Optimal train control via switched system dynamic optimization

Abstract

This paper considers an optimal train control problem with two challenging, non-standard constraints: a speed constraint that is piecewise-constant with respect to the train's position, and control constraints that are non-smooth functions of the train's speed. We formulate this problem as an optimal switching control problem in which the mode switching times are decision variables to be optimized, and the track gradient and speed limit in each mode are constant. Then, using control parameterization and time-scaling techniques, we approximate the switching control problem by a finite-dimensional optimization problem, which is still subject to the challenging speed limit constraint (imposed continuously during each mode) and the non-smooth control constraints. We show that the speed constraint can be transformed into a finite number of point constraints. We also show that the non-smooth control constraints can be approximated by a sequence of conventional (smooth) inequality constraints. The resulting approximate problem can be viewed as a nonlinear programming problem and solved using gradient-based optimization algorithms, where the gradients of the cost and constraint functions are computed via the sensitivity method. A case study using data for a real subway line shows that the proposed method yields a realistic optimal control profile without the undesirable control fluctuations that can occur with the pseudospectral method.