Abstract
We present a systematic approach to realize the precise observer-based trajectory tracking of the tip of a flexible beam using the energy-based port-Hamiltonian (PH) system representation. The first design steps are the structure-preserving spatial discretization by means of a pseudo-spectral method and the structure-preserving order reduction. The model structure is exploited for inversion-based feedforward control, and the control loop is closed via an observer for the friction torque and the state difference. Experimental results for the reference input and the disturbance response illustrate the quality of the design.
Zusammenfassung
Wir stellen ein systematisches Vorgehen vor, mit dem unter Ausnutzung der energiebasierten port-Hamiltonschen (PH) Systemdarstellung die präzise beobachterbasierte Trajektorienfolge der Spitze eines flexiblen Balkens realisiert wird. Entwurfsschritte sind zunächst die strukturerhaltende Ortsdiskretisierung mittels einer Pseudo-Spektralmethode sowie die ebenso strukturerhaltende Ordnungsreduktion. Die Modellstruktur wird für die inversionsbasierte Vorsteuerung ausgenutzt und der Regelkreis über einen Beobachter für das Reibmoment und die Zustandsdifferenz geschlossen. Messergebnisse zum Führungs- und Störverhalten illustrieren die Güte des Entwurfs.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: 260049780
Funding statement: This research was supported by Deutsche Forschungsgemeinsamschaft (DFG), project number 260049780.
About the authors
Mei Wang was a research assistant at the Chair of Automatic Control, Technical University of Munich, from 2015 to 2018. She was a member of the Energy-based Modeling and Control group and worked on the control of distributed parameter systems based on discretized port-Hamiltonian models.
Paul Kotyczka is adjunct teaching professor (Privatdozent) at the Chair of Automatic Control, Technical University of Munich, where he leads the Energy-based Modeling and Control group. He works on structured modeling, geometric numerical methods and nonlinear control design for multi-domain physical systems, with applications to mechatronic, robotic and process systems.
Appendix A Tables with parameters
The appendix contains tables with the parameters for the drive unit, the steel beam, the structure-preserving discretization and reduction, as well as the design of observers and controllers.
Parameters of the gear motor.
| Parameter | Value | Unit |
| 0.025 | m |
Parameters of the steel beam.
| Parameter | Value | Unit |
| Length | 1.20 | m |
| Height | 0.005 | m |
| Thickness | 0.03 | m |
| Volumetric mass density | 7856 | |
| Young’s modulus | 215 | |
| Poisson’s ratio | 0.28 | – |
| Shear correction factor | – |
Parameters for the spatial discretization and the order reduction of the beam model.
| Parameter | Value |
| Number of collocation points N | 9 |
| Order of the discretized model | 36 |
| Order of the reduced model r | 12 |
| Shift points |
Parameters of the LQG controller and PD controller.
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