A comparison between global and local orthogonal collocation methods for solving optimal control problems

A comparison is made between local and global orthogonal collocation methods for solving optimal control problems. The study is performed using an orthogonal collocation method called the Gauss pseudospectral method. In particular, this method is employed in two different ways. In the first approach, the method is applied globally and solutions are computed for various numbers of collocation points. In the second approach, the Gauss pseudospectral method is applied locally, meaning the problem is segmented, with each segment containing a small, fixed number of collocation points. Solutions are then computed for various numbers of equal-width segments. These two different approaches are then compared in terms of their accuracy and computational efficiency. The results of this study indicate that, for a given number of collocation points, global collocation is much more accurate than local collocation for smooth problems and can provide comparable results for problems with discontinuities. Furthermore, the results obtained in this paper indicate that for a desired accuracy on smooth problems the global approach is computationally more efficient. In order to substantiate the analysis, the comparison is performed on two examples with vastly different characteristics.