Abstract
In the last decade, pseudospectral methods have become popular for solving optimal control problems. Pseudospectral methods do not need prior knowledge about the optimal control structure and are thus very flexible for problems with complex path constraints, which are common in optimal train control, or train trajectory optimization. Practical optimal train control problems are nonsmooth with discontinuities in the dynamic equations and path constraints corresponding to gradients and speed limits varying along the track. Moreover, optimal train control problems typically include singular solutions with a vanishing Hessian of the associated Hamiltonian. These characteristics make these problems hard to solve and also lead to convergence issues in pseudospectral methods. We propose a computational framework that connects pseudospectral methods with Pontryagin’s Maximum Principle allowing flexible computations, verification and validation of the numerical approximations, and improvements of the continuous solution accuracy. We apply the framework to two basic problems in optimal train control: minimum-time train control and energy-efficient train control, and consider cases with short-distance regional trains and long-distance intercity trains for various scenarios including varying gradients, speed limits, and scheduled running time supplements. The framework confirms the flexibility of the pseudospectral method with regards to state, control and mixed algebraic inequality path constraints, and is able to identify conditions that lead to inconsistencies between the necessary optimality conditions and the numerical approximations of the states, costates, and controls. A new approach is proposed to correct the discrete approximations by incorporating implicit equations from the optimality conditions. In particular, the issue of oscillations in the singular solution for energy-efficient driving as computed by the pseudospectral method has been solved.
Highlights
Optimal control theory is widely applied in different fields to find the controls that minimize a cost functional subject to dynamic constraints, path constraints and boundary conditions
An indirect method derives the necessary optimality conditions based on PMP, which leads to a two-point boundary value problem (BVP) in the states and adjoint costates that needs to be solved by numerical methods (Ross, 2005)
In this paper we applied the pseudospectral method to two optimal train control problems with speed and time as state functions of distance
Summary
Optimal control theory is widely applied in different fields to find the controls that minimize a cost functional subject to dynamic constraints, path constraints and boundary conditions. An indirect method derives the necessary optimality conditions based on PMP, which leads to a two-point boundary value problem (BVP) in the states and adjoint costates that needs to be solved by numerical methods (Ross, 2005). We propose a computational evaluation framework where the PMP is applied to verify, validate and improve pseudospectral solutions to optimal train control problems. It is based on the costate approximations that the pseudospectral methods provide next to the controls and states. This reflects the traditional indirect optimal train control solution approach.
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