An algorithm for minimum-time feedforward control based on convexity

Starting from Pontryagin´s maximum principle (PMP), a geometric approach is presented in order to find the optimal control for dynamic systems with input constraints. The proposed algorithm works in the cases in which the reachable sets are convex. The approach is based on the PMP for which, in some cases, optimal solution can be generated with the knowledge of two parameters: the transition time, t*, and the final costate, q₁, which is the normal vector to the boundary of the set reachable at time t* at the final state. The devised algorithm is able to find the right values of t* and q₁ that guarantee to reach the final state x₁, through a geometric method that makes use of the convexity of system reachable sets. A convergence analysis is presented and the method is validated through simulations and experiments on three sample systems: a double order integrator, a mass on a cart, for which the reachable set is also represented, and the linearized model of a flexible joint device.