Stokes-Lagrange and Stokes-Dirac representations of \\( N \\)-dimensional port-Hamiltonian systems for modeling and control

Authors

Antoine Bendimerad-Hohl, Ghislain Haine, Laurent Lefèvre, Denis Matignon

Abstract

In this paper, we extended the port-Hamiltonian framework by introducing the concept of Stokes-Lagrange structure, which enables the implicit definition of a Hamiltonian over an \( N \)-dimensional domain and incorporates energy ports into the system. This new framework parallels the existing Dirac and Stokes-Dirac structures. We proposed the Stokes-Lagrange structure as a specific case where the subspace is explicitly defined via differential operators that satisfy an integration-by-parts formula. By examining various examples through the lens of the Stokes-Lagrange structure, we demonstrated the existence of multiple equivalent system representations. These representations provide significant advantages for both numerical simulation and control design, offering additional tools for the modeling and control of port-Hamiltonian systems.

Citation

• Journal: Communications in Analysis and Mechanics
• Year: 2025
• Volume: 17
• Issue: 2
• Pages: 474–519
• Publisher: American Institute of Mathematical Sciences (AIMS)
• DOI: 10.3934/cam.2025020

BibTeX

@article{Bendimerad_Hohl_2025,
  title={{Stokes-Lagrange and Stokes-Dirac representations of $ N $-dimensional port-Hamiltonian systems for modeling and control}},
  volume={17},
  ISSN={2836-3310},
  DOI={10.3934/cam.2025020},
  number={2},
  journal={Communications in Analysis and Mechanics},
  publisher={American Institute of Mathematical Sciences (AIMS)},
  author={Bendimerad-Hohl, Antoine and Haine, Ghislain and Lefèvre, Laurent and Matignon, Denis},
  year={2025},
  pages={474--519}
}

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References

• 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
• Gay-Balmaz, F. & Yoshimura, H. A Lagrangian variational formulation for nonequilibrium thermodynamics. Part II: Continuum systems. Journal of Geometry and Physics 111, 194–212 (2017) – 10.1016/j.geomphys.2016.08.019
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42, 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• Brugnoli, A., Haine, G. & Matignon, D. Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: An analytical viewpoint. CAM 15, 362–387 (2023) – 10.3934/cam.2023018
• Rashad, R., Califano, F., van der Schaft, A. J. & Stramigioli, S. Twenty years of distributed port-Hamiltonian systems: a literature review. IMA Journal of Mathematical Control and Information 37, 1400–1422 (2020) – 10.1093/imamci/dnaa018
• Beattie, C., Mehrmann, V., Xu, H. & Zwart, H. Linear port-Hamiltonian descriptor systems. Math. Control Signals Syst. 30, (2018) – 10.1007/s00498-018-0223-3
• Zwart, H. & Mehrmann, V. Abstract Dissipative Hamiltonian Differential-Algebraic Equations Are Everywhere. DAE Panel 2, (2024) – 10.52825/dae-p.v2i.957
• Mehrmann, V. & Unger, B. Control of port-Hamiltonian differential-algebraic systems and applications. Acta Numerica 32, 395–515 (2023) – 10.1017/s0962492922000083
• van der Schaft, A. & Maschke, B. Generalized port-Hamiltonian DAE systems. Systems & Control Letters 121, 31–37 (2018) – 10.1016/j.sysconle.2018.09.008
• van der Schaft, A. & Maschke, B. Dirac and Lagrange Algebraic Constraints in Nonlinear Port-Hamiltonian Systems. Vietnam J. Math. 48, 929–939 (2020) – 10.1007/s10013-020-00419-x
• Yaghi, M., Couenne, F., Galfré, A., Lefèvre, L. & Maschke, B. Port Hamiltonian formulation of the solidification process for a pure substance: A phase field approach*. IFAC-PapersOnLine 55, 93–98 (2022) – 10.1016/j.ifacol.2022.08.036
• Jacob, B. & Morris, K. On Solvability of Dissipative Partial Differential-Algebraic Equations. IEEE Control Syst. Lett. 6, 3188–3193 (2022) – 10.1109/lcsys.2022.3183479
• Heidari, H. & Zwart, H. Port-Hamiltonian modelling of nonlocal longitudinal vibrations in a viscoelastic nanorod. Mathematical and Computer Modelling of Dynamical Systems 25, 447–462 (2019) – 10.1080/13873954.2019.1659374
• Heidari, H. & Zwart, H. Nonlocal longitudinal vibration in a nanorod, a system theoretic analysis. Math. Model. Nat. Phenom. 17, 24 (2022) – 10.1051/mmnp/2022028
• Bendimerad-Hohl, A., Haine, G., Lefèvre, L. & Matignon, D. Implicit port-Hamiltonian systems: structure-preserving discretization for the nonlocal vibrations in a viscoelastic nanorod, and for a seepage model. IFAC-PapersOnLine 56, 6789–6795 (2023) – 10.1016/j.ifacol.2023.10.387
• Krhač, K., Maschke, B. & van der Schaft, A. Port-Hamiltonian systems with energy and power ports. IFAC-PapersOnLine 58, 280–285 (2024) – 10.1016/j.ifacol.2024.08.294
• Schöberl, M. & Schlacher, K. First-order Hamiltonian field theory and mechanics. Mathematical and Computer Modelling of Dynamical Systems 17, 105–121 (2011) – 10.1080/13873954.2010.537526
• Schoberl, M. & Siuka, A. Analysis and comparison of port-Hamiltonian formulations for field theories - demonstrated by means of the Mindlin plate. 2013 European Control Conference (ECC) 548–553 (2013) doi:10.23919/ecc.2013.6669137 – 10.23919/ecc.2013.6669137
• Schöberl, M. & Siuka, A. Jet bundle formulation of infinite-dimensional port-Hamiltonian systems using differential operators. Automatica 50, 607–613 (2014) – 10.1016/j.automatica.2013.11.035
• Bendimerad-Hohl, A., Matignon, D., Haine, G. & Lefèvre, L. On Stokes-Lagrange and Stokes-Dirac representations for 1D distributed port-Hamiltonian systems. IFAC-PapersOnLine 58, 238–243 (2024) – 10.1016/j.ifacol.2024.10.174
• Kurula, M., Zwart, H., van der Schaft, A. & Behrndt, J. Dirac structures and their composition on Hilbert spaces. Journal of Mathematical Analysis and Applications 372, 402–422 (2010) – 10.1016/j.jmaa.2010.07.004
• Ghislain Haine, G. H., Denis Matignon, D. M. & Anass Serhani, A. S. Numerical Analysis of a Structure-Preserving Space-Discretization for an Anisotropic and Heterogeneous Boundary Controlled $N$-Dimensional Wave Equation as a Port-Hamiltonian System. IJNAM 20, 92–133 (2023) – 10.4208/ijnam2023-1005
• Eringen, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics 54, 4703–4710 (1983) – 10.1063/1.332803
• Romano, G., Barretta, R., Diaco, M. & Marotti de Sciarra, F. Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. International Journal of Mechanical Sciences 121, 151–156 (2017) – 10.1016/j.ijmecsci.2016.10.036
• Preuster, T., Maschke, B. & Schaller, M. Jet space extensions of infinite-dimensional Hamiltonian systems. IFAC-PapersOnLine 58, 298–303 (2024) – 10.1016/j.ifacol.2024.08.297
• Schöberl, M. & Schlacher, K. Lagrangian and Port-Hamiltonian formulation for Distributed-parameter systems. IFAC-PapersOnLine 48, 610–615 (2015) – 10.1016/j.ifacol.2015.05.025
• van der Schaft, A. & Maschke, B. Differential operator Dirac structures. IFAC-PapersOnLine 54, 198–203 (2021) – 10.1016/j.ifacol.2021.11.078
• Gangi, A. F. Constitutive equations and reciprocity. Geophysical Journal International 143, 311–318 (2000) – 10.1046/j.1365-246x.2000.01259.x
• Mehrmann, V. & van der Schaft, A. Differential–algebraic systems with dissipative Hamiltonian structure. Math. Control Signals Syst. 35, 541–584 (2023) – 10.1007/s00498-023-00349-2
• Ortega, R., van der Schaft, A., Castanos, F. & Astolfi, A. Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems. IEEE Trans. Automat. Contr. 53, 2527–2542 (2008) – 10.1109/tac.2008.2006930
• Erbay, M., Jacob, B. & Morris, K. On the Weierstraß form of infinite-dimensional differential algebraic equations. J. Evol. Equ. 24, (2024) – 10.1007/s00028-024-01003-3
• Geometric Numerical Integration. Springer Series in Computational Mathematics (Springer-Verlag, 2006). doi:10.1007/3-540-30666-8 – 10.1007/3-540-30666-8
• Mindlin, R. D. Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates. Journal of Applied Mechanics 18, 31–38 (1951) – 10.1115/1.4010217
• Brugnoli, A., Alazard, D., Pommier-Budinger, V. & Matignon, D. Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates. Applied Mathematical Modelling 75, 940–960 (2019) – 10.1016/j.apm.2019.04.035
• Ponce, C., Wu, Y., Le Gorrec, Y. & Ramirez, H. A systematic methodology for port-Hamiltonian modeling of multidimensional flexible linear mechanical systems. Applied Mathematical Modelling 134, 434–451 (2024) – 10.1016/j.apm.2024.05.040
• Brugnoli, A., Alazard, D., Pommier-Budinger, V. & Matignon, D. Port-Hamiltonian formulation and symplectic discretization of plate models Part II: Kirchhoff model for thin plates. Applied Mathematical Modelling 75, 961–981 (2019) – 10.1016/j.apm.2019.04.036
• Kovetz, A. & Visscher, P. B. Electromagnetic Theory. American Journal of Physics 69, 829–830 (2001) – 10.1119/1.1371014
• Arnold, D. N. Finite Element Exterior Calculus. (2018) doi:10.1137/1.9781611975543 – 10.1137/1.9781611975543
• Buffa, A., Ammari, H. & Nédélec, J. C. A Justification of Eddy Currents Model for the Maxwell Equations. SIAM J. Appl. Math. 60, 1805–1823 (2000) – 10.1137/s0036139998348979
• Reis, T. & Stykel, T. Passivity, port-hamiltonian formulation and solution estimates for a coupled magneto-quasistatic system. EECT 12, 1208–1232 (2023) – 10.3934/eect.2023008
• Cardoso-Ribeiro, F. L., Haine, G., Le Gorrec, Y., Matignon, D. & Ramirez, H. Port-Hamiltonian formulations for the modeling, simulation and control of fluids. Computers & Fluids 283, 106407 (2024) – 10.1016/j.compfluid.2024.106407
• Haine, G. & Matignon, D. Incompressible Navier-Stokes Equation as port-Hamiltonian systems: velocity formulation versus vorticity formulation. IFAC-PapersOnLine 54, 161–166 (2021) – 10.1016/j.ifacol.2021.11.072
• Borja, P., Ferguson, J. & van der Schaft, A. Interconnection Schemes in Modeling and Control. IEEE Control Syst. Lett. 7, 2287–2292 (2023) – 10.1109/lcsys.2023.3286124
• Macchelli, A., van der Schaft, A. J. & Melchiorri, C. Multi-variable port Hamiltonian model of piezoelectric material. 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (IEEE Cat. No.04CH37566) vol. 1 897–902 – 10.1109/iros.2004.1389466
• Macchelli, A. Energy shaping of distributed parameter port-Hamiltonian systems based on finite element approximation. Systems & Control Letters 60, 579–589 (2011) – 10.1016/j.sysconle.2011.04.016