Causal Dynamical Systems: Irreducible Realizations

Abstract

Recently, much attention has been devoted to control systems described by the following data:

$$ {{dx} \over {dt}} = F(x,u) $$

((1))

$$ y = h(x) $$

((2))

where the state of the system x belongs to a differentiable manifold X , u belongs to a class of admissible inputs I , and the output map h takes values in a manifold Y ([2], [5], [9]). In this context it is generally assumed that for a variety of physical considerations, the state of the system x is not known directly but that it is only observed through the output h(x) . From this point of view, it appears natural to regard the system as an input -output relation S where (1) and (2) along with the set X where (1) is defined represent a state description for S . More precisely, if we denote the solution of (1) which corresponds to x ε X , and u ε I by π(x,u,t) (i.e., π(x,u,t) = x , and ${{\partial \pi } \over {\partial t}} = F(\pi ,u)$ , then (u,y) ε S if and only if y(t) = hπ(x,u,t) for some x ε X , and all t ≥ 0 . Thus, a state representation defines a system S , but corresponding to an S there may be many state representations.