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Computation of periodic solutions in maximal monotone dynamical systems with guaranteed consistency. (English) Zbl 1375.65087
Summary: In this paper, we study a class of set-valued dynamical systems that satisfy maximal monotonicity properties. This class includes linear relay systems, linear complementarity systems, and linear mechanical systems with dry friction under some conditions. We discuss two numerical schemes based on time-stepping methods for the computation of the periodic solutions when these systems are periodically excited. We provide formal mathematical justifications for the numerical schemes in the sense of consistency, which means that the continuous-time interpolations of the numerical solutions converge to the continuous-time periodic solution when the discretization step vanishes. The two time-stepping methods are applied for the computation of the periodic solution exhibited by a power electronic converter and the corresponding methods are compared in terms of approximation accuracy and computation time.
Reviewer: Reviewer (Berlin)
##### MSC:
 65K10 Numerical optimization and variational techniques 93C05 Linear systems in control theory 93C15 Control/observation systems governed by ordinary differential equations 93C10 Nonlinear systems in control theory 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory
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##### References:
 [1] Aubin, J. P.; Cellina, A., Differential inclusions, (1984), Springer-Verlag Berlin, Heidelberg [2] Minty, G., Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29, 341-346, (1962) · Zbl 0111.31202 [3] Brezis, H., Operateurs maximaux monotones, (1973), North-Holland/American Elsevier Amsterdam · Zbl 0252.47055 [4] Peypouquet, J.; Sorin, S., Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time, J. Convex Anal., 17, 3&4, 1113-1163, (2010) · Zbl 1214.47080 [5] van der Schaft, A. J.; Schumacher, J. M., Complementarity modeling of hybrid systems, IEEE Trans. Automat. Control, 43, 4, 483-490, (1998) · Zbl 0899.93002 [6] Heemels, W. P.M. H.; Schumacher, J. M.; Weiland, S., Linear complementarity systems, SIAM J. Appl. Math., 60, 1234-1269, (2000) · Zbl 0954.34007 [7] 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. Regul. Pap., 50, 1036-1046, (2003) · Zbl 1368.93368 [8] Brogliato, B., Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings, Systems Control Lett., 51, 343-353, (2004) · Zbl 1157.93455 [9] Shen, J.; Pang, J. S., Semicopositive linear complementarity systems, Internat. J. Robust Nonlinear, 17, 15, 1367-1386, (2007) · Zbl 1177.90388 [10] Vasca, F.; Iannelli, L.; Camlibel, M. K.; Frasca, R., A new perspective for modeling power electronics converters: complementarity framework, IEEE Trans. Power Electron., 24, 2, 456-468, (2009) [11] Johansson, K. H.; Rantzer, A.; Astrom, K. J., Fast switches in relay feedback systems, Automatica, 35, 539-552, (1999) · Zbl 0934.93033 [12] Pogromsky, A.; Heemels, W. P.M. H.; Nijmeijer, H., On solution concepts and well-posedness of linear relay systems, Automatica, 39, 2139-2147, (2003) · Zbl 1046.93029 [13] Heemels, W. P.M. H.; Schumacher, J. M.; Weiland, S., Projected dynamical systems in a complementarity formalism, Oper. Res. Lett., 27, 2, 83-91, (2000) · Zbl 0980.93031 [14] Nagurney, A.; Zhang, D., Projected dynamical systems and variational inequalities with applications, (1996), Kluwer Dordrecht [15] Brogliato, B.; Daniilidis, A.; Lemaréchal, C.; Acary, V., On the equivalence between complementarity systems, projected systems and differential inclusions, Systems Control Lett., 55, 45-51, (2006) · Zbl 1129.90358 [16] Sessa, V.; Iannelli, L.; Vasca, F., A complementarity model for closed-loop power converters, IEEE Trans. Power Electron., 29, 12, 6821-6835, (2014) [17] Iannelli, L.; Vasca, F.; Angelone, G., Computation of steady-state oscillations in power converters through complementarity, IEEE Trans. Circuits Syst. I. Regul. Pap., 58, 6, 1421-1432, (2011) [18] Vasca, F.; Iannelli, L., Dynamics and control of switched electronic systems, (2012), Springer-Verlag London [19] Brogliato, B., (Nonsmooth Impact Mechanics, Models, Dynamics and Control, vol. 220, (1996), Springer-Verlag London London) · Zbl 0861.73001 [20] Marques, M. D.P. M., (Differential Inclusions in Nonsmooth Mechanical Problems, Shocks and Dry Friction, vol. 7, (1993), Birkhuser Basel) · Zbl 0791.49029 [21] Liberzon, M. R., Essays on the absolute stability theory, Autom. Remote Control, 67, 10, 1610-1644, (2006) · Zbl 1195.34003 [22] Brogliato, B.; Heemels, W. P.M. H., Observer design for lur’e systems with multivalued mappings: A passivity approach, IEEE Trans. Automat. Control, 54, 8, 1996-2001, (2009) · Zbl 1367.93086 [23] Camlibel, M. K.; Schumacher, J. M., Math. Program., 157, 397, (2016) [24] Dontchev, A.; Lempio, F., Difference methods for differential inclusions: A survey, SIAM Rev., 4, 2, 263-294, (1992) · Zbl 0757.34018 [25] Stewart, D. E., Time-stepping methods and the mathematics of rigid body dynamics, (Guran, A.; Martins, J. A.C.; Klarbring, A., Impact and Friction, (1999), Birkhäuser), (Chapter 1) [26] 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. Regul. Pap., 49, 3, 349-357, (2002) · Zbl 1368.93279 [27] Han, L.; Tiwari, A.; Camlibel, M. 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) · Zbl 1203.65123 [28] Studer, C., Numerics of unilateral contacts and friction, vol. 47, (2009), Springer-Verlag Berlin, Heidelberg [29] Angeli, D., A Lyapunov approach to incremental stability properties, IEEE Trans. Automat. Control, 47, 3, 410-421, (2002) · Zbl 1364.93552 [30] Demidovich, B. P., Lectures on stability theory, (1967), Nauka Moscow, (in Russian) · Zbl 0155.41601 [31] Pavlov, A.; van de Wouw, N.; Nijmeijer, H., (Uniform Output Regulation of Nonlinear Systems: A Convergent Dynamics Approach, Systems and Control: Foundations and Applications (SC) Series, (2005), Birkhuser Basel) [32] Leine, R. I.; van de Wouw, N., Uniform convergence of monotone measure differential inclusions with application to the control of mechanical systems with unilateral constraints, Internat. J. Bifur. Chaos, 18, 5, 1435-1457, (2008) · Zbl 1147.34310 [33] M.K. Camlibel, N. van de Wouw, On the convergence of linear passive complementarity systems, in: Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, USA, 2007, pp. 5886-5891. [34] Sessa, V.; Iannelli, L.; Vasca, F.; Acary, V., A complementarity approach for the computation of periodic oscillations in piecewise linear systems, Nonlinear Dynam., 86, 2, 1255-1273, (2016) · Zbl 1355.93076 [35] W.P.M.H. Heemels, V. Sessa, F. Vasca, M.K. Camlibel, Time-stepping methods for constructing periodic solutions in maximally monotone set-valued dynamical systems, in: IEEE Conference on Decision and Control, Los Angeles, USA, 2014, pp. 3095-3100. [36] Cottle, R. W.; Pang, J.-S.; Stone, R. E., The linear complementarity problem, (1992), Academic Press, Inc. Boston · Zbl 0757.90078 [37] Acary, V.; Brogliato, B., Numerical methods for nonsmooth dynamical systems: applications in mechanics and electronics, vol. 35, (2008), Springer-Verlag London · Zbl 1173.74001 [38] Rockafellar, R. T.; Wets, R. J.-B., Variational analysis, (1998), Springer-Verlag Berlin, Heidelberg · Zbl 0888.49001 [39] Dirkse, S. P.; Ferris, M. C., The PATH solver: a non-monotone stabilization scheme for mixed complementarity problems, Optim. Methods Softw., 5, 123-156, (1995) [40] P. Bénilan, Equations d’Évolution dans un Espace de Banach Quelconque et Applications, Thèse, Orsay, 1972.
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