As of 2019, which discrete nonlinear, time-invariant systems with memory are considered "easy" to model and identify? 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?

Figure 2. A Wiener system is a cascade of an LTI system and a memoryless waveshaper.

Figure 3. A Wiener-Hammerstein system is a cascade of an LTI system, a memoryless waveshaper, and another LTI system.

Questions


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?
It is very strange phenomena that one object is completely dropped out of attention of researchers. It is Urysohn operator. First of all Urysohn is equivalent to multiple parallel Hammersteins and Urysohn followed by static nonlinearity is a model of any deterministic dynamic object, it maps any given input to any provided output. I obtained Ph.D. in modelling dynamic systems as Urysohn operators in 1991. Since that time I showed how to do that to many people, I published articles, I maintain the site, explaining it, and for this almost 20 years noone started using it, there is no single reference on my articles. People write 10,000 lines to identify Hammerstein, I show how to identify Urysohn by 10 lines of code, write here. Look 

for (int i = T - 1; i &lt; N; ++i) {
   double predicted = 0.0;
   for (int j = 0; j &lt; T; ++j) {
      predicted += U[(int)((x[i-j] - xmin) / deltaX), j];
   }
   double error = (y[i] - predicted) / T * learning_rate;
   for (int j = 0; j &lt; T; ++j) {
      U[(int)((x[i-j] - xmin) / deltaX), j] += error;
   }
}


That is it. It replace 10,000 line Hammerstein identification. It published, it showed, why noone use it I don't know. Google show my articles on "Urysohn identification" at the top. People don't search. 

Explanation to code. x[i], y[i] elements of input output arrays. U[k,m] elements of model matrix. T is one size of matrix U, another one is determined by deltaX, discretization interval, learning_rate is similar to LMS.