## 3 Implementation of the Soft Sensor in the Time Domain
### Application of the TVB Method to Obtain the Soft Sensor
As a result, \(K_{1}\) becomes:\[Q_{2,\rm dyn,1}(t_{\rm n})=\frac{1}{R_{\rm H}}\int_{0}^{t_{\rm n}}W_{1,\rm dyn }^{*}(t_{\rm n}-\tau)\frac{\partial\Delta p}{\partial t_{\rm n}}(\tau)d\tau= \frac{1}{R_{\rm H}}Y_{0}(t_{\rm n}). \tag{56}\]2 \[Q_{2,\rm dyn,1}(t_{\rm n})=\frac{1}{R_{\rm H}}\int_{0}^{t_{\rm n}}W_{1,\rm dyn }^{*}(t_{\rm n}-\tau)\frac{\partial\Delta p}{\partial t_{\rm n}}(\tau)d\tau= \frac{1}{R_{\rm H}}Y_{0}(t_{\rm n}). \tag{56}\]3  
It is important to note that the subsequent derivatives of the weighting function take the form of:  
\[Q_{2,\rm dyn,1}(t_{\rm n})=\frac{1}{R_{\rm H}}\int_{0}^{t_{\rm n}}W_{1,\rm dyn }^{*}(t_{\rm n}-\tau)\frac{\partial\Delta p}{\partial t_{\rm n}}(\tau)d\tau= \frac{1}{R_{\rm H}}Y_{0}(t_{\rm n}). \tag{56}\]4 \[Q_{2,\rm dyn,1}(t_{\rm n})=\frac{1}{R_{\rm H}}\int_{0}^{t_{\rm n}}W_{1,\rm dyn }^{*}(t_{\rm n}-\tau)\frac{\partial\Delta p}{\partial t_{\rm n}}(\tau)d\tau= \frac{1}{R_{\rm H}}Y_{0}(t_{\rm n}). \tag{56}\]5 (69) \[Q_{2,\rm dyn,1}(t_{\rm n})=\frac{1}{R_{\rm H}}\int_{0}^{t_{\rm n}}W_{1,\rm dyn }^{*}(t_{\rm n}-\tau)\frac{\partial\Delta p}{\partial t_{\rm n}}(\tau)d\tau= \frac{1}{R_{\rm H}}Y_{0}(t_{\rm n}). \tag{56}\]6  
Due to this repetitive structure, the general derivative (of order \(k\), where \(k\in\mathbb{N}\)) can be expressed as:  
\[Q_{2,\rm dyn,1}(t_{\rm n})=\frac{1}{R_{\rm H}}\int_{0}^{t_{\rm n}}W_{1,\rm dyn }^{*}(t_{\rm n}-\tau)\frac{\partial\Delta p}{\partial t_{\rm n}}(\tau)d\tau= \frac{1}{R_{\rm H}}Y_{0}(t_{\rm n}). \tag{56}\]7  
Inserting these definitions of the derivatives of the weighting functions into the Taylor expansion yields:  
\[Q_{2,\rm dyn,1}(t_{\rm n})=\frac{1}{R_{\rm H}}\int_{0}^{t_{\rm n}}W_{1,\rm dyn }^{*}(t_{\rm n}-\tau)\frac{\partial\Delta p}{\partial t_{\rm n}}(\tau)d\tau= \frac{1}{R_{\rm H}}Y_{0}(t_{\rm n}). \tag{56}\]8  
Additionally, it is noteworthy that the first term in this equation is \(Y_{0}(t_{\rm n})\), as defined earlier. \[Q_{2,\rm dyn,1}(t_{\rm n})=\frac{1}{R_{\rm H}}\int_{0}^{t_{\rm n}}W_{1,\rm dyn }^{*}(t_{\rm n}-\tau)\frac{\partial\Delta p}{\partial t_{\rm n}}(\tau)d\tau= \frac{1}{R_{\rm H}}Y_{0}(t_{\rm n}). \tag{56}\]9  
Interestingly, the terms within the brackets following \(Y_{0}(t_{\rm n})\) resemble the series approximation of an exponential term with a factor \(C\) in the exponent. Consequently, the bracket can be rewritten as an exponential term instead (in this case, \(C=n_{i}\)):  
\[Q_{2,\rm dyn,1}(t_{\rm n}+\Delta t_{\rm n})=\frac{1}{R_{\rm H}}\int_{0}^{t_{ \rm n}+\Delta t_{\rm n}}W_{1,\rm dyn}^{*}(t_{\rm n}+\Delta t_{\rm n}-\tau) \frac{\partial\Delta p}{\partial t_{\rm n}}(\tau)d\tau. \tag{57}\]0  
Both integral parts are now simplified and can be substituted back into Equation (58):  
\[Q_{2,{\rm dyn},1}(t_{\rm n}+\Delta t_{\rm n}) =\frac{1}{R_{\rm H}}[K_{1}+K_{2}],  
integral range from 0 to \(t_{\rm n}\), the length of the integral would increase indefinitely as time progresses, which would make the application of the soft sensor impracticable. With the implemented method, the analytical model can be used offline and online on a real fluid power system to constantly measure the volumetric flow rate. Especially for online use, the one-step integration method and the analytical model's simple computational operations, like adding exponential functions, underline the soft sensor's ability to perform real-time computation.