Solving mixed-integer optimal control problems by branch&bound: a case study from automobile test-driving with gear shift

Abstract

SUMMARYThe article discusses the application of the branch&bound method to a mixed integer non-linear optimization problem (MINLP) arising from a discretization of an optimal control problem with partly discrete control set. The optimal control problem has its origin in automobile test-driving, where the car model involves a discrete-valued control function for the gear shift. Since the number of variables in (MINLP) grows with the number of grid points used for discretization of the optimal control problem, the example from automobile test-driving may serve as a benchmark problem of scalable complexity. Reference solutions are computed numerically for two different problem sizes. A simple heuristic approach suitable for optimal control problems is suggested that reduces the computational amount considerably, though it cannot guarantee optimality anymore.

Avoidance Trajectories Using Reachable Sets and Parametric Sensitivity Analysis

“…Details on the model can be found in [5][6][7]. The dynamics are given by the following system of differential equations for the car's center of gravity (x, y) in the plane, the yaw angle ψ, the velocities v x and v y in x-and y-direction, respectively, the yaw angle rate w ψ , and the steering angle δ:…”

“…For further details please refer to [5][6][7]. For the following numerical computations we used realistic data for the various parameters involved in this model.…”

“…Unfortunately, these parameter values are proprietary and may not be published. For a different parameter set which is quite realistic please refer to [5].…”

Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control

“…A prominent example of this class is time-optimal car driving. In [16,17,21,35] numerical results for extended benchmark problems have been deduced. The outer convexification approach (Section 5.4) led to a tremendous speed-up compared to the published reference benchmark solution for a fixed control discretization grid by several orders of magnitude as shown in Table 1.…”

Block-structured quadratic programming for the direct multiple shooting method for optimal control

“…We finally apply the new complementary condensing approach for the first time to a vehicle control problem due to [13,14] in which the integer gear decision is convexified as first proposed for this problem in [23]. The obtained run times are compared to a general-purpose sparse symmetric indefinite factorization of the QP's KKT system using the HSL code MA57, as well as to the performance of the dense active set QP code QPOPT solving the condensed QPs obtained from the classical condensing algorithm.…”