Energy-Optimal Control of Discrete-Time Port-Hamiltonian Systems

Authors

Arijit Sarkar, Vaibhav Kumar Singh, Manuel Schaller, Karl Worthmann

Abstract

In this letter, we study the energy-optimal control of nonlinear port-Hamiltonian (pH) systems in discrete time. For continuous-time pH systems, energy-optimal control problems are strictly dissipative by design. This property, stating that the system to be optimized is dissipative with the cost functional as a supply rate, implies a stable long-term behavior of optimal solutions and enables stability results in predictive control. In this letter, we show that the crucial property of strict dissipativity is not straightforwardly preserved by any energy-preserving integrator such as the implicit midpoint rule. Then, we prove that discretizations via difference and differential representations lead to strictly dissipative discrete-time optimal control problems. Consequently, we rigorously show a stable long-term behavior of optimal solutions in the form of a manifold (subspace) turnpike property. Finally, we validate our findings using two numerical examples.

Citation

• Journal: IEEE Control Systems Letters
• Year: 2025
• Volume: 9
• Issue:
• Pages: 1526–1531
• Publisher: Institute of Electrical and Electronics Engineers (IEEE)
• DOI: 10.1109/lcsys.2025.3579391

BibTeX

@article{Sarkar_2025,
  title={{Energy-Optimal Control of Discrete-Time Port-Hamiltonian Systems}},
  volume={9},
  ISSN={2475-1456},
  DOI={10.1109/lcsys.2025.3579391},
  journal={IEEE Control Systems Letters},
  publisher={Institute of Electrical and Electronics Engineers (IEEE)},
  author={Sarkar, Arijit and Kumar Singh, Vaibhav and Schaller, Manuel and Worthmann, Karl},
  year={2025},
  pages={1526--1531}
}

Download the bib file

References

• Jacob, B. & Zwart, H. J. Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. (Springer Basel, 2012). doi:10.1007/978-3-0348-0399-1 – 10.1007/978-3-0348-0399-1
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. FnT in Systems and Control 1, 173–378 (2014) – 10.1561/2600000002
• Mehrmann, V. & Unger, B. Control of port-Hamiltonian differential-algebraic systems and applications. Acta Numerica 32, 395–515 (2023) – 10.1017/s0962492922000083
• Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38, 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
• Müller, M. A. & Worthmann, K. Quadratic costs do not always work in MPC. Automatica 82, 269–277 (2017) – 10.1016/j.automatica.2017.04.058
• van der Schaft, A. L2-Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer International Publishing, 2017). doi:10.1007/978-3-319-49992-5 – 10.1007/978-3-319-49992-5
• Sepulchre, R., Janković, M. & Kokotović, P. V. Constructive Nonlinear Control. Communications and Control Engineering (Springer London, 1997). doi:10.1007/978-1-4471-0967-9 – 10.1007/978-1-4471-0967-9
• Schaller, M. et al. Energy-optimal control of adaptive structures. at - Automatisierungstechnik 72, 107–119 (2024) – 10.1515/auto-2023-0090
• Schaller, M., Philipp, F., Faulwasser, T., Worthmann, K. & Maschke, B. Control of port-Hamiltonian systems with minimal energy supply. European Journal of Control 62, 33–40 (2021) – 10.1016/j.ejcon.2021.06.017
• Faulwasser, T., Maschke, B., Philipp, F., Schaller, M. & Worthmann, K. Optimal Control of Port-Hamiltonian Descriptor Systems with Minimal Energy Supply. SIAM J. Control Optim. 60, 2132–2158 (2022) – 10.1137/21m1427723
• Philipp, F. M., Schaller, M., Worthmann, K., Faulwasser, T. & Maschke, B. Optimal control of port-Hamiltonian systems: Energy, entropy, and exergy. Systems & Control Letters 194, 105942 (2024) – 10.1016/j.sysconle.2024.105942
• Zanon, M. & Faulwasser, T. Economic MPC without terminal constraints: Gradient-correcting end penalties enforce asymptotic stability. Journal of Process Control 63, 1–14 (2018) – 10.1016/j.jprocont.2017.12.005
• Şen, G. D., Schaller, M. & Worthmann, K. Stage‐cost design for optimal and model predictive control of linear port‐Hamiltonian systems: Energy efficiency and robustness. Proc Appl Math and Mech 23, (2023) – 10.1002/pamm.202300296
• Kotyczka, P. & Lefèvre, L. Discrete-time port-Hamiltonian systems: A definition based on symplectic integration. Systems & Control Letters 133, 104530 (2019) – 10.1016/j.sysconle.2019.104530
• Monaco, S. & Normand-Cyrot, D. Discrete-time state representations, a new paradigm. Perspectives in Control 191–203 (1998) doi:10.1007/978-1-4471-1276-1_14 – 10.1007/978-1-4471-1276-1_14
• Moreschini, A., Mattioni, M., Monaco, S. & Normand-Cyrot, D. Discrete port-controlled Hamiltonian dynamics and average passivation. 2019 IEEE 58th Conference on Decision and Control (CDC) 1430–1435 (2019) doi:10.1109/cdc40024.2019.9029809 – 10.1109/cdc40024.2019.9029809
• Faulwasser, T., Grüne, L. & Müller, M. A. Economic Nonlinear Model Predictive Control. FnT in Systems and Control 5, 224–409 (2018) – 10.1561/2600000014
• Karsai, A. Manifold turnpikes of nonlinear port-Hamiltonian descriptor systems under minimal energy supply. Math. Control Signals Syst. 36, 707–728 (2024) – 10.1007/s00498-024-00384-7
• Aronna, Conditions for singular optimal control of port-Hamiltonian systems. arXiv:2407.03213 (2024)
• Kölsch, L., Jané Soneira, P., Strehle, F. & Hohmann, S. Optimal control of port-Hamiltonian systems: A continuous-time learning approach. Automatica 130, 109725 (2021) – 10.1016/j.automatica.2021.109725
• Massaroli, S. et al. Optimal Energy Shaping via Neural Approximators. SIAM J. Appl. Dyn. Syst. 21, 2126–2147 (2022) – 10.1137/21m1414279
• Stegink, T., De Persis, C. & van der Schaft, A. A Unifying Energy-Based Approach to Stability of Power Grids With Market Dynamics. IEEE Trans. Automat. Contr. 62, 2612–2622 (2017) – 10.1109/tac.2016.2613901
• Gernandt, H. & Schaller, M. Port-Hamiltonian structures in infinite-dimensional optimal control: Primal–Dual gradient method and control-by-interconnection. Systems & Control Letters 197, 106030 (2025) – 10.1016/j.sysconle.2025.106030
• Grüne, L. & Pannek, J. Nonlinear Model Predictive Control. Communications and Control Engineering (Springer International Publishing, 2017). doi:10.1007/978-3-319-46024-6 – 10.1007/978-3-319-46024-6
• Grüne, L. & Müller, M. A. On the relation between strict dissipativity and turnpike properties. Systems & Control Letters 90, 45–53 (2016) – 10.1016/j.sysconle.2016.01.003
• Faulwasser, T., Flaßkamp, K., Ober-Blöbaum, S., Schaller, M. & Worthmann, K. Manifold turnpikes, trims, and symmetries. Math. Control Signals Syst. 34, 759–788 (2022) – 10.1007/s00498-022-00321-6
• Mehrmann, V. & Morandin, R. Structure-preserving discretization for port-Hamiltonian descriptor systems. 2019 IEEE 58th Conference on Decision and Control (CDC) 6863–6868 (2019) doi:10.1109/cdc40024.2019.9030180 – 10.1109/cdc40024.2019.9030180
• Xia, M., Antsaklis, P. J., Gupta, V. & Zhu, F. Passivity and Dissipativity Analysis of a System and Its Approximation. IEEE Trans. Automat. Contr. 62, 620–635 (2017) – 10.1109/tac.2016.2562919
• Moreschini, A., Bin, M., Astolfi, A. & Parisini, T. A Generalized Passivity Theory Over Abstract Time Domains. IEEE Trans. Automat. Contr. 70, 2–17 (2025) – 10.1109/tac.2024.3423510
• Moreschini, Modeling and control of discrete-time and sampled-data port-Hamiltonian systems. (2021)
• Moreschini, A., Monaco, S. & Normand-Cyrot, D. Gradient and Hamiltonian dynamics under sampling. IFAC-PapersOnLine 52, 472–477 (2019) – 10.1016/j.ifacol.2019.12.006
• Monaco, S., Normand-Cyrot, D. & Tiefensee, F. Sampled-Data Stabilization; A PBC Approach. IEEE Trans. Automat. Contr. 56, 907–912 (2011) – 10.1109/tac.2010.2101130