Properties of linear time-invariant multivariable systems subject to arbitrary output and state feedback

Abstract

It is shown that, for any linear time-invariant multivariable system which is both completely controllable and completely observable, almost all output feedback laws can be used to make the closed-loop system have distinct poles; i.e., the set of output feedback laws which fails to achieve this goal is either an empty set or a hypersurface in the parameter space. It is also shown that almost any output feedback law will make the closed-loop system's poles be disjoint from any given finite set of points on the complex plane. It is then shown that any controllable observable system can be made controllable and observable with respect to any nontrivial input and output by applying almost all output feedback laws about the original system. These results have immediate application in pole assignment, in the controllability of parallel-connected systems, and in the identification problem. In addition, it is shown that, for any linear system and for almost all state feedback laws, the closed-loop system has the property that all modes of the system, except possibly those corresponding to the uncontrollable-unobservable part of the system, are observed in the output.