## 4 The Control Strategy of the Semi-Active Energy Regenerative Suspension with an EHA  
The selection principle of switching the control critical point to control the force of the semi-active energy regenerative suspension with an EHA is expressed as follows:  
When \(\dot{x}_{2}\cdot(\dot{x}_{2}-\dot{x}_{1})>0\), the direction of the force applied to the spring mass by the actuator is opposite to the movement direction of the spring mass, but it has the same direction as the active control force, so it can change the external load resistance of the actuator to make the actual control output force be equal to the theoretical active control output force. Additionally, the theoretical active control output force is determined by the optimal control strategy. The optimal control objective of the semi-active energy regenerative suspension with an EHA is to make the car obtain higher comfort and handling stability, and reflecting in the amount of actual control is to reduce the acceleration of the sprung mass and tire dynamic load as much as possible, limit the range of the suspension's dynamic deflection, and reduce the possibility of suspension impact block. At the same time, it should not consume too much energy [20]. Based on the above considerations, the performance index function of a semi-active suspension output regulator can be written as follows:  
\[J=\int_{0}^{\infty}\Big{[}q_{1}\dot{x}_{2}^{2}+q_{2}(x_{2}-x_{1})^{2}+q_{3}(x _{1}-z)^{2}+rF^{2}\Big{]}\mathrm{d}t \tag{23}\]  
where \(q_{1}\) is the weighting coefficient of acceleration, \(q_{2}\) is the weighting coefficient of the suspension's dynamic deflection, \(q_{3}\) is the weighting coefficient of tire dynamic deformation, and \(r\) is the weighting coefficient of energy consumption. The above optimization index is expressed in matrix as:  
\[J=\int_{0}^{\infty}\Big{[}Y^{T}qY+F^{T}rF\Big{]}\mathrm{d}t \tag{24}\]  
where \(q\) is expressed as:  
\[q=\left[ \begin{array}{ccc}q_{1}&0&0\\ 0&q_{2}&0\\ 0&0&q_{3}\end{array} \right].\]  
Generally, the output regulator problem is converted to a state regulator problem. Taking the output equation \(Y=CX+DU\) into Equation (24), the quadratic performance index is given by:  
\[J=\int_{0}^{\infty}\Big{[}X^{T}QX+2X^{T}NF+F^{T}RF\Big{]}\mathrm{d}t \tag{25}\]  
where \(Q\) are the positive semidefinite symmetric weighting matrices of the state variables, \(N\) is the weighted matrix of two kinds of variables, and \(R\) is the positive definite symmetric weighting matrix of the control variables. Among them, \(Q=C^{T}qC,N=C^{T}qD,R=r+D^{T}qD\). The optimal control force \(F\), which serves for performance index \(J\), to be minimized is existent and unique, and it can be expressed as:  
\[F=-KX=-(B^{T}P+N^{T})X \tag{26}\]  
where \(P\) is a symmetric positive definite solution of the Riccati matrix equation. It meets the following equation:\[PA+A^{T}P-(PB+N)R^{-1}(B^{T}P+N^{T})+Q=0. \tag{27}\]  
The selection of the weighting factors in the performance index of the optimal control depends on practical experience. After repeated trials, the following options can be made: \(q_{1}=1.2\times 10^{5}\), \(q_{2}=1.65\times 10^{8}\), \(q_{3}=9.5\times 10^{9}\), \(r=1\). In order to obtain the feedback gain matrix \(K\), using the linear quadratic regulator function provided by Matlab software, the basic format is expressed as:  
\[(K,S,E)=LQR(A,B,Q,R,N) \tag{28}\]  
where \(S\) is the solution of the Riccati equation, and \(E\) is the system eigenvalue. When \(\dot{x}_{2}\cdot(\dot{x}_{2}-\dot{x}_{1})<0\), the direction of the force applied to the spring mass by the actuator has the same direction as the movement direction of the spring mass, but it is opposite to that of the active control force. In order to minimize this difference, at this moment the damping force, which is theoretically required to output from the suspension, is zero, but the actual output force is the inherent viscous damping force of the system, and its true value is still determined by Formula (17). At the same time, the motor load resistance is zero.