## 4 Slip Ratio Control Strategy Based on Dynamic Adaptive Compensation Sliding Mode Control  
### Design of the Slip Ratio Controller  
In the traditional design of wheel slip ratio sliding mode controllers, the control law for the sliding mode control output torque generally consists of two parts: the equivalent control \(T_{eq}\) and the switching robust control \(T_{sw}\). The equivalent control ensures that the system state remains on the sliding surface, while the switching control ensures that the system state does not leave the sliding surface. The switching function is designed as the error e between the actual slip ratio and the optimal slip ratio:  
\[S=e=\lambda_{i}-\lambda_{t} \tag{34}\]  
where \(\lambda_{t}\) is the optimal slip ratio. Setting \(\dot{S}=0\), the equivalent control \(T_{eq}\) is obtained from Equations (3) and (6):  
\[T_{eq}=F_{xt}R+\frac{J\dot{v}_{i}}{(1-\lambda_{i})R} \tag{35}\]  
To ensure that the system state can reach the sliding surface, the switching control is designed as follows:  
\[T_{sw}=\varepsilon\,\text{sgn}(S) \tag{36}\]  
where \(\varepsilon\) is the control gain and \(\varepsilon>0\), the sliding mode control law is:  
\[T=T_{eq}-T_{sw}=F_{xt}R+\frac{J\dot{v}_{i}}{(1-\lambda_{i})R}-\varepsilon\, \text{sgn}(S) \tag{37}\]  
To alleviate slip ratio oscillations caused by low adhesion and low signal-to-noise ratio at low speeds during vehicle start-up on icy and snowy roads, as well as chattering and steady-state errors in sliding mode control due to wheel speed signal delay and adhesion coefficient estimation errors, it is necessary to correct and compensate for the errors. This aims to improve the transient characteristics of sliding mode control, reduce system chattering, and decrease system response time. Redesign the integral sliding surface as:  
\[S(t)=c\int_{0}^{t}e(\tau)d\tau+e(t) \tag{38}\]  
Taking the derivative of the above equation, we obtain:  
\[\dot{S}(t)=ce(t)+\dot{\lambda}_{i} \tag{39}\]  
Next, we design a modified reaching law based on the hyperbolic tangent function. This reaching law introduces an integral term with a dynamic weight based on the error state to adaptively compensate for persistent errors and disturbances. The dynamic weight function \(\omega(t)\) will dynamically adjust according to the current state of the system. When the tracking error is large, it enhances the integral effect, helping to reduce the error quickly. When the error is small, it weakens the integral effect, avoiding over-regulation. The modified reaching law is given by the following equation:  
\[\left\{ \begin{array}{c}\dot{S}=-\varepsilon\,\text{tanh}\Big{(}\frac{S}{ \sigma}\Big{)}-kS-\int_{0}^{t}\omega(\tau)S(\tau)d\tau\\ \omega(t)=k_{\omega}\cdot exp^{-\beta|\varepsilon(t)|}\end{array} \right. \tag{40}\]  
where \(\varepsilon>0,\sigma>0,0<k_{\omega}<1,k>0,\beta\) is a positive tuning coefficient, and \(\text{tanh}\Big{(}\frac{S}{\sigma}\Big{)}\) is the hyperbolic tangent function, which is expressed as:\[\tanh\left(\frac{e^{\frac{\phi}{\sigma}}-e^{-\frac{\phi}{\sigma}}}{e^{\frac{\phi}{ \sigma}}+e^{-\frac{\phi}{\sigma}}}\right) \tag{41}\]  
Compared to the severe chattering problem in the control signal-switching process with the sign function, the saturation function can perform linear adjustment within the boundary layer but is still a discontinuous function. Therefore, a smooth and continuous hyperbolic tangent function is introduced to replace the original switching function, using the smoothness of this function to alleviate the chattering during control signal switching. Next, solve for the control law.