Learning Data-Driven PCHD Models for Control Engineering Applications

Authors

Annika Junker, Julia Timmermann, Ansgar Trächtler

Abstract

The design of control engineering applications usually requires a model that accurately represents the dynamics of the real system. In addition to classical physical modeling, powerful data-driven approaches are increasingly used. However, the resulting models are not necessarily in a form that is advantageous for controller design. In the control engineering domain, it is highly beneficial if the system dynamics is given in PCHD form (Port-Controlled Hamiltonian Systems with Dissipation) because globally stable control laws can be easily realized while physical interpretability is guaranteed. In this work, we exploit the advantages of both strategies and present a new framework to obtain nonlinear high accurate system models in a data-driven way that are directly in PCHD form. We demonstrate the success of our method by model-based application on an academic example, as well as experimentally on a test bed.

Keywords

PCHD; passivity; hybrid modeling; system identification; nonlinear control

Citation

• Journal: IFAC-PapersOnLine
• Year: 2022
• Volume: 55
• Issue: 12
• Pages: 389–394
• Publisher: Elsevier BV
• DOI: 10.1016/j.ifacol.2022.07.343
• Note: 14th IFAC Workshop on Adaptive and Learning Control Systems ALCOS 2022- Casablanca, Morocco, June 29 – July 01, 2022

BibTeX

@article{Junker_2022,
  title={{Learning Data-Driven PCHD Models for Control Engineering Applications}},
  volume={55},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2022.07.343},
  number={12},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Junker, Annika and Timmermann, Julia and Trächtler, Ansgar},
  year={2022},
  pages={389--394}
}

Download the bib file

References

• Anderson B (1968) A simplified viewpoint of hyperstability. IEEE Trans Automat Contr 13(3):292–294. https://doi.org/10.1109/tac.1968.1098910 – 10.1109/tac.1968.1098910
• Brunton SL, Brunton BW, Proctor JL, Kutz JN (2016) Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control. PLoS ONE 11(2):e0150171. https://doi.org/10.1371/journal.pone.0150171 – 10.1371/journal.pone.0150171
• Brunton SL, Proctor JL, Kutz JN (2016) Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc Natl Acad Sci USA 113(15):3932–3937. https://doi.org/10.1073/pnas.1517384113 – 10.1073/pnas.1517384113
• Byrnes CI, Isidori A, Willems JC (1991) Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems. IEEE Trans Automat Contr 36(11):1228–1240. https://doi.org/10.1109/9.100932 – 10.1109/9.100932
• Gillis N, Karow M, Sharma P (2019) Approximating the nearest stable discrete-time system. Linear Algebra and its Applications 573:37–53. https://doi.org/10.1016/j.laa.2019.03.014 – 10.1016/j.laa.2019.03.014
• Gillis N, Sharma P (2017) On computing the distance to stability for matrices using linear dissipative Hamiltonian systems. Automatica 85:113–121. https://doi.org/10.1016/j.automatica.2017.07.047 – 10.1016/j.automatica.2017.07.047
• Junker, Data-driven models for control engineering applications using the Koopman operator (accepted). (2022)
• Khalil, (2015)
• Koopman, Hamiltonian systems and transformations in Hilbert space. (1931)
• Korda M, Mezić I (2018) Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control. Automatica 93:149–160. https://doi.org/10.1016/j.automatica.2018.03.046 – 10.1016/j.automatica.2018.03.046
• Kotyczka, Parametrization of IDA-PBC by assignment of local linear dynamics. (2009)
• Mamakoukas, G., Abraham, I., and Murphey, T.D. (2020). Learning data-driven stable Koopman operators. arXiv: 2005.04291.
• Maschke B, Ortega R, Van Der Schaft AJ (2000) Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Trans Automat Contr 45(8):1498–1502. https://doi.org/10.1109/9.871758 – 10.1109/9.871758
• Maschke, Port-controlled Hamiltonian systems: Modelling origins and systemtheoretic properties. IFAC Proceedings (1992)
• Ortega R, van der Schaft A, Maschke B, Escobar G (2002) Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38(4):585–596. https://doi.org/10.1016/s0005-1098(01)00278-3 – 10.1016/s0005-1098(01)00278-3
• Popov, The solution of a new stability problem for controlled systems. Avtomat. i Telemekh. (1963)
• Popov, Hyperstability of control systems. Grundlehren der mathematischen Wis-senschaften. Editura Academiei (1973)
• Proctor JL, Brunton SL, Kutz JN (2016) Dynamic Mode Decomposition with Control. SIAM J Appl Dyn Syst 15(1):142–161. https://doi.org/10.1137/15m1013857 – 10.1137/15m1013857
• SCHMID PJ (2010) Dynamic mode decomposition of numerical and experimental data. J Fluid Mech 656:5–28. https://doi.org/10.1017/s0022112010001217 – 10.1017/s0022112010001217
• Sepulchre, (1997)
• Williams MO, Kevrekidis IG, Rowley CW (2015) A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition. J Nonlinear Sci 25(6):1307–1346. https://doi.org/10.1007/s00332-015-9258-5 – 10.1007/s00332-015-9258-5