# Algorithms Utilized for Creep Analysis in Torque Transducers for Wind Turbines
## 6 Approximate WTLS Solutions for Nonlinear Functions  
It turns out that in the case of the occurrence of generally different uncertainties for the time variable \(t_{i}\) and the measured signal \(y_{i}\) and correlations described by the covariance matrix, \(\mathbf{U_{T}}\)s nonlinear relationship \(y=f(t)\) can be represented in the form of a linear relationship \(y=\theta_{1}\xi(t,\,\beta)+\theta_{0}\), where the function \(\xi(t,\,\beta)\) is a transforming function, and \(\theta_{1}\), \(\theta_{0}\) and, in general, the vector \(\beta\) are the parameters of the fitting function, which result from the fitting of the function and the minimization of the criterion function, which approximately transforms to the form [29]:  
\[\phi_{ty}(\mathbf{\Delta t},\mathbf{\Delta Y})=[\mathbf{\Delta t},\mathbf{\Delta Y}]\mathbf{L}L^{- 1} \begin{bmatrix}\mathbf{U_{t}}&\mathbf{U_{tY}}\\ \mathbf{U_{tY}^{T}}&\mathbf{U_{Y}}\end{bmatrix} ^{-1}L^{-1}L \begin{bmatrix}\mathbf{\Delta t }\\ \mathbf{\Delta Y}\end{bmatrix} \approx[\mathbf{\Delta\xi},\mathbf{\Delta Y}] \begin{bmatrix} \mathbf{U_{\xi}}&\mathbf{U_{\xi Y}}\\ \mathbf{U_{\xi Y}^{T}}&\mathbf{U_{Y}}\end{bmatrix} ^{-1} \begin{bmatrix}\mathbf{\Delta\xi} \\ \mathbf{\Delta Y}\end{bmatrix} =\phi_{\xi y}(\mathbf{\Delta\xi},\mathbf{\Delta Y},\mathbf{\beta}) \tag{14}\]  
and diagonal matrix \(L\) size of \(2n\times 2n\) shall be determined by the first \(n\) elements as derivatives of functions \(\xi^{\prime}(t_{i},\,\,\beta)\) at subsequent measurement points, and the following \(n\), an identical diagonal value, is equal to a value of one. The laws of error and uncertainty propagation have been used by multiplying the first derivative of the transforming function, respectively, by error of time and uncertainty of time:  
\[\Delta\xi \approx\xi^{\prime}(t,\,\beta)\Delta t, \tag{15a}\] \[u(\xi) \approx\bigl{|}\xi^{\prime}(t,\,\beta)\bigr{|}u(t) \tag{15b}\]It follows that dependencies \(\phi_{\xi y}(\theta_{1})\) and \(\phi_{\xi y}(\theta_{0})\) for the selected parameter vector \(\mathbf{\beta}=[\beta_{1},\beta_{2}]\) are quasi-square and described by [22]:  
\[\phi_{\xi y}(\theta_{1})={\theta_{1}}^{2}\bigg{(}P_{tt}-\frac{P_{t}^{2}}{P} \bigg{)}+2\bigg{(}\frac{P_{t}P_{y}}{P}-P_{yy}\bigg{)}\theta_{1}+P_{yy}-\frac{P_ {y}^{2}}{S} \tag{16}\]  
where  
\[P=\mathbf{1}^{\mathbf{T}}\mathbf{U}_{\mathbf{x}\mathbf{c}\mathbf{f}}^{\mathbf{-1}}\mathbf{1}, \tag{17a}\] \[P_{t}=\mathbf{\xi}^{\mathbf{T}}\mathbf{U}_{\mathbf{x}\mathbf{c}\mathbf{f}}^{\mathbf{-1}}\mathbf{1}=\mathbf{1}^{ \mathbf{T}}\mathbf{U}_{\mathbf{x}\mathbf{c}\mathbf{f}}^{\mathbf{-1}}\mathbf{\xi},\] (17b) \[P_{y}=\mathbf{Y}^{\mathbf{T}}\mathbf{U}_{\mathbf{x}\mathbf{c}\mathbf{f}}^{\mathbf{-1}}\mathbf{1}=\mathbf{1}^{ \mathbf{T}}\mathbf{U}_{\mathbf{x}\mathbf{c}\mathbf{f}}^{\mathbf{-1}}\mathbf{Y},\] (17c) \[P_{tt}=\mathbf{\xi}^{\mathbf{T}}\mathbf{U}_{\mathbf{x}\mathbf{c}\mathbf{f}}^{\mathbf{-1}}\mathbf{\xi},\] (17d) \[P_{ty}=\mathbf{\xi}^{\mathbf{T}}\mathbf{U}_{\mathbf{x}\mathbf{c}\mathbf{f}}^{\mathbf{-1}}\mathbf{Y}=\mathbf{Y}^{ \mathbf{T}}\mathbf{U}_{\mathbf{x}\mathbf{c}\mathbf{f}}^{\mathbf{-1}}\mathbf{\xi},\] (17e) \[P_{yy}=\mathbf{Y}^{\mathbf{T}}\mathbf{U}_{\mathbf{x}\mathbf{c}\mathbf{f}}^{\mathbf{-1}}\mathbf{Y} \tag{17f}\]  
and effective inverse covariance matrix is given by:  
\[\mathbf{U}_{\mathbf{x}\mathbf{c}\mathbf{f}}^{\mathbf{-1}}=\mathbf{U}_{\mathbf{22}}-\Big{(}\mathbf{U}_{\mathbf{12} }^{\mathbf{T}}+\theta_{1}\mathbf{U}_{\mathbf{22}}\Big{)}\mathbf{V}^{\mathbf{-1}}(\mathbf{U}_{\mathbf{12} }+\theta_{1}\mathbf{U}_{\mathbf{22}}) \tag{18}\]  
and \(\mathbf{V}=\mathbf{U}_{\mathbf{11}}+\theta_{1}\big{(}\mathbf{U}_{\mathbf{12}}^{\mathbf{T}}+\mathbf{U}_\mathbf{{12}}\big{)}+{\theta_{1}}^{\mathbf{2}}\mathbf{U}_{\mathbf{22}}\) and matrixes \(\mathbf{U}_{\mathbf{11}},\mathbf{U}_{\mathbf{12}},\mathbf{U}_{\mathbf{22}}\) are the corresponding parts of the inverse covariance matrix \(\mathbf{U}_{\mathbf{TS}}^{\mathbf{-1}}=