## 4 Actuators/Sensors Position Configuration
### Location Optimization Design Based on Improved PSO  
After completing the derivation and design of the objective function for the position configuration of the active control device in the previous subsection, this subsection realizes the synchronous optimization of the sensor/actuator position coordinates based on the particle swarm optimization (PSO) algorithm. The core of particle swarm optimization algorithm design is the group sharing of individual information. Through the iterative interaction between individual optimal information and group optimality, the transformation from random disorder to reasonable order is realized, and the group optimal solution of individual combination is obtained. The mathematical understanding of the PSO algorithm is as follows  
1. At the initial moment, \(N\) random particles are put into the feasible domain, where the initial random particles have initial random positions and velocities within a limited range. The objective function value is calculated, and all particles move toward the point of maximum or minimum objective function value. 2. After the particle completes a displacement, the individual optimal value Sbest of a single particle is calculated, and the global optimal value Gbest of the group particles is calculated at the same time, and the next position and velocity update iteration of the particle group is guided according to Sbest and Gbest. The iterative update formula is as follows (30) and (31). The optimal value also becomes the fitness, which is used as the target value to measure the convergence of the algorithm. \[V_{n}^{k+1}=\phi V_{n}^{k}+c_{1}\left(P_{n}^{k}-X_{n}^{k}\right)+c_{2}\left(P_ {g}^{k}-X_{g}^{k}\right)\] (30) \[X_{n}^{k+1}=X_{n}^{k}+V_{n}^{k+1}\] (31)  
\(n\) represents the particle number, \(k\) corresponds to the number of update iterations, \(c_{1}\) and \(c_{2}\) are called individual learning factors and social learning factors, and their values are generally 2; \(\phi\) is an inertia factor, and its value can realize the dynamic adjustment of the iterative update process. \(P_{n}^{k}\) and \(P_{g}^{k}\) correspond to the individual optimal value and the group optimal value in the cycle. 3. After the particle completes a position update, the current objective function value (i.e., fitness) is calculated and obtained, and it is judged according to the condition of jumping out of the criterion to achieve convergence. The active control device's optimum configuration position at this moment is determined by the group coordinate combination. Body assembly is a irregular large-space structure, and it is difficult to obtain the energy correlation matrix required by the objective function through calculation and analysis alone. In this method, finite element/analytical method based on co-simulation is adopted to realize the optimization of active control devices. For the specific process, please refer to Figure 7.  
The value design of the inertia factor \(\omega\) in the speed update formula of PSO algorithm is improved to achieve a fast and stable iterative effect. A simple analysis shows that the \(\phi\) value is large at the initial stage of population iterative evolution, realizing the global rapid positioning of the optimal region, and the \(\omega\) value is rapidly reduced in the middle and late stages, realizing small-scale refined approximation. This paper proposes a linear adaptive PSO algorithm, which is as follows  
\[\phi= \begin{cases}\phi_{max}-(\phi_{max}-\phi_{1})f_{e}/f&k\geq n\\ \phi_{max}-(\phi_{max}-\phi_{2})k/k_{max}&k\leq n\end{cases}  \tag{32}\]The meanings of \(\omega_{max}\), \(f\), \(f_{e}\), \(k\) and \(k_{max}\) in the formula are the same as those in Equations (30) and (31). \(\omega_{1},\omega_{2}\) correspond to the minimum value of adaptive segmentation and the minimum value of linear segmentation, taking \(\phi_{max}=0.9\), \(\phi_{1}=0.3\), \(\phi_{2}=0.4\), \(k_{max}=300\).