# Kernel Identification of Non-Linear Systems with General Structure
## 6 Application in Testing of Chalcogenide Glasses with the Use of DSC Method
### Chalcogenide Glasses  
Materials with non-linear optical properties play a key role in frequency conversion and optical switching. One of the most promising materials in this area are chalcogenide glasses, because of good non-linear, passive and active properties. They are considered to be optical medium for the fibers of the 21 st century. Chalcogenide glass fibers transmit into the IR, hence they can have numerous potential applications in the civil, medical and military areas. The IR light sources, lasers and amplifiers developed using these phenomena will be very useful in many areas of industry. High-speed optical communication requires ultra-fast all-optical processing and switching capabilities. In DSC experiment energy (heat flow) in function of time or temperature could be established. The energy from an external source is required to set to zero the difference in temperature of the tested and reference samples. Both samples are heated or cooled in a controlled mode and both techniques enable the detection of thermal events observed in the physical or chemical transformation under the influence of the changing temperature in a specific manner. Owing to that, many thermodynamically important parameters can be established, e.g., a glass transition or softening temperature, melting temperature, and melting enthalpy. The results also allow observation of physical aging processes. The goal is to control temperature of heating module precisely and ensure linearity of it. It is planned to design Model Following Control (MFC) structure of system to optimize quality indexes of temperature controlling. Below we present the results of identification experiment.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\delta_{1}\) & \(\delta_{2}\) & \(\delta_{3}\) \\ \hline \(\tilde{R}_{s}\) & 0.0656 & 0.0279 & 0.0137 \\ BLA & 0.0660 & 0.0304 & 0.0176 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mean squared errors \(\delta_{s}\) of model outputs for \(s=1,2,3\), compared to best linear approximation (BLA).  
Figure 6: Three-dimensional regression-based model \(\tilde{R}_{3}\left(x_{0},x_{1},x_{2}\right)\).  
In Table 2 resulting mean squared errors for various scales \(s=1,2,3,4\) are shown. The strong point of the method is that asymptotically, as \(s(N)\to\infty\), the model becomes free of approximation error, on the contrary to linear or block-oriented representation.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\delta_{1}\) & \(\delta_{2}\) & \(\delta_{3}\) & \(\delta_{4}\) \\ \hline \(\hat{K}_{s}\) (presented method) & 1017 & 477 & 232 & 169 \\ BLA (Linear FIR(\(s\))) & 1710 & 1331 & 1165 & 1114 \\ Hammerstein polynomial (3 rd order + FIR(\(s\))) & 1102 & 553 & 296 & 202 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mean squared errors \(\delta_{s}\) of model outputs for \(s=1,2,3,4\).  
Figure 10: Experimental data—4-dimensional regression.  
Figure 9: Experimental data—3-dimensional regression.  
The results have been compared to FIR(\(s\)) linear model and parametric Hammerstein model. Regarding non-parametric modeling of Hammerstein systems, proposed by [PERSON] and [PERSON] in the 1980s [20], their algorithms suffer from correlation of input, and they are not applicable here. On the other hand, for parametric Hammerstein model, the results are strongly dependent on the arbitrarily selected basis functions of nonlinearity. In our experiment we applied 3 rd order polynomial function \(\mu()\) connected in cascade with FIR(s) linear dynamic filter. Table 2 shows that our method is more accurate, emphasizing that it works under mild prior restrictions.