# Active Optics and Aberration Correction Technology for Sparse Aperture Segmented Mirrors
## 3 Correction Quantity-Solving Method
### Improved Generalized Ridge Estimation  
Let the measurement parity model is given below.  
\[\left\{\begin{aligned} & L_{L}=A_{L}X_{L}+\Delta_{L}\\ E(\Delta_{L})=0,\text{cov}(\Delta_{L})=\sigma_{0}^{2}I\end{aligned}\right. \tag{29}\]  
where \(L_{L}\) is the observation vector of order \(n\times 1\); \(A_{L}=(a_{1},a_{2},\cdots,a_{n})^{\mathsf{T}}\) is the design matrix of order \(n\times t\); \(X_{L}\) is the parameter vector of order \(t\times 1\), \(\Delta_{L}\) is the observation error vector of order \(n\times 1\), and \(\sigma_{0}\) is the unit weight error. The corresponding error equation is given in the following equation.  
\[v=A_{L}\hat{X}_{L}-L_{L} \tag{30}\]  
where \(v\) is the residual vector, and \(\hat{X}_{L}\) is the estimated value of \(X_{L}\). The ordinary least squares estimation of \(X_{L}\)--which is referred to as the \(LS\) estimate--is given in Equation (31).  
\[\hat{X}_{LLS}=(A_{L}^{\mathsf{T}}A_{L})^{-1}A_{L}^{\mathsf{T}}L_{L} \tag{31}\]  
The corresponding influence function is defined in the following fashion.  
\[IF(L_{L}i;\hat{X}_{L})=-(A_{L}^{\mathsf{T}}A_{L})^{-1}a_{i}^{\mathsf{T}}v_{i} \tag{32}\]  
It can be seen that the influence function depends on three factors: the normal matrix \(A_{L}^{\mathsf{T}}A_{L}\), the row vector \(a_{i}\), and the residual vector \(v\). When the design matrix \(A_{L}\) contains complex covariance, which means the normal matrix \(A_{L}^{\mathsf{T}}A_{L}\) is sick, some of its eigenvalues \(\lambda_{j}\) are close to 0. Although \(\hat{X}_{LLS}\) has the smallest variance in the linear unbiased estimation, its mean square error \(MSE(\hat{X}_{LLS})=\sigma_{0}^{2}\sum\limits_{i=1}^{t}\frac{1}{\lambda_{i}}\) becomes large as a result.  
The solution accuracy of \(z\), \(\Delta z\) and \(f\) in Equations (16), (18) and (27) may be degraded due to the sickness of the normal matrices \(A^{\mathsf{T}}A\), \(\Delta U_{k}^{\mathsf{T}}\Delta U_{k}\) and \(E_{U}^{\mathsf{T}}E_{U}\) significantly when solving them using the ordinary least squares after segmented mirrors under load.  
It is crucial to address the effect of \(a_{i}\) in Equation (29) by considering them far away from each of the other points \(a_{j}\). First, an appropriate weight is added to \(a_{i}\). It is done by giving a smaller weight \(p_{i}\) to \(a_{i}\) when it is far away from the other \(a_{j}\). It is called \(GM\) estimation [16]. An improved generalized ridge estimation, also called \(GL\) estimation, is proposed to overcome the effect of normal matrix pathology on the estimation while resisting the effects of observation coarseness and high leverage points. It is done by combining \(LS\) estimation and \(GM\) estimation, establishing the following solution equation.  
\[\hat{X}_{GL}=\left(A_{L}^{\mathsf{T}}P_{L}A_{L}+\eta I\right)^{-1}A_{L}^{ \mathsf{T}}P_{L}L_{L} \tag{33}\]  
In the above equation, \(P_{L}=diag(p_{1},p_{2},\cdots,p_{n})\) is the weight matrix and \(\eta\) is the ridge parameter. Equation (33) is generally solved iteratively. If the parameter valuation at step \(l\) has been solved with a solution \(\hat{X}_{CL}^{(l)}\), observed residual \(v_{i}^{(l)}=A_{L}\hat{X}_{GL}^{(l)}-v_{i}^{(l-1)}\), and a ridge parameter \(\eta^{(l)}\), then the solution at step \(l+1\) is as follows.  
\[\hat{X}_{GL}^{(l+1)}=\left(A_{L}^{\mathsf{T}}P_{L}^{(l+1)}A_{L}+\eta^{(l+1)}I \right)^{-1}A_{L}^{\mathsf{T}}P_{L}^{(l+1)}v_{i}^{(l)} \tag{34}\]  
When the number of iterations is \(n\), the total observed residuals are calculated as \(v_{GL}=A_{L}\hat{X}_{GL}-L_{L}\), where \(\hat{X}_{GL}=\sum\limits_{l=1}^{n}\hat{X}_{GL}^{(l)}\), such that the estimated value of \(L_{L}\) is \(\hat{L}_{L}=A_{L}\hat{X}_{GL}\).  
Afterward, the accuracy evaluation function is solved as \(MSE(L_{L})\), similar to \(MSE(\hat{H})\) introduced in Section 2.2.1, \(MSE(\hat{U}_{k}^{\mathsf{T}})\) presented in Section 2.2.2, and \(MSE(\hat{U}_{D})\) discussed in Section 2.3.