Proportional observer design for port Hamiltonian systems using the contraction analysis approach

Authors

Saida Zenfari, Mohamed Laabissi, Mohammed Elarbi Achhab

Abstract

A simple proportional observer design method is presented for a class of linear port Hamiltonian systems. This observer design approach is based on the use of a powerful tool derived from continuum mechanics and differential geometry, known as contraction analysis. Under two verifiable assumptions, it is shown that error dynamics between the plant and the observer states converges exponentially to zero. Finally, our design method is applied to two physical systems arising from different domains: The DC motor and the RLC circuit.

Keywords

Port Hamiltonian systems; Observer design; Contraction analysis; Lyapunov equation; Quadratic Hamiltonian function

Citation

• Journal: International Journal of Dynamics and Control
• Year: 2022
• Volume: 10
• Issue: 2
• Pages: 403–408
• Publisher: Springer Science and Business Media LLC
• DOI: 10.1007/s40435-021-00830-3

BibTeX

@article{Zenfari_2021,
  title={{Proportional observer design for port Hamiltonian systems using the contraction analysis approach}},
  volume={10},
  ISSN={2195-2698},
  DOI={10.1007/s40435-021-00830-3},
  number={2},
  journal={International Journal of Dynamics and Control},
  publisher={Springer Science and Business Media LLC},
  author={Zenfari, Saida and Laabissi, Mohamed and Achhab, Mohammed Elarbi},
  year={2021},
  pages={403--408}
}

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References

• Duindam V, Macchelli A, Stramigioli S, Bruyninckx H (2009) Modeling and Control of Complex Physical Systems. Springer Berlin Heidelberg, Berlin, Heidelberg – 10.1007/978-3-642-03196-0
• Maschke BM, Van Der Schaft AJ, Breedveld PC (1992) An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators. Journal of the Franklin Institute 329(5):923–966. https://doi.org/10.1016/s0016-0032(92)90049-m – 10.1016/s0016-0032(92)90049-m
• Van Der Schaft A, Maschke B (1995) The Hamiltonian formulation of energy conserving physical systems with external ports. AEÜ Int J Electron Commun 49:362–371
• 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/10.1561/2600000002 – 10.1561/2600000002
• LOHMILLER W, SLOTINE J-JE (1998) On Contraction Analysis for Non-linear Systems. Automatica 34(6):683–696. https://doi.org/10.1016/s0005-1098(98)00019-3 – 10.1016/s0005-1098(98)00019-3
• Shim H, Seo JH, Teel AR (2003) Nonlinear observer design via passivation of error dynamics. Automatica 39(5):885–892. https://doi.org/10.1016/s0005-1098(03)00023-2 – 10.1016/s0005-1098(03)00023-2
• Venkatraman A, van der Schaft AJ (2010) Full-order observer design for a class of port-Hamiltonian systems. Automatica 46(3):555–561. https://doi.org/10.1016/j.automatica.2010.01.019 – 10.1016/j.automatica.2010.01.019
• Vincent B, Hudon N, Lefèvre L, Dochain D (2016) Port-Hamiltonian observer design for plasma profile estimation in tokamaks. IFAC-PapersOnLine 49(24):93–98. https://doi.org/10.1016/j.ifacol.2016.10.761 – 10.1016/j.ifacol.2016.10.761
• Wang Y, Ge SS, Cheng D (2005) Observer and observer-based H\(\infty\) control of generalized Hamiltonian systems. Sci China Ser F 48(2):211–224
• Bakhshande F, Söffker D (2015) Proportional-Integral-Observer: A brief survey with special attention to the actual methods using ACC Benchmark. IFAC-PapersOnLine 48(1):532–537. https://doi.org/10.1016/j.ifacol.2015.05.049 – 10.1016/j.ifacol.2015.05.049
• Borase RP, Maghade DK, Sondkar SY, Pawar SN (2020) A review of PID control, tuning methods and applications. Int J Dynam Control 9(2):818–827. https://doi.org/10.1007/s40435-020-00665-4 – 10.1007/s40435-020-00665-4
• LOHMILLER W, SLOTINE J-JE (1998) On Contraction Analysis for Non-linear Systems. Automatica 34(6):683–696. https://doi.org/10.1016/s0005-1098(98)00019-3 – 10.1016/s0005-1098(98)00019-3
• Lohmiller W, Slotine J-JE (2000) Control system design for mechanical systems using contraction theory. IEEE Trans Automat Contr 45(5):984–989. https://doi.org/10.1109/9.855568 – 10.1109/9.855568
• van der Schaft A (2000) L2 - Gain and Passivity Techniques in Nonlinear Control. Springer London, London – 10.1007/978-1-4471-0507-7
• Ortega R, García-Canseco E (2004) Interconnection and Damping Assignment Passivity-Based Control: A Survey. European Journal of Control 10(5):432–450. https://doi.org/10.3166/ejc.10.432-450 – 10.3166/ejc.10.432-450
• Khalil HK (2002) Nonlinear systems, 3rd edn. Prentice Hall, New York
• Van Der Schaft A, Maschke B (1995) The Hamiltonian formulation of energy conserving physical systems with external ports. AEÜ Int J Electron Commun 49:362–371
• 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/10.1561/2600000002 – 10.1561/2600000002
• Venkatraman A, van der Schaft AJ (2010) Full-order observer design for a class of port-Hamiltonian systems. Automatica 46(3):555–561. https://doi.org/10.1016/j.automatica.2010.01.019 – 10.1016/j.automatica.2010.01.019
• Medianu S, Lefèvre L (2021) Structural identifiability of linear Port Hamiltonian systems. Systems & Control Letters 151:104915. https://doi.org/10.1016/j.sysconle.2021.104915 – 10.1016/j.sysconle.2021.104915
• NAVARRO D, CORTES D, GALAZ-LARIOS M (2017) A Port-Hamiltonian Approach to Control DC-DC Power Converters. STUD INFORM CONTROL 26(3). https://doi.org/10.24846/v26i3y201702 – 10.24846/v26i3y201702
• Zhang M, Ortega R, Jeltsema D, Su H (2015) Further deleterious effects of the dissipation obstacle in control-by-interconnection of port-Hamiltonian systems. Automatica 61:227–231. https://doi.org/10.1016/j.automatica.2015.08.010 – 10.1016/j.automatica.2015.08.010