On the Behavior of Almost Ill-Posed Linear Discrete-Time Systems

The purpose of this note is to explore the limits on the behavior of a regulation scheme when the underlying model is not well controllable or observable. We examine what happens when the open loop system is close to uncontrollable or unobservable, either in the sense that some of the second order modes are small, or in the sense that some small perturbation to the coefficients in state-space realization will yield an uncontrollable or unobservable system. We show that in this situation, the system behaves very close to an uncontrollable and/or unobservable system, thus making the system undesirable, and we give analytic bounds on how close this behavior is.