**3. Modeling**

It was necessary to prepare a mathematical model to obtain the optimal estimations of the fuel flow. For this reason, it was worked using the measured values and keeping as reference value the fuel flow achieved by theoretical analysis that can be calculated either by Bernoulli or a calibrated sensor.

The Eq. (20) gives information of the estimated fuel flow "*Qest*", this result depends on the measured flows that are stored by the "*X*" matrix, and furthermore, this equation depends on the estimation coefficient "*β*".

$$Q\_{\text{ext}} = \mathbf{X}\boldsymbol{\beta} \tag{20}$$

On the other hand, the expression of **"***β***"** is replaced in Eq. (21); hence, it was possible to correlate the estimated fuel flow with a reference value of fuel flow [8, 10, 11]:

$$Q\_{\rm est} = X \left(X^T X\right)^{-1} X^T Q\_{\rm rcf} \tag{21}$$

Although fuel flow taken as a reference is given by Eq. (22), another theoretical model instead of the model chosen based on Bernoulli can be achieved, which is given by a model solved from Eq. (18). Even though, for Eqs. (18) and (22), there are parameters dependent on the geometry and material of the designed smart sensor, such as the constant "K" in Eq. (22), proportionate information of the geometrical and material characteristics of the passive components of the designed smart sensor:

$$Q\_{ref} = \left[\sqrt{\frac{\Delta \mathbf{P}}{\mathbf{K}}}\right] \tag{22}$$

In fact, the **Figure 3** depicts the designed algorithm for the smart sensor, in which "*Q mxn*" is the matrix information of the measured fuel flow in dimension "*mxn*," "*ΔPmxn*" is the difference of pressure matrix in dimension "*mx1*", the matrix of the internal smart sensor is given by "*Xmxn*", which needs the theoretical fuel flow "*Qtmxn*", as a consequence, the algorithm give optimal estimated fuel flow to the user by the matrix "*Qemxn*".
