Robust controllability assessment via semi-definite programming

This paper presents a new framework for systematic assessment of the controllability of uncertain linear time-invariant (LTI) systems. The objective is to evaluate controllability of an uncertain system with norm-bounded perturbation over the entire uncertain region. The method is based on a singular-value minimization problem, over the entire complex plane. To solve the problem, first, a necessary and sufficient condition is proposed to avert the difficulties of griding over the complex plane, and to verify the controllability of directed and undirected networks in a single step. Secondly, the results are utilized to formulate the problem as two Lyapunov-based linear matrix inequalities (LMIs) for undirected networks, and four LMIs, for directed ones. The proposed approach is subsequently extended to evaluate the maximum guaranteed distance to uncontrollability of a system through a quasi-convex optimization problem whose solution is guaranteed to be globally optimal. By duality, analogous results are established for robust observability of directed and undirected networks. The proposed framework is implemented efficiently using LMI tools, and provides a fast and reliable tool for the assessment of robust controllability (and by duality, robust observability). It is also extendable to robust control system design problems such as control node selection for uncertain systems.