In this paper, a pk-adaptive mesh refinement of pseudospectral method is proposed for solving optimal control problem by using collocation at Legendre-Gauss-Lobatto (LGL) points, motivated by reducing the redundant collocation points in the state-of-art mesh refinement methods to improve the time efficiency. The proposed method involves three phases, i.e., the determination of the polynomial degree, the determination of increasing intervals or nodes, and the optimization of the locations of segment breaks in each interval. First, determines the polynomial degree by the error estimation between the dynamics and the differentiation approximation of state variables according to the spectral matrix. Second, the maximum allowed polynomial degree in an interval is used to decide whether to segment interval or not. Third, the locations of segment points are obtained as the optimal design parameters of optimal control method. The terminology ‘‘pk-adaptive’’ or ‘‘p-then-k adaptive’’ is used because the polynomial degree is preferentially adaptive variation, then increases the segments by adding the optimal knots in each mesh interval. Finally, the residual of solutions, number of segments, number of nodes, CPU time, convergence of iteration, and parameters of the method have been analyzed in the comparing test to discuss the advantages of pk-adaptive mesh refinement. The discussions performed in two examples and demonstrated that the pk-adaptive method has the ability of optimizing nodes distribution to keep fewer nodes requirement and higher time efficiency than the hp- or ph-based pseudospectral methods while achieving the equivalent accuracy.

Introduction

Pseudospectral methods is widely used in the numerical solution of nonlinear optimal control problem [1], whose examples range from missiles’ dive phase trajectory maneuver [2], control of wave energy converters [3], trajectory optimization of boost-glide vehicle [4], trajectory design for lunar landing [5], etc. One of the key points for the wide application of pseudospectral method is the spectral accuracy of exponential convergence in differential approximation theory [6]. The selection of orthogonal basis functions and the orthogonal quadrature rules are two important factors to determine the nodes distribution for differential approximation with few discrete points [7]. Three orthogonal polynomials, i.e., Legendre [8], Chebyshev [9] and Laguerre [10], are commonly used as the basis functions. And three commonly used orthogonal quadrature rules are Gauss [11], Gauss-Radau [12], and Gauss-Lobatto [13]. By combining the above two factors, a pseudospectral method can be obtained to solve the nonlinear optimal control problems. For example, the LegendreGauss-Lobatto (LGL) pseudospectral method [14] combines Gauss-Lobatto quadrature with the Legendre polynomials according to the collocation points, which are known as LGL nodes. There is no definite evidence for the superiority of the selection of orthogonal basis functions between Legendre and Chebyshev, but Laguerre is the only one to be normally discussed for solving the infinite time problems [15]. Fahroo and Ross [16] discussed the application conditions of three orthogonal quadrature rules based on Legendre pseudospectral method and argued that Gauss-Lobatto should be used more in addition to special boundary problems.