# Self-Organized Fuzzy Neural Network Nonlinear System Modeling Method Based on Clustering Algorithm
## 4 Simulation Analysis
### Identification of Nonlinear Systems  
In this example, the proposed SOFNN-CA is applied to identify dynamic systems with the following input delays [14]:  
\[y(t+1)=0.72y(t)+0.025y(t-1)u_{1}(t-1)+0.01u_{1}^{2}(t-2)+0.2u_{1}(t-3). \tag{60}\]  
The initial state is \(y(1)=y(2)=y(3)=y(4)=0\), and \(u(t)=1.05\times\sin{(t/45)}\). Four variables \(y(t),y(t-1)\), \(u_{1}(t-1)\), and \(u_{1}(t-2)\) are selected as input samples, and e is the output samples, with a total of 1000 sets of data for training.The number of neurons after initialization using the k-means algorithm is six, and the thresholds are md-addth = 0.15, e-min = 0.2, den-th = 5, and S-th = 8, and the expected error \(e\) is 0.0001. After the training process, the following signal \(u(t)\) is used to test the performance of the proposed SOFNN-CA, in which there are 1000 sets of training samples. The comparison algorithms use the same training and test samples listed in Table 2, and the results are the means of multiple tests.  
\[u(t)=\begin{cases}\sin{(\pi t/25)}&t<250\\ 1.0&250\leq t<500\\ -1.0&500\leq t<750\\ 0.3\sin{(\pi t/25)}+0.1\sin{(\pi t/32)}+0.6\sin{(\pi t/10)}&750\leq y<1000\\ \end{cases}. \tag{61}\]  
The results are listed in the original papers.  
The test results are shown in Figure 9.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Rule Neurons** & **Testing RMSE** & **Testing APE** & **CPU Times (s)** \\ \hline SOFNN-CA & 8 & 0.0171 & 0.0490 & 5.11 \\ SOFNN-ALA [20] & 6 & 0.0297 & 0.0954 & 3.73 \\ SOFNN-AGA [10] & 6 & 0.0090 & 0.0464 & 13.10 \\ RSEFNN-LF [38] & 4 1  & 0.0280 1 & 0.0652 1 & 35.31 1 \\ FWNN [39] & 5 1  & 0.0280 1 & 0.0652 1 & 35.31 1 \\ FWNN [39] & 5 1  & 0.0280 1 & 0.0904 1 & 37.72 1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison results of different methods for identifying nonlinear system.  
Figure 9: (**a**) Training neuron change plot; (**b**) training RMSE plot; (**c**) test result plot; (**d**) test error plot.  
Figure 9d shows the test error of the neural network, which varies slightly with different input functions. Since the input data is a dynamically changing function, the error is larger in the function segmentation region than in other regions.  
It is shown in Table 2 that SOFNN-CA outperforms SOFNN-ALA in test accuracy, but increases in the number of neurons and training time. Due to the adaptive gradient descent algorithm, SOFNN-AGA requires more iterations under the same error standard as SOFNN-CA, and the training time is longer, while the training accuracy is almost the same.  
Compared to RSEFNN-LF and FWNN algorithms, RSEFNN-LF uses an auto-evolutionary method to generate fuzzy rules online and uses the gradient descent algorithm to update the parameters, which builds a small number of neurons but is inferior to the SOFNN-CA algorithm in accuracy and training time. FWNN added a wavelet function to the subsequent part of the rule, which achieved better experimental results, but the performance was slightly worse than SOFNN-CA. Therefore, the present algorithm has good performance in the dynamic recognition of nonlinear systems compared with the other existing algorithms.