## 2 Design of Active Fault-Tolerant Control System
### MFA Fault-Tolerant Control System Scheme  
The proposed method of the MIMO-MFA active fault-tolerant control system does not rely on accurate vehicle model information and only utilizes MIMO system information, which avoids the characteristics of complex vehicle models, such as nonlinearity and strong coupling, that may cause problems in the control system design. With reference to the longitudinal, lateral and yaw dynamic models of the vehicle system, combined with the steering and wheel dynamics model, we determine the input and output of the control system. The system inputs \(\mathbf{u}\) are \(T_{f1}\), \(T_{fr}\), \(T_{rl}\), \(T_{rr}\); the system outputs \(\mathbf{y}\) are \(\omega_{fl}\), \(\omega_{fr}\), \(\omega_{rl}\), \(\omega_{rr}\), \(\omega_{r}\). In order to design the MIMO-MFAC active fault tolerant controller, the following MIMO nonlinear discrete-time system is considered:  
\[\mathbf{y}(k+1)=f(\mathbf{y}(k)\dots\mathbf{y}(k-n_{y}),\mathbf{u}(k)\dots u(k-n_{u})), \tag{3}\]  
where \(\mathbf{u}(k)\in\mathrm{R}^{5}\) and \(\mathbf{y}(k)\in\mathrm{R}^{5}\) are the system inputs and outputs of time \(k\). Moreover, \(n_{y}\) and \(n_{u}\) are the unknown orders, while \(\mathbf{f}(\dots)=(f_{1}(\dots)\dots f_{5}(\dots))^{\mathrm{T}}\in\prod\limits_{ \mathrm{n}_{y}+\mathrm{u}_{u}+2}\mathrm{R}_{5}\mapsto\mathrm{R}_{5}\) is a nonlinear vector-valued function. The nonlinear system (3) is based on the following assumptions. Assumption 1: The partial derivatives of \(f_{i}(\dots)i=1,\dots,m\) with respect to the control input \(\mathbf{u}(k)\) are continuous. Assumption 2: System (3) is a generalized Lipschitz system; that is, for each \(k_{1}\
eq k_{2}\), \(k_{1},k_{2}\geq 0\) and \(\mathbf{u}(k_{1})\
eq\mathbf{u}(k_{2})\) have  
\[\Delta\mathbf{y}(k+1)=\mathbf{\Phi}_{c}(k)\Delta\mathbf{u}(k), \tag{4}\]  
where \(b>0\) is a constant. For the MIMO nonlinear system (Based on the algorithm design, the following is the MIMO-MFA active fault-tolerant control scheme for discrete-time MIMO nonlinear systems. \[\mathbf{\Phi}_{c}(k) = \mathbf{\Phi}_{c}(k-1)+\frac{\eta(\Delta\mathbf{y}(k)-\mathbf{\Phi}_{c}(k-1) \Delta\mathbf{u}(k-1))\Delta\mathbf{u}^{\text{T}}(k-1)}{\mu+\left\|\Delta\mathbf{u}(k-1) \right\|^{2}} \tag{6}\] \[\mathbf{u}(k) = \mathbf{u}(k-1)+\frac{\rho\mathbf{\hat{\Phi}}_{c}{}^{T}(k)(\mathbf{y}^{*}(k+1 )-\mathbf{y}(k))}{\lambda+\left\|\mathbf{\hat{\Phi}}_{c}(k)\right\|^{2}}, \tag{7}\]  
where \(\eta\in(0,2]\) is a step-sized constant, \(\mu>0\) is a weighting factor added to penalize the excessive changes in \(\mathbf{\hat{\Phi}}_{c}(k)\), \(\lambda>0\) is a penalty factor added to penalize the excessive changes in input and \(\rho\in(0,1]\) is a step-sized constant, which is added to make Equation (7) general and will be used in the stability proof later. Furthermore, \(\mathbf{y}^{*}(k+1)\) is the desired output signal, determined by the system and \(\mathbf{y}(k)\) is the output of the time \(k\). When the 4 WID electric vehicle drive system fails, the MIMO-MFA active fault tolerant controller corrects the vehicle attitude in terms of the following aspects. Firstly, based on the multi-input data wheel drive torque \(T_{(.)}\big{(}k-1)\),\(T_{(. )}(k-2)\) and front wheel angle \(\delta_{f}\big{(}k-1)\),\(\delta_{f}(k-2)\) and multi-output data of the actual wheel angular velocity \(\omega_{(.)}(k),\omega_{(. )}(k-1)\) and actual yaw rate \(\omega_{r}(k),\omega_{r}(k-1)\), our model estimates the pseudo-Jacobian matrix \(\mathbf{\Phi}_{c}(k)\) by using Equation (6). Then, according to the desired wheel angular velocity \(\omega_{(. )}{}^{*}(k+1)\), desired yaw rate \(\omega_{r}{}^{*}(k+1)\), actual wheel angular velocity \(\omega_{(. )}(k)\) and actual yaw rate \(\omega_{r}(k)\), we can obtain the error of the wheel angular velocity \(\omega_{(.)}{}^{*}\big{(}k+1)-\omega_{(. )}(k)\) and error of the yaw rate \(\omega_{r}{}^{*}(k+1)-\omega_{r}(k)\).