# Discrete-Time First-Order Plus Dead-Time Model-Reference Trade-off PID Control Design
## 5 Numerical Examples
### Comparison with the Conventional Discrete-Time Method  
The conventional discrete time design method [28] is compared with the proposed method. Here, two scenarios are conducted, in which a non-zero plant and a zero-included plant, respectively, are controlled. In the first simulation, we consider the following non-zero discrete time plant:  
\[P_{d1}(z^{-1})=\frac{0.05127z^{-1}}{1-0.9486z^{-1}}z^{-10} \tag{44}\]  
where Equation (44) is the discrete time representation of Equation (45) with a sampling interval of \(T_{s}=0.05\) s.  
\[P_{1}(s)=\frac{1}{0.95s+1}e^{-0.5s} \tag{45}\]  
Equation (44) has no zero since the continuous time dead-time is an integer multiple of the sampling interval. In the second simulation, the controlled discrete time system is given as follows:  
\[P_{d2}(z^{-1})=\frac{0.0201+0.02473z^{-1}}{1-0.9552z^{-1}}z^{-7} \tag{46}\]  
Figure 8: Results of the IMC-based method and the proposed servo and regulator optimization methods, where the system is disturbed by a unit step disturbance after 10 s; left: servo optimization; right: regulator optimization. where Equation (46) is the discrete time version of the continuous time system given by Equation (47) with a sampling interval of \(T_{s}=0.061\) s.  
\[P_{2}(s)=\frac{1}{1.33s+1}e^{-0.4s} \tag{47}\]  
Equation (46) has a zero since the dead-time in the continuous time model is not an integer multiple of the sampling interval. Since the discrete time system has a zero, the conventional method is not directly used. Therefore, Equation (46) is hereby approximated by the next discrete time system, and the conventional method is used:  
\[P_{d2}^{\prime}(z^{-1})=\frac{0.04483z^{-1}}{1-0.9552z^{-1}}z^{-7} \tag{48}\]  
In the simulations, the reference input is set to 1.0. Furthermore, the control input is disturbed by a unit step function signal after 10 s. The obtained PID parameters are shown in Tables 6-9. Using the obtained parameters, the discrete time models Equations (44) and (46) are controlled, respectively, and the output results are plotted in Figures 9 and 10. Furthermore, the obtained \(M_{s}\) value and index values \(J_{s}\) and \(J_{r}\) are also shown in Tables 6-9, where \(J_{s}\) denotes the SAE value while the control is not disturbed, and \(J_{r}\) denotes the SAE value while the control input is disturbed. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \(M_{s}^{d}\) & \(K_{p}\) & \(T_{i}\) & \(T_{d}\) & \(M_{s}\) & \(J_{s}\) & \(J_{r}\) \\ \hline  & 1.4 & 1.4664 & 1.4390 & 0.1009 & 1.4026 & 1.0088 & 0.9796 \\ Proposed & 1.6 & 1.9725 & 1.5788 & 0.1066 & 1.6010 & 0.8020 & 0.7978 \\ method & 1.8 & 2.3577 & 1.7146 & 0.1354 & 1.8014 & 0.7412 & 0.7236 \\  & 2.0 & 2.6043 & 1.8463 & 0.1550 & 2.0010 & 0.7320 & 0.7037 \\ \hline  & 1.4 & 1.3144 & 1.5050 & 0.1075 & 1.3571 & 1.1449 & 1.1407 \\ Conventional & 1.6 & 1.7619 & 1.6841 & 0.1194 & 1.5024 & 0.9553 & 0.9495 \\ method [28] & 1.8 & 2.0745 & 1.8297 & 0.1473 & 1.6576 & 0.8813 & 0.8737 \\  & 2.0 & 2.2804 & 1.9491 & 0.1662 & 1.8076 & 0.8537 & 0.8441 \\ \hline \end{tabular}
\end{table} 
Table 8: Proportional-integral-derivative parameters, \(M_{s}\), and SAE values in the servo design using Equations (46) and (48), respectively. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \(M_{s}^{d}\) & \(K_{p}\) & \(T_{i}\) & \(T_{d}\) & \(M_{s}\) & \(J_{s}\) & \(J_{r}\) \\ \hline  & 1.4 & 1.4332 & 0.7274  
In the case of the non-zero system Equation (44), \(J_{s}\), \(J_{r}\), and \(M_{s}\) obtained using the conventional method were close to those obtained using the proposed method. On the other hand, when the controlled plant was a zero-included system, Equation (46) was out of range of the conventional method. Therefore, the tracking performances using the conventional method were inferior to those using the proposed method, and hence, the SAE values using the conventional method were larger than those using the proposed method.