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Time‐optimal control of automobile test drives with gear shifts
Author(s) -
Kirches Christian,
Sager Sebastian,
Bock Hans Georg,
Schlöder Johannes P.
Publication year - 2009
Publication title -
optimal control applications and methods
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.458
H-Index - 44
eISSN - 1099-1514
pISSN - 0143-2087
DOI - 10.1002/oca.892
Subject(s) - benchmark (surveying) , discretization , equidistant , mathematical optimization , optimal control , initialization , mathematics , integer (computer science) , relaxation (psychology) , computer science , control theory (sociology) , control (management) , mathematical analysis , social psychology , geometry , geodesy , artificial intelligence , programming language , geography , psychology
We present a numerical method and results for a recently published benchmark problem ( Optim. Contr. Appl. Met. 2005; 26 :1–18; Optim. Contr. Appl. Met. 2006; 27 (3):169–182) 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 ( Optim. Contr. Appl. Met. 2005; 26 :1–18), for a fixed, equidistant control discretization grid and (b) As formulated in ( Optim. Contr. Appl. Met. 2006; 27 (3):169–182), taking into account free switching times. For the first case, we reproduce the results obtained in ( Optim. Contr. Appl. Met. 2005; 26 :1–18) 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 ( Optim. Contr. Appl. Met. 2006; 27 (3):169–182) we were 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). Copyright © 2009 John Wiley & Sons, Ltd.