IIR initial condition / state meaning I am studying filters, in particular the Infinite Impulse Response ones.
I digged into theory and practice and I gain some decent knowledge on the topic. However, I can't really figure out or find good resurces about the so called &quot;Initial conditions&quot; or &quot;State of the IIR&quot;.
So far I kind of understood that it is related to the fact that we are using a feedback on previous outputs (y(n)), but I couldn't find a clear and exhaustive explanation on the topic.
In the Digital Signal Processing, Proakis at page 91 of the book (107 if considering the PDF index) it says:

The term y(n_0 -1) is called the initial condition for the systems....

In Digital Signal Processing with Scipy at page 5, the author shows an example of a low pass IIR filter and shows how the initialization of the filter state changes the result of the output.
In the latter paper, the author uses 4 values as initial state, and I can't figure out why.
I'm having some problems in understanding the meaning of these initial conditions /state, where do they come from and how should I compute them. I know that Scipy and Matlab already provide functions to compute those states, but I would like to have a more deep knowledge and understanding of what's going on behind the &quot;scenes&quot;.
The states depend on the form of an IIR filter. Take second order filters for simplicity. The following picture shows the structure of a transposed direct form II filter.

The differential equation is
$y(n) = b_0  x(n) + s_1(n)$
and the filter states can be calculated by recurrence
$s_1(n) = b_1 x(n-1) + s_2(n-1) - a_1 y(n-1)$
$s_2(n) = b_2 x(n-1) - a_2 y(n-1)$
As for (transposed) direct form I, it should have a filter states with 4 elements. The following is the transposed direct form I implementation of a second order filter.