# Discrete-Time First-Order Plus Dead-Time Model-Reference Trade-off PID Control Design
## 5 Numerical Examples
### Comparison with the IMC-Based Method  
Consider the following discrete time system:  
\[P_{d}(z^{-1})=\frac{0.0100}{1-0.99z^{-1}}z^{-25} \tag{40}\]  
Figure 7: Output and input responses obtained using the proposed method for each \(M_{s}^{d}\) with model perturbation, where the system is disturbed after 15 s; left: servo optimization; right: regulator optimization.  
The system is the discrete time representation of the following continuous time system with a sampling interval of 0.01 s:  
\[P(s)=\frac{1}{s+1}e^{-0.25s} \tag{41}\]  
In the conventional discrete time trade-off design method [27], the IMC controller is designed as:  
\[C_{IMC}(z^{-1})=\frac{(1-a_{1})(1-\lambda_{c})}{b_{0}(1-\lambda_{c}z^{-1})} \tag{42}\]  
where \(\lambda_{c}\) is the trade-off design parameter. The obtained IMC controller is approximated by the following discrete time PID control law:  
\[C_{PID}(z^{-1})=K_{p}\left\{1+\frac{1}{T_{i}(1-z^{-1})}+(1-\alpha T_{d})\frac {T_{d}(1-z^{-1})}{1-\alpha T_{d}z^{-1}}\right\} \tag{43}\]  
where \(\lambda_{c}\) is designed in the range of [0.8,0.99]. However, the designed control systems are unstable when \(\lambda_{c}\leq 0.96\), and thus, the conventional control law is designed such that \(\lambda_{c}\) is set to 0.97, 0.98, and 0.99, respectively, where \(\alpha=0.1\). The designed PID parameters for the conventional method and the proposed method are shown in Table 5.  
The simulations were conducted using the conventional and proposed methods, where the reference input was set to 1.0, and the control input was disturbed by a unit step function after 30 s. The conventional method is compared with the proposed servo and regulator optimization methods in Figure 8. The obtained \(M_{s}\) values and the evaluated values \(J_{s}\) and \(J_{r}\) are also summarized in Table 5, where \(J_{s}\) denotes the SAE value from the start until 30 s, and \(J_{r}\) also denotes the SAE value from 30 s until the end. Table 5 shows that the conventional method provides a trade-off design by selecting \(\lambda_{c}\) even though no value is assigned to \(M_{s}\). Moreover, the tracking performances obtained using the proposed method are superior to those obtained using the conventional method.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \(\mathbf{K_{p}}\) & \(\mathbf{T_{i}}\) & \(\mathbf{T_{d}}\) & \(\mathbf{M_{s}}\) & \(\mathbf{J_{s}}\) & \(\mathbf{J_{r}}\) \\ \hline IMC-based & \(\lambda_{c}=0.99\) & 0.8248 & 103.10 & 4.6696 & 1.1625 & 1.2500 & 1.2493 \\ method [27] & \(\lambda_{c}=0.98\) & 1.3978 & 104.83 & 7.5346 & 1.2937 & 0.7533 & 0.7500 \\  & \(\lambda_{c}=0.97\) & 1.8184 & 106.07 & 9.4658 & 1.5659 & 0.5919 & 0.5833 \\ \hline  & \(M_{s}^{d}=1.4\) & 1.9120 & 1.1242 & 0.0606 & 1.4014 & 0.5936 & 0.5878 \\ Proposed & \(M_{s}^{d}=1.6\) & 2.5724 & 1.2107 & 0.0690 & 1.5919 & 0.4733 & 0.4705 \\ Servo & \(M_{s}^{d}=1.8\) & 3.0641 & 1.3062 & 0.0871 & 1.7869 & 0.4367 & 0.4260 \\  & \(M_{s}^{d}=2.0\) & 3.3823 & 1.4277 & 0.0984 & 1.9851 & 0.4313 & 0.4216 \\ \hline  & \(M_{s}^{d}=1.4\) & 1.9193 & 0.5000 & 0.1070 & 1.3999 & 0.8079 & 0.3352 \\ Proposed & \(M_{s}^{d}=1.6\) & 2.5319 & 0.4605 & 0.1079 & 1.5858 & 0.7646 & 0.2432 \\ Regulator & \(M_{s}^{d}=1.8\) & 3.0234 & 0.4928 & 0.1038 & 1.7778 & 0.6951 & 0.1975 \\  & \(M_{s}^{d}=2.0\) & 3.4407 & 0.5012 & 0.1069 & 1.9940 & 0.6642 & 0.1689 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Proportional-integral-derivative parameters, \(M_{s}\), and sum of absolute errors (SAE) values using the proposed and internal model control (IMC)-based methods for Equation (40).