Smooth nonlinearities (like in your case) can be modelled by Volterra series
You can see:
1. Identification using sweep sine
A. Farina, A. Bellini, E. Armelloni, Non-Linear
Convolution: A New Approach For The Auralization Of Distorting Systems, Proceedings
of the 110th AES Convention, 2001 May 12-15 Amsterdam, The Netherlands
2. Identification using MLS
MJ Reed and MOJ Hawksford. Identification of discrete Volterra series
using maximum length sequences. IEE Proc.-Circuits Devices Syst., 143(5):241-248,
October 1996.
For details on identification and modelling you can see book
VJ Mathews and GL.Sicuranza, Polynomial Signal Processing, New
York: John Wiley & Sons, Inc., 2000.