Finite Elements with Switch Detection for direct optimal control of nonsmooth systems

This paper introduces Finite Elements with Switch Detection (FESD), a numerical discretization method for nonsmooth differential equations. We consider the Filippov convexification of these systems and a transformation into dynamic complementarity systems introduced by Stewart (Numer Math 58(1):299–328, 1990). FESD is based on solving nonlinear complementarity problems and can automatically detect nonsmooth events in time. If standard time-stepping Runge–Kutta (RK) methods are naively applied to a nonsmooth ODE, the accuracy is at best of order one. In FESD, we let the integrator step size be a degree of freedom. Additional complementarity conditions, which we call cross complementarities, enable exact switch detection, hence FESD can recover the high order accuracy that the RK methods enjoy for smooth ODE. Additional conditions called step equilibration allow the step size to change only when switches occur and thus avoid spurious degrees of freedom. Convergence results for the FESD method are derived, local uniqueness of the solution and convergence of numerical sensitivities are proven. The efficacy of FESD is demonstrated in several simulation and optimal control examples. In an optimal control problem benchmark with FESD, we achieve up to five orders of magnitude more accurate solutions than a standard time-stepping approach for the same computational time.