线性—二次型最优控制问题的Chebyshev—Legendre拟谱方法

THE CHEBYSHEV-LEGENDRE PSEUDOSPECTRAL METHOD FOR SOLVING THE LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS

介绍了一种求解线性—二次型最优控制问题的拟谱方法.使用Legendre展开式逼近控制和状态函数,采用Chebyshev-Gauss-Lobatto(CGL)点作为插值点,对原问题进行离散,从而将最初的最优控制问题化归为一个与之等价的二次规划(QP)问题,对应QP问题的未知量分别为状态和控制函数的Legendre展开式系数.通过求解QP问题得到原问题的数值解.整个离散过程使用快速Legendre变换(FLT)以及相关的一些技巧,能方便计算出函数在各个CGL点上的函数值.数值实验结果表明用该方法解决这类最优控制问题的有效性和高精度.

A kind of spectral collocation method, the Chebyshev Legendre method, for solving the linear-quadratic optimal control problems is derived. In this method, the Legendre expansions are utilized to approximate the control and the state functions and the Chebyshev- Gauss-Lobatto （CGL） points are employed as the interpolation points. Thus the unknown variables of the equivalent quadratic programming problems are the Legendre expansion coefficients of both the state and the control functions. The function values on the CGL nodes are calculated via the fast Legendre transform （FLT）. In this way, the FLT can be used to save the CPU execution time. Some numerical examples are given by using the Chebyshev-Legendre method in order to show its efficiency and high accuracy.