Elsevier

Neurocomputing

Volume 332, 7 March 2019, Pages 351-359
Neurocomputing

Cooperative output regulation for nonlinear multi-agent systems described by T-S fuzzy models under jointly connected switching topology

https://doi.org/10.1016/j.neucom.2018.12.068Get rights and content

Abstract

This paper studies the problem of the cooperative output regulation for a class of nonlinear multi-agent systems described by T-S fuzzy models subject to the jointly connected switching topology. The considered switching topology permits the topology to be disconnected at any time interval which is weaker than the connected topology assumption. By using the parallel distributed compensation, the fuzzy internal reference model for each agent is constructed to estimate the exogenous signal, based on which, the distributed dynamic fuzzy output feedback controller is designed. The constructed fuzzy internal reference model is a switched nonlinear system where each subsystem may be unstable. An efficient condition is presented to ensure the solvability of the cooperative output regulation. An example of coupled pendula illustrates the validity of the result.

Introduction

The cooperative output regulation problem (CORP) is to design a controller to achieve asymptotic tracking and disturbance rejection subject to some exogenous signal for each agent in multi-agent systems (MASs). This problem contains some cooperative problems as its special cases such as leader-following consensus, output synchronization, and its applications cover a broad range of areas including engineering and sciences. Thus, the problem is more general and has been extensively addressed (see [1], [2], [3], [4] for example).

In cooperative control of MASs, the communication topology among agents is often switching due to the failed connections or the reconstructed connections. Under the switching topology, [5], [6] considers the consensus problem for the linear MASs with single integrators, where the topology is demanded to be connected at any time interval. In [7], the leader-following consensus by the distributed event-triggered scheme is addressed, whose topology is switching and is allowed to be disconnected at some time interval. The CORP is investigated [8], [9], which permits the topology to be disconnected at any time interval as long as the union of graph at finite time is connected. This type of the topology in [8], [9] is called the jointly connected topology. Note that the above investigations are mainly focused on the linear MASs. However, most physical systems come in the form of complex nonlinear systems [10]. To solve the CORP of the nonlinear MASs is nontrivial when the topology is switching. In [11], by employing the dwell time approach, the solvability condition is presented. The condition requires the topology to be connected at any time interval. Thus, the switching topology in [11] can be deemed as a particular case of the jointly connected switching topology. The CORP of the nonlinear MASs with the jointly connected topology is considered in [12], where the exosystem is still linear. We point out that the approach in [8], [9], [12] can hardly be applied to the nonlinear case since it depends on the expression of solutions of the linear time-invariant systems.

Fortunately, the T-S fuzzy model in [13], [14], [15], [16] provides a good approximation for the nonlinear systems by means of linear local subsystems. The model smoothly connects a finite number of linear subsystems by fuzzy membership functions. As pointed in [17], under the T-S fuzzy structures, the behavior of the nonlinear system is efficiently characterized and the simple property of the linear system is well inherited. The control design of the T-S fuzzy system is mainly based on the parallel distributed compensation of [18], where the local controllers are designed for each local model and then the global controller is constructed. For the fuzzy output regulation problem, several studies have been reported in [19], [20], [21]. These studies focus on the single systems. So far, to the best of our knowledge, there is no result about the CORP for the nonlinear MASs using T-S fuzzy models.

Motivated by the above discussions, this paper considers the CORP for the nonlinear MASs described by the T-S fuzzy models subject to the jointly connected switching topology via distributed dynamic fuzzy output feedback controller. Firstly, based on the parallel distributed compensation, the fuzzy internal reference model is presented to estimate the exogenous signal when only part of the agents can acquire the information of the exogenous signal. By applying the reference models, the distributed dynamic fuzzy output feedback controller is designed to guarantee the solvability of the CORP. Secondly, the fuzzy internal reference model is switched due to the jointly connected switching topology. It is a worse case that the resulted switched error system may be unstable since the corresponding graph is allowed to be disconnected at any time interval. Note that the arbitrary switching in [22], [23], the time-dependent switching in [24], [25], [26], [27], [28], [29], [30] and the state-dependent switching in [31], [32], [33] for the switched systems are all not applicable to the case. In fact, the arbitrary switching and the time-dependent switching both require some subsystems to be stable, and the state-dependent switching often can not guarantee the dwell time. Thirdly, the fuzzy switched internal reference model is a switched nonlinear system. In [12], [34], the cooperative problem under the jointly connected topology is also considered. But the approaches in [12], [34] can not be used to the nonlinear case, since these approaches rely on the expression of solutions of the linear system.

Notation: Throughout the paper, the notation is standard. The n-dimensional Euclidean space is denoted by Rn, and the set of real numbers is represented by R. Let ⊗ be the Kronecker product. Denote In the n × n identity matrix and ⋆ the term of symmetry. For a matrix P, its inverse and transpose are defined as P1 and PT respectively. λmin(P)(λmax(P)) represents the minimum (maximum) eigenvalue of the matrix. P > 0 illustrates that the matrix P is a positive definite matrix. diag(P1, P2, ⋅⋅⋅, Pn) means a block-diagonal matrix with matrices Pi, i=1,2,,n. ‖ · ‖ stands for the 2-norm for vectors or the induced 2-norm for matrices.

Section snippets

Graph theory

Denote G=(V,E) a graph composed by the node set V={1,2,,n} and the edge set EV×V. An edge (i,j)E represents that the node j can acquire the information of the node i. If (i,j)E implies (j,i)E, the graph G is known as a undirected graph. The neighbor set of the node i is described by Ni={jV|(j,i)E}. In the general, assume that the graph G does not have self-loops or repeated edges. It is called that the node ik is reachable from the node i1, if there exists {(i1,i2),(i2,i3),,(ik1,ik)}E

Controller design

In this section, under the jointly connected switching topology, by using the parallel distributed compensation and fuzzy internal reference models, we aim at designing the distributed dynamic fuzzy output feedback controller to ensure the solvability of the CORP for the MAS (5).

We consider the following controller for each agent:ui(t)=l=1rihil(zi(t))Kil(x^i(t)Πiωi(t))+Γiωi(t),x^˙i(t)=l=1rihil(zi(t)){Ailx^i(t)+Bilui(t)+Eilωi(t)+Jil[l=1rihil(zi(t))Cilx^i(t)yi(t)]},ω˙i(t)=l=1r0h0l(z0(t))Slωi

Example

This section gives an example to illustrate the validity of the result. We consider four coupled pendula shown in Fig. 1, each of which is described byxi1=xi2,xi2=glisin(xi1)+ui,yi=xi1,where xi=col{xi1,xi2} is the state of each agent, xi1 denotes the angle between the vertical axis and the pendulum bar, xi2 means the angular velocity, li denotes the length of the rod, and g is the gravity acceleration. By using the fuzzy modeling method in [14], (48) can be changed into:x˙i(t)=l=12hil(zi(t))[A

Conclusion

The CORP for nonlinear MASs described by the T-S fuzzy models under the switching topology has been considered. The fuzzy internal reference models have been designed for each agent to estimate the exogenous signal. Theorem 1 shows that the solvability condition, where the topology is allowed to be disconnected at any time interval.

Some interesting problems will be considered in the future research: (i) Recently, many results on MAS under intermittent communication, e.g. [3], [4], have been

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61773098 and the 111 Project (B16009).

Yajing Ma received her M.S. degree in the College of Mathematics and Information Science from Shaanxi Normal University, Xian, China. She is now pursuing the Ph.D. degree in College of Information Science and Engineering from Northeastern University, Shenyang, China. Her research interests include switched systems and multi-agent systems.

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  • Cited by (0)

    Yajing Ma received her M.S. degree in the College of Mathematics and Information Science from Shaanxi Normal University, Xian, China. She is now pursuing the Ph.D. degree in College of Information Science and Engineering from Northeastern University, Shenyang, China. Her research interests include switched systems and multi-agent systems.

    Jun Zhao was born in 1957. He received the B.S. and M.S. degrees in mathematics in 1982 and 1984 respectively, both from Liaoning University, China. He completed his Ph.D. in Control Theory and Applications in 1991 at Northeastern University, China. From 1992 to 1993 he was a postdoctoral fellow at the same University. Since 1994 he has been with the College of Information Science and Engineering, Northeastern University, China, where he is currently a professor. From February 1998 to February 1999, he was a senior visiting scholar at the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, USA. From November 2003 to May 2005, he was a Research Fellow at the Department of Electronic Engineering, City University of Hong Kong. From November 2006 to 2010, as a Fellow, he was with the Research School of Information Sciences and Engineering, The Australian National University. His main research interests include switched systems, hybrid control, nonlinear systems and robust control.

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