“Piecewise quadratic Lyapunov functions for stability verification of approximate explicit MPC”

Authors: Morten Hovd and Sorin Olaru,
Affiliation: NTNU, Department of Engineering Cybernetics and SUPELEC Systems Sciences (France)
Reference: 2010, Vol 31, No 2, pp. 45-53.

Keywords: Piecewise Quadratic Lyapunov function, Linear Matrix Inequalities, Model Predictive Control, Piecewise Affine Systems

Abstract: Explicit MPC of constrained linear systems is known to result in a piecewise affine controller and therefore also piecewise affine closed loop dynamics. The complexity of such analytic formulations of the control law can grow exponentially with the prediction horizon. The suboptimal solutions offer a trade-off in terms of complexity and several approaches can be found in the literature for the construction of approximate MPC laws. In the present paper a piecewise quadratic (PWQ) Lyapunov function is used for the stability verification of an of approximate explicit Model Predictive Control (MPC). A novel relaxation method is proposed for the LMI criteria on the Lyapunov function design. This relaxation is applicable to the design of PWQ Lyapunov functions for discrete-time piecewise affine systems in general.

PDF PDF (337 Kb)        DOI: 10.4173/mic.2010.2.1

DOI forward links to this article:
[1] W.P.M.H. Heemels, B.A.G. Genuit and L. Lu (2012), doi:10.1049/iet-cta.2010.0709
[2] Hoai-Nam Nguyen, Sorin Olaru and Morten Hovd (2012), doi:10.1080/00207179.2012.713516
[3] Ionela Prodan, Sorin Olaru, Ricardo Bencatel, João Borges de Sousa, Cristina Stoica and Silviu-Iulian Niculescu (2013), doi:10.1016/j.conengprac.2013.05.010
[4] Morten Hovd and Sorin Olaru (2013), doi:10.1016/j.automatica.2012.10.013
[5] S. Olaru, N. A. Nguyen, G. Bitsoris, P. Rodriguez-Ayerbe and M. Hovd (2013), doi:10.1109/ICSTCC.2013.6688989
[6] Insu Chang and Joseph Bentsman (2015), doi:10.1109/CDC.2015.7402821
[7] Yasuaki Oishi (2012), doi:10.3182/20120620-3-DK-2025.00174
[8] Richard Oberdieck, Nikolaos A. Diangelakis, Ioana Nascu, Maria M. Papathanasiou, Muxin Sun, Styliani Avraamidou and Efstratios N. Pistikopoulos (2016), doi:10.1016/j.cherd.2016.09.034
[9] Lipeng Wang, Zhi Zhang, Qidan Zhu and Ran Dong (2017), doi:10.1177/0954410017746432
[10] Lipeng Wang, Zhi Zhang and Qidan Zhu (2019), doi:10.1177/0959651819868039
[1] Bank, M., Guddat, J., Klatte, D., Kummer, B., Tammer, K. (1983). Non-linear Parametric Optimization, Birkhauser, Stuttgart, Germany.
[2] Bemporad, A. Filippi, C. (2003). Suboptimal explicit receding horizon control via approximate multiparametric quadratic programming, Journal of optimization theory and applications, 11.1:9--38 doi:10.1023/A:1023696221899
[3] Bemporad, A. Filippi, C. (2006). An algorithm for approximate multiparametric convex programming, Computational optimization and applications, 35:87--108 doi:10.1007/s10589-006-6447-z
[4] Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N. (2002). The explicit linear quadratic regulator for constrained systems, Automatica, 38:3--20 doi:10.1016/S0005-1098(01)00174-1
[5] Feng, G. (2002). Stability analysis of piecewise discrete-time linear systems, IEEE Trans. Autom. Contr., 47:1108--1112 doi:10.1109/TAC.2002.800666
[6] Ferrari-Trecate, G., Cuzzola, F.A., Mignone, D., Morari, M. (2002). Analysis of discrete-time piecewise affine and hybrid systems, Automatica, 38:2139--2146 doi:10.1016/S0005-1098(02)00142-5
[7] Gilbert, E. Tan, K. (1991). Linear systems with state and control constraints: The theory and application of maximal output admissible sets, IEEE Trans. Autom. Contr., 36:1008--1020 doi:10.1109/9.83532
[8] Hovd, M. Braatz, R.D. (2001). On the use of soft constraints in mpc controllers for plants with inverse response, In Preprints Dycops. Jejudo Island, Korea, pages 295--300.
[9] Hovd, M., Scibilia, F., Maciejowski, J.M., Olaru, S. (2009). Verifying stability of approximate explicit MPC, In Conference on Decision and Control. Shanghai, China doi:10.1109/CDC.2009.5400788
[10] Johansen, T.A. Grancharova, A. (2003). Approximate explicit constrained linear model predictive control via orthogonal search tree, IEEE Transactions on Automatic Control, 48:810--815 doi:10.1109/TAC.2003.811259
[11] Johansson, M. Rantzer, A. (1998). Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Transactions on Automatic Control, 43:555--559 doi:10.1109/9.664157
[12] Jones, C.N., Baric, M., Morari, M. (2007). Multiparametric linear programming with application to control, European Control Journal, 13:152--170 doi:10.3166/ejc.13.152-170
[13] Kvasnica, M. (2009). Real-time Model Predictive Control via Multi-Parametric Programming: Theory and Tools, VDM Verlag, Saarbruecken, Germany.
[14] Löfberg, J. Yalmip (2004). A toolbox for modeling and optimization in MATLAB, In Proceedings of the CACSD Conference. Taipei, Taiwan, http://control.ee.ethz.ch/joloef/yalmip.php.
[15] Qin, S.J. Badgwell, T.A. (2003). A survey of industrial model predictive control technology, Control Engineering Practice, pp. 733--764 doi:10.1016/S0967-0661(02)00186-7
[16] Rantzer, A. Johansson, M. (2000). Piecewise linear quadratic optimal control, IEEE Transactions on Automatic Control, 45:629--637. 10.1109/9.847100.
[17] Scibilia, F., Olaru, S., Hovd, M. (2009). Approximate explicit linear MPC via Delaunay tesselation, In European Control Conference. Budapest, Hungary.

  title={{Piecewise quadratic Lyapunov functions for stability verification of approximate explicit MPC}},
  author={Hovd, Morten and Olaru, Sorin},
  journal={Modeling, Identification and Control},
  publisher={Norwegian Society of Automatic Control}