**2. The System Description**

Consider the Hammerstein OEMA systems shown in Figure 1,

$$y\_k \quad = \frac{B(z)}{A(z)}\vec{u}\_k + D(z)v\_{k\prime} \tag{1}$$

$$d\_k \quad = \ c\_1 f\_1(u\_k) + c\_2 f\_2(u\_k) + \dots + c\_m f\_m(u\_k),\tag{2}$$

where {*uk*} and {*yk*} are the input and output sequences of the system, {*u*¯*k*} is the output sequence of the nonlinear block, and it can be represented as a linear combination of a known basis *f*(*uk*) := [ *f*1(*uk*), *f*2(*uk*), ··· , *fm*(*uk*)] with unknown coefficients *ci* (*i* = 1, 2, ··· , *<sup>m</sup>*), {*vk*} is a stochastic white noise sequence with zero mean and variance *σ*2, *<sup>A</sup>*(*z*), *<sup>B</sup>*(*z*) and *<sup>D</sup>*(*z*) are the polynomials in the unit backward shift operator *z*<sup>−</sup><sup>1</sup> [*z*<sup>−</sup><sup>1</sup>*yk* = *yk*−<sup>1</sup>], and defined as

$$\begin{aligned} A(z) &:= \quad 1 + a\_1 z^{-1} + a\_2 z^{-2} + \cdots + a\_{n\_d} z^{-n\_d}, \\ B(z) &:= \quad 1 + b\_1 z^{-1} + b\_2 z^{-2} + \cdots + b\_{n\_b} z^{-n\_b}, \\ D(z) &:= \quad 1 + d\_1 z^{-1} + d\_2 z^{-2} + \cdots + d\_{n\_d} z^{-n\_d}. \end{aligned}$$

Assume that the orders of these polynomials *na*, *nb* and *nd* are known and *uk* = 0, *yk* = 0 and *vk* = 0 for *k* 0.

**Figure 1.** The Hammerstein OEMA systems.

Define the intermediate variables *xk* and *wk* as follows:

$$\begin{aligned} x\_k &:= -\frac{B(z)}{A(z)} \vec{u}\_k \\ &= -[1 - A(z)]x\_k + B(z)\vec{u}\_k \\ &= -\vec{u}\_k - \sum\_{i=1}^{n\_k} a\_i x\_{k-i} + \sum\_{i=1}^{n\_k} b\_i \vec{u}\_{k-i}, \\ w\_k &:= -D(z)v\_k \\ &= -\sum\_{i=1}^{n\_d} d\_i v\_{k-i} + v\_k. \end{aligned} \tag{3}$$

Take the first variable *<sup>u</sup>*¯*k* on the right-hand side of (3) as a separated key-term. Based ontheprincipleofkey-termseparation[42,43], substituting*<sup>u</sup>*¯*k*in(2)into(3)gives

*i*=1

$$\mathbf{x}\_{k} \quad = \sum\_{i=1}^{m} c\_{i} f\_{i}(\mathbf{u}\_{k}) - \sum\_{i=1}^{n\_{a}} a\_{i} \mathbf{x}\_{k-i} + \sum\_{i=1}^{n\_{b}} b\_{i} \mathbf{d}\_{k-i}.\tag{5}$$

Define the following related parameter vectors:

$$\begin{split} \boldsymbol{\theta} &:= \quad \left[ \begin{array}{c} \boldsymbol{\theta}\_{\mathsf{s}} \\ d \end{array} \right] \in \mathbb{R}^{n} \; \mathsf{m} := n\_{a} + n\_{b} + n\_{d} + m \\ \boldsymbol{\theta}\_{\mathsf{s}} &:= \quad [a^{\mathsf{T}}, b^{\mathsf{T}}, \mathsf{c}^{\mathsf{T}}]^{\mathsf{T}} \in \mathbb{R}^{n\_{a} + n\_{b} + m} \\ \boldsymbol{a} &:= \quad [a\_{1}, a\_{2}, \cdots, a\_{n\_{a}}]^{\mathsf{T}} \in \mathbb{R}^{n\_{a}} \; \mathsf{m} := [b\_{1}, b\_{2}, \cdots, b\_{n\_{b}}]^{\mathsf{T}} \in \mathbb{R}^{n\_{b}} \; \mathsf{m} \\ \mathsf{c} &:= \quad [\mathsf{c}\_{1}, \mathsf{c}\_{2}, \cdots, \mathsf{c}\_{m}]^{\mathsf{T}} \in \mathbb{R}^{m} \; \mathsf{m} := [d\_{1}, d\_{2}, \cdots, d\_{n\_{d}}]^{\mathsf{T}} \in \mathbb{R}^{n\_{d}} \; \mathsf{m} \end{split}$$

and the information vectors:

$$\begin{split} \mathfrak{g}\_{k} &:= \left[ \begin{array}{c} \mathfrak{g}\_{s,k} \\ \mathfrak{g}\_{n,k} \end{array} \right] \in \mathbb{R}^{n}, \\ \mathfrak{g}\_{s,k} &:= \left[ \begin{array}{c} -\mathfrak{x}\_{k-1}, -\mathfrak{x}\_{k-2}, \cdot \cdot \cdot, -\mathfrak{x}\_{k-n\_{d}}, \mathfrak{a}\_{k-1}, \mathfrak{a}\_{k-2}, \cdot \cdot \cdot, \mathfrak{a}\_{k-n\_{b}}, f(\mathfrak{u}\_{k}) \right] \in \mathbb{R}^{n\_{d} + n\_{b} + m}, \\ \mathfrak{g}\_{n,k} &:= \left[ \mathfrak{v}\_{k-1}, \mathfrak{v}\_{k-2}, \cdot \cdot, \mathfrak{v}\_{k-n\_{d}} \right] \in \mathbb{R}^{n\_{d}}. \end{split} \end{split}$$

From (1)–(5), we have

$$\begin{aligned} \mathfrak{z}\_k &= \quad \mathfrak{x}\_k + \mathfrak{w}\_k \\ &= \quad \mathfrak{p}\_{\mathfrak{s},k}^\top \theta\_{\mathfrak{s}} + \mathfrak{p}\_{\mathfrak{n},k}^\top d + \upsilon\_k \\ &= \quad \mathfrak{p}\_k^\top \theta + \upsilon\_k. \end{aligned} \tag{6}$$

Equation (6) is the identification model of the Hammerstein OEMA system. Please note that the parameter vector *θ* contains all the parameters of the system in (1)–(2), and the parameters in the linear and nonlinear blocks are separated. This means there is no need to identify redundant parameters. This paper aims to present an AM-MIFSG algorithm for Hammerstein OEMA systems to improve the parameter estimation accuracy.
