A variable time transformation method for mixed‐integer optimal control problems

The article discusses a variable time transformation method for the approximate solution of mixed-integer non-linear optimal control problems (MIOCP). Such optimal control problems enclose real-valued and discrete-valued controls. The method transforms MIOCP using a discretization into an optimal control problem with only real-valued controls. The latter can be solved efficiently by direct shooting methods. Numerical results are obtained for a problem from automobile test-driving that involves a discrete-valued control for the gear shift of the car. The results are compared to those obtained by Branch&Bound and show a drastic reduction of computation time. This very good performance makes the suggested method applicable even for many discretization points. Copyright © 2006 John Wiley & Sons, Ltd.

[1]  K. Malanowski,et al.  Error bounds for euler approximation of a state and control constrained optimal control problem , 2000 .

[2]  M. Gerdts.  A moving horizon technique for the simulation of automobile test‐drives , 2003 .

[3]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2002, SIAM J. Optim..

[4]  A. Ioffe,et al.  Theory of extremal problems , 1979 .

[5]  B. T. Poljak,et al.  Lectures on mathematical theory of extremum problems , 1972 .

[6]  P. Gill,et al.  User's Guide for SOL/NPSOL: A Fortran Package for Nonlinear Programming. , 1983 .

[7]  K. Teo,et al.  Control parametrization enhancing technique for time optimal control problems , 1997 .

[8]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[9]  Kok Lay Teo,et al.  An optimal control approach to nonlinear mixed integer programming problems , 1998 .

[10]  M. Gerdts.  Direct Shooting Method for the Numerical Solution of Higher-Index DAE Optimal Control Problems , 2003 .

[11]  William W. Hager,et al.  Runge-Kutta methods in optimal control and the transformed adjoint system , 2000, Numerische Mathematik.

[12]  Volker Rehbock,et al.  Numerical procedure for solving a class of singular optimal control problems , 2004, Optim. Methods Softw..

[13]  Philip E. Gill,et al.  Practical optimization , 1981 .

[14]  Eugene Khmelnitsky,et al.  A Combinatorial, Graph-Based Solution Method for a Class of Continuous-Time Optimal Control Problems , 2002, Math. Oper. Res..

[15]  Kok Lay Teo,et al.  Control parametrization enhancing technique for optimal discrete-valued control problems , 1999, Autom..

[16]  K. Teo,et al.  The control parameterization enhancing transform for constrained optimal control problems , 1999, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[17]  M. Gerdts.  Solving mixed‐integer optimal control problems by branch&bound: a case study from automobile test‐driving with gear shift , 2005 .