Minimal Input and Output Selection for Stability of Systems With Uncertainties

Abstract

In networked control systems, selecting a subset of input and output nodes is a crucial step in designing a stabilizing controller. Most existing approaches to input and output selection focus on nominal systems with known parameters. For systems with uncertainties and time delays, current selection methods are based on exploiting the convexity after relaxing the original problem, which is inherently discrete, to continuous optimization forms, and hence lack optimality guarantees. This paper studies the problem of identifying the minimum-size sets of input and output nodes to guarantee stability of a linear system with uncertainties and time delays. We derive sufficient conditions to guarantee the existence of a stabilizing controller for an uncertain linear system, based on a subset of system modes lying within the controllability and observability subspaces induced by the selected inputs and outputs. We then formulate the problems of selecting minimum-size sets of input and output nodes to satisfy the derived conditions, and prove that they are equivalent to discrete optimization problems with bounded submodularity ratios. We develop polynomial-time selection algorithms with provable guarantees on the minimum number of inputs and outputs required. Our approach is applicable to various types of uncertainties, including additive uncertainty, multiplicative uncertainty, uncertain output delay, and structured uncertainty. In a numerical study, we test our approach for the wide-area damping control in power systems to ensure small signal stability. Our results are validated on the IEEE 39-bus test power system.