Is this controller technically nonlinear?

Question

Given a nonlinear dynamical system like

$$ \begin{split} \dot{x}_1 &= f_1(\mathbf{x}) \\ \dot{x}_2 &= f_2(\mathbf{x}) + g(\mathbf{x})u \,. \end{split} $$

Now assume I have a found a stabilizing nonlinear controller $u$ that contains $\dot{x}_1$ like (just an illustrative example):

$$ u = -k_1 x_1 - k_2 x_2 - f_1(\mathbf{x}) \,. $$

Since the control law $u$ contains the nonlinear function $f_1(\mathbf{x})$, it is a nonlinear control law.

However, what happens if I can actually measure $\dot{x}_1 = f_1(\mathbf{x})$? Then it would be just a linear combination of (measured) signals, is this still (technically) considered a nonlinear controller then? Or if I estimate $\dot{x}_1$ using a numerical derivative like

$$ D(s) = \frac{s}{T s + 1} $$

with $T$ small enough... $u$ still a nonlinear controller? Or a linear one?

Answer 1

The system $A(x, u)$ is linear with respect to the controller $u$ iff it satisfies the linearity arguments:

1) $y=A(x, u) \rightarrow \alpha y= A(x, \alpha u)$ and

2) $y_1=A(x, u_1), \ y_2=A(x, u_2) \rightarrow y_1+y_2=A(x,u_1+u_2) $