Abstract
Modelica models represent static or dynamic systems. Their outputs can be scalar (numbers) or time-dependent (time series). Most advanced mathematical methods for the analysis of numerical models cannot cope with functional outputs. This paper aims at showing an efficient method to reduce a time-dependent output to a few numbers. The Principal component analysis is a well-established method for dimension reduction and can be used to tackle this issue. It relies however on a linear hypothesis that limits its applicability. We adapt and implement an existing method called the auto-associative model, invented by Stéphane Girard, to overcome this shortcoming. The auto-associative model generalizes PCA, as it projects the data on a nonlinear (instead of linear) basis. It also provides physically interpretable data representations. The difference in efficiency between both methods is illustrated in a case study, the well-known bouncing ball model. We perform output reduction and reconstruction using both methods to compare the completeness of information kept throughout the dimension reduction process. Doi: 10.28991/HIJ-2021-02-01-01 Full Text: PDF
Highlights
The advent of the Functional Mock-Up Interface (FMI) and the emergence of associated tools considerably facilitated the analysis of Modelica models [1] with advanced mathematical methods
It relies on the hypothesis that the variables at hand can be aggregated into linear combinations, which is not true for many dynamic model outputs
We illustrate this issue on a simple case study and show how an alternative approach of more general applicability, the Auto-Associative Model (AAM), allows to overcome it
Summary
The advent of the Functional Mock-Up Interface (FMI) and the emergence of associated tools considerably facilitated the analysis of Modelica models [1] with advanced mathematical methods. Principal Component Analysis (PCA) is by far the most prominent method for dimension reduction. It relies on the hypothesis that the variables at hand can be aggregated into linear combinations, which is not true for many dynamic model outputs. Distances in large dimension spaces lose their discriminating power, especially when the component variables are correlated, which is especially true for discretised time functions. Model emulation ( known as meta-modelling or surrogate modelling) is another technique that cannot cope with high dimensional outputs It consists in substituting a CPU inexpensive mathematical approximation for a numerical model in order to achieve large sample size required for instance by some optimisation techniques, or for Bayesian parameter estimation, or to enable instantaneous interaction with the model. When there is no obvious candidate, principal component analysis allows to automatically build an adapted basis
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