Solution modules and system equivalence

In this paper we consider a system as a relation between input, internal variables and output. This relation is given by the solution space of the system's equations. For time invariant linear systems in differential operator representation the solution space carries a Ks]-module structure defined by the ordinary differential operator. This algebraic structure is exploited systematically to develop a self-contained theory of strict system equivalence in time domain.

The module of free motions is considered as space of initial conditions. An algebraic characterization of systems having the same solution space is presented. System homomorphisms are defined as special Ks] homomorphisms between the solution modules. Two systems are called system-equivalent, if there exists a system-isomorphism between their solution spaces. It turns out that, this concept coincides with Rosenbrock's concept of strict. system equivalence. It. is shown that further concepts and results of linear system theory (construction of a state-space model in time domain, controllability and observability criteria, uniqueness theorem of linear realization theory) can be derived within this framework.