Costate estimation for a multiple-interval pseudospectral method using collocation at the flipped legendre-Gauss-Radau points

Abstract

A multiple-interval pseudospectral method is presented for direct trajectory optimization and costate estimation of optimal control problems using collocation at the flipped Legendre-Gauss-Radau points. The recently developed global flipped Radau pseudospectral method is one of the special case. The necessary optimality conditions of the multiple-interval differential scaled optimal control problem are rigorously derived for the first time by applying the calculus of variations. Then, the costate and constraint multiplier estimates for the proposed method are derived in detail by comparing the discretized optimality conditions with the Karush-Kuhn-Tucker conditions of the nonlinear programming problem resulting from collocation. Furthermore, in order to improve the numerical stability and alleviate the computational error when calculating the state pseudospectral differentiation matrix, the barycentric Lagrange interpolation is adopted in the state approximation instead of the classic Lagrange interpolation without losing the computational efficiency. The method presented in this paper is applied to two benchmark optimal control problems and is compared with the de-facto standard pseudospectral methods. Extensive numerical results indicate that it has the ability to obtain highly accurate solutions to a wide variety of complex constrained optimal control problems.