Diophantine Equations for k-Step Ahead Prediction and Fixed Lag Smoothing: Recursive and Analytical Solutions

Lars Lindbom , Anders Ahlén and Mikael Sternad

Europoly Workshop, Glasgow, Scotland, April 15-16, 1999.

Abstract:

Diophantine equations constitute an important conceptual tool for solving problems which are formulated in the polynomial systems framework. Besides spectral factorization, Diophantine equations appear in most H2-optimal (MSE) solutions. It is therefore essential that the numerical procedures, used to solve these equations, provide adequate performance.

Although reliable and efficient algorithms exist, it is of great interest to take advantage of situations where analytical and recursive solutions can be obtained. For signals which can be modelled as multi-variable ARMA-processes, contaminated by either white or coloured noise, we present a set of predictors, filters and fixed-lag smoothers which can be obtained with a minimal use of numerical procedures.

If the noise is coloured, the Diophantine equation, associated with the signal estimation problem, has to be solved numerically for one value of the prediction horizon, or the smoothing lag. Once the solution is obtained, the whole set of predictors and smoothers, up to a predefined prediction horizon or smoothing lag, can be obtained by analytical expressions, based on the previously obtained numerical solution.

If the noise is white, no numerical solution at all will be required.

Related publications:

PhD Thesis by Lars Lindbom, deriving these results
and applying them to the design of adaptive filters.
Paper on design of adaptation algorithms, with a derivation and motivation of the equations that were presented here.
Book Chapter on H2-design of model-based nominal and robust filters.