View Full Version : PhD-thesis on Numerical Differentiation

Herman J. Woltring
12-07-1989, 05:47 AM
Dear Biomch-l readers,

Yesterday, I received a copy of Bengt Carlsson's PhD-thesis from Uppsala in
Sweden. This is the most recent result of a long-standing research programme
in optimal numerical differentiation of noisy data, originally started by Lars
Gustaffson & Haakan Lanshammar in the context of gait analysis and inverse
dynamic modelling (ENOCH - An integrated system for measurement and analysis of
human gait; PhD-thesis, Report UPTEC 7723R, Uppsala University, Uppsala, Sweden
1977). If you wonder why I should mention this: when I sent my own PhD-thesis
to Haakan in 1977, he replied with a letter explaining that he had valid rea-
sons for not being present at my public defense as apparent from his enclosure.
The enclosed thesis indicated that he was performing his public defense at the
same day as I was doing mine ...

Published versions of part of Haakan's work are available as

(1) On practical evaluation of differentiation techniques for human gait
analysis. Journal of Biomechanics 15(1982), 99-105.

(2) On precision limits for derivatives calculated from noisy data.
Journal of Biomechanics 15(1982), 459-470.

Hans Furn'ee's thesis recently announced on Biomch-L was largely influenced by
this work. One point of interest raised at Hans' public examination is the dif-
ference between accuracy and precision or resolution. Accuracy refers to bias-
es, i.e. the difference between a true value and an expected, observed value
(e.g., the mean of a large number of measurements); precision refers to the
(usually root-mean-square) difference between an observed and expected mean
value. It is precision which is addressed in Lanshammar's model for noise
influence assessment in differentiation, not accuracy. Thus, wide-band meas-
urement noise is accomodated by these models, but correlated skin-motion arte-
facts when measuring movement from externally affixed markers are not.

Bengt Carlsson informed me today that he has some copies of his thesis for
interested Biomch-L readers.

Herman J. Woltring, Eindhoven.



By: Bengt Carlsson, Automatic Control and Systems Analysis Group,
Department of Technology, Uppsala University, Box 534, S-751 21

Doctoral dissertation to be publicly examined in room 047 at the
Department of Technology, Uppsala University, on December 22, 1989
at 10.15 a.m., for the degree of Doctor of Technology.
Opponent: Professor Lennart Ljung, Linkoepings Univ., Sweden.


Carlsson, B., 1989. Digital differentiating filters and model based fault
detection. Acta Univ. Ups., Uppsala Dissertations from the Faculty of
Science 28}, 215pp., Uppsala. ISBN 91-554-2473-2.

This thesis treats two problems in digital signal processing:
Estimation of the derivative of a noise corrupted signal from discrete-time
measurements and the detection of faults in a dynamical system using an
estimated model.

The first part of the thesisis devoted to the design of digital differentiating
filters. General design strategies and characterizations of differentiating
filters are outlined. A design technique for constructing digital differentia-
ting filters based on stochastic signal and noise models is presented. IIR-
filters which minimize the mean square estimation error are calculated from a
spectral factorization and a linear polynomial equation. The filters can be
designed for prediction, filtering and smoothing problems. Furthermore, a
method for optimizing differentiating FIR-filters in the frequency domain is
presented. Illustrations and comparisons with other methods are provided.
The problem of tracking step changes in the true derivative is also studied.
Methods for detecting parameter changes are investigated and applied to the
differentiation problem. In a practical application, the temperature deriva-
tive of a reactor tank in a nuclear power plant is estimated by a simple
adaptive filter.

The second part of the thesis describes some approaches to off-line fault
detection based on estimated models. In practice, the estimated model is
subject to errors due to measurement noise and model misfit. Some statistical
tests based on standard asymptotic theory for system identification are com-
pared and illustrated. Two cases of model misfit are discussed: nonparametric
static nonlinearities and unmodelled linear dynamics. A proposal for a fault
detection procedure which accounts for undermodelling is given.

Key-words: Digital differentiators, Numerical differentiation,
Estimation, Filter design, Fault detection.