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2014 | Buch

Introduction to Synchronization in Nature and Physics and Cooperative Control for Multi-Agent Systems on Graphs Kapitel lesen Erstes Kapitel lesen

Cooperative Control of Multi-Agent Systems

Optimal and Adaptive Design Approaches

verfasst von: Frank L. Lewis, Hongwei Zhang, Kristian Hengster-Movric, Abhijit Das

Verlag: Springer London

Buchreihe : Communications and Control Engineering

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Cooperative Control of Multi-Agent Systems extends optimal control and adaptive control design methods to multi-agent systems on communication graphs. It develops Riccati design techniques for general linear dynamics for cooperative state feedback design, cooperative observer design, and cooperative dynamic output feedback design. Both continuous-time and discrete-time dynamical multi-agent systems are treated. Optimal cooperative control is introduced and neural adaptive design techniques for multi-agent nonlinear systems with unknown dynamics, which are rarely treated in literature are developed. Results spanning systems with first-, second- and on up to general high-order nonlinear dynamics are presented.

Each control methodology proposed is developed by rigorous proofs. All algorithms are justified by simulation examples. The text is self-contained and will serve as an excellent comprehensive source of information for researchers and graduate students working with multi-agent systems.

Inhaltsverzeichnis

Frontmatter
1. Introduction to Synchronization in Nature and Physics and Cooperative Control for Multi-Agent Systems on Graphs
Abstract
This chapter presents an overview of synchronization behavior in nature and social systems. It is seen that distributed decisions made by each agent in a group based only on the information locally available to it can result in collective synchronized motion of an overall group. The idea of a communication graph that models the information flows in a multi-agent group is introduced. Mechanisms are given by which decisions can be made locally by each agent and informed leaders can guide collective behaviors by interacting directly with only a few agents. Synchronization and collective behavior phenomena are discussed in biological systems, physics and chemistry, and engineered systems. The dependence of collective behaviors of a group on the type of information flow allowed between its agents is emphasized. Various different graph topologies are presented including random graphs, small-world networks, scale-free networks, and distance formation graphs. The early work in cooperative control systems on graphs is outlined.
Frank L. Lewis, Hongwei Zhang, Kristian Hengster-Movric, Abhijit Das
2. Algebraic Graph Theory and Cooperative Control Consensus
Abstract
Cooperative control studies the dynamics of multi-agent dynamical systems linked to each other by a communication graph. The graph represents the allowed information flow between the agents. The objective of cooperative control is to devise control protocols for the individual agents that guarantee synchronized behavior of the states of all the agents in some prescribed sense. In cooperative systems, any control protocol must be distributed in the sense that it respects the prescribed graph topology. That is, the control protocol for each agent is allowed to depend only on information about that agent and its neighbors in the graph. The communication restrictions imposed by graph topologies can severely limit what can be accomplished by local distributed control protocols at each agent. In fact, the graph topological properties complicate the design of synchronization controllers and result in intriguing behaviors of multi-agent systems on graphs that do not occur in single-agent, centralized, or decentralized feedback control systems.
Frank L. Lewis, Hongwei Zhang, Kristian Hengster-Movric, Abhijit Das

Local Optimal Design for Cooperative Control in Multi-Agent Systems on Graphs

Frontmatter
3. Riccati Design for Synchronization of Continuous-Time Systems
Abstract
This chapter studies cooperative tracking control of multi-agent dynamical systems interconnected by a fixed communication graph topology. Each agent or node is mathematically modeled by identical continuous linear time-invariant (LTI) systems, which includes the single-integrator and double-integrator as special cases. The communication network among the agents is described by a directed graph. A command generator or leader node generates the desired tracking trajectory to which all agents should synchronize. Only a few nodes are aware of information from the leader node. A locally optimal Riccati design approach is introduced here to synthesize the distributed cooperative control protocols. A framework for cooperative tracking control is proposed, including full state feedback control protocols, observer design, and dynamic output regulator control. The classical system theory notion of duality is extended to networked cooperative systems on graphs. It is shown that the local Riccati design method guarantees synchronization of multi-agent systems regardless of graph topology, as long as certain connectivity properties hold. This is formalized through the notion of synchronization region. It is shown that the Riccati design method yields unbounded synchronization regions and so achieves synchronization on arbitrary digraphs containing a spanning tree.
Frank L. Lewis, Hongwei Zhang, Kristian Hengster-Movric, Abhijit Das
4. Riccati Design for Synchronization of Discrete-Time Systems
Abstract
In this chapter, design methods are given for synchronization control of discrete-time multi-agent systems on directed communication graphs. The graph is assumed to have fixed topology and contain a spanning tree. The graph properties complicate the design of synchronization controllers due to the interplay between the eigenvalues of the graph Laplacian matrix and the required stabilizing gains. A method is given that decouples the design of the synchronizing feedback gains from the detailed graph properties. It is based on computation of the agent feedback gains using a local Riccati equation design. Conditions are given for synchronization based on the relation of the graph eigenvalues to a bounded circular region in the complex plane that depends on the agent dynamics and the Riccati solution. The notion of ‘synchronization region’ is used. Convergence to consensus and robustness properties are investigated. This chapter also investigates the design of distributed observers for identical agents using a local Riccati design. A cooperative observer design guaranteeing convergence of the estimates of all agents to their actual states is proposed. The notion of a convergence region for distributed observers on graphs is introduced.
Frank L. Lewis, Hongwei Zhang, Kristian Hengster-Movric, Abhijit Das
5. Cooperative Globally Optimal Control for Multi-Agent Systems on Directed Graph Topologies
Abstract
In Chaps. 3 and 4, we showed that locally optimal design in terms of Riccati equations can guarantee synchronization for cooperative multi-agents on graphs. In this chapter, we examine the design of distributed control protocols that solve global optimal problems for all the agents in the graph. In cooperative control systems on graphs, it turns out that local optimality for each agent and global optimality for all the agents are not the same. This chapter brings together stability and optimality theory to design distributed cooperative control protocols that guarantee synchronization and are also optimal with respect to a positive semidefinite global performance criterion. A common problem in optimal decentralized control is that global optimization problems generally require global information from all the agents, which is not available to distributed controllers. In cooperative control of multi-agent systems on graphs, each agent is only allowed to use distributed information that respects the graph topology, that is, information about itself and its neighbors. Global optimal control for multi-agent systems is complicated by the fact that the communication graph topology interplays with agent system dynamics.
Frank L. Lewis, Hongwei Zhang, Kristian Hengster-Movric, Abhijit Das
6. Graphical Games: Distributed Multiplayer Games on Graphs
Abstract
In this chapter, it is seen that distributed control protocols that both guarantee synchronization and are globally optimal for the multi-agent team always exist on any sufficiently connected communication graph if a different definition of optimality is used. To this end, we study the notion of Nash equilibrium for multiplayer games on graphs. This leads us to the idea of a new sort of differential game—graphical games. In graphical games, each agent has its own dynamics as well as its own local performance index. The dynamics and local performance indices of each agent are distributed; they depend on the state of the agent, the control of the agent, and the controls of the agent’s neighbors. We show how to compute distributed control protocols that guarantee global Nash equilibrium for multi-agent teams on any graph that has a spanning tree.
Frank L. Lewis, Hongwei Zhang, Kristian Hengster-Movric, Abhijit Das

Distributed Adaptive Control for Multi-Agent Cooperative Systems

Frontmatter
7. Graph Laplacian Potential and Lyapunov Functions for Multi-Agent Systems
Abstract
In this chapter we show that for networked multi-agent systems, there is an energy-like function, called the graph Laplacian potential, that depends on the communication graph topology. The Laplacian potential captures the notion of a virtual potential energy stored in the graph. We shall study the Laplacian potential for both undirected graphs and directed graphs. The Laplacian potential is further used here to construct Lyapunov functions that are suitable for the analysis of cooperative control systems on graphs. These Lyapunov functions depend on the graph topology, and based on them a Lyapunov analysis technique is introduced for cooperative multi-agent systems on graphs. Control protocols coming from such Lyapunov functions are distributed in form, depending only on information about the agent and its neighbors.
Frank L. Lewis, Hongwei Zhang, Kristian Hengster-Movric, Abhijit Das
8. Cooperative Adaptive Control for Systems with First-Order Nonlinear Dynamics
Abstract
In cooperative control systems, the control protocol for each agent is allowed to depend only on information about itself and its neighbors in the graph topology. We have confronted this problem for optimal cooperative control design in Part I of the book. In cooperative adaptive control systems, there is an additional problem. In adaptive control systems, the control law depends on unknown parameters that are tuned online in real time to improve the performance of the controller, whereas the challenge in cooperative adaptive control is to make sure that both the control protocols and the parameter tuning laws are distributed in terms of the allowed graph topology. That is, they are allowed to depend only on locally available information about the agent and its neighbors. We shall see in this chapter that the key to the design of distributed adaptive tuning algorithms is the selection of suitable Lyapunov functions that depend on the graph topology. This is closely connected to the selection of global performance indices that depend on the graph topology in Chap. 5. The basis for the selection of suitable graph-dependent Lyapunov functions was laid in the discussion on the graph Laplacian potential in Chap. 7.
Frank L. Lewis, Hongwei Zhang, Kristian Hengster-Movric, Abhijit Das
9. Cooperative Adaptive Control for Systems with Second-Order Nonlinear Dynamics
Abstract
In Chap. 8, we designed cooperative adaptive controllers for multi-agent systems having first-order nonlinear dynamics. In this chapter, we study adaptive control for cooperative multi-agent systems having second-order nonidentical nonlinear dynamics. The study of second-order and higher-order consensus is required to implement synchronization in most real-world applications such as formation control and coordination among unmanned aerial vehicles (UAVs), where both position and velocity must be controlled. Note that Lagrangian motion dynamics and robotic systems can be written in the form of second-order systems. Moreover, second-order integrator consensus design (as opposed to first-order integrator node dynamics) involves more details about the interaction between the system dynamics and control design problem and the graph structure as reflected in the Laplacian matrix. As such, second-order consensus is interesting because there one must confront more directly the interface between control systems and communication graph structure.
Frank L. Lewis, Hongwei Zhang, Kristian Hengster-Movric, Abhijit Das
10. Cooperative Adaptive Control for Higher-Order Nonlinear Systems
Abstract
Cooperative control on communication graphs for agents that have unknown nonlinear dynamics that are not the same is a challenge. The interaction of the communication graph topology with the agent’s system dynamics is not easy to investigate if the dynamics of the agents are heterogeneous, that is, not identical, since the Kronecker product cannot be used to simplify the analysis. Therefore, the intertwining of the graph structure with the local control design is more severe and makes the design of guaranteed synchronizing controls very difficult. That is to say, the communication graph structure imposes more severe limitations on the design of controllers for systems that have nonlinear and nonidentical dynamics, making it more challenging to guarantee the synchronization of all agents in the network.
Frank L. Lewis, Hongwei Zhang, Kristian Hengster-Movric, Abhijit Das
Backmatter
Metadaten
Titel
Cooperative Control of Multi-Agent Systems
verfasst von
Frank L. Lewis
Hongwei Zhang
Kristian Hengster-Movric
Abhijit Das
Copyright-Jahr
2014
Verlag
Springer London
Electronic ISBN
978-1-4471-5574-4
Print ISBN
978-1-4471-5573-7
DOI
https://doi.org/10.1007/978-1-4471-5574-4

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