On the (Internal) Symmetry Groups of Linear Dynamical Systems

Abstract

A time invariant linear dynamical system is a set of equations

$$\begin{array}{*{20}{c}} {{\rm{\dot x = Fx + Gu}}}\\ {{\rm{y = Hx}}}\\ {({\rm{continoustime}})} \end{array}(\Sigma )\begin{array}{*{20}{c}} {{\rm{x}}({\rm{t + 1}}){\rm{ = Fx}}({\rm{t}}){\rm{ + Gu}}({\rm{t}})}\\ {{\rm{y}}({\rm{t}}){\rm{ = Hx}}({\rm{t}})}\\ {({\rm{discretetime}})} \end{array}$$

where x ∈ X = IRⁿ, u ∈ U = IR^m, y ∈ Y = IR^p and where F, G, H are matrices with coefficients in IR of the dimensions n × n, n × m, p × n respectively. We speak then of a system of dimension n, dim(∑) = n, with m inputs and p outputs. Of cource the discrete time case also makes sense over any field k, (instead of IR). The spaces X, U, Y are respectively called state space, input space and output space. The usual picture is a “black box”.