## 4 Optimal Placement of PZT Actuators
### Implementation of Genetic Algorithm for Single-Objective Optimization
An iterative process for the above optimization problem can be described as follows. The GA is applied to determine the positions of the actuators. Then, \(\mathbf{U}(t_{i})\) and \(g(t_{i})\) at each time point are derived by solving the constrained quadratic programming problem (39). Subsequently, all \(g(t_{i})\) shall be substituted into Equation (47) to calculate the objective function \(F(\lambda,\mathbf{U})\). The penalty mechanism Equation (42) will also be used to calculate the fitness of the GA. The iteration of the GA will stop if the fitness value does not change significantly. * Numerical example  
The material thermal properties used in this simulation are shown in Table 3.  
Assume that the initial position of the reflecting surface coincides with the ideal position, which means \(\delta^{I}=\delta^{D}\). The surface RMS errors induced by the in-orbit thermal deformation during the day without any control are shown in Figure 10. The shaded area in Figure 10 represents the integral of the RMS error over time, which is used as the objective function for subsequent optimization. The small shaded area represents the integration from the RMS error \(y_{1}\) corresponding to time \(t_{1}\) to the RMS error \(y_{2}\) corresponding to time \(t_{2}\). The area of the shadow represents the objective function value that shall be minimized. When the iteration stops, the optimized placement of the actuators is obtained. The iteration curve of the optimization corresponding to \(N=35\) is shown in Figure 11. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & \multicolumn{2}{c}{**Value**} \\ \hline Materials & Single-fiber layer prepreg & PZT actuator (average) \\ Thermal Conductivity & 18.6 & 37.2 \\ (W/m K) & 850 & 460 \\ Specific Heat (/kg\(\cdot\)K) & \multirow{2}{*}{\(-1.01\times 10^{-6}\)} & \multirow{2}{*}{\(3.81\times 10^{-6}\)} & \multirow{2}{*}{\(8.00\times 10^{-6}\)} \\ CTE (/\({}^{\circ}\)C) & & \\ Density (kg/m\({}^{3}\)) & & 1510 & 5060 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Thermal properties. The iteration converges quickly, and the value of the objective function does not change when the number of generations is larger than 550. An actuator placement denoted by the binary vector \(\mathbf{\lambda}\) can be derived from the optimization. The optimization simulations are also studied when \(N\) equals 18, 48 and 54, respectively. Placement of the PZT actuators and the corresponding values of the objective functions are shown in Figure 12. Thick blue lines indicate where actuators are distributed, while thin lines indicate where there are no actuators. As shown in the figure, the placement of the actuators is symmetrical about a certain oblique axis with different quantity limitations of actuators. This is because the Sun is in the equatorial plane during the spring equinox. The external thermal loads on the reflector are symmetrical about a certain oblique axis during the circle. The optimized values of the objective function are 291.67, 253.03, 243.84, and 241.95 \(\upmu\)m\(\cdot\)h corresponding to the conditions of 18, 35, 48, and 54 actuators, respectively. The value of the objective function decreases with the increasing of the actuators. The curve of the objective function value with respect to the number of actuators is shown in Figure 13a. Figure 11: Iteration process of the Genetic Algorithm with Elite selection. Figure 10: In-orbit RMS error of the reflector during the spring equinox. numbers of actuators are shown in Figure 13b. It can be seen that the time and areas formed by the RMS errors become smaller as the number of actuators increases. However, the reduction becomes insignificant when the number of actuators is larger than 48. This phenomenon indicates that the actuators employed should not exceed a certain number.