## 3 Results
### Stage 3
#### 3.3.3 Graph Transfer Matrix vs. Optimum Control  
The responses to a step input with an amplitude of 72, of the transference matrix and with optimal control, can be seen below. By equaling the denominator to zero, the eigenvalues and the desired roots are obtained, in order to obtain control by assigning poles and obtain a faster behavior of the operation. In Figure 5, the x axis corresponds to the time in minutes, while the y axis represents the number of textile pieces obtained as a target. This means that the graph shows how the number of parts produced varies over time. Figure 5 shows a comparison between the transfer-matrix response and the optimal Ackermann control for a staggered input with an amplitude of 72. The transfer-matrix response is represented by the curve in blue, while the optimal Ackermann control response is represented by the curve in red. The graph shows that the optimal Ackermann control response is faster and has fewer oscillations than the transfer-matrix response. This is because the optimal Ackermann control uses complete system information to calculate the control action, while the transfer matrix only takes into account input and output information. In addition, it can be noticed that during the period of 0 to 20 min, the optimal control response presents a slight negative oscillation due to the state variable b being filled before the input c reaches its maximum value. However, after this period, the response is stabilized and the control of the output state variable of the simulation proposed in SD is achieved. In summary, the figure shows how optimal Ackermann control can improve the system response compared to the transfer matrix, allowing better control of the output state variable and greater efficiency in the proposed simulation process. Figure 4: State feedback with precompensation gain. Pole and Zero Diagram Transfer Function vs. Optimal Control  
The pole and zero diagram of a transfer function is a useful tool for analyzing and understanding the behavior of a control system. It allows visualization of the location of the poles and zeros in the complex plane, which provides important information about the stability and response of the system. Figure 6 shows the diagram of poles and zeros in the imaginary and the real plane of the transfer function. The following poles and zeros were identified. See Equations (45) and (46). \[\text{Poles}:\,-0.0371,\,-0.0279\text{ and }-0.0085 \tag{45}\]  
\[\text{Zeros}:\,0.0767\text{ and }-0.0117 \tag{46}\]  
Figure 5: Behavior of the transfer matrix vs. Ackerman control, before a step input. Figure 6: Diagram of poles and zeros of the transfer matrix. All poles have the imaginary part equal to zero and are located on the left side of the complex plane. Since the poles have the imaginary zero part and are on the left side of the complex plane, it can be concluded that they are stable. Figure 6 shows a positive zero, which does not directly affect the stability of the system, since its location does not determine the stability or instability of the system. However, zeros can affect the frequency response and form of the system response. Comparing the poles and zeros before and after applying the Ackermann control, we can see how they moved and the impact on the system response. Before the Ackermann control, the poles and zeros were as follows:  
\[\text{Poles}:\,-0.0371,\,-0.0279,\,-0.0085.\]  
\[\text{Zeros}:\,0.0767,\,-0.0117.\]  
After applying the Ackermann control, the poles and zeros changed to: see Equations (47) and (48). \[\text{Poles}:\,-0.0453,\,-0.0187,\,-0.0128 \tag{47}\]  
\[\text{Zeros}:\,0.0806,\,-0.0118 \tag{48}\]  
We note in Figure 7 that the poles moved to different locations in the complex plane. The absolute values of the poles increased slightly, indicating that the system became a little faster in its response. This may involve greater baseline tracking and faster response to changes in system conditions. Zeros also moved, although the changes were not as significant.