Competitive Robust Estimation for Uncertain Linear Dynamic Models

Abstract

In this paper, two types of robust linear estimation problems for dynamic channel model uncertainty are considered with the aim of characterizing (in computationally effective ways) competitive robust estimators, i.e., robust estimators that improve on the pointwise performance of minimax MSE (mean-squared error) estimators over the uncertain model set, at the expense of a moderate increase in the worst case MSE. The first one corresponds to the minimization of the worst case value of an approximate-regret function defined by a quadratic approximation of the “lower MSE envelope” on the uncertain model set. The second one corresponds to minimizing the nominal MSE error while ensuring that the worst case estimation error does not exceed a prescribed value. For uncertain classes defined by $H_{2}$ -norm balls, it is shown that these two types of estimation problems can be recast as “semidefinite programming problems (SDPs, for short).” Numerical examples are presented for both the case of linear, finite-dimensional model classes (FIRs of a given length) and the case of nonparametric uncertain sets of causal, real-rational frequency-responses, suggesting that these two types of estimators can be attractive alternatives to the min–max MSE estimator. For the case of spectral-norm (in the finite-dimensional case) or $H_{\infty }$ -norm (in the nonparametric case), the worst case MSE and approximate-regret for each candidate estimator are replaced by upper bounds obtained by Lagrangian relaxation and (somewhat conservative) versions of the estimation problems previously mentioned are posed. It is shown that these problems can also be recast as SDPs.