Symbolic Transformations of Dynamic Optimization Problems

Fredrik Magnusson
Department of Automatic Control, Lund University, Lund, Sweden

Karl Berntorp
Department of Automatic Control, Lund University, Lund, Sweden

Björn Olofsson
Department of Automatic Control, Lund University, Lund, Sweden

Johan Åkesson
Department of Automatic Control, Lund University, Lund, Sweden/Modelon AB, Ideon Science Park, Lund, Sweden

Ladda ner artikelhttp://dx.doi.org/10.3384/ecp140961027

Ingår i: Proceedings of the 10th International Modelica Conference; March 10-12; 2014; Lund; Sweden

Linköping Electronic Conference Proceedings 96:107, s. 1027-1036

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Publicerad: 2014-03-10

ISBN: 978-91-7519-380-9

ISSN: 1650-3686 (tryckt), 1650-3740 (online)


Dynamic optimization problems involving differential-algebraic equation (DAE) systems are traditionally solved while retaining the semi-explicit or implicit form of the DAE. We instead consider symbolically transforming the DAE into an ordinary differential equation (ODE) before solving the optimization problem using a collocation method. We present a method for achieving this; which handles DAE-constrained optimization problems. The method is based on techniques commonly used in Modelica tools for simulation of DAE systems.

The method is evaluated on two industrially relevant benchmark problems. The first is about vehicletrajectory generation and the second involves startup of power plants. The problems are solved using both the DAE formulation and the ODE formulation and the performance of the two approaches is compared. The ODE formulation is shown to have roughly three times shorter execution time. We also discuss benefits and drawbacks of the two approaches.


Dynamic optimization; symbolic transformations; causalization; collocation


[1] F. Casella, F. Donida, and J. Åkesson, “Objectoriented modeling and optimal control: A case study in power plant start-up,” in 18th IFAC World Congress, (Milano, Italy), Aug. 2011.

[2] P.-O. Larsson, J. Åkesson, and N. Andersson, “Economic cost function design and grade change optimization for a gas phase polyethylene reactor,” in 50th IEEE Conf. Decision and Control and European Control Conference, (Orlando, FL), Dec. 2011.

[3] K. Berntorp, B. Olofsson, K. Lundahl, B. Bernhardsson, and L. Nielsen, “Models and methodology for optimal vehicle maneuvers applied to a hairpin turn,” in Am. Control Conf. (ACC), (Washington, DC), pp. 2139–2146, 2013.

[4] B. Olofsson, H. Nilsson, A. Robertsson, and J. Åkesson, “Optimal tracking and identification of paths for industrial robots,” in 18th IFAC World Congress, (Milano, Italy), Aug. 2011.

[5] F. E. Cellier and E. Kofman, Continuous System Simulation. New York, NY: Springer, Mar. 2006.

[6] B. Bachmann, L. Ochel, V. Ruge, M. Gebremedhin, P. Fritzson, V. Nezhadali, L. Eriksson, and M. Sivertsson, “Parallel multipleshooting and collocation optimization with OpenModelica,” in 9th Int. Modelica Conf., (Munich, Germany), Sept. 2012.

[7] R. Franke, “Formulation of dynamic optimization problems using Modelica and their efficient solution,” in 2nd Int. Modelica Conf., (Oberpfaffenhofen, Germany), Mar. 2002.

[8] R. Tarjan, “Depth-first search and linear graph algorithms,” SIAM J. Computing, vol. 1, no. 2, pp. 146–160, 1972.

[9] L. T. Biegler, Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes. MOS-SIAM Series on Optimization, Philadelphia, PA: Mathematical Optimization Society and the Society for Industrial and Applied Mathematics, 2010.

[10] J. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming: Second Edition. Advances in Design and Control, Philadelphia, PA: Society for Industrial and Applied Mathematics, 2010.

[11] F. Magnusson and J. Åkesson, “Collocation methods for optimization in a Modelica environment,” in 9th Int. Modelica Conf., (Munich, Germany), Sept. 2012.

[12] J. Åkesson, K.-E. Årzén, M. Gäfvert, T. Bergdahl, and H. Tummescheit, “Modeling and optimization with Optimica and JModelica.org—languages and tools for solving large-scale dynamic optimization problem,” Computers and Chemical Engineering, vol. 34, pp. 1737–1749, Nov. 2010.

[13] J. Åkesson, “Optimica—an extension of Modelica supporting dynamic optimization,” in 6th Int. Modelica Conf., (Bielefeld, Germany), Mar. 2008.

[14] A. Wächter and L. T. Biegler, “On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming,” Mathematical Programming, vol. 106, no. 1, pp. 25–57, 2006.

[15] J. Andersson, J. Åkesson, and M. Diehl, “CasADi – A symbolic package for automatic differentiation and optimal control,” in Recent Advances in Algorithmic Differentiation, Lecture Notes in Computational Science and Engineering, Berlin, Germany: Springer, 2012.

[16] B. Olofsson, K. Lundahl, K. Berntorp, and L. Nielsen, “An investigation of optimal vehicle maneuvers for different road conditions,” in 7th IFAC Symp. Advances in Automotive Control (AAC), (Tokyo, Japan), pp. 66–71, 2013.

[17] K. Lundahl, K. Berntorp, B. Olofsson, J. Åslund, and L. Nielsen, “Studying the influence of roll and pitch dynamics in optimal road-vehicle maneuvers,” in 23rd Int. Symp. Dynamics of Vehicles on Roads and Tracks (IAVSD), (Qingdao, China), 2013.

[18] R. Rajamani, Vehicle Dynamics and Control. Berlin Heidelberg: Springer-Verlag, 2006.

[19] K. Berntorp, “Derivation of a six degrees-offreedom ground-vehicle model for automotive applications,” Technical Report ISRN LUTFD2/TFRT--7627--SE, Department of Automatic Control, Lund University, Sweden, Feb. 2013.

[20] H. B. Pacejka, Tire and Vehicle Dynamics. Oxford, United Kingdom: Butterworth-Heinemann, 2nd edition ed., 2006.

[21] J. Wong, Theory of Ground Vehicles. Hoboken, NJ: John Wiley & Sons, 2008.

[22] HSL, “A collection of fortran codes for large scale scientific computation.” http://www.hsl.rl.ac.uk, 2013.

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