3种伪谱最优控制方法的积分形式及统一性证明

Integral form and equivalence proof of three pseudospectral optimal control methods

为了进一步提高伪谱最优控制方法的计算精度,削弱微分形式伪谱法对状态变量近似误差的放大幅度,研究基于积分形式的伪谱最优控制方法.依次给出3种伪谱法的积分伪谱离散形式,证明当Lagrange多项式对状态变量的近似误差等于零时,Gauss伪谱法和Radau伪谱法的积分形式与微分形式是等价的,而Legendre伪谱法的积分形式与微分形式是不等价的,并分析了其不等价的原因.

In order to further improve the accuracy of the pseudospectral optimal control method, and weaken the amplification of approximation errors of state variables in the differential pseudospectral method, the integral pseudospectral optimal control method is studied. The integral pseudospectral discrete forms of three pseudospectral methods are presented,which are Legendre pseudospectral method, Gauss pseudospectral method and Radau pseudospectral method, respectively.When the approximation errors of Lagrange polynomials for state variables are equal to zero, it is proved that the differential and integral forms of Gauss pseudospectral method and Radau pseudospectral method are equivalent, but that of Legendre pseudospectral method is not equivalent, and the reason of nonequivalence is also analyzed.