Recently I have read a paper where an LTI system of the form
$$ \begin{align} \dot{x}_p &= A_p x_p + B_p u \\ y &= C_p x_p \end{align}\tag{1} $$
for the control plant was considered. In order to analyze the system for different reference inputs, the author used the notation of what they called exosystem defined as:
$$ \begin{align} \dot{x}_e &= A_e x_e \\ r &= C_e x_e \end{align}\tag{2} $$
which serves as a reference generator. Assume a P controller with $u = K(r - y)$, then they combine $(1)$ and $(2)$ to
$$ \begin{align} \dot{x} &= A x \\ y &= C x \end{align} \tag{3} $$
with $x = [x_p^T \, x_e^T]^T$ and $A, C$ the resulting matrices from the feedback interconnection of $(1)$ and $(2)$. So far so good.
But now it is stated that such an exosystem can be used to generate all of the common input functions (step, ramp, sine, etc.) However I cannot see how the system $(2)$ is able to generate even just a simple step for example as it by itself, does not have any input.
Question: How can such an exosystem used to bring a system like $(1)$ to a form like $(3)$ given a specific input (e.g. step)? And how is the exosystem supposed to generate such input functions?
If you set the initial conditions at $t=0$ of $x_p$ to zero, then system $(3)$ for $t>0$ behaves the same as when the input from $(2)$ would have been multiplied by a step function. It can be noted that the initial conditions of $x_e$ are not zero, unless a $r=0$ is desired. So for example to generate a step or a ramp then the reference should be $r=1$ or $r=t$ respectively which can be obtained by choosing appropriate $A_e$, $C_e$ and $x_e(0)$, namely after multiplying these with a step function you get the desired inputs.
If you want $r$ to be an $n$th order polynomial in time then you could for example choose $A_e$ to be a Jordan block of size $n+1$ with eigenvalue zero. For example when $n=2$ one gets
$$ x_e(t)=e^{A_e\,t} x_e(0)= \begin{bmatrix} 1 & t & \tfrac12 t^2 \\ 0 & 1 & t \\ 0 & 0 & 1 \end{bmatrix}x_e(0) $$
Now by choosing $x_e(0)$ and $C_e$ one can get any linear combination of $1$, $t$ and $t^2$. For polynomial times sinusoids you would need to use completely imaginary values or real Jordan blocks for the eigenvalues instead of zero.