A ph mesh refinement method for optimal control

• Published in: Optimal control applications & methods Vol. 36; no. 4; pp. 398 - 421
• Main Authors: Patterson, Michael A., Hager, William W., Rao, Anil V.
• Format: Journal Article
• Language: English
• Published: Glasgow Blackwell Publishing Ltd 01.07.2015; Wiley Subscription Services, Inc
• Subjects: Approximation; collocation; Computational efficiency; Estimates; Gaussian quadrature; Intervals; mesh refinement; Optimal control; Permissible error; Polynomials; Tolerances; variable-order
• ISSN: 0143-2087; 1099-1514
• DOI: 10.1002/oca.2114

SummaryA mesh refinement method is described for solving a continuous‐time optimal control problem using collocation at Legendre–Gauss–Radau points. The method allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. First, a relative error estimate is derived based on the difference between the Lagrange polynomial approximation of the state and a Legendre–Gauss–Radau quadrature integration of the dynamics within a mesh interval. The derived relative error estimate is then used to decide if the degree of the approximating polynomial within a mesh should be increased or if the mesh interval should be divided into subintervals. The degree of the approximating polynomial within a mesh interval is increased if the polynomial degree estimated by the method remains below a maximum allowable degree. Otherwise, the mesh interval is divided into subintervals. The process of refining the mesh is repeated until a specified relative error tolerance is met. Three examples highlight various features of the method and show that the approach is more computationally efficient and produces significantly smaller mesh sizes for a given accuracy tolerance when compared with fixed‐order methods. Copyright © 2014 John Wiley & Sons, Ltd.