# PID Controller Parameter Tables for Time-Delayed Systems Optimized Using Hill-Climbing
## 3 The Hill-Climbing Method for Calculating the PID Controller Parameters
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**PT1** & **+/\(\boldsymbol{-2}\)** & **+/\(\boldsymbol{-3}\)** & **+/\(\boldsymbol{-5}\)** & **+/\(\boldsymbol{-10}\)** \\ \hline IAE & Kp-Ks = 10 & Kp-Ks = 10 & Kp-Ks = 10 & Kp-Ks = 10 \\ IAE & Ti = 3.1\(\cdot\)T1 & Ti = 2\(\cdot\)T1 & Ti = 1.3\(\cdot\)T1 & Ti = 1\(\cdot\)T1 \\ IAE & Td = 0 (PI) & Td = 0 (PI) & Td = 0 (PI) & Td = 0 (PI) \\ \hline ITAE & Kp-Ks = 9.3 & Kp-Ks = 9.5 & Kp-Ks = 9.1 & Kp-Ks = 10 \\ ITAE & Ti = 2.9\(\cdot\)T1 & Ti = 1.9\(\cdot\)T1 & Ti = 1.2\(\cdot\)T1 & Ti = 1\(\cdot\)T1 \\ ITAE & Td = 0 (PI) & Td = 0 (PI) & Td = 0 (PI) & Td = 0 (PI) \\ \hline ISE & Kp-Ks = 10 & Kp-Ks = 10 & Kp-Ks = 9.8 & Kp-Ks = 10 \\ ISE & Ti = 2.7\(\cdot\)T1 & Ti = 1.6\(\cdot\)T1 & Ti = 1.5\(\cdot\)T1 & Ti = 0.2\(\cdot\)T1 \\ ISE & Td = 0 (PI) & Td = 0 (PI) & Td = 0 (PI) & Td = 0 (PI) \\ \hline
**PT2, Tg/Tu = 9.71** & **+/\(\boldsymbol{-2}\)** & **+/\(\boldsymbol{-3}\)** & **+/\(\boldsymbol{-5}\)** & **+/\(\boldsymbol{-10}\)** \\ \hline IAE & Kp-Ks = 10 & Kp-Ks = 10 & Kp-Ks = 10 & Kp-Ks = 10 \\ IAE & Ti = 9.6\(\cdot\)T1 & Ti = 7.3\(\cdot\)T1 & Ti = 5.6\(\cdot\)T1 & Ti = 3.7\(\cdot\)T1 \\ IAE & Td = 0.3\(\cdot\)T1 & Td = 0.3\(\cdot\)T1 & Td = 0.3\(\cdot\)T1 & Td = 0.2\(\cdot\)T1 \\ \hline ITAE & Kp-Ks = 10 & Kp-Ks = 10 & Kp-Ks = 9.6 & Kp-Ks = 9.8 \\ ITAE & Ti = 9.6\(\cdot\)T1 & Ti = 7.3\(\cdot\)T1 & Ti = 5.4\(\cdot\)T1 & Ti = 4.7\(\cdot\)T1 \\ ITAE & Td = 0.3\(\cdot\)T1 & Td = 0.3\(\cdot\)T1 & Td = 0.3\(\cdot\)T1 & Td = 0.3\(\cdot\)T1 \\ \hline ISE & Kp-Ks = 10 & Kp-Ks = 10 & Kp-Ks = 10 & Kp-Ks = 10 \\ ISE & Ti = 9.7\(\cdot\)T1 & Ti = 7.3\(\cdot\)T1 & Ti = 5.1\(\cdot\)T1 & Ti = 4.6\(\cdot\)T1 \\ ISE & Td = 0.2\(\cdot\)T1 & Td = 0.2\(\cdot\)T1 & Td = 0.2\(\cdot\)T1 & Td = 0.1\(\cdot\)T1 \\ \hline
**PT3, Tg/Tu = 3.61** & **+/\(\boldsymbol{-2}\)** & **+/\(\boldsymbol{-3}\)** & **+/\(\boldsymbol{-5}\)** & **+/\(\boldsymbol{-10}\)** \\ \hline IAE & Kp-Ks = 5.4 & Kp-Ks = 7 & Kp-Ks = 8.4 & Kp-Ks = 10 \\ IAE & Ti = 9.4\(\cdot\)T1 & Ti = 10\(\cdot\)T1 & Ti = 9.8\(\cdot\)T1 & Ti = 9.7\(\cdot\)T1 \\ IAE & Td = 0.7\(\cdot\)T1 & Td = 0.7\(\cdot\)T1 & Td = 0.7\(\cdot\)T1 & Td = 0.7\(\cdot\)T1 \\ \hline ITAE & Kp-Ks = 5.4 & Kp-Ks = 7 & Kp-Ks = 8.2 & Kp-Ks = 10 \\ ITAE & Ti = 9.4\(\cdot\)T1 & Ti = 10\(\cdot\)T1 & Ti = 9.6\(\cdot\)T1 & Ti = 9.7\(\cdot\)T1 \\ ITAE & Td = 0.7\(\cdot\)T1 & Td = 0.7\(\cdot\)T1 & Td = 0.7\(\cdot\)T1 & Td = 0.7\(\cdot\)T1 \\ \hline ISE & Kp-Ks = 6.1 & Kp-Ks = 8.1 & Kp-Ks = 10 & Kp-Ks = 10 \\ ISE & Ti = 10\(\cdot\)T1 & Ti = 9.8\(\cdot\)T1 & Ti = 10\(\cdot\)T1 & Ti = 7.8\(\cdot\)T1 \\ ISE & Td = 0.6\(\cdot\)T1 & Td = 0.6\(\cdot\)T1 & Td = 0.6\(\cdot\)T1 & Td = 0.6\(\cdot\)T1 \\ \hline
**PT4, Tg/Tu = 3.14** & **+/\(\boldsymbol{-2}\)** & **+/\(\boldsymbol{-3}\)** & **+/\(\boldsymbol{-5}\)** & **+/\(\boldsymbol{-10}\)** \\ \hline IAE & Kp-Ks = 2 & Kp-Ks = 2.9 & Kp-Ks = 3.3 & Kp-Ks = 3.3 \\ IAE & Ti = 5.2\(\cdot\)T1 & Ti = 6.5\(\cdot\)T1 & Ti = 7.1\(\cdot\)T1 & Ti = 6.9\(\cdot\)T1 \\ IAE & Td = 1.1\(\cdot\)T1 & Td = 1.2\(\cdot\)T1 & Td = 1.3\(\cdot\)T1 & Td = 1.3\(\cdot\)T1 \\ \hline ITAE & Kp-Ks = 1.9 & Kp-Ks = 2.4 & Kp-Ks = 2.3 & Kp-Ks = 2.1 \\ ITAE & Ti = 5.7\(\cdot\)T1 & Ti = 5.9\(\cdot\)T1 & Ti = 5.7\(\cdot\)T1 & Ti = 5.5\(\cdot\)T1 \\ ITAE & Td = 1.1\(\cdot\)T1 & Td = 1.2\(\cdot\)T1 & Td = 1.2\(\cdot\)T1 & Td = 1.1\(\cdot\)T1 \\ \hline ISE & Kp-Ks = 2.8 & Kp-Ks = 3.6 & Kp-Ks = 4.9 & Kp-Ks = 5.2 \\ ISE & Ti = 6.6\(\cdot\)T1 & Ti = 7.7\(\cdot\)T1 & Ti = 7.1\(\cdot\)T1 & Ti = 7.7\(\cdot\)T1 \\ ISE & Td = 1.2\(\cdot\)T1 & Td = 1.2\(\cdot\)T1 & Td = 1.4\(\cdot\)T1 & Td = 1.4\(\cdot\)T1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: PID parameters for minimized ITAE, IAE and ISE criteria. Ks is the static gain, T\({}_{1}\) is the time constant and n is the number of PT1 elements connected in series. Ks, T\({}_{1}\) and n can be calculated, for example, from Figure 2 and Table 1, or by using other methods.