论文标题
使用集成残差方法和柔性网络解决非平滑溶液的最佳控制问题
Solving optimal control problems with non-smooth solutions using an integrated residual method and flexible mesh
论文作者
论文摘要
最佳控制问题的解决方案也可能是不连续的，即使定义问题的所有功能都是平稳的。当数值计算解决这些问题的解决方案时，这可能会造成困难。尽管常规数值方法假设状态和输入轨迹是连续，可区分或平滑的，但我们的方法能够通过引入时间网状节点作为决策变量来捕获解决方案中的不连续性。与固定的时机方法相比，这允许相同数量的网格节点获得更高的精度解决方案。此外，我们建议首先解决一系列适当定义的最小二乘问题，以确保动态方程中的误差低于给定的公差。然后将成本功能最小化受到动态方程残差的不平等约束。我们证明了我们对具有颤抖解决方案的最佳控制问题的实施。解决这样的问题很困难，因为该解决方案涉及降低持续时间的许多切换。该仿真显示了柔性网格如何捕获解决方案中存在的不连续性并随着网格间隔的数量的增加而获得超线性收敛。
Solutions to optimal control problems can be discontinuous, even if all the functionals defining the problem are smooth. This can cause difficulties when numerically computing solutions to these problems. While conventional numerical methods assume state and input trajectories are continuous and differentiable or smooth, our method is able to capture discontinuities in the solution by introducing time-mesh nodes as decision variables. This allows one to obtain a higher accuracy solution for the same number of mesh nodes compared to a fixed time-mesh approach. Furthermore, we propose to first solve a sequence of suitably-defined least-squares problems to ensure that the error in the dynamic equation is below a given tolerance. The cost functional is then minimized subject to an inequality constraint on the dynamic equation residual. We demonstrate our implementation on an optimal control problem that has a chattering solution. Solving such a problem is difficult, since the solution involves infinitely many switches of decreasing duration. This simulation shows how the flexible mesh is able to capture discontinuities present in the solution and achieve superlinear convergence as the number of mesh intervals is increased.
