数值求解最优控制问题中的精确导数计算方法

EXACT DERIVATIVE CALCULATION FOR SOLVING OPTIMAL CONTROL PROBLEM NUMERICALLY

针对常微分方程最优控制问题的数值求解,提出了一种自动计算性能指标梯度的方法.该方法通过显式数值积分方法求解状态方程,将最优控制的性能指标看作控制参数向量的显式函数,并根据链式求导法则计算其梯度.针对欧拉法和四阶龙格库塔法,推导了下一时刻状态对当前状态及控制导数的解析表达式.在求解状态方程的过程中,计算并采用行压缩方式存储了这些数值.然后在计算性能指标时反向计算,获得性能指标的精确梯度,并将其用于基于梯度的最优控制求解.通过计算实例验证了所计算导数的准确性,并在求解最优控制问题时与前向差分梯度方法进行了比较,求解结果表明了该方法的有效性.

An automatic approach is presented for calculating the gradient of per- formance in solving optimal control problem （OCP） governed by ordinary differential equations. The state equations are integrated explicitly, and the performance of OCP is regarded as an explicit function with respect to control parameters, the gradient of which can be calculated by using chain rule of derivation. The derivatives of states on next step with respect to current states and controls are given analytically for Euler method and 4th-order Runge-Kutta method. During the procedure of state equation integrating, these derivatives are calculated numerically and stored in row compressed form. The exact gradient of performance are computed backward time with the stored derivatives and can be used for solving OCP based on gradient method. An example is given for verify the accuracy of the derivatives provided by the exact derivative method. The results are compared to the gradient calculation method of forward difference in solving the OCP, which demonstrated the effectiveness of the presented method.