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Trajectory control of an elastic beam based on port-Hamiltonian numerical models

Trajektorienfolgeregelung eines elastischen Balkens auf Basis Port-Hamiltonscher numerischer Modelle
  • Mei Wang

    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.

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    and Paul Kotyczka

    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.

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.

Award Identifier / Grant number: 260049780

Funding statement: This research was supported by Deutsche Forschungsgemeinsamschaft (DFG), project number 260049780.

About the authors

Mei Wang

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

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.

Table 1

Parameters of the gear motor.

ParameterValueUnit
IM2.29·104kgm2
Ih1.69·104kgm2
cS5·104Nm/rad
rh0.025m

Table 2

Parameters of the steel beam.

ParameterValueUnit
Length1.20m
Height0.005m
Thickness0.03m
Volumetric mass density7856kgm3
Young’s modulus215GPa
Poisson’s ratio0.28
Shear correction factor56

Table 3

Parameters for the spatial discretization and the order reduction of the beam model.

ParameterValue
Number of collocation points N9
Order of the discretized model 4N36
Order of the reduced model r12
Shift points s016,18.43,115.6

Table 4

Parameters of the LQG controller and PD controller. Qoa, Qob and Qos: observer weight matrices for the (modeling) state variables of the drive (a), the beam model (b) as well as the disturbance (s), respectively. Q˜cl and Q˜ch: controller weight matrices for the first and second eigenmodes (l) and the remaining high-frequency eigenmodes (h), respectively.

Table 4 Parameters of the LQG controller and PD controller. Qoa{\boldsymbol{Q}_{o}^{a}}, Qob{\boldsymbol{Q}_{o}^{b}} and Qos{\boldsymbol{Q}_{o}^{s}}: observer weight matrices for the (modeling) state variables of the drive (a), the beam model (b) as well as the disturbance (s), respectively. Q˜cl{\tilde{\boldsymbol{Q}}_{c}^{l}} and Q˜ch{\tilde{\boldsymbol{Q}}_{c}^{h}}: controller weight matrices for the first and second eigenmodes (l) and the remaining high-frequency eigenmodes (h), respectively.

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Received: 2020-11-06
Accepted: 2021-02-16
Published Online: 2021-05-27
Published in Print: 2021-06-25

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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