Robust fault-tolerant cooperative control of multi-agent systems: A constructive design method

doi:10.1016/j.jfranklin.2015.05.031

Journal of the Franklin Institute

Volume 352, Issue 10, October 2015, Pages 4045-4066

https://doi.org/10.1016/j.jfranklin.2015.05.031 Get rights and content

Abstract

This paper investigates the distributed cooperative controller design problem for multi-agent systems in the presence of actuator faults. Both the partial loss of actuator effectiveness and the actuator bias faults are considered. A constructive design method is presented to achieve the synchronization and/or tracking control of multi-agent systems. The control protocol for each agent consists of three modules: the actuator fault detection module, the nominal control module, and the auxiliary control module. The auxiliary control module is activated as soon as the actuator faults are detected. The control protocols are locally distributed in the sense that the control modules designed for each agent only require the information of itself and its neighbors. The loss of symmetry in the digraph Laplacian matrix is also considered. The analysis is suitable for the multi-agent systems with a general directed communication graph. The proposed fault-tolerant control schemes are applied in the formation control of multi-robot systems. Numerical simulation results are presented to show the effectiveness of the proposed control methods.

Introduction

Distributed cooperative control of multi-agent systems has received significant attention from the control community due to its potential applications in many areas, such as distributed control of networked systems, formation control of unmanned aerial/ground vehicles, and cooperative control of distributed generator systems. Distributed cooperative control aims at designing appropriate controllers by using the information of each agent and its neighbors such that all the networked agents achieve the coordination requirements. Fruitful results have already appeared in the literature, for example, consensus [1], [2], [3], [4], formation control [5], [6], synchronization [7], [8], and target tracking control [9], [10], [11].

The development of advanced control technologies and the explosion in computation and communication capacities enable the applications of cooperative control systems. Compared to a single agent that performs certain task, greater efficiency and capability can be achieved by a group of agents operating in a coordinated fashion. On the other hand, advances in computer, communication, and control technologies have increased the complexity of engineering systems. Sometimes, a fault in the systems may deteriorate the systems performance or even cause catastrophic accidents. Thus, the reliability is one key objective for the design of engineering systems. It is required that the systems operate safely even in the presence of faults. In general, the faults in the systems are difficult to foresee. Thus, fault tolerance control is viewed as one of the most promising control technologies that are capable of maintaining stable and acceptable performance of the systems in the presence of unexpected faults. The fault tolerance control approaches can be classified into two classes: the passive approach and the active approach. The passive approaches use unchangeable controllers throughout the healthy and faulty cases [12], [13]. In contrast to the passive fault-tolerant control approach, the active approaches compensate for the faults either by selecting a pre-computed control strategy or by synthesizing a new control scheme on-line as soon as a diagnostic algorithm has detected the presence of a fault [14], [15], [16], [17], [18], [19]. Although the fault tolerance control of centralized systems has been extensively investigated, it is only in recent years that the fault-tolerant cooperative control of distributed systems with actuator faults receives some researchers’ attention. The work in [20] investigated the problem of fault tolerant control in cooperative interception. The work in [21] studied the mean-square consensus problem for a class of multi-agent systems subject to noise perturbations and actuator failures, where the stability analysis requires that the actuator failures are globally synchronous. In [22], the authors provided a fault-tolerant strategy for a class of multi-agent systems such that the states of all agents reach a common target point in spite of agent faults. The work in [23] presented a novel cooperative fault-tolerant fuzzy control scheme for multi-agent systems with the actuator bias faults. In [24], the authors proposed a decentralized fault-tolerant synchronization strategy for a group of networked satellites with actuator faults. The work in [25] provided the performance analysis of a team of unmanned vehicles subject to actuator faults.

In this paper, we provide a constructive method to design the robust fault-tolerant control scheme for multi-agent systems subject to actuator faults. An actuator fault detection algorithm is first presented. Then, a robust fault-tolerant protocol is proposed to achieve the synchronization and/or tracking control of multi-agent systems. The works in [20], [21], [22], [25] investigated the passive fault-tolerant control methods. We discuss the active fault-tolerant control methods. The control protocol of agents will be reconfigured after an actuator fault is detected. Although the authors in [23], [24] also discussed the active fault-tolerant methods, they only discussed the cooperative control problem with bias-type actuator faults. We consider the cooperative control problem with the bias-type actuator faults and the partial loss of actuator effectiveness. Since a large class of mechanical system can be transformed into the second-order form by using the feedback linearization techniques, we focus on the second-order multi-agent systems in this paper and design locally distributed control algorithm for each agent by using the information of itself and its neighbors only. The contributions of this paper can be summarized as follows.

(1) A constructive design method is presented to achieve the cooperative control of multi-agent systems in the presence of actuator faults. The control protocol for each agent consists of three modules: the actuator fault detection module, the nominal control module, and the auxiliary control module. The fault detection module monitors the actuator online. The nominal control module is designed for the agent with healthy actuator. The auxiliary control module is activated as soon as the actuator faults are detected.

(2) The proposed fault-tolerant control protocols are locally distributed in the sense that the control modules designed for each agent only require the information of itself and its neighbors. The control methods do not require the symmetry in the digraph Laplacian matrix. The analysis is suitable for the multiagent systems with a general directed communication graph.

The rest of the paper is organized as follows. Section 2 gives the problem formulation and preliminaries. Section 3 provides a fault detection algorithm. The robust fault-tolerant synchronization and tracking controller design methods are given in 4 Fault-tolerant synchronization controller design, 5 Fault-tolerant cooperative tracking controller design, respectively. The numerical simulations are presented in Section 6. Section 7 draws the conclusion.

Section snippets

Graph theory

Consider a digraph $G = (V, E)$ with a nonempty finite set of n nodes $V = (1, \dots, n)$ and a set of edges or arcs $E \subseteq V \times V$ . We assume that the graph has no self-loops, i.e., $(i, i) \notin E$ , $\forall i$ . Denote the adjacency matrix as $A = [a_{ij}]$ with $a_{ij} > 0$ if $(j, i) \in E$ and $a_{ij} = 0$ otherwise. The set of neighbors of a node i is $N_{i} = {j : (j, i) \in E}$ , i.e., the set of nodes with arcs incoming to i. Define the weighted in-degree of node i as $d_{i} = \sum_{j \in N_{i}} a_{ij}$ . The degree matrix D of G is $D = diag {d_{i}}$ . The Laplacian matrix of G is defined as $L = D - A$ . By

Actuator fault detection algorithm

We design an observer for each agent i as follows: $[\begin{matrix} {\dot{\hat{x}}}_{i} \\ {\dot{\hat{ν}}}_{i} \end{matrix}] = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] [\begin{matrix} {\hat{x}}_{i} \\ {\hat{ν}}_{i} \end{matrix}] + [\begin{matrix} 0 \\ 1 \end{matrix}] u_{i} + Γ_{i} ([\begin{matrix} x_{i} \\ ν_{i} \end{matrix}] - [\begin{matrix} {\hat{x}}_{i} \\ {\hat{ν}}_{i} \end{matrix}])$ where $Γ_{i} \in R^{2 \times 2}$ is an observer gain matrix. Let $e_{i} = [\begin{matrix} x_{i} \\ ν_{i} \end{matrix}] - [\begin{matrix} {\hat{x}}_{i} \\ {\hat{ν}}_{i} \end{matrix}], A_{i} = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] and B_{i} = [\begin{matrix} 0 \\ 1 \end{matrix}] .$ Assume that the actuator i is faulty. Thus, we have ${\dot{e}}_{i} = (A_{i} - Γ_{i}) e_{i} + B_{i} ((b_{i} - 1) u_{i} + ω_{i}) .$ Let $P_{i} \in R^{2 \times 2}$ be a positive definite matrix. The observer gain matrix Γ_i is designed such that $Q_{i} = - P_{i} (A_{i} - Γ_{i}) - {(A_{i} - Γ_{i})}^{T} P_{i} - 2 P_{i} B_{i} B_{i}^{T} P_{i} > 0 .$

Remark 3

We first choose the gain matrix Γ_i. Then, we check the feasibility of the linear inequality matrix $[\begin{matrix} - P \end{matrix}$

Fault-tolerant synchronization controller design

Let the reference trajectory of the ith agent be denoted by $x_{i 0} (t)$ . The position tracking error between the coordinates of the ith agent and its desired coordinates is defined as $ϱ_{i} (t) = x_{i 0} (t) - x_{i} (t)$ The filtered tracking error is defined according to $r_{i} (t) = {\dot{ϱ}}_{i} (t) + η_{i} ϱ_{i} (t)$ where η_i is a positive design parameter. Let $f_{i} (y_{i}) = {\ddot{x}}_{i 0} (t) + η_{i} {\dot{ϱ}}_{i} (t)$ with $y_{i} = {({\dot{x}}_{i}, {\dot{x}}_{i 0}, {\ddot{x}}_{i 0})}^{T}$ . Note that $x_{i 0} (t)$ is the prescribed desired trajectory for the ith agent. Thus, y_i is a known local variable.

Differentiating r_i(t) and

Fault-tolerant cooperative tracking controller design

In this section, we investigate the robust fault-tolerant cooperative tracking control problem for multi-agent systems with a general digraph communication topology. Assume that the desired trajectory is provided by a leader agent labeled as 0. The communication topology among the $n + 1$ agents is a digraph $\bar{G} = (\bar{V}, \bar{E})$ with $\bar{V} = {0, 1, \dots, n}$ . Denote $C = diag {c_{1}, c_{2}, \dots, c_{n}}$ and $c = {(c_{1}, c_{2}, \dots, c_{n})}^{T}$ with $c_{i} > 0 (i = 1, 2, \dots, n)$ if $0 \in N_{i}$ and $c_{i} = 0$ otherwise. For the digraph $\bar{G}$ , the Laplacian matrix can be written as $\bar{L} = [\begin{matrix} 0 & 0_{1 \times n} \\ - c \end{matrix}$

Simulation example

In this section, the fault-tolerant cooperative control schemes developed in the previous sections are applied to solve the formation control problem of a multiple unicycle robot network with actuator faults. The dynamics model of each unicycle robot is shown as follows: ${\dot{x}}_{i} = v_{i} \cos (θ_{i}) {\dot{y}}_{i} = v_{i} \sin (θ_{i}) {\dot{θ}}_{i} = χ_{i} {\dot{v}}_{i} = \frac{f_{i}}{m_{i}} {\dot{χ}}_{i} = \frac{τ_{i}}{J_{i}}$ where $(x_{i}, y_{i})$ is the Cartesian position of the robot center, θ_i is the steering angle, v_i is the transition speed, χ_i is the rotation speed, m_i represents the mass, J_i denotes the

Conclusion

This paper investigated the robust fault-tolerant synchronizing and tracking control problem for second-order multi-agent systems with actuator faults. A constructive design method of robust fault-tolerant control schemes was presented. An online actuator fault detection algorithm was first proposed. Then, the distributed control protocols were designed to achieve the state synchronizing or trajectory tracking of multi-agent systems with actuator bias faults and the partial loss of

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 61273108, the Major State Basic Research Development Program 973 (No. 2012CB215202), the Fundamental Research Funds for the Central Universities (106112013CDJZR175501), and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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Robust fault-tolerant cooperative control of multi-agent systems: A constructive design method

Abstract

Introduction

Section snippets

Graph theory

Actuator fault detection algorithm

Fault-tolerant synchronization controller design

Fault-tolerant cooperative tracking controller design

Simulation example

Conclusion

Acknowledgments

J. Frankl. Inst.

J. Frankl. Inst.

Automatica

Syst. Control Lett.

Automatica

J. Frankl. Inst.

J. Frankl. Inst.

J. Frankl. Inst.

Syst. Control Lett.

J. Frankl. Inst.

Automatica

Syst. Control Lett.

Distributed Consensus in Multi-vehicle Cooperative Control: Theory and Applications

Coordination of groups of mobile autonomous agents using nearest neighbor rules

IEEE Trans. Autom. Control

Information flow and cooperative control of vehicle formations

IEEE Trans. Autom. Control