On the singular periodic Riccati equation: Theoretical results and an application to a prediction problem

With reference to periodic discrete-time systems, the solution of the periodic Riccati equation is given in the general case of singular and nearly-singular problems. To this aim, the concept of Periodic Spectral Interactor Matrix Operator (PSIMO) is introduced. The PSIMO is a synthetic tool to represent the invariant zeros structure at infinity (delays) of periodic systems in an operatorial fashion. It enables one to transform a singular system into a non-singular one with identical spectral properties. This concept is useful to characterise the family of periodic solutions of the singular periodic Riccati equation arising from the singular prediction problem. The developed theory is proved to be very convenient in the solution of singular and nearly-singular periodic prediction problems.