Exponential Stability and Tuning for a Class of Mechanical Systems

Authors

Carmen Chan-Zheng, Pablo Borja, Nima Monshizadeh, Jacquelien M.A. Scherpen

Abstract

In this paper, we prove the exponential stability property of a class of mechanical systems represented in the port-Hamiltonian framework. To this end, we propose a Lyapunov candidate function different from the Hamiltonian of the system. Moreover, we study how the proposed analysis can be used to determine the exponential stability and the rate of convergence of some (nonlinear)-mechanical systems stabilized by a passivity-based control technique, namely, PID passivity-based control. We implement such a control approach to stabilize a three-degree-of-freedom robotic arm at the desired equilibrium point to illustrate the mentioned analysis.

Citation

• Journal: 2021 European Control Conference (ECC)
• Year: 2021
• Volume:
• Issue:
• Pages: 1875–1880
• Publisher: IEEE
• DOI: 10.23919/ecc54610.2021.9655218

BibTeX

@inproceedings{Chan_Zheng_2021,
  title={{Exponential Stability and Tuning for a Class of Mechanical Systems}},
  DOI={10.23919/ecc54610.2021.9655218},
  booktitle={{2021 European Control Conference (ECC)}},
  publisher={IEEE},
  author={Chan-Zheng, Carmen and Borja, Pablo and Monshizadeh, Nima and Scherpen, Jacquelien M.A.},
  year={2021},
  pages={1875--1880}
}

Download the bib file

References

• Delgado, S. & Kotyczka, P. Overcoming the Dissipation Condition in Passivity-based Control for a class of mechanical systems. IFAC Proceedings Volumes 47, 11189–11194 (2014) – 10.3182/20140824-6-za-1003.00499
• nesterov, Introductory Lectures on Convex Optimization A Basic Course (2013)
• Boyd, S. & Vandenberghe, L. Convex Optimization. (2004) doi:10.1017/cbo9780511804441 – 10.1017/cbo9780511804441
• Horn, R. A. & Johnson, C. R. Matrix Analysis. (2012) doi:10.1017/cbo9781139020411 – 10.1017/cbo9781139020411
• khalil, Nonlinear Systems (2002)
• Zhang, M., Borja, P., Ortega, R., Liu, Z. & Su, H. PID Passivity-Based Control of Port-Hamiltonian Systems. IEEE Trans. Automat. Contr. 63, 1032–1044 (2018) – 10.1109/tac.2017.2732283
• Borja, P., Ortega, R. & Scherpen, J. M. A. New Results on Stabilization of Port-Hamiltonian Systems via PID Passivity-Based Control. IEEE Trans. Automat. Contr. 66, 625–636 (2021) – 10.1109/tac.2020.2986731
• Romero, J. G., Donaire, A., Ortega, R. & Borja, P. Global stabilisation of underactuated mechanical systems via PID passivity-based control. Automatica 96, 178–185 (2018) – 10.1016/j.automatica.2018.06.040
• Ghorbel, F., Srinivasan, B. & Spong, M. W. On the positive definiteness and uniform boundedness of the inertia matrix of robot manipulators. Proceedings of 32nd IEEE Conference on Decision and Control 1103–1108 doi:10.1109/cdc.1993.325355 – 10.1109/cdc.1993.325355
• rijs, Philips experimental robot arm: User instructor manual. Koninklijke Philips Electronics N V Eindhoven (2010)
• Venkatraman, A., Ortega, R., Sarras, I. & van der Schaft, A. Speed Observation and Position Feedback Stabilization of Partially Linearizable Mechanical Systems. IEEE Trans. Automat. Contr. 55, 1059–1074 (2010) – 10.1109/tac.2010.2042010
• 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
• Ferguson, J., Donaire, A. & Middleton, R. H. Kinetic-Potential Energy Shaping for Mechanical Systems With Applications to Tracking. IEEE Control Syst. Lett. 3, 960–965 (2019) – 10.1109/lcsys.2019.2919842
• Romero, J. G., Ortega, R. & Sarras, I. A Globally Exponentially Stable Tracking Controller for Mechanical Systems Using Position Feedback. IEEE Trans. Automat. Contr. 60, 818–823 (2015) – 10.1109/tac.2014.2330701
• Venkatraman, A., Ortega, R., Sarras, I. & van der Schaft, A. Control of underactuated mechanical systems: Observer design and position feedback stabilization. 2008 47th IEEE Conference on Decision and Control 4969–4975 (2008) doi:10.1109/cdc.2008.4738912 – 10.1109/cdc.2008.4738912
• Gómez-Estern, F. & Van der Schaft, A. J. Physical Damping in IDA-PBC Controlled Underactuated Mechanical Systems. European Journal of Control 10, 451–468 (2004) – 10.3166/ejc.10.451-468
• 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
• duindam, Modeling and control of complex physical systems: the port-Hamiltonian approach. (2009)
• Acosta, J. A., Panteley, E. & Ortega, R. A new strict Lyapunov function for fully-actuated mechanical systems controlled by IDA-PBC. 2009 IEEE International Conference on Control Applications 519–524 (2009) doi:10.1109/cca.2009.5280704 – 10.1109/cca.2009.5280704