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^[3] ^[4] with pseudospectral method to generate outputs in the so-called flat space.

Concept

Because the differentiation matrix, $D$ , in a pseudospectral method is square, higher-order derivatives of any polynomial, $y$ , can be obtained by powers of $D$ ,

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

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

{\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,

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

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]

References

1 2 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.

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Flat pseudospectral method

Concept

See also

References