Cooperative control for consensus of multi-agent systems with actuator faults☆
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: is the identity matrix. Let be norm of vectors. is the L-norm of an matrix , i.e., . denotes the Kronecker product of two matrices. denotes the eigenvalues of A. Let and be the eigenvalues of matrix A and B, define the distance between and as , where the permutation is taken over 1 to n.
Section snippets
Problem formulation
Let us consider a group of N agents modeled as follows:where are the state and input of the i-th agent. A is a constant matrix of appropriate dimensions. In contrast to most of the existing literatures on distributed control of MAS, actuator faults are considered here which are described by the input matrix . Similar to [7], [17], it can be expressed aswhere B is a constant matrix of full column
Fault-tolerant consensus tracking control
An active fault accommodate control strategy is proposed to handle the faults in (2), of which the block diagram is depicted in Fig. 1. It can be seen that the actuator faults are estimated on-line firstly and the fault estimation result is denoted by . Then a reconfiguration block is established, which is located between the plant and controller. Assume that is obtained by certain means which will be discussed in the later part of the paper. The virtual actuator technique is
Actuator fault estimation
Several methods have been proposed to estimate the actuator faults. In [23], the actuator effectiveness estimation problem is formulated as an augmented state Kalman filter problem. The solution is derived in the form of two-stage Kalman filter. In [24], an adaptive observer is proposed to estimate the unknown fault parameters. In [25], a sliding mode observer is presented to reconstruct the faults from the signal called as the equivalent output injection. Here we adopt the sliding mode
Simulation results
In this paper, we consider a group of linearized fourth-order lateral F-8 aircraft models which can be described by Eqs. (1), (2), (3) with the following parameters [27]
The model has four states and two inputs with denoting
Conclusion
This paper investigates the consensus tracking problem of MAS in the presence of outage and partial loss of effectiveness types of actuator faults. A distributed control strategy is developed based on the virtual actuator to achieve fault tolerant consensus tracking. The virtual actuator is built with fault estimation results. It is proved that the fault-free agents can asymptotically track the leader’s state and the faulty agents can track the leader’s state with bounded tracking errors in the
Acknowledgement
This work is supported by National Nature Science Foundation under Grant 61203068, 61290324, and 61021063.
BO Zhou received his Bachelor’s degree from the Huazhong University of Science and Technology, China in 2007. He is currently pursuing his Ph.D. in Tsinghua University, China. His main research interests include fault tolerant control of multi-agent systems and fault diagnosis of industrial processes.
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BO Zhou received his Bachelor’s degree from the Huazhong University of Science and Technology, China in 2007. He is currently pursuing his Ph.D. in Tsinghua University, China. His main research interests include fault tolerant control of multi-agent systems and fault diagnosis of industrial processes.
Wei Wang received her Ph.D. from Nanyang Technological University Singapore, in November 2011. She is currently a Lecturer with the Department of Automation at Tsinghua University, China. Her research interests include adaptive control, fault tolerant control, distributed cooperative control, state observer design and applications related to mechanical systems.
Hao Ye received his Bachelors and Doctoral degrees in Automation from Tsinghua University, China, in 1992 and 1996, respectively. He has been with the Department of Automation, Tsinghua University since 1996. He is currently a Professor of the Department of Automation, Tsinghua University. He is mainly interested in fault detection and diagnosis of dynamic systems.
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