Optimal, Worst Case Filter Design via Convex Optimization

Abstract

The problem of model based signal estimation is fundamental to control theory and signal processing, and several approaches have been developed in last decades, for instance, the Kalman filter (H₂ optimal filtering) [1], H_∞ optimal filtering [15] and set-membership approach [4]. The performance of these optimal filters degrades in the presence of model/plant mismatch. Robust filter techniques have been studied to relieve this situation, and numerous papers on this subject have appeared, namely, [9, 24] and references therein, and the text by I.R. Petersen and A.V. Savkin [21] is a comprehensive collection of Riccati based (H₂ , H_∞ and set-membership) approaches. Most of these results are characterized by first upper-bounding the performance objective, then selecting filter parameters to minimize the upper bound. There is little quantitative analysis on the conservativeness introduced by the use of upper bounds. Usually, these bounds guarantee performance not just over all fixed values of the uncertainty, but over time-varying uncertainty as well. Hence, if the actual uncertainty model is time-invariant, these design methods may be conservative. A recent paper by Geromel et al. ([11]) considers systems with time-invariant uncertainty, and the H₂ bound they optimize partially exploits the time-invariance of the uncertainty.