Degree of controllability for linear unstable systems

Abstract

The degree of controllability is a measure to check how controllable a given system is, which makes up for the weak points of traditional methods for checking controllability. There are many definitions of the degree of controllability, and the most widely used measure is represented by controllability Gramian. This measure has physical meaning of minimum input energy to change the states. However, it is hard to use this method for unstable systems because the steady-state value of Gramian cannot be defined. Therefore, the degree of controllability for unstable systems is proposed in this paper. The definition of the measure is minimum control input energy to change the state from an arbitrary initial condition to an arbitrary final condition. The derivation result shows that the measure is the sum of two terms, one represented by Gramian of a stable subsystem and the other represented by Gramian of a subsystem that has mirror image poles of an anti-stable subsystem. These Gramians can be obtained by solving two Lyapunov equations. A simple numerical example shows the usefulness of the proposed measure of degree of controllability.