论文标题
带有开关检测的有限元素，可直接对非滑动系统的最佳控制
Finite Elements with Switch Detection for Direct Optimal Control of Nonsmooth Systems
论文作者
论文摘要
本文引入了使用开关检测（FESD）的有限元素，这是一种非平滑微分方程的数值离散方法。我们考虑[Stewart，1990]引入的动态互补系统的Filippov凸化以及转换为动态互补系统。 FESD基于解决非线性互补性问题，并且可以自动在及时检测非平滑事件。如果将标准的时间稳定runge-kutta（RK）方法天真地应用于非平滑颂歌，则精度充其量是Order One。在FESD中，我们让集成商的步进大小是一定程度的自由度。我们称之为交叉互补性的其他互补条件，启用了精确的开关检测，因此FESD可以恢复RK方法对光滑ode享受的高阶精度。称为步骤平衡的其他条件才能使步长大小仅在发生开关时发生变化，从而避免了虚假的自由度。 FESD方法的收敛结果是得出的，溶液的局部唯一性和数值灵敏度的收敛性得到了证明。在几个模拟和最佳控制示例中证明了FESD的功效。在使用FESD的最佳控制问题基准中，我们最多可以在相同的计算时间内实现高出五个数量级的准确解决方案。
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, 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.
