## 7 Simulation Results
### Example 2  
Consider the unstable process with delay given by  
\[G(s)=\frac{1}{(s-0.5)(s+1)}e^{-1.4s}. \tag{30}\]  
For this example, \(b_{1}=1\), \(a=0.5\), \(b=1\), and \(\tau=1.4\) are the parameters of the process. Thus, by substituting and in condition of Theorem 1, we obtain  
\[\tau <3.236.\]  
This means that the maximal delay-time size allowed by the proposed control strategy is \(\tau<3.236\), and since the delay time of the process (\(\tau=1.4\)) is less than this bound,  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Nominal Case**} & \multicolumn{4}{c}{**Robust Analisis (+200\% Plant Uncertainties)**} \\ \hline
**System** & \multicolumn{3}{c}{\(\frac{Y(s)}{U(s)}=\frac{3.433}{(103.1s-1)}e^{-20s}\).} & \multicolumn{3}{c}{\(\frac{Y(s)}{U(s)}=\frac{3.433}{(346.3s-1)}e^{-20s}\).} \\ \hline  & \multicolumn{4}{c}{**Control Action \(u(t)\)**} \\ \hline
**Tuning methods** & **ISE** & **ITSE** & **IAE** & **ITAE** & **ISE** & **ITSE** & **IAE** & **ITAE** \\ \hline Proposed strategy & 217 & \(1.087\times 10^{5}\) & 408.1 & \(1.584\times 10^{5}\) & 3067 & \(4.156\times 10^{5}\) & 874.1 & \(3.13\times 10^{5}\) \\ \hline GP-based approach & 265.5 & \(1.116\times 10^{5}\) & 394 & \(1.6\times 10^{5}\) & 1279 & \(4.515\times 10^{5}\) & 874.8 & \(3.17\times 10^{5}\) \\ \hline  & \multicolumn{4}{c}{**Output \(y(t)\)**} \\ \hline
**Tuning methods** & **ISE** & **ITSE** & **IAE** & **ITAE** & **ISE** & **ITSE** & **IAE** & **ITAE** \\ \hline Proposed strategy & 717.7 & \(2.585\times 10^{5}\) & 695.7 & \(2.51\times 10^{5}\) & 814.9 & \(2.65\times 10^{5}\) & 726.1 & \(2.51\times 10^{5}\) \\ \hline GP based approach & 725.5 & \(2.  
therefore the proposed control strategy can stabilize this process. Then, Corollary 1 can be used to obtain the values of \(g_{1}\) and \(g_{2}\). For the simulation, \(g_{1}=0.5\) and \(g_{2}=0.03\) are used. Further, for the simulation experiment, the parameters of the control \(C(s)\) given by (22) are chosen as \(k_{p}=27.29\), \(k_{i}=9.10\), and \(k_{d}=18.19\). An initial condition of \(0.1\) is used for each of the plant states. A step input reference, \(R(s)\), with an amplitude \(5\), is set. Figure 8 shows the behavior of the estimation error, and we see that \(e_{\omega}(t)=0\) in the steady state.  
To test the robustness with respect to uncertainty in the delay time, the analysis presented in Section 6 is used. The interval of time delays that guarantees closed-loop stability is given by \(1.33<\tau_{0}<1.467\) where the nominal value of the delay is \(\tau=1.4\). This is shown in Figure 9. Figure 10 shows the response of the system to uncertainties in the model parameters, and it can be seen that the system is stable with a parameter \(a=0.5\) greater than \(+34\%\) of its nominal value of the unstable pole. Figure 11 shows the response in which the tracking of the steady state reference can be observed. Likewise, it shows that the system is robust to some uncertainty with respect to the nominal delay (\(\tau=1.4\)).  
Figure 8: Convergence of the estimated signal from example 2.  
Figure 9: Stability region for \(\tau\) and \(\tau_{0}\) for example 2.  
Finally, Figure 12 shows the output response when white noise effects are considered at the output. In this figure, it can be observed that system stability prevails in the face of such effects.