## 4 Optimal Placement of PZT Actuators
### Implementation of Genetic Algorithm for Multi-Objective Optimization
A flowchart of the algorithm is shown in Figure 14. A non-dominated solution filter is designed to reserve a certain number of non-dominated solutions. The reserved non-dominated solutions are chosen to yield 30-40% of the population. Based on the distance between the solutions, a filter is designed to reserve a certain number of solutions. It can be expressed as \[\left\{ \begin{array}{ll}min\Big{(}\sqrt{\left(f_{1}(X_{i})-f_{1}(X_{j}) \right)^{2}+\left(f_{2}(X_{i})-f_{2}(X_{j})\right)^{2}}\Big{)}\Big{|}j=1,2, \cdots k\&j\
eq i\ >\epsilon,reserveX_{i}\\ else,\quad\quad\text{delete}\ \ X_{i}\\ \end{array} \right.\] (54) Here, \(\epsilon\) is an adjustable value of the minimum distance between \(f_{2}(X_{i})\) and \(f_{2}(X_{j})\). The filter can remove crowded individuals and scatter reserved solutions in the solution domain. An optimization flow chart is shown in Figure 14.
* Numerical examples  
The loads in case 1 are also used in this case. The results of the method for the weighted summation of objectives are obtained to compare with those of the modified algorithm. A normalized weighted-sum function of the objectives can be denoted as  
\[f(X)=\alpha_{1}\overline{f_{1}(X)}+\alpha_{2}\overline{f_{2}(X)}, \tag{55}\]  
Here, \(\alpha_{1}\) and \(\alpha_{2}\) are coefficients of weight; \(\overline{f_{1}(X)}\) is the normalized RMS error and \(\overline{f_{2}(X)}\) is the normalized energy consumption, as defined by Equation (42). Setting \(\alpha_{1}=\alpha_{2}=0.5\) implies that the weights of the two objective functions are equal. The allowed number of actuators is first set to be 72. The Pareto front of the non-dominated solutions of the two methods can be obtained. Figure 14: Flow chart for the multi-objective optimization of PZT actuators. After 1000 iterations, the Pareto fronts of the non-dominated solutions derived by the two methods are shown in Figure 15a-c, which show 40 non-dominated solutions reserved through the filter using the modified method. As shown in (a), more solutions are biased toward the objective of energy consumption and are not uniformly distributed. From (b) and (c), the number of non-dominated solutions of the modified method is more than that of the weighted-sum objective method and the solutions are distributed more widely. Table 4 lists the solutions of a single objective optimization and 15 non-dominated solutions using the modified GA.  
As shown in Table 4, the energy consumption reduces as the residual RMS error increases. This implies that the two objectives are contradictory to each other. When the residual RMS error is 5.2 \(\upmu\)m, the corresponding energy consumption is 7.8 in the Pareto set. Compared with the result of the single RMS error optimization, the residual RMS error is 4.4 \(\upmu\)m, but the energy consumption is 45.4. The residual RMS error of the multi-objective optimization increased by 15.4% but the energy consumption reduced by 82.8%. The actuator voltages of these two solutions are shown in Figure 16.  
As shown in Figure 16, the amplitude of the voltage distribution of the solution in the multi-objective optimal control (the red line) is much smaller than that of the single-objective optimal control (the black line). The maximum voltage in the multi-objective control is 79.8 V. However, nine actuators reached the maximum voltage amplitude of 150 V in the single-objective optimal control. The voltage range of the multi-objective optimal control is smaller than that of the single-objective optimal control. The allowed number of actuators significantly affects the multi-objective optimal control. Fewer actuators imply less alternative displacement patterns in the control process. When the number of allowed actuators is 30, 50, and 72, respectively, the corresponding Pareto frontiers are shown in Figure 17.