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PhD-thesis on Numerical Differentiation

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  • PhD-thesis on Numerical Differentiation

    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
    Uppsala, Sweden. EMAIL: BC@SIGURD.SUNET.SE

    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.