A geometric approach to minimum-time control based on convexity

Starting from Pontryagin´s Maximum Principle (PMP), a geometric approach is presented in order to find the optimal control for multivariable systems with input constraints. The proposed algorithm works in all the cases in which the reachable sets are convex. The proposed approach is based on the PMP according to which every optimal solution can be generated with the knowledge of two parameters: the transition time t*, and the final costate q, 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 q that guarantee to reach the final state x₁, through a geometric method that make use of the hypothesis of convexity of system reachable sets. A convergence analysis is presented and the method is validated through simulations and experimental result on two sample systems: a double order integrator, for which the reachable set is also represented, and a rotary flexible joint device.