Abstract
The problem of modeling a linear dynamic system is discussed and a novel approach to automatically combine black-box and white-box models is introduced. The solution proposed in this contribution is based on the usage of regularized finite-impulse-response (FIR) models. In contrast to classical gray-box modelling, which often only optimizes the parameters of a given model structure, our approach is able to handle the problem of undermodeling as well. Therefore, the amount of trust in the white-box or gray-box model is optimized based on a generalized cross-validation criterion. The feasibility of the approach is demonstrated with a pendulum example. It is furthermore investigated, which level of prior knowledge is best suited for the identification of the process.
Zusammenfassung
Als Problemstellung wird die Modellierung linearer dynamischer Systeme betrachtet. Hierzu wird ein neuer Ansatz zur automatischen Kombination von datenbasierten und physikalischen Modellen vorgestellt. Der in diesem Aufsatz vorgeschlagene Ansatz basiert auf der regularisierten Schätzung endlicher Impulsantwortmodelle. Im Gegensatz zur klassischen Kombination physikalischer und datenbasierter Modelle, bei denen einzelne Parameter des physikalischen Modells basierend auf Eingangsdaten geschätzt werden, ist unser Ansatz in der Lage, ebenso Fehler in der Modellstruktur des physikalischen Modells zu berücksichtigen. Dafür wird ein Strafterm für die Abweichung vom physikalischen Modell bei der Modellierung des datenbasierten Modells berücksichtigt und die Höhe des Strafterms auf Basis des verallgemeinerten Kreuzvalidierungskriteriums optimiert. Die Machbarkeit des Ansatzes wird an dem Beispiel eines an einer Feder befestigten beweglichen Pendels illustriert. Weiterhin wird untersucht, welcher Umfang an Vorwissen zur Identifikation des Prozesses am Besten geeignet ist.
About the authors
Tobias Münker received his B. Sc. from Universität Siegen in 2010 and his M. Sc. from TU Darmstadt in 2012. After 3 years of industry experience, he has been working towards his Ph. D. under the supervision of Prof. Nelles since 2015. His main research interests are new techniques for the identification of linear and nonlinear systems.
Timm J. Peter graduated with a Master of Science degree from Universität Siegen in 2018. After finishing his masters thesis about regularized FIR models he joined the working group Automatic Control – Mechatronics of Prof. Nelles as a research assistant. His research topics focus on new techniques for linear and nonlinear system identification.
Oliver Nelles is Professor at the University of Siegen in the Department of Mechanical Engineering and chair of Automatic Control – Mechatronics. He received his doctor’s degree in 1999 at the Technical University of Darmstadt. His key research topics are nonlinear system identification, dynamics representations, design of experiments, metamodeling and local model networks.
AppendixParameters of the physical model
For completeness, the parameters of the physical model are listed below. The values from Tab. 3 allow the calculation of the coefficients in Tab. 4.
Physical constants for the cart and pendulum.
| Constant | Value |
| Spring Rate | |
| Motor Armature Resistance | |
| Motor Armature inductance | |
| Motor Torque Constant | |
| Motor Efficiency | |
| Back-Electromotive-Force Constant | |
| Gear Ratio | |
| Planetary Gearbox Efficiency | |
| Rotor Moment of Inertia | |
| Cart Mass | |
| Cart Weight Mass | |
| Motor Pinion Radius | |
| Motor Pinion Number of Teeth | |
| Position Pinion Number of Teeth | |
| Position Pinion Radius | |
| Rack Pitch | |
| Cart Travel | |
| Cart Encoder Resolution | |
| Pendulum Encoder Resolution | |
| Damping Coefficient Engine | |
| Acceleration of Gravity | |
| Cart Mass Total | |
| Pendulum Mass | |
| Pendulum Full Length | |
| Pendulum Distance Center of Gravity | |
| Pendulum Moment of Inertia | |
| Damping Coefficient Pendulum |
Formulas for the computation of the white-box-model coefficients.
| Constant | Formula |
| Denominator | |
| Denominator | |
| Engine Coefficient | |
| Engine Coefficient | |
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