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.
Unlike the
Legendre pseudospectral method The Legendre pseudospectral method for optimal control problems is based on Legendre polynomials. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross. A basic version of the Legendre pseudospectral was origin ...
, 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.,
and another by Fahroo and Ross.
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.
This breakthrough result opened the door for a covector mapping theorem for Chebyshev PS methods.
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
proposed using
Chebyshev polynomials
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebys ...
to solve
optimal control problems; however, none of these methods belong to the class of
pseudospectral method
Pseudo-spectral methods, also known as discrete variable representation (DVR) methods, are a class of numerical methods used in applied mathematics and scientific computing for the solution of partial differential equations. They are closely r ...
s.
See also
*
Legendre pseudospectral method The Legendre pseudospectral method for optimal control problems is based on Legendre polynomials. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross. A basic version of the Legendre pseudospectral was origin ...
*
Ross–Fahroo pseudospectral methods
*
Ross–Fahroo lemma
*
Bellman pseudospectral method
*
DIDO
Dido ( ; , ), also known as Elissa ( , ), was the legendary founder and first queen of the Phoenician city-state of Carthage (located in modern Tunisia), in 814 BC.
In most accounts, she was the queen of the Phoenician city-state of Tyre (t ...
References
{{DEFAULTSORT:Pseudospectral Optimal Control
Optimal control
Numerical analysis
Control theory