论文标题
通过Koopman Lifts实现全球最佳控制
Towards Global Optimal Control via Koopman Lifts
论文作者
论文摘要
本文介绍了一个框架，用于解决所有最小化器满足Pontryagin必要的最佳条件的假设，以解决时间自主的非线性无限视野最佳控制问题。详细说明，我们使用来自符号几何学领域的方法来分析库普曼操作员的特征值，该操作员将Pontryagin的微分方程提升到适当定义的无限尺寸符号空间。这具有一个优势，即可以使用光谱分析领域的方法来表征全球最佳控制定律。然后，通过计算通过将Pontryagin-Koopman运算符将矩阵获得的特征值和特征向量计算在有限的尺寸空间中获得的矩阵的特征值和特征向量，从而获得了用于非线性系统最佳反馈定律的数值方法。我们通过计算范德尔控制系统的最佳非线性反馈定律的准确近似值来说明这种方法的有效性，该法无法通过线性控制定律稳定。
This paper introduces a framework for solving time-autonomous nonlinear infinite horizon optimal control problems, under the assumption that all minimizers satisfy Pontryagin's necessary optimality conditions. In detail, we use methods from the field of symplectic geometry to analyze the eigenvalues of a Koopman operator that lifts Pontryagin's differential equation into a suitably defined infinite dimensional symplectic space. This has the advantage that methods from the field of spectral analysis can be used to characterize globally optimal control laws. A numerical method for constructing optimal feedback laws for nonlinear systems is then obtained by computing the eigenvalues and eigenvectors of a matrix that is obtained by projecting the Pontryagin-Koopman operator onto a finite dimensional space. We illustrate the effectiveness of this approach by computing accurate approximations of the optimal nonlinear feedback law for a Van der Pol control system, which cannot be stabilized by a linear control law.
