“Parameter-dependent PWQ Lyapunov function stability criteria for uncertain piecewise linear systems”

Authors: Morten Hovd and Sorin Olaru,
Affiliation: NTNU, Department of Engineering Cybernetics and CentraleSupelec, France
Reference: 2018, Vol 39, No 1, pp. 15-21.

Keywords: Lyapunov stability; Piecewise linear/affine dynamics; Uncertain systems; Linear matrix inequalities

Abstract: The calculation of piecewise quadratic (PWQ) Lyapunov functions is addressed in view of stability analysis of uncertain piecewise linear dynamics. As main contribution, the linear matrix inequality (LMI) approach proposed in (Johansson and Rantzer, 1998) for the stability analysis of PWL and PWA dynamics is extended to account for parametric uncertainty based on a improved relaxation technique. The results are applied for the analysis of a Phase Locked Loop (PLL) benchmark and the ability to guarantee a stability region in the parameter space well beyond the state of the art is demonstrated.

PDF PDF (307 Kb)        DOI: 10.4173/mic.2018.1.2

DOI forward links to this article:
[1] Jose Renato Campos, Edvaldo Assuncao, Geraldo Nunes Silva, Weldon Alexander Lodwick and Ulcilea Alves Severino Leal (2022), doi:10.1007/s00500-022-06958-4
[2] Leila Gharavi, Bart De Schutter and Simone Baldi (2023), doi:10.1016/j.ifacol.2023.10.369
[3] L. Cabral, G. Valmorbida and J.M. Gomes da Silva (2023), doi:10.1016/j.ifacol.2023.10.1626
References:
[1] Akre, J.-M., Juillard, J., Galayko, D., and Colinet, E. (2012). Akre, J, -M., Juillard, J., Galayko, D., and Colinet, E. Synchronization analysis of networks of self-sampled all-digital phase-locked loops. IEEE Transactions on Circuits and Systems -I. 4:708--720. doi:10.1109/TCSI.2011.2169745
[2] Akre, J.M., Juillard, J., Olaru, S., and Galayko, D. (2010). Akre, J, M., Juillard, J., Olaru, S., and Galayko, D. Determination of the behavior of self-sampled digital phase-locked loops. In 53rd IEEE Int. MWSCAS. pages 1089--1092. doi:10.1109/MWSCAS.2010.5548840
[3] Bemporad, A., Morari, M., Dua, V., and Pistikopoulos, E.N. (2002). Bemporad, A, , Morari, M., Dua, V., and Pistikopoulos, E.N. The explicit linear quadratic regulator for constrained systems. Automatica. 38:3--20. doi:10.1016/S0005-1098(01)00174-1
[4] Breiman, L. (1993). Breiman, L, Hinging hyperplanes for regression, classification, and function approximation. Information Theory, IEEE Transactions on. 39(3):999--1013. doi:10.1109/18.256506
[5] Daafouz, J. and Bernussou, J. (2001). Daafouz, J, and Bernussou, J. Parameter dependent lyapunov functions for discrete time systems with time varying parametric uncertainties. Systems & Control Letters. pages 355--359. doi:10.1016/S0167-6911(01)00118-9
[6] DiBernardo, M., Budd, C., Champneys, A., and Kowalczyk, P. (2007). DiBernardo, M, , Budd, C., Champneys, A., and Kowalczyk, P. Piecewise smooth dynamical systems: theory and applications Applied mathematical sciences, vol. 163. 2007. doi:10.1007/978-1-84628-708-4
[7] Feng, G. (2002). Feng, G, Stability analysis of piecewise discrete-time linear systems. IEEE Trans. Autom. Contr.. 47:1108--1112. doi:10.1109/TAC.2002.800666
[8] Ferrari-Trecate, G., Cuzzola, F.A., Mignone, D., and Morari, M. (2002). Ferrari-Trecate, G, , Cuzzola, F.A., Mignone, D., and Morari, M. Analysis of discrete-time piecewise affine and hybrid systems. Automatica. 38:2139--2146. doi:10.1016/S0005-1098(02)00142-5
[9] Heemels, W., DeSchutter, B., and Bemporad, A. (2001). Heemels, W, , DeSchutter, B., and Bemporad, A. Equivalence of hybrid dynamical models. Automatica. 37(7):1085--1091. doi:10.1016/S0005-1098(01)00059-0
[10] Hovd, M. and Olaru, S. (2013). Hovd, M, and Olaru, S. Relaxing pwq lyapunov stability criteria for pwa systems. Automatica. 49(2):667--670. doi:10.1016/j.automatica.2012.10.013
[11] Iervolino, R., Vasca, F., and Iannelli, L. (2015). Iervolino, R, , Vasca, F., and Iannelli, L. Cone-copositive piecewise quadratic lyapunov functions for conewise linear systems. Automatic Control, IEEE Transactions on. 60(11):3077--3082. doi:10.1109/TAC.2015.2409933
[12] Johansson, M. (2003). Johansson, M, Piecewise linear control systems: a computational approach. Springer Verlag. doi:10.1007/3-540-36801-9
[13] Johansson, M. and Rantzer, A. (1998). Johansson, M, and Rantzer, A. Computation of piecewise quadratic lyapunov functions for hybrid systems. IEEE Transactions on Automatic Control. 43:555--559. doi:10.1109/9.664157
[14] Lazar, M. (2006). Lazar, M, Model predictive control of hybrid systems: Stability and robustness. Ph.D. thesis, Technische Universiteit Eindhoven, Department of Electrical Engineering. .
[15] Oliveira, R., deOliveira, M.C., and Peres, P. (2008). Oliveira, R, , deOliveira, M.C., and Peres, P. Convergent lmi relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent lyapunov functions. Systems & Control Letters. pages 680--689. doi:10.1016/j.sysconle.2008.01.006
[16] Oliveira, R. and Peres, P. (2005). Oliveira, R, and Peres, P. Lmi conditions for robust stability analysis based on polynomially parameter-dependent lyapunov functions. Systems & Control Letters. pages 52--61. doi:10.1016/j.sysconle.2005.05.003
[17] Rantzer, A. and Johansson, M. (2000). Rantzer, A, and Johansson, M. Piecewise linear quadratic optimal control. IEEE Transactions on Automatic Control. 45:629--637. doi:10.1109/9.847100
[18] Rodrigues, L. and Boyd, S. (2005). Rodrigues, L, and Boyd, S. Piecewise-affine state feedback for piecewise-affine slab systems using convex optimization. Systems & Control Letters. 54(9):835--853. doi:10.1016/j.sysconle.2005.01.002
[19] Sontag, E. (1981). Sontag, E, Nonlinear regulation: The piecewise linear approach. Automatic Control, IEEE Transactions on. 26(2):346--358. doi:10.1109/TAC.1981.1102596


BibTeX:
@article{MIC-2018-1-2,
  title={{Parameter-dependent PWQ Lyapunov function stability criteria for uncertain piecewise linear systems}},
  author={Hovd, Morten and Olaru, Sorin},
  journal={Modeling, Identification and Control},
  volume={39},
  number={1},
  pages={15--21},
  year={2018},
  doi={10.4173/mic.2018.1.2},
  publisher={Norwegian Society of Automatic Control}
};