An overview of three pseudospectral methods for the numerical solution of optimal control problems

Divya Garg (1) , Michael Patterson (2) , William Hager (3) , Anil Rao (4) , David R. Benson (5) , Geoffrey T Huntington (6)

1 IHMC - Florida Institute for Human and Machine Cognition [Pensacola]
2 BEAGLE - Artificial Evolution and Computational Biology
3 UF|Math - Department of Mathematics [Gainesville]
4 UF|MAE - Department of Mechanical and Aerospace Engineering [Gainesville]
5 The Charles Stark Draper Laboratory, Inc. [Cambridge]
6 Blue Origin [Kent]

An overview is presented of three different pseudospectral methods based on collocation at Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and Legendre-Gauss-Lobatto (LGL) points. For each of the schemes presented here, (1) the state at the final time can be expressed in terms of a quadrature rule associated with the collocation points, (2) the state at the initial time is approximated by interpolation, and (3) the control and the state are approximated at the collocation points. The LG-based and LGR-based schemes presented here employ polynomials to approximate the state that are the same degree as the number of collocation points. In the corresponding LGL scheme, the state approximation is a polynomial that is one degree lower than the number of collocation points. Each of these scheme can be expressed in either a differential or an integral formulation. The LG and LGR differentiation and integration matrices are invertible, and the differential and integral versions are equivalent. The LGL differentiation matrix is singular and the equivalence between the differential and integral version is lost. For each scheme, the transformation between the KKT multipliers of the discrete nonlinear programming problem and costates of the continuous optimal control problem is developed. The LGL collocation is the only scheme for which the differentiation matrices for the state and the costate dynamics are the same. Two examples are used to assess the accuracy and features of each collocation scheme.

Domaines: Traitement du signal et de l'image [eess.SP]

Divya Garg, Michael Patterson, William Hager, Anil Rao, David R. Benson, et al.. An overview of three pseudospectral methods for the numerical solution of optimal control problems. 2017. ⟨hal-01615132⟩