# Online SOPDT Model Identification Method Using a Relay
## 3 Simulation Results and Discussion
### Robustness  
Let us assume that the measurements carried out during the identification test were subject to greater error than was apparent from the capabilities of the method used. To investigate what effect this would have on the accuracy of the model in example 1, let us assume that the error values of the measurements denoting steady-state gain, time delay, and ultimate frequency without and with the integral term in the feedback were equal:  
(Case 1) \(K_{1\text{e}}=90\%\cdot K\), \(\theta_{1\text{e}}=120\%\cdot\theta\), \(\omega_{1\text{e}}=120\%\cdot\omega_{1}\), \(\omega_{2\text{e}}=120\%\cdot\omega_{2}\);  
(Case 2) \(K_{2\text{e}}=110\%\cdot K\), \(\theta_{2\text{e}}=80\%\cdot\theta\), \(\omega_{2\text{e}}=80\%\cdot\omega_{1}\), \(\omega_{2\text{e}}=80\%\cdot\omega_{2}\). The following models were obtained from these measurements:  
\[G_{1\text{e}}(s)=\frac{1.8\,e^{-3.6s}}{3.01s^{2}+11.19s+1} \tag{37}\]  
\[G_{2\text{e}}(s)=\frac{2.2\,e^{-2.4s}}{57.47s^{2}+13.47s+1} \tag{38}\]  
The differences between the responses of the plant and the models determined from measurements subject to errors are shown in Figure 8. However, what effect these dif \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(K_{m}\) & \(\theta\) & \(T_{1}\) & \(T_{2}\) & \(I\) \\ \hline Plant & 1 & 2 & 1 & 10 & 0 \\ \hline [PERSON] and [PERSON] & 1.05 & 1.814 & 1.217 & 9.766 & 0.4603 \\ \hline [PERSON], [PERSON], and [PERSON] & 0.853 & 2.0 & 1.15 & 7.416 & 1.122 \\ \hline [PERSON], [PERSON], and [PERSON] & 0.9923 & 2.0 & 1.0105 & 9.899 & 0.05974 \\ \hline [PERSON], [PERSON], and [PERSON] & 0.998 & 2.0 & 1.044 & 9.14 & 0.07201 \\ \hline Presented method & 0.996 & 2.01 & 0.992 & 9.775 & 0.03274 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of parameters of the plant and models obtained by considered methods. Figure 7: Nyquist plots of the plant from example 2 and selected models. ferences have on the variation of the controlled variable will depend on several factors, including the structure of the control system and the control rule. When a controller's tuning is aimed at preserving higher robustness, the effect on the decrease in control performance will be low. The presented method of online relay identification allowed for precisely determining the model of a time-delayed second-order inertial plant. Of course, the correct result will be obtained only when the measurement variables required for the calculations, such as the ultimate frequencies of the closed system and the system with an added integrator, static gain, and dead time, are accurately measured. When the measurements of these variables contain some errors, the obtained model will differ from the mathematical description of the real object. Measurement noise falling into the high-frequency range of the signal spectrum is a common problem that arises during process identification. [PERSON] proposed [5] the introduction of an integral filter into the loop. Such a noise-cancellation technique requires two relay experiments: one to design the integral filter and another to identify the process model parameters. Another approach used to eliminate limit cycle noise via a wavelet decomposition technique was presented by [PERSON] and coworkers [19]. During the identification process, they applied a wavelet transform to extract a noise-free signal. While the methods for determining the static gain mentioned in Section 2.2 are robust to both load disturbances and measurement noise, the most accurate method for determining the delay time based on the shift of the plant's response to the input is not robust. It is based on zero-crossing detectors, which, in the presence of noise, i.e., signals that are inadequate for the actual value of the controlled variable, could distort the measurement.