## 4 Actuators/Sensors Position Configuration  
### Controllability/Observability Criterion for Multimodal Composite Control Improvement  
Unlike traditional linear quadratic regulator (LQR) optimal control based on full feedback, the purpose of this article is to provide a technical basis for active control of vibration, noise, and sound quality of the cab structure by optimizing the optimal placement of actuators/sensors. The position configuration of the active control device mainly involves two aspects, one is the reasonable optimization objective function design, and the other is the precise iterative optimization algorithm selection. In this section, taking the degenerate structure as an example, the objective function is solved based on the linear transformation of the structural dynamic equation and the coordinate coupling, and combined with the improved controllability/observability criterion for multimodal composite control, the objective function design of the device position configuration optimization is realized. Furthermore, the final control position of the system is obtained based on the optimization of the particle swarm optimization (PSO) algorithm. In the traditional observability/controllability control concept, the purpose of observability and controllability is to make the investigated system visible, changeable, and evaluable at the digital level, that is, to realize the competent qualitative measurement of the objective structure or system. In this section, based on the multidimensional variable and multimodal characteristic operating conditions existing in the irregular structure of  
Figure 5: Partial mode shapes: (a) 20 Hz; (b) 45 Hz; (c) 75 Hz; (d) 157 Hz. large space, the controllability/observability criterion for multimodal composite control is improved and designed. On the state equation and its coordinate decoupling formula, combined with the derivation of the system response and the observable/controllable state equation, a reasonable optimization objective function is obtained to achieve the optimal configuration. Similarly, taking flexible simplified body structure plate as an example, according to the Ritz formula, the displacement of the structure in the time series is  
\[w(p,t)=\sum\limits_{i=1}^{S}\eta_{i}(p)x_{i}(t)=\eta^{T}x \tag{12}\]  
where \(\eta_{i}(p)\) represents the Ritz function at point \(p\), corresponding to the product of the eigenfunctions of the beam in the direction of the plate boundary condition: \(x_{i}(t)\) is the generalized displacement at time \(t\). The influence of a small number of actuators/sensors on the additional mass and modal shape of the structure can be ignored. Combined with the mass matrix \(M\) and stiffness matrix \(K\), the expressions of kinetic energy and potential energy are  
\[ \begin{cases}E_{k}=\frac{1}{2}\dot{x}^{T}M\dot{x}\\ E_{v}=\frac{1}{2}\dot{x}^{T}K\dot{x}\end{cases}  \tag{13}\]  
The structure's vibration equation is derived to be expressed as following the Lagrange theory. \[M\dot{x}+Kx=Q \tag{14}\]  
where \(Q\) represents the modal vector force. Solve Equation (14) to obtain the system eigenvector matrix \(\varphi\) and eigenfrequency \(\omega_{i}\), then rewrite the above structural vibration equation to get  
\[\ddot{y}+diag(\omega_{i}^{2})y=\varphi^{T}Q \tag{15}\]  
Combining with Equation (13), it can be deduced that the state equations of the system kinetic energy and potential energy are expressed as  
\[ \begin{cases}\begin{array}{c}E_{k}=\frac{1}{2}\sum\limits_{i=1}^{\infty} \dot{y}_{i}^{2}(t)\\ E_{v}=\frac{1}{2}\sum\limits_{i=1}^{\infty}\omega_{i}^{2}y_{i}^{2}(t)\end{array} \end{cases}  \tag{16}\]  
The physical meaning of the above formula can be understood as: the total energy of the vibration system can be decomposed into the sum of the energy contributed by each order mode.