D-stability and robust stability conditions for LTI systems with generalized frequency variables

A class of large-scale systems with decentralized information structures such as multi-agent dynamical systems can be represented by a linear system with a generalized frequency variable. This paper is concerned with D-stability and robust stability conditions for such systems. We first briefly review stability conditions for the systems, and a motivating example which shows why we need to investigate the D-stability conditions is provided. We then propose a unified treatment of D-stability region and a unified D-stability condition is derived, which is confirmed by a numerical example. The second topic is on robust stability, where we consider three different types of multiplicative perturbations and derive necessary and sufficient conditions for the robust stability.