Observer Design for a Class of Discrete Port Hamiltonian Systems

Authors

Saida Zenfari, Mohamed Laabissi, Mohammed Elarbi Achhab

Abstract

In this paper, a simple observer design method is presented for a class of discrete-time port Hamiltonian systems. The proposed observer is full order, and it is a copy of the original system dynamics with a corrective term. The suggested design methodology benefits from the port Hamiltonian framework properties. Based on a convenient assumption and exploiting the fact that the observer is structure preserving, the error between the plant and the observer converges exponentially to zero. The key tool in achieving our goal is the contraction analysis method. The observer design method is illustrated in the RLC circuit. We show that the presented approach may be applied in the inverted pendulum example. For both examples, some simulation results are presented.

Keywords

Discrete port Hamiltonian systems; Observer design; Contraction analysis; Lyapunov equation

Citation

• Journal: Journal of Control, Automation and Electrical Systems
• Year: 2023
• Volume: 34
• Issue: 5
• Pages: 963–970
• Publisher: Springer Science and Business Media LLC
• DOI: 10.1007/s40313-023-01017-1

BibTeX

@article{Zenfari_2023,
  title={{Observer Design for a Class of Discrete Port Hamiltonian Systems}},
  volume={34},
  ISSN={2195-3899},
  DOI={10.1007/s40313-023-01017-1},
  number={5},
  journal={Journal of Control, Automation and Electrical Systems},
  publisher={Springer Science and Business Media LLC},
  author={Zenfari, Saida and Laabissi, Mohamed and Achhab, Mohammed Elarbi},
  year={2023},
  pages={963--970}
}

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References

• 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
• Aoues, S., Eberard, D. & Marquis-Favre, W. Discrete IDA-PBC design for 2D port-Hamiltonian systems. IFAC Proceedings Volumes vol. 46 134–139 (2013) – 10.3182/20130904-3-fr-2041.00088
• B Biedermann. Biedermann, B., & Meurer, T. (2021). Observer design for a class of nonlinear systems combining dissipativity with interconnection and damping assignment. International Journal of Robust and Nonlinear Control, 3102, 188–193. (2021)
• Goldberg, M. & Zwas, G. On matrices having equal spectral radius and spectral norm. Linear Algebra and its Applications vol. 8 427–434 (1974) – 10.1016/0024-3795(74)90076-7
• Itoh, T. & Abe, K. Hamiltonian-conserving discrete canonical equations based on variational difference quotients. Journal of Computational Physics vol. 76 85–102 (1988) – 10.1016/0021-9991(88)90132-5
• Jouffroy, J. (2006). Some ancestors of contraction analysis. In Proceedings of the 44th IEEE conference on decision and control (pp. 5450–5455).
• P Kotyczka. Kotyczka, P., & Wang, M. (2015). Dual observer-based compensator design for linear port-hamiltonian systems. European Control Conference, 5, 2908–2913. (2015)
• Laila, D. S. & Astolfi, A. DISCRETE-TIME IDA-PBC DESIGN FOR SEPARABLE HAMILTONIAN SYSTEMS. IFAC Proceedings Volumes vol. 38 838–843 (2005) – 10.3182/20050703-6-cz-1902.00540
• DS Laila. Laila, D. S., & Astolfi, A. (2006). Discrete-time ida-pbc design for underactuated hamiltonian control systems. American Control Conference, 5, 188–193. (2006)
• 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
• Maschke, B. M., Van Der Schaft, A. J. & Breedveld, P. C. An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators. Journal of the Franklin Institute vol. 329 923–966 (1992) – 10.1016/s0016-0032(92)90049-m
• 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
• 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
• Pfeifer, M., Caspart, S., Strehle, F. & Hohmann, S. Full-Order Observer Design for a Class of Nonlinear Port-Hamiltonian Systems. IFAC-PapersOnLine vol. 54 149–154 (2021) – 10.1016/j.ifacol.2021.11.070
• Rojas, M., Granados-Salazar, C. & Espinosa-Pérez, G. Observer Design for a Class of Nonlinear Hamiltonian Systems. IFAC-PapersOnLine vol. 54 125–130 (2021) – 10.1016/j.ifacol.2021.11.066
• Stramigioli, S., Secchi, C., van der Schaft, A. J. & Fantuzzi, C. Sampled data systems passivity and discrete port-Hamiltonian systems. IEEE Transactions on Robotics vol. 21 574–587 (2005) – 10.1109/tro.2004.842330
• 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
• Zenfari, S., Laabissi, M. & Achhab, M. E. Proportional observer design for port Hamiltonian systems using the contraction analysis approach. International Journal of Dynamics and Control vol. 10 403–408 (2021) – 10.1007/s40435-021-00830-3