## 5 Discussion  
The analytical soft sensor analysis based on two pressure signals demonstrated its promising ability to determine high-frequency flow rates. The model accurately computes the various flow rates using pressure signal information. Due to its analytical nature, the computation is fast and not delayed by iterative numerical calculations, unlike the distributed-parameter simulation model, DSHplus. Nevertheless, the results obtained from the soft sensor show good agreement with the numerically computed values from the simulation model. A soft sensor based on the HP law exhibits high percentile errors (up to 3800%) due to its neglect of transient effects. The analytical model exhibits a small mean error of less than 0.15% for all test cases, with most mean errors below 0.045%. The standard deviation of the error further highlights the good performance of the soft sensor, with values ranging from 1.3% to 0.59%. The analytical model can accurately compute the correct flow rate for sine waves up to 1000 Hz, with a mean error of 0.14% and a standard deviation of 1.3%. This high accuracy for transient pulsations represents a promising application of the soft sensor for the investigation of axial piston pumps, which exhibit pressure ripples in the range of 1000 Hz. Furthermore, the computation of volumetric flow rate based on the pressure difference allows for power monitoring of a pump, enabling the characterization of pulsation and a pump's performance. Regarding the sum of sine waves, the model provides precise values for several overlaying sine waves. For both cases, the mean error is below 0.04%, with a mean error of 0.0049% for the sum of sines with 1, 10, 20, and 40 Hz, and 0.043% for the sum of sines with 10, 100, 200, and 400 Hz. The standard deviation values of 0.067% and 1.3% further underline the high accuracy of the soft sensor. These results demonstrate the soft sensor's ability to capture different levels of transient effects in a system, enabling the detailed investigation of a hydraulic system's dynamic behavior. Furthermore, the knowledge of pressure and volumetric flow rates for arbitrary systems allows for calculating hydraulicpower for hydraulic systems with varying behavior. This is further highlighted by the good results of the sawtooth test case, with a mean error of 0.0043% and a standard deviation of 0.11%. Regarding the step function, the soft sensor exhibits good agreement with the distributed-parameter simulation, with a mean error of 0.0051% and a standard deviation of the error of 0.059%. This accuracy allows the sensor to be applied to systems where sudden pressure changes, e.g., due to valve closing, resulting in a water hammer, are investigated. Notably, the step function contains a bandwidth of frequencies, which the analytical soft sensor accurately captures. Significantly, the investigation of the flow rate directly after the pressure step shows the model's ability to capture compression phenomena, as demonstrated by the stair-like increase in the volumetric flow rate. The novel soft sensor exhibits good agreement across all investigated test cases, considering various frequency ranges and transient pressure signals. In comparison, the HP-based soft sensor exhibits errors of up to 3800% and a mean error of 29% for the lowest frequency sine wave. This highlights the analytical model's advancement over the typically applied HP law. The demonstrated precision at high frequencies and compressible flow conditions underscores the soft sensor's potential for deployment in various industrial applications, such as condition monitoring and control, where real-time and accurate computation of crucial system parameters is essential. Additionally, the sensor's ability to track pressure and flow rate enables the calculation of hydraulic power, which can be utilized for predictive maintenance, offering more profound insights into system performance.