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## The challenge of cascading operators

Template for challenges.
Cascading operators are connected sequentially and output of the previous is an input of the next. The challenging task is to identify entire chain as dynamic object without recording of intermediate signals, having only first input and final output for the chain of operators.

It was already shown in Cascade article that two sequential Urysohns are generic enough for modeling of any deterministic dynamic object. Some approaches to modeling of such complicated objects are published in references of Cascade article , but generically the problem is not yet solved.

The sequence of dynamic blocks (both linear and nonlinear) is not only a mathematical description for convenience. It may be a model only feature, but not necessarily. It may be a matter of the physical object. For example, a car. If the road is straight and driver only applies random pushes to accelerator, the car reacts by changing the speed. But internally there is chain of objects connected sequentially: the engine, the transmission, the car wheels and road. In the modern cars the angular velocity of the engine and speed is shown in the front panel, so intermediate parameter is registered, but it is not always like that. Each physical object may have internal elements interacting in a complicated way and their internal actions and reactions can't be observed.

Only two sequential linear objects is another linear object, but two sequential Hammersteins is not a Hammerstein and two sequential Wieners is not a Wiener. Similarly, two sequential Urysohns can't be described by a single Urysohn.

The coding template at the link provides a challenging data generator. It generates two sequential Urysohns, input signal and computes output. It is prepared for the convenience of those researchers who like challenges and wish to build the model adequately converting input into output.

For the provided code, when apply single Urysohn to a data obtained from two sequential Urysohns, the average error is near 11 percent. This code is provided to those who wish to try other models and identification methods, for example Voltera LMS, kernel LMS, neuron network or point cloud. The authors of this site have their own ideas, they will be published when matured.