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A complementarity approach for the computation of periodic oscillations in piecewise linear systems. (English) Zbl 1355.93076
Summary: Piecewise linear (PWL) systems can exhibit quite complex behaviours. In this paper, the complementarity framework is used for computing periodic steady-state trajectories belonging to linear time-invariant systems with PWL, possibly set-valued, feedback relations. The computation of the periodic solutions is formulated in terms of a mixed quadratic complementarity problem. Suitable anchor equations are used as problem constraints in order to determine the unknown period and to fix the phase of the steady-state oscillation. The accuracy of the complementarity problem solution is shown through numerical investigations of stable and unstable oscillations exhibited by practical PWL systems: a neural oscillator, a deadzone feedback system, a stick-slip system, a repressilator and a relay feedback system.

93B52 Feedback control
34A36 Discontinuous ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
Full Text: DOI
[1] Stern, T.E.: Piecewise-linear network theory. Technical report, PhD thesis, MIT, Research Laboratory of Electronics, Cambridge, MA, USA (1956)
[2] Camlibel, MK; Heemels, WPMH; Schaft, AJ; Schumacher, JM, Switched networks and complementarity, IEEE Trans. Circuits Syst. I Reg. Pap., 50, 1036-1046, (2003) · Zbl 1032.93501
[3] Acary, V; Bonnefon, O; Brogliato, B, Time-stepping numerical simulation of switched circuits within the nonsmooth dynamical systems approach, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst., 29, 1042-1055, (2010)
[4] Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems. Springer, Berlin (2008) · Zbl 1173.74001
[5] Melin, J; Hultgren, A; Lindstrom, T, Two types of limit cycles of a resonant converter modelled by three-dimensional system, Nonlinear Anal. Hybrid Syst., 2, 1275-1286, (2008) · Zbl 1163.93312
[6] Yang, X-S; Chen, G, Limit cycle and chaotic invariant sets in autonomous hybrid planar systems, Nonlinear Anal. Hybrid Syst., 2, 952-957, (2008) · Zbl 1218.34013
[7] Aprille, T; Trick, T, A computer algorithm to determine the steady-state response of nonlinear oscillators, IEEE Trans. Circuit Theory, 19, 354-360, (1972) · Zbl 0341.65058
[8] Flieller, D; Riedinger, P; Louis, JP, Computation and stability of limit cycles in hybrid systems, Nonlinear Anal. Theory Methods Appl., 64, 352-367, (2006) · Zbl 1096.34020
[9] Li, D; Xu, J, A new method to determine the periodic orbit of a nonlinear dynamic systems and its period, Eng. Comput., 20, 316-322, (2005)
[10] Leine, R.I., Nijmeijer, H.: Dynamics and Bifurcations of Non-Smooth Mechanical Systems. Springer, London (2004) · Zbl 1068.70003
[11] Theodosiou, C; Pournaras, A; Natsiavas, S, On periodic steady state response and stability of Filippov-type mechanical models, Nonlinear Dyn., 66, 355-376, (2011) · Zbl 1331.70015
[12] Bizzarri, F; Brambilla, A; Gajani, GS, Steady state computation and noise analysis of analog mixed signal circuits, IEEE Trans. Circuits Syst. I Reg. Pap., 59, 541-554, (2012)
[13] Leine, RI; Campen, DH; Kraker, A, Stick-slip vibrations induced by alternate friction models, Nonlinear Dyn., 16, 41-54, (1998) · Zbl 0908.70021
[14] Vrande, BL; Campen, DH; Kraker, A, An approximate analysis of dry-friction-induced stick-slip vibrations by smoothing procedure, Nonlinear Dyn., 19, 157-169, (1999) · Zbl 0966.70013
[15] Thota, P; Dankowicz, H, TC-HAT (\(\hat{TC}\)): a novel toolbox for the continuation of periodic trajectories in hybrid dynamical systems, SIAM J. Appl. Dyn. Syst., 7, 1283-1322, (2008) · Zbl 1192.34004
[16] Lu, C-J; Lin, Y-M, A modified incremental harmonic balance method for rotary periodic motions, Nonlinear Dyn., 66, 781-788, (2011)
[17] Bonnin, M; Gilli, M; Civalleri, PP, A mixed time-frequency-domain approach for the analysis of a hysteretic oscillator, IEEE Trans. Circuits Syst. II Exp. Br., 52, 525-529, (2005)
[18] Brambilla, A; Gruosso, G; Storti Gajani, G, MTFS mixed time-frequency method for the steady-state analysis of almost-periodic nonlinear circuits, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst., 31, 1346-1355, (2012)
[19] Duarte, FB; Tenreiro Machado, J, Fractional describing function of systems with Coulomb friction, Nonlinear Dyn., 56, 381-387, (2009) · Zbl 1204.70026
[20] Huang, Y-J; Wang, Y-J, Steady-state analysis for a class of sliding mode controlled systems using describing function method, Nonlinear Dyn., 30, 223-241, (2002) · Zbl 1016.93016
[21] Engelberg, S, Limitations of the describing function for limit cycle prediction, IEEE Trans. Autom. Control, 47, 1887-1890, (2002) · Zbl 1364.93334
[22] Vasca, F; Iannelli, L; Camlibel, MK, A new perspective for modeling power electronics converters: complementarity framework, IEEE Trans. Power Electron., 24, 456-468, (2009)
[23] Schumacher, JM, Complementarity systems in optimization, Math. Program., 101, 263-296, (2004) · Zbl 1076.90060
[24] Sessa, V., Iannelli, L., Vasca, F.: A complementarity model for closed-loop power converters. IEEE Trans. Power Electron, 29(12), 6821-6835 (2014)
[25] Schaft, AJ; Schumacher, JM, Complementarity modeling of hybrid systems, IEEE Trans. Autom. Control, 43, 483-490, (1998) · Zbl 0899.93002
[26] Heemels, WPMH; Schutter, B; Bemporad, A, Equivalence of hybrid dynamical model, Automatica, 37, 1085-1091, (2001) · Zbl 0990.93056
[27] Iannelli, L., Vasca, F.: Computation of limit cycles and forced oscillations in discrete-time piecewise linear feedback systems through a complementarity approach. In: 47th IEEE Conference on Decision and Control, pp. 1169-1174, Cancun, Mexico (2008) · Zbl 1402.92194
[28] Iannelli, L., Vasca, F., Sessa, V.: Computation of limit cycles in Lur’e systems. In: American Control Conference, pp. 1402-1407, San Francisco, CA, USA (2011)
[29] Sessa, V., Iannelli, L., Vasca, F.: Mixed linear complementarity problems for the analysis of limit cycles in piecewise linear systems. In: The 51st IEEE Conference on Decision and Control, pp. 1023-1028, Maui, Hawaii, USA (2012) · Zbl 1331.70015
[30] Sessa, V., Iannelli, L., Acary, V., Brogliato, B., Vasca, F.: Computing period and shape of oscillations in piecewise linear Lur’e systems: a complementarity approach. In: The 52nd IEEE Conference on Decision and Control, pp. 4680-4685, Florence, Italy (2013) · Zbl 1016.93016
[31] Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Operations Research. Springer, New York (2003) · Zbl 1062.90002
[32] Dirkse, SP; Ferris, MC, The PATH solver: a non-monotone stabilization scheme for mixed complementarity problems, Optim. Methods Softw., 5, 123-156, (1995)
[33] Cottle, R., Pang, J., Stone, R.: The Linear Complementarity Problem, 2nd edn. Academic Press, Cambridge (2009) · Zbl 1192.90001
[34] Gowda, MS; Pang, J, Stability analysis of variational inequalities and nonlinear complementarity problems, via the mixed linear complementarity problem and degree theory, Math. Oper. Res., 19, 831-879, (1994) · Zbl 0821.90114
[35] Farkas, M.: Periodic Motions. Series in Applied Mathematical, Sciences edn. Springer, New York (1994) · Zbl 0805.34037
[36] Doedel, EJ, AUTO: a program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30, 1265-1284, (1981) · Zbl 0511.65064
[37] Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods. Wiley, Chichester (1995) · Zbl 0848.34001
[38] Seydel, R.U.: Practical Bifurcation and Stability Analysis. Interdisciplinary Applied Mathematics, 3rd edn. Springer, New York (1988) · Zbl 0652.34059
[39] Acary, V; Vasca, F (ed.); Iannelli, L (ed.), Time-stepping via complementarity, (2012), London
[40] Han, L; Tiwari, A; Camlibel, MK; Pang, J-S, Convergence of time-stepping for passive and extended linear complementarity systems, SIAM J. Numer. Anal., 47, 3768-3796, (2009) · Zbl 1203.65123
[41] Pang, J-S; Stewart, DE, Differential variational inequalities, Math. Program., 113, 345-424, (2008) · Zbl 1139.58011
[42] Matsuoka, K, Sustained oscillations generated by mutually inhibiting neurons with adaptation, Biol. Cybern., 52, 367376, (1985) · Zbl 0574.92013
[43] Goncalves, JM, Region of stability for limit cycles in piecewise linear systems, IEEE Trans. Autom. Control, 50, 1877-1882, (2005) · Zbl 1365.93278
[44] Hu, T; Thibodeau, T; Teel, TR, A unified Lyapunov approach to analysis of oscillations and stability for systems with piecewise linear elements, IEEE Trans. Autom. Control, 55, 2864-2869, (2010) · Zbl 1368.93374
[45] Khalil, H.K.: Nonlinear Systems. Prentice Hall, New Jersey (2002) · Zbl 1003.34002
[46] Heemels, W.P.M.H., Camlibel, M.K., Schumacher, J.M.: A time-stepping method for relay systems. In: 39th IEEE Conference on Decision and Control, pp. 4461-4466, Sydney, Australia (2000)
[47] Astrom, KJ; Canudas De Wit, C, Revisiting the lugre friction model, IEEE Control Syst. Mag., 28, 101-114, (2008) · Zbl 1395.74065
[48] Tonnelier, A, Cyclic negative feedback systems: what is the chance of oscillation?, Bull. Math. Biol., 76, 1155-1193, (2014) · Zbl 1297.92035
[49] Acary, V; Jong, H; Brogliato, B, Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems, Phys. D Nonlinear Phenom., 269, 103-119, (2014) · Zbl 1402.92194
[50] Bernardo, M., Budd, C., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems, vol. 163. Springer, London (2008) · Zbl 1146.37003
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