{\cal S}$ . Node $v_j$ is the neighbor of node $v_i$ if $(v_i,v_j)\in{\cal S}$ . The neighbor set of the node $v_i$ is given by ${\cal N}_i = \{v_j\in{\cal N}:(v_i,v_j)\in{\cal S}\}$ . The adjacency matrix ${\cal A} = [a_{ij}]\in {\mathbb R}^{N\times N}$ is denoted as $a_{ii} = 0$ , $a_{ij} = 0$ if $(v_i,v_j)\not\in{\cal S}$ and $a_{ij} = 1$ if $(v_i,v_j)\in{\cal S}$ . The definition of Laplacian matrix ${\cal L}$ is ${\cal L} = {\rm diag}\{\phi_1,\ldots,\phi_N\}-{\cal A}$ with $\phi_i = \sum\nolimits_{j = 1}^Na_{ji}$ . Let ${\cal M} = {\rm diag}\{a_{10}, \ldots, a_{N0}\}$ , where $a_{i0} = 1$ represents that the exosystem can send information to subsystem $i$ and $a_{i0} = 0$ otherwise. Define ${\cal H} = {\cal L}+{\cal M}$ .

Assumption 3: The directed graph ${\cal G}$ contains a directed spanning tree and the exosystem is the root.

B Fault Model

Similar to [27], this paper considers the following actuator faults. $u_{ij}(t)$ and $u^{F_h}_{ij}(t)$ represent respectively the input and the output under the fault model $h$ ( $h = 1,\ldots,H$ ) for the actuator $j$ ( $j = 1,\ldots,m_i$ ) of subsystem $i$ ( $i = 1,\ldots,N$ ). It is defined by

where $H$ represents the total of fault models. The unknown constant $\rho ^h_{ij}$ satisfies $0\leq\underline\rho^{h}_{ij} \leq \rho^{h}_{ij}\leq \overline\rho^{h}_{ij}\leq1$ . $\omega_{ij}^h(t)$ is a bounded time-varying function.

We can summarize the model (5) as follows: 1) $\underline\rho^{h}_{ij} = \overline\rho^{h}_{ij} = 1$ and $\omega_{ij}^h(t) = 0$ mean the fault-free; 2) $0< \underline\rho^{h}_{ij}\leq \overline\rho^{h}_{ij}< 1$ and $\omega_{ij}^h(t) = 0$ signify the loss-of-effectiveness fault; 3) $\rho^{h}_{ij} = 0$ and $\omega_{ij}^h(t) = 0$ represent the outage fault; 4) $\rho^{h}_{ij} = 0$ and $\omega_{ij}^h(t)\neq 0$ represent the bias fault.

Denote

where $u^{F_h}_{i}(t)\! = \!col\{u^{F_h}_{i1}(t)$ , $\ldots,u^{F_h}_{im_i}(t)\}$ , $u_i(t) = col\{u_{i1}(t),\ldots, u_{im_i}(t)\}$ , $\Lambda_i^{h} = {\rm diag}\{\rho^{h}_{i1},\ldots,$ $\rho^{h}_{im_i}\}$ and $\omega_i^h(t) = col\{\omega_{i1}^h,\ldots,\omega_{im_i}^h\}$ . Finally, the fault model is presented as

where $\Lambda_i = {\rm diag}\{\rho_{i1},\ldots,\rho_{im_i}\}$ , $\Lambda_i\in \{\Lambda_i^1,\ldots,\Lambda_i^H\}$ and $\omega_i\in \{\omega_i^1, \ldots, \omega_i^H\}$ .

Assumption 4: The uncertain parameter $G_i(t)$ and actuator bias fault signal $\omega_i(t)$ are bounded, i.e., there exist $\bar g_i>0$ and $\bar w_i>0$ such that $||G_i(t)||\leq \bar g_i$ and $||\omega_i(t)||\leq \bar w_i$ .

Assumption 5: rank $(B_i\Lambda_i)$ = rank $(B_i)$ for any $\Lambda_i\in \{\Lambda_i^1,$ $\ldots,\Lambda_i^H\}$ , $i = 1,\ldots,N$ . rank $(B_i)$ represents the rank of matrix $B_i$ .

Lemma 1 [27]: If Assumption 5 holds, there exists positive constants $\mu_1$ , $\mu_2$ , $\ldots,$ and $\mu_N$ such that

Lemma 2 [23]: If Assumptions 1, 2 and 5 are satisfied, then there exist matrices $X_i$ and fault-dependent matrices $U_i^{\Lambda_i}$ satisfy the following regulator equations

Lemma 3 [28]: Consider the system:

Let $l_1,l_2,\ldots,l_r$ be positive constants such that $s^n+l_rs^{r-1}+ \cdots+l_2s+l_1$ is Hurwitz. There exists positive constant $\varepsilon<1$ such that, for any $\kappa\in (1-\varepsilon,1)$ , the above system is globally finite-time stability under the following controller

where $\kappa_i$ satisfies

with $\kappa_{r+1} = 1$ and $\kappa_r = \kappa$ .

Remark 2: Assumption 4 has been used in some existing fault-tolerant control results [23] and [27]. From Lemma 4 in [23], we know that a sufficient condition, which ensures that the cooperative fault-tolerant output regulation problem is solvable, is that Assumption 4 holds.

C Cooperative Fault-Tolerant Output Regulation Problem

The control objective is to propose a distributed adaptive robust fault-tolerant control method for MASs (1)–(3) with system uncertainties and mismatched coupling uncertainties among subsystems under the influence of actuator faults (6) such that

1) The regulated outputs satisfy $\lim\nolimits_{t\rightarrow\infty}e_i(t) = 0$ , for $i = 1,2,\ldots,N$ .

2) The closed-loop MASs are asymptotically stable when $v = 0$ .

Remark 3: Different from the existing cooperative fault-tolerant results [18] and [19], the dynamics of the MASs considered in this paper contain nonvanishing signal $v(t)$ and the unknown interconnection $\Phi_i(e,v)$ in (2). To eliminate the influence of the signal $v(t)$ and $\Phi_i(e,v)$ , a framework of COR theory and cyclic-small-gain techniques is introduced in this paper. Besides, compared with [18] and [19], the outage faults, which may bring difficulties to theoretical analysis and often occur in practical applications, is considered in this paper.

III. Main Results

A Distributed Finite-Time Observers Design

In order to estimate the state of exosystem (1), we design the distributed finite-time observers for subsystems $i = 1,2,\ldots,N$

where $\bar S = TST^{-1}$ , $\bar T\! = \!{\rm diag}\{T_{1},\ldots,T_{r}\}$ , and $T$ is given in Lemma 2 of [23]. $t_i$ is the time constant, which is given by

with

$\nu_i(t)\! = \!col\{\nu_{i1}(t),\ldots,$ $\nu_{ir}(t)\}\!\in {\mathbb R}^r$ is the input vector which is given by

where parameters $\alpha_{jk}$ are given in Lemma 2 of [23] and positive scalars $\beta_{ij}^k$ and $\theta_{ij}^k$ are given in Lemma 4 and ${\zeta}_{ij}(t) = col\{{\zeta}_{ij}^1(t),\ldots,{\zeta}_{ij}^{p_j}(t)\}\in {\mathbb R}^{p_j}$ . The dynamic of ${\zeta}_i(t)\! = \!col\{{\zeta}_{i1}(t), \ldots,{\zeta}_{ir}(t)\}$ is described by

where $\varphi_i(t) = \sum_{j = 0}^Na_{ij}c_{ij}(t)[(\zeta_i(t)-\zeta_j(t))\!-\!(\eta_i(t)-\eta_j(t))]$ and $\eta_0(t) = Tv(t)$ and

with

$\iota$ is an arbitrary positive constant and $c\geq{1}/{\lambda_1}$ . $\bar T$ is given in the proof of Lemma 2 of [23]. $R>0$ satisfies a Riccati equation $R\bar S+\bar S^TR-2R\bar T\bar T^TR = -Q_1$ , where $Q_1$ is a any positive definite matrix.

One can have the following Lemma 4 to ensure the finite-time estimation.

Lemma 4: Choose $\beta_{ij}^1,\ldots,\beta_{ij}^{p_j}$ such that the polynomial $s^{p_j}+\beta_{ij}^{p_j}s^{p_j-1}+\cdots+\beta_{ij}^2s+\beta_{ij}^1$ is Hurwitz. If $\theta_{ij}^k$ is chosen as

with $\theta_{ij}^{p_j+1} = 1$ , $\theta_{ij}^{p_j} = \theta_{ij}$ and $\theta_{ij}\in(1-\varepsilon,1)$ ( $\varepsilon$ is given in Lemma 3). Then, the state $v_i(t)$ with $v_i(t) = T^{-1}\eta_i(t)$ converges to $v(t)$ after finite-time $T_0$ , i.e., $v_i(t) \equiv v(t)$ for any $t\geq T_0$ .

Proof: See Appendix. ■

Remark 4: Different from the existing finite-time observers in [23], the developed distributed finite-time observers can deal with more general directed network topology, which contains the undirected network topology as a special case. Interestingly, a new variable is introduced in the observers to identify which subsystem accurately estimates the state of the exosystem.

Remark 5: In the distributed finite-time observers (8), the matrix $S$ of the exosystem is known for all subsystems. However, it is only assumed that the neighbors of the exosystem know this matrix in [24]. To solve the problem, a data-driven distributed learning approach in [29] can be used to estimate the matrix $S$ for all subsystems.

Remark 6: Different from [30], in which the distributed cooperative filters are designed for MASs to estimate the state, the objective of designing distributed finite-time observers is to estimate the trajectories of the exosystem. Based on the developed observers, the fault-tolerant controller will be designed in next subsection to eliminate the influence of the nonvanishing signal $v(t)$ for each subsystem.

B The Fault-Tolerant Controller Design

Incorporating finite-time observers (8), the fault-tolerant controller is designed as follows

where

$\bar{ x}_i(t) = x_i(t)-X_iv_i(t)$ , $\hat K_i(t) = col\{ \hat K_{i}^1(t)$ , $\ldots,\hat K_{i}^{m_i}(t)\}$ and $\hat U_i(t) = col\{ \hat U_{i}^1(t),\ldots,\hat U_{i}^{m_i}(t)\}$ . $\hat K_{i}^j(t)\in {\mathbb R}^{1\times n_i}$ and $\hat U_{i}^j(t)\in {\mathbb R}^{1\times q}$ are updated by

where $b_{ij}$ is the $j$ th column of matrix $B_i$ . $\gamma_{ij1}$ and $\gamma_{ij2}$ are arbitrary positive constants. $\hat { {w}}_i$ and $\hat { {g}}_i$ are the estimation of $\bar w_i$ and $\bar g_i$ and is updated by

where $\hat w_i(0)$ and $\hat g_i(0)$ are nonnegative and $\gamma_{i}$ and $\gamma_{gi}$ are arbitrary positive constants. Besides, $P_i>0$ is determined by

where $\pi = \max_{i = 1,2,\ldots,N} ||C_i^TC_i||$ , $\beta_i>0$ is a design parameter.

Remark 7: Different from the existing cooperative fault-tolerant output regulation result [23], the more general heterogeneous MASs with actuator bias fault $\omega_i(t)$ and system uncertainty $\Delta A_i$ are considered in this paper. To solve the problem, compensation terms $u_{i2}(t)$ and $u_{i3}(t)$ with adaptive parameters $\hat \omega_i(t)$ and $\hat g_i(t)$ are introduced, respectively.

C Stability Analysis

Let $\bar x_i(t) = x_i(t)-X_iv(t)$ . From the MASs (1)–(3) and the fault model (6), we have

Using $\Delta A_i = B_iG_i$ , we further have

According to the regulator equations (7), it yields

Define $\bar A_i = A_i+B_i\Lambda_i K_i$ . It is derived that

Then, a sufficient condition on the COR for the MASs (1)-(3) with actuator faults (6) is presented in the following Theorem.

Theorem 1: For given $\beta_i>0$ where $\beta_i$ satisfies the following cyclic-small-gain condition

If there exists $P_i>0$ such that (23) is satisfied, then the cooperative fault-tolerant output regulation problem can be solved by using the finite-time observer (8) and fault-tolerant controller (15) with adaptive laws (19)–(21).

Proof: Construct the Lyapunov functional

where

and

$\tilde K_{i}^j = \hat K_{i}^j- K_{i}^j$ , $\tilde U_{i}^j = \hat U_{i}^j- U_{i}^j$ , $\tilde w_{i} = \hat w_{i}- \bar w_{i}/\mu_i$ and $\tilde g_{i} = \hat g_{i}- \bar g_{i}/\mu_i$ . $K_i = [K_{i}^1,\ldots,K_{i}^{m_i}] = -\frac{1}{\mu_i}B_i^TP_i$ and $U_i = [U_{i}^1,\ldots,U_{i}^{m_i}] = U_i^{\Lambda_i}$ . The definition of $\mu_i$ is given in Lemma 1. From Lemma 3, we know $v_i\equiv v$ for $t\geq T_0$ . Thus, we discuss the case for $t\geq T_0$ in the following. The derivative of $V_i$ along (24) is given as

By using the fault-tolerant controller (15), we further have

Using the Young’s inequality, it yields that

From $|\Phi_i(e,v)|\leq d_i|e|$ and using the regulator equation (7), it yields that

where $\bar x = col\{\bar x_1,\bar x_2,\ldots,\bar x_N\}$ . Besides, the fact $e_i = C_ix_i+ F_iv = C_i\bar x_i$ is used. Substituting (29) and (30) into (28), we have

Using Assumption 4 and the adaptive updated law (21), we have

Using Assumption 4 and the adaptive updated law (22), we have

Using the adaptive laws (19) and (20), it yields

Substituting (32) – (34) into (31), we have

From the definition of $\bar A_i = A_i+B_i\Lambda_i K_i$ and $K_i = -\frac{1}{\mu_i}B_i^TP_i$ , we have

By applying the Schur complement lemma to (23), we have

Substituting (36) and (37) into (35), we have

Hence,

Using the cyclic-small-gain condition (25), we have

By using the Barbalat’s lemma in [31], it is deduced that $\lim\nolimits_{t\rightarrow\infty}||\bar x(t)|| = 0$ . By using the regulator equations (7), we further show that

■

Remark 8: Different from our previous work [25], in which only the matched coupling uncertainties are considered, the more general unmatched coupling uncertainties are considered in this paper. The major difficulties met when deriving the current results can be summarized: 1) whether the cooperative fault-tolerant output regulation problem can be solved under the more general unmatched coupling uncertainties? 2) if 1) is solved, then how to obtain a new circle-small-gain condition? To overcome the difficulties, a novel sufficient condition (23) with cyclic-small-gain condition (25) is proposed.

Remark 9: In the inequality condition (23), the Lyapunov matrix $P_i$ cannot be solved directly because the term $P_iB_iB_i^TP_i$ is a nonlinear term. To solve the problem, the linear matrix inequality (41) can be achieved by using the following mathematical derivation: 1) multiplying ${\rm diag}\{X_i,I\}$ ( $X_i = P_i^{-1}$ ) on the left and right in (23); 2) using the well known Schur complement lemma.

Remark 10: In the fault-tolerant controller (17), when ${t}$ goes to infinity, the function $e^{-t}$ will tend to zero, which means that the chattering phenomenon of ${u_i}$ may occur in simulation. To eliminate chattering, the well-known boundary layer approach in [32] and [33] is introduced as follows

where $\varepsilon$ is a sufficiently small constant. This method can be also used for $u_{i3}$ .

IV. Simulation Results

In this section, we present an example from [25] to demonstrate the efficiency of the proposed method. The network topology graph ${\cal G}$ is depicted in Fig. 1. Considering the following MASs (1) – (3) with

Figure  1.  The network topology.

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The considered system uncertainties are chosen as $\Delta A_i = B_i\sin(t)$ for $i = 1,2,3,4$ . In this simulation, subsystems 1–3 are fault-free and the actuator fault of the $4$ th subsystem is set as

which implies that the actuator is loss-of-effectiveness at $t = 50$ s.

The initial states of the systems and the adaptive distributed observer are set as $v(0) = [1\; -1]^T$ , $v_i(0) = i\cdot[1\; -1]^T$ , $x_{i}(0) = i\cdot[0.1\; 0.1]^T$ , $\hat K_{i}^1(0) = \hat K_{i}^2(0) = i$ , $\hat U_{i}^1(0) = \hat U_{i}^2(0) = i$ , ${\hat{ w}}_i(0) = i$ , $i = 1,\ldots,4$ .

The regulated outputs $e_i(t)$ with the developed method and the method in [25] are shown in Figs. 2 and 3, which illustrates that, in the presence of actuator faults, the regulated outputs converge to zero by using the developed method, while they cannot converge to zero by using the method in [25]. The trajectories of adaptive parameters $\hat K_{i}^1(t)$ , $\hat K_{i}^2(t)$ , $\hat U_{i}^1(t)$ , $\hat U_{i}^2(t)$ , ${\hat{ w}}_i(t)$ , and ${\hat{ g}}_i(t)$ are given in Figs. 4-9, which demonstrate that all these signals are bounded by using the developed method.

Figure  2.  Trajectories of $e_{i}$ under the proposed method.

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Figure  3.  Trajectories of $e_{i}$ under the method in [25].

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Figure  4.  Trajectories of $\hat K_{i}^1$

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Figure  5.  Trajectories of $\hat K_{i}^2$ .

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Figure  6.  Trajectories of $\hat U_{i}^1$ .

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Figure  7.  Trajectories of $\hat U_{i}^2$ .

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Figure  8.  Trajectories of adaptive parameters $\hat w_{i}$ .

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Figure  9.  Trajectories of adaptive parameters $\hat g_{i}$

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V. Conclusion

This paper has proposed a novel distributed adaptive robust fault-tolerant control method to deal with the COR problem for heterogeneous MASs with matched system uncertainties and mismatched coupling uncertainties among subsystems under the influence of actuator faults. A new fault-tolerant controller has been proposed to compensate for the influence of matched system uncertainties and actuator faults. Further, a sufficient condition based on linear matrix inequality technique has been provided to guarantee the solvability of the considered problem under the influence of mismatched coupling uncertainties. It has been shown that the COR problem can be solved via the developed method by using the Lyapunov stability theory and cyclic-small-gain theorem.

AppendixProof of Lemma 4

Now we prove via mathematical induction.

1) Consider the subsystems which can receive the information from the exosystem. Define $\tau_i = \zeta_i-\tilde \eta_i$ , where $\tilde\eta_i = col\{\tilde\eta_{i1},\ldots,\tilde\eta_{ir}\} = \eta_i-\eta_0$ is the observation error. From (1) and (8), the dynamics of $\tilde\eta_i$ are

According to (12) and (43), we have

where $\tilde \eta_0 = 0$ . Let $\tau = [\tau_1,\ldots,\tau_N]^T$ . Accordingly,

Let $W = \tau_i^T R\tau_i$ . The derivative of $W$ is given by

Applying Theorem 4.2 in [34], it can be shown that $\tau_i$ converges to zero after finite-time.

From (43) and the definitions of $\tilde\eta_i$ , $\bar S$ , $\bar T$ and $\nu_i$ , one obtains

Substituting the value of of $\nu_{ij}$ in (11) into (44), it shows that

Therefore, after finite-time, (45) will reduce to

According to Lemma 3, we have that ${\tilde\eta}_{ij}$ converges to zero after finite time, which together with the conclusion of that $\tau_i$ converges to zero after finite-time yields that $\eta_i$ converges to $\eta_0$ in finite-time.

2) Consider the $j$ th subsystem which can obtain the information from the $i$ -th subsystem, which converges to $\eta_0$ in finite-time. By using the similar method, we can obtain that $\eta_j$ converges to $\eta_0$ in finite-time.

Summing up 1) and 2), we can complete the proof. ■