Local Discretizations

Direct collocation methods that use local information to approximate the functions associated with an optimal control problem are well established [1, 2].

Sometimes, it may be convenient for users to compare the performance and solutions obtained by means of the pseudospectral methods implemented in PSOPT , with local discretization methods. Also, if a given problem cannot be solved by means of a pseudospectral discretization, the user has the option to try the local discretizations implemented in PSOPT, which include the trapezoidal and Hermite-Simpson transcriptions. PSOPT also implements the local mesh refinement method described by Betts [1] which uses these two local transcriptions.

An advantage of using a local discretization method as opposed to a pseudospectral discretization method, is that the resulting Jacobian and Hessian matrices needed by the NLP solver are more sparse with local methods, which facilitates the NLP solution. This becomes more noticeable as the number of grid points increases. The disadvantage of using a local method is that the spectral accuracy in the discretization of the differential constraints offered by pseudospectral methods is lost. Moreover, the accuracy of Gauss type integration employed in pseudospectral methods is also lost if pseudospectral grids are not used. Note also that local mesh refinement methods such as the one proposed by Betts [1], which is implemented in PSOPT, help to improve the accuracy of the solution by concentrating more grid points in areas of greater activity of the function.

The trapezoidal method has an accuracy of O(h2), while the Hermite-Simpson method has an accuracy of O(h4) [1,3], where h is the local interval between grid points. Both the trapezoidal and Hermite-Simpson discretization methods are widely used in computational optimal control, and have solved many challenging problems [1, 2]. When the user selects the trapezoidal or Hermite-Simpson discretizations, and if the initial grid points are not provided, the grid is started with equal spacing between grid points. In these two cases any integrals associated with the problem are computed using the trapezoidal and Simpson quadrature method, respectively. Additionally, an option is provided to use a differentiation matrix based on the central difference method (which has an accuracy of O(h2)) in conjunction with pseudospectral grids. The central differences option uses either the LGL or the Chebyshev points and Gauss-type quadrature.

[1] J.T. Betts. "Practical Methods for Optimal Control and Estimation Using Nonlinear Programming". SIAM, 2010.

[2] J. T. Betts, “Survey of Numerical Methods for Trajectory Optimization”,  AIAA Journal of Guidance, Control and Dynamics, Vol. 21, No. 2, March-April 1998, pp. 193-207
[3] A. Engelsone, S.L. Campbell J.T. Betts. "Order of Convergence in the Direct Transcription Solution of Optimal Control Problems". Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005