## 3 Finite-Time Disturbance Observer and Controller Design
### FTDO-Based Sliding Mode Controller Design and Stability Analysis  
A novel FTDO-based sliding mode controller for a vehicle platoon affected by matched and mismatched disturbances is developed in this section to accomplish the platoon control goals. Define the sliding mode surface as  
\[s_{i}(t)=e_{i}(t)+\int_{0}^{t}\ell_{i}e_{i}(\tau)d\tau \tag{23}\]  
where \(\ell_{i}>0\). To ensure string stability, a coupled sliding mode surface is designed as [16]  
\[S_{i}(t)=\left\{ \begin{array}{ll}\gamma s_{i}(t)-s_{i+1}(t),&i=1,\ldots,N-1 \\ \gamma s_{i}(t),&i=N\end{array} \right. \tag{24}\]  
where \(\gamma>0\). Then, from (24), we have  
\[S(t)=\text{Y}s(t) \tag{25}\]  
where \(S(t)=\left[S_{1}(t)S_{2}(t)\cdots S_{n}(t)\right]^{T}\),\(s(t)=\left[s_{1}(t)s_{2}(t)\cdots s_{n}(t)\right]^{T}\), and  
\[\text{Y}= \begin{bmatrix}\gamma&-1&\cdots&0&0\\ 0&\gamma&-1&\cdots&0\\ &&\ddots&&\\ 0&0&\cdots&\gamma&-1\\ 0&0&\cdots&0&\gamma\end{bmatrix} \]  
As \(\gamma\
eq 0\), Y is an invertible matrix. According to (24), \(S_{i}(t)\) converges to a bounded neighborhood of zero if, and only if, \(s_{i}(t)\) converges to a bounded neighborhood of zero at the same time, and vice versa [35]. Differentiating \(S_{i}\), one can obtain  
\[\dot{S}_{i}(t)=\left\{ \begin{array}{ll}&\gamma\{v_{0}(t)-v_{i}(t)-\varpi_{i 1}(t)-h_{i}[u_{i}(t)+\varpi_{i2}(t)]+\ell_{i}e_{i}(t)\\ &-A_{i}(t)\},i=1\\ &\gamma\{v_{i-1}(t)+\varpi_{i-1,1}(t)-v_{i}(t)-\varpi_{i1}(t)-h_{i}[u_{i}(t)+ \varpi_{i2}(t)]\\ &+\ell_{i}e_{i}(t)-A_{i}(t)\},i\in\{2,...,N-1\}\\ &\gamma\{v_{i-1}(t)+\varpi_{i-1,1}(t)-v_{i}(t)-\varpi_{i1}(t)-h_{i}[u_{i}(t)+ \varpi_{i2}(t)]\\ &+\ell_{i}e_{i}(t)\},i=N\end{array} \right. \tag{26}\]  
where \(A_{i}(t)=\dfrac{1}{\gamma}(\dot{e}_{i+1}+\ell_{i+1}e_{i+1})\), \(i\in\{1,2,\cdots,N-1\}\). The novel FTDO-based sliding mode controller is designed as  
\[u_{i}(t)= \begin{cases}-\dot{\varpi}_{i2}(t)-\dot{\varpi}_{i1}(t)+\dfrac{1}{h _{i}}\Big{[}\ell_{i}e_{i}(t)+\dfrac{1}{\gamma}(k_{1}|S_{i}(t)|^{1/2}\operatorname {sgn}(S_{i}(t))+\int_{0}^{t}k_{2}\operatorname{sgn}(S_{i}(\tau))d\tau)\\ +v_{0}(t)-\Big{(}v_{i}(t)+\dot{\varpi}_{i1}(t)\Big{)}-A_{i}(t)\Big{]},i=1\\ -\dot{\varpi}_{i2}(t)-\dot{\varpi}_{i1}(t)+\dfrac{1}{h_{i}}\Big{[}\ell_{i}e_{i }(t)+\dfrac{1}{\gamma}(k_{1}|S_{i}(t)|^{1/2}\operatorname{sgn}(S_{i}(t))+\int_ {0}^{t}k_{2}\operatorname{sgn}(S_{i}(\tau))d\tau)\\ +v_{i-1}(t)+\dot{\varpi}_{i-1,1}(t)-\Big{(}v_{i}(t)+\dot{\varpi}_{i1}(t)\Big{)} -A_{i}(t)\Big{]},i=2,\ldots,N\end{cases}  \tag{27}\]where \(k_{1}>0\) and \(k_{2}>0\); \(\hat{\omega}_{2}(t)\) and \(\hat{\phi}_{i1}(t)\) are the feedforward compensation terms, constructed by utilizing the estimation of mismatched and matched disturbances. To provide a clearer explanation of the structure of the controller (27), we present its design block diagram as shown in Figure 3.  
**Theorem 2**. : _Consider the vehicle dynamics \((4)\), the leader \((5)\), and the FTDO \((12)\) under Assumption \(1\). The FTDO-based sliding mode controller (27) can guarantee that \(\epsilon_{i}(t)\) for \(i\in\{1,2,\ldots,N\}\) converges to a bounded neighborhood of zero, while the string stability of the entire vehicle platoon can also be ensured when \(0<\gamma\leq 1\)._  
Proof of Theorem 2.: The proof procedure first analyzes the state boundedness of \(\epsilon_{i}(t)\) for \(t\in\{0,T_{d}\}\), and then addresses the global stabilization of \(\epsilon_{i}(t)\) in the second part. Part 1 ( State boundedness for \(t\in[0,T_{d})\)): According to \((4)\) and \((6)\), the differential of \(\epsilon_{i}(t)\) is  
\[\dot{\epsilon}_{i}(t)= \begin{cases}v_{0}(t)-v_{i}(t)-\varpi_{i1}(t)-h_{i}(u_ {i}(t)+\varpi_{i2}(t)\\ +\dot{\varpi}_{i1}(t)),i=1\\ v_{i-1}(t)+\varpi_{i-1,1}(t)-v_{i}(t)-\varpi_{i1}(t)-h_{i}(u_{i}(t)\\ +\varpi_{i2}(t)+\dot{\varpi}_{i1}(t)),i=2,\ldots,N\end{cases}  \tag{28}\]  
Substituting (27) into (28), one can obtain  
\[\dot{\epsilon}_{i}(t)=