## 4 Optimal Placement of PZT Actuators
### Basic Strategies of Genetic Algorithm for Actuator Placement Optimization  
The Genetic Algorithm uses binary numbers to discretize and encode the optimizing variables. To model an actuator placement optimization problem, a certain number of candidate positions are prepared for the selection of the PZT actuators. Here, the total number of candidate positions is 72. A representation of the candidate positions and its physical meaning are shown in Figure 3, where the bold lines and thin lines represent one actuator placed and not placed at this candidate position, respectively. Thus, a 72-bit binary number \(\lambda\) is used to encode the actuator positions, \"0\" and \"1\" indicate no actuator and one actuator at a candidate position, respectively. The voltage applied to each actuator is a continuous variable in the optimization process. The four basic steps of the Genetic Algorithm are selection, crossover, mutation and reorganization. For this problem, some modifications have been made to improve the iteration efficiency. The Roulette and Elite selection methods are employed for the selection, where Elite selection entails making the best individual in the current generation not participate in the crossover and mutation but directly reserve to the next generation. Then, adaptive crossover and mutation operators [32] are employed to act on the individuals selected. The probability of crossover and mutation can be written as  
\[P_{c}=\left\{ \begin{aligned} & P_{c1}-\frac{(P_{c1}-P_{c2})(f^{ -}f_{ave})}{f_{max}-f_{ave}}\\ & P_{c1}\end{aligned} \right\} \begin{aligned} & f^{\prime}>f_{ave}\\ & f^{\prime}\leq f_{ave}\end{aligned} , \tag{40}\]  
Figure 3: The candidate positions and representation of the actuators. \[P_{m}= \begin{cases}P_{c1}-\frac{(P_{m1}-P_{m2})(f^{\prime}-f_{mev})}{f_{max}-f_{mev} }&\quad\begin{cases}f>f_{ave}\\ f\leq f_{ave}\end{cases},\end{cases}  \tag{41}\]  
where \(P_{c1}\), \(P_{c2}\) are the minimum and maximum probabilities of crossover; \(P_{m1}\) and \(P_{m2}\) are the minimum and maximum probabilities of mutation, respectively; \(f\) is the fitness of an individual; \(f^{\prime}\) is the larger one of fitness values in the two cross individuals; \(f_{max}\) is the maximum value of fitness in each generation; \(f_{ave}\) is the average value of fitness in each generation; and \(P_{c}\) and \(P_{m}\) are the final crossover and mutation probability, respectively. To ensure the number of actuators is less than the allowed number in the final solutions, an adaptive penalty function is proposed, to be added to the original objective function f to form a new objective function. Therefore, a modified objective function can be derived from Equation (31) and an exterior penalty function. The modified objective function can be written as Equation (42). \[P_{c}= \begin{cases}f(\mathbf{\lambda},\mathbf{U})&\|\mathbf{\lambda}\|_{1}\leq N\\ f(\mathbf{\lambda},\mathbf{U})+f^{\text{penalty}}(g)[max(\|\mathbf{\lambda}\|_{1}-N,0)]^{2 }\|\mathbf{\lambda}\|_{1}>N\end{cases} . \tag{42}\]  
The dynamic penalty factor can be expressed as Equation (43). \[f^{\text{penalty}}(g)=R\Big{[}1+\varepsilon-sin\Big{(}\frac{\pi g}{2G}\Big{)} \Big{]}. \tag{43}\]  
\(\mathbf{U}\) is the voltage vector. \(N\) is the maximum number of actuators allowed for shape control. \(R\) is a scaling parameter that can be chosen according to the original objective function, \(G\) is the total number of generations, and \(g\) is the number of the current generation. Equation (43) indicates that when the number of actuators does not exceed the allowed value, the modified function value equals the original function value. However, when the number of actuators exceeds the allowed value, the more the number exceeds it, the larger the modified objective function value. \(\varepsilon\) is another adjustable parameter that causes the value of the penalty function to be greater than zero when the number of allowed actuators is exceeded.