There are several types of discrete nonlinear time-invariant systems with memory ("NTIM") which are considered "easy" to model and identify. Any such system can be represented using a Volterra series, but Volterra kernels become very difficult to model for any more than quadratic or perhaps cubic nonlinearities.
However, for certain types of NTIM systems, there exist clear methods to completely measure and replicate some arbitrary unknown system, typically via measurement of certain test signals.
One good example are those systems for which each Volterra kernel has nonzero coefficients only on the diagonal. These are sometimes called "Farina," "diagonal-Volterra," or "Generalized Hammerstein" systems (Fig. 1). Given some suitable smoothness assumptions, these can be perfectly modeled using dynamic convolution, which is a variant of regular convolution that has a different impulse response for each scaled impulse. As a result, an arbitrary system of this type can be perfectly characterized by simply measuring the impulse response for each scaled impulse. Some info on that in this related answer: Dynamic convolution vs Volterra series. (There are also techniques involving synchronized sine sweeping)
Figure 1. A diagonal Volterra series system, which has nonzero Volterra coefficients only on the diagonal of each Volterra kernel, is equivalent (given suitable smoothness assumptions) to a sum of cascades of a memoryless waveshaper (illustrated as a sigmoid) and a linear time-invariant (LTI) system.
I am wondering if there exists any larger set of nonlinear systems that is also considered easy to model. There also exist the Wiener (Fig. 2) and Wiener-Hammerstein systems (Fig. 3) , which are not diagonal-Volterra; are those easy to model? And all of these systems seem to be subsets of "NARMAX" systems; how easy are they to model? Is there any easily modelable type of system which generalizes both the Weiner-Hammerstein and diagonal-Volterra systems?
- As of 2019, which nonlinear time-invariant systems with memory are clearly easy to model and profile?
- Is there any type of nonlinear system, which includes all the Volterra-diagonal ones as a subset, which is easy to model?
- Is there some type of system which includes both the Volterra-diagonal and the Weiner-Hammerstein systems, and is easy to model?
- All of these systems are a subset of the "NARMAX" nonlinear models - are those easy to model?