Periodic systems: a sequence algebra approach to realisation, parameterisation and topology

Abstract

A characterisation of discrete time linear periodic systems is presented, based on a general framework using cyclic projection operators on sequences. Two known liftings of periodic systems to a time invariant one, the monodromy and the cyclic representation, are readily derived in this framework. This approach also leads to the definition of the operational transfer function for such systems, and more generally, the operational transfer inclusion. It is shown that the cyclic realisation allows the multiplexing of individual realisations of a periodic system. These descriptions are useful in the realisation problem and the search for canonical forms. Parameterisations for the class of reachable systems are recalled, and their geometric and topological features are illustrated.

Keywords:

• periodic systems
• realisation theory
• state-space methods
• parameterisation
• canonical forms

Notes

The indexing here is different from the one used in Helmke and Verriest (2011)

To simplify notation, we shall denote the identity operator on simply by I, when is understood.

Iterations of varying dimension and forming consistent matrix products were considered in Helmke and Verriest (2011).