Port-Hamiltonian Systems Theory and Monotonicity

Authors

M. K. Camlibel, A. J. van der Schaft

Abstract

The relationship of the theory of port-Hamiltonian systems with the mathematical concept of monotonicity is explored. The earlier introduced notion of incrementally port-Hamiltonian systems is extended to systems defined with respect to maximal cyclically monotone relations, together with their generating convex functions. This gives rise to interesting subclasses of incrementally port-Hamiltonian systems, with examples stemming from physical systems modeling as well as from convex optimization. Furthermore, it is shown how cyclical monotonicity for Dirac structures is equivalent to separability. An in-depth treatment is given of the composition of maximal monotone and maximal cyclically monotone relations, where in the latter case the resulting maximal cyclically monotone relation is shown to be computable through the use of generating functions. The results on compositionality are employed for steady-state analysis and for a convex optimization approach to the computation of the equilibria of interconnected incrementally port-Hamiltonian systems. Finally, the relation to incremental and differential passivity is discussed, and it is shown how incrementally port-Hamiltonian systems with strictly convex Hamiltonians are equilibrium independent passive.

Citation

• Journal: SIAM Journal on Control and Optimization
• Year: 2023
• Volume: 61
• Issue: 4
• Pages: 2193–2221
• Publisher: Society for Industrial & Applied Mathematics (SIAM)
• DOI: 10.1137/22m1503749

BibTeX

@article{Camlibel_2023,
  title={{Port-Hamiltonian Systems Theory and Monotonicity}},
  volume={61},
  ISSN={1095-7138},
  DOI={10.1137/22m1503749},
  number={4},
  journal={SIAM Journal on Control and Optimization},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={Camlibel, M. K. and van der Schaft, A. J.},
  year={2023},
  pages={2193--2221}
}

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References

• Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
• van der Schaft A. J., -Gain and Passivity Techniques in Nonlinear Control, Communications and Control Engineering (2017)
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. (2014) doi:10.1561/9781601987877 – 10.1561/9781601987877
• Chaffey T, Sepulchre R (2024) Monotone One-Port Circuits. IEEE Trans Automat Contr 69(2):783–796. https://doi.org/10.1109/tac.2023.3274690 – 10.1109/tac.2023.3274690
• Desoer, C. A. & Vidyasagar, M. Feedback Systems. (2009) doi:10.1137/1.9780898719055 – 10.1137/1.9780898719055
• Parikh N., Proximal Algorithms (2013)
• Brézis H., Operateurs Maximaux Monotones (1973)
• Bauschke, H. H. & Combettes, P. L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics (Springer New York, 2011). doi:10.1007/978-1-4419-9467-7 – 10.1007/978-1-4419-9467-7
• Rockafellar, R. T. & Wets, R. J. B. Variational Analysis. Grundlehren der mathematischen Wissenschaften (Springer Berlin Heidelberg, 1998). doi:10.1007/978-3-642-02431-3 – 10.1007/978-3-642-02431-3
• Camlibel, K., Iannelli, L. & Tanwani, A. Convergence of proximal solutions for evolution inclusions with time-dependent maximal monotone operators. Mathematical Programming vol. 194 1017–1059 (2021) – 10.1007/s10107-021-01666-7
• Camlibel MK, van der Schaft AJ (2013) Incrementally port-Hamiltonian systems. In: 52nd IEEE Conference on Decision and Control. IEEE, pp 2538–2543 – 10.1109/cdc.2013.6760262
• Camlibel, M. K. & Schumacher, J. M. Linear passive systems and maximal monotone mappings. Mathematical Programming vol. 157 397–420 (2015) – 10.1007/s10107-015-0945-7
• Bürger, M., Zelazo, D. & Allgöwer, F. Duality and network theory in passivity-based cooperative control. Automatica vol. 50 2051–2061 (2014) – 10.1016/j.automatica.2014.06.002
• Hines, G. H., Arcak, M. & Packard, A. K. Equilibrium-independent passivity: A new definition and numerical certification. Automatica vol. 47 1949–1956 (2011) – 10.1016/j.automatica.2011.05.011
• 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
• Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
• van der Schaft A. J., Archiv für Elektronik und Übertragungstechnik (1995)
• Mehrmann V., Math. Control Signals Systems (2023)
• van der Schaft, A. & Mehrmann, V. Linear port-Hamiltonian DAE systems revisited. Systems & Control Letters vol. 177 105564 (2023) – 10.1016/j.sysconle.2023.105564
• Ryu, E. K. & Yin, W. Large-Scale Convex Optimization. (2022) doi:10.1017/9781009160865 – 10.1017/9781009160865
• Arrow K. J., Studies in Linear and Non-linear Programming (1958)
• Stegink, T., De Persis, C. & van der Schaft, A. A Unifying Energy-Based Approach to Stability of Power Grids With Market Dynamics. IEEE Transactions on Automatic Control vol. 62 2612–2622 (2017) – 10.1109/tac.2016.2613901
• 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
• Khatri, C. G. & Mitra, S. K. Hermitian and Nonnegative Definite Solutions of Linear Matrix Equations. SIAM Journal on Applied Mathematics vol. 31 579–585 (1976) – 10.1137/0131050
• Cervera, J., van der Schaft, A. J. & Baños, A. Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica vol. 43 212–225 (2007) – 10.1016/j.automatica.2006.08.014
• Bertsekas D. P., Convex Optimization Theory (2009)
• Bregman, L. M. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics vol. 7 200–217 (1967) – 10.1016/0041-5553(67)90040-7
• Pavlov, A. & Marconi, L. Incremental passivity and output regulation. Systems & Control Letters vol. 57 400–409 (2008) – 10.1016/j.sysconle.2007.10.008
• Angeli, D. A Lyapunov approach to incremental stability properties. IEEE Transactions on Automatic Control vol. 47 410–421 (2002) – 10.1109/9.989067
• F. Forni and R. Sepulchre , On differentially dissipative dynamical systems, in 9th IFAC Symposium on Nonlinear Control Systems, Toulouse, France, 2013, pp. 4–6.
• van der Schaft AJ (2013) On differential passivity. IFAC Proceedings Volumes 46(23):21–25. https://doi.org/10.3182/20130904-3-fr-2041.00008 – 10.3182/20130904-3-fr-2041.00008
• Forni F, Sepulchre R, van der Schaft AJ (2013) On differential passivity of physical systems. In: 52nd IEEE Conference on Decision and Control. IEEE, pp 6580–6585 – 10.1109/cdc.2013.6760930
• Variational and Hamiltonian Control Systems. Lecture Notes in Control and Information Sciences (Springer Berlin Heidelberg, 1987). doi:10.1007/bfb0042858 – 10.1007/bfb0042858
• Kulkarni VV, Safonov MG (2003) Incremental positivity non-preservation by stability multipliers. In: Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228). IEEE, pp 33–38 – 10.1109/cdc.2001.980064
• LOHMILLER, W. & SLOTINE, J.-J. E. On Contraction Analysis for Non-linear Systems. Automatica vol. 34 683–696 (1998) – 10.1016/s0005-1098(98)00019-3