making a function non-linear using a Lagrangian function

Question

How Is this formula a Lagrangian function ? And how can a non-linear element be added to a function using this "Lagrangian function"

This is where i got this

In order to improve the performance of a PID controller, a non-linear element will be added to the controller. The non-linear effect is to be added in the controller gain. In this case the gains are to be a function of the error, using a Lagrangian function as shown above.

Answer 1

Look up lagrange polynomials on http://en.wikipedia.org/wiki/Lagrange_polynomial, and you will see that this is the general form of a lagrange polynomial or "lagrange function".

Also, to generate a lagrange polnomial use use data points, if you have $n$ data points $(x_1,y_1),...,(x_n,y_n)$, you will generate an $n$-th order polynomial approximation.

But if we know the function we are trying to approximate is linear i.e $f(x)=ax+b$, and we interpolate it with two data points, we will obtain a quadratic approximation, so we will have added a non linear term.