Design of optimal reduced order filter for unstable discrete time systems

The paper is concerned with H₂ optimal reduced-order filtering for a linear discrete time signal model. The signal model is allowed to be unstable. 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 H₂ norm. The authors first derive a parameterisation of a set of filters of fixed order which lead to stable filtering error transfer functions. Such a parameterisation is given in terms of an arbitrary orthogonal projection matrix. As a consequence, the problem of minimising the H₂ norm of the filtering error transfer function over the set of reduced-order filters is formulated as an equivalent unconstrained parametric optimisation problem over a compact manifold. Two gradient-based algorithms are then proposed to compute an optimal reduced-order filter. The good properties of the algorithms, including the convergence property, are established theoretically as well as demonstrated numerically with an example