# Robust Control with Uncertain Disturbances for Vehicle Drift Motions
## 3 Robust Control in Drift Conditions  
This section designs the controller based on robust with uncertain external disturbances to realize drift motions. The control system is described as state space representation with the linearized tire model. The whole equations of UniTire are used in the simulation model to sufficiently describe tire characteristics and the tire model is linearized in the control system to simplify the controller state-space expressions. The linearized tire model and the affine expression can optimize the controller design in [29]. According to UniTire, the tire longitudinal and lateral forces are related to the tire slip ratios, the tire slip angle, the wheel center velocity, and the tire vertical force. The tire longitudinal slip ratio has a greater influence on the tire longitudinal force and the tire slip angle affects the tire lateral force relatively large. The affine functions of the tire longitudinal and lateral forces are shown as Equation (23) in Figure 3.  
\[\left\{ \begin{array}{c}F_{xi}\approx\widetilde{F}_{xi}=\overline{C}_{xi}( \kappa_{i}-\overline{\kappa}_{i})+\overline{F}_{xi}\\ F_{yi}\approx\widetilde{F}_{yi}=\overline{C}_{yi}(\alpha_{i}-\overline{\alpha}_ {i})+\overline{F}_{yi}\end{array} \right. \tag{23}\]  
where \(\kappa_{i}=-(v_{xi}-\omega_{i}r_{ei})/v_{xi}=S_{xi}/(S_{xi}-1)\) is the TYDEX longitudinal slip ratio, and \(\overline{\kappa}_{i}\), \(\overline{\alpha}_{i}\), \(\overline{C}_{xi}\), \(\overline{C}_{yi}\), \(\overline{F}_{xi}\), and \(\overline{F}_{yi}\) are the known slip ratio, slip angle, slip stiffness, cornering stiffness, longitudinal force, and lateral force of the approximation point respectively. Combining Equations (1) and (23) and the drift motion mechanism analysis, the vehicle state space representation is \(\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}\). Considered with uncertain external disturbances a control system can be described as:  
\[\left\{ \begin{array}{c}\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}+\mathbf{d}\\ \mathbf{y}=\mathbf{x}\end{array} \right. \tag{24}\]  
where \(\mathbf{x}=\left[v-v^{e},\beta-\beta^{e},\gamma-\gamma^{e},\phi-\phi^{e}\right]^{T}\), \(\mathbf{u}=\left[\delta_{f}-\delta_{f}^{e},\kappa_{f}-\kappa_{f}^{e},\kappa_{r}- \kappa_{r}^{e}\right]^{T}\) and \(\mathbf{y}\) are the state, input and output vectors of the system respectively, and the coefficient matrices \(\mathbf{A}\) and \(\mathbf{B}\) are the Jacobian matrices with respect to the state and input vectors. The uncertain external disturbance \(\mathbf{d}\) is defined as:  
\[\mathbf{d}=\mathbf{B}\mathbf{d}_{e} \tag{25}\]  
where \(\mathbf{d}_{e}=\left[d_{\delta},d_{f},d_{r}\right]^{T}\), and \(d_{\delta}\), \(d_{f}\) and \(d_{r}\) are uncertain external disturbances related to three inputs respectively. The robust control law is designed as:  
\[\mathbf{u}=\mathbf{K}\mathbf{x} \tag{26}\]  
where \(\mathbf{K}\) is the control gain matrix. The cost function is shown as:  
\[\frac{\int_{0}^{t}\mathbf{y}^{T}\mathbf{y}\mathbf{d}t}{\int_{0}^{t}d_{e}^{T}d_{e}\mathbf{d}t} \leq\rho^{2} \tag{27}\]  
where \(\mathbf{\rho}\) is the scalar to express the anti-disturbance capability and \(\rho>0\). Supposing that the control system is internally stable, the Lyapunov stability is satisfied and the Lyapunov function and the derived function are selected as:  
\[\mathbf{L}=\mathbf{x}^{T}\mathbf{P}\mathbf{x} \tag{28}\]  
\[\dot{\mathbf{L}}=\dot{\mathbf{x}}^{T}\mathbf{P}\mathbf{x}+\mathbf{x}^{T}\mathbf{P}\dot{\mathbf{x}} \tag{29}\]  
where \(\mathbf{P}\) is the positive definite symmetric matrix. To obtain the gain matrix under the Lyapunov stability, the inequation Equation (30) needs to be satisfied and Equation (31) is established based on Equations (24)-(26), (29) and the two definitions \(\mathbf{\lambda}=\left[\mathbf{x}^{T}~{}~{}\mathbf{d}_{e}^{T}\right]^{T}\) and \(\mathbf{Q}=\mathbf{P}(\mathbf{A}+\mathbf{B}\mathbf{K})\):  
Figure 3: UniTire time model with affine approximation: (**a**) longitudinal force; (**b**) lateral force.