Input signal rescaling block

Question

1. The input signal is limited and lies in range $[-U;+U]$. $U$ - unknown.
2. The output signal must be rescaled to be in the range $[-1;+1]$, i.e. is $+1$ if the input is at steady-state $+U$, and $-1$ if the input at steady-state $-U$.
3. Between $-1$ and $+1$ the output signal must also occupy some rescaled value.

What can be used as a block $ ??? $ for such a conversion? Is there a linear / non-linear filter that does this, and such and which is described by the differential equation?

Remarks: Red line, just my fantasy about how the input signal changes