## 4 Parameter-Identification-Based Configuration Design
### Parameter Identification of Configuration
Consequently, our goal is to find such a transformation matrix \(g(k_{1},k_{2})\) that, with its treatment, we arrive at  
\[ \begin{cases}\left(\mathbf{e}_{1}g(k_{1},k_{2})\mathbf{r}_{D_{1}}\right)^{2}+\left(\bm {e}_{2}g(k_{1},k_{2})\mathbf{r}_{D_{1}}\right)^{2}-\left(\frac{D_{1}}{2}\right)^{ 2}=0,\\ \mathbf{e}_{3}g(k_{1},k_{2})\mathbf{r}_{C_{1}}-\frac{r_{\mathrm{c}}^{2}}{4\bar{q}}-h=0, \end{cases}  \tag{11}\]  
where \(\mathbf{e}_{i}\), \(i=1,2,3\), represents a vector whose \(i\)-th component is \(1\) and whose remaining components are all \(0\). It is fortunate that Equation (11) happens to have solutions. However, the process of error compensation is conducted according to our expectation, so there may not exist any solutions to Equation (11). Considering this situation, it is preferable to convert Equation (11) into an optimization model. Foremost, the error function can be expressed as follows:  
\[E(k_{1},k_{2})=\left[\left(\mathbf{e}_{1}g(k_{1},k_{2})\mathbf{r}_{D_{1}}\right)^{2}+ \left(\mathbf{e}_{2}g(k_{1},k_{2})\mathbf{r}_{D_{1}}\right)^{2}-\left(\frac{D_{\mathrm{ f}}}{2}\right)^{2}\right]^{2}+\left[\mathbf{e}_{3}g(k_{1},k_{2})\mathbf{r}_{C_{1}}-\frac{r_{ \mathrm{c}}^{2}}{4\bar{q}}-h\right]^{2}, \tag{12}\]  
where \(E(k_{1},k_{2})\) represents the residual produced by the left-hand side of Equation (11), which is a function of \(k_{1}\) and \(k_{2}\). Secondly, as the inner optimization model, it can be formulated as follows:  
\[ \begin{cases}\text{find}&(k_{1},k_{2}),\\ \text{min}&E(k_{1},k_{2}),\\ \text{s.t.}&0<k_{1}<1,0<k_{2}<1,\end{cases}  \tag{13}\]  
If the minimum of the optimization problem in Equation (13) is \(0\) or a value considerably approximating \(0\), we obtain the precise solution of Equation (11); otherwise, we obtain a least-square solution. In pursuit of the same pose as that described in Figure 6c, compensation parameter \(\beta\) is adjusted to maximize the distance from point \(B_{1}\) to the \(Z\)-axis, thereby satisfying the GC 4; this fact will be verified in Section 4.4. This distance maximization can also be regarded as an optimization problem, so the objective function shall be established first. For a given \(\beta\), its corresponding transformation matrix is obtained and denoted as \(\mathbf{T}(\beta)\), so the distance denoted as \(g_{1}(\beta)\) can be formulated as:  
\[g_{1}(\beta)=\|\mathbf{Diag}(1,1,0,0)\mathbf{T}(\beta)r_{B_{1}}\|_{2}, \tag{14}\]  
where \(\mathbf{Diag}(\cdot)\) represents the diagonal matrix. Therefore, the outer optimization model can be described as  
\[ \begin{cases}\text{find}&\beta,\\ \text{max}&g_{1}(\beta),\\ \text{s.t.}&\frac{-2\pi}{n}<\beta<\frac{2\pi}{n},\end{cases}  \tag{15}\]  
where the maximum of the model is a function of \(D_{\text{a}}\), denoted as \(g_{2}(D_{\text{a}})\). Eventually, finding the maximum aperture diameter \(D_{\text{a}}\) is equivalent to solving the following equation:  
\[g_{2}(D_{\text{a}})=\frac{s}{\sin\frac{\pi}{n}}. \tag{16}\]  
After solving Equation (16), the maximum aperture diameter is denoted as \(D_{\text{a}}^{*}\), the resulting matrix is denoted as \(\mathbf{T}^{*}\), the residual error is denoted as \(\mathbf{L}^{*}\) and the corresponding compensation parameters are denoted as \(k_{1}^{*}\), \(k_{2}^{*}\) and \(\beta^{*}\), respectively. Furthermore, the resulting RSE, known as the simulated RSE and denoted as \(\zeta_{\text{s}}\), is equivalent to \(D_{\text{a}}^{*}/D_{\text{f}}\). To sum up, the process of parameter identification is to solve a two-level nested optimization model (Equations (13) and (15)) in combination with an equation (Equation (16)), and its pseudocode is presented as follows. Thereinto, the inputs are the structural parameter values of the antenna and the outputs are the maximum aperture diameter, transform matrix and identified parameters.