Identification of nonlinear stochastic systems: Strongly consistent estimates for function values, gradients and system orders

The system, for which the current output is a nonlinear function f(·) of its past inputs and outputs of a fixed period of time superimposed by a random noise, is called the nonlinear ARX (NARX) system. It greatly extends the linear ARX systems and describes a large class of dynamic phenomena. Similar to the linear case, the most remote past input and output the current output will depend on, define the system order. The system order of NARX systems may change from place to place. The paper proposes the estimates for the values of f(·), its gradients, and orders and proves the strong consistency of the estimates. A numerical example is demonstrated, which is consistent with the theoretical analysis.