Vehicular Optimal Control

Chapter 9 deals with the solution of minimum-time and minimum-fuel vehicular optimal control problems. These problems are posed as fuel usage optimization problems under a time-of-arrival constraint, or minimum-time problems under a fuel usage constraint. The first example considers three variants of a simple fuel usage minimization problem under a time-of-arrival constraint. The first variant is worked out theoretically, and serves to highlight several of the structural features of these problems; the other two more complicated variants are solved numerically.The second example is also a multi-stage fuel usage minimization problem under a timeof- arrival constraint.More complicated track and vehicle models are then employed; the problem is solved numerically. The third problem is a lap time minimization problem taken from Formula One and features a thermoelectric hybrid powertrain. The fourth and final problem is a minimum-time closed-circuit racing problem featuring a racing motorcycle and rider.

Design and Control of Nonlinear Mechanical Systems for Minimum Time

This paper presents an integrated methodology for optimal design and control of nonlinear flexible mechanical systems, including minimum time problems. This formulation is implemented in an optimum design code and it is applied to the nonlinear behavior dynamic response. Damping and stiffness characteristics plus control driven forces are considered as decision variables. A conceptual separation between time variant and time invariant design parameters is presented, this way including the design space into the control space and considering the design variables as control variables not depending on time. By using time integrals through all the derivations, design and control problems are unified. In the optimization process we can use both types of variables simultaneously or by interdependent levels. For treating minimum time problems, a unit time interval is mapped onto the original time interval, then treating equally time variant and time invariant problems. The dynamic response and its sensitivity are discretized via space-time finite elements, and may be integrated either by at-once integration or step-by-step. Adjoint system approach is used to calculate the sensitivities.

A Fast and Robust Algorithm for General Inequality/Equality Constrained Minimum-Time Problems

This work has developed a new robust and reliable O(N) algorithm for solving general inequality/equality constrained minimum-time problems. To our knowledge, no one has ever applied an O(N) algorithm for solving such minimum time problems. Moreover, the algorithm developed here is new and unique and does not suffer the inevitable ill-conditioning problems that pre-existing O(N) methods for inequality-constrained problems do. Herein we demonstrate the new algorithm by solving several cases of a tip path constrained three-link redundant robotic arm problem with torque bounds and joint angle bounds. Results are consistent with Pontryagin’s Maximum Principle. We include a speed/robustness/complexity comparison with a sequential quadratic programming (SQP) code. Here, the O(N) complexity and the significant speed, robustness, and complexity improvements over an SQP code are demonstrated. These numerical results are complemented with a rigorous theoretical convergence proof of the new O(N) algorithm.