On Invariants, Canonical Forms and Moduli for Linear, Constant, Finite Dimensional, Dynamical Systems

Abstract

A linear, constant, finite dimensional dynamical system is thought of as being represented by a triple of matrices (F,G,H), where F is an n × n matrix, G an n × m matrix, and H an p × n matrix; i.e. there are m inputs, p outputs and the state space dimension is n. The dynamical system itself is

$$ \mathop {\text{x}}\limits^. = {\text{Fx}}\;{\text{ + }}\;{\text{Gu,}}\;{\text{y}}\;{\text{ = }}\;{\text{Hx}} $$

((1.1))

or, if one prefers discrete time systems

$$ {{\text{x}}_{{\text{t}} + 1}} = {\text{F}}{{\text{x}}_{\text{t}}}\;{\text{ + }}\;{\text{G}}{{\text{u}}_{\text{t}}}{\text{,}}\;{{\text{y}}_{\text{t}}}\;{\text{ = }}\;{\text{H}}{{\text{x}}_{\text{t}}} $$

((1.2))

A change of coordinates in state space changes the triple of matrices (F,G,H) into the triple (SFS^-1, SG, HS^-1). Let DS denote the space of all triples (F,G,H); i.e. DS is affine space of dimension np + n² + nm.