## 2 Mechanism Model Construction and Identification Data Collection
### Construction of Longitudinal Tire Force Estimation Mechanism Model  
By observing Equations (4) and (5), it can be seen that in the mathematical mechanism model involving the relationship between motor drive and tire longitudinal force, longitudinal force is the input of the system model from the perspective of control theory. In order to estimate the longitudinal force of the tire, it is necessary to convert the longitudinal force from an input quantity to a state quantity. To construct a tire longitudinal force estimation model, Equations (4) and (5) can be expressed as the following system equations:  
\[\left\{ \begin{array}{l}\dot{x}=Ax+Bu+Dd\\ y=Cx\end{array} \right., \tag{6}\]  
where \(x\), \(u\), \(d\), and \(y\) are the system state variables, known inputs, unknown inputs, and measured values, respectively, and it can be written as \(x= \begin{bmatrix}i&\omega\end{bmatrix} ^{T}\), \(u=U\), \(d=F_{x}\), \(y= \begin{bmatrix}i&\omega\end{bmatrix} ^{T}\). In addition, the parameter matrix can be denoted as \(A= \begin{bmatrix}-\frac{R}{L}&-\frac{K_{x}}{L}\\ \frac{K_{x}}{J}&-\frac{1}{J}\end{bmatrix} \), \(B= \begin{bmatrix}\frac{1}{L}\\ 0\end{bmatrix} \), \(D= \begin{bmatrix}0\\ -\frac{J}{J}\end{bmatrix} \), \(C= \begin{bmatrix}1&0\\ 0&1\end{bmatrix} \). By analyzing the dynamic Equation (6) of the electrically driven wheel, it can be concluded that there are unmeasured longitudinal force input parameters of the tire in the system model. To effectively obtain the dynamic characteristics of the unmeasurable  
Figure 3: Chassis dynamometer test data. (**a**) Motor current, (**b**) motor speed, (**c**) motor voltage, (**d**) longitudinal tire force. variable, it is necessary to establish a decoupling mechanism between the state parameters and external inputs to derive the dynamic relationship of tire force. This scheme is based on the electric drive wheel architecture to construct a Luenberger state observer and then uses the state extension method to embed the longitudinal force parameters into the system state space. A tire force dynamic identification model is constructed by introducing a proportional integral observer with an integral compensation mechanism, ultimately achieving input variable decoupling and joint estimation of mechanical parameters. The extended state observation system after algorithm reconstruction can be expressed as  
\[\left\{ \begin{array}{l}\dot{\hat{x}}=A\hat{x}+Bu+K_{g}(y-\hat{y})+D\hat{d}\\ \dot{\hat{d}}=K_{i}(y-\hat{y})\\ \dot{y}=C\hat{x}\end{array} \right., \tag{7}\]  
where \(\hat{x}\) represents system state estimation, \(\hat{d}\) represents unknown input estimation, \(K_{g}\) represents observer gain matrix, and \(K_{i}\) represents the integral gain matrix of longitudinal tire force. By analyzing the structure of the integral observer constructed by Equation (7), it can be concluded that the existing observer only has an integral gain matrix configured for compensating unknown inputs. This architecture exhibits good estimation accuracy when dealing with slowly changing system state variables, but its tracking performance deteriorates significantly when the system state exhibits significant dynamic characteristics. To enhance the response speed and tracking accuracy of the observer to dynamic processes, this study introduces a proportional gain matrix into the original architecture and constructs an improved proportional integral observer structure. Its mathematical expression is  
\[\left\{ \begin{array}{l}\dot{\hat{x}}=A\hat{x}+Bu+K_{g}(y-\hat{y})+D\hat{d}\\ \dot{\hat{d}}=K_{i}(y-\hat{y})+K_{p}\Big{(}\dot{y}-\dot{\hat{y}}\Big{)}\\ \dot{y}=C\hat{x}\end{array} \right., \tag{8}\]  
where \(K_{p}\) is the proportional gain matrix of longitudinal tire force. The mechanism model for estimating longitudinal force in tires can be constructed in the following form by combining Equation (8)  
\[\left\{