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?


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) $

  • $\begingroup$ I don't mean the system itself but just the controller. With using for example state feedback linearization, some nonlinear systems can be exactly rendered linear - the controller itself might be nonlinear nevertheless. $\endgroup$
    – SampleTime
    Jun 25 '17 at 15:46
  • $\begingroup$ Well, you design the control law that contains a nonlinear function. The control is thus nonlinear. Then you approximate the nonlin. f-n with a linear one and the control becomes linear. There is nothing inherently linear or nonlinear in control, it depends on the way you write it. $\endgroup$
    – Dmitry
    Jun 26 '17 at 8:14
  • $\begingroup$ I don't think this is linear approximation of the nonlinear controller. By making $T$ arbitrary small, the approximation error gets arbitrary small (assuming no measurement noise). Therefore, if anything, it should be a nonlinear approximation of the nonlinear controller? $\endgroup$
    – SampleTime
    Jun 27 '17 at 17:45
  • 1
    $\begingroup$ You can say that you approximate (using $f(x)=\dot{x} \sim D(s)x$) or reformulate (using $\dot{x}=f(x)$) the nonlinear static control by a linear dynamic one. The feedback control becomes dynamic as you either use the derivative of the state or transform the state by a dynamic block. $\endgroup$
    – Dmitry
    Jun 29 '17 at 9:45

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