## 5 Neural Control Strategies
### Inverse Neural Controller
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Network** & **MSE Step** & **MSE Sinusoid** & **MSE Sawtooth** & **Total** \\ \hline
1 th5s5d & \(2.1019\times 10^{-5}\) & \(1.5618\times 10^{-4}\) & \(3.0189\times 10^{-4}\) & \(4.7908\times 10^{-4}\) \\ \hline
1 th10s5d & \(2.0237\times 10^{-5}\) & \(1.5752\times 10^{-4}\) & \(3.0275\times 10^{-4}\) & \(4.8050\times 10^{-4}\) \\ \hline
1 th15s5d & \(2.7159\times 10^{-5}\) & \(1.5577\times 10^{-4}\) & \(3.0127\times 10^{-4}\) & \(4.8420\times 10^{-4}\) \\ \hline
1 th5s10d & \(9.8642\times 10^{-5}\) & \(2.6887\times 10^{-4}\) & \(5.2893\times 10^{-4}\) & \(8.9644\times 10^{-4}\) \\ \hline
1 th10s10d & \(9.8300\times 10^{-5}\) & \(2.7111\times 10^{-4}\) & \(5.2967\times 10^{-4}\) & \(8.9908\times 10^{-4}\) \\ \hline
1 th15s10d & \(1.9305\times 10^{-2}\) & \(2.8199\times 10^{-4}\) & \(2.5666\times 10^{-2}\) & \(4.5253\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: MSE obtained by the inverse model control networks for the different reference signals. We verify that the inverse model network can stabilize the system, however it presents a steady-state error in all reference signals. In order to improve further the system response, we can insert an adaptive block whose output is added to the output of the inverse model network, defined by expression:  
\[B=\beta\sum_{i=0}^{k}(r(i)-z(i)), \tag{32}\]  
where \(\beta\) is an adjustable gain between 0 and 1 [35]. Note that because the system has a negative gain, the tuning parameter \(\beta\) has also a negative gain. It is expected that the addition of this scheme will eliminate the steady-state error. When implementing the adaptive block and testing the system, we obtained the responses in Figure 18. It is clear that the steady-state error that the inverse model network entails was eliminated with the addition of the adaptive block. Figure 17: Responses of the real system when controlled by the 1 th5s5d network. Figure 18: System response with the inverse model controllers. For comparison of the controllers we compute the performance indexes of the Integral Square Error (ISE), the Integral Absolute Error (IAE), and the Integral Time Absolute Error (ITAE), as:  
\[ISE=\int_{0}^{T_{sim}}e^{2}(t)dt, \tag{33}\]  
\[IAE=\int_{0}^{T_{sim}}|e(t)|dt, \tag{34}\]  
\[ITAE=\int_{0}^{T_{sim}}t|e(t)|dt. \tag{35}\]  
where \(e(t)\) is the error between the reference signal and the actual position response of the levitation system. We use T\({}_{sim}=40\) s that proved to be enough in order to compare the responses of the different controllers. Table 9 shows the values of ISE, IAE, and ITAE for the inverse model network with and without the adaptive block. As expected, due to the steady-state error, the inverse model without adaptive block has the worst performance indexes. Figure 19 illustrates the responses of the system when controlled with the lead-lag compensator and the inverse model network with the adaptive block. We clearly see the better response of the inverse controller in tracking the reference signals, also shown by the much lower values of ISE, IAE, and ITAE indexes presented in Table 10.  
Figure 19: Comparison of system responses with lead–lag compensator and the inverse model with adaptive block. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & **Step** & **Sine** & **Saw** & **Total** \\ \hline \multirow{2}{*}{ISE} & Inverse Model & 2.2681 & 1.4103 & 2.2240 & 5.9024 \\ \cline{2-5}  & Inverse Model + Adaptive Block & 0.1166 & 0.1819 & 0.1884 & 0.4869 \\ \hline \multirow{2}{*}{IAE} & Inverse Model & 9.3312 & 6.8233 & 8.8432 & 24.9977 \\ \cline{2-5}  & Inverse Model + Adaptive Block & 1.0882 & 2.0457 & 1.6444 & 4.7783 \\ \hline \multirow{2}{*}{ITAE} & Inverse Model & 185.3760 & 139.8848 & 165.4900 & 490.7508 \\ \cline{2-5}  & Inverse Model + Adaptive Block & 18.2365 & 38.6153 & 29.8916 & 85.7434 \\ \hline \hline \end{tabular}
\end{table}