# Optimal Robust PID Control for First- and Second-Order Plus Dead-Time Processes
## 4 Numerical Simulation
### Comparison with Conventional Methods  
The proposed method is compared with conventional methods: the ZN method [2], conventional SOPDT-based PI and PID control [18], and the conventional SOPDT-based 2 DoFI control [19]. As a controlled plant, consider the following transfer function:  
\[P(s)=\frac{1}{(s+1)(0.62s+1)}e^{-1.5s}. \tag{27}\]  
The reference input is 1.0, the control input is disturbed by the step signal with amplitude 0.5 after 150 s, and the controlled plant is perturbed to Equation (28) after 300 s.  
\[P^{\prime}(s)=\frac{1.4}{(1.2s+1)(0.36\times 1.2s+1)}e^{-1.2s}. \tag{28}\]  
In the design of the conventional SOPDT model-based methods [18; 19], the robust stability is designed with \(M_{s}^{d}=1.6\), and \(a=0.5\) is used to decide the PID parameters because \(a=0.62\) in Equation (27) is not directly used. In the proposed method, \(M_{s}^{d}\) is also set to 1.6 for comparison with the conventional methods. In the conventional [18] and the proposed trade-off design methods, the controller parameters are designed in the servo and regulation modes, respectively. Furthermore, only in the conventional 2 DoF design method [19], 2 DoF parameter \(\beta\) is used in the following control law:  
\[U(s) = K_{p}\bigg{(}\beta R(s)-Y(s)+\frac{1}{TiS}(R(s)-Y(s)).\bigg{)} \tag{29}\]  
The calculated controller parameters based on Equation (27) are shown in Table 9.  
Figure 15: Enlarged views of Figure 14.  
The control results are shown in Figure 16, and the enlarged views of the tracking, regulation, and model perturbation trajectories are also shown in Figure 17. Using the conventional [18; 19] and proposed SOPDT-based design methods, the closed-loop system stabilizes even when the plant is perturbed after 300 s because the stability margin is designed. On the other hand, using the ZN method, the closed-loop system is unstable after plant perturbation, although it can be stabilized until plant perturbation. The obtained control results are evaluated by the IAE, as shown in Table 10. In both the servo performance from to 0 s to 150 s and the regulation performance from 150 s to 300 s, the IAE of the ZN method is the largest among the considered methods because the plant output vibrates and because the settling time is the longest among the considered design methods. Table 10 shows that the trade-off between the servo and regulation performances is achieved using the conventional [18] and proposed trade-off methods because in the servo evaluation period (0 s to 150 s), the IAE using the servo-optimized controllers, i.e., 1 DoF PI (servo), 1 DoF PID (servo), and the proposed 1 DoF PID (servo), is smaller than that obtained using the regulation-optimized controllers, i.e., 1 DoF PI (reg), 1 DoF PID (reg), and the proposed 1 DoF PID (reg). On the other hand, in the regulation evaluation period (150 s to 300 s), the IAE using the regulation-optimized controllers is superior to that obtained using the servo-optimized controllers. Figure 16: Plant outputs of the proposed and conventional methods. \begin{table}
\begin{tabular}{c c c c c} \hline
**Design Method** & \(\mathbf{K_{p}}\) & \(\mathbf{T_{i}}\) & \(\mathbf{T_{d}}\) & \(\mathbf{\beta}\) \\ \hline ZN [2] & 1.52 & 3.44 & 0.859 & – \\
1 DoF PI (servo) [18] & 0.469 & 1.92 & – & – \\
1 DoF PI (reg) [18] & 0.488 & 2.09 & – & – \\
1 DoF PID (servo) [18] & 0.633 & 2.18 & 0.642 & – \\
1 DoF PID (reg) [18] & 0.610 & 1.68 & 0.774 & – \\
2 DoF PI [19] & 0.431 & 1.66 & – & 0.983 \\ Proposed 1 DoF PID (servo) & 0.670 & 2.04 & 0.567 & – \\ Proposed 1 DoF PID (reg) & 0.665 & 1.87 & 0.582 & – \\ \hline \end{tabular}
\end{table} 
Table 9: Proportional-integral-derivative (PID) control system parameters. In the servo evaluation period, both of the proposed controllers, i.e., the proposed 1 DoF PID (servo) and the proposed 1 DoF PID (reg), are superior to all of the conventional controllers.