**6. Examples**

Consider the following Hammerstein OEMA system:

$$\begin{array}{rcl} y\_k &=& \frac{B(z)}{A(z)} \bar{u}\_k + D(z)v\_{k\prime} \\ A(z) &=& 1 + a\_1 z^{-1} + a\_2 z^{-2} = 1 + 0.45z^{-1} + 0.56z^{-2}, \\ B(z) &=& 1 + b\_1 z^{-1} + b\_2 z^{-2} = 1 + 0.25z^{-1} - 0.35z^{-2}, \\ D(z) &=& 1 + d\_1 z^{-1} = 1 - 0.54z^{-1}, \\ \bar{u}\_k &=& c\_1 u\_k + c\_2 u\_k^2 + c\_3 u\_k^3 = 0.52u\_k + 0.54u\_k^2 + 0.82u\_k^3 \\ \theta &=& [a\_1 a\_2 b\_1 b\_2, b\_2 c\_1, c\_2 c\_3, d\_2]^\top = [0.45, 0.56, 0.25, -0.35, 0.52, 0.54, 0.82, -0.54]^\top. \end{array}$$

In this example, the input {*uk*} is a persistently excited signal sequence and {*vk*} is a white noise sequence with zero mean and variances *σ*<sup>2</sup> = 0.802. The data length is taken as *L* = 4000, where the first 3500 samples are assigned for system identification and the remaining 500 samples are assigned for prediction and validation. The details are as follows.

1. Firstly, applying the AM-MISG algorithm and the AM-MIFSG algorithm with *α* = 0.94 to estimate the parameters of considered system. Tables 1 and 2 show the parameter estimates and their errors with *p* = 1, 2, 4 and 6. Figures 2 and 3 indicate the parameter estimation errors *δ* := *θ*ˆ *k* − *θ*/*θ* versus *k*.

**Figure 2.** The AM-MISG estimation error *δ* versus *k* with *p* = 1, 2, 4 and 6 and the AM-MIFSG estimation error *δ* versus *k* with *p* = 2 and 6.

**Figure 3.** The AM-MIFSG estimation error *δ* versus *k* with *p* = 1, 2, 4 and 6.

**Table 1.** The AM-MISG estimates and errors *p* = 1, 2, 4 and 6.



**Table 2.** The AM-MIFSG estimates and errors with *p* = 1, 2, 4 and 6.

2. Secondly, to validate the influence of the fraction order *α*, in the AM-MIFSG algorithm, we take *p* = 5 and 6, and *α* =0.80, 0.90 and 0.92, respectively, the simulation results are shown in Tables 3 and 4, and Figures 4 and 5.

3. In the end, a different data set (*Le* = 500 samples from *k* = 3501 to 4000) and the estimated model obtained by the AM-MIFSG algorithm with *p* = 6 and *α* = 0.92 are used for model validation. The predicted output and true output are plotted in Figure 6 from *k* = 3501 to 3700 and Figure 7 from *k* = 3501 to 4000, where the average predicted output error is

$$\delta\_{\mathfrak{c}} = \frac{1}{L\_{\mathfrak{c}}} \left[ \sum\_{k=3501}^{4000} [\mathfrak{g}\_k - \mathfrak{g}\_k]^2 \right]^{1/2} = 0.0658\_{\text{eq}}$$

and the dots line is the output *y*ˆ*k* of the estimated model and the solid line is the true output *yk*.

From Tables 1–4 and Figures 2–7, we can draw the following conclusions: (1) with the innovation length *p* increases, both the AM-MISG and the AM-MIFSG algorithm can give higher parameter estimation accuracy; (2) in general, the AM-MIFSG algorithm has a faster convergence rate than the AM-MISG algorithm in the same situation, and the introduction of the fractional-order can improve the parameter estimation accuracy; (3) the convergence rate of the AM-MIFSG increases as the fractional-order *α* increases, the *α* within the range of [0.90, 0.95] seems to be an appropriate choice which can give better estimation results for the Hammerstein output-error systems; (4) the estimated model obtained by the AM-MIFSG algorithm can well capture system dynamics.


**Table 3.** The AM-MIFSG estimates and errors with *α* = 0.80, 0.90 and 0.92 (*p* = 5).

**Table 4.** The AM-MIFSG estimates and errors with *α* = 0.80, 0.90 and 0.92 (*p* = 6).


 **Figure 4.** The AM-MIFSG estimation error *δ* versus *k* with *α* = 0.80, 0.90 and 0.92 (*p* = 5).

 **Figure 5.** The AM-MIFSG estimation error *δ* versus *k* with *α* = 0.80, 0.90 and 0.92 (*p* = 6).

Solid line: the true output *yk*, dots: the predicted output *y*<sup>ˆ</sup>*k*. **Figure6.**Thepredictedoutput*y*ˆ*k*and trueoutput*yk*from*k*=3501to3700.

Solid line: the true output *yk*, dots: the predicted output *y*<sup>ˆ</sup>*k*. 

**Figure 7.** The predicted output *y*ˆ*k*and true output *yk*from *k* = 3501 to 4000.
