I am a beginner in Koopman operators，and I want to apply it to the robotics. But I think the Koopman operator approximated by EDMD is more like a fitting tool. Because we always choose some dictionary functions $\Phi=\left \{ \phi_1(x),\phi_2(x),\dots,\phi_n(x) \right \} $, and the approximation of koopman operator is $K$ that satisfy $\Phi_{k+1}=K\Phi_{k}$, taking $\phi_1$ as an example, we get $$\phi_1(x_{k+1})=K_{11}\phi_1(x_k)+K_{12}\phi_2(x_k)+\ldots+K_{1n}\phi_1(x_k).$$ Doesn't it seem like a curve fitting?
Taking a simple discrete nonlinear system as an example, $$\begin{bmatrix}x_{1,k+1} \\x_{2,k+1} \end{bmatrix}=\begin{bmatrix}2*x_{1,k} \\x_{2,k}-x^2_{1,k} \end{bmatrix} $$.
- consider the fitting method
we will choose the basis function(like dictionary functions)$\Phi=\left \{ x_{1,k},x_{2,k},x^2_{1,k}\right\} $, and we get $$\begin{bmatrix}x_{1,k+1} \\x_{2,k+1} \end{bmatrix}=\begin{bmatrix} a &b &c \\ d &e &f \\ \end{bmatrix}\begin{bmatrix}x_{1,k} \\x_{2,k} \\x^2_{1,k} \end{bmatrix} $$. And we can solve the fitting problem and get $a=2,b=0,c=0,d=0,e=1,f=-1$.
- consider EDMD method $$\begin{bmatrix}x_{1,k+1} \\x_{2,k+1} \\x^2_{1,k+1} \end{bmatrix} =\begin{bmatrix} a &b &c \\ d &e &f \\ g &h &i \end{bmatrix}\begin{bmatrix}x_{1,k} \\x_{2,k} \\x^2_{1,k} \end{bmatrix} $$.And we can use the EDMD method and get $a=2,b=0,c=0,d=0,e=1,f=-1,g=0,h=0,i=4$.
I am confused about the modeling effect of the EDMD method and the fitting method seem similar(may be not for controlling, but the effect also doesn't seem very good).
I don't think this will have a very good effect in application, if we can't find the "best" dictionary functions(for EDMD method) or the basis functions(for fitting method)，and this situation is very common in applications. I may have a misunderstanding about the application of the Koopman operator, and I hope you can help me.