Flat pseudospectral method explained
The flat pseudospectral method is part of the family of the Ross–Fahroo pseudospectral methods introduced by Ross and Fahroo.[1] [2] The method combines the concept of differential flatness with pseudospectral optimal control to generate outputs in the so-called flat space.[3] [4]
Concept
Because the differentiation matrix,
, in a pseudospectral method is square, higher-order derivatives of any polynomial,
, can be obtained by powers of
,
&=DY\
\ddoty&=D2Y\\
&{} \vdots\\
y(\beta)&=D\betaY
\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,
\ldots,y(\beta))\
u&=b(y,
\ldots,y(\beta)
\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
Notes and References
- Ross, I. M. and Fahroo, F., “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., “A Unified Framework for Real-Time Optimal Control,” Proceedings of the IEEE Conference on Decision and Control, Maui, HI, December, 2003.
- Fliess, M., Lévine, J., Martin, Ph., and Rouchon, P., “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., “Configuration flatness of Lagrangian systems underactuated by one control” SIAM Journal on Control and Optimization, 36, 164,1998.
