Port-Hamiltonian structures in infinite-dimensional optimal control: Primal–Dual gradient method and control-by-interconnection

Authors

Hannes Gernandt, Manuel Schaller

Abstract

In this note, we consider port-Hamiltonian structures in numerical optimal control of ordinary differential equations. By introducing a novel class of nonlinear monotone port-Hamiltonian (pH) systems, we show that the primal–dual gradient method may be viewed as an infinite-dimensional nonlinear pH system. The monotonicity and the particular block structure arising in the optimality system is used to prove exponential stability of the dynamics towards its equilibrium, which is a critical point of the first-order optimality conditions. Leveraging the port-based modeling, we propose an optimization-based controller in a suboptimal receding horizon control fashion. To this end, the primal–dual gradient based optimizer-dynamics is coupled to a pH plant dynamics in a power-preserving manner. We show that the resulting model is again monotone pH system and prove that the closed-loop exhibits local exponential convergence towards the equilibrium.

Keywords

control-by-interconnection, monotone operator, optimal control, passivity, port-hamiltonian system, primal–dual gradient method

Citation

• Journal: Systems & Control Letters
• Year: 2025
• Volume: 197
• Issue:
• Pages: 106030
• Publisher: Elsevier BV
• DOI: 10.1016/j.sysconle.2025.106030

BibTeX

@article{Gernandt_2025,
  title={{Port-Hamiltonian structures in infinite-dimensional optimal control: Primal–Dual gradient method and control-by-interconnection}},
  volume={197},
  ISSN={0167-6911},
  DOI={10.1016/j.sysconle.2025.106030},
  journal={Systems \& Control Letters},
  publisher={Elsevier BV},
  author={Gernandt, Hannes and Schaller, Manuel},
  year={2025},
  pages={106030}
}

Download the bib file

References

• Hauschild, Port-Hamiltonian modeling of district heating networks. (2020)
• Fiaz S, Zonetti D, Ortega R, Scherpen JMA, van der Schaft AJ (2013) A port-Hamiltonian approach to power network modeling and analysis. European Journal of Control 19(6):477–485. https://doi.org/10.1016/j.ejcon.2013.09.00 – 10.1016/j.ejcon.2013.09.002
• Gernandt H, Hinsen D (2024) Stability and Passivity for a Class of Distributed Port-Hamiltonian Networks. SIAM J Control Optim 62(6):2936–2962. https://doi.org/10.1137/22m153917 – 10.1137/22m1539174
• Gernandt H, Haller FE, Reis T, Schaft AJ van der (2021) Port-Hamiltonian formulation of nonlinear electrical circuits. Journal of Geometry and Physics 159:103959. https://doi.org/10.1016/j.geomphys.2020.10395 – 10.1016/j.geomphys.2020.103959
• Rashad, Port-Hamiltonian modeling of ideal fluid flow: Part I. Foundations and kinetic energy. J. Geom. Phys. (2021)
• Reis, Port-Hamiltonian formulation of Oseen flows. (2024)
• Clemens, Structural aspects of electromagneto-quasistatic field formulations of darwin-type derived in the port-Hamiltonian system framework. IEEE Trans. Magn. (2024)
• Reis T, Stykel T (2023) Passivity, port-hamiltonian formulation and solution estimates for a coupled magneto-quasistatic system. EECT 12(4):1208–1232. https://doi.org/10.3934/eect.202300 – 10.3934/eect.2023008
• Ramirez H, Le Gorrec Y (2022) An Overview on Irreversible Port-Hamiltonian Systems. Entropy 24(10):1478. https://doi.org/10.3390/e2410147 – 10.3390/e24101478
• Jacob, (2012)
• Skrepek, Well-posedness of linear first order port-Hamiltonian systems on multidimensional spatial domains. Evol. Equ. Control. Theory (2021)
• Le Gorrec Y, Zwart H, Maschke B (2005) Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM J Control Optim 44(5):1864–1892. https://doi.org/10.1137/04061167 – 10.1137/040611677
• Mehrmann V, Unger B (2023) Control of port-Hamiltonian differential-algebraic systems and applications. Acta Numerica 32:395–515. https://doi.org/10.1017/s096249292200008 – 10.1017/s0962492922000083
• Kotyczka P, Maschke B, Lefèvre L (2018) Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems. Journal of Computational Physics 361:442–476. https://doi.org/10.1016/j.jcp.2018.02.00 – 10.1016/j.jcp.2018.02.006
• van der Schaft A, Jeltsema D (2014) Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control 1(2–3):173–378. https://doi.org/10.1561/260000000 – 10.1561/2600000002
• Ortega R, van der Schaft A, Maschke B, Escobar G (2002) Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38(4):585–596. https://doi.org/10.1016/s0005-1098(01)00278- – 10.1016/s0005-1098(01)00278-3
• Breiten T, Karsai A (2023) Structure-preserving \(\mathcal{H}_{\infty}\) control for port-Hamiltonian systems. Systems & Control Letters 174:105493. https://doi.org/10.1016/j.sysconle.2023.10549 – 10.1016/j.sysconle.2023.105493
• Doganay, (2023)
• Schaller M, Philipp F, Faulwasser T, Worthmann K, Maschke B (2021) Control of port-Hamiltonian systems with minimal energy supply. European Journal of Control 62:33–40. https://doi.org/10.1016/j.ejcon.2021.06.01 – 10.1016/j.ejcon.2021.06.017
• Faulwasser T, Maschke B, Philipp F, Schaller M, Worthmann K (2022) Optimal Control of Port-Hamiltonian Descriptor Systems with Minimal Energy Supply. SIAM J Control Optim 60(4):2132–2158. https://doi.org/10.1137/21m142772 – 10.1137/21m1427723
• Schaller, Energy-optimal control of adaptive structures. At Autom. (2024)
• Stegink, Port-Hamiltonian formulation of the gradient method applied to smart grids. IFAC Pap. (2015)
• Cherukuri A, Mallada E, Cortés J (2016) Asymptotic convergence of constrained primal–dual dynamics. Systems & Control Letters 87:10–15. https://doi.org/10.1016/j.sysconle.2015.10.00 – 10.1016/j.sysconle.2015.10.006
• Camlibel MK, van der Schaft AJ (2023) Port-Hamiltonian Systems Theory and Monotonicity. SIAM J Control Optim 61(4):2193–2221. https://doi.org/10.1137/22m150374 – 10.1137/22m1503749
• Grüne, (2017)
• Rawlings, (2017)
• Scokaert POM, Mayne DQ, Rawlings JB (1999) Suboptimal model predictive control (feasibility implies stability). IEEE Trans Automat Contr 44(3):648–654. https://doi.org/10.1109/9.75136 – 10.1109/9.751369
• Diehl M, Bock HG, Schlöder JP (2005) A Real-Time Iteration Scheme for Nonlinear Optimization in Optimal Feedback Control. SIAM J Control Optim 43(5):1714–1736. https://doi.org/10.1137/s036301290240071 – 10.1137/s0363012902400713
• Zanelli A, Tran-Dinh Q, Diehl M (2021) A Lyapunov function for the combined system-optimizer dynamics in inexact model predictive control. Automatica 134:109901. https://doi.org/10.1016/j.automatica.2021.10990 – 10.1016/j.automatica.2021.109901
• Yoshida K, Inoue M, Hatanaka T (2019) Instant MPC for Linear Systems and Dissipativity-Based Stability Analysis. IEEE Control Syst Lett 3(4):811–816. https://doi.org/10.1109/lcsys.2019.291809 – 10.1109/lcsys.2019.2918095
• Karapetyan, On the finite-time behavior of suboptimal linear model predictive control. (2023)
• Häberle, Non-convex feedback optimization with input and output constraints. IEEE Control. Syst. Lett. (2020)
• Pham TH, Vu NMT, Prodan I, Lefèvre L (2022) A combined Control by Interconnection—Model Predictive Control design for constrained Port-Hamiltonian systems. Systems & Control Letters 167:105336. https://doi.org/10.1016/j.sysconle.2022.10533 – 10.1016/j.sysconle.2022.105336
• Vu, Port-Hamiltonian observer for state-feedback control design. (2023)
• Ekeland, (1999)
• Barbu, (2010)
• Grüne, (2015)
• Kato N (1995) A principle of linearized stability for nonlinear evolution equations. Trans Amer Math Soc 347(8):2851–2868. https://doi.org/10.1090/s0002-9947-1995-1290722- – 10.1090/s0002-9947-1995-1290722-8
• Curtain, (1995)
• Schiela A (2013) A concise proof for existence and uniqueness of solutions of linear parabolic PDEs in the context of optimal control. Systems & Control Letters 62(10):895–901. https://doi.org/10.1016/j.sysconle.2013.06.01 – 10.1016/j.sysconle.2013.06.013