# A Novel Complex-Valued Encoding Grey Wolf Optimization Algorithm
## 3 Complex-Valued Encoding Grey Wolf Optimization Algorithm (CGWO)
### CGWO Algorithm
As the results of the mean and STD show, CGWO performs better than the other algorithms on 30-D, 50-D multimodal benchmark functions. For \(F_{9}\) on 200-D, 300-D, 400-D, the GWO has better value than the CGWO. For \(F_{12}\) on 10-D, the GGSA has best result. For \(F_{14}\) on 100-D, the GWO is slightly better than the CGWO. For \(F_{16}\) on 100-D, CGWO and GGSA have very close results. This table also shows that CGWO provides better results than the other algorithms across the majority of dimensions on 10-D, 100-D, 200-D, 300-D, and 400-D. Moreover, the \(p\) values of \(F_{9}\sim F_{16}\) with different dimensions reported in Tables 5-11 are less than 0.05 across the majority of dimensions, which is strong evidence against null hypothesis. So, this evidence demonstrates that the results of CGWO are statistically significant and do not occur by coincidence. Figure 3 shows the averaged convergence curves of multimodal benchmark functions with 30-D over 20 independent runs. As shown in Figure 2, the convergence rate of CGWO in all cases is better than the other algorithms. Figure 4 depicts the anova tests between CGWO and the other algorithms on multimodal functions with 30-D. As shown, the more stable performance belongs to CGWO. According to Tables 4-11 and Figures 3 and 4 it can be stated that CGWO is capable of avoiding local minima with good convergence speed and stable property when dealing with multimodal benchmark functions. According to this comprehensive comparative study and discussion, we state that this algorithm has merit compared to the other algorithms. The next section inspects the performance of the CGWO in solving the IIR model identification problem. Figure 2: ANOVA tests of ABC, GGSA, GWO, DGWO, and CGWO on unimodal functions with Dim = 30. Figure 3: Convergence behavior of ABC, GGSA, GWO, DGWO, and CGWO on multimodal functions with Dim = 30. Figure 4: _Cont_

## 5 IIIR Model Identification

System identification is an effective method to obtain the mathematical model of an unknown system by analyzing its input and output values. In system identification, the use of infinite impulse response (IIR) has been widely studied since many problems encountered in signal processing can be characterized as a system identification problem (Figure 5). In this case, an optimization algorithm tries to iteratively adjust the parameters until the error between the output of the candidate model and the actual output of the plant is minimized. The use of IIR models for identification is preferred over their equivalent FIR (finite impulse response) models since the former produce more accurate models of physical plants for real world applications [32]. Moreover, IIR models are able to use fewer model parameters to meet the performance specifications. The fitness function, which is used in the article, is calculated as follows [33]. In an IIR system, the transfer function is provided as follows:

\[\frac{Y(z)}{X(z)}=\frac{b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+...+b_{m}z^{-m}}{1+a_{1} z^{-1}+a_{2}z^{-2}+...+a_{n}z^{-n}} \tag{17}\]

Figure 4: ANOVA tests of ABC, GGSA, GWO, DGWO, and CGWO on multimodal functions with Dim = 30. Table 3.0 Block diagram of IIR system identification. where \(m\) and \(n\) are the number of numerator and denominator coefficients of the transfer function, respectively, \(a_{i}\)\((i\in[1,\,...\,,\,n])\) is the pole of the IIR model, \(b_{j}(j\in[1,\,...\,,\,m])\) is the zero parameters of the IIR model. Equation (17) is expressed in another form as follows:

Table 3.1

where \(u(t)\) indicates the \(t\)th input of the system, \(y(t)\) indicates the \(t\)th output of the system. Hence, the set of unknown parameters that simulates the IIR system is indicated by \(\theta=\{a_{1},...,a_{n},b_{0},...,b_{m}\}\). The number of unknown parameters of \(\theta\) is \((n+m+1)\). Correspondingly, the search space \(S\) of IIR feasible values for \(\theta\) is \(\Re^{(n+m+1)}\). According to Figure 5, the output of the plant is \(d(t)\), while the output of the IIR filter is \(y(t)\).