# Algorithms Utilized for Creep Analysis in Torque Transducers for Wind Turbines
## 4 The Proposed Model  
The linear or nonlinear functions of time are described by the polynomial expansion of a Taylor series \(y=f(t,c)=c_{0}+c_{1}t+c_{2}t^{2}+\ldots+c_{m}t^{m}\) and the match parameter vector \(c=[c_{0},c_{1},c_{2},\ldots,c_{m}]^{\mathsf{T}}\). This means that in order to determine the fit vector, the following are generally required: \(m\)-nonlinear equations [30; 31; 32; 33; 34; 35; 36; 37]. This expansion also represents the application of many complex elementary functions describing a given physical process. In our case, it is a process stretched over time of \"creeping\" a physical quantity to be measured, in this case the moment of force, resulting, inter alia, from the inertia of the processes occurring during the measurement on the measuring standard. The basic tool used to fit a nonlinear relationship to measurement points is the least squares method. In the general approach, it is the WTLS method that takes into account all the generally different uncertainties of the coordinates of the points and the various correlations between the coordinates of the measurement points. The parameters of this schematic view are determined from the vector equation for the least squares method, taking into account not only the coordinates of the measured measurement points but also the uncertainty of the signal and time, as well as all possible correlations between them related to environmental conditions and the properties of the measuring system. The general optimization condition boils down to maximizing the greatest likelihood function, which means minimizing the dimensionless criterion function with the most general equation for the least squares method (WTLS):  
\[\phi_{ty}(\mathbf{\Delta t},\mathbf{\Delta Y})=[\mathbf{\Delta t},\mathbf{\Delta Y}]\mathbf{U_{TS} }^{-1} \begin{bmatrix}\mathbf{\Delta t}\\ \mathbf{\Delta Y}\end{bmatrix} =[\mathbf{\Delta t},\mathbf{\Delta Y}] \begin{bmatrix}\mathbf{U_{ t}}&\mathbf{U_{tY}}\\ \mathbf{U_{tY}}^{T}&\mathbf{U_{Y}}\end{bmatrix} ^{-1} \begin{bmatrix}\mathbf{\Delta t}\\ \mathbf{\Delta Y}\end{bmatrix} \to min \tag{3}\]  
where \(\mathbf{U_{t}},\mathbf{U_{Y}},\mathbf{U_{tY}}\) are covariance matrixes that are single parts of the covariance matrix \(\mathbf{U_{TS}}= \begin{bmatrix}\mathbf{U_{t}}&\mathbf{U_{tY}}\\ \mathbf{U_{tY}^{T}}&\mathbf{U_{Y}}\end{bmatrix} \) appropriate for the time variable \(t\) and the measured signal \(y\) in \(n\)-points taking into account autocorrelations and a covariance matrix taking into account the effects of cross correlation between time and signal coordinates, while \(\mathbf{\Delta t},\mathbf{\Delta Y}\) denote vectors containing deviation values for time \(\Delta t_{1},\ldots,\Delta t_{n}\) and for the signal \(\Delta y_{1}=f(t_{1}+\Delta t_{1},\mathbf{c})-f(t_{1},\mathbf{c})\), \(\ldots,\Delta y_{n}=f(t_{n}+\Delta t_{n},\mathbf{c})-f(t_{n},\mathbf{c})\) at subsequent measuring points \(t_{1},\ldots,t_{n}\). In the special case of no correlation, the dependence of the minimum criterion function (3) means a minimum of:  
\[\phi_{ty}(\mathbf{\Delta t},\mathbf{\Delta Y})=\sum_{i=1}^{n}\frac{{\Delta{t_{i}}^{2 }}}{u^{2}(t_{i})}+\frac{{\Delta{y_{i}}^{2}}}{u^{2}(y_{i})}\to min \tag{4}\]  
where \(u^{2}(t_{i})\) and \(u^{2}(y_{i})\) are variances (squared uncertainties) for time deviations \(\Delta t_{i}\) and for signal deviations \(\Delta y_{i}\). The most commonly used method of least squares is the OLSs (ordinary least squares) or special case of TLSs (total least squares for identical values of uncertainties of time and signal) without taking into account the uncertainty of the abscissa variable in this case of time \(u(t)=0\), assuming that all signal uncertainties are the same \(u^{2}(y_{i})=\text{const}\).