Cooperative control for consensus of multi-agent systems with actuator faults

Abstract

Consensus tracking problem in multi-agent systems with both outage and partial loss of effectiveness types of actuator faults is investigated in this paper. By adopting the virtual actuator technique based on the obtained fault estimates, the effects of possible actuator faults can be effectively compensated in each individual agent. Based on this, a distributed control strategy is developed by using locally available information. It is shown that the tracking error of each fault-free agent can converge to zero and the tracking errors of the faulty agents can remain bounded if the fault estimates are accurate. Moreover, a sufficient condition for achieving bounded tracking errors for all the followers is derived in terms of the fault estimation error. The sliding mode observer is utilized to estimate the faults. Simulation results are given to show the effectiveness of the proposed approach.

Introduction

Cooperative control of multi-agent systems (MAS) has received substantive research in the past two decades due to its wide potential applications in unmanned ground/air vehicles [1], satellite formation [2], distributed filtering, sensor fusion [3] and so forth. The main challenges in MAS control can be summarized as follows. (i) The centralized information is difficult to be collected by all the agents as they are sometimes distributed distantly in space. This motivates the investigation of distributed (cooperative) control strategies. (ii) Various uncertainties such as unknown intrinsic parameters [4], external disturbances [5] as well as possible faults in both agents and communication links inevitably exist and require to be well handled to maintain desired control performances.

Fruitful results have been reported on centralized fault tolerant control (FTC) including fault detection/diagnosis (FDD) and fault accommodation control. It should be noted that although fault tolerance is also vitally important for guaranteeing the safety of controlled MAS, this area has not yet been extensively explored. Some preliminary results are reviewed here. In [6], a semi-decentralized optimal control strategy is proposed for a group of agents modeled by single integrator dynamics. The robustness of the control scheme against time-invariant actuator faults is analyzed. However, the details about how the actuator faults can be accommodated are not discussed. In [7], a bank of unknown input observers are developed in each agent to detect and isolate the faults in the interconnected second-order systems. It is not clear whether the results are applicable to deal with actuator faults. In [8], velocity synchronization for networked Euler–Lagrange systems subject to Partial Loss of Effectiveness (PLOE) type and additive type of actuator faults is explored. In [9], a robust adaptive control method is proposed to obtain a desired formation for multi-robot systems. However, the control scheme is only applicable to handle PLOE type of actuator faults. And the results in [8], [9] are both obtained in the context of undirected communication topology. In [10], [11], the consensus problem for MAS with additive actuator faults is explored. It is worth noting that the control laws in [8], [9], [10], [11] may suffer from severe chattering phenomenon due to the adoption of sign function. There are also some literatures considering the faults of transmitting or receiving mechanisms in agents, examples can be found in [12], [13]. In summary, none of the aforementioned work study the distributed control of MAS with general linear node dynamics and suffering PLOE and outage types of actuator faults in a directed communication topology, which is the main motivation of our work.

In this paper, we investigate the problem of distributed fault accommodation control of MAS for achieving consensus tracking in the directed communication topology. In contrast to aforementioned results on FTC of MAS, the main features of our problem formulation are summarized as follows. (i) The agent dynamics is not restricted to be integrator models; (ii) Both of the outage and PLOE types of actuator faults are considered and allowed to occur simultaneously; (iii) The communication topology is allowed to be directed; and (iv) The control scheme avoids the use of sign function, thus is more practical to be implemented. The occurrence of actuator faults results in networks of heterogeneous agents. To handle such a problem, the virtual actuator approach [14][15] is applied in each follower based on the fault estimation results. By transmitting the “modified” state among the agents, we show that the neighbor-based control scheme in [16] which is originally developed for homogeneous case can still be applied in solving our problem. Moreover, our proposed approach can hide the faults in the faulty followers from being seen by other followers in a group. Therefore, the tracking errors of the faulty agents can be prevented from spreading to the healthy ones. It is shown that if the obtained fault estimates are accurate, the boundedness of all closed-loop signals in the group of agents can be ensured and the tracking error of each fault-free agent will converge to zero with the proposed scheme. In addition, the effect of fault estimation error on the tracking errors of all the followers is also studied and a sufficient condition for achieving bounded tracking errors is derived. In this paper, the sliding mode observer is presented as a suggesting tool for estimating the faults. Simulation results are given to show the effectiveness of the proposed approach.

The rest of the paper is organized as follows. Section 2 presents the mathematical models of the agents and the communication topology. In Section 3, the fault accommodate control strategy is described in detail. Section 4 provides the sliding mode observer as an alternative tool for fault estimation. The simulation results are presented in Section 5 to demonstrate the performance of the proposed control scheme. Section 6 finishes the paper with conclusions.

$•$ Notations: $I_N$ is the $N\times N$ identity matrix. Let $\Vert ·\Vert$ be $l_2$ norm of vectors. $\Vert ·{\Vert }_L$ is the L-norm of an $m\times n$ matrix $A=\left(a_{\mathit{ij}}\right)$, i.e., ${‖A‖}_L={\sum }_{i\text{,}j}\left|a_{\mathit{ij}}\right|$. $\otimes$ denotes the Kronecker product of two matrices. $\mathit{eig}(A)$ denotes the eigenvalues of A. Let ${\alpha }_1\text{,}\dots \text{,}{\alpha }_n$ and ${\beta }_1\text{,}\dots \text{,}{\beta }_n$ be the eigenvalues of matrix A and B, define the distance between $\mathit{eig}(A)$ and $\mathit{eig}(B)$ as $d\left(\mathit{eig}(A)\text{,}\mathit{eig}(B)\right)={\min }_{\sigma }{\max }_i\left(\left|{\alpha }_i-{\beta }_{\sigma (i)}\right|\right)$, where the permutation $\sigma$ is taken over 1 to n.