Robust Linear Estimation with Non-parametric Uncertainty: Average and Worst-case Performance (Full Version)

TL;DR

This paper develops robust linear estimators that balance worst-case and average performance under non-parametric uncertainty, formulated as semi-definite programs, to reduce conservatism in traditional minimax approaches.

Contribution

It introduces a new class of estimators that trade off worst-case robustness for improved average performance, extending robust estimation methods to non-parametric uncertainty sets.

Findings

SDP formulations for estimator design problems

Illustration of reduced conservatism in simple examples

Potential for improved performance over traditional minimax estimators

Abstract

In this paper, two types of linear estimators are considered for three related estimation problems involving set-theoretic uncertainty pertaining to $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ balls of frequency-responses. The problems at stake correspond to robust $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ in the face of non-parametric "channel-model" uncertainty and to a nominal $\mathcal{H}_{\infty}$ estimation problem. The estimators considered here are defined by minimizing the worst-case squared estimation error over the "uncertainty set" and by minimizing an average cost under the constraint that the worst-case error of any admissible estimator does not exceed a prescribed value. The main point is to explore the derivation of estimators which may be viewed as less conservative alternatives to minimax estimators, or in other words, that allow for trade-offs between worst-case performance and better performance over "large" subsets of the uncertainty set. The "average costs" over $\mathcal{H}_{2}-$signal balls are obtained as limits of averages over sets of finite impulse responses, as their length grows unbounded. The estimator design problems for the two types of estimators and the three problems addressed here are recast as semi-definite programming problems (SDPs, for short). These SDPs are solved in the case of simple examples to illustrate the potential of the "average cost/worst-case constraint" estimators to mitigate the inherent conservatism of the minimax estimators.