## 2 Problem Formulation
### Definitions and Lemmas  
**Definition 1** ([33]).: _If for any \(\varepsilon>0\), there exists \(\varsigma>0\) such that \(\parallel e_{i}(0)\parallel_{\infty}<\varsigma\Rightarrow sup_{i}\parallel e_ {i}(\cdot)\parallel_{\infty}<\varepsilon\), the origins of \((\ref{eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq  
_where \(y(\cdot)\) is a continuous bounded function. The origin is a finite-time convergent equilibrium of \((\ref{eq:1})\) if an open neighborhood \(U\subseteq D\) of the origin and a function \(T_{z}:U\setminus\{0\}\to[0,\infty)\) exist, such that for every initial point \(z_{0}\in U\setminus\{0\}\), the solution trajectory \(z(t,z_{0})\) is well defined for all \(t\in[0,T(z_{0}))\) and \(\lim_{t\to T(z_{0})}z(t,z_{0})=0\). The function \(T_{z}(z_{0})\) is known as the convergence-time function. The origin is considered as a finite-time equilibrium if it is both finite-time convergent and Lyapunov stable. If \(U=D=\mathbb{R}^{n}\), the origin is a globally finite-time stable equilibrium._  
**Definition 3** ([35]).: _The origins of \((\ref{eq:1})\) with vehicle dynamics \((\ref{eq:2})\) are regarded as strong string stable, if \(|e_{N}(t)|\leq|e_{N-1}(t)|\leq\cdots\leq|e_{1}(t)|\) such that the error transfer function \(|G_{i}(s)|=|E_{i+1}(s)/E_{i}(s)|\leq 1\) is satisfied, where \(E_{i}(s)\) is the Laplace transform of \(e_{i}\)._  
**Lemma 1** ([36]).: _Consider the following differentiator_  
\[\dot{w}_{0}(t)= -\lambda_{0}|w_{0}(t)|^{n/(n+1)}\operatorname{sgn}\left(w_{0}(t) \right)-\varrho_{0}w_{0}(t)+w_{1}(t)\] \[\dot{w}_{1}(t)= -\lambda_{1}\big{|}w_{1}(t)-\dot{w}_{0}(t)|^{(n-1)/n}\operatorname {sgn}(w_{1}(t)-\dot{w}_{0}(t))-\varrho_{1}\big{(}w_{1}(t)-\dot{w}_{0}(t)\big{)} +w_{2}(t)\] \[\dot{w}_{n-1}(t)= -\lambda_{n-1}|w_{n-1}(t)-w_{n-2}(t)|^{1/2}\operatorname{sgn} \left(w_{n-1}(t)-\dot{w}_{n-2}(t)\right)- \tag{9}\] \[\varrho_{n-1}(w_{n-1}-\dot{w}_{n-2})+w_{n}\] \[\dot{w}_{n}(t)= -\lambda_{n}\operatorname{sgn}(w_{n}(t)-\dot{w}_{n-1}(t))-\varrho _{n}(w_{n}(t)-\dot{w}_{n-1}(t))-1/L_{i}\phi(t)\]  
_where \(w_{0},\ldots,w_{n}\) are the state variables; \(\lambda_{i}>0\) and \(\varrho_{i}>0\); The disturbance term \(\phi(t)\) satisfies \(|\phi(t)|\leq L_{i}\), where \(L_{i}>0\), for \(i\in\{1,2,\ldots,n\}\). Then, the differentiator converges to a bounded neighborhood of zero for \(t<T_{d}\)._  
**Lemma 2** ([30]).: _There are \(\chi(t)\in\mathbb{R}\) and \(\zeta(t)\in\mathbb{R}\), if \(\mathfrak{V}_{1}>0\) and \(\mathfrak{V}_{2}>0\), then_  
\[|\chi(t)|^{\mathfrak{V}_{1}}|\zeta(t)|^{\mathfrak{V}_{2}}\leq\frac{\mathfrak{V }_{1}|\chi(t)|^{\mathfrak{V}_{1}+\mathfrak{V}_{2}}}{\mathfrak{V}_{1}+\mathfrak{ V}_{2}}+\frac{\mathfrak{V}_{2}|\zeta(t)|^{\mathfrak{V}_{1}+\mathfrak{V}_{2}}}{ \mathfrak{V}_{1}+\mathfrak{V}_{2}} \tag{10}\]  
**Lemma 3** ([37]).: _The super-twisting algorithm (STA) is described as_  
\[\dot{p}_{1}(t) = -\kappa_{1}|p_{1}(t)|^{1/2}\operatorname{sgn}(p_{1}(t))+p_{2}(t) \tag{11}\] \[\dot{p}_{2}(t) = -\kappa_{2}\operatorname{sgn}(p_{1}(t))\]  
_where \(\kappa_{1}>0\), \(\kappa_{2}>0\); \(p_{1}(t)\) and \(p_{2}(t)\) are the state variables. The trajectories of \((\ref{eq:1})\) converge to the origin \(p=0\) at \(T_{s}(p_{0})\leq\frac{2}{\mathfrak{V}}V^{1/2}(p_{0})\), where \(p_{0}\) is the initial state, \(\vartheta=\frac{\lambda_{\min}^{1/2}(p)\lambda_{\min}(\{Q\})}{\lambda_{\max} \{P\}}\), and \(V(p)=\zeta^{T}P\zeta\) with_  
\[\zeta^{T} =[|p_{1}|^{1/2}\operatorname{sign}(p_{1}),p_{2}]\] \[P =\tfrac{1}{2}\Bigg{[}\begin{array}{cc}4\kappa_{2}+\kappa_{1}^{2} &-\kappa_{1}\\ -\kappa_{1}&2\end{array}\] \[Q =\tfrac{1}{2}\kappa_{1}\Bigg{[}\begin{array}{cc}2\kappa_{2}+ \kappa_{1}^{2}&-\kappa_{1}\\ -\kappa_{1}&1\end{array}\]