Parametrization and optimization of reduced-order filters

This paper deals with the design of reduced-order filters for linear signal models to ensure that the transfer function from the noise to the filtering error is not only stable but also has a minimum H₂ norm. One of the major results in the paper is the discovery and 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 and a given full-order filter. As a consequence, the problem of minimizing the H₂ norm of the filtering error transfer function over the set of reduced-order filters can be formulated as an equivalent unconstrained parametric optimization problem over a compact manifold. A gradient flow-based algorithm is proposed to find an optimal solution to the problem. Nice properties of the algorithm including the convergence property are established theoretically as well as demonstrated numerically with an example