Distributed adaptive output feedback consensus protocols for linear systems on directed graphs with a leader of bounded input

doi:10.1016/j.automatica.2016.07.041

Automatica

Volume 74, December 2016, Pages 308-314

https://doi.org/10.1016/j.automatica.2016.07.041 Get rights and content

Abstract

This paper studies output feedback consensus protocol design problems for linear multi-agent systems with directed graphs containing a leader whose control input is nonzero and bounded. We present novel distributed adaptive output feedback protocols to achieve leader–follower consensus for any directed graph containing a directed spanning tree with the leader as the root. The proposed protocols are independent of any global information of the graph and can be constructed as long as the agents are stabilizable and detectable.

Introduction

Over the past decade, the consensus control problem of multi-agent systems has emerged as a focal research topic in the field of control, due to its various applications to, e.g., UAV formation flying, multi-point surveillance, and distributed reconfigurable sensor networks (Antonelli, 2013, Olfati-Saber et al., 2007, Ren et al., 2007). Considerable work from different perspectives has been conducted on consensus and other related cooperative control problems; see the recent works (Antonelli, 2013, Lewis et al., 2014, Li et al., 2010, Olfati-Saber et al., 2007, Ren et al., 2007), and the references therein. Existing consensus algorithms can be essentially divided into two broad categories, namely, consensus without a leader (i.e., leaderless consensus) and consensus with a leader, whereas the latter is also called leader–follower consensus or distributed tracking. In a leader–follower consensus problem, it is often the case that the leader may need to implement its own control actions to achieve certain objectives, e.g., to reach a desirable consensus trajectory or to avoid hazardous obstacles. Compared to leaderless consensus problem, an additional difficulty arises with leader–follower consensus, in which one must deal with the effect of the leader’s control input.

A central task in consensus studies is to design distributed consensus protocols based on solely the local information of each agent and its neighbors to ensure that the states of the agents reach an agreement. In most of the previous works on consensus, (e.g., Li et al., 2010, Seo, Shim, & Back, 2009, Zhang, Lewis, & Das, 2011, Yu, Chen, Cao, & Kurths, 2010), the design of the consensus protocols requires the knowledge of certain global information of the communication graph in terms of the nonzero eigenvalues of the corresponding Laplacian matrix, implying that such consensus protocols in essence cannot be determined in a distributed manner. Distributed consensus protocols, nevertheless, can be developed by implementing adaptive laws to dynamically update the coupling weights of neighboring agents, thus removing the aforementioned requirement on the global eigenvalue information. Such adaptive consensus protocols are proposed in Li, Wen, Duan, and Ren (2015), Li, Ren, Liu, and Fu (2013) and Li, Ren, Liu, and Xie (2013b) for linear multi-agent systems. Similar adaptive schemes are presented in DeLellis, diBernardo, and Garofalo (2009) for synchronization of Lipschitz-type complex networks. Note that the adaptive protocols in DeLellis et al. (2009), Li et al. (2013b) and Li et al. (2013) are applicable to only undirected graphs and the protocols in Li et al. (2015) work for general directed graphs, which however rely on the relative states of neighboring agents. How to design distributed adaptive output feedback consensus protocols using local output information appears much more challenging and remains to be an open issue.

The aforementioned works are concerned with the leaderless consensus problem or distributed tracking problem for the case where the leader has zero control input. The distributed tracking problem in the presence of a leader having a nonzero control input is generally more difficult and has been addressed in Cao and Ren (2012), Dimarogonas, Tsiotras, and Kyriakopoulos (2009),Li et al. (2013b), Li, Liu, Ren, and Xie (2013a) and Mei, Ren, and Ma (2012). In particular, the authors in Cao and Ren (2012) present nonsmooth controllers for first-order and second-order integrators in the absence of velocity or acceleration measurements. The authors in Dimarogonas et al. (2009) and Mei et al. (2012) address a distributed coordinated tracking and containment control problem for multiple Euler–Lagrange systems with one or more dynamic leaders. Distributed static and adaptive protocols are given in Li et al. (2013b) and Li et al. (2013a) for general linear multi-agent systems with a leader of bounded control input. It is worth noting that one common assumption in Cao and Ren (2012),Li et al. (2013b) and Li et al. (2013a) is that the subgraph among the followers is undirected. The case where this subgraph is directed remains unsolved for general linear multi-agent systems. The main obstacle lies in the unpleasant interrelations between the nonlinear functions used to deal with the leader’s control input and the directed subgraph among followers.

In this paper, we address the distributed adaptive output feedback consensus protocol design problem for general linear multi-agent systems with directed communication graphs. In this setting, the relative states of neighboring agents are not available, but only local output information is accessible. Note that simply combining the techniques for the state feedback case (e.g., those proposed in Li et al., 2015) and distributed adaptive observer-type protocols (e.g., in Li et al., 2013b) for undirected graphs will not yield a distributed adaptive output feedback consensus protocol applicable to general directed graphs, nor by a routine modification or extension. The main reason is that the monotonically increasing functions introduced in Li et al. (2015), when used for observer-type adaptive protocols in Li et al. (2013b), will still depend on the relative states of neighboring agents.

Partly inspired by the observer structure proposed in Huang (2015), Su and Huang (2012) and Wieland, Sepulchre, and Allgower (2011), in this paper we develop a distributed continuous adaptive output feedback protocol, which includes continuous nonlinear functions to deal with the effect of the leader’s nonzero control input. It is shown that the continuous adaptive protocol can ensure the ultimate boundedness of the consensus error and the adaptive gains. The upper bound of the consensus error is explicitly derived, which can be made satisfactorily small by appropriately tuning the design parameters. A distributed discontinuous adaptive output feedback protocol is also presented, to achieve leader–follower consensus for any directed graph containing a directed spanning tree with the leader as the root. The protocols designed in this paper exchange the local estimates among neighboring agents via the communication graph and implement adaptive laws to update the time-varying coupling weights among the agents. As such, these two adaptive protocols use only the local output information and operate in a distributed manner over directed communication graphs. Unlike the protocols in the previous works (Cao and Ren, 2012, Li et al., 2013a, Li et al., 2013b), the adaptive protocols proposed herein appear to be the first available for linear multi-agent systems with general directed graphs and a leader of nonzero control input.

Section snippets

Problem statement

Consider a group of $N + 1$ identical agents with general linear dynamics described by $\begin{matrix} {\dot{x}}_{i} = A x_{i} + B u_{i}, \\ y_{i} = C x_{i}, i = 0, 1, \dots, N, \end{matrix}$ where $x_{i} \in R^{n}$ is the state vector, $y_{i} \in R^{m}$ the measured output vector, $u_{i} \in R^{p}$ the control input vector of the $i$ th agent, respectively, and $A$ , $B$ and $C$ are known constant matrices with compatible dimensions.

The information exchange among the $N + 1$ agents is governed by a directed graph $G = (V, E)$ , where $V = {0, \dots, N}$ is the set of nodes (each node represents an agent) and $E \subseteq V \times V$ denotes the set of

Adaptive output feedback consensus protocols

Based on the relative estimates of the states of neighboring agents, we propose the following distributed continuous adaptive controller to each follower: $\begin{matrix} {\dot{v}}_{i} & = & A v_{i} + B u_{i} + F (C v_{i} - y_{i}), \\ {\dot{w}}_{i} & = & A w_{i} + (d_{i} + ρ_{i}) F C ψ_{i} - β B h_{i} (B^{T} Q ψ_{i}), \\ u_{i} & = & K (v_{i} - w_{i}) - β h_{i} (B^{T} Q η_{i}), \\ {\dot{d}}_{i} & = & - φ_{i} (d_{i} - 1) + ψ_{i}^{T} C^{T} C ψ_{i}, i = 1, \dots, N, \end{matrix}$ where $v_{i} \in R^{n}$ is the estimate of the state of the $i$ th follower, ${\dot{v}}_{0} = A v_{0} + B u_{0} + F (C v_{0} - y_{0})$ , $w_{0} = v_{0}$ , $ψ_{i} ≜ \sum_{j = 0}^{N} a_{i j} (w_{i} - w_{j})$ , $η_{i} ≜ \sum_{j = 0}^{N} a_{i j} (v_{i} - v_{j})$ , $ρ_{i} = ψ_{i}^{T} S ψ_{i}$ , $S > 0$ , $Q > 0$ , $β$ is a positive constant, $d_{i}$ denotes the time-varying coupling weight associated

Conclusion

In this paper, we have addressed the distributed output feedback consensus protocol design problem for linear multi-agent systems with directed graph and a leader of bounded input. We have presented novel distributed adaptive output feedback consensus protocols, which solved the leader–follower consensus problem over directed graphs when the leader is driven by a bounded control input. The adaptive output feedback protocols presented in this paper are independent of any global information of

Yuezu Lv received his B.S. degree in College of Engineering from Peking University, Beijing, China, in 2013. He is now a Ph.D. candidate in Department of Mechanics and Engineering Science, College of Engineering at Peking University. He is a finalist for Zhang Si-Ying (CCDC) Outstanding Youth Paper Award in 2015. His research interests include cooperative control of multi-agent systems, adaptive control and robust control of uncertain systems.

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Zhongkui Li is currently an Assistant Professor with the Department of Mechanics and Engineering Science, College of Engineering, Peking University, China. He received the B.S. degree in space engineering from the National University of Defense Technology, China, in 2005, and his Ph.D. degree in dynamics and control from Peking University, China, in 2010. He held visiting positions in City University of Hong Kong, China, and Nanyang Technological University, Singapore.

He was the recipient of the State Natural Science Award of China (Second Prize) in 2015, the Yang Jiachi Science and Technology Award in 2015, the IEEE CSS Beijing Chapter Young Author Prize in 2013, the National Excellent Doctoral Thesis Award of China in 2012, and the Natural Science Award (First Prize) from Ministry of Education of China in 2011. His coauthored papers received the 2013 IET Control Theory & Applications Premium (Best Paper) Award and the 2009–2011 Best Paper Award of Journal of Systems Science & Complexity. He is an author of a book “Cooperative Control of Multi-Agent Systems: A Consensus Region Approach” (CRC press, 2014) and has published a number of journal papers. His current research interests include cooperative control of multi-agent systems and networked control systems. He is an associate editor of the Conference Editorial Board of the IEEE Control Systems Society.

Zhisheng Duan received the M.S. degree in Mathematics from Inner Mongolia University, China and the Ph.D. degree in Control Theory from Peking University, China in 1997 and 2000, respectively. From 2000 to 2002, he worked as a Post-Doctor in Peking University. Since 2008, he has been a full Professor with Peking University. He received the 2001 Chinese Control Conference Guan-ZhaoZhi Award and the 2011 first class award in Natural Science from China Ministry of Education. He also obtained the outstanding youth research fund of Natural Science of Foundation of China in 2012. He is currently a Cheung Kong Scholar Chair Professor of China Ministry of Education and an executive council member of Chinese Association of Automation. His research interests include robust control, stability of interconnected systems, flight control, and analysis and control of complex dynamical networks.

Jie Chen is a Chair Professor in the Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China. He received the B.S. degree in aerospace engineering from Northwestern Polytechnic University, Xian, China in 1982, the M.S.E. degree in electrical engineering, the M.A. degree in mathematics, and the Ph.D. degree in electrical engineering, all from The University of Michigan, Ann Arbor, Michigan, in 1985, 1987, and 1990, respectively.

Prior to joining City University, he was with School of Aerospace Engineering and School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia from 1990 to 1993, and with University of California, Riverside, California from 1994 to 2014, where he was a Professor and served as Professor and Chair for the Department of Electrical Engineering from 2001 to 2006. His main research interests are in the areas of linear multivariable systems theory, system identification, robust control, optimization, time-delay systems, networked control, and multi-agent systems. He is the author of two books, respectively, (with G. Gu) Control-Oriented System Identification: An H-infinity Approach (Wiley- Interscience, 2000), and (with K. Gu and V.L. Kharitonov) Stability of Time-Delay Systems (Birkhauser, 2003).

An elected Fellow of IEEE, Fellow of AAAS, Fellow of IFAC and a Yangtze Scholar/Chair Professor of China, Dr. Chen was a recipient of 1996 US National Science Foundation CAREER Award, 2004 SICE International Award, and 2006 Natural Science Foundation of China Outstanding Overseas Young Scholar Award. He served on a number of journal editorial boards, as an Associate Editor and a Guest Editor for the IEEE Transactions on Automatic Control, a Guest Editor for IEEE Control Systems Magazine, an Associate Editor for Automatica, and the founding Editor-in-Chief for Journal of Control Science and Engineering. He is currently an IEEE Control Systems Society (CSS) Distinguished Lecturer and serves as an Associate Editor for SIAM Journal on Control and Optimization. He was a member on IEEE CSS Board of Governors in 2014 and has served as IEEE CSS Chapter Activities Chair since 2015.

^☆: This work was supported in part by the National Natural Science Foundation of China under grants 61473005, 11332001, 61225013, 61528301, and in part by the Hong Kong RGC under the project CityU 111613, CityU 11200415. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Wei Ren under the direction of Editor Christos G. Cassandras.

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Brief paperDistributed adaptive output feedback consensus protocols for linear systems on directed graphs with a leader of bounded input☆

Abstract

Introduction

Section snippets

Problem statement

Adaptive output feedback consensus protocols

Conclusion

Automatica

Systems and Control Letters

Automatica

Automatica

Automatica

Automatica

Interconnected dynamic systems: An overview on distributed control

IEEE Control Systems Magazine

Matrix mathematics: Theory, facts, and formulas

Distributed coordinated tracking with reduced interaction via a variable structure approach

IEEE Transactions on Automatic Control

Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems

IEEE Transactions on Automatic Control

Nonlinear systems

Brief paper
Distributed adaptive output feedback consensus protocols for linear systems on directed graphs with a leader of bounded input☆