Convexification of semi-activity constraints applied to minimum-time optimal control for vehicles with semi-active limited-slip differential

Semi-active actuators provide a good compromise between low energy consumption and high performance. Thus, they are deployed in many engineering applications, often combined with other actuators into complex systems requiring an integrated control concept for optimal performance. Optimal control can be used to objectively evaluate the performance of such systems as well as to deduce optimal control input trajectories and optimal passive system designs. We present a novel approach which enables considering a broad class of semi-active actuators in optimal control problems via convex sets. This procedure is exemplarily depicted for semi-active limited-slip differentials which are used in automotive applications for lateral torque distribution. The performance benefit gained by installing a semi-active limited-slip differential at the rear axle of a vehicle is objectively quantified by numerically computing time-optimal trajectories on a racetrack via direct optimal control with Hermite-Simpson collocation. Although the overall problem remains nonconvex for this particular application, this procedure is a first step towards a fully convex implementation. By iteratively increasing the upper boundary for the differential torque in multiple optimisations, we identify the smallest upper differential torque boundary for optimal laps and determine the lap time sensitivity regarding this limit.