Chebyshev pseudospectral method
The Chebyshev pseudospectral method for optimal control problems is based on Chebyshev polynomials of the first kind. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross.{{cite journal | last1 = Ross | first1 = I. M. | last2 = Karpenko | first2 = M. | year = 2012 | title = A Review of Pseudospectral Optimal Control: From Theory to Flight | journal = Annual Reviews in Control | volume = 36 | issue = 2| pages = 182–197 | doi=10.1016/j.arcontrol.2012.09.002}} Unlike the Legendre pseudospectral method, the Chebyshev pseudospectral (PS) method does not immediately offer high-accuracy quadrature solutions. Consequently, two different versions of the method have been proposed: one by Elnagar et al.,{{cite journal | last1 = Elnagar | first1 = G. | last2 = Kazemi | first2 = M. A. | year = 1998 | title = Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems | journal = Computational Optimization and Applications | volume = 11 | issue = 2| pages = 195–217 | doi = 10.1023/A:1018694111831 | s2cid = 30241469 }} and another by Fahroo and Ross.{{cite journal | last1 = Fahroo | first1 = F. | last2 = Ross | first2 = I. M. | year = 2002 | title = Direct trajectory optimization by a Chebyshev pseudospectral method | url =https://zenodo.org/record/1235943 | journal = Journal of Guidance, Control, and Dynamics | volume = 25 | issue = 1| pages = 160–166 | doi=10.2514/2.4862| bibcode = 2002JGCD...25..160F }} The two versions differ in their quadrature techniques. The Fahroo–Ross method is more commonly used today due to the ease in implementation of the Clenshaw–Curtis quadrature technique (in contrast to Elnagar–Kazemi's cell-averaging method). In 2008, Trefethen showed that the Clenshaw–Curtis method was nearly as accurate as Gauss quadrature.
{{cite journal | last1 = Trefethen | first1 = Lloyd N. | author-link = Lloyd N. Trefethen | year = 2008 | title = Is Gauss quadrature better than Clenshaw–Curtis? | journal = SIAM Review | volume = 50 | issue = 1| pages = 67–87 | doi=10.1137/060659831| bibcode = 2008SIAMR..50...67T | citeseerx = 10.1.1.468.1193 }} This breakthrough result opened the door for a covector mapping theorem for Chebyshev PS methods.{{cite journal | last1 = Gong | first1 = Q. | last2 = Ross | first2 = I. M. | last3 = Fahroo | first3 = F. | year = 2010 | title = Costate Computation by a Chebyshev Pseudospectral Method | journal = Journal of Guidance, Control, and Dynamics | volume = 33 | issue = 2| pages = 623–628 | doi=10.2514/1.45154| bibcode = 2010JGCD...33..623G | hdl = 10945/48187 | s2cid = 55780038 | hdl-access = free }} A complete mathematical theory for Chebyshev PS methods was finally developed in 2009 by Gong, Ross and Fahroo.Q. Gong, I. M. Ross and F. Fahroo, A Chebyshev Pseudospectral Method for Nonlinear Constrained Optimal
Control Problems, Joint 48th IEEE Conference on Decision and Control and
28th Chinese Control Conference
Shanghai, P.R. China, December 16–18, 2009
Other Chebyshev methods
The Chebyshev PS method is frequently confused with other Chebyshev methods. Prior to the advent of PS methods, many authors{{cite journal | last1 = Vlassenbroeck | first1 = J. | last2 = Dooren | first2 = R. V. | year = 1988 | title = A Chebyshev technique for solving nonlinear optimal control problems | journal = IEEE Transactions on Automatic Control | volume = 33 | issue = 4| pages = 333–340 | doi=10.1109/9.192187}} proposed using Chebyshev polynomials to solve optimal control problems; however, none of these methods belong to the class of pseudospectral methods.
See also
References
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