A complete set of independent decoupling invariants†

Abstract

The systems considered in the paper are m-input m-output, linear, time invariant decouplable systems S = (A, B, C) which can be described by the set of differential equations $ where A∊R n×n , B∊R n×m and C∊R m×n . In § 1 the decoupling problem is stated and some results relevant to our discussion are reviewed. In § 2 we analyse the effect of an integrator decoupling feedback law (IDF) upon the controllability and the observability of the system under consideration, showing that observability is preserved under IDF if and only if $. Furthermore it is shown that if a decouplable system satisfies this condition, then it is controllable and observable. In § 3 we give our main result : for the class Ω of decouplable systems which preserve observability under IDF, the set of integers {di } ; i = 1, 2,…, m is a complete set of independent invariants under feedback and similarity transformations. Finally a canonical form (unique) is given for the elements of Ω, completely determined by the set {di } ; i = 1,…, m.