The Classification of Linear State Variable Control Laws

Abstract

Two fundamental classes of control laws for linear time-invariant systems were introduced by Kalman [4]. Purely feed-forward control laws do not alter the open loop eigenvalues and purely feedback control laws do not alter the cyclic structure of the open loop system matrix. Here the decomposition of an arbitrary control law into the sum of three laws, two from one of the classes and one from the other, is obtained. The uniqueness of the decomposition is studied. The notion of a covariant control law is introduced to give a decomposition of control laws related to the invariant description of reachable linear systems given by Popov [9]. Two applications of covariant control laws are illustrated, including their use in obtaining maximally unobservable canonical forms for linear multivariable systems under an equivalence relation induced by control laws and state-space basis transformations.