suppose I have a system like
$$ \begin{split} \dot{x}_1 &= -5x_1 + x_2 + u\\ \dot{x}_2 &= -x_2 \end{split} $$
then its pretty easy to derive a static state feedback controller since the system is linear. I would like to use this as a simple example for tracking control to understand how to apply the same techniques for nonlinear systems.
However, I have some understanding problems with what to do if I want to do tracking control. I would then define the error dynamics as (suppose I want to track $y = x_1$):
$$ e = y_{\text{ref}} - y \rightarrow \dot{e} = \dot{y}_{\text{ref}} - \dot{x}_1 $$
So, two questions:
1.) In litrature, the term $\dot{y}_{\text{ref}}$ is often omited (or better: set to zero). I guess this comes from an assumption that the reference does not change "fast", is this true?
2.) In the error dynamics, when inserting the equation for $\dot{x}_1$, I get:
$$ \dot{e} = \dot{y}_{\text{ref}} + 5x_1 - x_2 - u $$
How to proceed from there? Do I have to replace $x_1$ in terms of $e$? And what happens with $x_2$? Since $x_2$ doesn't appear in the error equation, how is this state included into the tracking control problem?
1) Derivative of a constant$=0$. When we say $\dot{y}_{\text{ref}}=0$, it means that the reference is constant, and that implies reference is not varying with respect to time. Example ${y}_{\text{ref}}(t)=2$, $t\ge0$.
2) You need to track $x_1$ right? such as say $y=x_1=y_{\rm ref}=2$. For that, we have, as you derived, $\dot{e}=\dot{y}_{\text{ref}}+5x_1-x_2-u(t)$. For tracking a constant output, the term $\dot{e}=5x_1-x_2-u(t)$ needs to be decreasing. That is as $t\to\infty$, ${e}\to 0$. Finding a $u(t)$ that ensures that the error will go to zero is what you need to do. One method is to find \begin{align}u=&-[k_1~k_2]\left[ \begin{array}{c} (x_1-y_{\rm ref}) \\ x_2 \\ \end{array} \right]\nonumber\\=&k_1e-k_2x_2. \end{align}
Find $K=[k_1~k_2]$ for stabilizing(regulating) the system, may be by pole placement.