Uppsala universitet


Signals and Systems, Uppsala University

Researchers: Mikael Sternad, Anders Ahlén .

Previous Ph.D students: Lars-Johan Brännmark , Kenth Öhrn , Lars Lindbom and Simon Widmark .

We are never able to guarantee in advance that any model-based algorithm will work in practice. Irrespective of how the models used for signal processing and control are obtained, they will be imperfect.

A way to reduce the risk somewhat is to construct reliable, robust, filters and controllers. The design is then based not only on nominal design models, but also on specified sets of deviations between the models and the true systems. Such sets are called error models, or uncertainty models.

We have spent considerable effort on robust design, focusing on methods to handle spectral uncertainty in linear models of dynamic systems. Techniques have been developed for Wiener filtering, Kalman filtering, feedforward control and for the design of adaptation laws.

A Probabilistic Robust Design of Filters and of Feedforward Controllers

Primarily, we have refined a methodology for robust design based on probabilistic descriptions of model errors. The PhD Thesis by Kenth Öhrn contains a complete description of the methodology. The aim is to obtain filters and controllers which are insensitive to spectral uncertainties. A guiding principle is that large but unlikely model uncertainties should be taken into account, but they should not be allowed to dominate the design. Therefore, we minimize the mean square estimation (or control) error, averaged with respect to possible model errors.

This is in contrast to minimax design which optimizes the worst case performance. A minimax design requires model error sets with "hard" guaranteed bounds, which rarely exist in practice. It also often results in a significant loss of performance in situations where the model is nearly correct.

[Plot of MSE versus Rho, for different filters]
In the example above, five different filters are applied to a measurement signal. The filters are designed in different ways, based on a linear design model with an uncertain parameter (rho). The parameter may have values from -0.24 to +0.24 with equal probability. The estimation error variance (MSE) is plotted for different true values of the parameter:

  • The dash-dotted horizontal line represents the trivial zero estimate (always estimating the signal to be zero).
  • A nominal Wiener or Kalman estimator designed for rho=0 (dashed) could perform much worse.
  • A filter based on the probabilistic robust design (solid) will provide the minimal average MSE.
  • A minimax design (upper dotted) instead minimizes the worst performance, at rho=-0.24, but turns out to be rather conservative.
  • Finally, the lower dotted line indicates the performance attainable for a known parameter value.

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Uncertainty Modelling

Technically, we represent signal and system parameter deviations as random variables, with known covariances. These covariances could either be estimated from data, or be used as robustness ``tuning knobs". A robust design is then obtained by minimizing the squared estimation or control error, averaged both with respect to model errors and with respect to the noise.

<cite> u(k) = </cite> [ <cite> F<sub>o</sub> + Delta F </cite> ] 
   <cite> e<sub>u</sub>  </cite>
The uncertainty model can, for example, be used to describe the effect of slow time-variations. This is useful in equalizer design for time-varying radio channels. The error models can capture properties of parametric as well as nonparametric (frequency domain) uncertainties.

It is also of interest to investigate the properties of models and filters based on short data records, such as the training sequences used in digital mobile radio communications. This has been done in the Licentiate Thesis by S. Bigi.

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Robust Design

One focus of the research has been to develop a polynomial equations approach for robust design, based on averaged spectral factorizations and averaged Diophantine equations. Linear time-invariant dynamic systems are then described by transfer functions, parameterized by polynomial (matrix) fractions. As error models, we use sets of additive transfer functions, having stochastic numerators and fixed denominators. The robust design then turns out to be no more complicated than the design of an ordinary Wiener filter or polynomial LQG regulator.

State-space design of robustified Kalman estimators has been considered as well as the related robustification of Kalman-based adaptation algorithms. The robust design will then imply an expansion of the state vector and a modification of the covariance matrices used in the filter design.

Robust design along the principles outlined above has been applied in MAP deconvolution of signals for ultrasonic nondestructive evaluation of materials. Robustness against inaccurate impulse responses or position errors in the multiple transducer setup was treated by letting the model of the unknown system belong to an uncertainty set of possible models. See the PhD Thesis by Tomas Olofsson.

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Robust Model-based Design of Adaptive Filters

Adaptive algorithms are used for adjusting the coefficients of models, filters and of controllers. In our research on adaptive filtering, we have developed low-complexity algorithms which track fast time-variations of signal dynamics.

A key concept is to utilize ARMA-like models for the assumed dynamics of the time variations. The performance of the resulting adaptation laws is sometimes sensitive to the underlying model assumptions. As outlined in the PhD Thesis by Lars Lindbom, a probabilistic robust design can be utilized to lower the sensitivity.

See also a Conference Paper on this subject at the IFAC Workshop on Adaptation and Learning in Control and Signal Processing, Como, Italy, August 2001.

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Robust Model-based Design of Audio Precompensation Filters

In audio signal processing, it is of interest to design pre-compensation filters that counteract undesired influences of the loudspeaker dynamic and undesired room acoustics of the listening room. In model-based design, robust feedforward control as described above can then be used to take modelling errors into account.

An additional problem is that the acoustic response is measured in a finite number of measurement points, while we wish to control the properties of the sound field also in-between the measurement points. Stochastic uncertainty modelling can then be used to describe the expected deviations of the impulse responses in such locations from the measured impulse responses at the measurement points. The extended design model can then be used in a linear-quadratic optimization of a feedforward precompensation filter.

This application and design is described in the paper L-J. Brännmark, "Robust Audio Precompensation with Probabilistic Modeling of Transfer Function Variability", 2009 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA 2009), Oct. 2009 New York City, NY.
In Pdf (650K).

Main References

PhD Thesis by Simon Widmark (2018)
PhD Thesis by Lars-Johan Brännmark (2011)
PhD Thesis by Kenth Öhrn (1996)
Licentiate Thesis by Stefano Bigi (1995)
Book chapter in Grimble and Kucera eds. (1996)
Multivariable robust filtering and open-loop control (IEEE AC 95)
Scalar polynomial filtering and open-loop control (Automatica 93)
State-space observer design (ECC 95)
Robust equalizers for time-varying channels (NRS 93)
Robust DFE's, or decision feedback equalizers (ICASSP 93)
Controller design: Improved performance robustness of feedforward controllers (Reglermöte 92)