## 3 MFAC Method Based on Genetic Algorithm  
Because the vertical and pitch accelerations of the vehicle body, along with the vertical acceleration of the seat \((z_{s}^{\prime\prime},q_{s}^{\prime\prime},z_{a}^{\prime\prime})\), jointly affect the driver's ride comfort [36], we define \(y=(z_{s}^{\prime\prime},q_{s}^{\prime\prime},z_{a}^{\prime\prime})^{T}\) and \(y=(u_{f},u_{r},u_{a})^{T}\) as the output and control vectors of the active HVS system coupled with NESI. Taking \(y_{d}=(z_{sd}^{\prime\prime},q_{sd}^{\prime\prime},z_{ad}^{\prime\prime})^{T}=(0,0, 0)^{T}\) as the control expectation, an MFAC method based on the genetic algorithm is designed and the block diagram is shown in Figure 2.  
Lots of of I/O data are generated when vehicles drive on different pavement surfaces; based on these I/O data, the active HVS system coupled with NESI can be represented as a general discrete-time nonlinear system [37], which is given as  
\[y(k+1)=f\big{(}y(k),\cdots y\big{(}k-n_{y}\big{)},u(k),\cdots u(k-n_{u})\big{)} \tag{11}\]  
where \(k\) is the sampling moment; \(u(k)\) and \(y(k)\) are the inputs and outputs of the system at moment \(k\); \(f(\cdots)=[f_{1}(\cdots),\cdots,f_{m}(\cdots)]^{T}\in\prod\limits_{n_{y}+n_{u} +2}R^{m}\mapsto R^{m}\) denotes the unknown nonlinear vector function. \(H_{L_{y},L_{u}}(k)=\big{[}y(k),\cdots y\big{(}k-L_{y}+1\big{)},u(k),\cdots u( k-L_{u}+1)\big{]}^{T}\) is defined as all input data of time window \([k-L_{u}+1,k]\) associated with all output data of time window \([k-L_{y}+1,k]\)[38]. The following assumptions are made for a reasonable dynamic linearization of the system. **Assumption 1**. : _There exist continuous partial derivatives of \(f(\cdots)\) related to \(u(k)\)._  
**Assumption 2**. : _On the control domain, when \(|u(k)|\
eq 0\), the control system that needs to be dynamically linearized satisfies the generalized Lipschitz continuity condition, which has_  
\[|\Delta y(k+1)|\leq a\Big{|}\Delta H_{L_{y},L_{u}}(k)\Big{|} \tag{12}\]  
_where \(a>0\) is a constant; \(\Delta y(k)=y(k)-y(k-1)\); \(\Delta H_{L_{y},L_{u}}(k)=H_{L_{y},L_{u}}(k)-H_{L_{y},L_{u}}(k-1)\)._  
**Lemma 1**. : _For system (11) satisfying Assumptions 1 and 2, when \(0\leq L_{y}\leq n_{y}\), \(0\leq L_{u}\leq n_{u}\), \(\Delta H_{L_{y},L_{u}}(k)\
eq 0\), there must exist a pseudo Jacobian matrix \(\Phi(k)\) such that the system (11) can be changed to_  
\[\Delta y(k+1)=y(k+1)-y(k)=\Phi^{T}{}_{f,L_{y},L_{u}}(k)\Delta H_{L_{y},L_{u}}(k) \tag{13}\]  
_where \(\Phi^{T}{}_{f,L_{y},L_{u}}(k)=\Big{[}\Phi_{1}(k),\cdots,\Phi_{L_{y}+L_{u}}(k) \Big{]}\) is bounded for any time \(k\)._  
For ease of calculation, taking \(L_{y}=L_{u}=3\), we have  
\[\Phi^{T}{}_{f,L_{y},L_{u}}(k)=[\Phi_{1}(k),\Phi_{2}(k),\Phi_{3}(k),\Phi_{4}(k ),\Phi_{5}(k),\Phi_{6}(k)] \tag{14}\]  
\[\Delta H_{L_{y},L_{u}}(k)=[\Delta y(k),\Delta y(k-1),\Delta u(k),\Delta u(k-1) ]^{T} \tag{15}\]  
Figure 2: Block diagram of MFAC method based on genetic algorithm. By bringing Equations (15) and (14) into Equation (13), the dynamical model can be transformed into  
\[ \begin{array}{l}\Delta y(k+1)=\Phi_{1}(k)\Delta y(k)+\Phi_{2}(k)\Delta y(k-1)+ \Phi_{3}(k)\Delta y(k-2)\\ +\Phi_{4}(k)\Delta u(k)+\Phi_{5}(k)\Delta u(k-1)+\Phi_{6}(k)\Delta u(k-2)\end{array}  \tag{16}\]  
where \(\Phi_{1}(k),\Phi_{2}(k),\Phi_{3}(k),\Phi_{4}(k),\Phi_{5}(k),\Phi_{6}(k)\) is the partial derivative of the system. To prevent instability in the control system due to excessively large control inputs and the resulting significant errors, the following criterion function for control inputs is introduced:  
\[J(u(k))=|y_{d}(k+1)-y(k+1)|^{2}+\lambda|u(k)-u(k-1)|^{2} \tag{17}\]  
where \(y_{d}(k+1)\) is the desired value; \(\lambda\) is a variable coefficient utilized as a weighting factor to constrain variations in the control inputs. Carrying Equation (16) into Equation (17), deriving derivation for \(u(k)\) and making \(u(k)^{\prime}=0\), the MFAC rate can be obtained as follows:  
\[