Efficient ST Techniques for Nonlinear Optimal Control Analyses Using a Pseudospectral Framework

In this brief, numerical techniques for an efficient implementation of the spline transcription (ST) method by using the pseudospectral framework are proposed. It intends to develop a new approach to the ST method, analyses with which can be carried out with nearly the same level of computational burden without resorting to a specific type of collocation points as in the pseudospectral method. Therefore, the piecewise continuous polynomial interpolation is converted into a global spline interpolation, which has the same forms as the Lagrange interpolation and is used to derive the formulas for the partial integration and the differentiation. Using these formulas, the nonlinear optimal control problem can be transcribed in a similar manner as in the pseudospectral method and analyzed with numerical techniques and mathematical theories established well for the pseudospectral method. Also, this brief shows the Lagrange and tension spline interpolation can be used for the proposed methods. Numerical accuracy and efficiency of the proposed method are accessed through its applications to the solution of two nonlinear optimal control problems on uniform nodes. The predicted results of states, controls, and costates are well correlated with the optimal solutions. Finally, this brief recommends the use of advanced grid topologies to enhance both accuracy and convergence by using the flexibility in node selections with the proposed methods.