Port-Hamiltonian Nonlinear Systems
Authors
Abstract
Quite often, control theory takes the state space model of the to-be-controlled system for granted. In contrast, port-Hamiltonian systems theory bridges the gap between modeling and control of physical systems. It provides a unified framework for the modeling of complex multiphysics systems. At the same time it offers powerful tools for analysis and control by identifying the underlying physical structure, as reflected in, e.g., energy balance and other conserved quantities. This leads to control schemes that exploit the physical structure, instead of compensating for it. As a result, the derived control laws tend to be simple, physically interpretable, and robust with respect to physical parameter variations. In this paper, after introducing port-Hamiltonian systems, the focus is on ‘control by interconnection’ for set-point stabilization of nonlinear physical systems. Most of this theory is well-established, but novel developments using ‘energy ports’ instead of ‘power ports’ are also included. In this paper, after introducing port-Hamiltonian systems, the focus is on ‘control by interconnection’ for set-point stabilization of nonlinear physical systems. Most of this theory is well-established but novel developments using ‘energy ports’ instead of ‘power ports’ are also included.
Keywords
Control by interconnection; Energy ports; Energy shaping; Energy-Casimir method; Input-output Hamiltonian systems; Lyapunov functions; Modeling for control; Multiphysics systems; Negative imaginary systems; Network modeling; Passivity; Port-Hamiltonian systems; Stabilization
Citation
- ISBN: 9780128035818
- Publisher: Elsevier
- DOI: 10.1016/b978-0-443-14081-5.00148-3
BibTeX
@inbook{van_der_Schaft_2025,
title={{Port-Hamiltonian Nonlinear Systems}},
ISBN={9780128035818},
DOI={10.1016/b978-0-443-14081-5.00148-3},
booktitle={{Reference Module in Materials Science and Materials Engineering}},
publisher={Elsevier},
author={van der Schaft, Arjan},
year={2025}
}
References
- Angeli, D. Systems With Counterclockwise Input–Output Dynamics. IEEE Transactions on Automatic Control vol. 51 1130–1143 (2006) – 10.1109/tac.2006.878747
- Angeli, D. On systems with counter-clock-wise input/output dynamics. 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601) 2527-2532 Vol.3 (2004) doi:10.1109/cdc.2004.1428820 – 10.1109/cdc.2004.1428820
- Borja, P., Ferguson, J. & van der Schaft, A. Interconnection Schemes in Modeling and Control. IEEE Control Systems Letters vol. 7 2287–2292 (2023) – 10.1109/lcsys.2023.3286124
- Brockett, Control theory and analytical mechanics. (1977)
- Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
- Crouch, Variational and Hamiltonian Control Systems. (1993)
- Dalsmo, M. & van der Schaft, A. On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM Journal on Control and Optimization vol. 37 54–91 (1998) – 10.1137/s0363012996312039
- Dorfman, (1987)
- Duindam, (2009)
- Folkertsma, G. A. & Stramigioli, S. Energy in Robotics. Foundations and Trends® in Robotics vol. 6 140–210 (2017) – 10.1561/2300000038
- Golo, Hamiltonian formulation of bond graphs. (2003)
- Krhač, K., Maschke, B. & van der Schaft, A. Port-Hamiltonian systems with energy and power ports. IFAC-PapersOnLine vol. 58 280–285 (2024) – 10.1016/j.ifacol.2024.08.294
- Lanzon, A. & Petersen, I. R. Stability Robustness of a Feedback Interconnection of Systems With Negative Imaginary Frequency Response. IEEE Transactions on Automatic Control vol. 53 1042–1046 (2008) – 10.1109/tac.2008.919567
- Marsden, (1999)
- Nijmeijer, (2016)
- Ortega, Putting energy back in control. Control Systems Magazine (2001)
- Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica vol. 38 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
- Ortega, R., van der Schaft, A., Castanos, F. & Astolfi, A. Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 53 2527–2542 (2008) – 10.1109/tac.2008.2006930
- Paynter, (1960)
- Feedback Control of Negative-Imaginary Systems. IEEE Control Systems vol. 30 54–72 (2010) – 10.1109/mcs.2010.937676
- Stramigioli, S., Maschke, B. & van der Schaft, A. Passive Output Feedback and Port Interconnection. IFAC Proceedings Volumes vol. 31 591–596 (1998) – 10.1016/s1474-6670(17)40401-0
- van der Schaft, A. Classical Thermodynamics Revisited: A Systems and Control Perspective. IEEE Control Systems vol. 41 32–60 (2021) – 10.1109/mcs.2021.3092809
- van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
- Schaft, A. J. Hamiltonian dynamics with external forces and observations. Mathematical Systems Theory vol. 15 145–168 (1981) – 10.1007/bf01786977
- van der Schaft, A. J. Observability and Controllability for Smooth Nonlinear Systems. SIAM Journal on Control and Optimization vol. 20 338–354 (1982) – 10.1137/0320026
- van der Schaft, Port-Hamiltonian systems. (2009)
- van der Schaft, (2017)
- van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
- van der Schaft, A. J. & Maschke, B. M. Port-Hamiltonian Systems on Graphs. SIAM Journal on Control and Optimization vol. 51 906–937 (2013) – 10.1137/110840091
- van der Schaft, A. & Jeltsema, D. On Energy Conversion in Port-Hamiltonian Systems. 2021 60th IEEE Conference on Decision and Control (CDC) 2421–2427 (2021) doi:10.1109/cdc45484.2021.9683292 – 10.1109/cdc45484.2021.9683292
- van der Schaft, A. J. Positive feedback interconnection of Hamiltonian systems. IEEE Conference on Decision and Control and European Control Conference 6510–6515 (2011) doi:10.1109/cdc.2011.6160395 – 10.1109/cdc.2011.6160395
- van der Schaft, A. Interconnections of input-output Hamiltonian systems with dissipation. 2016 IEEE 55th Conference on Decision and Control (CDC) 4686–4691 (2016) doi:10.1109/cdc.2016.7798983 – 10.1109/cdc.2016.7798983
- Vu, N. M. T., Lefèvre, L. & Maschke, B. A structured control model for the thermo-magneto-hydrodynamics of plasmas in tokamaks. Mathematical and Computer Modelling of Dynamical Systems vol. 22 181–206 (2016) – 10.1080/13873954.2016.1154874
- Xiong, J., Petersen, I. R. & Lanzon, A. A Negative Imaginary Lemma and the Stability of Interconnections of Linear Negative Imaginary Systems. IEEE Transactions on Automatic Control vol. 55 2342–2347 (2010) – 10.1109/tac.2010.2052711