**Page description appears here**

“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.

     Valid XHTML 1.0 Strict

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, Joo 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

[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] Lfberg, 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}


May 2016: MIC reaches 2000 DOI Forward Links. The first 1000 took 34 years, the next 1000 took 2.5 years.

July 2015: MIC's new impact factor is now 0.778. The number of papers published in 2014 was 21 compared to 15 in 2013, which partially explains the small decrease in impact factor.

Aug 2014: For the 3rd year in a row MIC's impact factor increases. It is now 0.826.

Dec 2013: New database-driven web-design enabling extended statistics. Article number 500 is published and MIC reaches 1000 DOI Forward Links.

Jan 2012: Follow MIC on your smartphone by using the RSS feed.


July 2011: MIC passes 1000 ISI Web of Science citations.

Mar 2010: MIC is now indexed by DOAJ and has received the Sparc Seal seal for open access journals.

Dec 2009: A MIC group is created at LinkedIn and Twitter.

Oct 2009: MIC is now fully updated in ISI Web of Knowledge.