PID Passivity-Based Control of Port-Hamiltonian Systems

Authors

Meng Zhang, Pablo Borja, Romeo Ortega, Zhitao Liu, Hongye Su

Abstract

In this note, we address the problem of stabilization of port-Hamiltonian systems via the ubiquitous proportional-integral-derivative (PID) controller. The design is based on passivity theory, hence the first step is to identify all passive outputs of the system, which is the first contribution of the paper. Adding a PID around this signal ensures that the closed-loop system is \( {\mathcal L}_2 \)-stable for all positive PID gains. Global stability (and/or global attractivity) of a desired constant equilibrium is also guaranteed for a new class of systems for which a Lyapunov function can be constructed. A second contribution is to prove that this class—that is identified via some easily verifiable integrability conditions—is strictly larger than the ones previously reported in the literature. Comparisons of the proposed PID controller with control-by-interconnection passivity-based control are also discussed.

Citation

• Journal: IEEE Transactions on Automatic Control
• Year: 2018
• Volume: 63
• Issue: 4
• Pages: 1032–1044
• Publisher: Institute of Electrical and Electronics Engineers (IEEE)
• DOI: 10.1109/tac.2017.2732283

BibTeX

@article{Zhang_2018,
  title={{PID Passivity-Based Control of Port-Hamiltonian Systems}},
  volume={63},
  ISSN={1558-2523},
  DOI={10.1109/tac.2017.2732283},
  number={4},
  journal={IEEE Transactions on Automatic Control},
  publisher={Institute of Electrical and Electronics Engineers (IEEE)},
  author={Zhang, Meng and Borja, Pablo and Ortega, Romeo and Liu, Zhitao and Su, Hongye},
  year={2018},
  pages={1032--1044}
}

Download the bib file

References

• venkatraman, Energy shaping of port-Hamiltonian systems by using alternate passive outputs. Proc Eur Control Conf (2009)
• van der Schaft A (2016) Interconnections of input-output Hamiltonian systems with dissipation. In: 2016 IEEE 55th Conference on Decision and Control (CDC). IEEE, pp 4686–4691 – 10.1109/cdc.2016.7798983
• van der schaft, Port-Hamiltonian Systems Theory An Introductory Overview (2014)
• van der schaft, Gain and Passivity Techniques in Nonlinear Control (2016)
• 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
• Zhang F, Trentelman HL, Scherpen JMA (2016) Robust cooperative output regulation of heterogeneous Lur’e networks. Intl J Robust & Nonlinear 27(16):3061–3078. https://doi.org/10.1002/rnc.3725 – 10.1002/rnc.3725
• vidyasagar, Nonlinear Systems Analysis (1993)
• Ferguson J, Middleton RH, Donaire A (2015) Disturbance rejection via control by interconnection of port-Hamiltonian systems. In: 2015 54th IEEE Conference on Decision and Control (CDC). IEEE, pp 507–512 – 10.1109/cdc.2015.7402279
• 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
• Hernandez-Gomez M, Ortega R, Lamnabhi-Lagarrigue F, Escobar G (2010) Adaptive PI Stabilization of Switched Power Converters. IEEE Trans Contr Syst Technol 18(3):688–698. https://doi.org/10.1109/tcst.2009.2023669 – 10.1109/tcst.2009.2023669
• Horn RA, Johnson CR (1985) Matrix Analysis. Cambridge University Press – 10.1017/cbo9780511810817
• Jayawardhana B, Ortega R, García-Canseco E, Castaños F (2007) Passivity of nonlinear incremental systems: Application to PI stabilization of nonlinear RLC circuits. Systems & Control Letters 56(9-10):618–622. https://doi.org/10.1016/j.sysconle.2007.03.011 – 10.1016/j.sysconle.2007.03.011
• khalil, Nonlinear Systems (2002)
• Maschke B, Ortega R, Van Der Schaft AJ (2000) Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Trans Automat Contr 45(8):1498–1502. https://doi.org/10.1109/9.871758 – 10.1109/9.871758
• Meza C, Biel D, Jeltsema D, Scherpen JMA (2012) Lyapunov-Based Control Scheme for Single-Phase Grid-Connected PV Central Inverters. IEEE Trans Contr Syst Technol 20(2):520–529. https://doi.org/10.1109/tcst.2011.2114348 – 10.1109/tcst.2011.2114348
• (2001) Putting energy back in control. IEEE Control Syst 21(2):18–33. https://doi.org/10.1109/37.915398 – 10.1109/37.915398
• Ortega R, Jeltsema D, Scherpen JMA (2003) Power shaping: A new paradigm for stabilization of nonlinear RLC circuits. IEEE Trans Automat Contr 48(10):1762–1767. https://doi.org/10.1109/tac.2003.817918 – 10.1109/tac.2003.817918
• Sepulchre R, Janković M, Kokotović PV (1997) Constructive Nonlinear Control. Springer London, London – 10.1007/978-1-4471-0967-9
• Castaños F, Jayawardhana B, Ortega R, García-Canseco E (2009) Proportional Plus Integral Control for Set-Point Regulation of a Class of Nonlinear RLC Circuits. Circuits Syst Signal Process 28(4):609–623. https://doi.org/10.1007/s00034-009-9103-x – 10.1007/s00034-009-9103-x
• Sanders SR, Verghese GC (1992) Lyapunov-based control for switched power converters. IEEE Trans Power Electron 7(1):17–24. https://doi.org/10.1109/63.124573 – 10.1109/63.124573
• Castaños F, Ortega R (2009) Energy-balancing passivity-based control is equivalent to dissipation and output invariance. Systems & Control Letters 58(8):553–560. https://doi.org/10.1016/j.sysconle.2009.03.007 – 10.1016/j.sysconle.2009.03.007
• Donaire A, Junco S (2009) On the addition of integral action to port-controlled Hamiltonian systems. Automatica 45(8):1910–1916. https://doi.org/10.1016/j.automatica.2009.04.006 – 10.1016/j.automatica.2009.04.006
• spivak, A Comprehensive Introduction to Differential Geometry (1999)
• Cisneros R, Pirro M, Bergna G, Ortega R, Ippoliti G, Molinas M (2015) Global tracking passivity-based PI control of bilinear systems: Application to the interleaved boost and modular multilevel converters. Control Engineering Practice 43:109–119. https://doi.org/10.1016/j.conengprac.2015.07.002 – 10.1016/j.conengprac.2015.07.002
• Donaire A, Ortega R, Romero JG (2016) Simultaneous interconnection and damping assignment passivity-based control of mechanical systems using dissipative forces. Systems & Control Letters 94:118–126. https://doi.org/10.1016/j.sysconle.2016.05.006 – 10.1016/j.sysconle.2016.05.006
• Donaire A, Mehra R, Ortega R, Satpute S, Romero JG, Kazi F, Singh NM (2016) Shaping the Energy of Mechanical Systems Without Solving Partial Differential Equations. IEEE Trans Automat Contr 61(4):1051–1056. https://doi.org/10.1109/tac.2015.2458091 – 10.1109/tac.2015.2458091
• Borja P, Cisneros R, Ortega R (2016) A constructive procedure for energy shaping of port—Hamiltonian systems. Automatica 72:230–234. https://doi.org/10.1016/j.automatica.2016.05.028 – 10.1016/j.automatica.2016.05.028
• desoer, Feedback Systems Input–Output Properties (1975)
• Åström KJ, Hägglund T (2001) The future of PID control. Control Engineering Practice 9(11):1163–1175. https://doi.org/10.1016/s0967-0661(01)00062-4 – 10.1016/s0967-0661(01)00062-4
• Ortega R, van der Schaft A, Castanos F, Astolfi A (2008) Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems. IEEE Trans Automat Contr 53(11):2527–2542. https://doi.org/10.1109/tac.2008.2006930 – 10.1109/tac.2008.2006930
• Ortega R, Romero JG (2012) Robust integral control of port-Hamiltonian systems: The case of non-passive outputs with unmatched disturbances. Systems & Control Letters 61(1):11–17. https://doi.org/10.1016/j.sysconle.2011.09.015 – 10.1016/j.sysconle.2011.09.015
• 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
• rao, Generalized Inverse of Matrices and its Applications (1971)
• (2016) Adaptation Is Unnecessary in \(L_{1}\)-Adaptive Control: What Makes an Adaptive Controller “Adaptive”?. IEEE Control Syst 36(1):47–52. https://doi.org/10.1109/mcs.2015.2495023 – 10.1109/mcs.2015.2495023
• Romero JG, Ortega R, Donaire A (2016) Energy Shaping of Mechanical Systems via PID Control and Extension to Constant Speed Tracking. IEEE Trans Automat Contr 61(11):3551–3556. https://doi.org/10.1109/tac.2016.2521725 – 10.1109/tac.2016.2521725
• Romero JG, Donaire A, Ortega R (2013) Robust energy shaping control of mechanical systems. Systems & Control Letters 62(9):770–780. https://doi.org/10.1016/j.sysconle.2013.05.011 – 10.1016/j.sysconle.2013.05.011