Distributed nonlinear control algorithms for network consensus

doi:10.1016/j.automatica.2008.01.011

Automatica

Volume 44, Issue 9, September 2008, Pages 2375-2381

https://doi.org/10.1016/j.automatica.2008.01.011 Get rights and content

Abstract

In this paper, we develop a thermodynamic framework for addressing consensus problems for nonlinear multiagent dynamical systems with fixed and switching topologies. Specifically, we present distributed nonlinear static and dynamic controller architectures for multiagent coordination. The proposed controller architectures are predicated on system thermodynamic notions resulting in controller architectures involving the exchange of information between agents that guarantee that the closed-loop dynamical network is consistent with basic thermodynamic principles.

Introduction

Modern complex dynamical systems are highly interconnected and mutually interdependent, both physically and through a multitude of information and communication networks. Distributed decision-making for the coordination of networks of dynamic agents involving information flow can be naturally captured by graph-theoretic notions. These dynamical network systems cover a very broad spectrum of applications including cooperative control of unmanned air vehicles (UAV’s), autonomous underwater vehicles (AUV’s), distributed sensor networks, air and ground transportation systems, swarms of air and space vehicle formations, and congestion control in communication networks, to cite but a few examples. Hence, it is not surprising that a considerable research effort has been devoted to the control of networks and control over networks in recent years (Jadbabaie et al., 2003, Olfati-Saber and Murray, 2004, Ren and Beard, 2005, Tanner et al., 2007).

A key application area of multiagent network coordination within aerospace systems is cooperative control of vehicle formations using distributed and decentralized controller architectures. Distributed control refers to a control architecture wherein the control is distributed via multiple computational units that are interconnected through information and communication networks, whereas decentralized control refers to a control architecture wherein local decisions are based only on local information. Vehicle formations are typically dynamically decoupled, that is, the motion of a given agent or vehicle does not directly affect the motion of the other agents or vehicles. The multiagent system is coupled via the task which the agents or vehicles are required to perform.

In many applications involving multiagent systems, groups of agents are required to agree on certain quantities of interest. In particular, it is important to develop information consensus protocols for networks of dynamic agents wherein a unique feature of the closed-loop dynamics under any control algorithm that achieves consensus is the existence of a continuum of equilibria representing a state of equipartitioning or consensus. Under such dynamics, the limiting consensus state achieved is not determined completely by the dynamics, but depends on the initial system state as well. For such systems possessing a continuum of equilibria, semistability (Bhat and Bernstein, 2003a, Bhat and Bernstein, 2003b), and not asymptotic stability, is the relevant notion of stability. Semistability is the property whereby every trajectory that starts in a neighborhood of a Lyapunov stable equilibrium converges to a (possibly different) Lyapunov stable equilibrium. From a practical viewpoint, it is not sufficient to only guarantee that a network converges to a state of consensus since steady-state convergence is not sufficient to guarantee that small perturbations from the limiting state will lead to only small transient excursions from a state of consensus. It is also necessary to guarantee that the equilibrium states representing consensus are Lyapunov stable, and consequently, semistable.

Using graph-theoretic notions, in this paper we develop control algorithms for addressing consensus problems for nonlinear multiagent dynamical systems with fixed and switching topologies. The proposed controller architectures are predicated on the recently developed notion of system thermodynamics (Haddad, Chellaboina, & Nersesov, 2005) resulting in controller architectures involving the exchange of information between agents that guarantee that the closed-loop dynamical network is consistent with basic thermodynamic principles. The proposed controllers use undirected and directed graphs to accommodate for a full range of possible graph information topologies without limitations of bidirectional communication.

Section snippets

The consensus problem in dynamical networks

In this paper, we use undirected and directed graphs to represent a nonlinear dynamical network and present solutions to the consensus problem for nonlinear networks with both graph topologies (or information flows) (Olfati-Saber & Murray, 2004). Specifically, let $G = (V, E, A)$ be a weighted directed graph (or digraph) denoting the dynamical network (or dynamic graph) with the set of nodes (or vertices) $V = {1, \dots, q}$ involving a finite nonempty set denoting the agents, the set of edges $E \subseteq V \times V$ involving a

Distributed nonlinear control algorithms for consensus

In this section, we develop a thermodynamically motivated information consensus framework for multiagent nonlinear systems that achieve semistability and state equipartition. Specifically, consider $q$ continuous-time integrator agents with dynamics ${\dot{x}}_{i} (t) = u_{i} (t), x_{i} (0) = x_{i 0}, t \geq 0,$ where for each $i \in {1, \dots, q}$ , $x_{i} (t) \in R$ denotes the information state and $u_{i} (t) \in R$ denotes the information control input for all $t \geq 0$ . The nonlinear consensus protocol is given by $u_{i} (t) = \sum_{j = 1, j \neq i}^{q} ϕ_{i j} (x_{i} (t), x_{j} (t)),$ where $ϕ_{i j} (\cdot, \cdot)$ , $i, j$

Network consensus with switching topology

Communication links among multiagent systems are often unreliable due to multipath effects and exogenous disturbances leading to dynamic information exchange topologies. In this section, we develop a switched consensus protocol to achieve agreement over a network with switching topology. In contrast to the static controllers addressed in Jadbabaie et al. (2003), Olfati-Saber and Murray (2004), and Ren and Beard (2005), the proposed controller is a dynamic compensator. This controller

Conclusion

This paper presents a thermodynamic framework for addressing consensus problems for multiagent dynamical systems. Specifically, nonlinear static and dynamic network protocols are designed that guarantee convergence to Lyapunov stable equilibria for a network of dynamic agents with undirected and directed information flows as well as fixed and switching topology. Our analysis relies on several tools from algebraic graph theory and system thermodynamics. Future extensions will focus on

Acknowledgement

This research was supported in part by the Air Force Office of Scientific Research under Grant FA9550-06-1-0240.

Qing Hui received the B. Eng. degree in aerospace engineering from the National University of Defense Technology, Changsha, China, and the M. Eng. degree in automotive engineering from the Tsinghua University, Beijing, China, in 1999 and 2002 respectively. In 2005 he received an M. S. degree in applied mathematics from the Georgia Institute of Technology, Atlanta. In 2002 he joined the School of Aerospace Engineering, Georgia Institute of Technology, where he is currently working toward a Ph.D.

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Dr. Wassim M. Haddad received the B.S., M.S., and Ph.D. degrees in mechanical engineering from the Florida Institute of Technology, Melbourne, FL in 1983, 1984, and 1987, respectively, with specialization in dynamical systems and control. From 1987 to 1994 he served as a consultant for the Structural Controls Group of the Government Aerospace Systems Division, Harris Corporation, Melbourne, FL. In 1988 he joined the faculty of the Mechanical and Aerospace Engineering Department at Florida Institute of Technology where he founded and developed the Systems and Control Option within the graduate program. Since 1994 he has been a member of the faculty in the School of Aerospace Engineering at Georgia Institute of Technology where he holds the rank of Professor. Dr. Haddad’s research contributions in linear and nonlinear dynamical systems and control are documented in over 470 archival journal and conference publications. His recent research is concentrated on nonlinear robust and adaptive control, nonlinear dynamical system theory, largescale systems, hierarchical nonlinear switching control, analysis and control of nonlinear impulsive and hybrid systems, system thermodynamics, thermodynamic modeling of mechanical and aerospace systems, network systems, nonlinear analysis and control for biological and physiological systems, and active control for clinical pharmacology. Dr. Haddad is an NSF Presidential Faculty Fellow, a member of the Academy of Nonlinear Sciences, and a coauthor of the books Hierarchical Nonlinear Switching Control Design with Applications to Propulsion Systems (Springer-Verlag, 2000), Thermodynamics: A Dynamical Systems Approach (Princeton University Press, 2005), Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control (Princeton University Press, 2006), and Nonlinear Dynamical Systems and Control: A Lyapunaov-Based Approach (Princeton University Press, 2008).

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Andrey V. Savkin under the direction of Editor Ian Petersen.

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Brief paperDistributed nonlinear control algorithms for network consensus☆

Abstract

Introduction

Section snippets

The consensus problem in dynamical networks

Distributed nonlinear control algorithms for consensus

Network consensus with switching topology

Conclusion

Acknowledgement

Nonlinear Analysis: RWA

Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions

ESIAM Control Optimisation Calculus Variations

Nonnegative matrices in the mathematical sciences

Matrix mathematics

Nontangency-based Lyapunov tests for convergence and stability in systems having a continuum of equilibra

SIAM Journal on Control and Optimisation

Optimization and nonsmooth analysis

Brief paper
Distributed nonlinear control algorithms for network consensus☆