论文标题
一种时间冻结的方法，用于使用状态跳跃的非平滑微分方程的数值最佳控制
A Time-Freezing Approach for Numerical Optimal Control of Nonsmooth Differential Equations with State Jumps
论文作者
论文摘要
我们为非平滑差分方程式提供了一种新颖的重新进行状态跳跃的重新进行，它可以使其更容易的模拟和在最佳控制问题中使用，而无需使用整数变量。主要思想是引入一个辅助微分方程来模仿状态跳跃图。因此，还引入了时钟状态，该状态不会在辅助系统的运行时间内发展。当时钟状态不断发展时，与零件相对应的轨迹的碎片恢复了以跳跃的方式恢复原始系统的解决方案。我们的重新制作导致非平滑的普通微分方程，其中不连续性首次是轨迹的衍生物，而不是轨迹本身。这类系统更易于在理论上和数字上处理。我们提供数值示例，证明在模拟和最佳控制中易于使用该重新印象。在最佳控制示例中，非线性编程（NLP）求解器的单个调用产生与多阶段公式相同的解决方案，而无需通过枚举或启发式学探索最佳阶段数量。
We present a novel reformulation of nonsmooth differential equations with state jumps which enables their easier simulation and use in optimal control problems without the need of using integer variables. The main idea is to introduce an auxiliary differential equation to mimic the state jump map. Thereby, also a clock state is introduced which does not evolve during the runtime of the auxiliary system. The pieces of the trajectory that correspond to the parts when the clock state was evolving recover the solution of the original system with jumps. Our reformulation results in nonsmooth ordinary differential equations where the discontinuity is in the first time derivative of the trajectory, rather than in the trajectory itself. This class of systems is easier to handle both theoretically and numerically. We provide numerical examples demonstrating the ease of use of this reformulation in both simulation and optimal control. In the optimal control example a single call of a nonlinear programming (NLP) solver yields the same solution as a multi-stage formulation, without the need for exploring the optimal number of stages by enumeration or heuristics.
