Cooperative guaranteed cost fault-tolerant control for multi-agent systems with time-varying actuator faults

doi:10.1016/j.neucom.2016.06.022

Neurocomputing

Volume 214, 19 November 2016, Pages 382-390

https://doi.org/10.1016/j.neucom.2016.06.022 Get rights and content

Highlights

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The impacts of actuator faults on leaderless consensus are studied.
•
The impacts of both actuator faults and uncertainties on leader-following consensus are investigated.
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Based on the relative state information of neighboring agents, a distributed cooperative guaranteed cost controller is designed.
•
A general approach is developed to simultaneously consider consensus regulation performances and control energy costs.

Abstract

This paper investigates the guaranteed cost consensus problem for multi-agent systems with actuator faults/uncertainties. For the leaderless case with actuator faults, a global performance index is constructed by all states and control inputs of all agents, where consensus regulation performances and control energy costs are considered simultaneously. Then, based on the relative state information of neighboring agents, a distributed cooperative guaranteed cost controller is designed, which not only makes the consensus problem solvable but also provides an upper bound of the given global performance index even in the presence of actuator faults. Extensions to the leader-following case with actuator faults and uncertainties are further studied. Finally, two simulation examples are provided to show the effectiveness of the proposed approach.

Introduction

In the last two decades, the cooperative control of networked multi-agent systems (MASs) has received considerable attention due to its potential applications in broad areas including spacecraft formation flying, mobile robots, distributed sensor networks, and so on. Consensus is one of the fundamental problems in coordination control. Consensus problem is concerned with finding a controller with their neighbor information such that all agents reach an agreement on certain quantities of interest.

Motivated by the pioneering works of [1], [2], a general framework of the consensus problem for first-order MASs was proposed in [3]. Since then, the consensus problem has been widely studied by numerous researchers from different perspectives, e.g., see survey articles [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. Generally speaking, the existing consensus algorithms can be broadly classified into two categories: leaderless consensus (i.e., consensus without a leader) and leader-following consensus (i.e., consensus with a leader). For the leaderless consensus problem, also referred as cooperative regulation, distributed consensus protocols are designed for each agent such that all the states of agents are eventually converge to an unprescribed common value. On the other hand, the leader-following consensus problem allows all agents to track the desired trajectory formed by the leader. The leader-following consensus is also called cooperative tracking. It is worth mentioning that control energy costs were not taken into account in the above literatures. In general, in practical control design, consensus regulation performances and control energy costs should be considered simultaneously, e.g., see survey articles [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. The approaches proposed in [17], [18], [19], [20], [21], [22], [23] are based on individual agent cost optimization for achieving some global goals. Global performance index is minimized in a distributed manner [24], [25]. Xi et al. [26], [27] investigated the guaranteed cost consensus problem for linear MASs with switching topologies.

Moreover, actuator failures generally result in poor system performance or even cause the instability. Although the research on fault-tolerant control (FTC) design for dynamic systems has attracted considerable attention [28], [29], [30], [31], [32] due to the safety demand of real-time systems over the past few decades, it is only in recent years that considerable research efforts have been made with respect to the FTC of MASs [33], [34], [35], [36], [37], [38], [39], [40]. To name a few, by designing distributed adaptive laws to compensate the effects of actuator faults and uncertainties, a fully distributed consensus protocol was developed in [38] for MASs. A cooperative FTC control approach was proposed in [39] for a group of high-order nonidentical nonlinear systems with actuator faults and unreliable interconnections. Most recently, Deng et al. [40] investigated the cooperative adaptive output feedback control problem for nonlinear multi-agent systems with actuator failures. Furthermore, the robustness of consensus protocol with respect to uncertainties is also a critical issue. Therefore, it is highly desirable to develop a distributed consensus protocol for MASs with actuator faults/uncertainties. Even though the perfect consensus cannot be achieved under the effects of actuator faults/uncertainties, it is always desirable to design a MAS which is not only stable but also guarantees an adequate level of performance. From the above discussions, we will encounter the following problem: Can the agents achieve guaranteed cost consensus even in the presence of actuator faults/uncertainties by using distributed adaptive protocol? In the literature, there is no result concerning this question. This motivates the present study.

This paper is concerned with the guaranteed cost consensus problem of MASs subject to actuator faults/uncertainties for both the cases without and with a leader. The undirected connected graph with a fixed topology is considered. For the leaderless case with actuator faults, based on all states and control inputs of all agents, a global performance index is formulated. Furthermore, a distributed cooperative guaranteed cost controller, relying on the relative state information of neighboring agents, is proposed. The proposed protocol not only makes the leaderless consensus problem solvable but also provides an upper bound of the given global performance index. Extensions to the leader-following case with actuator faults and uncertainties are further studied. Compared with the existing works about consensus, the main contributions of this paper are as follows:

(1)
The current paper investigates the impacts of actuator faults on leaderless consensus, as well as the impacts of both actuator faults and uncertainties on leader-following consensus.
(2)
A general approach is developed to simultaneously consider consensus regulation performances and control energy costs, which potentially contributes both the theoretical framework and the applications for MASs.

The rest of this paper is organized as follows. Section 2 introduces problem statement and preliminaries. Guaranteed cost fault-tolerant controller designs for both the cases without and with a leader are presented in Sections 3 and 4, respectively. Simulation examples are given to illustrate the theoretical results in Section 5. Finally, the conclusions end the paper in Section 6.

Notations: For a matrix A, A^T, $kernel A$ , $λ_{\min} (A)$ and $λ_{> 0, \min} (A)$ denote its transpose, null space (or kernel), minimal singular value and minimal nonzero singular value, respectively. The notion $A > 0$ ( $A \geq 0$ ) means that A is a symmetric positive definite (positive semi-definite) matrix of appropriate dimension. A diagonal matrix with $▵_{1}, ▵_{2}, \dots, ▵_{n}$ on its main diagonal is denoted as $diag {▵_{1}, ▵_{2}, \dots, ▵_{n}}$ . $0_{m}$ refers to the zero matrix with m×m dimensions, I_n denotes the identity matrix with n×n dimensions, $1_{N}$ denotes an N-dimensional column vector with all components 1, and $∥ \cdot ∥$ denotes Euclidean norm of vectors or matrices.

Section snippets

Graph theory

A directed graph $G$ is a pair $(V, E)$ , where $V = {ν_{1}, ν_{2}, \dots, ν_{N}}$ is a nonempty finite set of nodes and $E \subseteq V \times V$ is a set of edges, where an edge is represented by an ordered pair of distinct nodes. It is assumed that the graph is simple, i.e., no repeated edges and self-loops $(ν_{i}, ν_{i}) \notin E, \forall i$ . For an edge $(ν_{i}, ν_{j})$ , node ν_i is called the parent node, node ν_j the child node, and ν_i is a neighbor of ν_j. The set of neighbors of node ν_i is denoted as $N_{i} = {ν_{j} | (ν_{j}, ν_{i}) \in E}$ . Define the connectivity matrix as $A = [a_{ij}]$ with

Guaranteed cost fault-tolerant controller design: leaderless case

In this section, guaranteed cost fault-tolerant controller design for the leaderless consensus problem is considered. The dynamics of ith agent are described by (6). It is assumed that only relative state information of its neighbors can be used for the controller design. The objective is to construct a suitable distributed guaranteed cost controller such that the following two aspects are satisfied:

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$∥ x_{i} (t) - x_{j} (t) ∥ \to 0$ as $t \to \infty, \forall i, j = 1, 2, \dots, N$ .
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The global quadratic performance index is within a

Guaranteed cost fault-tolerant controller design: leader-following case

In this section, we study the leader-following consensus problem, as well as the robustness of the distributed consensus protocol to uncertainties. The dynamics of ith follower are described by (1) with actuator faults (5) and uncertainties added. That is, ${\dot{x}}_{i} (t) = {Ax}_{i} (t) + H (t, x_{i} (t)) + B (I_{m} - Λ_{i} (t)) u_{i} (t), 0_{m} \leq Λ_{i} (t) \leq \bar{Λ} < I_{m} .$ where $0_{m} \leq Λ_{i} (t) \leq \bar{Λ} < I_{m}$ i, x_i(t), u_i(t), A, B, $Λ_{i} (t)$ and $\bar{Λ}$ are the same as those defined in (6). $H (t, x_{i} (t)) : [0, + \infty) \times R^{n} \to R^{n}$ represents the uncertainty of the system dynamics and it is assumed

Simulation studies

In this section, two examples will be given to illustrate the effectiveness of the proposed guaranteed cost FTC protocols.

Example 1 Guaranteed cost consensus without a leader

Consider a network of six B747-100/200 aircrafts, with the communication structure given in Fig. 1. The differential equations of each agent are borrowed from NASA [49]. For design purposes, only the first four states have been retained, i.e., pitch rate, true airspeed, angle of attack and pitch angle. The inputs are elevator deflection, total thrust and horizontal

Conclusions

Guaranteed cost consensus problem for multi-agent systems with actuator faults/uncertainties has been studied. For the leaderless case with actuator faults, a distributed cooperative guaranteed cost controller has been designed based on the relative state information of neighboring agents. Extensions to the leader-following case with actuator faults and uncertainties have been investigated as well. It has been proved that the proposed consensus protocol not only makes the consensus problem

Acknowledgments

This work was supported in part by the Funds of National Science of China (Grant nos. 61273148 and 61420106016), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant no. 201157), the Fundamental Research Funds for the Central Universities (Grant no. N150406002) and the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant no. 2013ZCX01).

Chun-Hua Xie received the B.S. degree in detection, guidance and control technology from North University of China, Taiyuan, China, in 2012, and the M.S. degree in navigation, guidance and control from Northeastern University, Shenyang, China, in 2014. Currently, he is pursuing the Ph.D. degree in control theory and control engineering from Northeastern University, Shenyang, China. His current research interests include adaptive robust control, fault-tolerant control, fault diagnosis, and

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Guang-Hong Yang received the B.S. and M.S. degrees from Northeast University of Technology, Liaoning, China, in 1983 and 1986, respectively, and the Ph.D. degree in Control Engineering from Northeastern University, China (formerly, Northeast University of Technology), in 1994. He was a Lecturer/Associate Professor with Northeastern University from 1986 to 1995. He joined the Nanyang Technological University in 1996 as a Postdoctoral Fellow. From 2001 to 2005, he was a Research Scientist/Senior Research Scientist with the National University of Singapore. He is currently a Professor at the College of Information Science and Engineering, Northeastern University. His current research interests include fault-tolerant control, fault detection and isolation, non-fragile control systems design, and robust control. Dr. Yang is an Associate Editor for the International Journal of Control, Automation, and Systems (IJCAS), the International Journal of Systems Science (IJSS), the IET Control Theory & Applications, and the IEEE Transactions on Fuzzy Systems.

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