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State jumps and bifurcations


Switching circuits are encountered in various applications from power converters to signal processing. To
simplify the analysis, the switching elements are typically taken as ideal elements. Such ideal switching behavior may cause:
  • state jumps

  • bifurcations


  • Cem Goknar, Dogus University, Istanbul Turkey
  • Kanat Camlibel, Univeristy of Groningen, The Netherlands
  • Stephan Trenn, University of Kaiserslautern, Germany


For discontinuities in the state variables (typically voltages across capacitors and currents through inductors), a good deal of literature has been devoted to the problem of state reinitialization, i.e., determining the state after a discontinuity. We study the state reinitilization problem for electrical networks consisting of linear passive elements, independent voltage/current sources, and ideal switches. We employ the so-called linear switched systems framework in which these circuits can be analyzed for any given switch configuration. After providing a complete characterization of admissible inputs and consistent initial states with respect to a switch configuration, we introduce a new state reinitialization rule that is based on energy minimization at the time of switching.

Inspired by nonlinear phenomena exhibited by power electronics converters, we study the classification of non-standard bifurcations in piecewise smooth feedback systems. Bifurcations involving fixed points in maps, equilibria and limit cycles in flows are considered with particular attention to: border collisions of fixed points in maps; non-smooth bifurcations of equilibria, grazing bifurcations and sliding bifurcations of limit cycles in flows. The aim is to describe existing and novel results to form the basis of a consistent theory of bifurcations in such systems. In so doing, a novel approach to classify non-smooth bifurcations of equilibria in flows can be proposed.

PhD Theses
  • ongoing
    • Carmen Pedicini, PhD student in Automatic Control, University of Sannio, "Averaging for switched systems", co-tutor prof. Luigi Iannelli.
  • 2007
    • Roberto Frasca, PhD in Automatic Control, University of Sannio, "Modeling and simulation of switched electrical networks: A complementarity systems approach", co-tutor prof. Luigi Iannelli.

Laurea Magistralis Theses
  • 2011
    1. Maurizio Zoppi, LS Ingegneria dell'Automazione, ottobre 2011, "Complementary model and minimum energy control for switched capacitor converters", 110 e lode, co-relatore ing. Gianluca Angelone.

Laurea Theses

  • 2012
    1. Giuliano Costantini, L Ingegneria Informatica, luglio 2012, "State jumps in electrical circuits", 110 e lode.


  • 2010
  • 2008
    • R. Frasca, K. Camlibel , C. I. Goknar, L. Iannelli, F. Vasca, "State Jump Rules in Linear Passive Networks with Ideal Switches and Sources", IEEE International Symposium on Circuits and Systems, Seattle, Washington, May 2008, p. 540-543.
  • 1999
    • M. di Bernardo, K. H. Johansson, F. Vasca, "Sliding Orbits and Their Bifurcations in Relay Feedback Systems", 38th IEEE Conference on Decision and Control, Phoenix, Arizona, December 1999, p. 708-713.
    • M. di Bernardo, A. R. Champneys, C. J. Budd, F. Vasca, "Sliding Orbits and Double Spiral Bifurcation Diagrams in Power Electronics DC/DC Converters", IEEE International Symposium on Circuits And Systems, Orlando, Florida, vol. V, May 1999, p. 262-265.
  • 1997
    • M. di Bernardo, E. Fossas, G. Olivar, F. Vasca, "Secondary Bifurcations and High Periodic Orbits in Voltage Controlled Buck Converter", International Journal of Bifurcation and Chaos, vol. 7, no. 11, November 1997, p. 2755-2771.
    • M. di Bernardo, F. Garofalo, L. Glielmo, F. Vasca, "Analysis of Chaotic Buck, Boost and Buck-Boost Converters through Switching Maps", 28th IEEE Power Electronics Specialists Conference, St. Louis, Missouri, June 1997, p. 754-760.

  • 1996
    • M. di Bernardo, F. Garofalo, L. Glielmo, F. Vasca, "Nonlinear Phenomena in Pulse Width Modulated Feedback Controlled Systems", 35th IEEE Conference on Decision and Control, Kobe, Japan, December 1996, p. 2161-2166.
    • M. di Bernardo, F. Garofalo, L. Glielmo, F. Vasca, "Quasi-Periodic Behaviors in DC/DC Converters", 27th IEEE Power Electronics Specialists Conference, Baveno, Italy, June 1996, p. 1376-1381.