Authors

Fernando Castaños, Bayu Jayawardhana, Romeo Ortega, Eloísa García-Canseco

Abstract

In this paper we identify graph-theoretic conditions which allow us to write a nonlinear RLC circuit as port-Hamiltonian with constant input matrices. We show that under additional monotonicity conditions on the network’s components, the circuit enjoys the property of relative passivity, an extended notion of classical passivity. The property of relative passivity is then used to build simple, yet robust and globally stable, proportional plus integral controllers.

Keywords

Nonlinear networks; Passivity; Port-Hamiltonian systems; Stability; Stabilization

Citation

  • Journal: Circuits, Systems, and Signal Processing
  • Year: 2009
  • Volume: 28
  • Issue: 4
  • Pages: 609–623
  • Publisher: Springer Science and Business Media LLC
  • DOI: 10.1007/s00034-009-9103-x

BibTeX

@article{Casta_os_2009,
  title={{Proportional Plus Integral Control for Set-Point Regulation of a Class of Nonlinear RLC Circuits}},
  volume={28},
  ISSN={1531-5878},
  DOI={10.1007/s00034-009-9103-x},
  number={4},
  journal={Circuits, Systems, and Signal Processing},
  publisher={Springer Science and Business Media LLC},
  author={Castaños, Fernando and Jayawardhana, Bayu and Ortega, Romeo and García-Canseco, Eloísa},
  year={2009},
  pages={609--623}
}

Download the bib file

References

  • Bernstein, G. M. & Lieberman, M. A. A method for obtaining a canonical Hamiltonian for nonlinear LC circuits. IEEE Transactions on Circuits and Systems vol. 36 411–420 (1989) – 10.1109/31.17588
  • Blankenstein, G. Geometric modeling of nonlinear RLC circuits. IEEE Transactions on Circuits and Systems I: Regular Papers vol. 52 396–404 (2005) – 10.1109/tcsi.2004.840481
  • Brayton, R. K. & Moser, J. K. A theory of nonlinear networks. I. Quarterly of Applied Mathematics vol. 22 1–33 (1964) – 10.1090/qam/169746
  • Brayton, R. K. & Moser, J. K. A theory of nonlinear networks. II. Quarterly of Applied Mathematics vol. 22 81–104 (1964) – 10.1090/qam/169747
  • Byrnes, C. I., Priscoli, F. D. & Isidori, A. Output Regulation of Uncertain Nonlinear Systems. Systems & Control: Foundations & Applications (Birkhäuser Boston, 1997). doi:10.1007/978-1-4612-2020-6 – 10.1007/978-1-4612-2020-6
  • Cahill, L. On the Selection of State Variables for Nonlinear RLC Networks. IEEE Transactions on Circuit Theory vol. 16 553–555 (1969) – 10.1109/tct.1969.1083016
  • Chua, L. & Green, D. Graph-theoretic properties of dynamic nonlinear networks. IEEE Transactions on Circuits and Systems vol. 23 292–312 (1976) – 10.1109/tcs.1976.1084220
  • Chua, L. & McPherson, J. Explicit topological formulation of Lagrangian and Hamiltonian equations for nonlinear networks. IEEE Transactions on Circuits and Systems vol. 21 277–286 (1974) – 10.1109/tcs.1974.1083849
  • C.A. Desoer. C.A. Desoer, E.S. Kuh, Basic Circuit Theory (McGraw-Hill Kogausha, Tokyo, 1969) (1969)
  • C.A. Desoer. C.A. Desoer, M. Vidyasagar, Feedback Systems: Input–Output Properties (Academic Press, New York, 1975) (1975)
  • Duerbaum, T., Kuebrich, D. & Schetters, K. Sensitivity Analysis of Air Gap Size of Non Linear Inductance in Passive Mains Harmonic Reduction Circuits. IEEE 36th Conference on Power Electronics Specialists, 2005. 619–623 doi:10.1109/pesc.2005.1581690 – 10.1109/pesc.2005.1581690
  • Gluskin, E. A nonlinear resistor and nonlinear inductor using a nonlinear capacitor. Journal of the Franklin Institute vol. 336 1035–1047 (1999) – 10.1016/s0016-0032(99)00029-0
  • Hiriart-Urruty, J.-B. & Lemaréchal, C. Convex Analysis and Minimization Algorithms I. Grundlehren der mathematischen Wissenschaften (Springer Berlin Heidelberg, 1993). doi:10.1007/978-3-662-02796-7 – 10.1007/978-3-662-02796-7
  • Jayawardhana, B., Ortega, R., Garcia-Canseco, E. & Castanos, F. Passivity of Nonlinear Incremental Systems: Application to PI Stabilization of Nonlinear RLC Circuits. Proceedings of the 45th IEEE Conference on Decision and Control 3808–3812 (2006) doi:10.1109/cdc.2006.377132 – 10.1109/cdc.2006.377132
  • Jeltsema, D. & Scherpen, J. M. A. A dual relation between port-Hamiltonian systems and the Brayton–Moser equations for nonlinear switched RLC circuits. Automatica vol. 39 969–979 (2003) – 10.1016/s0005-1098(03)00070-0
  • H.K. Khalil. H.K. Khalil, Nonlinear Systems (Prentice-Hall, Upper Saddle River, 1996) (1996)
  • B. Maschke, Interconnexion et structure des systèmes hamiltoniens commandés: Une approche réseau. Université Paris-Sud XI, Orsay, France, Tech. Rep. (August 1998). Mémoire présenté pour obtenir l’Habilitation á diriger les recherches
  • Maschke, B. M., van der Schaft, A. J. & Breedveld, P. C. An intrinsic Hamiltonian formulation of the dynamics of LC-circuits. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications vol. 42 73–82 (1995) – 10.1109/81.372847
  • Roska, T. The limits of modeling of nonlinear circuits. IEEE Transactions on Circuits and Systems vol. 28 212–216 (1981) – 10.1109/tcs.1981.1084974
  • J.L. Salle. J.L. Salle, S. Lefschetz, Stability by Liapunov’s Direct Method with Applications (Academic Press, New York, 1961) (1961)
  • Sangiovanni-Vincentelli, A. & Wang, Y. On equivalent dynamic networks: Elimination of capacitor loops and inductor cutsets. IEEE Transactions on Circuits and Systems vol. 25 174–177 (1978) – 10.1109/tcs.1978.1084448
  • Stern, T. On the Equations of Nonlinear Networks. IEEE Transactions on Circuit Theory vol. 13 74–81 (1966) – 10.1109/tct.1966.1082545
  • Trajkovic, L. & Willson, A. N. Negative differential resistance in two-transistor one-ports with no internal sources. 1988., IEEE International Symposium on Circuits and Systems 747–750 doi:10.1109/iscas.1988.15033 – 10.1109/iscas.1988.15033
  • 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