Spectral Algorithm for Pseudospectral Methods in Optimal Control

By Qi Gong, Fariba Fahroo, I. Michael Ross. Journal of Guidance Control and Dynamics, 2008

198 citations recorded by the source corpus.

Recent convergence results with pseudospectral methods are exploited to design a robust, multigrid, spectral algorithm for computing optimal controls. The design of the algorithm is based on using the pseudospectral differentiation matrix to locate switches, kinks, corners, and other discontinuities that are typical when solving practical optimal control problems. The concept of pseudospectral knots and Gaussian quadrature rules are used to generate a natural spectral mesh that is dense near the points of interest. Several stopping criteria are developed based on new error-estimation formulas and Jackson's theorem. The sequence is terminated when all of the convergence criteria are satisfied. Numerical examples demonstrate the key concepts proposed in the design of the spectral algorithm. Although a vast number of theoretical and algorithmic issues still remain open, this paper advances pseudospectral methods along several new directions and outlines the current theoretical pitfalls in computation and control.

Subfields: Aerospace Engineering, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/3666c7aa9e340898eed87090f7ca81b2fc1ccadf

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Citations in the corpus

Riemann–Stieltjes Optimal Control Problems for Uncertain Dynamic Systems (2015, unknown): “(44) are called the spectral coefficients [17,18,21].”
Robust Reentry Guidance of a Reusable Launch Vehicle Using Model Predictive Static Programming (2014, methodology): “An alternate promising technique that leads to computationally efficient solutions of optimal control problems is the “pseudospectral method” [19,20], which has also been used for reentry guidance problems.”
Generalized Model Predictive Static Programming and Angle-Constrained Guidance of Air-to-Ground Missiles (2014, methodology): “A notatble computationally efficient solution of optimal control problems is the pseudospectral method [14,15].”
Homotopic approach and pseudospectral method applied jointly to low thrust trajectory optimization (2012, methodology): “Actually it is noted by many researchers [33,35, 40,45] that the discontinuous bang–bang controls will be smeared or diffused if the pseudospectral method is applied directly to the fuel-optimal problem.”
A review of pseudospectral optimal control: From theory to flight (2012, unknown): “This concept is part of a broader spectral algorithm [22, 52, 54, 19] wherein additional conditions are imposed such as a dual stopping condition for the adjoint covector derived by an application of the CMP.”
Density Functions for Mesh Refinement in Numerical Optimal Control (2011, methodology): “A mesh refinement method following this philosophy has been proposed in [4].”