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]

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]

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