Cooperative Control of Linear Systems

Abstract

In this chapter, the cooperative control problem of linear systems is studied by first analyzing properties of linear cooperative systems. The results on multiplicative sequence of non-negative matrices from Chapter 4 naturally render the cooperative stability condition for linear systems. Based on the condition, a class of linear cooperative controls are designed, and cooperative controllability can be determined by the cumulative connective property of sensing/communication network topologies over time. As applications, cooperative control is used to solve the problems of consensus, rendezvous, tracking of a virtual leader, formation control, and synchronization. The issue of maintaining network connectivity is also addressed. In the cooperative control problem, information are collected or transmitted through a sensing/communication network, and the cooperative control law is designed to account for topological changes of the network. Availability of the feedback needed for implementing decentralized controls is not continuous or continual or even predictable, which is the main distinction from a standard control problem. This distinction is obviously desirable for applications and positive for implementation, but it makes stability analysis much more involved. To establish the connection back to the standard methods and to describe the cumulative effect of topology changes, an average system is introduced and derived for cooperative systems. The average system enables us to determine Lyapunov function and control Lyapunov function for cooperative systems, although common Lyapunov function is not expected to exist in general. Robustness of cooperative systems against measurement noises and disturbance is also studied.