广义代价随机最优控制的路径积分形式
Path integral formulation of stochastic optimal control with generalized costs
摘要：
路径积分控制是用蒙特卡罗方法求解一类随机最优控制问题的一种方法。MC方法避免了对HJB方程的域的全局网格的需要，并且因此，路径积分控制原则上适用于中等到大维度的控制问题。然而，路径积分控制能够解决的问题的类别由对费用函数、噪声协方差阵和控制输入矩阵的要求限定。我们放宽了对费用函数的要求，通过引入一个新的状态，代表了一个增广的运行成本。在我们的新公式中，费用函数可以包含随机积分项和线性控制费用，这在工程、经济和金融等应用中是重要的。我们找到了一个有效的数值实现我们的无网格MC方法，并证明其性能和实用性的例子，从分层电力负荷管理。我们的例子之一的尺寸是大到足以使经典的基于网格的HJB求解器不切实际。
英文摘要：
Path integral control solves a class of stochastic optimal control problems with a Monte Carlo (MC) method for an associated Hamilton-Jacobi-Bellman (HJB) equation. The MC approach avoids the need for a global grid of the domain of the HJB equation and, therefore, path integral control is in principle applicable to control problems of moderate to large dimension. The class of problems path integral control can solve, however, is defined by requirements on the cost function, the noise covariance matrix and the control input matrix. We relax the requirements on the cost function by introducing a new state that represents an augmented running cost. In our new formulation the cost function can contain stochastic integral terms and linear control costs, which are important in applications in engineering, economics and finance. We find an efficient numerical implementation of our grid-free MC approach and demonstrate its performance and usefulness in examples from hierarchical electric load management. The dimension of one of our examples is large enough to make classical grid-based HJB solvers impractical.
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