## 3 Design of ACM Controller
### LQR Optimal Control  
The control object of the LQR algorithm is a linear system given in the form of state space. Its objective function is a quadratic function with input object state and control input [26]. The advantage of LQR algorithm lies in the fact that according to the variables to be optimized in the actual model, while taking into account the system performance and control energy, the optimal control law of stabilized linear feedback is obtained, which is easy to form a closed-loop optimal control, so that the comprehensive performance of the system is optimized. The linear mathematical model of ACM system of Section 2 is converted into the form of state equation and output equation. The engine vibration \(\mathbf{u}_{1}\), control input variable \(\mathbf{u}_{2}\), state variable \(\mathbf{x}\) and output variable \(\mathbf{y}\) are as below:  
\[\left\{ \begin{array}{l}\mathbf{u}_{1}=f_{1},\ \ \ \mathbf{u}_{2}=i\\ \mathbf{x}=\left[y_{1}\ \hat{y}_{1}\ y_{3}\ \hat{y}_{3}\right]^{\mathrm{T}}\\ \mathbf{y}=\hat{f}_{5}\end{array} \right. \tag{24}\]  
In addition, we can get the state equation and the output equation \(\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}_{2}+\mathbf{E}\mathbf{u}_{1}\), \(\mathbf{y}=\mathbf{C}\mathbf{x}+\mathbf{D}\mathbf{u}_{2}\). Details as follows:\[f_{5}=\left[ \begin{array}{cc}k_{1}+A_{1}K_{1}(A_{1}-A_{3})&c_{1}&k_{3}-A_{3}K_{1 }(A_{1}-A_{3})&c_{3}\end{array} \right]\left[ \begin{array}{c}y_{1}\\ \dot{y}_{1}\\ y_{3}\\ \end{array} \right]+[-k_{\text{M}}]i \tag{26}\]  
The goal of active control is to minimize the force transmitted to the chassis while ensuring that the engine side displacement and active force are not too great. Therefore, the objective function can be expressed as:  
\[J=\int_{0}^{\infty}\left(qf_{5}^{2}+ri^{2}\right)\text{d}t \tag{27}\]  
where \(q\) and \(r\) are weighting coefficient, and their values are determined according to different requirements of the ACM performance index. When we set the state variable weighting matrix \(\mathbf{Q}=[q]\), the control input variable weighting matrix \(\mathbf{R}=[r]\), the objective function becomes  
\[J=\int_{0}^{\infty}\left(\mathbf{y}^{\text{T}}\mathbf{Q}\mathbf{y}+\mathbf{u}_{2}^{\text{T}} \mathbf{R}\mathbf{u}_{2}\right)\text{d}t \tag{28}\]  
Furthermore, \(\mathbf{y}=\mathbf{C}\mathbf{x}+\mathbf{D}\mathbf{u}_{2}\), set \(\mathbf{Q}_{\text{n}}=\mathbf{C}^{\text{T}}\mathbf{Q}\mathbf{C}\), \(\mathbf{N}_{\text{n}}=\mathbf{C}^{\text{T}}\mathbf{Q}\mathbf{D}\), \(\mathbf{R}_{\text{n}}=\mathbf{R}+\mathbf{D}^{\text{T}}\mathbf{Q}\mathbf{D}\), the objective function can be expressed as  
\[J=\int_{0}^{\infty}\left(\mathbf{x}^{\text{T}}\mathbf{Q}_{\text{n}}\mathbf{x}+2\mathbf{x}^{ \text{T}}\mathbf{N}_{\text{n}}\mathbf{u}+\mathbf{u}^{\text{T}}\mathbf{R}_{\text{n}}\mathbf{u} \right)\text{d}t \tag{29}\]  
Set the optimal control amount \(\mathbf{u}_{2}^{*}(t)=\mathbf{K}\mathbf{x}\), the feedback matrix can be expressed as, \(\mathbf{K}=-\mathbf{R}_{\text{n}}^{-1}(\mathbf{N}_{\text{n}}^{\text{T}}+\mathbf{B}^{\text{T}} \mathbf{P})\), \(\mathbf{P}\) is the solution of Riccati equation \(\mathbf{A}^{\text{T}}\mathbf{P}+\mathbf{P}\mathbf{A}-(\mathbf{P}\mathbf{B}+\mathbf{N}_{\text{n}})\mathbf{R}_{ \text{n}}^{-1}(\mathbf{B}^{\text{T}}\mathbf{P}+\mathbf{N}_{\text{n}}^{\text{T}})+\mathbf{Q}_{ \text{n}}=0\). When solving the Riccati equation, there is usually more than one solution that fits the equation itself, while there is only one symmetric positive definite solution which makes the system asymptotically stable and satisfies the condition of constructing the optimal state feedback gain matrix. The LQR optimal control function is only related to \(\mathbf{A}\), \(\mathbf{B}\), \(\mathbf{Q}\), \(\mathbf{R}\), where \(\mathbf{A}\), \(\mathbf{B}\) are known, the performance of the system depends on the values of the weighting matrices \(\mathbf{Q}\), \(\mathbf{R}\). Set \(\mathbf{R}\) to be the identity matrix. After repeated debugging, \(\mathbf{Q}_{\text{n}}\) is a second-order unit matrix in this paper.