## 7 Simulation Results  
### Example 1  
Consider the following system with delay, which corresponds to an unstable chemical reactor reported in [22],  
\[G(s)=\frac{3.433}{103.1s-1}e^{-20s}. \tag{29}\]  
Figure 4: Stability region for \(\tau\) and \(\tau_{0}\) for the characteristic Equation (28).  
In [22], a generalized predictor (GP) scheme is presented. The prefilters used in [22] are given by \(K(z)=\frac{3.295-3.253}{z-1}\), \(K_{f}(z)=\frac{0.4599z-0.4446}{z-0.9887}\), and \(F_{k}(z)=\frac{0.006231}{(z-0.9938)}\) which are derived from \(K(s)=\frac{3.29(43.87s+1)}{43.87s},K_{f}(s)=3.29\frac{20s+1}{43.87s+1}\), and \(F_{k}(s)=\frac{1}{(80s+1)}\), respectively, by considering a sampling time \(T=0.5s\).  
The GP scheme has performed better than many previous approaches, so the observer-predictor scheme presented in this work is compared with the GP scheme. For this example, \(a=0.0096\), \(b=0.033\), and \(\tau=20\) are the parameters of the process. Thus, using the observer-predictor scheme presented in this work, it is possible to stabilize the process given by (29) due to the condition of Theorem 1 being satisfied. Using the Corollary 1, the value \(g_{1}=0.7\) is chosen; this value guarantees the estimation of the signal of interest. Additionally, to improve the output response, a two-degree-of-freedom controller is used, which is defined by \(U(s)=C_{DOF-PID}R(s)-C_{PID}(s)Y_{f}(s)\) where \(C_{2 DOF-PID}(s)=\alpha k_{p}+\frac{k_{i}}{s}+\beta k_{d}s\), with \(k_{p}=14.679\), \(k_{d}=6.99\), \(k_{i}=1.398\), \(\alpha=0.6\), and \(\beta=0.9\).  
For the simulation, a step-type input \(R(s)\) of amplitude 1, initial conditions for the plant state of 0.01 and a perturbation acting at 250s are considered. Figure 5 shows the output response of both strategies considering white noise effects; it is observed that both strategies show a stable, steady-state output. It is also important to note that the scheme proposed in this work improves the time to reach a steady state and shows better performance with respect to disturbance rejection.  
In Figure 6, the behavior of the strategy reported in [22] and the proposed observer-predictor scheme under uncertainties in the plant are shown; the uncertainty is set at the unstable pole of the system with \(+200\%\) of its nominal value.  
Finally, Table 1 presents the evaluation results for different performance indicators: the integral quadratic error criterion (ISE), the integral of quadratic error multiplied by time (ITSE), the integral absolute error (IAE), and the integral of absolute error multiplied by time (ITAE). It is observed that the results of the quantitative evaluation of the strategy proposed in this work have better results in the performance indexes ITSE and ITAE for the control action \(u(t)\) and in the performance indexes ISE, ITSE, and ITAE for the output signal \(y(t)\) (nominal values) with respect to the work presented in [22]. For simulations where uncertainties in the plant are considered, the evaluation results show that the proposed observer-predictor scheme performance indexes ITSE, IAE, and ITAE are better in the control action \(u(t)\); likewise, for the output signal \(y(t)\) it is observed that in all the performance indexes, the proposed strategy presents minor error magnitude with respect to [22].  
On the other hand, Figure 7 shows the output responses of the proposed observer-predictor scheme and the Filtered Smith Predictor reported in [23] with its respective parameters given by \(C(s)=\frac{3.29(43.87s+1)}{43.87s}\), \(F(s)=\frac{20s+1}{43.87s+1}\), and \(F_{r}(s)=\frac{(20s+1)^{2}(93.16s+1)}{(26s+1)^{2}(43.87s+1)}\). It is observed that under non-zero initial conditions, the output response of the strategy presented in [23] is unstable, while the response of the observer-proposed scheme is stable.  
Figure 5: System output response with perturbation of example 1.