Optimal filter reduction for unstable discrete-time systems

This paper deals with the H₂ optimal reduced order filtering for a linear unstable discrete-time signal model. The objective is to obtain a reduced order filter that not only gives rise to a stable filtering error transfer function but also minimises the H ₂ norm. We first derive a parametrization of a set of filters of fixed order which lead to stable filtering error transfer functions. Such a parametrization is given in terms of an arbitrary orthogonal projection matrix. As a consequence, the problem of minimizing the H₂ norm of the filtering error transfer function over the set of reduced order filters is formulated as an equivalent unconstrained parametric optimization problem over a compact manifold. Two gradient based algorithms are then proposed to compute an optimal reduced order filter