# Output is derivative of state variable

Given simple system of ODE.

$$\begin{cases} \dot{x_1}=-x_1+u \\ \dot{x_2}=-x_2-x_1 \end{cases}$$

As an output, I want to use $$y=\dot{x_1}$$.

But when I use the AsymptoticOutputTracker command, I get the error: AsymptoticOutputTracker::drtrz: The direct transmission matrix {{1/Subscript[\[FormalX], 2][t]},{0},{0}} is not a zero matrix. (code in Mathematica for example).

   asys = AffineStateSpaceModel[{x1'[t] == -x1[t] + u[t],
x2'[t] == -x2[t] - x1[t]}, {{x1[t], 1}, {x2[t], 0}}, {u[
t]}, {(u[t] - x1[t])}, t] // Simplify

pars1 = {Subscript[r, 1] -> 0, Subscript[p, 1] -> -1};

fb = AsymptoticOutputTracker[asys, {Subscript[r, 1]}, {Subscript[p, 1]}] // Simplify;


I got the idea to add a differentiating filter:

$$\begin{cases} \dot{x_1}=-x_1+u \\ \dot{x_2}=-x_2-x_1 \\ \frac{1}{k}\dot{X}+X=\dot{x_1} \end{cases}$$

And now $$y=X \approx \dot{x_1}$$ and $$k>>1$$.

How to transform state-space and get $$y=\dot{x_1}$$?

• Have you tried it with u[t]-x1[t] as the output? Jun 13 at 10:20
• @LouisB please see my edit
– dtn
Jun 13 at 10:27
• @dtn Is this optimization problem? Where is your code? Jun 13 at 11:00
• @AlexTrounev asys = AffineStateSpaceModel[{x1'[t] == -x1[t] + u[t], x2'[t] == -x2[t] - x1[t]}, {{x1[t], 1}, {x2[t], 0}}, {u[ t]}, {(u[t] - x1[t])}, t] // Simplify pars1 = {Subscript[r, 1] -> 0, Subscript[p, 1] -> -1}; fb = AsymptoticOutputTracker[asys, {Subscript[r, 1]}, {Subscript[p, 1]}] // Simplify;
– dtn
Jun 13 at 11:01
• @AlexTrounev This is me studying nonlinear control theory, section on asymptotic output tracking.
– dtn
Jun 13 at 11:02