Port-Hamiltonian descriptor systems are relative generically controllable and stabilizable

Authors

Achim Ilchmann, Jonas Kirchhoff, Manuel Schaller

Abstract

The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and of Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2022) on relative generic controllability of linear differential-algebraic equations. We extend the result from general, unstructured differential-algebraic equations to differential-algebraic equations of port-Hamiltonian type. We derive results on relative genericity. These findings are the basis for characterizing relative generic controllability of port-Hamiltonian systems in terms of dimensions. A similar result is proved for relative generic stabilizability.

Keywords

Differential-algebraic equation; Port-Hamiltonian system; Controllability; Stabilizability; Genericity; Relative genericity; 34A09; 93B05

Citation

• Journal: Mathematics of Control, Signals, and Systems
• Year: 2025
• Volume: 37
• Issue: 1
• Pages: 23–59
• Publisher: Springer Science and Business Media LLC
• DOI: 10.1007/s00498-024-00392-7

BibTeX

@article{Ilchmann_2024,
  title={{Port-Hamiltonian descriptor systems are relative generically controllable and stabilizable}},
  volume={37},
  ISSN={1435-568X},
  DOI={10.1007/s00498-024-00392-7},
  number={1},
  journal={Mathematics of Control, Signals, and Systems},
  publisher={Springer Science and Business Media LLC},
  author={Ilchmann, Achim and Kirchhoff, Jonas and Schaller, Manuel},
  year={2024},
  pages={23--59}
}

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References

• Achleitner, F., Arnold, A. & Mehrmann, V. Hypocoercivity and controllability in linear semi‐dissipative Hamiltonian ordinary differential equations and differential‐algebraic equations. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik vol. 103 (2021) – 10.1002/zamm.202100171
• Berger, T. & Reis, T. Controllability of Linear Differential-Algebraic Systems—A Survey. Surveys in Differential-Algebraic Equations I 1–61 (2013) doi:10.1007/978-3-642-34928-7_1 – 10.1007/978-3-642-34928-7_1
• Bernstein, D. S. Scalar, Vector, and Matrix Mathematics. (2018) doi:10.1515/9781400888252 – 10.1515/9781400888252
• G Birkhoff. Birkhoff G, MacLane S (1988) Algebra, 3rd edn. Chelsea, New York (1988)
• 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
• 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
• J Dugundji. Dugundji J (1970) Topoloy, 5th edn. Allyn and Bacon, Boston (1970)
• Faulwasser T, Kirchhoff J, Mehrmann V, Philipp F, Schaller M, Worthmann K (2023) Hidden regularity in singular optimal control of port-Hamiltonian systems
• T Faulwasser. Faulwasser T, Maschke B, Philipp F, Schaller M, Worthmann K (2021) Control of port-Hamiltonian systems with minimal energy supply. Eur J Control 60:33–40 (2021)
• Faulwasser, T., Maschke, B., Philipp, F., Schaller, M. & Worthmann, K. Optimal Control of Port-Hamiltonian Descriptor Systems with Minimal Energy Supply. SIAM Journal on Control and Optimization vol. 60 2132–2158 (2022) – 10.1137/21m1427723
• H Federer. Federer H (1969) Geometric measure theory. Springer, Berlin (1969)
• Fischer, G. Lineare Algebra. (Vieweg+Teubner Verlag, 2005). doi:10.1007/978-3-322-93424-6 – 10.1007/978-3-322-93424-6
• Fuhrmann, P. A. A Polynomial Approach to Linear Algebra. Universitext (Springer New York, 2012). doi:10.1007/978-1-4614-0338-8 – 10.1007/978-1-4614-0338-8
• Hauschild, S.-A. et al. Port-Hamiltonian Modeling of District Heating Networks. Differential-Algebraic Equations Forum 333–355 (2020) doi:10.1007/978-3-030-53905-4_11 – 10.1007/978-3-030-53905-4_11
• Hinrichsen, D. & Oeljeklaus, E. The set of controllable multi-input systems is generically convex. Mathematics of Control, Signals, and Systems vol. 31 265–278 (2019) – 10.1007/s00498-019-0243-7
• Hinrichsen, D. & Oeljeklaus, E. Are delay-differential systems generically controllable? Mathematics of Control, Signals, and Systems vol. 34 679–714 (2022) – 10.1007/s00498-022-00329-y
• Horn, R. A. & Johnson, C. R. Matrix Analysis. (2012) doi:10.1017/cbo9781139020411 – 10.1017/cbo9781139020411
• Ilchmann, A. & Kirchhoff, J. Differential-algebraic systems are generically controllable and stabilizable. Mathematics of Control, Signals, and Systems vol. 33 359–377 (2021) – 10.1007/s00498-021-00287-x
• Ilchmann, A. & Kirchhoff, J. Relative genericity of controllablity and stabilizability for differential-algebraic systems. Mathematics of Control, Signals, and Systems vol. 35 45–76 (2022) – 10.1007/s00498-022-00332-3
• Jacob, B. & Zwart, H. J. Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. (Springer Basel, 2012). doi:10.1007/978-3-0348-0399-1 – 10.1007/978-3-0348-0399-1
• RE Kalman. Kalman RE (1960) Contributions to the theory of optimal control. Bol Soc Matem Mexico II Ser 5:102–119 (1960)
• Kalman RE (1961) On the general theory of control systems. In: Proceedings of the first international congress on automatic control, Moscow 1960, pp 481–493. Butterworth’s, London
• Kirchhoff, J. Linear Port-Hamiltonian Systems Are Generically Controllable. IEEE Transactions on Automatic Control vol. 67 3220–3222 (2022) – 10.1109/tac.2021.3098176
• 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
• Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
• EB Lee. Lee EB, Markus L (1967) Foundations of optimal control theory. Wiley, New York (1967)
• Louati, H. et al. Network‐Based Modeling of Transport Phenomena in Solid and Fluid Phases of Open‐Cell Foams: Construction of Graphs. Advanced Engineering Materials vol. 22 (2020) – 10.1002/adem.201901468
• Maschke, B. M. & van der Schaft, A. J. PORT-CONTROLLED HAMILTONIAN SYSTEMS: MODELLING ORIGINS AND SYSTEMTHEORETIC PROPERTIES. Nonlinear Control Systems Design 1992 359–365 (1993) doi:10.1016/b978-0-08-041901-5.50064-6 – 10.1016/b978-0-08-041901-5.50064-6
• van der Schaft, A. & Maschke, B. Dirac and Lagrange Algebraic Constraints in Nonlinear Port-Hamiltonian Systems. Vietnam Journal of Mathematics vol. 48 929–939 (2020) – 10.1007/s10013-020-00419-x
• Mehl, C., Mehrmann, V., Ran, A. C. M. & Rodman, L. Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations. Linear Algebra and its Applications vol. 435 687–716 (2011) – 10.1016/j.laa.2010.07.025
• Mehl, C., Mehrmann, V. & Wojtylak, M. Distance problems for dissipative Hamiltonian systems and related matrix polynomials. Linear Algebra and its Applications vol. 623 335–366 (2021) – 10.1016/j.laa.2020.05.026
• Mehrmann, V. & Unger, B. Control of port-Hamiltonian differential-algebraic systems and applications. Acta Numerica vol. 32 395–515 (2023) – 10.1017/s0962492922000083
• 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
• Oxtoby, J. C. Measure and Category. Graduate Texts in Mathematics (Springer US, 1971). doi:10.1007/978-1-4615-9964-7 – 10.1007/978-1-4615-9964-7
• 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 vol. 37 1400–1422 (2020) – 10.1093/imamci/dnaa018
• Reich, S. & Zaslavski, A. J. Genericity in Nonlinear Analysis. Developments in Mathematics (Springer New York, 2014). doi:10.1007/978-1-4614-9533-8 – 10.1007/978-1-4614-9533-8
• M Reid. Reid M (1998) Undergraduate algebraic geometry. Cambridge University Press, Cambridge (1998)
• Rockafellar, R. T. Convex Analysis. (1970) doi:10.1515/9781400873173 – 10.1515/9781400873173
• van der Schaft, A. L2 - Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer London, 2000). doi:10.1007/978-1-4471-0507-7 – 10.1007/978-1-4471-0507-7
• 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
• Vovk, V. Itô Calculus without Probability in Idealized Financial Markets*. Lithuanian Mathematical Journal vol. 55 270–290 (2015) – 10.1007/s10986-015-9280-1
• Warsewa, A., Böhm, M., Sawodny, O. & Tarín, C. A port-Hamiltonian approach to modeling the structural dynamics of complex systems. Applied Mathematical Modelling vol. 89 1528–1546 (2021) – 10.1016/j.apm.2020.07.038
• Wonham, W. M. Linear Multivariable Control. Lecture Notes in Economics and Mathematical Systems (Springer Berlin Heidelberg, 1974). doi:10.1007/978-3-662-22673-5 – 10.1007/978-3-662-22673-5