论文标题
线性约束的线性二次调节器从内核方法的角度
Linearly-constrained Linear Quadratic Regulator from the viewpoint of kernel methods
论文作者
论文摘要
线性二次调节剂问题在最佳控制中至关重要，并且自控制理论的开始以来就进行了研究。然而，当它包含仿射状态的约束时，从经典的``最大原则''观点仍然非常具有挑战性。在这项研究中，我们介绍了矩阵价值的再现核如何允许替代观点。我们表明，与线性动力学配对的二次物镜编码相关的内核，从而定义了受控轨迹的希尔伯特空间。借助内核形式主义，我们引入了加强的连续凸优化问题，可以通过有限的维求解器准确解决，哪些解决方案是限制因素的内部。在完善时间散布网格时，可以任意将该解决方案接近状态约束的线性二次调节器的解决方案。我们说明了该方法在路径规划问题上的实现。
The linear quadratic regulator problem is central in optimal control and was investigated since the very beginning of control theory. Nevertheless, when it includes affine state constraints, it remains very challenging from the classical ``maximum principle`` perspective. In this study we present how matrix-valued reproducing kernels allow for an alternative viewpoint. We show that the quadratic objective paired with the linear dynamics encode the relevant kernel, defining a Hilbert space of controlled trajectories. Drawing upon kernel formalism, we introduce a strengthened continuous-time convex optimization problem which can be tackled exactly with finite dimensional solvers, and which solution is interior to the constraints. When refining a time-discretization grid, this solution can be made arbitrarily close to the solution of the state-constrained Linear Quadratic Regulator. We illustrate the implementation of this method on a path-planning problem.
