## 2 Performance Indicators of System Response
### Additional Requirements' Formulation  
One of the most important aspects in selecting control system parameters is the feasibility of the parameters in a real plant. This aspect is strictly dependent on the plant and the control structure. In this part of the paper, the authors present few possibilities to determine the constraints to obtain the feasible control structure parameters. The basic feasibility problems are related to the control signal's chattering caused by high control structure parameter values. To solve it, the researchers are using a simple penalty function related to control signal changes during the evaluation [9; 18; 21; 22; 26]:  
\[\text{PF}_{\Delta u}=\int_{0}^{T}\left(\frac{du(t)}{dt}\right)^{2}dt \tag{11}\]  
A similar approach has been proposed in [12]. Instead of calculating the sum of changes in the control signal, the threshold value related to the laboratory stand was used. In such a case, the number of violations of the threshold value is considered. If the counted number of violations exceeds the limit considered, the examined solution is rejected. In this approach, the objective function does not require the weighting factor. However, expert knowledge of the analyzed system is required to determine the threshold values of the control signal derivative. Another method is to set a lower bound for the rise-time indicator. The lower the dynamic of the control system, the lower the control signal, and it has a \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **IAE** & **ITAE** & **ISE** & **ITSE** & **ISTSE** \\ \hline Rise Time (SI\({}_{\text{RT}}\)) & \(\bullet\bullet\bullet\circ\circ\) & \(\bullet\bullet\bullet\bullet\bullet\) & \(\bullet\circ\circ\circ\) & \(\bullet\bullet\circ\circ\) & \(\bullet\bullet\bullet\bullet\circ\) \\ \hline Settling Time (SI\({}_{\text{ST}}\)) & \(\bullet\bullet\bullet\circ\circ\) & \(\bullet\circ\circ\circ\circ\circ\) & \(\bullet\bullet\bullet\bullet\bullet\) & \(\bullet\bullet\circ\circ\circ\) & \(\bullet\bullet\bullet\circ\circ\) \\ \hline Percentage Overshoot (SI\({}_{\text{PO}}\)) & \(\bullet\bullet\bullet\circ\circ\) & \(\bullet\circ\circ\circ\circ\circ\) & \(\bullet\bullet\bullet\bullet\bullet\) & \(\bullet\bullet\bullet\circ\) & \(\bullet\bullet\circ\circ\circ\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the impact of integral quality indicators to step response indicators. Low value: \(\bullet\circ\circ\circ\circ\). High value: \(\bullet\bullet\bullet\bullet\bullet\). smaller value and a smoother shape. It can be simply defined using the following formula for minimization:  
\[\text{PF}_{\text{RT min}}=\max\left(0,\text{ }\text{SI}_{\text{RT}}^{\text{min}}- \text{SI}_{\text{RT}}\right) \tag{12}\]  
Due to the use of the function \(\max\), if the current rise time is less than the minimum value, the penalty factor will be greater than zero (\(\text{PF}_{\text{RT min}}>0\)). Otherwise, it will be equal to zero and will not affect the optimization process. To define the required rise time, the penalty function for minimization is as follows:  
\[\text{PF}_{\text{RT required}}=\left|\text{SI}_{\text{RT}}^{\text{ required}}-\text{SI}_{\text{RT}}\right| \tag{13}\]  
The absolute difference between the required value and the current value of rising time causes the optimization algorithm to try to minimize this penalty function, so these values will be close. To set the upper bound for the percentage overshoot, the following penalty function should be minimized:  
\[\text{PF}_{\text{PO max}}=\max(0,\text{ }\text{SI}_{\text{PO}}-\text{SI}_{ \text{PO}}^{\text{max}}) \tag{14}\]  
The higher percentage of overshoot than the maximum values will cause the penalty function to be greater than zero (\(\text{PF}_{\text{PO max}}>0\)). Therefore, the percentage overshoot will be minimized to not exceed this value. However, the final value of the percentage overshoot may not be close to the maximum value.