# Improved ADRC-Based Autonomous Vehicle Path-Tracking Control Study Considering Lateral Stability
## 3 Improved ADRC
### Stability Analysis of IESO  
To analyze the stability of IESO, define the observer error quantity \(e=\left(\begin{array}{cc}e_{1}&e_{2}&e_{3}\end{array}\right)^{T}\), and let \(e_{1}=z_{1}-x_{1}\), \(e_{2}=z_{2}-x_{2}\), \(e_{3}=z_{3}-x_{3}\), assuming \(x_{3}=d_{0}\) is a constant, the error equations (Equations (10) and (11)) can be obtained as  
\[\left\{\begin{array}{l}e_{1}=z_{1}-y\\ \dot{e}_{1}=e_{2}-\beta_{01}e_{1}\\ \dot{e}_{2}=e_{3}-\beta_{02}Fal_{1}(e_{1})\\ \dot{e}_{3}=-\beta_{03}Fal_{2}(e_{1})\end{array}\right. \tag{16}\]  
Rewriting these equations into the form of a state space, we can obtain  
\[\dot{e}=-G(e)e \tag{17}\]  
where \(G(e)=\left(\begin{array}{ccc}\beta_{01}&-1&0\\ \beta_{02}\frac{Fal_{1}(e_{1})}{e_{1}}&0&-1\\ \beta_{03}\frac{Fal_{2}(e_{1})}{e_{1}}&0&0\end{array}\right)\), and also we can see that \(\frac{Fal_{1}(e_{1})}{e_{1}}>0\), \(\frac{Fal_{2}(e_{1})}{e_{1}}>0\).  
**Lemma 1 (2009).** If there exists a matrix \(H=\left(\begin{array}{ccc}h_{11}&h_{12}&h_{13}\\ -h_{12}&h_{22}&h_{23}\\ -h_{13}&-h_{23}&h_{33}\end{array}\right)\), and the principal diagonal elements of \(H\) are positive such that the matrix \(HG(e)\) is positive definite symmetric, then the zero solution of the system (17) is Lyapunov asymptotically stable.  
According to Lemma 1, it is necessary to construct the matrix \(H\), which satisfies the conditions such that the system is asymptotically stable; calculating \(HG(e)\) yields  
\[HG(e)=\left(\begin{array}{ccc}H_{11}&-h_{11}&-h_{12}\\ H_{21}&h_{12}&-h_{22}\\ H_{31}&h_{13}&h_{23}\end{array}\right) \tag{18}\]  
where \(H_{11}=\beta_{01}h_{11}+\beta_{02}\frac{Fal_{1}(e_{1})}{e_{1}}h_{12}+\beta_{0 3}\frac{Fal_{2}(e_{1})}{e_{1}}h_{13}\), \(H_{21}=-\beta_{01}h_{12}+\beta_{02}\frac{Fal_{1}(e_{1})}{e_{1}}h_{22}\), \(+\beta_{03}\frac{Fal_{2}(e_{1})}{e_{1}}h_{23}\), \(H_{31}=-\beta_{01}h_{13}-\beta_{02}\frac{Fal_{1}(e_{1})}{e_{1}}h_{23}+\beta_{0 3}\frac{Fal_{2}(e_{1})}{e_{1}}h_{33}\).  
According to Equation (18), \(HG(e)\) is a positive definite matrix equivalent to  
\[\left\{\begin{array}{l}H_{31}=-h_{12},H_{21}=-h_{11},h_{13}=-h_{22}\\ \Delta_{1}=H_{11}>0\\ \Delta_{2}=\left|\begin{array}{ccc}H_{11}&-h_{11}\\ H_{21}&h_{12}\end{array}\right|>0\\ \Delta_{3}=\left|\begin{array}{ccc}H_{11}&-h_{11}&-h_{12}\\ H_{21}&h_{12}&-h_{22}\\ H_{3 where \(h_{12}=\frac{\beta_{02}\frac{Fd_{1}(e_{1})}{\alpha_{1}}}{\beta_{01}\beta_{02}\frac {Fd_{1}(e_{1})}{\alpha_{1}}-\beta_{03}\frac{Fd_{2}(e_{1})}{\alpha_{1}}}+\varphi _{1},h_{23}=\frac{1}{\beta_{01}\beta_{02}\frac{Fd_{1}(e_{1})}{\alpha_{1}}-\beta _{03}\frac{Fd_{2}(e_{1})}{\alpha_{1}}}+\varphi_{2},\varepsilon\) is a sufficiently small positive number; \(\varphi_{1}\), \(\varphi_{2}\) are values that are infinitely close to zero with respect to \(\varepsilon\); and \(Fd_{1}(e_{1})\), \(Fd_{2}(e_{1})\) are determined by the corresponding parameters.  
Therefore, the Lyapunov function is defined as  
\[\begin{array}{ll}V&=\int_{0}^{t}\big{(}HG(e)e,\dot{e}\big{)}d\tau+C\\ &=\int_{0}^{t}\Bigl{\{}-(\beta_{01}e_{1}-e_{2})^{2}-\varepsilon(\beta_{02}Fd_ {1}(e_{1})-e_{3})^{2}-\varepsilon(\beta_{03}Fd_{2}(e_{1})+d_{0})^{2}\Bigr{\}}d \tau+C\end{array} \tag{21}\]  
where the integrand is bounded and \(C\) is a positive number. It is obvious to choose an appropriate positive number \(C\) so that \(V>0\).  
The derivative of the Lyapunov function is  
\[\dot{V}=-(\beta_{01}e_{1}-e_{2})^{2}-\varepsilon(\beta_{02}Fd_{1}(e_{1})-e_{3 })^{2}-\varepsilon(\beta_{03}Fd_{2}(e_{1})+d_{0})^{2}\leq 0 \tag{22}\]  
Thus, this error system in Equation (17) is asymptotically stable at its equilibrium point.