## 3 Adjustment Method for the Pre-Stress Deviation  
### Fundamental Theory for the Adjustment of Pre-Stress Deviation  
The adjustment of the pre-stress deviation is generally conducted by correcting the adjustment members. During the correction procedure, the adjustment members are monitored to guarantee the accurate correction value, and the optimized result of the correction is to make the pre-stress free from deviation. Suppose the number of adjustment members is \(k\). In order to achieve \(\delta\mathbf{t}_{q}=0\), adjustment members must generate a length change, \(\delta\mathbf{e}_{0}^{k}\), and meet the following condition:  
\[-\delta t_{\mathbf{q}}=\mathbf{S}_{tq}^{k}\delta\mathbf{e}_{0}^{k} \tag{10}\]  
where \(\mathbf{S}_{tq}^{k}\) is the force sensitivity matrix for the adjustment members. If the values of the length change, \(\delta\mathbf{e}_{0}^{k}\), of the adjustment members can be obtained on the basis of the monitored values of the pre-stress deviation, \(\delta t_{q}\), then the obtained \(\delta\mathbf{e}_{0}^{k}\) can be applied to correct the pre-stress deviation. To obtain \(\delta\mathbf{e}_{0}^{k}\), \(\delta t_{q}\) in Equation (10) is added to the force sensitivity matrix, \(\mathbf{S}_{tq}^{k}\), to compose an augmented matrix, \(\left[\mathbf{S}_{tq}^{k}|\delta t_{q}\right]\). Based on the matrix theory, the value of adjustment, \(\delta\mathbf{e}_{0}^{k}\), can be calculated by analysing the relationship between the rank \(r_{k}\) of the matrix \(\mathbf{S}_{tq}^{k}\) and the rank \(r_{k}\prime\) of the augmented matrix \(\left[\mathbf{S}_{tq}^{k}|\delta t_{q}\right]\).  
1. If \(r_{k}\prime=r_{k}<k\), then multiple solutions of \(\delta\mathbf{e}_{0}^{k}\) are present for Equation (10), and not all the adjustment members are needed to correct the pre-stress deviation. If \(r_{k}\prime=r_{k}=q<k\), then only \(q\) adjustment members are needed.
2. If \(r_{k}\prime=r_{k}=k\), then it means only one solution for \(\delta\mathbf{e}_{0}^{k}\) is available for Equation (10), and the unique solution can be obtained by \[\delta\mathbf{e}_{0}^{k}=-{\left[\mathbf{S}_{tq}^{k}\right]}^{+}\delta t_{q}\] (11) Equation (11) indicates that \(k\) adjustment members are adequate to obtain a precise adjustment values of the pre-stress deviation. Moreover, the number of monitored cables, \(q\), does not exceed the number of adjustment members, \(k\). The correction values for the pre-stress deviation, \(\delta\mathbf{k}^{k}\), can be obtained by substituting the values of the length change, \(\delta\mathbf{e}_{0}^{k}\), into Equation (10).
3. If \(r_{k}\prime=k+1\), then it means no accurate solution for \(\delta\mathbf{e}_{0}^{k}\) is available for Equation (10); namely, the pre-stress deviation cannot be corrected only by correcting the \(k\) adjustment members. Extra adjustment members are needed to precisely correct the pre-stress deviation, and the least square estimation is used to obtain approximate \(\delta\mathbf{e}_{0}^{k}\).  
\[{\left(\delta\mathbf{e}_{0}^{k}\right)}^{\#}={\left[(\mathbf{S}_{tq}^{k})^{\text{T}} \left(\mathbf{S}_{tq}^{k}\right)\right]}^{-1}{\left(\mathbf{S}_{tq}^{k}\right)}^{ \text{T}}\delta t_{q} \tag{12}\]  
Then, the correction values for the pre-stress deviation, \(\delta\mathbf{k}^{k}\), can also be calculated by substituting this estimate \({\left(\delta\mathbf{e}_{0}^{k}\right)}^{\#}\) into Equation (10).