Elsevier

Automatica

Volume 50, Issue 2, February 2014, Pages 499-506
Automatica

Brief paper
Global consensus for discrete-time multi-agent systems with input saturation constraints

https://doi.org/10.1016/j.automatica.2013.11.008Get rights and content

Abstract

In this paper, we consider the global consensus problem for discrete-time multi-agent systems with input saturation constraints under fixed undirected topologies. We first give necessary conditions for achieving global consensus via a distributed protocol based on relative state measurements of the agent itself and its neighboring agents. We then focus on two special cases, where the agent model is either neutrally stable or a double integrator. For the neutrally stable case, any linear protocol of a particular form, which solves the consensus problem for the case without input saturation constraints, also solves the global consensus problem for the case with input saturation constraints. For the double integrator case, we show that a subset of linear protocols, which solve the consensus problem for the case without saturation constraints, also solve the global consensus problem for the case with input saturation constraints. The results are illustrated by numerical simulations.

Introduction

In recent years, the distributed coordination of a multi-agent system (MAS) has received substantial attention due to its wide application areas, including consensus computation (Bai et al., 2011, Jadbabaie et al., 2003, Olfati-Saber and Murray, 2004, Tsitsiklis, 1984), synchronization (Wu & Chua, 1995), distributed processing (Lynch, 1996), and network flow control (Low et al., 2002, Wen and Arcak, 2004). When it comes to the consensus problem, each agent has to implement a distributed protocol based on the limited information about itself and its neighboring agents.

The design of consensus protocols can be generally divided into two categories depending on whether the agent models are continuous-time or discrete-time. Much attention has been devoted to the continuous-time case. The existing works here can be categorized into two directions depending whether the agent models are identical or not. The consensus problem for homogeneous networks (i.e., networks where the agent models are identical) has been considered (e.g.,  Li, Du et al., 2011, Li et al., 2010, Olfati-Saber and Murray, 2004, Ren and Beard, 2005, Scardovi and Sepulchre, 2009, Seo et al., 2009, Seyboth et al., 2013, Shi and Hong, 2009, Xiao and Wang, 2008, Yang et al., 2011, Yu et al., 2010), while the consensus problem for heterogeneous networks (i.e., networks where the agent models are non-identical) has been a recent focus (e.g.,  Grip et al., 2012, Wieland et al., 2011, Zhao et al., 2011). The studies on the discrete-time case are rather limited, but some results can be found in (e.g., Blondel, Hendrickx, Olshevsky, & Tsitsiklis, 2005; Jadbabaie et al., 2003; Moreau, 2005; Ren & Beard, 2005; Tuna, 2008; You & Xie, 2011; Zhang & Tian, 2009).

Most consensus literature does not consider the case where the agents are subject to input saturation. However, in almost every physical application, the actuator has bounds on its input, and thus actuator saturation is important to study. The protocol design for achieving consensus for the case with input saturation constraints is a challenging problem, and only few results are available for continuous-time agent models (e.g.,  Cortés, 2006, Du et al., 2013, Li, Xiang et al., 2011, Meng et al., 2013, Yang, Stoorvogel, Grip et al., in preparation). For the single integrator case, Li, Xiang et al. (2011) showed that any linear protocol based on the relative state information, which solves the consensus problem for the case without input saturation constraints under fixed directed network topologies, also solves the global consensus problem in the presence of input saturation constraints. Meng et al. (2013) proposed a linear protocol based on the relative state information to solve the global consensus problem for an MAS with input saturation constraints under fixed undirected network topologies and time varying network topologies. Yang, Stoorvogel, Grip et al. (in preparation) studied semi-global regulation of output synchronization for heterogeneous networks under fixed directed network topologies.

To the best of the authors’ knowledge, all the existing works on the consensus problem for an MAS with input saturation constraints are restricted to continuous-time agent models. This motivates us to consider the consensus problem for the case where the agents models are discrete-time, as such models are relevant for many practical sampled-data systems. As a first step, in this paper, we assume that the network topology is fixed and undirected. This paper may be seen as a continuation of the work of Meng et al. (2013). We extend their continuous-time results for fixed topologies to a discrete-time setting. The extension is considerable. First, Meng et al. (2013) considered the leader–follower case while we consider the leaderless case. Second, we use a completely new set Lyapunov stability theory argument.

The remainder of the paper is organized as follows. In Section  2, some preliminaries and notations are introduced. In Section  3, we first formulate the global consensus problem with input saturation constraints, and then give necessary conditions for achieving global consensus under fixed undirected topologies. In Sections  4 Neutrally stable agent model, 5 Double integrator agent model, we consider the case where the agent model is neutrally stable and a double integrator, respectively. Simulation examples are presented in Section  6 followed by conclusions.

Section snippets

Preliminaries and notations

In this paper, we assume that the network topology among the agents is described by a fixed undirected weighted graph G=(V,E,A), with the set of agents V={1,,N}, the set of undirected edges EV×V, and a weighted adjacency matrix A=[aij]RN×N, where aij>0 if and only if (j,i)E and aij=0 otherwise. In this paper, we also assume that aij=aji for all i,jV, and that there are no self-loops, i.e., aii=0 for iV. The set of neighboring agents of agent i is defined as Ni={jV|aij>0}. A path from

Problem formulation

We consider an MAS of N identical discrete-time agents xi(k+1)=Axi(k)+Bσ(ui(k)),iV, where xi(k)Rn,ui(k)Rm, σ(ui(k))=[σ1(ui,1(k));σ1(ui,2(k));;σ1(ui,m(k))], and each σ1(u) is the standard saturation function σ1(u)={1if  u>1,uif  |u|1,1if  u<1. The only information available for agent i comes from the network. In particular, agent i receives a linear combination of its own state relative to that of neighboring agents, i.e., ζi(k)=jNiaij(xi(k)xj(k)). Our goal is to design distributed

Neutrally stable agent model

In this section, we consider the case where the agent model (1) is open-loop neutrally stable.

Under Assumption 1, there exists a non-singular state transformation T1, such that A=T1[Ac00As]T,B=T1[BcBs], where ATcAc=I, As is the Schur stable (i.e., all its eigenvalues are within the unit circle), and the pair (Ac,Bc) is controllable.

As shown in You and Xie (2011), the asymptotically stable modes can be ignored since we can set the corresponding gain matrix to zero. Thus, without loss of

Double integrator agent model

In this section, we consider the case where the agent model (1) is a double integrator.

Assumption 4

The matrices A and B are of the form A=[1101],B=[01].

Let us first recall the following result which gives a necessary and sufficient condition on the feedback gain parameters for achieving consensus without input saturation constraints.

Lemma 2

Xie & Wang, 2012

Consider an MAS of N agents described by[xi(k+1)vi(k+1)]=A[xi(k)vi(k)]+Bui(k),iV.Assume that   Assumption 2, Assumption 4   are satisfied. Then the protocol   (2)   with K=

Illustrative example

In this section, we illustrate our results on global consensus with input saturation constraints for a network with N=7 double integrators, whose topology is given in Fig. 1. Choose α=0.07 and β=0.15 such that the condition (10) is satisfied. The simulation results shown in Fig. 2 confirm the results of Theorem 2.

Conclusions and future work

This paper considered the global consensus problem for an MAS of discrete-time identical linear agents, where the agent dynamics are either neutrally stable or a double integrator, with input saturation constraints under fixed undirected network topologies. Extensions to directed topologies and time-varying topologies are currently under investigation. Another interesting topic is to consider heterogeneous networks.

Tao Yang received the B.S. degree in Computer Science from the Harbin University of Science and Technology in July 2003, the M.S. degree with distinction in Control Engineering from City University, London, in November 2004, and the Ph.D. degree in Electrical Engineering from Washington State University in August 2012. He is currently a Postdoctoral Researcher at the ACCESS Linnaeus Centre, Royal Institute of Technology (KTH), Sweden. His research interests are multi-agent systems, constrained

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    Tao Yang received the B.S. degree in Computer Science from the Harbin University of Science and Technology in July 2003, the M.S. degree with distinction in Control Engineering from City University, London, in November 2004, and the Ph.D. degree in Electrical Engineering from Washington State University in August 2012. He is currently a Postdoctoral Researcher at the ACCESS Linnaeus Centre, Royal Institute of Technology (KTH), Sweden. His research interests are multi-agent systems, constrained control, and switched linear systems.

    Ziyang Meng received the B.S. degree from the Huazhong University of Science and Technology, Wuhan, China, in 2006 and the Ph.D. degree from Tsinghua University, Beijing, China, in 2010. From 2008 to 2009, he was an exchange Ph.D. student supported by China Scholarship Council with the Department Electrical and Computer Engineering, Utah State University, Logan. He was a Postdoctoral Researcher from 2010 to 2012 with Shanghai Jiao Tong University, Shanghai, China. He is currently a Postdoctoral Researcher with the KTH Royal Institute of Technology, Sweden. His research interests include cooperative control of distributed multi-agent systems and spacecraft attitude determination and control.

    Dimos V. Dimarogonas was born in Athens, Greece, in 1978. He received the Diploma in Electrical and Computer Engineering in 2001 and the Ph.D. in Mechanical Engineering in 2006, both from the National Technical University of Athens (NTUA), Greece. Between May 2007 and February 2009, he was a Postdoctoral Researcher at the Automatic Control Laboratory, School of Electrical Engineering, ACCESS Linnaeus Center, Royal Institute of Technology (KTH), Stockholm, Sweden. Between February 2009 and March 2010, he was a Postdoctoral Associate at the Laboratory for Information and Decision Systems (LIDS) at the Massachusetts Institute of Technology (MIT), Boston, MA, USA. He is currently an Assistant Professor at the Automatic Control Laboratory, ACCESS Linnaeus Center, Royal Institute of Technology (KTH), Stockholm, Sweden.

    His current research interests include Multi-Agent Systems, Hybrid Systems and Control, Robot Navigation and Networked Control. He was awarded a Docent in Automatic Control from KTH in 2012. He serves in the Editorial Board of Automatica and the IET Control Theory and Applications and is a member of IEEE and the Technical Chamber of Greece.

    Karl H. Johansson is Director of the KTH ACCESS Linnaeus Centre and Professor at the School of Electrical Engineering, Royal Institute of Technology, Sweden. He is a Wallenberg Scholar and has held a six-year Senior Researcher Position with the Swedish Research Council. He is the Director of the Stockholm Strategic Research Area ICT The Next Generation. He received M.Sc. and Ph.D. degrees in Electrical Engineering from Lund University. He has held visiting positions at UC Berkeley (1998–2000) and the California Institute of Technology (2006–2007). His research interests are in networked control systems, hybrid and embedded system, and applications in smart transportation, energy, and automation systems. He has been a member of the IEEE Control Systems Society Board of Governors and the Chair of the IFAC Technical Committee on Networked Systems. He has been on the Editorial Boards of several journals, including Automatica, IEEE Transactions on Automatic Control, and IET Control Theory and Applications. He has been Guest Editor for special issues, including the one on “Wireless Sensor and Actuator Networks” of IEEE Transactions on Automatic Control 2011. He was the General Chair of the ACM/IEEE Cyber-Physical Systems Week 2010 in Stockholm and IPC Chair of many conferences. He has served on the Executive Committees of several European research projects in the area of networked embedded systems. In 2009, he received the Best Paper Award of the IEEE International Conference on Mobile Ad-hoc and Sensor Systems. In 2009, he was also awarded Wallenberg Scholar, as one of the first ten scholars from all sciences, by the Knut and Alice Wallenberg Foundation. He was awarded an Individual Grant for the Advancement of Research Leaders from the Swedish Foundation for Strategic Research in 2005. He received the triennial Young Author Prize from IFAC in 1996 and the Peccei Award from the International Institute of System Analysis, Austria, in 1993. He received Young Researcher Awards from Scania in 1996 and from Ericsson in 1998 and 1999. He is a Fellow of the IEEE.

    This work has been supported in part by the Knut and Alice Wallenberg Foundation and the Swedish Research Council. The material in this paper was partially presented at the 2013 European Control Conference (ECC’13), July 17–19, 2013, Zurich, Switzerland. This paper was recommended for publication in revised form by Associate Editor David Angeli under the direction of Editor Andrew R. Teel.

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