MIT Department of Electrical Engineering & Computer Science

Topics in Robust Estimation and Adaptive Filtering

Babak Hassibi
Stanford University

Abstract

Robust estimation is concerned with the design of estimators that have acceptable performance in the face of model uncertainties and lack of statistical information, and can be considered an outgrowth and extension of (the now classical) LQG theory which assumed perfect models and complete statistical knowledge. It has particular significance in adaptive signal processing where one needs to cope with time-variation of system parameters and to compensate for lack of a priori knowledge of the statistics of the input data and disturbances.

We have recently studied adaptive filtering using the so-called H-infinity approach to robust estimation and have shown that the celebrated LMS (Least-Mean Squares) adaptive algorithm is H-infinity optimal. This result solves the long standing issue of finding a rigorous basis for LMS (which was long thought to be an approximate least-squares solution). It also suggests some further ramifications, such as the design of robust adaptive filters with more desirable tracking properties, as well as some directions for further research, such as the mixed H2/H-infinity problem.

Despite the "fundamental differences" between the philosophies of the H-infinity and LQG approaches to control and estimation, there are striking "formal similarities" between the controllers and estimators obtained from these two methodologies. In an attempt to explain these similarities, we shall describe a new approach to H-infinity estimation (and control), different from the existing (e.g., interpolation-theoretic-based, game-theoretic-based, etc.) approaches, that is based upon setting up estimation (and control) problems, not in the usual Hilbert space of random variables, but in an indefinite (so-called Krein) space.

The Krein space formulation provides a unified approach for problems in LQG, H-infinity, risk-sensitive, and game-theoretic estimation and control, and therefore allows one to use the insight obtained from over three decades of work in traditional LQG theory to obtain new results in these other areas. We shall mention a few of these generalizations here, such as (possibly) numerically superior square-root algorithms and fast Chandrasekhar algorithms for H-infinity problems, and some new investigations on the asymptotic behaviour of H-infinity filters.