## 3 Proposed Controller Design and Stability Analysis
### Construction of AFSEFBC  
In this subsection, the AFSEFBC is designed to attain satisfactory tracking efficiency. The tracking state error vector of the SBW system, \(\phi(t)=\left[\phi_{1}(t),\,\phi_{2}(t),\,\phi_{3}(t)\right]^{T}\), is presented by  
\[\begin{cases}\phi_{1}(t)=y_{d}(t)-x_{1}(t)\\ \phi_{2}(t)=\dot{y}_{d}(t)-\dot{x}_{1}(t)=\dot{y}_{d}(t)-x_{2}(t)\\ \phi_{3}(t)=\ddot{y}_{d}(t)-\ddot{x}_{1}(t)=\ddot{y}_{d}(t)-x_{3}(t)\end{cases} \tag{26}\]  
where \(y_{d}\) stands for the desirable reference signal. Based on (26) and networked time-delayed SBW system (22), the tracking error dynamic is determined as  
\[\begin{cases}\dot{\phi}_{1}(t)=\phi_{2}(t)\\ \dot{\phi}_{2}(t)=\phi_{3}(t)\\ \dot{\phi}_{3}(t)=\ddot{y}_{d}(t)-f_{0}(t)-b_{0}u(t)-\zeta(x,t).\end{cases} \tag{27}\]  
It is noteworthy that the tracking error, \(\phi_{1}(t)\), is formally defined in (26), the estimation error, \(e_{1}(t)\), is precisely delineated in (24), and the lumped uncertainty is exhibited in (6). Furthermore, the lumped uncertainty naturally resides within the plant model, whereas the estimation error serves as the foundation for designing the AESO, and the tracking error is utilized in the design of the AFSEFBC.  
According to the AESO (24), the AFSEFBC, \(u(t)\), for the dynamical error system in (27) is designed in (24),  
\[u(t)=\frac{1}{b_{0}}\big{[}\,\ddot{y}_{d}(t)+\lambda_{1}\big{(}\phi_{1}(t)) \phi_{1}(t)+\lambda_{2}(\phi_{1}(t))\dot{\phi}_{2}(t)+\,\lambda_{3}(\phi_{1}(t ))\dot{\phi}_{3}(t)-f_{0}(z_{i},t)-z_{4}(t)\big{]} \tag{28}\]  
where \(\dot{\phi}_{2}(t)=\dot{y}_{d}(t)-z_{2}(t)\) and \(\dot{\phi}_{3}(t)=\dot{y}_{d}(t)-z_{3}(t)\) are, respectively, the estimates of velocity and acceleration tracking error and variables \(z_{2}\) and \(z_{3}\) are the states estimated by the AESO. Here, the variable gains are designed as  
\[\begin{cases}\lambda_{1}(\phi_{1}(t))=(\omega_{c}+g_{c}(\phi_{1}(t)))^{3}\\ \lambda_{2}(\phi_{2}(t))=3(\omega_{c}+g_{c}(\phi_{1}(t)))^{2}\\ \lambda_{3}(\phi_{3}(t))=3(\omega_{c}+g_{c}(\phi_{1}(t)))\end{cases} \tag{29}\]  
and  
\[g_{c}(\phi_{1}(t))=\eta_{c}|\phi_{1}(t)| \tag{30}\]  
where \(\omega_{c}>0\) is the basic control bandwidth and \(\eta_{c}>0\) is the control accuracy factor, which is used to increase the tracking accuracy of the AFSEFBC.  
**Remark 2**: _It is worth pointing out that the proposed adaptive extended state observer AESO in (24) and the proposed adaptive full state error feedback control AFSEFBC (28) work together in the novel adaptive ADRC approach for the time-delayed SBW dynamics. For controlling the steering angle of the SBW as the desired driver hand-wheel angle, the proposed control process starts with preparing the estimation of unmeasured states and the lumped disturbances by the AESO, in which those estimates are required in designing the control law of the AFSEFBC (28). In addition, the AESO is employed to estimate the lumped disturbance that is compensated for in the control law in (28). With the aid of estimations, the proposed AESO-based AFSEFBC is then synthesized to constitute the AADRC scheme, which is used to deal with time-varying disturbances and time delays._  
To further help understand the proposed control scheme and ease the practical design process, the corresponding flowchart is presented in Figure 3.