## 4 Optimal Placement of PZT Actuators
### Implementation of Genetic Algorithm for Single-Objective Optimization
In-orbit thermal load  
Large space antennas orbiting the Earth always suffer from severe thermal effects, including an extremely cold black background, strong solar radiations, and complex heat fluxes of the Earth's infrared radiation and albedo. Complex thermal effects yield complicated thermal deformation modes that correspond to deteriorated antenna performance owing to the reduction in the reflector surface shape accuracy. However, the optimal placement of actuators based on in-orbit thermal distortions has not been addressed previously. In this case, the placement optimization of actuators is investigated based on the surface errors induced by in-orbit thermal deformation. To improve the control efficiency under in-orbit thermal effects, a typical day needs to be selected to develop and demonstrate this method. Assuming a reflector is operated in a geostationary orbit, typical dates are the spring equinox, autumn equinox, summer solstice, and winter solstice. During the spring equinox, a cycling reflector is under the most severe thermal effects, especially when flying into and out of the umbra area of Earth. Significant changes in solar radiation lead to large temperature gradients across the reflector [34]. Thus, the corresponding thermal distortion is more severe than any other days. A schematic of the in-orbit thermal condition during the spring equinox is shown in Figure 9.  
To maintain the high surface accuracy of the reflector, the arrangement of a limited number of actuators must be optimized to minimize the RMS error during a cycling period. The change in the temperature distribution is assumed to be much slower than the responses of the adjustment process induced by the actuators. Thus, an optimizing objective function that is an integration of the discrete RMS errors in a time span is proposed. It can be expressed as  
\[F(\mathbf{\lambda},\mathbf{U}(t))=\int_{t_{s}}^{t_{c}}\left\|\left[\delta^{I}-\delta^{ D}+\
abla\delta_{v}(\mathbf{\lambda},\mathbf{U}(t))+\
abla\delta_{t}(\mathbf{T}(t))\right] \right\|_{2}, \tag{46}\]  
where \(\mathbf{U}(t)\) denotes the voltage vector that varies with time, \(\mathbf{T}(t)\) is the nodal temperature vector, and ts and te are the starting and ending times of the circle, respectively. \(\
abla\delta_{v}\) and \(\
abla\delta_{t}\) are the nodal displacement vectors induced by voltages applied and temperature  
Figure 9: Schematic of in-orbit thermal condition during the spring equinox. distribution of the reflector at time point t. By dividing the integration interval with respect to time t into n equal sections with a midpoint and using the Simpson integration formula in each section, the objective function can be expressed as  
Figure 5:0  
The equivalent RMS error at time \(t_{i}\) can be expressed as Equation (48). Figure 5:1  
\(\Delta t=t_{i+1}-t_{i}\) is a small-time span. Minimizing this objective function can effectively improve the control efficiency of the orbital process. An optimization formulation can be written as  
\[ \begin{cases}\min\,F(\lambda,\mathbf{U})=\frac{\Delta t}{6}\sum\limits_{i=1}^{n} \biggl{[}g(t_{i})+4g(\frac{t_{i}+t_{i+1}}{2})+g(t_{i+1})\biggr{]}\\ \text{s.t. }g(t_{i})=\left\|\delta^{I}-\delta^{D}+\
abla\delta_{v}(\lambda,\mathbf{U}(t_ {i}))+\
abla\delta_{t}(\mathbf{T}(t_{i}))\right\|_{2}.\\ \mathbf{K}_{mm}\
abla\delta_{v}(\lambda,\mathbf{U}(t_{i}))+\mathbf{K}_{mp}\mathbf{U}(t_{i})=F \\ \max|\mathbf{U}(t_{i})|\leq U_{max}\end{cases} . \tag{49}\]  
NX TMG (originally integrated in I-DEAS) has been used to construct an in-orbit thermal model for the reflector's time-dependent temperature distributions calculation on the day of the spring equinox. The distribution of the reflector nodal temperature is calculated at each time point. So, the corresponding thermal deformations at each time point \(\
abla\delta_{t}(\mathbf{T}(t_{i}))\) can thus be obtained. Then, they can be substituted to Equation (48) for calculating the voltages and equivalent RMS error \(g(t_{i})\).