The K method

The loop is solved by means of the implicit function theorem.

$$g(\mbox{\boldmath${\xi}$},\Delta p)=f\left( \mbox{\boldmath${K}$} \Delta p+ \mbox{\boldmath${\xi}$} \right) -\Delta p =0$$

If $g$ satisfies the hypotheses of the theorem, then

$\Delta p=f \left( \mbox{\boldmath${K}$}\Delta p+\mbox{\boldmath${\xi}$} \righ... ...gmapsto \hspace{.5 cm}\Delta p=\bar{f}\left( \mbox{\boldmath${\xi}$} \right)$

At each step

• $\mbox{\boldmath${\xi}$}(n)$ is computed from from inputs and past values of $x_r, \dot{x}_r$.
• $\Delta p (n)$ is computed from $\mbox{\boldmath${\xi}$}(n)$.

Function $\bar{f}$ can be stored as a precomputed table. No iterative solvers are needed.