
RESEARCH ON POLYNOMIAL METHODS
FOR THE H2 OPTIMAL
DESIGN OF FILTERS AND CONTROLLERS
Signals and Systems, Uppsala University
Researchers:
Mikael Sternad,
Anders Ahlén
.
Previous Ph.D. students:
LarsJohan Brännmark
,
Adrian Bahne
,
Annea Barkefors
,
Kenth Öhrn
,
Lars Lindbom
,
Claes Tidestav
,
Erik Lindskog
,
Simon Widmark.
The use of linear models, in the form of impulse responses,
transfer functions or statespace models, is a central feature
of modern signal processing and control theory.
When such models are available, a wide variety of
powerful methods can be called upon for analysis
and design.
In particular,
Wiener and Kalman techniques for modelbased filter design
are versatile tools.
Wiener methods are based on inputoutput models and
frequency domain design; Kalman design is based on
statespace models and Riccati equations.
The Polynomial Approach to the Design
of Linear Filters and Controllers

The polynomial approach is an algebraic development of the
Wiener methodology.
Polynomial methods were originally developed with control applications in mind, but have turned out to be very useful
also within digital signal processing and communications.
In a polynomial framework, numerators and
denominators of transfer functions are handled separately,
as polynomials. The design equations for linear filters
and controllers then take the form of
linear polynomial equations (Diophantine equations),
polynomial spectral factorizations and coprime factorizations.
Polynomial techniques are a useful
complement to statespace methodologies.
Over a span of several years, we have found the polynomial
approach to be convenient when deriving new types of filters.
It has also helped us to understand how various properties of
models influence the resulting designs.
With a polynomial framework,
the influence of the model structure on qualitative filter
properties becomes more transparent
than when using statespace models.
The use of filters in inputoutput form furthermore provide immediate
physical insight: A quick inspection of poles and zeros
roughly indicate what properties can be expected.
State space techniques are often more convenient
when models have timevarying dynamics.
Furthermore,
a drawback with polynomial methods, as compared to
statespace based techniques, has previously
been their higher numerical
sensitivity in some (highorder) problems.
However, numerically stable implementations,
for example in the
polynomial toolbox for use with Matlab,
are now available.
For example, we have routinely calculated
Wiener inverses for systems of order 3000 in our work
loudspeaker precompensation.
Polnomial methods for fedforward controller
design and robust design form a cornerstone of our research
on audio signal processing.
Here, we have e.g. generaliszed the polynomial methods for
feedforward control and Wiener filtering to work under
quadratic (power) constraints
on filters, filtering errors and
control errors, and also under
timedomain constraints
on the "preringing" of impulse responses.
Research Results:

We have developed orthogonalitybased
methods for deriving polynomial design equations,
for nominal and robust
filters
as well as for
LQGcontrollers.
These and related tools have been utilized in the design of optimal
and robust filters and controllers, mainly based on
quadratic criteria and on stochastic discretetime signal models.
In our investigations,
we strive for a minimal number of
numerically wellbehaved design equations.
(In particular, the numerically sensitive operation
of coprime factorizations is avoided in multivariable problems.)
Such aspects are of particular importance in online
applications, such as
adaptive
filtering and control,
and in our development of design tools for
robust
filtering and control,
as exemplified by the
PhD Thesis by Kenth Öhrn.
The solution of Diophantine equations becomes
trivial in some types of problems;
in others, such as the design of
Decision Feedback Equalizers ,
spectral factorizations can be avoided.
The Decision feedback equalizer solution has found use
in our research on digital mobile communications, see
the PhD theses by
Claes Tidestav
and by
Erik Lindskog.
An application where the use of polynomial methods has turned
out to be fruitful is the construction of
adaptation laws
for tracking
timevarying parameters of linear regression models.
A family of algorithms with low computational complexity,
and close to optimal performance, has been presented in the
PhD Thesis
by Lars Lindbom.
The use of polynomial methods have here turned out to be
fruitful in the analysis, as well as in the design,
of adaptation laws.
Links to our main references in which polynomial methodologies are
developed, described and utilized
are listed below.

Summaries in Book Chapters:

H2 design of nominal and robust discretetime filters (1996)

Derivation
and design of Wiener filters using polynomial equations (1994)

LQ controller design and selftuning control (1993)

Signal Processing:

Causal IIR precompensator filters subject to quadratic constraints, in IEEE TASLP 2018.

Direct and Iterative solution of Diophantine
filter design equations.

Multivariable robust filtering
and openloop control (IEEE AC 95)

Robust filtering based on probabilistic
descriptions of model errors (Automatica 93)

Wiener filter design
based on polynomial equations (IEEE SP 91)

Optimal differentiation based on stochastic signal models
(IEEE SP 91)

Adaptive deconvolution based on spectral decomposition
(SPIE Conf. 91)

Scalar deconvolution filters, predictors and smoothers
(IEEE ASSP 89)

Audio Systems:
(See also our
audio research page)

PhD Thesis by LarsJohan Brännmark 2011
on robust sound field control for audio reproduction.

Licentiate Thesis by Annea Barkefors 2014
on active noise control.

PhD Thesis by Adrian Bahne 2014
on multichannel audio signal processing.

PhD Thesis by Simon Widmark 2018
on causal MMSE filters for personal audio.
 Adaptation and tracking:
 Paper 1
on design of general constantgain adaptation algorithms.
 Paper 2
on analysis of stability and performance.
 Paper 3
on the Wiener LMS adaptation algorithm (a special case).
 Paper 4
on a case study on DAMPS 1900 channels.
 Conference paper
at the European Control Conference, 2001.
 Digital Communications:

PhD Thesis by Claes Tidestav, on the multivariable
Decision Feedback Equalizer for
multiuser detection and interference rejection.

PhD Thesis by Erik Lindskog, on
spacetime processing and equalization for
wireless communications.

Paper on the structure and design of realizable
MIMO Decision Feedback Equalizers (IEEESP)

Databased DFE design, investigated in the Licentiate Thesis of Stefano Bigi.

Robust DFE's, or decision feedback equalizers (ICASSP 93)

Decision Feedback Equalizers for IIR channels with colored noise
(IEEE IT 90)
 Control Systems Design:

Antiwindup compensator design for multivariable systems

A derivation methodology for polynomialLQ controller design
(IEEE AC 93)

Selftuning LQG regulators with disturbance measurement
feedforward
(IJC 91)

Scalar LQG regulators with
disturbance measurement feedforward (Automatica 88)
