Discrete-Time, Robust Wiener Filtering with Non-Parametric Spectral Uncertainty

In this paper, a discrete-time, robust Wiener filtering problem is approached along the lines pursued in [4] for the continuous-time case. The robust, multivariate filtering problem considered here involves non-parametric spectral uncertainty defined by the so-called spectral band and generalized-variance constraints. The approach in question centers on computing upper and lower bounds on the minimum worst-case performance achieved with causal filters, together with filters which attain such bounds, on the basis of semi-definite linear programming problems (SDLP, for short). Upper bounds are obained from a Lagrangean duality formulation. Relying on finite-dimensional, linearly-parametrized classes of dynamic multipliers and filter transfer functions, the computation of progressively tighter upper bounds together with causal filters which achieve them is reduced to solving SDLPs. Analogously, on the basis of the min-max theorem and relying on similar classes of rational power spectral densities, the computation of lower bounds together with the corresponding filters is also shown to he equivalent to solving SDLPs. Combining these results, causal filters can be obtained whose worst-case, least squares performance can be certified to be close to the optimal one, as illustrated in a simple numerical example.