On the minimum number of general or dedicated controllers required for system controllability

Abstract

The paper determines the minimum number of general or dedicated controllers which are required to guarantee the controllability of a class of dynamical systems. For this purpose, we begin by reformulating Kalman controllability condition for linear systems ( A , B ) in the context of the similarity equivalence class associated with A , and then we provide new al- ternative characterizations of such controllability condition based on the cases when ma- trix A is in Jordan or Frobenius form, so that theoretical aspects and computational re- quirements are comparatively commented. In the second part of the paper, given a system state matrix A , we solve the optimal design problem of determining the minimum number of controllers (columns of B ) required for making ( A , B ) controllable, resorting also to the Jordan and Frobenius canonical forms in the similarity equivalence class associated with A . These canonical forms are proven to be also fundamental to provide bounds on the minimum number of dedicated controllers (corresponding to columns of B with a single non-zero element) required to make ( A , B ) controllable. Some examples serve to illustrate the fundamental results.