# Improved ADRC-Based Autonomous Vehicle Path-Tracking Control Study Considering Lateral Stability
## 3 Improved ADRC
### Classic ESO Analysis and Improvement  
Antidisturbance capability is the core performance of ADRC, which is achieved by estimating the real-time action value of disturbance internal and external in the system through the ESO and giving compensation in the feedback to eliminate the effect of disturbance. According to the actual problem that is the subject this paper, a single input and single output system is discussed below, assuming that a second-order system is represented as follows [23; 24]:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=f(\cdot)+bu\\ y=x_{1}\end{array} \right. \tag{9}\]  
where \(f(\cdot)\) is the total disturbance of the system, which can be seen as an extended state variable represented by \(x_{3}\). Therefore, according to the ADRC control theory, we can obtain the extended system:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=x_{3}+bu\\ \dot{x}_{3}=\dot{f}(\cdot)\\ y=x_{1}\end{array} \right. \tag{10}\]  
Assuming that the estimated value of the extended state variables are \(Z=[z_{1},z_{2},z_{3}]^{T}\), the extended state observer is constructed according to Equation (10), as shown in Equation (11), so that we can estimate the total unknown disturbance, that is, \(f(\cdot)=z_{3}\). Therefore, the nonlinear ESO can be expressed as  
Figure 2: The structure of IADRC. \[\left\{ \begin{array}{l}e=z_{1}-y\\ \dot{z}_{1}=z_{2}-\beta_{01}e\\ \dot{z}_{2}=z_{3}-\beta_{02}fal(e,a_{1})+bu\\ \dot{z}_{3}=-\beta_{03}fal(e,a_{2})\end{array} \right. \tag{11}\]  
where \(\beta_{01}\), \(\beta_{02}\), \(\beta_{03}\) are the observer gains in the ESO, respectively, and \(fal\) is the function used to implement the ESO nonlinearity, and can be expressed as  
\[fal(e,a_{i},\delta)=\left\{ \begin{array}{l}\frac{e}{a^{2}-a_{i}},\ |e|\leq\delta\\ |e|^{a_{i}^{2}}\text{sgn}(e),\ |e|>\delta\end{array} \right. \tag{12}\]  
where \(i=1,2\), \(a_{i}\), \(\delta\) represent the piecewise function parameters. In this function, the magnitude of \(a\) determines the degree of nonlinearity of the function; the larger the value of \(a\), the stronger the degree of linearity of the function, and vice versa, the weaker the degree of nonlinearity. Parameter \(\delta\) is the piecewise point of the piecewise function, which determines the length of the linear interval. In the process of control system design, the selection of the value for \(\delta\) is important, and may lead to the jitter phenomenon of disturbance estimation, which further leads to jitter in the control input signal. The function in Equation (12) is an odd function symmetric about the origin, so it may be useful to analyze the situation when \(\mathbf{e}>0\), and the same situation when \(\mathbf{e}<0\). The derivative of the function at the piecewise point \(\delta\) is obtained as  
\[fal^{\prime}(e,a_{i},\delta)=\left\{ \begin{array}{l}\delta^{a_{i}-1},\ 0< \mathbf{e}\leq\delta\\ a_{i}e^{a_{i}-1},\ \mathbf{e}>\delta\end{array} \right. \tag{13}\]  
when \(\mathbf{e}=\)\(\delta\), then \(a_{i}e^{a_{i}-1}\
eq\delta^{a_{i}-1}\). From Equation (13) and Figure 3, it can be seen that the derivative of the \(fal\) function at the piecewise point is not continuous, and there is an abrupt change at \(\pm\delta\), which does not have good continuity and smoothness, resulting in the estimated disturbance value being insufficiently smooth, which in turn may lead to the phenomenon of jitter in the control input. To solve this problem, and simultaneously guaranteeing the characteristics of small gain at large error and large gain at small error, a novel continuous nonlinear function named \(Fal\), as shown in Equation (14), is proposed to replace the traditional piecewise function:  
\[ \begin{array}{rl}Fal_{i}(e)&=\lambda_{i}\big{|}\text{arsh}(e)\big{|}^{a_{i} }\text{arctan}(\gamma_{i}e)\\ &=\lambda_{i}\big{|}\text{ln}(e+\sqrt{e^{2}+1})\big{|}^{a_{i}}\text{arctan}( \gamma_{i}e)\end{array}  \tag{14}\]  
where \(\lambda_{i}\), \(a_{i}\), and \(\gamma_{i}\) represent the novel nonlinear function parameters.