Essential Uncontrollability of Discrete Linear, Time-Invariant, Dynamical Systems

Abstract

The concept of a "best approximating m-dimensional subspace" for a given set of vectors in Rn is introduced. Such a subspace is easily described in terms of the eigenvectors of an associated Gram matrix. This technique is used to approximate an achievable set for a discrete, linear, time invariant dynamical system. This approximation characterizes the part of the state space that may be reached using modest levels of control. If the achievable set can be closely approximated by a proper subspace of Rn, then the system is "essentially uncontrollable". The notion finds application in studies of failure-tolerant systems, and in decoupling.