Flat pseudospectral method

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, ${\displaystyle D}$, in a pseudospectral method is square, higher-order derivatives of any polynomial, ${\displaystyle y}$, can be obtained by powers of ${\displaystyle D}$,

${\displaystyle {\begin{aligned}{\dot {y}}&=DY\\{\ddot {y}}&=D^{2}Y\\&{}\ \vdots \\y^{(\beta )}&=D^{\beta }Y\end{aligned}}}$

where ${\displaystyle Y}$ is the pseudospectral variable and ${\displaystyle \beta }$ is a finite positive integer. By differential flatness, there exists functions ${\displaystyle a}$ and ${\displaystyle b}$ such that the state and control variables can be written as,

${\displaystyle {\begin{aligned}x&=a(y,{\dot {y}},\ldots ,y^{(\beta )})\\u&=b(y,{\dot {y}},\ldots ,y^{(\beta +1)})\end{aligned}}}$

The combination of these concepts generates the flat pseudospectral method; that is, x and u are written as,

${\displaystyle x=a(Y,DY,\ldots ,D^{\beta }Y)}$

${\displaystyle u=b(Y,DY,\ldots ,D^{\beta +1}Y)}$

Thus, an optimal control problem can be quickly and easily transformed to a problem with just the Y pseudospectral variable. ^[1]

See also

• Ross' π lemma
• Ross–Fahroo lemma
• Bellman pseudospectral method

References

1. 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.
2. 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.
3. 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.
4. Rathinam, M. and Murray, R. M., “Configuration flatness of Lagrangian systems underactuated by one control” SIAM Journal on Control and Optimization, 36, 164,1998.