Stability Analysis of Linear Systems with Generalized Frequency Variables and Its Applications to Formation Control

A linear system with a generalized frequency variable denoted by G(s) is a system which is given by replacing transfer function´s ´s´ variable in the original system G₀(s) with a rational function ´Phi(s)´, i.e., G(s) is defined by Go(Phi(s)). A class of large-scale systems with decentralized information structures such as multi-agent systems can be represented by this form. In this paper, we investigate fundamental properties of such a system in terms of controllability, observability, and stability. Specifically, we first derive necessary and sufficient conditions that guarantee controllability and observability of the system Q(s) based on those of subsystems Go(s) and 1/Phi(s). Then we present Nyquist-type stability criterion which can be reduced to a linear matrix inequality (LMI) feasibility problem. Finally, we apply the results to stability analysis of a class of formation control and confirm the effectiveness of the approach as a general framework which can unify variety of results in the field.