Validation of Dynamic Models in the Time-Scale Domain

[+] Author and Article Information
James R. McCusker

Department of Mechanical and Industrial Engineering, University of Massachusetts, Amherst, MA 01003

Kourosh Danai1

Department of Mechanical and Industrial Engineering, University of Massachusetts, Amherst, MA 01003danai@ecs.umass.edu

David O. Kazmer

Department of Plastics Engineering, University of Massachusetts, Lowell, MA 01854


Corresponding author.

J. Dyn. Sys., Meas., Control 132(6), 061402 (Oct 28, 2010) (9 pages) doi:10.1115/1.4002479 History: Received September 26, 2008; Revised April 16, 2010; Published October 28, 2010; Online October 28, 2010

Model validation is the procedure whereby the fidelity of a model is evaluated. The traditional approaches to dynamic model validation consider model outputs and observations as time series and use their similarity to assess the closeness of the model to the process. A common measure of similarity between the two time series is the cumulative magnitude of their difference, as represented by the sum of squared (or absolute) prediction error. Another important measure is the similarity of shape of the time series, but that is not readily quantifiable and is often assessed by visual inspection. This paper proposes the continuous wavelet transform as the framework for characterizing the shape attributes of time series in the time-scale domain. The feature that enables this characterization is the multiscale differential capacity of continuous wavelet transforms. According to this feature, the surfaces obtained by certain wavelet transforms represent the derivatives of the time series and, hence, can be used to quantify shape attributes, such as the slopes and slope changes of the time series at different times and scales (frequencies). Three different measures are considered in this paper to quantify these shape attributes: (i) the Euclidean distance between the wavelet coefficients of the time series pairs to denote the cumulative difference between the wavelet coefficients, (ii) the weighted Euclidean distance to discount the difference of the wavelet coefficients that do not coincide in the time-scale plane, and (iii) the cumulative difference between the markedly different wavelet coefficients of the two time series to focus the measure on the pronounced shape attributes of the time series pairs. The effectiveness of these measures is evaluated first in a model validation scenario where the true form of the process is known. The proposed measures are then implemented in validation of two models of injection molding to evaluate the conformity of shapes of the models’ pressure estimates with the shapes of pressure measurements from various locations of the mold.

Copyright © 2010 by American Society of Mechanical Engineers
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Figure 1

Measured pressure from the mold during an injection molding cycle shown with its estimates by two different models

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Figure 2

Surfaces in the time-scale domain of the Gaussian-smoothed measured pressure in Fig. 1 shown in the top left, and the Gauss and Mexican hat WTs of the measured pressure in the top middle and top right plots, respectively. The bottom plots represent the time series associated with a slice of each surface at the first scale to represent the differential relationship between the surfaces.

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Figure 3

Top view of binary wavelet coefficients, depicted as images, to illustrate the characteristic difference between the Euclidean distance and weighted Euclidean distance

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Figure 4

Regions of marked deviation between the Gauss wavelet coefficients of the measured pressure and its estimates in Fig. 1

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Figure 5

Differential wavelet coefficients, |W{y}−W{ŷ}|, of the measured pressure and its estimates at the marked regions in Fig. 4

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Figure 6

A nonlinear two compartment model of drug kinetics in human body. The circles in this figure depict the compartments.

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Figure 7

Instrumented ASTM test mold and a set of measured pressures from one of the experiments

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Figure 8

Pressure values obtained at five different locations of the mold, shown together with their estimated counterparts by model 3




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