Demonstrating equivalence between PNLSS and Volterra models for some SISO block-oriented models

This work aims to establish a relationship between the Polynomial NonLinear State Space (PNLSS) model and the Volterra model, focusing on three standard block-oriented models: Wiener, Hammerstein and WienerHammerstein. Initially, we generate input/output data for these block-oriented models in PNLSS form, utilizing
the known PNLSS parameters, while considering a specified memory length and order of nonlinearity. Subsequently, we estimate the Volterra kernel of the same data using a linear system solver, assuming the availability of model order and memory length information. The symmetric Volterra kernel tensor encompasses various combinations of PNLSS parameters specific to each block-oriented model. We extract the PNLSS parameters and utilize them to regenerate the output signal. Both analytical and numerical approaches are employed to demonstrate an equivalence between the two models. Due to the limited knowledge of the MultipleInput Multiple-Output (MIMO) block-oriented models, particularly concerning the dimensions of intermediate signals (i.e., signals between the blocks), our investigation is focused solely on Single-Input Single-Output (SISO) systems.