Periodic Solutions of Matrix Riccati Equation in Optimal Filtering of Nonstabilizable Periodic Systems

Abstract

This paper investigates the steady state periodic solutions of the matrix Riccati difference equation in optimal filtering of discrete time linear systems having periodic system matrices. Special emphasis is given to systems not necessarily stabilizable in the filtering sense and the question addressed is the existence and uniqueness of a steady state periodic nonnegative definite solution which gives rise to an asymptotically stable filter. The convergence of the solution of the Riccati difference equation to a steady state periodic solution is also discussed. The results presented here generalize recent results on nonstabilizable discrete time-invariant systems to the context of discrete time-varying periodic systems. This extension means that the results have many potential applications including the optimal filtering problem for periodic systems having only measurement noise and for periodic systems having purely deterministic disturbances such as sinusoids.