Abstract
Abstract The regulator problem is solvable for a linear dynamical system Σ \Sigma if and only if Σ \Sigma is both pole assignable and state estimable. In this case, Σ \Sigma is a canonical system (i.e., reachable and observable). When the ring R R is a field or a Noetherian total ring of fractions the converse is true. Commutative rings which have the property that the regulator problem is solvable for every canonical system (RP-rings) are characterized as the class of rings where every observable system is state estimable (SE-rings), and this class is shown to be equal to the class of rings where every reachable system is pole-assignable (PA-rings) and the dual of a canonical system is also canonical (DP-rings).
Highlights
The mathematical theory of systems is a theory about mathematical models of real-life systems, which are often given by functions that describe the time dependence of a point in some state-space
This paper deals with the study of regulation of linear systems over commutative rings by means of dynamic compensators
When does this result generalize to commutative rings? That is to say, we are interested in describing conditions for a commutative ring R in order that any canonical system over R admits a dynamic compensator
Summary
The mathematical theory of systems is a theory about mathematical models of real-life systems, which are often given by functions that describe the time dependence of a point in some state-space. This paper deals with the study of regulation of linear systems over commutative rings by means of dynamic compensators. It is clear from the aforementioned definition and the aforementioned construction that a dynamic compensator can be built from linear system Σ = (A, B, C) if and only if Σ is both pole-assignable and state-estimable. Notions of reachability and observability are reviewed These conditions on a system (A, B, C) will be shown to be necessary for pole-assignability and state-estimability, and to build dynamic compensators. It is clear that every canonical system over a field admits dynamic compensator When does this result generalize to commutative rings?
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