RESEARCH ON ROBUST FILTERING AND CONTROL
Signals and Systems, Uppsala University
Anders Ahlén ,
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.
PhD Thesis by Kenth Öhrn
contains a complete description of the methodology.
The aim is to obtain filters and controllers which are insensitive to
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.
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
- 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.
Top of page
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.
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.
Top of page
One focus of the research has been to develop a
polynomial equations approach for robust design,
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
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.
PhD Thesis by Tomas Olofsson.
Top of page
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
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
by Lars Lindbom, a
probabilistic robust design can be utilized to lower the
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.
Top of page
Robust Model-based Design of Audio Precompensation Filters
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
"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).
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
State-space observer design (ECC 95)
Robust equalizers for time-varying channels (NRS 93)
Robust DFE's, or decision feedback equalizers (ICASSP 93)
Improved performance robustness of feedforward controllers