## 3 Actuators/Sensors Quantity Configuration  
To simplify the body structure of the commercial cab, the optimum number of sensors/actuators required for active vibration control of the structure is investigated through the modal energy response of the control system in the frequency domain of 0-200 Hz. The number of actuators/sensors should be selected according to reasonable evaluation indexes. With flexible simplified body structure plate as the object, the objective function \(J\) can be designed by referring to the dynamic equation and the mode equation, so that it can achieve the purpose of controlling structural vibration quickly and input the smallest control force or control energy as possible. \[J=\int\limits_{0}^{\infty}[(Z^{T})_{1\times 2m}Q_{2m\times 2m}z_{2m\times 1}+( \mu^{T})_{1\times r}R_{r\times r}\mu_{r\times 1}]dt \tag{1}\]  
\(Q_{2m\times 2m}\) and \(R_{r\times r}\) are the weighting coefficient matrices, which are adjusted according to the control objectives of the vibration system. \(\mu_{r\times 1}\) corresponds to the external input excitation force, and \(z_{2m\times 1}\) is the state variable of the controlled object. The above adjustable parameter variables can reasonably configure the active control effect and control input of the flexible plate structure. According to optimal control theory, the solution of the objective function \(J\) can be expressed as  
\[ \begin{cases}u(t)_{r\times 1}=-S_{r\times 2m}z(t)_{2m\times 1}\\ S_{r\times 2m}=\left(R^{-1}\right)_{r\times r}\left(B^{T}\right)_{r\times 2m} P_{2m\times 2m}\end{cases}  \tag{2}\]  
where matrix \(P_{2m\times 2m}\) is calculated by equation [30]. \[ \begin{split}\left(A^{T}\right)_{2m\times 2m}P_{2m\times 2m}+P_{2m \times 2m}A_{2m\times 2m}+Q_{2m\times 2m}\\ -P_{2m\times 2m}B_{2m\times r}\left(R^{-1}\right)_{r\times r}\left(B^{T} \right)_{r\times 2m}P_{2m\times 2m}=0\end{split}  \tag{3}\]  
in which  
\[ \begin{split} A_{2m\times 2m}=\begin{bmatrix}0&I\\ -\left[\omega_{i}\right]&0\end{bmatrix},\;\;B_{2m\times r}=\begin{bmatrix} 0_{m\times r}\\ b_{m\times r}\end{bmatrix},\\ b_{m\times r}=\left[m_{ii}\right]_{m\times m}^{-1}\left(\phi^{T}\right)_{m \times m}L_{n\times r},\;\;\left[m_{ii}\right]_{m\times m}=\left(\phi^{T} \right)_{m\times n}M_{n\times n}\phi_{n\times m}\end{split} \]  
where the term \(\phi\) refers to a modal matrix containing \(m\)-order eigenvalues. Modal decoupling is applied to the corresponding excitation force of active control to obtain the mode control force vector \(f_{2m\times 1}\), whose energy correlation matrix \(G_{2m\times 2m}\) can be expressed as  
\[f_{2m\times 1}=B_{2m\times r}\mu_{r\times 1} \tag{4}\]  
\[G_{2m\times 2m}=\int\limits_{0}^{\infty}f_{2m\times 1}\left(f^{T}\right)_{1 \times 2m}dt=B_{2m\times r}\left(\int\limits_{0}^{\infty}\mu_{r\times 1}\left(\mu^{T} \right)_{1\times r}dt\right)\left(B^{T}\right)_{r\times 2m} \tag{5}\]  
Substitute the solution of the objective function \(J\) to the formula above. \[G_{2m\times 2m}=H_{2m\times 2m}Z_{2m\times 2m}\left(H^{T}\right)_{2m\times 2m} \tag{6}\]  
The terms \(H_{2m\times 2m}\) and \(Z_{2m\times 2m}\) in the formula are  
\[H_{2m\times 2m}=B_{2m\times r}\left(R^{-1}\right)_{r\times r}\left(B^{T} \right)_{r\times 2m}P_{2m\times 2m} \tag{7}\]\[Z_{2m\times 2m}=\int\limits_{0}^{\infty}Z_{2m\times 1}\Big{(}Z^{T}\Big{)}_{1 \times 2m}dt \tag{8}\]  
The decoupled mode control force is an \(m\)-order vector. The eigenvalues of the energy correlation matrix \(G_{2m\times 2m}\) can be expressed as  
\[\Lambda_{G}=\{\lambda_{1},\lambda_{2},\lambda_{3},\ldots,\lambda_{m},0_{m+1},0 _{m+2},0_{m+3},\ldots,0_{2m}\} \tag{9}\]  
All non-zero eigenvalues in the eigenvector \(\Lambda_{G}\) correspond to the number of modal control forces that need to be input in the active control of structure-acoustic vibration, and its amplitude is the input energy of the actuator. In the 0-200 Hz frequency band, there are many modal eigenvalues in the body structure.