# PID Controller Parameter Tables for Time-Delayed Systems Optimized Using Hill-Climbing
## 5 Applications for the Use of the Table: PID-Controlled PT3 and PT5  
To show the correct use of the table, in the following, two examples were calculated and simulated. The controller structure was always in accordance with Figure 1 and the table values of Section 4. The two limitations must also be observed. Both the anti-windup and the controller output limitation were assumed to be the same. With the parameter sets according to Section 4, however, the anti-windup of the integral component is not actually necessary. This is never activated in the concrete examples, due to the controller output limitation and the use of the optimal parameters. Since it can occur in practice, for various reasons, that the actual value does not reach the setpoint, the anti-windup is nevertheless inserted here. Example 1, control of a PT3 system (Tg/Tu = 4.61): for a PT3 system with T\({}_{1}\) = 1 s and Ks = 1, as well as an assumed controller output limitation of \(+/-2\), the following PID parameters are read from the table. For IAE and ITAE:  
\[\text{Kp}=\frac{5.4}{\text{Ks}}=5.4\quad\text{Ti}=9.4\cdot\text{T}_{1}=9.4\; \text{s}\quad\text{Td}=0.7\cdot\text{T}_{1}=0.7\;\text{s}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**PT5, Tg/Tu = 2.44** & \(+/-2\) & \(+/-3\) & \(+/-5\) & \(+/-10\) \\ \hline IAE & Kp-Ks = 1.7 & Kp-Ks = 1.8 & Kp-Ks = 1.8 & Kp-Ks = 1.7 \\ IAE & Ti = 5.8-T1 & Ti = 5.9-T1 & Ti = 5.8-T1 & Ti = 5.5-T1 \\ IAE & Td = 1.6-T1 & Td = 1.6-T1 & Td = 1.6-T1 & Td = 1.6-T1 \\ \hline ITAE & Kp-Ks = 1.4 & Kp-Ks = 1.4 & Kp-Ks = 1.4 & Kp-Ks = 1.4 \\ ITAE & Ti = 5.3-T1 & Ti = 5.2-T1 & Ti = 5.2-T1 & Ti = 5.0-T1 \\ ITAE & Td = 1.4-T1 & Td = 1.4-T1 & Td = 1.4-T1 & Td = 1.4-T1 \\ \hline ISE & Kp-Ks = 1.9 & Kp-Ks = 2.6 & Kp-Ks = 2.5 & Kp-Ks = 2.5 \\ ISE & Ti = 5.9-T1 & Ti = 6.5-T1 & Ti = 6.3-T1 & Ti = 6.1-T1 \\ ISE & Td = 1.7-T1 & Td = 1.8-T1 & Td = 1.8-T1 & Td = 1.8-T1 \\ \hline
**PT6, Tg/Tu = 2.03** & \(+/-2\) & \(+/-3\) & \(+/-5\) & \(+/-10\) \\ \hline IAE & Kp-Ks = 1.3 & Kp-Ks = 1.3 & Kp-Ks = 1.3 & Kp-Ks = 1.3 \\ IAE & Ti = 5.9-T1 & Ti = 5.8-T1 & Ti = 5.8-T1 & Ti = 5.6-T1 \\ IAE & Td = 1.9-T1 & Td = 1.9-T1 & Td = 1.9-T1 & Td = 1.9-T1 \\ \hline ITAE & Kp-Ks = 1.1 & Kp-Ks = 1.1 & Kp-Ks = 1.1 & Kp-Ks = 1.1 \\ ITAE & Ti = 5.5-T1 & Ti = 5.5-T1 & Ti = 5.4-T1 & Ti = 5.3-T1 \\ ITAE & Td = 1.7-T1 & Td = 1.7-T1 & Td = 1.7-T1 & Td = 1.7-T1 \\ \hline ISE & Kp-Ks = 1.8 & Kp-Ks = 1.8 & Kp-Ks = 1.8 & Kp-Ks = 1.8 \\ ISE & Ti = 6.8-T1 & Ti = 6.5-T1 & Ti = 6.5-T1 & Ti = 6.3-T1 \\ ISE & Td = 2.1-T1 & Td = 2.1-T1 & Td = 2.1-T1 & Td = 2.1-T1 \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._For ISE:  
\[\text{Kp}=\frac{6.1}{\text{Ks}}=6.1\quad\text{Ti}=10\cdot\text{T}_{1}=10\text{ s}\quad\text{Td}=0.6\cdot\text{T}_{1}=0.6\text{ s}\]  
The response of the closed-loop system to a unit step, according to Figure 1, is shown in Figure 7 with these parameters. Example 2, control of a real PT5 system, i.e., a temperature chamber: this example shows the application with \(\text{Ks}\
eq 1\) and \(\text{T}_{1}\
eq 1\), as well as the correct use of the table with the controller output limitation. At this point, attention is also drawn to the special feature that the table values can be scaled with \(\text{T}_{1}\) and \(\text{Ks}\). This makes them usable for many applications. A thermal actuator heats a small heating chamber. For a biological experiment, the chamber should be heated from an initial temperature of 20 \({}^{\circ}\)C (outside temperature) to 40 \({}^{\circ}\)C. The temperature sensor in the chamber measures differentially in comparison to the outside temperature and has a linear characteristic with 1 V/4\({}^{\circ}\) K. The actuator is a coil, the power of which is linearized with an internal circuit and an amplifier. At the stationary end value, it heats the chamber at 6\({}^{\circ}\) K/1 V. The block diagram is shown in Figure 8.  
For physical reasons, in a thermal system the heating reacts with a time delay.