Chapter 2
Symmetry, Stability and Scalability
People today see not the moon the ancients saw, and yet this moon shone once upon the ancients.
—Li Bai (701–762), ‘Inquiring of the moon, with wine in hand'
This chapter introduces general models of distributed control systems based on graph theory and linear system theory. The symmetry property of distributed control systems in the frequency domain is defined and investigated. Classic results of the frequency-domain test of the stability of multivariable feedback control systems are reviewed with the emphasis on their extension to the semi-stability and the scalability for distributed control systems.
2.1 System Model
2.1.1 Graph-Based Model of Distributed Control Systems
Consider a distributed control system with n subsystems, each of which is also called an agent in this book. Let the model of the dynamics of the ith agent be given by the following state-space model
where 
, 
and 
denote the state, input and output, respectively, of the ith agent, Ti is the input delay. Note that in distributed control systems input delays are usually ...
