## 3 High-Dynamic Atmospheric Turbulence Compensation Algorithm
### Capturing Aberrations  
In the previous subsection, we apply the Gauss-Newton method to compute controlling voltages for the electrodes of wavefront correctors. However, an input is supposed to be provided as the target of iteration which is the \(\mathbf{offset}_{Target}\) in Equation (9). In this subsection, we introduce our method of providing input for the Gauss-Newton method. For single DM AO architecture, all workloads are put on the only wavefront corrector. Traditional iterative wavefront correction algorithms building on the relationship between controlling voltages and spots offsets linearly combine current spots offsets, wavefront residual of the previous turn and model value as \(\mathbf{offset}_{Target}\). The algorithm captures accurate wavefront aberrations and according to [29], the wavefront residual remains under \(\frac{1}{8}\)_wavelength_ after compensation. Nevertheless, in double wavefront correctors architecture, distribution of the workload on FSM and DM separately determines the precision of the compensated results. In order to divide the labor of woofer and tweeter which are the FSM and DM in our context, the Lagrange-multiplier method [23, 24] multiplied a multiplier \(\lambda\) to the left of the response matrix of tweeeter as shown in Equation (15)  
\[\mathbf{offset}=P_{twoofer}\times\mathbf{s}_{twoofer}+\lambda P_{tweeter}\times\mathbf{s}_ {tweeter} \tag{15}\]  
where \(P_{twoofer}\) and \(P_{tweeter}\) are response matrices for the woofer and tweeter, while \(\mathbf{s}_{waofer}\) and \(\mathbf{s}_{tweeter}\) are actuator vectors for the woofer and tweeter. The basic solving process are expressed in Equation (16)  
\[ \begin{bmatrix}P_{twoofer}^{T}P_{twoofer}&\lambda P_{twoofer}^{T}P_{tweeter }\\ \lambda P_{tweeter}^{T}P_{twoofer}&\lambda^{2}P_{tweeter}^{T}P_{tweeter}\\ \end{bmatrix}\begin{bmatrix}\mathbf{s}_{twoofer}\\ \mathbf{s}_{tweeter}\\ \end{bmatrix} = \begin{bmatrix}P_{twoofer}^{T}\\ \lambda P_{tweeter}^{T}\\ \end{bmatrix} \mathbf{offset} \tag{16}\]  
However, by marking \([P_{woofer}\lambda P_{tweeter}]\) as \(P_{global}\), the irreversibility of \(P_{global}^{T}P_{global}\) determines that a damped least-square method is needed for solving actuator vectors as shown in Equation (17). \[ \begin{bmatrix}\mathbf{s}_{woofer}\\ \mathbf{s}_{tweeter}\\ \end{bmatrix} = \begin{bmatrix}P_{woofer}^{T}P_{woofer}+\beta_{1}&\lambda P_{ twoofer}^{T}P_{tweeter}\\ \lambda P_{tweeter}^{T}P_{twoofer}&\lambda^{2}P_{tweeter}^{T}P_{tweeter}+ \beta_{2}\\ \end{bmatrix} ^{-1} \begin{bmatrix}P_{woofer}^{T}\\ \lambda P_{tweeter}^{T}\\ \end{bmatrix} \mathbf{offset} \tag{17}\]  
where \(\beta_{1}\) and \(\beta_{2}\) are damping parameter matrices. In Equation (17), \(\lambda,\beta_{1}\) and \(\beta_{2}\) are manually set. The empirically selected \(\lambda\) can suppress coupling between the woofer and tweeter. However, Equation (15) is neither a necessary nor sufficient condition for decoupling high and low-order aberrations. By contrast, in our FSM-DM cascaded AO architecture, the workload of FSM and DM consists of decoupled current spots offsets, decoupled wavefront residual of the previous turn and model value. Model value can be computed with coefficient matrices and current controlling voltages. Decoupled current spots offsets and decoupled wavefront residual of the previous turn are acquired from current spots offsets and wavefront residual. By changing them to wavefront slopes using Equation (1) the coefficients of Zernike polynomials can be fitted with Equation (2). Subsequently, after separated Zernike coefficients performing the inverse process of Equations (1) and (2) in turn, the decoupled current spots offsets, and the decoupled wavefront residual of the previous turn can be gained. It should be noted that, the current spots offsets are calculated from the current wavefront on SHWS while the wavefront residuals are acquired from the compensated wavefront output of the previous turn on SHWS.