Suppose I have a digital filter implemented in Direct Form II. How do I initialize the state of the filter as if the input $x[n]$ had a fixed value $x_0$ for all $n<0$?
The difference equations for this filter are:
y[n] = b0 w[n] + b1 w[n-1] + b2 w[n-2] w[n] = x[n] - a1 w[n-1] - a2 w[n-2]
To achieve steady-state,
w[n] == w[n-1] == w[n-2]. Call this value
w. Solving the second difference equation, we find
w = x/(1 + a1 + a2), where x is the steady-state input value.