## 3 The Conveyor Speed Control System with Fuzzy Controller
In turn, the model of the waste product collection process is a model with no delays, and provides the controller with information about the facility's reaction in advance by time T.  
This research presents an experiment in which the traditional PI controller was replaced with a fuzzy controller. The experiment was carried out in two stages: in the first stage, a fuzzy controller with a PI structure was used (Figure 4a), and in the second stage, a PD structure was used (Figure 4b). Figure 3: Time process for conveyor drive power consumption. In the system, a model of the following structure was used [7]:  
\[ \begin{array}{l}y_{1}(i)=\frac{11,27\text{+}11,27\text{z}^{-1}}{1-0.426\text{z} ^{-1}+0.3759\text{z}^{-2}-0.9496\text{z}^{-3}}u_{1}(i)\\ \qquad+\frac{2888-368\text{z}^{-1}}{1-0.2933\text{z}^{-1}-0.9704\text{z}^{-2}+0. 2859\text{z}^{-3}}u_{2}(i)\\ \qquad+\frac{1+2.571\text{z}^{-1}\text{+}3.073\text{z}^{-2}+0.5148\text{z}^{-3}} {1+0.9318\text{z}^{-1}+0.3576\text{z}^{-2}-1.169\text{z}^{-3}-0.328\text{z}^{- 4}}e(i)\end{array}  \tag{3}\]  
where its terms are defined as follows:  
\(u_{1}(i)\)--discrete sequence of opening values of the discharge flaps,  
\(u_{2}(i)\)--discrete sequence of conveyor speeds,  
\(y_{1}(i)\)--discrete sequence of values of the average material weight on the conveyor,  
\(e(i)\)--white noise. These are the following linguistic variables of the process state:  
\[e(n)=M_{z}(n)-M_{y}(n) \tag{4}\]  
Figure 4: Control system diagram with fuzzy controller: (**a**) PI structure, (**b**) PD structure, \(M_{z}\)—set value of the transported product weight [kg], \(M_{p}\)—sum of degrees of opening the discharge flaps in the jig [%], \(M_{y}\)—current weight of transported heavy product [kg]. \[\delta e(n)=\sum_{i=(n-5)}^{n}e(i) \tag{5}\]  
\[\Delta e(n)=e(n)-e(n-1) \tag{6}\]  
The output of the controller \(y(n)\) is the increase in conveyor speed, expressed using the frequency of the drive power supply, and described by a two-dimensional control surface. \[y=f(e,\ \delta e)\ \text{lub}\ y=f(e,\ \Delta e) \tag{7}\]  
Differentiation or integration of the control error is made at each time step before being fed into the fuzzy part of the controller. The normalized membership functions of the controller are shown in Figure 5.  
Figure 5: Cont. In relation to the output linguistic variable y, the used triangular membership functions are distributed unevenly in the normalized interval \([-1,1]\). In this study, each fuzzy rule takes the following form:  
\[\text{IF ($e$ is $Ei$) AND (($\delta e$ is $E_{i}^{\prime}$) or ($\Delta e$ is $E_{i}^{\prime}$)) THEN ($y$ is $Y_{i}$) END\_IF} \tag{8}\]  
where the values are defined as follows:  
\(E_{i}\), \(E_{i}^{\prime}\)--linguistic values defined as fuzzy sets from the universe \(E\)\(i\)\(E^{\prime}\). On the basis of the implemented functions and observations of the dynamic characteristics of the controlled process, as well as on the basis of preliminary tests analyses, fuzzy rule matrixes were established, and are presented in Table 1.  
A graphical interpretation of this structure is shown in Figure 6.  
The developed rules contain information about the controller's behavior for various states of input signals. These rules are the basis for the reasoning process, i.e., assessing the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(\Delta y\)} & \multicolumn{5}{c}{\(E\)} \\ \cline{2-7}  & & **1** & **2** & **3** & **4** & **5** \\ \hline \multirow{4}{*}{\(\text{E}^{\prime}\)} & 1 & 1 & 1 & 2 & 3 & 3 \\ \cline{2-7}  & 2 & 1 & 1 & 3 & 3 & 3 \\ \cline{2-7}  & 3 & 1 & 2 & 3 & 4 & 4 \\ \cline{2-7}  & 4 & 1 & 3 & 3 & 5 & 5 \\ \cline{2-7}  & 5 & 2 & 2 & 4 & 5 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Rule matrix. Figure 6: Control surface output of the fuzzy logic controller with a PI structure. degree of fulfilment of the premises of each rule. This degree, unlike the rules of classical logic, can take not only the value 0 or 1, but also fractional values in the range \([0,1]\).