Flat pseudospectral method
{{Short description|Optimal control technique}}
The flat pseudospectral method is part of the family of the Ross–Fahroo pseudospectral methods introduced by Ross and Fahroo. Ross, I. M. and Fahroo, F., “[https://core.ac.uk/download/pdf/36722493.pdf Pseudospectral Methods for the Optimal Motion Planning of Differentially Flat Systems],” IEEE Transactions on Automatic Control, Vol.49, No.8, pp. 1410–1413, August 2004. Ross, I. M. and Fahroo, F., “[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.966.7574&rep=rep1&type=pdf A Unified Framework for Real-Time Optimal Control],” Proceedings of the IEEE Conference on Decision and Control, Maui, HI, December, 2003. The method combines the concept of differential flatness with pseudospectral optimal control to generate outputs in the so-called flat space. Fliess, M., Lévine, J., Martin, Ph., and Rouchon, P., “[https://www.researchgate.net/profile/Philippe_Martin/publication/292446786_Flatness_and_defect_of_nonlinear_systems_Introductory_theory_and_applications/links/56b1cd1908ae5ec4ed49eea5.pdf Flatness and defect of nonlinear systems: Introductory theory and examples],” International Journal of Control, vol. 61, no. 6, pp. 1327–1361, 1995. Rathinam, M. and Murray, R. M., “[https://authors.library.caltech.edu/28012/1/960006.pdf Configuration flatness of Lagrangian systems underactuated by one control]” SIAM Journal on Control and Optimization, 36, 164,1998.
Concept
Because the differentiation matrix, , in a pseudospectral method is square, higher-order derivatives of any polynomial, , can be obtained by powers of ,
:
\begin{align}
\dot y &= D Y \\
\ddot y & = D^2 Y \\
&{} \ \vdots \\
y^{(\beta)} &= D^\beta Y
\end{align}
where is the pseudospectral variable and is a finite positive integer.
By differential flatness, there exists functions and such that the state and control variables can be written as,
:
\begin{align}
x & = a(y, \dot y, \ldots, y^{(\beta)}) \\
u & = b(y, \dot y, \ldots, y^{(\beta + 1)})
\end{align}
The combination of these concepts generates the flat pseudospectral method; that is, x and u are written as,
:
:
Thus, an optimal control problem can be quickly and easily transformed to a problem with just the Y pseudospectral variable.
See also
References
{{Reflist}}
{{DEFAULTSORT:Pseudospectral Optimal Control}}