Abstract
A numerical algorithm for approximate multi-parametric nonlinear programming (mp-NLP) is developed. The algorithm locally approximates the mp-NLP with a multi-parametric quadratic program (mp-QP). This leads to an approximate mp-NLP solution that is composed from the solution of a number of mp-QP solutions. The method allows approximate solutions to nonlinear optimization problems to be computed as explicit piecewise linear functions of the problem parameters. In control applications such as nonlinear constrained model predictive control this allows efficient online implementation in terms of an explicit piecewise linear state feedback without any real-time optimization.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Keywords
- Model Predictive Control
- Sequential Quadratic Programming
- Binary Search Tree
- Nonlinear Model Predictive Control
- Polyhedral Region
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Bemporad, A., Filippi, C.: Suboptimal explicit RHC via approximate multiparametric quadratic programming. J. Optimization Theory and Applications 117, 9–38 (2003)
Bemporad, A., Filippi, C.: An algorithm for approximate multiparametric convex programming. Computational Optimization and Applications 35, 87–108 (2006)
Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N.: The explicit linear quadratic regulator for constrained systems. Automatica 38, 3–20 (2002)
Bertsekas, D.P., Tsitsiklis, J.N.: Neuro-dynamic programming. Athena Scientific, Belmont (1998)
Domínguez, L.F., Narciso, D.A., Pistikopoulos, E.N.: Recent advances in multiparametric nonlinear programming. Computers and Chemical Engineering 34, 707–716 (2010)
Domínguez, L.F., Pistikopoulos, E.N.: Quadratic approximation algorithm for multiparametric nonlinear programming problems. Technical report. Imperial College London (2009)
Fiacco, A.V.: Introduction to sensitivity and stability analysis in nonlinear programming. Academic Press, Orlando (1983)
Gravdahl, J.T., Egeland, O.: Compressor surge control using a close-coupled valve and backstepping. In: Proceedings of the American Control Conference, Albuquerque, NM, vol. 2, pp. 982–986 (1997)
Grancharova, A., Johansen, T.A.: Approximate explicit model predictive control incorporating heuristics. In: Proceedings of IEEE International Symposium on Computer Aided Control System Design, Glasgow, Scotland, U.K., pp. 92–97 (2002)
Grancharova, A., Johansen, T.A., Tøndel, P.: Computational aspects of approximate explicit nonlinear model predictive control. In: Findeisen, R., Allgöwer, F., Biegler, L. (eds.) Assessment and Future Directions of Nonlinear Model Predictive Control. LNCIS, vol. 358, pp. 181–192. Springer, Heidelberg (2007)
Greitzer, E.M.: Surge and rotating stall in axial flow compressors, part i: Theoretical compression system model. J. Engineering for Power 98, 190–198 (1976)
Guddat, J., Guerra Vazquez, F., Jongen, H.T.: Parametric optimization: Singularities, pathfollowing and jumps. Wiley (1990)
Johansen, T.A.: On multi-parametric nonlinear programming and explicit nonlinear model predictive control. In: Proceedings of the IEEE Conference on Decision and Control, Las Vegas, NV, vol. 3, pp. 2768–2773 (2002)
Johansen, T.A., Grancharova, A.: Approximate explicit constrained linear model predictive control via orthogonal search tree. IEEE Trans. Automatic Control 48, 810–815 (2003)
Johansen, T.A.: Approximate explicit receding horizon control of constrained nonlinear systems. Automatica 40, 293–300 (2004)
Kojima, M.: Strongly stable stationary solutions in nonlinear programs. In: Robinson, S.M. (ed.) Analysis and Computation of Fixed Points, pp. 93–138. Academic Press, London (1980)
Levitin, E.S.: Perturbation theory in mathematical programming. Wiley (1994)
Mangasarian, O.L., Rosen, J.B.: Inequalities for stochastic nonlinear programming problems. Operations Research 12, 143–154 (1964)
Narciso, D.: Developments in nonlinear multiparametric programming and control. PhD thesis. London, U.K. (2009)
Nocedal, J., Wright, S.J.: Numerical optimization. Springer, New York (1999)
Parisini, T., Zoppoli, R.: A receding-horizon regulator for nonlinear systems and a neural approximation. Automatica 31, 1443–1451 (1995)
Parisini, T., Zoppoli, R.: Neural approximations for multistage optimal control of nonlinear stochastic systems. IEEE Trans. on Automatic Control 41, 889–895 (1996)
Pistikopoulos, E.N., Georgiadis, M.C., Dua, V.: Multi-parametric programming: Theory, algorithms, and applications. Wiley-VCH (2007)
Ralph, D., Dempe, S.: Directional derivatives of the solution of a parametric nonlinear program. Mathematical Programming 70, 159–172 (1995)
Rockafellar, R.T.: Convex analysis. Princeton University Press, New Jersey (1970)
Spjøtvold, J., Kerrigan, E.C., Jones, C.N., Tøndel, P., Johansen, T.A.: On the facet-to-facet property of solutions to convex parametric quadratic programs. Automatica 42, 2204–2209 (2006)
Tøndel, P., Johansen, T.A., Bemporad, A.: Evaluation of piecewise affine control via binary search tree. Automatica 39, 743–749 (2003)
Tøndel, P., Johansen, T.A., Bemporad, A.: An algorithm for multi-parametric quadratic programming and explicit MPC solutions. Automatica 39, 489–497 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag GmbH Berlin Heidelberg
About this chapter
Cite this chapter
Grancharova, A., Johansen, T.A. (2012). Explicit NMPC Using mp-QP Approximations of mp-NLP. In: Explicit Nonlinear Model Predictive Control. Lecture Notes in Control and Information Sciences, vol 429. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28780-0_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-28780-0_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-28779-4
Online ISBN: 978-3-642-28780-0
eBook Packages: EngineeringEngineering (R0)