On the regulator problem for linear systems over rings and algebras. (English) Zbl 1475.93031
Summary: The regulator problem is solvable for a linear dynamical system \(\Sigma\) if and only if \(\Sigma\) is both pole assignable and state estimable. In this case, \(\Sigma\) is a canonical system (i.e., reachable and observable). When the ring \(R\) 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).
MSC:
| 93B25 | Algebraic methods |
| 13P25 | Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.) |
Keywords:
linear systems over commutative rings; regulator problem; duality principle; pole assignmentReferences:
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