## 4 Design of Adaptive PID Controller Based on ISSA
### Traditional PID Controller
The proportional term generates the control output by adjusting the error magnitude, the integral term eliminates steady-state errors by accumulating the error over time, and the derivative term predicts the future trend based on the rate of change of the  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**Function Name**} & \multicolumn{1}{c}{**PSO**} & \multicolumn{1}{c}{**SA**} & \multicolumn{1}{c}{**WOA**} & \multicolumn{1}{c}{**SSA**} & \multicolumn{1}{c}{**ISSA**} \\ \hline F1 (Sphere) & Mean & \(5.30\times 10^{4}\) & \(2.62\times 10^{3}\) & \(3.05\times 10^{-31}\) & \(7.83\times 10^{-178}\) & \(0.00\times 10^{0}\) \\  & Std & \(6.77\times 10^{3}\) & \(1.59\times 10^{3}\) & \(5.52\times 10^{-31}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) \\ \hline F2 (Schwefel 2.22) & Mean & \(8.94\times 10^{11}\) & \(5.09\times 10^{1}\) & \(4.00\times 10^{-17}\) & \(2.48\times 10^{-91}\) & \(3.16\times 10^{-207}\) \\  & Std & \(1.47\times 10^{12}\) & \(1.47\times 10^{1}\) & \(7.53\times 10^{-17}\) & \(6.58\times 10^{-91}\) & \(0.00\times 10^{0}\) \\ \hline F3 (Schwefeel 1.2) & Mean & \(1.00\times 10^{5}\) & \(1.44\times 10^{4}\) & \(7.55\times 10^{3}\) & \(3.40\times 10^{-288}\) & \(0.00\times 10^{0}\) \\  & Std & \(4.01\times 10^{4}\) & \(4.71\times 10^{3}\) & \(7.22\times 10^{3}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) \\ \hline F4 (Schwefel 2.21) & Mean & \(7.99\times 10^{1}\) & \(3.34\times 10^{1}\) & \(3.62\times 10^{-2}\) & \(6.37\times 10^{-72}\) & \(1.66\times 10^{-203}\) \\  & Std & \(5.41\times 10^{0}\) & \(3.43\times 10^{0}\) & \(5.25\times 10^{-2}\) & \(1.80\times 10^{-71}\) & \(0.00\times 10^{0}\) \\ \hline F5 (Generalized Rosenbrock) & Mean & \(1.60\times 10^{8}\) & \(4.68\times 10^{5}\) & \(5.09\times 10^{-1}\) & \(1.52\times 10^{3}\) & \(8.55\times 10^{-4}\) \\  & Std & \(6.47\times 10^{7}\) & \(3.18\times 10^{5}\) & \(4.39\times 10^{-1}\) & \(1.43\times 10^{3}\) & \(1.42\times 10^{-3}\) \\ \hline F6 (Step Function) & Mean & \(5.54\times 10^{4}\) & \(2.26\times 10^{3}\) & \(6.97\times 10^{-3}\) & \(1.28\times 10^{-6}\) & \(7.49\times 10^{-7}\) \\  & Std & \(9.81\times 10^{3}\) & \(7.29\times 10^{2}\) & \(4.18\times 10^{-3}\) & \(1.05\times 10^{-6}\) & \(1.13\times 10^{-6}\) \\ \hline F7 (Quartic) & Mean & \(8.15\times 10^{1}\) & \(7.03\times 10^{-1}\) & \(2.21\times 10^{-3}\) & \(2.93\times 10^{4}\) & \(9.42\times 10^{-5}\) \\  & Std & \(2.92\times 10^{1}\) & \(2.24\times 10^{-1}\) & \(3.81\times 10^{-3}\) & \(1.98\times 10^{4}\) & \(1.03\times 10^{-4}\) \\ \hline F8 (Schwefel 2.26) & Mean & \(-4.28\times 10^{3}\) & \(-7.65\times 10^{3}\) & \(-1.22\times 10^{4}\) & \(-7.46\times 10^{3}\) & \(-1.00\times 10^{4}\) \\  & Std & \(6.15\times 10^{2}\) & \(6.01\times 10^{2}\) & \(7.75\times 10^{2}\) & \(1.56\times 10^{3}\) & \(1.30\times 10^{3}\) \\ \hline F9 (Rastrigin) & Mean & \(4.28\times 10^{2}\) & \(1.17\times 10^{2}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) \\  & Std & \(2.16\times 10^{1}\) & \(1.99\times 10^{1}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) \\ \hline F10 (Ackley) & Mean & \(1.99\times 10^{1}\) & \(1.23\times 10^{1}\) & \(1.17\times 10^{-14}\) & \(4.44\times 10^{-16}\) & \(4.44\times 10^{-16}\) \\  & Std & \(8.55\times 10^{-2}\) & \(2.11\times 10^{0}\) & \(7.42\times 10^{-15}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) \\ \hline F11 (Griewank) & Mean & \(4.93\times 10^{2}\) & \(2.66\times 10^{2}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) \\  & Std & \(9.11\times 10^{1}\) & \(1.09\times 10^{1}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) & \(0.00\times 10^{0}\) \\ \hline F12 (Generalized Penalized Function1) & Mean & \(3.30\times 10^{8}\) & \(5.67\times 10^{3}\) & \(9.67\times 10^{-4}\) & \(3.63\times 10^{-7}\) & \(1.81\times 10^{-7}\) \\  & Std & \(1.35\times 10^{8}\) & \(1.01\times 10^{4}\) & \(9.37\times 10^{-4}\) & \(2.53\times 10^{-7}\) & \(2.06\times 10^{-7}\) \\ \hline F13 (Generalized Penalized Function2) & Mean & \(6.59\times 10^{8}\) & \(3.63\times 10^{5}\) & \(1.03\times 10^{-2}\) & \(5.04\times 10^{-6}\) & \(1.04\times 10^{-6}\) \\  & Std & \(2.85\times 10^{8}\) & \(5.43\times 10^{5}\) & \(1.15\times 10^{-2}\) & \(5.44\times 10^{-6}\) & \(1.12\times 10^{-6}\) \\ \hlineerror, thereby suppressing oscillations and enhancing system stability.