Adaptive Mesh Refinement Method for Optimal Control Using Decay Rates of Legendre Polynomial Coefficients

Abstract

An adaptive mesh refinement method for solving optimal control problems is described. The method employs orthogonal collocation at Legendre-Gauss-Radau points. Accuracy in the method is achieved by adjusting the number of mesh intervals, the polynomial degree within each mesh interval, and, when possible, reducing the mesh size. The decision to increase the degree of the polynomial within a mesh interval or to create new mesh intervals is based on the decay rate of the coefficients of a Legendre polynomial approximation of the state as a function of the index of the Legendre polynomial expansion. The polynomial degree in a mesh interval is increased if the Legendre polynomial coefficient decay rate exceeds a user-specified threshold. Otherwise, the mesh interval is divided into subintervals. The method developed in this brief is then demonstrated on two examples, one of which is a practical problem in aircraft performance optimization. It is found that the approach developed in this brief is more efficient, is simpler to implement, and requires the specification of fewer user-defined parameters when compared with recently developed adaptive mesh refinement methods for optimal control.

Mesh Refinement Method for Solving Bang-Bang Optimal Control Problems Using Direct Collocation

“…For problems requiring more than a single mesh refinement to meet accuracy tolerance, the bang-bang mesh refinement method will utilize the hp-adaptive method described in Ref. [32] to further refine the phases not meeting the specified mesh accuracy tolerance. It is noted that any of the four previously developed hp-adaptive mesh refinement methods may be paired with the bang-bang mesh refinement method.…”

“…In an hp method, both the number of mesh intervals and the degree of the approximating polynomial within each mesh interval are allowed to vary. While hp methods were originally developed as finite-element methods for solving partial differential equations [26,27,28,29,30], over the past several years hp methods have been developed for solving optimal control problems [15,16,19,31,32]. The methods described in Refs.…”

“…Moreover, the approach of this paper is designed to be used in conjunction with a previously developed hp mesh refinement method such as those described in Refs. [15,16,19,31,32]. First, a solution is obtained on a coarse mesh.…”

“…Section 6 demonstrates the performance of the mesh refinement method of this paper when solving bang-bang optimal control problems as compared to the four previously developed hp-adaptive mesh refinement methods of Refs. [19,15,31,32]. Section 7 provides a discussion of both the approach and the results.…”

Mesh adaptation in direct collocated nonlinear model predictive control

“…These include pseudospectral methods 9 and adaptive wavelet methods. 10 In the work of Liu et al, 11 the decay rates of Lengendre polynomial coefficients act as an estimate of the state approximation error.…”