## 1 Introduction
Process measurements are generally subject to random or systematic errors, often violating conservation laws [8; 9]. DR in steady-state systems is an important process used in many industries to improve the accuracy and reliability of information collected from systems operating in equilibrium. In a steady-state system, variables such as flow, temperature, and pressure remain constant over time. Data from sensors and instruments may contain errors due to calibration issues or external conditions. The goal of reconciliation is to correct these errors so that the data are more accurate and useful for decision-making. The reconciliation process involves several steps. First, a mathematical model based on physical principles must be developed to describe the relationship between the process variables. Next, measurement data are collected from different sources, such as flow metersand sensors. From these data, statistical methods are used to estimate the true values of the variables, minimizing the differences between the measured data and those predicted by the model. After this estimation, it is essential to analyze the differences (residuals) to identify measurement errors or anomalies. Based on this analysis, the measured values are adjusted to reflect the reconciled estimates. The corrected data are then checked against known standards or additional measurements to validate their accuracy. Finally, the reconciled data are presented clearly, including the adjustments made and their implications for system performance. The benefits of DR are significant. Increased accuracy results in more reliable data, improving system representation. In addition, data quality leads to more efficient operation control, optimizing resource allocation and reducing waste. Reliable data also increase the confidence in analysis and reporting, facilitating better decisions. Finally, the reconciliation process can reveal issues that need attention, driving continuous improvement. In summary, DR in steady-state systems is essential to ensuring accurate and reliable data in industries such as chemical processing, oil and gas, and manufacturing. Organizations can improve their operations and achieve better results by correcting measurement errors and validating information. On the other hand, DR in dynamic systems also involves validating consistency and completeness but focuses on information in constantly changing environments, such as monitoring platforms and databases that receive input in real time. In this context, reconciliation is essential to ensure data accuracy in operations. DR was chosen over other filtering techniques because it utilizes process model constraints, such as the degree of statistical dependence between variables, to obtain estimates of process variables. These estimates are obtained by adjusting process measurements to satisfy the constraints. DR improves the measurement accuracy by reducing random errors and leveraging redundant information from the system. Reconciled estimates are more accurate than experimental estimates, as DR exploits the redundancy property of measurements to minimize errors through weighted least squares optimization [10]. The reconciliation algorithms use multiple redundant measurements in experiments, which are multiple measurements that provide essentially the same information, \(m_{i}^{e}\), to achieve a unique reconciled value, a reconciled measurement, \(m^{c}\), that better represents the population. On the other hand, the measurement uncertainty, that is, the variability associated with a measurement, is minimized, and is represented as \(u_{m^{c}}\)[11]. The typical procedure for implementing DR involves the following steps: (i) Definition of the reference model, i.e., a common structure and understanding of a specific system. In this step, it is assumed that the mathematical model is ideal, free from gross or systematic errors, mistakes, or inaccuracies in measurements, and follows a normal distribution. It also assumes independence between the measurements [12].