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Time-optimal control of automobile test drives with gear shifts. (English) Zbl 1204.49033

Summary: We present a numerical method and results for a recently published benchmark problem [(*) M. Gerdts, “Solving mixed-integer optimal control problems by Branch&Bound: a case study from automobile test-driving with gear shift”, Optim. Contr. Appl. Met. 26, 1–18 (2005); (**) “A variable time transformation method for mixed-integer optimal control problems, Optim. Contr. Appl. Met. 27, No. 3, 169–182 (2006)] in mixed-integer optimal control. The problem has its origin in automobile test-driving and involves discrete controls for the choice of gears. Our approach is based on a convexification and relaxation of the integer controls constraint. Using the direct multiple shooting method we solve the reformulated benchmark problem for two cases: (a) As proposed in (*)), for a fixed, equidistant control discretization grid and (b) As formulated in (**) with a speed-up of several orders of magnitude compared with the Branch&Bound approach applied there (taking into account precision and the different computing environments). For the second case we optimize the switching times and propose to use an initialization based on the solution of (a). Compared with (*) we are able to reduce the overall computing time considerably, applying our algorithm. We give theoretical evidence on why our convex reformulation is highly beneficial in the case of time-optimal mixed-integer control problems as the chosen benchmark problem basically is (neglecting a small regularization term).

MSC:

49M37 Numerical methods based on nonlinear programming
90C11 Mixed integer programming
93A30 Mathematical modelling of systems (MSC2010)
90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut

Software:

DAESOL-II; Mintoc; SODAS
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Full Text: DOI

References:

[1] Gerdts, Solving mixed-integer optimal control problems by Branch&Bound: a case study from automobile test-driving with gear shift, Optimal Control Applications and Methods 26 pp 1– (2005)
[2] Gerdts, A variable time transformation method for mixed-integer optimal control problems, Optimal Control Applications and Methods 27 (3) pp 169– (2006)
[3] Sager, Numerical Methods for Mixed-integer Optimal Control Problems (2005) · Zbl 1094.65512
[4] Sager, Recent Advances in Optimization: Proceedings of the 12th French-German-Spanish Conference on Optimization pp 269– (2006)
[5] Sager, Proceedings OR2006 pp 37– (2007)
[6] Sager S, Reinelt G, Bock H. Direct methods with maximal lower bound for mixed-integer optimal control problems. Mathematical Programming, 2008. Published online at: http://dx.doi.org/10.1007/s10107-007-0185-6 on 14 August 2007, URL http://dx.doi.org/10.1007/s10107-007-0185-6. · Zbl 1160.49032
[7] Antsaklis P, Koutsoukos X. On hybrid control of complex systems: a survey. Third International Conference ADMP’98, Automation of Mixed Processes: Dynamic Hybrid Systems, Reims, France, March 1998, 1998; 1-8. · Zbl 1087.93025
[8] Oldenburg, Mixed logic dynamic optimization applied to batch distillation process design, AIChE Journal 49 (11) pp 2900– (2003)
[9] Oldenburg, Logic-based Modeling and Optimization of Discrete-continuous Dynamic Systems, Fortschritt-Berichte VDI Reihe 3, Verfahrenstechnik 830 (2005)
[10] Sussmann H. A maximum principle for hybrid optimal control problems. Conference Proceedings of the 38 th IEEE Conference on Decision and Control, Phoenix, 1999. · Zbl 0967.49016
[11] Shaikh M. Optimal control of hybrid systems: theory and algorithms. Ph.D. Thesis, Department of Electrical and Computer Engineering, McGill University, Montreal, Canada 2004. URL: http://www.cim.mcgill.ca/msshaikh/.
[12] Shaikh, On the hybrid optimal control problem: theory and algorithms, IEEE Transactions on Automatic Control 52 pp 1587– (2007) · Zbl 1366.93061
[13] Attia S, Alamir M, Canudas de Wit C. Sub optimal control of switched nonlinear systems under location and switching constraints. IFAC World Congress, Prague, Czech Republic, 2005.
[14] Alamir M, Attia SA. On solving optimal control problems for switched hybrid nonlinear systems by strong variations algorithms. Sixth IFAC Symposium, NOLCOS, Stuttgart, Germany, 2004.
[15] Terwen S, Back M, Krebs V. Predictive powertrain control for heavy duty trucks. Proceedings of IFAC Symposium in Advances in Automotive Control, Salerno, Italy, 2004; 451-457.
[16] Chachuat, Global methods for dynamic optimization and mixed-integer dynamic optimization, Industrial and Engineering Chemistry Research 45 (25) pp 8373– (2006)
[17] Kawajiri, Sixteenth European Symposium on Computer Aided Process Engineering and Ninth International Symposium on Process Systems Engineering pp 131– (2006)
[18] Bock H, Longman R. Computation of optimal controls on disjoint control sets for minimum energy subway operation. Proceedings of the American Astronomical Society. Symposium on Engineering Science and Mechanics, Taiwan, 1982.
[19] Plitt K. Ein superlinear konvergentes Mehrzielverfahren zur direkten Berechnung beschränkter optimaler Steuerungen. Master’s Thesis, Universität Bonn, 1981.
[20] Bock, Proceedings 9th IFAC World Congress pp 243– (1984)
[21] Biegler, Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation, Computers and Chemical Engineering 8 pp 243– (1984)
[22] Binder, Online Optimization of Large Scale Systems: State of the Art pp 295– (2001) · doi:10.1007/978-3-662-04331-8_18
[23] Till, Applied hybrid system optimization: an empirical investigation of complexity, Control Engineering 12 pp 1291– (2004)
[24] Pacejka, The magic formula tyre model, Vehicle System Dynamics 21 pp 1– (1993) · doi:10.1080/00423119208969994
[25] Sager S. MIOCP benchmark site. Available from: http://mintoc.de.
[26] Leineweber, An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization (Parts I and II), Computers and Chemical Engineering 27 pp 157– (2003)
[27] Diehl, IWR-Preprint 2001-25 (2001)
[28] Bauer I, Bock H, Schlöder J. DAESOL-a BDF-code for the numerical solution of differential algebraic equations. Internal Report, IWR, SFB 359, Universität Heidelberg, 1999.
[29] Albersmeyer, Modeling, Simulation and Optimization of Complex Processes: Proceedings of the International Conference on High Performance Scientific Computing pp 15– (2008) · doi:10.1007/978-3-540-79409-7_2
[30] Petzold, Sensitivity analysis of differential-algebraic equations and partial differential equations, Computers and Chemical Engineering 30 pp 1553– (2006)
[31] Leineweber, Fortschritt-Berichte VDI Reihe 3 613 (1999)
[32] Kaya, Computations and time-optimal controls, Optimal Control Applications and Methods 17 pp 171– (1996) · Zbl 0861.49008
[33] Kaya, A computational method for time-optimal control, Journal of Optimization Theory and Applications 117 pp 69– (2003) · Zbl 1029.49029
[34] Maurer, Optimization methods for the verification of second-order sufficient conditions for bang-bang controls, Optimal Control Methods and Applications 26 pp 129– (2005)
[35] Potschka, Diplomarbeit (2006)
[36] Gerdts, Direct shooting method for the numerical solution of higher index DAE optimal control problems, Journal of Optimization Theory and Applications 117 (2) pp 267– (2003) · Zbl 1033.65046
[37] Bryson, Applied Optimal Control (1975)
[38] Pontryagin, The Mathematical Theory of Optimal Processes (1962) · Zbl 0112.05502
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