Stabilization of Port-Hamiltonian Systems Based on Shifted Passivity via Feedback

Authors

Chengshuai Wu, Arjan van der Schaft, Jian Chen

Abstract

In this article, stabilization of port-Hamiltonian (pH) systems is studied in the context of shifted passivity. The pH systems, in general, are not shifted passive with respect to the given Hamiltonian and output. Therefore, we consider enforcing the property of shifted passivity by designing a state feedback, namely, shifted passivity via feedback. This introduced concept brings more insight into the stabilization of pH systems, and a set of partial differential equations is derived such that its solution specifies a state feedback achieving shifted passivity via feedback. Based on the framework of shifted passivity via feedback, a group of sufficient conditions is given such that a proposed proportional control with respect to a designed power shaping output can achieve closed-loop asymptotic stability. Some extended control design is provided to relax the required conditions.

Citation

• Journal: IEEE Transactions on Automatic Control
• Year: 2021
• Volume: 66
• Issue: 5
• Pages: 2219–2226
• Publisher: Institute of Electrical and Electronics Engineers (IEEE)
• DOI: 10.1109/tac.2020.3005156

BibTeX

@article{Wu_2021,
  title={{Stabilization of Port-Hamiltonian Systems Based on Shifted Passivity via Feedback}},
  volume={66},
  ISSN={2334-3303},
  DOI={10.1109/tac.2020.3005156},
  number={5},
  journal={IEEE Transactions on Automatic Control},
  publisher={Institute of Electrical and Electronics Engineers (IEEE)},
  author={Wu, Chengshuai and van der Schaft, Arjan and Chen, Jian},
  year={2021},
  pages={2219--2226}
}

Download the bib file

References

• Ortega, R., Jeltsema, D. & Scherpen, J. M. A. Power shaping: A new paradigm for stabilization of nonlinear RLC circuits. IEEE Transactions on Automatic Control vol. 48 1762–1767 (2003) – 10.1109/tac.2003.817918
• Acosta, J. A., Ortega, R., Astolfi, A. & Mahindrakar, A. D. Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one. IEEE Transactions on Automatic Control vol. 50 1936–1955 (2005) – 10.1109/tac.2005.860292
• Ortega, R. & García-Canseco, E. Interconnection and Damping Assignment Passivity-Based Control: A Survey. European Journal of Control vol. 10 432–450 (2004) – 10.3166/ejc.10.432-450
• Jayawardhana, B., Ortega, R., García-Canseco, E. & Castaños, F. Passivity of nonlinear incremental systems: Application to PI stabilization of nonlinear RLC circuits. Systems & Control Letters vol. 56 618–622 (2007) – 10.1016/j.sysconle.2007.03.011
• Monshizadeh, N., Monshizadeh, P., Ortega, R. & van der Schaft, A. Conditions on shifted passivity of port-Hamiltonian systems. Systems & Control Letters vol. 123 55–61 (2019) – 10.1016/j.sysconle.2018.10.010
• Ferguson, J., Middleton, R. H. & Donaire, A. Disturbance rejection via control by interconnection of port-Hamiltonian systems. 2015 54th IEEE Conference on Decision and Control (CDC) 507–512 (2015) doi:10.1109/cdc.2015.7402279 – 10.1109/cdc.2015.7402279
• Bregman, L. M. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics vol. 7 200–217 (1967) – 10.1016/0041-5553(67)90040-7
• Maschke, B., Ortega, R. & Van Der Schaft, A. J. Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Transactions on Automatic Control vol. 45 1498–1502 (2000) – 10.1109/9.871758
• fulwani, Mitigation of Negative Impedance Instabilities in DC Distribution Systems A Sliding Mode Control Approach (2016)
• He, W., Ortega, R., Machado, J. E. & Li, S. An Adaptive Passivity‐Based Controller of a Buck‐Boost Converter with a Constant Power Load. Asian Journal of Control vol. 21 581–595 (2018) – 10.1002/asjc.1751
• Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Energy-shaping of port-controlled Hamiltonian systems by interconnection. Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304) vol. 2 1646–1651 – 10.1109/cdc.1999.830260
• Wu, C., Chen, J., Xu, C. & Liu, Z. Real-Time Adaptive Control of a Fuel Cell/Battery Hybrid Power System With Guaranteed Stability. IEEE Transactions on Control Systems Technology vol. 25 1394–1405 (2017) – 10.1109/tcst.2016.2611558
• Ortega, R., van der Schaft, A. J. & Maschke, B. M. Stabilization of port-controlled Hamiltonian systems via energy balancing. Lecture Notes in Control and Information Sciences 239–260 (1999) doi:10.1007/1-84628-577-1_13 – 10.1007/1-84628-577-1_13
• Zhang, M., Borja, P., Ortega, R., Liu, Z. & Su, H. PID Passivity-Based Control of Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 63 1032–1044 (2018) – 10.1109/tac.2017.2732283
• Ortega, R., van der Schaft, A., Castanos, F. & Astolfi, A. Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 53 2527–2542 (2008) – 10.1109/tac.2008.2006930
• Castaños, F., Ortega, R., van der Schaft, A. & Astolfi, A. Asymptotic stabilization via control by interconnection of port-Hamiltonian systems. Automatica vol. 45 1611–1618 (2009) – 10.1016/j.automatica.2009.03.015
• 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
• van der Schaft, A. L2-Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer International Publishing, 2017). doi:10.1007/978-3-319-49992-5 – 10.1007/978-3-319-49992-5
• García-Canseco, E., Jeltsema, D., Ortega, R. & Scherpen, J. M. A. Power-based control of physical systems. Automatica vol. 46 127–132 (2010) – 10.1016/j.automatica.2009.10.012
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. (2014) doi:10.1561/9781601987877 – 10.1561/9781601987877
• 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
• Sbarbaro, D. & Ortega, R. Averaging level control: An approach based on mass balance. Journal of Process Control vol. 17 621–629 (2007) – 10.1016/j.jprocont.2007.01.005
• farina, Positive Linear Systems Theory and Applications (2011)
• Nijmeijer, H. & van der Schaft, A. Nonlinear Dynamical Control Systems. (Springer New York, 1990). doi:10.1007/978-1-4757-2101-0 – 10.1007/978-1-4757-2101-0
• Cheng, D., Astolfi, A. & Ortega, R. On feedback equivalence to port controlled Hamiltonian systems. Systems & Control Letters vol. 54 911–917 (2005) – 10.1016/j.sysconle.2005.02.005
• Berman, A. & Plemmons, R. J. Nonnegative Matrices in the Mathematical Sciences. (1994) doi:10.1137/1.9781611971262 – 10.1137/1.9781611971262