Abstract
To embrace a broader class of input-output models in linear system theory, it is necessary to extend the notions devised in the context of stronger algebraic structures (vector spaces) to those in weaker structures.
In this paper, the system theory-structure theory and realization theory-of linear dynamical systems, previously developed over a field, are extended to a principal ideal domain. It includes the system theory over integers as a special case.
The systems under discussion here are linear, constant and discrete-time dynamical systems over a principal ideal domain A of the form x(t+1)=Fx(t)+Gu(t) y(t)=Hx(t) whereX(t)∈An, u(t)∈Am, y(t)∈Ap, and F, G, H are matrices over A.
The concepts of reachability, observability, minimality, canonical system and realization are difined just as in the case when A is a field. Some of the results are analogous to those of linear systems over a field and others are not.
Main differences:
(1) The duality between reachability and observability does not hold.
(2) The minimal system and the canonical system are not equivalent.
(3) The state space can not be decomposed by reachability as a direct sum.
Analogous results:
(4) The state space decomposition by observability is possible.
(5) For any system, we can find a canonical system which has the same input-output relations.
(6) Ho's realization algorithm can be generalized (with minor modification) to a principal ideal domain.
