**2. Analysis of the optimal transduction design**

Optimization analysis through MF is briefly studied in this chapter, in order to find the best solution for the data interpretation from the designed intelligent sensor, whereby the theoretical models of the physical process and analytic interpretation of the experimental results give the support to achieve the costing function by multivariable systems, because of the correlation among the internal variables of the system with the costing function.

Therefore, it was analyzed by polynomial equations as it is described by the general model in Eq. (1), for which "*Dn*" is the derivative *<sup>∂</sup><sup>n</sup> <sup>∂</sup>tn*, *y t*ð Þ and it is the variable output matrix, "*u t*ð Þ" is the variable input matrix, "*e t*ð Þ" is the variable error matrix, "*a*" and "*b*" are the adaptive coefficients of the system [3, 12].

$$D^n \boldsymbol{\jmath}(t) + \sum\_{j=1}^n a\_j D^{n-j} \boldsymbol{\jmath}(t) = \sum\_{j=1}^n b\_j D^{n-j} \boldsymbol{u}(t) + \boldsymbol{e}(t) \tag{1}$$

Where solution error analysis, "*e t*ð Þ", is discrete error, and "*V*" keeps the Fourier series coefficients, which is given by the Eq. (2) [3, 12].

$$\epsilon\_n(m) = \sum\_{k=m}^{n+m} a(k, m, \theta\_a) \ V(k) \tag{2}$$

Furthermore, *α* is the frequency parameter function given by Eq. (3) [3, 12].

$$a(k, m, \theta\_d) = \mathcal{C}\_{k-m} \sum\_{j=0}^{n} a\_j (jkw\_0)^{n-j} \tag{3}$$

For which, the nonlinear model for the error analysis is given by the Eq. (4) [3, 12].

*Perspective Chapter: Optimal Analysis for the Correlation between Vibration and Temperature… DOI: http://dx.doi.org/10.5772/intechopen.107622*

$$\sum\_{j=0}^{n\_1} \sum\_{k=1}^{n\_2} \mathbf{g}\_j(\theta) F\_{jk}(u, y) P\_{jk}(p) E\_k(u, y) = \mathbf{0} \tag{4}$$

Therefore, the costing function is given by Eq. (5) [3, 12].

$$J(\theta) = \sum\_{j=0}^{n\_1} \sum\_{k=0}^{n\_1} r\_{jk} \mathbf{g}\_j(\theta) \mathbf{g}\_k(\theta) \tag{5}$$

Also, according to get parameters of the main model, it was achieved the derivation showed by Eq. (6) [3, 12].

$$\frac{\partial \mathcal{J}}{\partial \theta} = \left(\Upsilon - \Gamma \theta\right)^{T} \mathcal{W}^{-1} (\Upsilon - \Gamma \theta) \tag{6}$$

Where parameters are showed in Eq. (7), as the dependence on the adaptive coefficients, in which Υ is the response matrix, Γ is the internal variables matrix, *W* is the weight matrix and *θ* is the sensor parameters matrix [3, 12].

$$\boldsymbol{\theta} = \left(\boldsymbol{\Gamma}^T \boldsymbol{\mathcal{W}}^{-1} \boldsymbol{\Gamma}\right)^{-1} \boldsymbol{\Gamma}^T \boldsymbol{\mathcal{W}}^{-1} \boldsymbol{\Upsilon} \tag{7}$$

The interpretation and scheme for optimization is depicted by **Figure 3**, in which are presented 3 paths C1 (green color curve), C2 (violet color curve) and C3 (red color curve) according to achieve the position B from the position A. C2 represents the theoretical path such as the theoretical variable of a process, C3 represents the experimental path due to an experimental data, therefore C1 is the optimal path owing to achieve the position B.
