Optimal control of thermodynamic port-Hamiltonian Systems

Authors

Bernhard Maschke, Friedrich Philipp, Manuel Schaller, Karl Worthmann, Timm Faulwasser

Abstract

We consider the problem of minimizing the entropy, energy, or exergy production for state transitions of irreversible port-Hamiltonian systems subject to control constraints. Via a dissipativity-based analysis we show that optimal solutions exhibit the manifold turnpike phenomenon with respect to the manifold of thermodynamic equilibria. We illustrate our analytical findings via numerical results for a heat exchanger.

Keywords

port-Hamiltonian systems; irreversible thermodynamic systems; optimal control; manifold turnpike

Citation

• Journal: IFAC-PapersOnLine
• Year: 2022
• Volume: 55
• Issue: 30
• Pages: 55–60
• Publisher: Elsevier BV
• DOI: 10.1016/j.ifacol.2022.11.028
• Note: 25th International Symposium on Mathematical Theory of Networks and Systems MTNS 2022- Bayreuth, Germany, September 12-16, 2022

BibTeX

@article{Maschke_2022,
  title={{Optimal control of thermodynamic port-Hamiltonian Systems}},
  volume={55},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2022.11.028},
  number={30},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Maschke, Bernhard and Philipp, Friedrich and Schaller, Manuel and Worthmann, Karl and Faulwasser, Timm},
  year={2022},
  pages={55--60}
}

Download the bib file

References

• Alonso, Process systems, passivity and the second law of thermodynamics. Computers & Chemical Engineering (1996)
• Alonso, A. A., Ydstie, B. E. & Banga, J. R. From irreversible thermodynamics to a robust control theory for distributed process systems. Journal of Process Control vol. 12 507–517 (2002) – 10.1016/s0959-1524(01)00017-8
• Altaner, B. Nonequilibrium thermodynamics and information theory: basic concepts and relaxing dynamics. Journal of Physics A: Mathematical and Theoretical vol. 50 454001 (2017) – 10.1088/1751-8121/aa841d
• Couenne, F., Jallut, C., Maschke, B., Breedveld, P. C. & Tayakout, M. Bond graph modelling for chemical reactors. Mathematical and Computer Modelling of Dynamical Systems vol. 12 159–174 (2006) – 10.1080/13873950500068823
• Eberard, D., Maschke, B. M. & van der Schaft, A. J. An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes. Reports on Mathematical Physics vol. 60 175–198 (2007) – 10.1016/s0034-4877(07)00024-9
• Faulwasser, Manifold turnpikes, trims, and symmetries. Mathematics of Control, Signals, and Systems (2022)
• Faulwasser, T., Flaßkamp, K., Ober-Blöbaum, S. & Worthmann, K. A Dissipativity Characterization of Velocity Turnpikes in Optimal Control Problems for Mechanical Systems. IFAC-PapersOnLine vol. 54 624–629 (2021) – 10.1016/j.ifacol.2021.06.125
• Faulwasser, T. & Grüne, L. Turnpike properties in optimal control. Handbook of Numerical Analysis 367–400 (2022) doi:10.1016/bs.hna.2021.12.011 – 10.1016/bs.hna.2021.12.011
• Faulwasser, Optimal control of port-Hamiltonian descriptor systems with minimal energy supply. Accepted for publication in SIAM. Journal on Control and Optimization (2021)
• Favache, A., Dochain, D. & Maschke, B. An entropy-based formulation of irreversible processes based on contact structures. Chemical Engineering Science vol. 65 5204–5216 (2010) – 10.1016/j.ces.2010.06.019
• Favache, A., Dos Santos Martins, V. S., Dochain, D. & Maschke, B. Some Properties of Conservative Port Contact Systems. IEEE Transactions on Automatic Control vol. 54 2341–2351 (2009) – 10.1109/tac.2009.2028973
• García-Sandoval, J. P., Hudon, N. & Dochain, D. Generalized Hamiltonian representation of thermo-mechanical systems based on an entropic formulation. Journal of Process Control vol. 51 18–26 (2017) – 10.1016/j.jprocont.2016.09.011
• García-Sandoval, J. P., Hudon, N., Dochain, D. & González-Álvarez, V. Stability analysis and passivity properties of a class of thermodynamic processes: An internal entropy production approach. Chemical Engineering Science vol. 139 261–272 (2016) – 10.1016/j.ces.2015.07.039
• Hoang, H., Couenne, F., Jallut, C. & Le Gorrec, Y. Lyapunov-based control of non isothermal continuous stirred tank reactors using irreversible thermodynamics. Journal of Process Control vol. 22 412–422 (2012) – 10.1016/j.jprocont.2011.12.007
• JOHANNESSEN, E. Minimum entropy production rate in plug flow reactors: An optimal control problem solved for SO2 oxidation. Energy vol. 29 2403–2423 (2004) – 10.1016/j.energy.2004.03.033
• Macki, (2012)
• Maschke, B. & Schaft, A. van der. Structure preserving feedback of port-thermodynamic systems. IFAC-PapersOnLine vol. 52 418–423 (2019) – 10.1016/j.ifacol.2019.11.816
• Maschke, B., Philipp, F., Schaller, M., Worthmann, K. & Faulwasser, T. Optimal control of thermodynamic port-Hamiltonian Systems. IFAC-PapersOnLine vol. 55 55–60 (2022) – 10.1016/j.ifacol.2022.11.028
• Öttinger, H. C. Nonequilibrium thermodynamics for open systems. Physical Review E vol. 73 (2006) – 10.1103/physreve.73.036126
• Philipp, F., Schaller, M., Faulwasser, T., Maschke, B. & Worthmann, K. Minimizing the energy supply of infinite-dimensional linear port-Hamiltonian systems. IFAC-PapersOnLine vol. 54 155–160 (2021) – 10.1016/j.ifacol.2021.11.071
• Ramírez, H., Le Gorrec, Y., Maschke, B. & Couenne, F. On the passivity based control of irreversible processes: A port-Hamiltonian approach. Automatica vol. 64 105–111 (2016) – 10.1016/j.automatica.2015.07.002
• Ramirez, H., Maschke, B. & Sbarbaro, D. Feedback equivalence of input–output contact systems. Systems & Control Letters vol. 62 475–481 (2013) – 10.1016/j.sysconle.2013.02.008
• Ramirez, H., Maschke, B. & Sbarbaro, D. Irreversible port-Hamiltonian systems: A general formulation of irreversible processes with application to the CSTR. Chemical Engineering Science vol. 89 223–234 (2013) – 10.1016/j.ces.2012.12.002
• Ramirez, H., Maschke, B. & Sbarbaro, D. Modelling and control of multi-energy systems: An irreversible port-Hamiltonian approach. European Journal of Control vol. 19 513–520 (2013) – 10.1016/j.ejcon.2013.09.009
• Ruszkowski, M., Garcia‐Osorio, V. & Ydstie, B. E. Passivity based control of transport reaction systems. AIChE Journal vol. 51 3147–3166 (2005) – 10.1002/aic.10543
• Sangi, R. & Müller, D. Application of the second law of thermodynamics to control: A review. Energy vol. 174 938–953 (2019) – 10.1016/j.energy.2019.03.024
• Schaft, About some system-theoretic properties of port-thermodynamic systems. (2019)
• Schaller, M., Philipp, F., Faulwasser, T., Worthmann, K. & Maschke, B. Control of port-Hamiltonian systems with minimal energy supply. European Journal of Control vol. 62 33–40 (2021) – 10.1016/j.ejcon.2021.06.017
• Sieniutycz, S. Hamilton–Jacobi–Bellman framework for optimal control in multistage energy systems. Physics Reports vol. 326 165–258 (2000) – 10.1016/s0370-1573(99)00116-7
• Van der Schaft, A. & Maschke, B. Geometry of Thermodynamic Processes. Entropy vol. 20 925 (2018) – 10.3390/e20120925
• Wang, Stabilization of control contact systems. (2015)
• Ydstie, B. E. & Alonso, A. A. Process systems and passivity via the Clausius-Planck inequality. Systems & Control Letters vol. 30 253–264 (1997) – 10.1016/s0167-6911(97)00023-6