# Identification, Control, and Characterization of Peristaltic Pumps in Hemodialysis Machines
## 3 Analysis of Results
### Validation
Minor discrepancies in accuracy can result in cumulative errors in flow rate predictions in real-world conditions. The lower RMSE of polynomial estimation in comparison to linear and potential methods demonstrates its superior capacity to capture subtle non-linearities in the speed-flow relationship. This advantage is of particular importance in practical settings, where even minor errors can have a significant impact on flow regulation, particularly in applications that require precise control. Therefore, while all methods perform well theoretically, the slight RMSE advantage of polynomial estimation becomes important in experimental conditions. In light of the aforementioned considerations and the specific results presented in Table 4, the polynomial function is deemed to be the optimal method. Figure 11: Linear, potential, and polynomial estimation of the experimental characterization data in Table 2. Caption: Comparison of linear, polynomial, and power function estimations of experimental data for peristaltic pump flow rate. The polynomial model demonstrates the best fit with minimal error, highlighting the limitations of linear and power functions in capturing the system’s dynamics. In order to validate the volumetric flow of our control and characterization proposal, and once the appropriate function relating RPM to mL/min was selected, a comparison was made with the volumetric flow of a commercial hemodialysis machine used as a reference. The commercial machine was configured under the same conditions as ours, using a blood circuit of the same diameter and brand, as well as the same filter. Twenty-four different setpoints were measured within a range of 20 to 600 mL/min, which aligns with the operating range of the reference machine. Additionally, the machine's resolution of 5 mL was maintained. For each measurement, the average error (calculated as the difference between the setpoint and the measured value) and the ratio (representing the proportion between the setpoint and the measured value) were calculated, as shown in Table 5.  
A review of Table 5 reveals that the reference hemodialysis machine exhibits an average error of approximately 8.9929 mL/min, whereas our control system demonstrates a lower average error of 4.0000 mL/min. This discrepancy serves to illustrate the enhanced precision of our control system in maintaining the specified flow rate across the tested range. Specifically, the reference machine exhibits a range of errors between 4 and 20 mL/min, with larger deviations observed at higher flow rates. In contrast, our control system demonstrates a consistent ability to limit deviations to below 10 mL/min at the majority of set points. With regard to the ratio values, the control system once again demonstrates superior performance in comparison to the reference machine. The commercial machine has an average ratio of approximately 0.9723, while the control system achieves a ratio closer to 0.9947, approaching the ideal value of 1. This closer alignment demonstrates that the  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Range**} & \multirow{2}{*}{ \begin{tabular}{c} **Setpoint** \\ **(mL/min)** \\ \end{tabular}  } & \multicolumn{2}{c}{ \begin{tabular}{c} **Commercial** \\ **Hemodialysis** \\ **Machine Flow** \\ **Rate** \\ \end{tabular}  } & \multicolumn{2}{c}{ \begin{tabular}{c} **Error** \\ **(Commercial** \\ **Handodialysis** \\ **Machine)** \\ \end{tabular}  } & \multicolumn{2}{c}{\ \begin{tabular}{c} **Ratio** \\ **(Commercial** \\ **Control)** \\ **Machine)** \\ \end{tabular}  } & \multicolumn{2}{c}{ \begin{tabular}{c} **Ratio (Our** \\ **Hemodialysis** \\ **Machine)** \\ \end{tabular}