RESEARCH ON POLYNOMIAL METHODS FOR THE H-2 OPTIMAL DESIGN OF FILTERS AND CONTROLLERS

Signals and Systems, Uppsala University

Researchers: Mikael Sternad, Anders Ahlén .

Previous Ph.D. students: Lars-Johan Brännmark , Adrian Bahne , Annea Barkefors , Kenth Öhrn , Lars Lindbom , Claes Tidestav , Erik Lindskog , Simon Widmark.

In particular, Wiener and Kalman techniques for model-based filter design are versatile tools. Wiener methods are based on input-output models and frequency domain design; Kalman design is based on state-space 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 state-space 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 state-space models. The use of filters in input-output 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 time-varying dynamics. Furthermore, a drawback with polynomial methods, as compared to state-space based techniques, has previously been their higher numerical sensitivity in some (high-order) 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 pre-compensation.

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 time-domain constraints on the "pre-ringing" of impulse responses.

Research Results:

We have developed orthogonality-based methods for deriving polynomial design equations, for nominal and robust filters as well as for LQG-controllers. 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 discrete-time signal models.

In our investigations, we strive for a minimal number of numerically well-behaved design equations. (In particular, the numerically sensitive operation of coprime factorizations is avoided in multivariable problems.) Such aspects are of particular importance in on-line 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 time-varying 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:

H-2 design of nominal and robust discrete-time filters (1996)

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

LQ controller design and self-tuning 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 open-loop 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 Lars-Johan 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 constant-gain 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 D-AMPS 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 space-time processing and equalization for wireless communications.

Paper on the structure and design of realizable MIMO Decision Feedback Equalizers (IEEE-SP)

Data-based 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:

Anti-windup compensator design for multivariable systems

A derivation methodology for polynomial-LQ controller design (IEEE AC 93)

Self-tuning LQG regulators with disturbance measurement feedforward (IJC 91)

Scalar LQG regulators with disturbance measurement feedforward (Automatica 88)