A Contribution to the Study of Periodic Systems in the Behavioral Approach


José Carlos Aleixo1 — Jan Willem Polderman2 — Paula Rocha3

1 Department of Mathematics University of Beira Interior Rua Marquês d’Ávila e Bolama 6201-001 Covilhã – Portugal jcaleixo@mat.ubi.pt

2 Department of Applied Mathematics University of Twente P.O. Box 217 7500 AE Enschede – The Netherlands j.w.polderman@math.utwente.nl

3 Department of Mathematics University of Aveiro 3810-193 Aveiro – Portugal procha@mat.ua.pt

ABSTRACT. In this paper we obtain new results on periodic kernel representations and propose a definition of “image" representation for periodic behaviors. Further, we characterize controllability and autonomicity in representation terms. We also show that the concept of free variable used in the time-invariant case cannot be carried over to the periodic case in a straightforward manner and introduce a new concept of variable freeness (P-periodic freeness). This allows us to define input/output structures for periodic behaviors.

KEYWORDS: Periodic systems; behaviors; representations

1. Introduction

In this paper we present new results on the behavioral theory of linear periodically time-varying systems based on the framework developed by Kuijper and Willems, see [KUI 97] and the references therein. This approach uses a technique known as lifting, that associates to each periodic behavior a time-invariant one. This allows to derive many results for periodic systems based on the existing ...

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