Flat pseudospectral method

The flat pseudospectral method is part of the family of the Ross-Fahroo pseudospectral methods introduced by Ross and Fahroo. The method combines the concept of differential flatness with pseudospectral methods 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 $D$,

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

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,

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

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 easiy transformed to a problem with just the Y pseudospectral variable.

See also

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