Acta Applicandae Mathematica

, Volume 88, Issue 1, pp 1–45 | Cite as

An Algebraic Interpretation to the Operator-Theoretic Approach to Stabilizability. Part I: SISO Systems

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Abstract

The purpose of this paper is to show that a duality exists between the fractional ideal approach [23, 26] and the operator-theoretic approach [4, 6, 8, 9, 33, 34] to stabilization problems. In particular, this duality helps us to understand how the algebraic properties of systems are reflected by the operator-theoretic approach and conversely. In terms of modules, we characterize the domain and the graph of an internally stabilizable plant or that of a plant which admits a (weakly) coprime factorization. Moreover, we prove that internal stabilizability implies that the graph of the plant and the graph of a stabilizing controller are direct summands of the global signal space. These results generalize those obtained in [6, 8, 9, 33, 34]. Finally, we exhibit a class of signal spaces over which internal stabilizability is equivalent to the existence of a bounded inverse for the linear operator mapping the errors e1 and e2 of the closed-loop system to the inputs u1 and u2.

Keywords

linear control theory fractional representation to analysis and synthesis problems theory of fractional ideals module duality operator-theoretic approach behavioural approach 
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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.CAFEINRIA Sophia AntipolisSophia Antipolis CedexFrance

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