Control Design for a Class of Discrete-Time Port-Hamiltonian Systems
Authors
Abstract
This article aims at extending the continuous-time energy-shaping plus damping injection control design technique to deal with a class of nonlinear, discrete-time port-Hamiltonian systems. For such systems, the gradient of the Hamiltonian function in the continuous-time dynamics is replaced by a discrete gradient, thus leading to a state equation in implicit form. Its well-posedness is studied both in the autonomous and nonautonomous cases to determine when the dynamical equation admits a solution for the next state. Based on this analysis, the extension of the energy-shaping plus damping injection control methodology is discussed. At first, it is supposed that the control action depends on the discrete gradient of an energy function. Then, this hypothesis is removed, and an algebraic solution to the matching equation is proposed to enlarge the class of stabilizing controllers.
Citation
- Journal: IEEE Transactions on Automatic Control
- Year: 2023
- Volume: 68
- Issue: 12
- Pages: 8224–8231
- Publisher: Institute of Electrical and Electronics Engineers (IEEE)
- DOI: 10.1109/tac.2023.3292180
BibTeX
@article{Macchelli_2023,
title={{Control Design for a Class of Discrete-Time Port-Hamiltonian Systems}},
volume={68},
ISSN={2334-3303},
DOI={10.1109/tac.2023.3292180},
number={12},
journal={IEEE Transactions on Automatic Control},
publisher={Institute of Electrical and Electronics Engineers (IEEE)},
author={Macchelli, Alessandro},
year={2023},
pages={8224--8231}
}
References
- Maschke, Port-controlled Hamiltonian systems: Modeling origins and system theoretic properties. Proc. Nonlinear Control Syst. Proc. 3rd IFAC Symp. (1992)
- 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
- Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
- 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
- Fujimoto, K., Sakurama, K. & Sugie, T. Trajectory tracking control of port-controlled Hamiltonian systems via generalized canonical transformations. Automatica vol. 39 2059–2069 (2003) – 10.1016/j.automatica.2003.07.005
- Feng, K. & Qin, M. Symplectic Geometric Algorithms for Hamiltonian Systems. (Springer Berlin Heidelberg, 2010). doi:10.1007/978-3-642-01777-3 – 10.1007/978-3-642-01777-3
- Talasila, V., Clemente-Gallardo, J. & van der Schaft, A. J. Discrete port-Hamiltonian systems. Systems & Control Letters vol. 55 478–486 (2006) – 10.1016/j.sysconle.2005.10.001
- Ruth, R. D. A Can0nical Integrati0n Technique. IEEE Transactions on Nuclear Science vol. 30 2669–2671 (1983) – 10.1109/tns.1983.4332919
- Gonzalez, O. & Simo, J. C. On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry. Computer Methods in Applied Mechanics and Engineering vol. 134 197–222 (1996) – 10.1016/0045-7825(96)01009-2
- Marsden, J. E. & West, M. Discrete mechanics and variational integrators. Acta Numerica vol. 10 357–514 (2001) – 10.1017/s096249290100006x
- Kotyczka, P. & Lefèvre, L. Discrete-time port-Hamiltonian systems: A definition based on symplectic integration. Systems & Control Letters vol. 133 104530 (2019) – 10.1016/j.sysconle.2019.104530
- Kotyczka, P. & Thoma, T. Symplectic discrete-time energy-based control for nonlinear mechanical systems. Automatica vol. 133 109842 (2021) – 10.1016/j.automatica.2021.109842
- Gonzalez, O. Time integration and discrete Hamiltonian systems. Journal of Nonlinear Science vol. 6 449–467 (1996) – 10.1007/bf02440162
- Quispel, G. R. W. & Turner, G. S. Discrete gradient methods for solving ODEs numerically while preserving a first integral. Journal of Physics A: Mathematical and General vol. 29 L341–L349 (1996) – 10.1088/0305-4470/29/13/006
- Gören-Sümer, L. & Yalçιn, Y. Gradient Based Discrete-Time Modeling and Control of Hamiltonian Systems. IFAC Proceedings Volumes vol. 41 212–217 (2008) – 10.3182/20080706-5-kr-1001.00036
- Monaco, S., Normand-Cyrot, D. & Tiefensee, F. Nonlinear port controlled Hamiltonian systems under sampling. Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference 1782–1787 (2009) doi:10.1109/cdc.2009.5399866 – 10.1109/cdc.2009.5399866
- Celledoni, Energy-preserving and passivity-consistent numerical discretization of port-Hamiltonian systems. (2017)
- Aoues, S., Di Loreto, M., Eberard, D. & Marquis-Favre, W. Hamiltonian systems discrete-time approximation: Losslessness, passivity and composability. Systems & Control Letters vol. 110 9–14 (2017) – 10.1016/j.sysconle.2017.10.003
- 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
- Moreschini, A., Monaco, S. & Normand-Cyrot, D. Gradient and Hamiltonian dynamics under sampling. IFAC-PapersOnLine vol. 52 472–477 (2019) – 10.1016/j.ifacol.2019.12.006
- Moreschini, A., Mattioni, M., Monaco, S. & Normand-Cyrot, D. Stabilization of Discrete Port-Hamiltonian Dynamics via Interconnection and Damping Assignment. IEEE Control Systems Letters vol. 5 103–108 (2021) – 10.1109/lcsys.2020.3000705
- Riis, E. S., Ehrhardt, M. J., Quispel, G. R. W. & Schönlieb, C.-B. A Geometric Integration Approach to Nonsmooth, Nonconvex Optimisation. Foundations of Computational Mathematics vol. 22 1351–1394 (2021) – 10.1007/s10208-020-09489-2
- Aoues, S., Eberard, D. & Marquis-Favre, W. Discrete IDA-PBC control law for Newtonian mechanical port-Hamiltonian systems. 2015 54th IEEE Conference on Decision and Control (CDC) 4388–4393 (2015) doi:10.1109/cdc.2015.7402904 – 10.1109/cdc.2015.7402904
- Nunna, K., Sassano, M. & Astolfi, A. Constructive Interconnection and Damping Assignment for Port-Controlled Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 60 2350–2361 (2015) – 10.1109/tac.2015.2400663
- 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
- Harten, A., Lax, P. D. & Leer, B. van. On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws. SIAM Review vol. 25 35–61 (1983) – 10.1137/1025002
- Brouwer, L. E. J. �ber Abbildung von Mannigfaltigkeiten. Mathematische Annalen vol. 71 97–115 (1911) – 10.1007/bf01456931
- Sümer, L. G. & Yalçin, Y. A Direct Discrete-time IDA-PBC Design Method for a Class of Underactuated Hamiltonian Systems. IFAC Proceedings Volumes vol. 44 13456–13461 (2011) – 10.3182/20110828-6-it-1002.01187
- Leoni, G. A First Course in Sobolev Spaces. Graduate Studies in Mathematics (2017) doi:10.1090/gsm/181 – 10.1090/gsm/181
- Anstreicher, K. M. & Wright, M. H. A Note on the Augmented Hessian When the Reduced Hessian is Semidefinite. SIAM Journal on Optimization vol. 11 243–253 (2000) – 10.1137/s1052623499351791