## 7 Simulation Results
### Example 3  
Consider the unstable high-order system with delay previously studied in [19],  
\[\frac{Y(s)}{U(s)}=\frac{0.2}{(s-0.2)(s+0.5)(s+2)}e^{-0.5s}. \tag{31}\]  
Figure 11: Output response in example 2.  
Figure 12: Output response to noise effects of example 2.  
Figure 10: Output response to plant uncertainty in example 2.  
In [19], an Active Disturbance Rejection Control (ADRC) is presented with a two-degree of freedom scheme. The proposed control and observer gains for the ADRC are \(K_{o}=[9406.895\ 998.3894\ 56.6806\ 1]\) and \(L_{o}=[9406.895\ 998.3894\ 56.6806\ 1]^{T}\). Likewise for the two-degree of freedom strategy, the controller \(\mathcal{C}=(2.07s+1)(5s-1)/(s+1)^{2}\) and \(N=\mathrm{e}^{-0.939s}/(s+1)^{2}\) are proposed.  
On the other hand, for this example, \(b_{1}=0.5\), \(b_{2}=2\), \(a=0.2\), \(b=0.2\), and \(\tau=0.5\) are the parameters of the process. Considering the observer-predictor presented in this paper, the stability condition given in Theorem 3 is \(\tau<8.3\), which is satisfied. Using Corollary 1, we calculate the values of \(g_{1}\) and \(g_{2}\). In this example, the values of \(g_{1}=10\) and \(g_{2}=-5\) are used. For the control \(C(s)\) given by Equation (22), the parameters used are \(k_{p}=3.94\), \(k_{l}=0.32\), and \(k_{d}=6.57\).  
Additionally, in order to achieve a better response, a two-degree of freedom controller of PID type [31] defined by \(U(s)=C_{2 DOF-PID}R(s)-C_{PID}(s)Y_{f}(s)\) is considered, where \(C_{2 DOF-PID}(s)=\alpha k_{p}+\frac{k_{i}}{s}+\beta k_{d}s\), with \(k_{p}=3.94\), \(k_{d}=6.57\), \(k_{i}=0.32\), \(\alpha=0.8\), and \(\beta=0.8\).  
For the simulation, a unitary step input reference \(R(s)=1/s\) and a perturbation of the actuators at 40 s is used. The output response of the control scheme proposed in [19] and the output response of the observer-predictor scheme presented in this work are shown in Figure 13 to noise effects. It can be seen that in the presence of a perturbation acting at 40 s, both strategies allow perturbation rejection and reference tracking. It is important to highlight that in [19], necessary and sufficient conditions are not provided to stabilize systems with time delay, while in the observer-predictor presented in this work, it is possible to control the system of example 2 with a delay \(\tau<8.3\).  
Figure 14 shows the output response under an uncertainty in the delay \(\tau\). It can be seen that with a delay uncertainty of \(+100\%\), both strategies still have a stable output response. However, at a delay uncertainty \(\tau\) of \(+120\%\), it can be seen in Figure 15 that the strategy presented in [19] is no longer stable, while the proposed observer-predictor scheme preserves a stable response. Figure 16 shows the control signal \(u(t)\) response under an uncertainty in the delay \(\tau\).  
Finally, Table 2 shows the evaluation results for different performance indices. It can be observed that the observer-predictor scheme presented in this work has better performance indexes in the ITSE, IAE, and ITAE for the control action \(u(t)\) and ISE, ITSE, IAE, and ITAE indexes for the output signal \(y(t)\) (nominal case) with respect to the work reported in [19]. When a delay-time uncertainty is considered in the system, the proposed observer-predictor scheme has better performance indexes ITSE, IAE, and ITAE in the control action \(u(t)\) and the indexes ISE, ITSE, IAE, and ITAE for the output signal \(y(t)\) with respect to the work presented in [19].  
Figure 13: Output response to noise effects of example 3.  
Figure 16: Control response of example 3.  
Figure 14: Stable system output response with time delay uncertainty \(\tau\) of example 3.  
Figure 15: Output response of example 3.