## 4 Actuators/Sensors Position Configuration
### Controllability/Observability Criterion for Multimodal Composite Control Improvement
Under the same excitation state, when the diagonal elements of the matrix are the largest, the total vibration response of the system reaches the maximum value. The engineering significance of the controllability/observability criterion for multimodal composite control can be understood as: in a certain frequency band, apply equal-amplitude excitation to the structure or system at a fixed position to obtain the sensor position that can detect the maximum vibration response; or make the actuator apply a constant value excitation force, change the actuator position, and obtain the actuator position combination that shows the peak vibration response of the structure or system. Its core point lies in the compound vector control of multimodal mode shapes. The schematic diagram for easy understanding is shown in Figure 6.  
On the above derivation, the diagonal elements of the Gramian matrix are equivalently replaced by the system response correlations, and the configuration objective function for multimodal composite control is designed as follows  
\[Obj=trace(E)\times\sqrt[2n]{det(E)}/\sigma(E) \tag{29}\]  
\(E=diag(p_{i},p_{i})\ i=1,2\ldots n\), at each modal frequency position, for the respective configuration of the actuator/sensor, \(p_{i}\) represents the single actuation/vibration response of each sensor position, respectively, the total vibrational energy of the system excited by the device. Meanwhile, \(n\) represents the number of actuators/sensors, \(trace(E)\) corresponds to the trace of matrix \(E\), \(\sqrt[2n]{det(E)}\) refers to the geometric mean of energy, and \(\sigma(E)\) is the energy variance. The optimal position of the actuator/sensor can be obtained by obtaining the maximum value of the objective function \(Obj\) through reasonable iterative optimization. The selection of the optimal position is based on the comprehensive influence of the multiorder modes. The variation trend of the products \(trace(E)\), \(\sqrt[2n]{det(E)}\), \(1/\sigma(E)\) in the objective function is consistent with the optimal solution, and the energy summation process of \(trace(E)\) ignores the more the effect of small components, \(\sqrt[2n]{det(E)}\) combined with the geometric mean solution of the product, involves all components while maximizing the influence of large components, and \(1/\sigma(E)\) avoids the appearance of extreme values. Simultaneously, the synchronization of the eigenvalues of the Gramian matrix derived from the observability and controllability enables the actuators/sensors to achieve synchronous optimal configuration, this method considers the vector representation of the vibration velocity of multiorder modes at the same position and in different directions, and adopts the method of applying equal-amplitude excitation to the structure or system at a fixed position to obtain the combination of sensor extreme values, which improves the operability of the method and obtains the response under the optimal precise configuration. Figure 6: Schematic diagram of the optimal configuration objective function for multimodality. To sum up, the objective function designed by this configuration criterion comprehensively considers the energy components of multiorder modes, and effectively realizes the optimal configuration of actuators/sensors with irregular structures in large spaces.