# A Novel Complex-Valued Encoding Grey Wolf Optimization Algorithm
## 3 Complex-Valued Encoding Grey Wolf Optimization Algorithm (CGWO)
### CGWO Algorithm
The error \(e(t)\) indicates the difference between the actual system and its model yields. Here the error is \(e(t)=d(t)-y(t)\). Therefore, the problem of IIR model identification can be solved as a minimization problem of function. Additionally, the function can be molded as follows:

Table 3.2

where \(W\) indicates the number of the samples used in the simulation, and the aim is to minimize the function \(f(\theta)\) through adjusting \(\theta\). When the error function \(f(\theta)\) reaches its minimum value, the optimal model \(\theta^{*}\) or solution is obtained. The \(\theta^{*}\) is shown as follows:

Table 3.3

### IIR Model Identification Result

The result is reported considering a superior-order plant through a high-order IIR model. The unknown plant \(H_{p}\) and the IIR model \(H_{M}\) hold the following transfer function [27]:

Table 3.4

Table 3.5

Since the plant is a sixth-order system and the IIR model a fourth-order system, the error surface \(f(\theta)\) is multimodal. A white sequence with 100 samples has been used as input. The result of the experience over 20 consecutive experiments is shown in Tables 5 and 6. Table 13 shows the best parameter values (ABP). Table 14 indicates the average \(f(\theta)\) value (AVE) and the standard deviation (STD). Figure 6 provides a comparison between CGWO and the other algorithms for IIR system identification. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithms** & **AVE** & **STD** \\ \hline CGWO & 3.4013E\(-\)03 & 2.1644E\(-\)03 \\ DGWO & 1.2513E\(-\)01 & 7.1059E\(-\)02 \\ GWO & 5.5317E\(-\)03 & 4.9962E\(-\)03 \\ GGSA & 1.2537E\(-\)02 & 2.4702E\(-\)03 \\ ABC & 4.7898E\(-\)02 & 1.2004E\(-\)02 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: The average \(f(\theta)\) value (AVE) and the standard deviation (STD). Figure 6: Comparison between CGWO and the other algorithms for IIR system identification. According to the AVE and STD indexes in Table 6, the CGWO algorithm finds better results than the other algorithms. The results show that CGWO provides better precision (AVE value) and robustness (STD). As shown in Figure 6, the CGWO has the fastest convergence rate. In conclusion, the results show that the proposed method successfully outperforms the other algorithms in a majority of benchmark functions. In addition, IIR model identification shows that CGWO is able to obtain very competitive results. Therefore, as shown in the results of this comparative study, the proposed method has merit in the field of evolutionary algorithms and optimization. ## 6 Conclusions

In this work, the idea of complex-valued encoding is introduced into GWO, and a complex-valued version of GWO called CGWO was proposed based on complex-valued encoding. The proposed functions justified its excellent property on 16 benchmark functions in terms of improved avoidance of local minima and increased convergence speed. Furthermore, a complex system identification problem been employed to test the performance of the proposed method. The results verified the superior performance of CGWO on IIR model identification when compared to the other algorithms because of its improved ability. For future studies, it is recommended to apply the CGWO to solve more real-world engineering problems. This work is supported by National Science Foundation of China under Grants No. 61463007; 6153008. Project of Guangxi University for Nationalities Science Foundation under Grant No. 2012 MDZD037. **Author Contributions:** [PERSON] designed algorithm; [PERSON] and [PERSON] performed the experiments; [PERSON] wrote the paper. **Conflicts of Interest:** The authors declare no conflict of interest. ## References

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