## 2 SBW Plant Dynamical Modeling
### Time-Delay Approximation by Taylor Series  
To cope with the time delays that uncertain SBW system (9) suffers, this SBW plant under the overall input delay term \(b_{0}\)\(u(t-\tau)\) is approximated by utilizing first-order Taylor series expansion. The detailed approximation of the time-delayed SBW system is elaborately given herein. We recall the acceleration state of System Dynamics (9) as follows:  
\[\dot{x}_{2}(t)=a_{20}x_{2}(t)+b_{0}u(t-\tau)+D(x,t). \tag{10}\]  
Taking the Laplace transformation of the above Dynamics (10) leads to  
\[X_{1}(S)S^{2}=-a_{20}X_{1}(S)S+b_{0}u(s)e^{-\tau S}+D(S) \tag{11}\]  
By combining the terms with \(X_{1}(S)\) in (11), we have  
\[(S^{2}+a_{20}S)X_{1}(S)=b_{0}u(s)e^{-\tau S}+D(S) \tag{12}\]  
which is divided by \(e^{-\tau S}\) to have the following dynamics:  
\[(S^{2}+a_{20}S)e^{\tau S}X_{1}(S)=b_{0}u(s)+D(S)e^{\tau S} \tag{13}\]  
where time delay operator \(e^{-\tau S}\) is approximated using the following first-order Taylor series expansion:  
\[e^{-\tau S}=1/(1+\tau S). \tag{14}\]  
Accordingly, the simplification of the above formula is rendered as follows:  
\[(S^{2}+a_{20}S)(1+\tau S)X_{1}(S)=b_{0}u(s)+D(S)(1+\tau S). \tag{15}\]  
For further simplification, Dynamics (15) become  
\[\Big{[}(\tau S^{3}+(1+a_{20}\tau)S^{2}+a_{20}S\Big{]}X_{1}(S)=b_{0}u(S)+D(S)+ \tau SD(S). \tag{16}\]  
The final dynamics of the Laplace transformation in (16) can be rearranged as follows:  
\[\tau S^{3}X_{1}(S)=-\Big{[}(1+a_{20}\tau)S^{2}+a_{20}S\Big{]}X_{1}(S)+b_{0}u( S)+D(S)+\tau SD(S). \tag{17}\]  
Taking the Laplace inverse of the above Dynamics (17) results in  
\[\tau\,\ddot{x}_{1}(t)= -\,\ddot{x}_{1}(t)-a_{20}\tau\,\ddot{x}_{1}(t)-a_{20}\dot{x}_{1}( t)+b_{0}u(t)+D(t)+\tau D(t). \tag{18}\]  
With considering the variation of time delay \(\Delta\tau\) in Dynamical System (18), we obtain  
\[(\tau_{0}+\Delta\tau)\,\ddot{x}_{1}(t)= -\,\ddot{x}_{1}(t)-a_{20}(\tau_{0}+\Delta\tau)\dot{x}_{1}(t)-a_{2 0}\dot{x}_{1}(t)+b_{0}u(t)+D(t)+\tau\dot{D}(t) \tag{19}\]  
which can be represented by  
\[X_{1}(S)S^{2}=-a_{20}X_{1}(S)S+b_{0}u(s)e^{-\tau S}+D(S) \tag{11}\]0 \[X_{1}(S)S^{2}=-a_{20}X_{1}(S)S+b_{0}u(s)e^{-\tau S}+D(S) \tag{11}\]1Based on the dynamics in (20), we can reformulate the high-order derivative system dynamics as follows:  
\[X_{1}(S)S^{2}=-a_{20}X_{1}(S)S+b_{0}u(s)e^{-\tau S}+D(S) \tag{11}\]2 \[X_{1}(S)S^{2}=-a_{20}X_{1}(S)S+b_{0}u(s)e^{-\tau S}+D(S) \tag{11}\]3  
To this end, SBW system (9) can be expressed as third-order dynamics that can be written in the following compact state space form:  
\[ \begin{cases}\dot{x}_{1}(t)=x_{2}(t)\\ \dot{x}_{2}(t)=x_{3}(t)\\ \dot{x}_{3}(t)=f_{0}(x,t)+b_{0}u(t)+\zeta(x,t)\\ y=x_{1}(t)\end{cases}  \tag{22}\]  
where  
\[f_{0}(x,t)=-\frac{(1+a_{20}\tau_{0})}{\tau_{0}}x_{3}(t)-\frac{a_ {20}}{\tau_{0}}x_{2}(t)\] \[\zeta(x,t)=-\frac{\Delta\tau}{\tau_{0}}\dot{x}_{3}(t)-\frac{a_{2 0}\Delta\tau}{\tau_{0}}x_{3}(t)+\frac{1}{\tau_{0}}D(t)+\frac{\tau}{\tau_{0}} \dot{D}(t)\]  
and \(x=\big{[}x_{1}=\theta_{s},x_{2}=\dot{\theta}_{s},x_{3}=\ddot{\theta}_{s}\big{]}\) is the new state vector of approximated SBW system (22). This paper aims to develop a control methodology for Dynamic Model (22) to regulate the angular position of the front wheel, \(\theta_{s}\), to be a desired range of the hand wheel so that steering responses can still be satisfied regardless of the uncertain dynamics, disturbances, and input and output delays in the transmission mechanism. Two factors make control construction and stability analysis challenging: (1) the dimensionality of the SBW is higher in a third-order system due to the adoption of the Taylor approach, (2) the SBW is subject to uncertain communication delays in addition to parametric uncertainties and external disturbances, and (3) only the output angular position of the SBW plant \(x_{1}\) can be measured from a position encoder that is time-delayed by a certain amount. **Assumption 1**.