Uppsala universitet
Design of Multivariable Cautious Discrete-time Wiener Filters:
A Probabilistic Approach

Kenth Öhrn

PhD Thesis, Uppsala University, 248pp, ISBN 91-506-1157-7, May 1996.

Chapter 1 of the Thesis available in Pdf.

Paper copies of the whole thesis can be obtained from Ylva Johansson, Signals and Systems Group, Uppsala University, Box 534, SE-75121 Uppsala, Sweden.

Can uncertain dynamic models be used for linear filter design in a systematic way? The answer is yes, if appropriate descriptions of the model uncertainty are available. The thesis develops a design method based on probabilistic descriptions of the uncertainty, and on the minimization of averaged mean square error criteria.

A new approach to robust filtering, prediction, smoothing and open-loop control 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. Modelling errors are described by sets of time-invariant 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 H-2 norm, averaged with respect to the assumed model errors. A polynomial matrix solution, based on an averaged spectral factorization and a Diophantine equation, is derived. The robust filters are referred to as cautious filters. The filters turn out to be not more complicated to design than the ordinary filters.

The main effort is put into the design of cautious multivariable Wiener filters. However, also robust open-loop control, such as the design of robust multivariable feedforward regulators, decoupling and model matching filters is considered. Furthermore, robust Kalman filters, predictors and smoothers for state estimation are designed in the same spirit, using time-invariant probabilistic error models. These state estimators can be used as adaptation algorithms for tracking the parameters of linear regression models. A systematic way is thus obtained for utilizing uncertain a priori information in the design of Kalman-based adaptation laws.

1. Summary and Introduction
2. Probabilistic Error Models
3. Obtaining Error Models
4. Robust Multivariable H2 Estimation
5. Robust Multivariable H2 Feedforward Control
6. Robust State Estimation
7. A Concluding Example
8. Conclusions

Related publications:
Paper in IEEE Trans. AC 1995, describing the multivariable Wiener solution.
State-space design and comparison to minimax H-2, European Control Conf. 1995.
SISO filtering, feedforward control and uncertainty modelling, Automatica 1993.
Robust decision feedback equalizers , IEEE ICASSP'93.
PhD thesis by Lars Lindbom on model-based design of adaptive filters.

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