An algorithm to compute the distance to uncontrollability of linear time-invariant control systems

Abstract

In this study, we determine how far a Linear Dynamic System is from the nearest uncontrollable system. We will call this quantity "The Distance to Uncontrollability". Estimating this distance, not only do we know if a given linear dynamical system is controllable or uncontrollable, but in the case of a controllable system, we also know how far it is from being uncontrollable. This could be found useful by a control engineer for example, in making a decision to insert additional controls to the system design. As it turns out, the estimation of the "distance to uncontrollability" is equivalent to determining the global minimum of a certain function. In this work, we will examine some already existing algorithms and will present a Two-Phase Algorithm that will combine a novel search algorithm, termed the Density Search Algorithm and the Tunneling Algorithm [21] for the computation of this global minimum.