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Passivity-Based Control of Mechanical Systems

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Feedback Stabilization of Controlled Dynamical Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 473))

Abstract

Stabilization of mechanical systems by shaping their energy function is a well-established technique whose roots date back to the work of Lagrange and Dirichlet. Ortega and Spong in 1989 proved that passivity is the key property underlying the stabilization mechanism of energy shaping designs and the, now widely popular, term of passivity-based control (PBC) was coined. In this chapter, we briefly recall the history of PBC of mechanical systems and summarize its main recent developments. The latter includes: (i) an explicit formula for one of the free tuning gains that simplifies the computations, (ii) addition of PID controllers to robustify and make constructive the PBC design and to track ramp references, (iii) use of PBC to solve the position feedback global tracking problem, and (iv) design of robust and adaptive speed observers.

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Notes

  1. 1.

    See Remark 5.6 of [10].

  2. 2.

    To avoid cluttering the notation we call this additional signal also u.

  3. 3.

    Recall that \({d \nabla V_d \over dt}=\nabla ^2 V_d M^{-1}p\).

  4. 4.

    To avoid cluttering we use the same symbol to denote the energy function in all cases.

  5. 5.

    See Sect. 7.5.1 for a robust adaptive version of the I&I momenta observer.

  6. 6.

    The notation for the momenta is different from the one used in Sect. 7.4, but is consistent with the one used in [56].

  7. 7.

    See Eqs. (6) and (7) of [67] for the definition of these symbols and the proof of this fact.

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Acknowledgements

R. Ortega is supported by Government of Russian Federation (grant 074-U01, GOSZADANIE 2014/190 (project 2118)), the Ministry of Education and Science of Russian Federation (project 14.Z50.31.0031). The work of J.G. Romero is supported by a public grant overseen by the French National Research Agency (ANR) as part of the Investissement d’Avenir program, through the iCODE Institute, research project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02.

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Authors and Affiliations

  1. Laboratoire des Signaux et Systèmes, CNRS–SUPELEC, Plateau du Moulon, 91192, Gif–sur–Yvette, France

    Romeo Ortega

  2. PRISMA Lab, Dipartimento di Ingegneria Elettrica e Tecnologie dell’Informazione, Università di Napoli Federico II, Via Claudio 21, 80125, Naples, Italy

    Alejandro Donaire

  3. School of Engineering, The University of Newcastle, Callaghan, Australia

    Alejandro Donaire

  4. Departamento Académico de Sistemas Digitales, Instituto Tecnológico Autónomo de México-ITAM, Rio Hondo No. 1, 01080, Distrito Federal, Mexico

    Jose Guadalupe Romero

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  1. Romeo Ortega
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  3. Jose Guadalupe Romero
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Correspondence to Romeo Ortega .

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  1. Centre Automatique et Systèmes CAS, MINES ParisTech, PSL Research University, Centre Automatique et Systèmes CAS, Paris Cedex 06, France

    Nicolas Petit

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Ortega, R., Donaire, A., Romero, J.G. (2017). Passivity-Based Control of Mechanical Systems. In: Petit, N. (eds) Feedback Stabilization of Controlled Dynamical Systems. Lecture Notes in Control and Information Sciences, vol 473. Springer, Cham. https://doi.org/10.1007/978-3-319-51298-3_7

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