Precise Configuring of Actuators/Sensors for Active Control of Sound Quality in Cabs with Modal Vibration Energy and LA-PSO

Abstract

Active control of structural modal vibration is an effective strategy to enhance the sound quality of cabs in commercial vehicles. However, accurate determination of the positioning and quantity of modal active control sensors and actuators is crucial for cabs with intricate structures, owing to the presence of multiorder modes and their coupling. The study presented herein focuses on the cab of a commercial vehicle and contemplates the features of the irregular large-space structure of the cab. By capitalizing on the modal frequency and mode shape of the cab, utilizing the piezoelectric control principle and modal vibration energy as the assessment index, an advanced multimode composite control criterion is postulated to ascertain the configuration of primary sensors and actuators. The particle swarm optimization (PSO) objective function is constructed to accomplish the optimal position matching of the actuator/sensor, using the multimodal surface velocity vector of the vibration sensor as the core parameter. Furthermore, an improved linear adaptive particle swarm optimization (LA-PSO) technique is advanced to satisfy the requirements of optimal convergence performance and accuracy of the complex cab structure. The optimization culminates in a 9 × 9 multichannel active control scheme for determining the optimal position of the actuators/sensors. This investigation provides a technical foundation for the active control of sound quality in automotive cabs and presents an innovative method for implementing effective noise control systems in large-scale machinery and equipment.

1. Introduction

The most significant element of the sound quality of a commercial vehicle’s cab is its modal vibration, which includes the frequency characteristics of the vibration and the distribution of its vibration energy, especially for new-energy commercial vehicles, where there is no engine to act as the primary sound source and the interior noise or sound quality is mostly derived from the structural vibration of the body. Conventional acoustic and vibration control primarily emphasizes passive control, including sound absorption/vibration isolation design, vibration damping/vibration isolation design, and the prudent use of impedance mufflers [1,2,3]. The passive control scheme primarily has a good suppression effect on medium- and high-frequency vibration and noise, but the control effect is less suitable for structural modal vibration dominated by low- and medium-frequency vibration. It is also difficult for the passive control to improve the vibrating acoustic radiation quality and acoustic comfort requirements of the commercial vehicle cab [4,5,6]. Consequently, the active acoustic vibration control method is used to add a new vibration signal or sound signal to the original structural system, which is the secondary force source or secondary sound source, and combined with the real-time response algorithm control so that the original vibration signal and the secondary vibration signal of the evaluation monitoring point cancel and interfere with each other to achieve the aim of suppressing structural vibration and improving the quality of vibration and acoustic radiation in the vehicle [7,8].

Among the most important pieces of hardware for the active control of structural noise and vibration are sensors and actuators. The actuators provide sound or vibration signals as secondary excitation loading terminals, and the sensors receive them as signal monitoring terminals [9]. The quantity and position of the secondary force or sound source of the actuator can be changed by the adjustable control algorithm to achieve active control of vibration and noise for the acoustic radiation brought on by structural modal vibration [10]. A key element in determining the efficiency of the system’s sound quality management is the proper matching of the number and position of actuators/sensors, particularly for complicated structures with significant spatial irregularities such as commercial vehicle cabins. The inherent performance of the system can become uncontrollable due to improper quantity allocation and site selection, which will have the opposite impact on acoustic and vibration suppression [11,12].

The controlled structure and control system have an impact on the number of active control devices, and it is easier to optimize objects with a small acoustic area and a basic structure. However, the control effect has not worked out well yet for irregular acoustic domain situations with numerous noise sources and complex structures [13,14]. Currently, research on actuator/sensor count configurations is still scarce. On the other hand, scholars have thoroughly studied the best sensor/actuator configuration, including the theoretical optimization of the objective function and criteria, sane selection, and the creation of multiobjective optimization algorithms [15,16,17].

Scholars have been very interested in enhancing the efficiency, real time, and adaptability of the control algorithm since it directly influences the effect of active control of sound quality. Adaptively tracking noise signals, the ANE algorithm enhances sound quality in accordance with various requirements [18,19]. However, the FELMS algorithm’s control system is easier than the determination of the gain coefficient, which is more involved and harder to implement in hardware [20,21,22]. Instead of utilizing adaptive algorithms to change filters, the ANN technique uses neural networks to do so. It can more effectively handle non-linear difficulties and is still in the research stage [23,24]. The majority of the aforementioned algorithms only take into account single-channel scenarios; therefore, more research into multichannel control algorithms is necessary as control requirements increase.

Although there has been some progress in the current research on active vibration control and some reports on the configuration method of sensors/actuators, the majority of the current research explores straightforward structures such as flat plates and cylinder shells with the primary goal of noise reduction [25,26,27,28,29]. The setup of active control sensors/actuators for sound quality is highly complex for the vibration mode and active control of the sound quality of irregular and complex structures in large spaces, such as the cab of commercial vehicles, due to the combination and allocation of modal vibration frequency and energy. Hence, the existing body of research in this area is comparatively sparse. This paper proposes an improved observability/controllability criterion for multimodal composite control using the cab body structure of commercial vehicles as an example, taking into account the characteristics of irregular large-space structures, based on structural vibration modal characteristics and the piezoelectric control principle. It also establishes a particle swarm optimization algorithm with a multimodal surface velocity vector as the core parameter to extract the position of the cab. An enhanced linear adaptive particle swarm optimization technique is suggested in order to satisfy the convergence and accuracy requirements of complicated structures.

2. Structural Characteristics of Commercial Vehicle Cabs

The heavy-duty commercial vehicle model studied in this paper is a purely electric semi-trailer, manufactured in China. Complete vehicle outline size 7390 × 2525 × 3715 (mm) with a total mass of 25,000 (kg) is the main new-energy tractor on the market. The cab body of the commercial vehicle involved in this study has a irregular large-space structure, which has specified complexity in size, structure, number of connected parts, etc. This complexity necessitates targeted treatment in the design of active vibration control. The complex features of irregular large-space structures are mainly reflected in two aspects—coupling between global modes and local modes and the spatial complexity of quantity and location allocation.

The cab assembly is comprised of hundreds of parts. The simplified body structure also includes the floor, the front inner surface, the ceiling, and the side inner surface. These parts are connected by welding or bolts to form a body assembly, as shown in Figure 1 and Figure 2. Simultaneously, a single structure, such as the complex structure of the front wall itself, has many large mounting holes or grooves. For the mode response of common engineering structures, there is a local mode response only in a local area of the system assembly. For the whole structure, when the contribution of the local mode to the overall structural response tends to zero, then considering the active vibration control of the whole structure, the influence of the local mode can be ignored. Therefore, irregular large-space structures present features of small-interval multimodes in the process of modal analysis. Global mode and local mode coupling exists and needs to be distinguished from structural response contribution and energy proportion.

On the spatial complexity of quantity allocation and location allocation in irregular large-space structures, generally, active control device configurations are designed for a simple plane or surface, which can be optimized with two dimensions of variables. As a three-dimensional irregular space structure, the body structure has a non-negligible vibration response on multiplanes, which requires optimizing configuration with at least three-dimensional position variables.

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.

$$J=\underset{0}{\overset{\infty }{\int }}[{(Z^T)}_{1\times 2m}{Q}_{2m\times 2m}{z}_{2m\times 1}+{\left({\mu }^T\right)}_{1\times r}{R}_{r\times r}{\mu }_{r\times 1}]dt$$

(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

$$\{\begin{array}{c}u{\left(t\right)}_{r\times 1}=-S_{r\times 2m}z{\left(t\right)}_{2m\times 1}\\ S_{r\times 2m}={\left(R^{-1}\right)}_{r\times r}{\left(B^T\right)}_{r\times 2m}{P}_{2m\times 2m}\end{array}$$

(2)

where matrix $P_{2m\times 2m}$ is calculated by equation [30].

$$\begin{gathered} {\left(A^T\right)}_{2m\times 2m}{P}_{2m\times 2m}+P_{2m\times 2m}{A}_{2m\times 2m}+Q_{2m\times 2m} \\ -P_{2m\times 2m}{B}_{2m\times r}{\left(R^{-1}\right)}_{r\times r}{\left(B^T\right)}_{r\times 2m}{P}_{2m\times 2m}=0 \end{gathered}$$

(3)

in which

$$\begin{gathered} A_{2m\times 2m}=\left[\begin{array}{cc}0& I\\ -\left[{\omega }_i\right]& 0\end{array}\right], B_{2m\times r}=\left[\begin{array}{c}{0}_{m\times r}\\ b_{m\times r}\end{array}\right], \\ {b}_{m\times r}={\left[m_{ii}\right]}_{m\times m}^{-1}{\left({\varphi }^{\mathrm{T}}\right)}_{m\times m}{L}_{n\times r}, {\left[m_{ii}\right]}_{m\times m}={\left({\varphi }^T\right)}_{m\times n}{M}_{n\times n}{\varphi }_{n\times m} \end{gathered}$$

where the term $\varphi$ 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

$$f_{2m\times 1}=B_{2m\times r}{\mu }_{r\times 1}$$

(4)

$$G_{2m\times 2m}=\underset{0}{\overset{\infty }{{\displaystyle \int }}}{f}_{2m\times 1}{\left(f^T\right)}_{1\times 2m}dt=B_{2m\times r}\left(\underset{0}{\overset{\infty }{{\displaystyle \int }}}{\mu }_{r\times 1}{\left({\mu }^T\right)}_{1\times r}dt\right){\left(B^T\right)}_{r\times 2m}$$

(5)

Substitute the solution of the objective function J to the formula above.

$$G_{2m\times 2m}=H_{2m\times 2m}{Z}_{2m\times 2m}{\left(H^T\right)}_{2m\times 2m}$$

(6)

The terms $H_{2m\times 2m}$ and $Z_{2m\times 2m}$ in the formula are

$$H_{2m\times 2m}=B_{2m\times r}{\left(R^{-1}\right)}_{r\times r}{\left(B^T\right)}_{r\times 2m}{P}_{2m\times 2m}$$

(7)

$$Z_{2m\times 2m}=\underset{0}{\overset{\infty }{{\displaystyle \int }}}{Z}_{2m\times 1}{\left(Z^T\right)}_{1\times 2m}dt$$

(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

$${\mathsf{\Lambda }}_G=\left\{{\lambda }_1,{\lambda }_2,{\lambda }_3,\dots ,{\lambda }_m,0_{m+1},0_{m+2},0_{m+3},\dots ,0_{2m}\right\}$$

(9)

All non-zero eigenvalues in the eigenvector ${\mathsf{\Lambda }}_G$ 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. Under the same control excitation force, the modal frequency point with a larger eigenvalue can provide more control energy. Similarly, the modal frequency where the larger eigenvalue is located is the sensitive response position of the system vibration.

To sum up, according to the eigenvalues in the energy correlation matrix $G_{2m\times 2m}$, the n-order modal frequency that satisfies a certain energy ratio of $\delta$ can be selected to realize the active control of acoustic and vibration of the structure. Considering different vibration systems, the value of $\delta$ is generally 90% or higher as follows

$$\frac{{{\displaystyle \sum }}_{i=1}^n{\lambda }_i}{{{\displaystyle \sum }}_{i=1}^m{\lambda }_i}>\delta$$

(10)

The ranking calculation obtains the minimum value of n when the above equation is satisfied. Consequently, it can be obtained that under certain operating conditions, the proportion of system vibration response energy exceeds $\delta$ minimum number of modes. This quantity is the minimum number of control modes to achieve efficient active sound and vibration suppression for a vibration system.

$$\frac{{{\displaystyle \sum }}_{i=1}^n{E}_i}{{{\displaystyle \sum }}_{i=1}^m{E}_i}>\delta$$

(11)

The size of ${\lambda }_i$ in the above formula corresponds to the proportion of the energy contribution of the i-th modal force of the structure to the vibration response of the system. For the multidegree of freedom and irregular large-space complex structures involved in this paper, ${\lambda }_i$ is not easy to obtain by computational simulation. Based on the existing conclusions, applying equal-intensity excitation in a certain frequency range to obtain the vibration energy of the vibration system at each modal frequency, and adopting the same principle to achieve sorting and statistics, as shown in Equation (11). $E_i$ corresponds to the i-th vibration energy from large to small at each modal position, and its value is the sum of the kinetic energy and elastic strain energy of the structure at the modal position.

Using this evaluation index, the reasonable configuration of the number of sensors/actuators in complex model control can be realized, and the optimal control order and corresponding modal frequency of the investigated structure under active acoustic and vibration control can be determined.

Base on the modal shape analysis, in accordance with the above structural model and excitation loading conditions, the frequency domain simulation analysis is carried, and the modal energy value of each frequency point is obtained through the acceleration surface integral. The results are shown in Figure 3, the modal energy is composed of elastic energy and kinetic energy. The above energy data are statistically transformed and filtered to obtain the modal energy distribution of each order as shown in Figure 4.

From Figure 4, in the 0–200 Hz frequency band, after applying a specific excitation, the various order modes of the body structure can be excited to a certain extent, among which the modal frequency positions correspond to 20 Hz, 45 Hz, 75 Hz and 157 Hz. Resonance energy is generated, and its mode shape is shown in Figure 4, which is close to the previous modal shape analysis results. Simultaneously, combined with Equation (11), the modal energy results are sorted and processed, and the energy proportions of multiorder modes are obtained as shown in

Table 1. Energy ratio of each order mode.

Eigenfrequency	Corresponding Energy	Energy Ratio
157	40.563	65.9580%
87	10.4412	16.9781%
45	3.58916	5.8362%
20	1.65848	2.6968%
116	1.55857	2.5343%
143	1.06384	1.7299%
172	0.96976	1.5769%
15	0.60876	0.9899%
87	0.555772	0.9037%
191	0.48967	0.7962%

.

On the data in the above table, the sum of the modal energy ratios at the frequency positions of 157 Hz, 75 Hz, 45 Hz and 20 Hz reaches 91.4691%, which meets the control criteria for selecting the objective function and criteria based on the quantity. Vibration pickup and actuation are realized based on the four frequency positions, and the noise and vibration of the body structure can be efficiently controlled. As shown in Figure 5, considering the modal shapes of each order, there are six peak areas and three peak areas at the ceiling and rear surrounds at 45 Hz and 75 Hz, respectively, and the distribution is similar to the modal shapes at 157 Hz and 20 Hz. Therefore, nine actuators/sensors are selected to achieve active acoustic and vibration suppression of the body structure by controlling the above-mentioned four vibration modes.

4. Actuators/Sensors Position Configuration

4.1. 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 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

$$w\left(p,t\right)={\displaystyle \sum }_{i=1}^S{\eta }_i\left(p\right)x_i\left(t\right)={\eta }^{T}x$$

(12)

where ${\eta }_i\left(p\right)$ 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: $x_i\left(t\right)$ 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

$$\{\begin{array}{c}{E}_k=\frac{1}{2}{\dot{x}}^{T}M\dot{x}\\ E_v=\frac{1}{2}{\dot{x}}^{T}K\dot{x}\end{array}$$

(13)

The structure’s vibration equation is derived to be expressed as following the Lagrange theory.

$$M\dot{x}+Kx=Q$$

(14)

where $Q$ represents the modal vector force. Solve Equation (14) to obtain the system eigenvector matrix $\phi$ and eigenfrequency ${\omega }_i$, then rewrite the above structural vibration equation to get

$$\ddot{y}+diag\left({\omega }_i^2\right)y={\phi }^{T}Q$$

(15)

Combining with Equation (13), it can be deduced that the state equations of the system kinetic energy and potential energy are expressed as

$$\{\begin{array}{c}{E}_k=\frac{1}{2}{\displaystyle {\displaystyle \sum }_{i=1}^{\infty }}{\dot{y}}_i^2\left(t\right)\\ E_v=\frac{1}{2}{\displaystyle {\displaystyle \sum }_{i=1}^{\infty }}{\omega }_i^2{y}_i^2\left(t\right)\end{array}$$

(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.

Assume that the first m-order modal matrix z of the system is as follows, and at the same time, its modal expression in the form of state space is

$$z={\left[{\dot{y}}_1,{\omega }_1{y}_1,{\dot{y}}_2,{\omega }_2{y}_2,\dots ,{\dot{y}}_m,{\omega }_m{y}_m\right]}^T$$

(17)

$${\dot{z}}_{2m\times 1}=A_{2m\times 2m}{z}_{2m\times 1}+B_{2m\times r}{\mu }_{r\times 1}$$

(18)

where

$$z_{2m\times 1}=\left\{\begin{array}{c}y\\ \dot{y}\end{array}\right\}$$

(19)

Based on the conversion relationship between the electrical signal and the force signal of the piezoelectric effect of the material, combined with Equation (18), the state space expression of the flexible plate can be obtained.

$$\{\begin{array}{c}\dot{z}=Az+B\mu \\ v=Cz\end{array}$$

(20)

In the formula, A corresponds to the inherent characteristics of the system itself such as modal frequency and damping ratio, B corresponds to the form and position of the actuator, and C corresponds to the form and position of the sensor. As shown in the following Equation (21), A is represented as a DIAG diagonal matrix $diag(A_i)$, b_i represents the matrix ${\phi }^{T}Q$ in the i-th row vector, and its length n corresponds to the number of actuators. $c_{ij}$ refers to the characteristic constant of the j-th sensor of the i-th order modal displacement of the flexible plate.

$$\begin{gathered} A_i=\left[\begin{array}{cc}-2{\xi }_i{\omega }_i& -{\omega }_i\\ {\omega }_i& 0\end{array}\right],B={\left(b_1,0,b_2,0,\dots ,b_m,0\right)}^T, \\ C=\left[\begin{array}{ccccccc}0& c_{11}& 0& c_{12}& \dots & 0& c_{1n}\\ 0& c_{21}& 0& c_{22}& \dots & 0& c_{2n}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& c_{m1}& 0& c_{m2}& \dots & 0& c_{mn}\end{array}\right] \end{gathered}$$

(21)

In the instantaneous excitation condition, the vibration response of each sensor position at time $T_0$ can be expressed as

$$J={{\displaystyle \int }}_0^{T_0}{v}^T\left(t\right)v\left(t\right)dt$$

(22)

Referring to the [31], the maximum energy in the time domain can be written as

$$J={\left(e^{A{t}_f}{z}_{0+}-z_{t_f}\right)}^T{W}_o{\left(t_f\right)}^{-1}\left(e^{A{t}_f}{z}_{0+}-z_{t_f}\right)$$

(23)

Term $W_o$ is the observability Gramian matrix, and when the time series is positive infinity, $W_o$ results

$$W_o={{\displaystyle \int }}_0^{\infty }{e}^{A^T\tau }{C}^{T}C{e}^{A\tau }d\tau$$

(24)

Minimizing the W_o norm yields the sensor position that maximizes the vibration response $J$ and $W_o$ satisfies the Lyapunov equation.

$$A{W}_o+W_o{A}^T=-M^{T}M$$

(25)

Solve then can get

$$W_o=diag\left(\frac{c_n}{4{\xi }_i{\omega }_i},\frac{c_n}{4{\xi }_i{\omega }_i}\right)$$

(26)

The reasoning analysis process based on the observability Gramian matrix shows that the matrix describes the vibration response of each vibration modal frequency point at the sensor configuration position. For systems or structures with multiorder dispersed modal frequencies, the sensor position is different, and the matrix data change. When the diagonal elements of the matrix are the largest, the sensor combination can obtain the largest system response.

The position selection of the actuator is similarly to that of the sensor. It tries to maximize the input energy of the system, and derives the controllability of the system in turn, the Gramian matrix and its solution in the form of a diagonal matrix.

$$W_c={{\displaystyle \int }}_0^{\infty }{e}^{A^T\tau }B{B}^T{e}^{A\tau }d\tau$$

(27)

$$W_c=diag\left(\frac{{\beta }_n}{4{\xi }_i{\omega }_i},\frac{{\beta }_n}{4{\xi }_i{\omega }_i}\right)$$

(28)

${\xi }_i$ represented as the damping rate of the i-th mode, ${\omega }_i$ represents the i-th mode frequency, $c_n$ is the vibration response of the n-th sensor, and ${\beta }_n$ is the vibration response of the n-th actuator.

In the same way, for systems or structures oriented to multiorder dispersed modal frequencies, the response of the system represented by the matrix is different depending on the position of the actuator. 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

$$Obj=trace\left(E\right)\times \sqrt[2n]{det\left(E\right)}/\sigma \left(E\right)$$

(29)

$E=diag\left(p_i,p_i\right) i=1,2\dots n$, 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, $trace\left(E\right)$ corresponds to the trace of matrix $E$, $\sqrt[2n]{\det \left(E\right)}$ refers to the geometric mean of energy, and $\sigma \left(E\right)$ 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), $\sqrt[2n]{\det \left(E\right)}$, $1/\mathsf{\sigma }\left(E\right)$ in the objective function is consistent with the optimal solution, and the energy summation process of $trace\left(E\right)$ ignores the more the effect of small components, $\sqrt[2n]{\det \left(E\right)}$ combined with the geometric mean solution of the product, involves all components while maximizing the influence of large components, and $1/\mathsf{\sigma }\left(E\right)$ 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.

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.

4.2. 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.

$$V_n^{k+1}=\varphi V_n^k+c_1\left(P_n^k-X_n^k\right)+c_2\left(P_g^k-X_g^k\right)$$

(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; $\varphi$ 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 $\varphi$ 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

$$\varphi =\{\begin{array}{c}{\varphi }_{max}-\left({\varphi }_{max}-{\varphi }_1\right)f_e/f k\ge n\\ {\varphi }_{max}-\left({\varphi }_{max}-{\varphi }_2\right)k/k_{max} k\le n\end{array}$$

(32)

The meanings of ${\omega }_{max}, f, f_{\mathrm{e}}, k$ 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 ${\varphi }_{max}=0.9, {\varphi }_1=0.3, {\varphi }_2=0.4, k_{max}=300$. The linear adaptive PSO algorithm adopts the form of piecewise function, that is, when the number of iterations is less than n times, the value of $\omega$ decreases linearly. When the number of iterations is greater than N, the value of $\varphi$ is adjusted in real time with the fitness correlation. The function segment points are matched and adjusted according to the engineering complexity, which is taken as n = 50.

The cycle termination condition of population evolution is designed as Equation (33), that is, the standard deviation of the optimal fitness that satisfies more than 90 iterations and 30 consecutive iterations is less than 0.001.

$$k\ge 90\&std\left(y\left(k-30:k\right)\right)<0.001$$

(33)

The modal concentration plane in the search space is divided into nine regions, as seen in Figure 8. On the one hand, it prevents the concentration of ideal values, and on the other, it increases the algorithm’s accuracy and search effectiveness.

Figure 8 shows the plan division of the position selection points, where Figure 8a–c, represent the division of the rear inner surface at 20 Hz, 45 Hz and 75 Hz, while Figure 8d–f, represent the division of the ceiling at 45 Hz, 75 Hz and 157 Hz, respectively. Using the improved LA-PSO algorithm, the active control device’s optimal position matching is performed, and the optimal coordinate combination as shown in Figure 9 is obtained, meaning that the objective function value is maximized when the 9 actuators/sensors are arranged at the corresponding coordinates. The system achieves the best active control result.

Figure 9 shows the final actuator/sensor coordinates and corresponding modal shapes, where Figure 9a–c represent the division of the rear inner surface at 20 Hz, 45 Hz and 75 Hz, while Figure 9d–f represent the division of the ceiling at 45 Hz, 75 Hz and 157 Hz, respectively. The diagram illustrates the coincidence of the coordinate points of the device and the vibration modes of each order, in which the position of the yellow aperture is the position of the active control device, and its spatial geometric coordinate combination is shown in

Table 2. Joint simulation results—optimal coordinate combination of devices.

Number	X/m	Y/m	Z/m
1	−1.4681	0.0000	0.5958
2	−0.4824	0.0000	0.5959
3	−0.9750	0.0000	0.7254
4	−0.5401	−0.4207	0.0000
5	−0.9972	−0.2644	0.0000
6	−1.6115	−0.2419	0.0000
7	−0.5012	−0.7888	0.0000
8	−1.0770	−0.7489	0.0000
9	−1.6836	−0.6757	0.0000

. The analysis shows that the position of the device is close to the peak points of the main mode shapes of the body roof and rear, and there is a strong correlation between the combination distribution and the four mode shapes. The rationality of the selection of active control devices for multiorder modal compound control is verified from the point of view of geometry.

In summary, through the improved LA-PSO adaptive optimization algorithm, combined with co-simulation, the actuator/sensor position combination with the highest efficiency for active acoustic and vibration suppression of the body structure is obtained, and a 9 × 9 multichannel is constructed. The multichannel control system is the premise of the establishment of the active control system.

5. Conclusions

Listed below are this article’s main innovations:

• The cab of the commercial vehicles has a three-dimensional irregular space structure with multiorder modes, coupled overall and local modes, and a non-negligible vibration response in multiple planes. Three-dimensional position variables are necessary for the sensor/actuator to be configured in the best way possible.
• The structural dynamic equation and the objective function of the energy correlation matrix are calculated and analyzed in order to establish an evaluation index focused on modal vibration energy, which is then used to estimate the number of sound quality active control sensors/actuators. Through frequency domain energy analysis, four primary vibration modes that require control were identified and determined by the ranking of evaluation indicators. Nine appropriate actuators/sensors were selected for the four primary mode shapes in order to achieve effective control of multimodal vibration in the cab.
• The characteristics of multimodal composite control of the cabin are combined with the improved observability/controllability criteria to build a particle swarm optimization technique using multimodal surface velocity vector equivalent displacement as the core parameter. The ideal arrangement of 9 $\times$ 9 multichannel commercial vehicle cab sensors/actuators was accomplished with the help of the LA-PSO adaptive optimization technique of co-simulation.