Reduction of Band Matrices in Large Dynamic Systems Controllability and Observability

Abstract

An approach is proposed for linear stationary dynamical system with controllability and observability band matrices making it possible to simplify procedures for evaluating controllability and observability of this system. The obtained results are based on the fact that the controllability and observability criteria of a dynamic system are equivalent due to their required and sufficient properties; therefore, any transformations of one criterion not violating the conditions of necessity and sufficiency could be reduced to transformations in a sense equivalent to the initial transformations. The Popov --- Belevich --- Hautus transformations of the controllability and observability criteria were taken as a basis, and then results of such transformations were correctly extended to the band criteria. It was proved that the controllability and observability analysis of a linear stationary system with a large number of the state dimensions was reduced to studying the matrices rank of a much smaller size. The proposed approach is based on the existence condition for a numerical matrix of a certain rank of the nondegenerate matrices that satisfy certain transformations. The corresponding controllability and observability theorems for the stationary dynamical systems were provided. It was shown that for systems with one input and one output, the controllability and observability analysis was reduced to the analysis of scalars