## 3 Non-Self-Regulating Process Model
\[G_{\text{C}}(s)=K_{c}\bigg{(}1+\frac{1}{T_{i}s}\bigg{)}\Bigg{(}\frac{1+T_{d}s} {1+\frac{T_{d}}{N}s}\bigg{)}, \tag{4}\]0 Simulation: Selected sets for step response and disturbance reaction for the I2 PD system in \[G_{\text{C}}(s)=K_{c}\bigg{(}1+\frac{1}{T_{i}s}\bigg{)}\Bigg{(}\frac{1+T_{d}s} {1+\frac{T_{d}}{N}s}\bigg{)}, \tag{4}\]1 (**a**) The step response for the setting _Skogestad._ (**b**) The step response after PID optimization. (**c**) The step response in the case of the MPID controller for \(\gamma_{1}=2\). (**d**) The step response in the case of the MPID controller for \(\gamma_{1}=0\). The simulation-derived illustrations are also labeled with a test code identifying the trial (e.g., code MPID_multi47_02a_9). Figure 5a shows the waveforms for the reference system, for the controller selected according to (5). Next, a series of set-optimization tests were performed for the reference system, with different configurations to improve the quality indices. For the optimization of only controller settings, waveforms were obtained, as shown in Figure 5b. In the next step, the PID controller was completed by adding an FIR unit to its input, and optimization of the controller parameters and weights was carried out for different values of delays defined as multiples of the calculation step: \(\{2,3,7,11,20\}\) and for different parameters of the quality function (8). The example results for parameter optimization are shown in Figure 5c,d. These are the waveforms obtained for a delay of \(2\cdot T_{s}\) and \(11\cdot T_{s}\), respectively. For the presented series of tests, these were the best results. A summary of the quality indices obtained in the tests is presented in \[G_{\text{C}}(s)=K_{c}\bigg{(}1+\frac{1}{T_{i}s}\bigg{)}\Bigg{(}\frac{1+T_{d}s} {1+\frac{T_{d}}{N}s}\bigg{)}, \tag{4}\]2 Settling times were determined for the 2 % tolerance band. The first row shows the results for the reference system (Figure 5a), where row two shows the results for the system after PID controller optimization (Figure 5b). The third and fourth rows show the results for the MPID system for \(\gamma_{1}=2\) (Figure 5c) and \(\gamma_{1}=0\) (Figure 5d), respectively. The first column shows the value of the quality index obtained after optimization (7), and column two shows the overshoot value as a percentage. Column three shows the minimum value of the output in response to the disturbance (the higher the value, the better). The fourth column shows the settling time. The last column shows the response time to a disturbance (the recovery time). It can be seen that optimizing the controller parameters according to the criterion reduces the value of the quality index by more than two times, but this comes at the expense of increased overshoot, increased settling time, and very poor oscillation damping. A comparable quality index value is provided by the use of the FIR for the \(\gamma_{1}=2\) parameter, which further reduces overshoot by more than two times and increases oscillation damping. As a result, the responses to step changes in the reference values or disturbances are smoother. When the FIR is used for the \(\gamma_{1}=0\) parameter, a significant reduction in the quality index value is achieved, nearly twice as small compared to PID settings (7), and clearly better quality parameters, such as the overshoot, settling time, and reaction to disturbance (maximum error and recovery time). Since a significant improvement in quality indices was obtained for such a simple model, the concept was then tested for a model of a drive with a PMSM, and in the next step, the concept was verified for a laboratory system with a PMSM.