## 3 Non-Self-Regulating Process Model  
The solution proposed in this paper was verified by simulation for an I2 PD-type object described by the following transfer function:  
\[G_{\text{OB1}}(s)=K\frac{e^{-s\tau}}{s^{2}}, \tag{3}\]  
where \(K\) is the gain of the object, and \(\tau\) is the time delay. In the preliminary studied system \(K=1\), \(\tau=0.5\,s\). The example values of the above parameters were selected according to the parameters of the real system--the commercial quadcopter's position control using visual feedback from [39]. For the considered type of object (3), there are various ready-made procedures for calculating the controller parameters. For the system with a single controller \(C\) (Figure 4), a serial PID controller structure with derivative prefiltering was selected as the reference circuit. The settings were chosen according to the method proposed by [PERSON] in Section 4.5.2 of the book [21]. \[G_{\text{C}}(s)=K_{c}\bigg{(}1+\frac{1}{T_{i}s}\bigg{)}\Bigg{(}\frac{1+T_{d}s} {1+\frac{T_{d}}{N}s}\bigg{)}, \tag{4}\]  
where  
\[K_{c}=\frac{0.0625}{K\tau^{2}},T_{i}=8\tau,T_{d}=8\tau,N=\infty. \tag{5}\]  
Due to the problem of the numerical convergence of calculations, \(N~{}=~{}100\) was used in the simulation. The control system with settings (4) and (5) is the reference system against which the results obtained for the proposed controller system will be compared. The modified controller with an FIR of the form shown in Figure 3 in the tests replaces the \(C\) module in Figure 4. The setpoint filter \(F\) is of the form:  
\[F(s)=\frac{1}{\left(T_{F}s+1\right)^{2}}~{}, \tag{6}\]  
where \(T_{F}=0.5\,s\) is the time constant. This value was selected according to \(\tau\). The tests consisted of determining the quality index of the step response for a change in the setpoint at time \(t=0\,s\) from \(0\) to \(1\), and at time \(t=25\,s\), there is a step change in the disturbance from \(0\) to a value of \(-0.2\). The value of the disturbance was chosen to obtain a clear waveform on a suitable scale for plotting. For the purpose of comparing the optimization results (after the learning process), we used the quality index _IAE (Integral Absolute Error)_, which has the form:  
\[J_{1}=\int_{0}^{\infty}\lvert\varepsilon_{c}(t)\rvert dt, \tag{7}\]  
where \(\varepsilon_{c}\) is the control error. During the optimization process, a quality index of the following form was used:  
\[J_{2}=\int_{0}^{\infty}\lvert\varepsilon_{c}(t)\rvert dt+\gamma_{1}\cdot\int_{0 }^{\infty}\lvert\varepsilon_{int}(t)\rvert dt, \tag{8}\]  
where \(\varepsilon_{int}\) is an additional \"virtual\" error:  
\[\varepsilon_{int}(t)=INT_{1\,ref}(t)-INT_{1}(t). \tag{9}\]  
The addition of this component allows the optimization process to achieve a smoother response to disturbances and changes in the setpoint. The value denoted as \(\gamma_{1}\) is a weighting factor. The auxiliary variables \(INT_{1\,ref}\) and \(INT_{1}\) (Figure 4) are the output of the controller \(C\) and the output of the first integral element, respectively. The error \(\varepsilon_{int}\) is called virtual because the system does not have a controller for the \(INT_{1}\) variable. Regardless of the value of \(\gamma_{1}\) used in the search for the optimalparameters, the value of the quality index (labeled as \(q\)) in Figure 5 is shown for the quality function (7). In the studied system, the calculation step is \(T_{s}=$33\,\mathrm{ms}$\). For the presented assumptions, simulations were performed for the reference system, then for the reference system after optimization of the controller settings, and, finally, for optimization of the weights and controller settings for different parameters of the quality function. Figure 5 shows a series of step responses along with the response to a step change in disturbance for a comparison of the reference system, before and after optimization. Figure 4: General view of the control structure for a simple object.