# Algorithms Utilized for Creep Analysis in Torque Transducers for Wind Turbines
## 6 Approximate WTLS Solutions for Nonlinear Functions
\begin{bmatrix}\mathbf{U}_{\mathbf{f}}&\mathbf{U}_{\mathbf{f}\mathbf{y} }\\ \mathbf{U}_{\mathbf{f}\mathbf{y}}^{\mathbf{-1}}&\mathbf{U}_{\mathbf{f}\mathbf{y}}\end{bmatrix} ^{-1}= \begin{bmatrix} \mathbf{U}_{\mathbf{11}}&\mathbf{U}_{\mathbf{12}}\\ \mathbf{U}_{\mathbf{12}}^{\mathbf{T}}&\mathbf{U}_{\mathbf{22}}\end{bmatrix} \) and are functions of only one parameter \(\theta_{1}\) or \(\theta_{0}=(P_{y}-\theta_{1}P_{t})\big{/}P\), and these functions have quasi vertices [28]. By determining for a series of points within the accepted test interval \(\theta_{1L}<\theta_{1H}\) for selected values of vector parameters \(\mathbf{\beta}\) values of the local minimum and then looking for the value of the vector \(\mathbf{\beta}\) (the easiest one-dimensional vector), it is possible to minimize the criterion function by obtaining a global minimum at the points: \(\theta_{1 min}\), \(\mathbf{\beta}_{min}\). Algorithm Schema for the Method WTLSThe algorithm is implemented in four steps:  
1. For a given fitting function, e.g., \(y=a_{3}t^{3}+a^{2}t^{2}+a_{1}t+a_{0}\), we create a function \(\xi(t,\beta)\), in this case, \(\xi(t,\,[\beta_{1},\,\beta_{2}])=t^{3}+\beta_{1}t^{2}+\beta_{2}t\), where \(\beta_{1}=a_{2}/a_{3}\) and \(\beta_{2}=a_{1}/a_{3}\) by setting initial values \(\beta_{1}\) and \(\beta_{2}\), which we calculate numerically with fixed steps \(h_{\beta 1}\) i \(h_{\beta 2}\) so that these values are in the ranges of \(\beta_{1L}\leq\beta_{1}\leq\beta_{1H}\) and \(\beta_{2L}\leq\beta_{2}\leq\beta_{2H}\);
2. Then, on the basis of the given covariance matrix \(\mathbf{U}_{\mathbf{TS}}\), we determine the matrix product \(\mathbf{L}\)\(\mathbf{U}_{\mathbf{TS}}\mathbf{L}\), where \(n\) first diagonal elements of a matrix \(\mathbf{L}\) are defined by \(\xi^{\prime}(t_{i},\,\beta)=3t_{i}^{2}+2\beta_{1}t_{i}+\beta_{2}\);
3. At the end, we designate a series of points \(\theta_{1j}\) and specify the values of the \(\theta_{1}\) with a step \(h_{\theta 1}\) from the selected range \(\theta_{1L}\leq\theta_{1j}\leq\theta_{1H}\) (\(j=1,\,\ldots,\,M\)) = [\((\theta_{1H}-\theta_{1L})/h_{\theta 1}\)] and calculate characteristics \(\phi_{\xi y}(\theta_{1})\) and the resulting characteristics of the \(\phi_{\xi y}(\theta_{0})\), whereby the points \(\theta_{1j}\) should be selected in such a way that they make visible a quasi-minimum (quasi vertex);
4. Repeat the above points one by one, first adjusting the \(\beta_{1}\), and next \(\beta_{2}\) or vice versa, until the global minimum of the criterion function is obtained \(\phi_{y\xi\min}\) for fitting parameters \(a_{3}=\theta_{1\min}\), \(a_{2}=\theta_{1\min}\)\(\beta_{1\min}\), \(a_{1}=\theta_{1\min}\)\(\beta_{2\min}\) and \(a_{0}=\theta_{0\min}\). In the case of a simplified third-degree fitting function \(y=b_{3}t^{3}+b_{1}t+b_{0}\) function \(\xi(t,\,\beta_{1})=t^{3}+\beta_{1}t\) and then the values of the adjusted parameters due to the presence of only one parameter \(\beta_{1}\) shall be, respectively, calculated from \(b_{3}=\theta_{1\min}\), \(b_{1}=\theta_{1\min}\)\(\beta_{1\min}\) and \(b_{0}=\theta_{0\min}\). Exemplary characteristics \(\phi_{\xi y}(\theta_{1})\) and \(\phi_{\xi y}(\theta_{0})\) are below in Figure 5.