Time-Stepping via Complementarity

  • Vincent Acary
Part of the Advances in Industrial Control book series (AIC)


In this chapter, the numerical time integration methods for switched electronic circuits are described with a focus on the event-capturing time-stepping schemes based on the complementarity theory. After briefly introducing the strengths and weaknesses of various simulation approaches (hybrid, regularised and non-smooth), the mathematical nature of solutions for dynamical complementarity systems are discussed in view of numerical time-integration. Then the formulation of the time-stepping methods via complementarity will be described. For each class of solutions, a suitable method is provided, and its properties are illustrated on simple electrical circuits. Some implementation details are then explained. Especially, the complementarity solvers that are used at each time-step are described recalling the main families of available solvers. Some insights on the software implementation are also given. Finally, numerical applications and examples on more realistic circuits are considered. We will mainly focus on the architecture of a direct current–direct current (DC–DC) buck power converter.


Variational Inequality Central Processing Unit Time Local Truncation Error Buck Converter Spice Simulator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The author would like to warmly thank Pascal Denoyelle for his contribution in the earlier version of this work and his two main co-workers on this project Olivier Bonnefon and Bernard Brogliato. The authors acknowledge Michael Ferris (University Wisconsin–Madison) for providing us with the PATH solver. Part of this work has been supported by the ANR project VAL-AMS (ANR-06-SETI-018-01).


  1. 1.
    Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics. Lecture Notes in Applied and Computational Mechanics, vol. 35. Springer, Berlin (2008) MATHGoogle Scholar
  2. 2.
    Acary, V., Brogliato, B.: Implicit Euler numerical scheme and chattering-free implementation of sliding mode systems. Syst. Control Lett. 59(5), 284–295 (2010) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Acary, V., Pérignon, F.: Siconos: A software platform for modeling, simulation, analysis and control of nonsmooth dynamical systems. Simul. News Eur. 17(3–4), 19–26 (2007) Google Scholar
  4. 4.
    Acary, V., Brogliato, B., Goeleven, D.: Higher order Moreau’s sweeping process: Mathematical formulation and numerical simulation. Math. Program. 113(1), 133–217 (2008) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Acary, V., Bonnefon, O., Brogliato, B.: Improved circuit simulator. Patent number 09/02605 (2009) Google Scholar
  6. 6.
    Acary, V., Bonnefon, O., Brogliato, B.: Time-stepping numerical simulation of switched circuits with the nonsmooth dynamical systems approach. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 29(7), 1042–1055 (2010) CrossRefGoogle Scholar
  7. 7.
    Acary, V., Brogliato, B., Orlov, Y.: Chattering-free digital sliding-mode control with state observer and disturbance rejection. IEEE Trans. Autom. Control (2011). doi: 10.1109/TAC.2011.2174676. The Research Report RR-7326, INRIA (2010) is available as a preprint at Google Scholar
  8. 8.
    Acary, V., Bonnefon, O., Brogliato, B.: Nonsmooth Modeling and Simulation for Switched Circuits. Lecture Notes in Electrical Engineering, vol. 69. Springer, Berlin (2011) MATHCrossRefGoogle Scholar
  9. 9.
    Bächle, S., Ebert, F.: Element-based topological index reduction for differential-algebraic equations in circuit simulation. Technical Report Preprint 05-246 (Matheon), Inst. f. Mathematik, TU Berlin (2005) Google Scholar
  10. 10.
    Bächle, S., Ebert, F.: Graph theoretical algorithms for index reduction in circuit simulation. Technical Report Preprint 05-245 (Matheon), Inst. f. Mathematik, TU Berlin (2005) Google Scholar
  11. 11.
    Bastien, J., Schatzman, M.: Numerical precision for differential inclusions with uniqueness. Math. Model. Numer. Anal. 36(3), 427–460 (2002) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Billups, S.C., Dirkse, S.P., Ferris, M.C.: A comparison of large scale mixed complementarity problem solvers. Comput. Optim. Appl. 7, 3–25 (1997) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Biolek, D., Dobes, J.: Computer simulation of continuous-time and switched circuits: Limitations of SPICE-family programs and pending issues. In: Proc. of the International Conference Radioelektronika, Brno, Czech Republic, pp. 1–11 (2007) CrossRefGoogle Scholar
  14. 14.
    Brogliato, B., Goeleven, D.: Well-posedness, stability and invariance results for a class of multivalued Lur’e dynamical systems. Nonlinear Anal. 74(1), 195–212 (2011) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Brogliato, B., Thibault, L.: Well-posedness results for non-autonomous complementarity systems. J. Convex Anal. 17(3–4), 961–990 (2010) MathSciNetMATHGoogle Scholar
  16. 16.
    Camlibel, M.K., Heemels, W.P.M.H., Schumacher, J.M.: Consistency of a time-stepping method for a class of piecewise-linear networks. IEEE Trans. Circuits Syst. I 49(3), 349–357 (2002) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Camlibel, M.K., Heemels, W.P.M.H., Schumacher, J.M.: On linear passive complementarity systems. Eur. J. Control 8(3), 220–237 (2002) CrossRefGoogle Scholar
  18. 18.
    Camlibel, M.K., Heemels, W.P.M.H., van der Schaft, A.J., Schumacher, J.M.: Switched networks and complementarity. IEEE Trans. Circuits Syst. I 50(8), 1036–1046 (2003) CrossRefGoogle Scholar
  19. 19.
    Cao, M., Ferris, M.C.: A pivotal method for affine variational inequalities. Math. Oper. Res. 21(1), 44–64 (1996) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Chung, H.S.H., Ioinovici, A.: Fast computer aided simulation of switching power regulators based on progressive analysis of the switches’ state. IEEE Trans. Power Electron. 9(2), 206–212 (1994) CrossRefGoogle Scholar
  21. 21.
    Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955) MATHGoogle Scholar
  22. 22.
    Cottle, R.W., Pang, J., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Boston (1992) MATHGoogle Scholar
  23. 23.
    Elmqvist, H., Mattsson, S.E., Otter, M.: Object-oriented and hybrid modeling in Modelica. J. Eur. Syst. Autom. 35(4), 395–404 (2001) Google Scholar
  24. 24.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, vols. I & II. Springer, New York (2003) Google Scholar
  25. 25.
    Frasca, R., Camlibel, M.K., Goknar, I.C., Vasca, F.: State discontinuity analysis of linear switched systems via energy function optimization. In: Proc. of the IEEE International Symposium on Circuits and Systems, Seattle, Washington, USA, pp. 540–543 (2008) Google Scholar
  26. 26.
    Frasca, R., Camlibel, M.K., Goknar, I.C., Iannelli, L., Vasca, F.: Linear passive networks with ideal switches: Consistent initial conditions and state discontinuities. IEEE Trans. Circuits Syst. I 57(12), 3138–3151 (2010) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53(1–3), 99–110 (1992) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Greenhalgh, S., Acary, V., Brogliato, B.: Preservation of the dissipativity properties of a class of nonsmooth dynamical systems with the (θ,γ)-algorithm. Research Report RR-7632, INRIA (2011). URL
  29. 29.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-algebraic Problems, 2nd edn. Series in Computational Mathematics, vol. 14. Springer, London (1996) MATHGoogle Scholar
  30. 30.
    Han, L., Tiwari, A., Camlibel, K., Pang, J.S.: Convergence of time-stepping schemes for passive and extended linear complementarity systems. SIAM J. Numer. Anal. 47(5), 3768–3796 (2009) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Heemels, W.P.M.H., Camlibel, M.K., Schumacher, J.M.: A time-stepping method for relay systems. In: Proc. of the IEEE Conference on Decision and Control, Sydney, Australia, pp. 461–466 (2000) Google Scholar
  32. 32.
    Heemels, W.P.M.H., Schumacher, J.M., Weiland, S.: Linear complementarity systems. SIAM J. Appl. Math. 60, 1234–1269 (2000) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Springer, Heidelberg (1993) Google Scholar
  34. 34.
    Iannelli, L., Vasca, F., Camlibel, M.K.: Complementarity and passivity for piecewise linear feedback systems. In: Proc. of the IEEE Conference on Decision and Control, San Diego, California, USA, pp. 4212–4217 (2006) CrossRefGoogle Scholar
  35. 35.
    Isidori, A.: Nonlinear Control Systems, 3rd edn. Springer, London (1995) MATHGoogle Scholar
  36. 36.
    Leenaerts, D.M.W., Bokhoven, W.M.V.: Piecewise Linear Modeling and Analysis. Kluwer Academic Publishers, Dordrecht (1998) Google Scholar
  37. 37.
    Leenarts, D.M.: On linear dynamic complementarity systems. IEEE Trans. Circuits Syst. I 46(8), 1022–1026 (1999) CrossRefGoogle Scholar
  38. 38.
    Luca, T.D., Facchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program. 75(3), 407–439 (1996) MATHCrossRefGoogle Scholar
  39. 39.
    Maffezzoni, P., Codecasa, L., D’Amore, D.: Event-driven time-domain simulation of closed-loop switched circuits. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 25(11), 2413–2426 (2006) CrossRefGoogle Scholar
  40. 40.
    Marques, M.D.P.M.: Differential Inclusions in Nonsmooth Mechanical Problems. Shocks and Dry Friction. Progress in Nonlinear Differential Equations and Their Applications, vol. 9. Birkhäuser, Boston (1993) MATHGoogle Scholar
  41. 41.
    Mayaram, K., Lee, D.C., Moinian, D.A., Roychowdhury, J.: Computer-aided circuit analysis tools for RFIC simulation: Algorithms, features, and limitations. IEEE Trans. Circuits Syst. II 47(4), 274–286 (2000) CrossRefGoogle Scholar
  42. 42.
    Moreau, J.J.: Evolution problem associated with a moving convex set in a Hilbert space. J. Differ. Equ. 26, 347–374 (1977) MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Moreau, J.J.: Bounded variation in time. In: Moreau, J.J., Panagiotopoulos, P.D., Strang, G. (eds.) Topics in Nonsmooth Mechanics, pp. 1–74. Birkhäuser, Basel (1988) Google Scholar
  44. 44.
    Pang, J.S., Stewart, D.: Differential variational inequalities. Math. Program. 113(2), 345–424 (2008) MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Pogromski, A.Y., Heemels, W.P.M.H., Nijmeijer, H.: On solution concepts and well-posedness of linear relay systems. Automatica 39, 2139–2147 (2003) CrossRefGoogle Scholar
  46. 46.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) MATHGoogle Scholar
  47. 47.
    Sargent, R.W.H.: An efficient implementation of the Lemke algorithm and its extension to deal with upper an lower bounds. Math. Program. Stud. 7, 36–54 (1978) MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    van Bokhoven, W.M.G.: Piecewise linear analysis and modelling. Ph.D. thesis, Technical University of Eindhoven, TU/e (1981) Google Scholar
  49. 49.
    van Bokhoven, W.M.G., Jess, J.A.G.: Some new aspects of P and P 0 matrices and their application to networks with ideal diodes. In: Proc. of the IEEE International Symposium on Circuits and Systems, New York, USA, pp. 806–810 (1978) Google Scholar
  50. 50.
    van Eijndhoven, W.M.G.: A piecewise linear simulator for large scale integrated circuits. Ph.D. thesis, Technical University of Eindhoven, TU/e (1984) Google Scholar
  51. 51.
    van Stiphout, M.T.: Plato—a piecewise linear analysis for mixed-level circuit simulation. Ph.D. thesis, Technical University of Eindhoven, TU/e (1990) Google Scholar
  52. 52.
    Vandenberghe, L., Moor, B.L.D., Vandewalle, J.: The generalized linear complementarity problem applied to the complete analysis of resistive piecewise-linear circuits. IEEE Trans. Circuits Syst. 36(11), 1382–1391 (1989) CrossRefGoogle Scholar
  53. 53.
    Yuan, F., Opal, A.: Computer methods for switched circuits. IEEE Trans. Circuits Syst. I 50(8), 1013–1024 (2003) CrossRefGoogle Scholar
  54. 54.
    Zhu, D.L., Marcotte, P.: Modified descents methods for solving the monotone variational inequality problem. Oper. Res. Lett. 14(2), 111–120 (1993) MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Zhu, D.L., Marcotte, P.: An extended descent framework for monotone variational inequalities. J. Optim. Theory Appl. 80(2), 349–366 (1994) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.INRIA Rhône-Alpes, Centre de recherche GrenobleSt Ismier CedexFrance

Personalised recommendations