Authors

Bastian Biedermann, Thomas Meurer

Abstract

A flatness-based approach for motion planning for a class of boundary controlled port-Hamiltonian systems with distributed parameters is presented. The goal is to achieve open-loop output tracking or finite-time transitions between steady states or operating profiles. Introducing new (fictious) boundary conditions in terms of so-called flat outputs, the port-Hamiltonian system is reformulated as a Cauchy problem in the spatial domain. The parametrization of any system variable and input by the flat output is computed using two different solution approaches. By assigning a suitable desired trajectory for the flat output, the input parametrization yields the feed forward control law to solve the motion planning task. The presented theory is applied to the wave equation with spatially varying parameters and is evaluated by numerical calculations and simulations.

Keywords

Motion Planning; Port-Hamiltonian System; Boundary Control; Trajectory Planning; Distributed Parameter System; Partial Differential Equation; Wave Equation

Citation

  • Journal: IFAC-PapersOnLine
  • Year: 2020
  • Volume: 53
  • Issue: 2
  • Pages: 7710–7715
  • Publisher: Elsevier BV
  • DOI: 10.1016/j.ifacol.2020.12.1516
  • Note: 21st IFAC World Congress- Berlin, Germany, 11–17 July 2020

BibTeX

@article{Biedermann_2020,
  title={{Motion planning for a class of boundary controlled 1D port-Hamiltonian systems}},
  volume={53},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2020.12.1516},
  number={2},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Biedermann, Bastian and Meurer, Thomas},
  year={2020},
  pages={7710--7715}
}

Download the bib file

References

  • Abate, J. & Whitt, W. A Unified Framework for Numerically Inverting Laplace Transforms. INFORMS Journal on Computing vol. 18 408–421 (2006) – 10.1287/ijoc.1050.0137
  • Andrej, J. & Meurer, T. Flatness-based constrained optimal control of reaction-diffusion systems. 2018 Annual American Control Conference (ACC) 2539–2544 (2018) doi:10.23919/acc.2018.8431201 – 10.23919/acc.2018.8431201
  • DaCunha, J. J. Transition matrix and generalized matrix exponential via the Peano-Baker series. Journal of Difference Equations and Applications vol. 11 1245–1264 (2005) – 10.1080/10236190500272798
  • Duindam, Port-Hamiltonian systems. (2009)
  • Dunbar, Motion Planning for a nonlinear Stefan Problem. European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations (2003)
  • Fliess, M., Levine, J., Martin, P. & Rouchon, P. A Lie-Backlund approach to equivalence and flatness of nonlinear systems. IEEE Transactions on Automatic Control vol. 44 922–937 (1999) – 10.1109/9.763209
  • FLIESS, M., LÉVINE, J., MARTIN, P. & ROUCHON, P. Flatness and defect of non-linear systems: introductory theory and examples. International Journal of Control vol. 61 1327–1361 (1995) – 10.1080/00207179508921959
  • Horváth, G., Horváth, I., Almousa, S. A.-D. & Telek, M. Numerical inverse Laplace transformation using concentrated matrix exponential distributions. Performance Evaluation vol. 137 102067 (2020) – 10.1016/j.peva.2019.102067
  • Jacob, (2012)
  • Laroche, B., Martin, P. & Rouchon, P. Motion planning for the heat equation. International Journal of Robust and Nonlinear Control vol. 10 629–643 (2000) – 10.1002/1099-1239(20000715)10:8<629::aid-rnc502>3.0.co;2-n
  • Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
  • Lynch, Flatness-based boundary control of a class of quasilinear parabolic distributed parameter systems. International Journal of Control (2002)
  • Meurer, T. Flatness-based trajectory planning for diffusion–reaction systems in a parallelepipedon—A spectral approach. Automatica vol. 47 935–949 (2011) – 10.1016/j.automatica.2011.02.004
  • Meurer, (2012)
  • Meurer, T. & Kugi, A. Trajectory Planning for Boundary Controlled Parabolic PDEs With Varying Parameters on Higher-Dimensional Spatial Domains. IEEE Transactions on Automatic Control vol. 54 1854–1868 (2009) – 10.1109/tac.2009.2024572
  • Meurer, T. & Zeitz, M. Feedforward and Feedback Tracking Control of Nonlinear Diffusion−Convection−Reaction Systems Using Summability Methods. Industrial & Engineering Chemistry Research vol. 44 2532–2548 (2004) – 10.1021/ie0495729
  • Petit, N. & Rouchon, P. Dynamics and solutions to some control problems for water-tank systems. IEEE Transactions on Automatic Control vol. 47 594–609 (2002) – 10.1109/9.995037
  • Rathgeber, Firedrake: automating the finite element method by composing abstractions. ACM Trans. Math. Softw. (2016)
  • Rothfuß, (1997)
  • Rothfuss, R., Rudolph, J. & Zeitz, M. Flatness based control of a nonlinear chemical reactor model. Automatica vol. 32 1433–1439 (1996) – 10.1016/0005-1098(96)00090-8
  • Rudolph, J. & Woittennek, F. Motion planning and open loop control design for linear distributed parameter systems with lumped controls. International Journal of Control vol. 81 457–474 (2008) – 10.1080/00207170701561435
  • Schorkhuber, B., Meurer, T. & Jungel, A. Flatness of Semilinear Parabolic PDEs—A Generalized Cauchy–Kowalevski Approach. IEEE Transactions on Automatic Control vol. 58 2277–2291 (2013) – 10.1109/tac.2013.2256013
  • van der Schaft, A. Port-Hamiltonian systems: an introductory survey. Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006 1339–1365 (2007) doi:10.4171/022-3/65 – 10.4171/022-3/65
  • Woittennek, On flatness and controllability of simple hyperbolic distributed parameter systems. IFAC Proceedings (2011)