A probabilistic approach to multivariable robust filtering and open-loop control

A new approach to robust filtering, prediction, and smoothing of discrete-time signal vectors is presented. Linear time-invariant filters are designed to be insensitive to spectral uncertainty in signal models. The goal is to obtain a simple design method, leading to filters which are not overly conservative. Modeling errors are described by sets of models, parameterized by random variables with known covariances. These covariances could either be estimated from data or be used as robustness “tuning knobs.” A robust design is obtained by minimizing the ℋ₂-norm or, equivalently, the mean square estimation error, averaged with respect to the assumed model errors. A polynomial solution, based on an averaged spectral factorization and a unilateral Diophantine equation, is derived. The robust estimator is referred to as a cautious Wiener filter. It turns out to be only slightly more complicated to design than an ordinary Wiener filter. The methodology can be applied to any open-loop filtering or control problem. In particular, we illustrate this for the design of robust multivariable feedforward regulators, decoupling and model matching filters