A Hermite‐Lobatto Pseudospectral Method for Optimal Control

A pseudospectral (PS) method based on Hermite interpolation and collocation at the Legendre-Gauss-Lobatto (LGL) points is presented for direct trajectory optimization and costate estimation of optimal control problems. A major characteristic of this method is that the state is approximated by the Hermite interpolation instead of the commonly used Lagrange interpolation. The derivatives of the state and its approximation at the terminal time are set to match up by using a Hermite interpolation. Since the terminal state derivative is determined from the dynamic, the state approximation can automatically satisfy the dynamic at the terminal time. When collocating the dynamic at the LGL points, the collocation equation for the terminal point can be omitted because it is constantly satisfied. By this approach, the proposed method avoids the issue of the Legendre PS method where the discrete state variables are over-constrained by the collocation equations, hence achieving the same level of solution accuracy as the Gauss PS method and the Radau PS method, while retaining the ability to explicitly generate the control solution at the endpoints. A mapping relationship between the Karush-Kuhn-Tucker multipliers of the nonlinear programming problem and the costate of the optimal control problem is developed for this method. The numerical example illustrates that the use of the Hermite interpolation as described leads to the ability to produce both highly accurate primal and dual solutions for optimal control problems.