# Identification, Control, and Characterization of Peristaltic Pumps in Hemodialysis Machines
## 2 Methodology
### Auxiliary Methodologies for System Approximation Verification  
The efficacy of the proposed first-order system approximation about the sampled data is evaluated using the functions defined in the following equations, which employ the optimization methods proposed in [18; 19; 20; 21]. The aforementioned analysis yielded the comprehensive evaluation detailed in Table 1  
A number of tools are available for the qualitative verification and validation of control systems. This section presents four approaches to verification through the calculation of definite integrals, as illustrated in the following equations. The op \begin{table}
\begin{tabular}{c c c c c} \hline
**Methodology for Absolute** & **Data** & **First Order** & **Error** &
\begin{tabular}{c} **\%Approximation** \\ **Error of the System** \\ \end{tabular} \\ \hline Integral Absolute Error IAE & 2383 & 2394 & \(-10.9300\) & 0.4586\% \\ \hline Integral Square Error ISE & \(7.706\times 10^{4}\) & \(7.973\times 10^{4}\) & \(-2671\) & 3.4670\% \\ \hline Integral Time Absolute Error ITAE & 286.2 & 286.4 & \(-0.1534\) & 0.0536\% \\ \hline Integral Time Square Error ITSE & 7740 & 7722 & 18.16 & 0.2345\% \\ \hline \end{tabular}
\end{table} 
Table 1: Percentage of approximation error: data vs. proposed first-order plant. Figure 4: First-order system identification. **Caption:** Adjustment of the first-order transfer function using experimental data obtained using the non-linear least squares method. The function accurately represents the system dynamics with an RMSE of 1.897 RPM. classified into four categories. The Integral Absolute Error (IAE), the Integral Squared Error (ISE), the Integral Time-weighted Absolute Error (ITAE), and the Integral Time-weighted Squared Error (ITSE) are integral-based error metrics. The Integral Absolute Error (IAE) [21] is a metric utilized in control systems to assess a controller's performance based on the accumulated error over time. It is defined as the integral of the absolute value of the error between the reference signal and the system output signal, as expressed in Equation (7). In contrast to metrics such as the Integral Squared Error (ISE), which assigns greater significance to more significant errors, the IAE assigns equal weight to all errors, rendering it a valuable tool for minimizing the overall system error without unduly emphasizing more significant errors. \[\text{IAE}=\int_{0}^{T}\lvert e(t)\rvert \tag{7}\]  
The Integral of Squared Error (ISE) [22; 23] is a performance metric employed in analyzing control systems to quantify the accumulated error over time, emphasizing more significant errors. The Integral of Squared Error (ISE) is the integral of the square of the error between the reference signal and the system output signal, as expressed in Equation (8). The ISE assigns a greater penalty to more significant errors due to the use of the squared error, making it particularly useful in applications where the minimization of significant deviations from the reference signal is desired. This metric is frequently employed in the design and tuning of controllers to optimize system performance and enhance stability. \[\text{ISE}=\int_{0}^{T}\lvert e(t)\rvert^{2} \tag{8}\]  
The Integral of Time-weighted Absolute Error (ITAE) [18; 19] is a performance metric utilized in the design of control systems for evaluating error as a function of time. The Integral of Time-weighted Absolute Error (ITAE) is defined as the integral of the product of time and the absolute value of the error between the reference signal and the system output signal, as illustrated in Equation (9). The ITAE assigns greater significance to errors that manifest later, thereby imposing a more substantial penalty on prolonged errors. This encourages a more expeditious correction of errors, rendering this metric particularly useful in applications where both accuracy and rapid system response are desired.