## 3 Finite-Time Disturbance Observer and Controller Design
### Finite-Time Disturbance Observer Design  
Design the novel FTDO for \(i\)th vehicle as  
\[\dot{\mathbf{x}}_{i}(t)= v_{i}(t)+\mu_{i1}(t) \tag{12}\] \[\mu_{i1}(t)= -\lambda_{0}L_{i}^{1/3}\big{|}\dot{x}_{i}(t)-x_{i}(t)\big{|}^{2/3} \operatorname{sgn}\big{(}\hat{x}_{i}(t)-x_{i}(t)\big{)}\] \[-\varrho_{0}\big{(}\hat{x}_{i}(t)-x_{i}(t)\big{)}+\hat{\omega}_{i 1}(t)\] \[\hat{\mathbf{\omega}}_{i_{1}}(t)= -\lambda_{1}L_{i}^{1/2}\big{|}\hat{\omega}_{i1}(t)-\mu_{i1}(t) \big{|}^{1/2}\operatorname{sgn}(\hat{\omega}_{i1}(t)-\mu_{i1}(t))\] \[-\varrho_{1}(\hat{\omega}_{i1}(t)-\mu_{i1}(t))+\hat{\mathbf{\omega}}_ {i1}(t)\] \[\dot{\hat{\mathbf{\omega}}}_{i1}(t)= -\lambda_{2}L_{i}\operatorname{sgn}(\hat{\mathbf{\omega}}_{i1}(t)- \dot{\hat{\mathbf{\omega}}}_{i1}(t))\] \[-\varrho_{2}(\hat{\omega}_{i1}(t)-\dot{\hat{\mathbf{\omega}}}_{i1}(t))\] \[\dot{\hat{v}}_{i}(t)= u_{i}(t)+\mu_{i2}(t)\] \[\mu_{i2}(t)= -\lambda_{3}L_{i}^{1/2}\big{|}\hat{v}_{i}(t)-v_{i}(t)\big{|}^{1/2 }\operatorname{sgn}(\hat{v}_{i}(t)-v_{i}(t)\] \[-\varrho_{3}(\hat{v}_{i}(t)-v_{i}(t))+\hat{\mathbf{\omega}}_{i2}(t))\] \[\mu_{i1}(t)= -\lambda_{0}L_{i}^{1/3}\big{|}\dot{x}_{i}(t)-x_{i}(t)\big{|}^{2/3} \operatorname{sgn}\big{(}\hat{x}_{i}(t)-x_{i}(t)\big{)}\]0 \[\mu_{i1}(t)= -\lambda_{0}L_{i}^{1/3}\big{|}\dot{x}_{i}(t)-x_{i}(t)\big{|}^{2/3} \operatorname{sgn}\big{(}\hat{x}_{i}(t)-x_{i}(t)\big{)}\]1  
where \(\lambda_{i}>0\), \(L_{i}>0\), and \(\varrho_{i}>0\), for \(i\in\{1,2,3,4\}\); \(\hat{x}_{i}(t)\) represents the estimate of the position; \(\hat{\mathbf{\omega}}_{i1}(t)\) is the estimate of \(\hat{\mathbf{\omega}}_{i1}(t);\hat{\mathbf{\omega}}_{i1}(t)\) is the estimate of \(\hat{\mathbf{\omega}}_{i1}(t);\hat{\mathbf{\omega}}_{i}(t)\) and \(\hat{\mathbf{\omega}}_{i2}(t)\) represent the estimates of the velocity and \(\omega_{i2}(t)\), respectively. Define the estimation errors of states and disturbances as follows: \(\hat{x}_{i}(t)=\frac{\hat{x}_{i}(t)-x_{i}(t)}{L_{i}}\hat{v}_{i}(t)=\frac{\hat{ v}_{i}(t)-v_{i}(t)}{L_{i}}\), \(v_{i1}(t)=\frac{\hat{\mathbf{\omega}}_{i1}(t)-\hat{\mathbf{\omega}}_{i1}(t)}{L_{i}}\), and \(\hat{\mathbf{\omega}}_{ik}(t)=\frac{\hat{\mathbf{\omega}}_{ik}(t)-\hat{\mathbf{\omega}}_{ ik}(t)}{L_{i}}\), for \(k=1,2\). To provide a detailed explanation of the FTDO design process, we present its design block diagram as shown in \[\mu_{i1}(t)= -\lambda_{0}L_{i}^{1/3}\big{|}\dot{x}_{i}(t)-x_{i}(t)\big{|}^{2/3} \operatorname{sgn}\big{(}\hat{x}_{i}(t)-x_{i}(t)\big{)}\]2  
**Theorem 1**. : _Consider the vehicle dynamics (4) under Assumption \(1\). The proposed FTDO in (12) is finite-time stable in the sense that there exists a time \(T_{d}\) such that \(\hat{x}_{i}(t),\hat{v}_{i}(t),\hat{\mathbf{\omega}}_{ik}(t)\), and \(v_{i1}(t)\) converge to a bounded neighborhood of zero, \(\forall t\geq T_{d}\)._  
Proof of Theorem 1.: Taking the derivative of \(\hat{x}_{i}(t)\) according to (4) and (12), we can obtain  
\[\mu_{i1}(t)= -\lambda_{0}L_{i}^{1/3}\big{|}\dot{x}_{i}(t)-x_{i}(t)\big{|}^{2/3} \operatorname{sgn}\big{(}\hat{x}_{i}(t)-x_{i}(t)\big{)}\]3 \[\mu_{i1}(t)= -\lambda_{0}L_{i}^{1/3}\big{|}\dot{x}_{i}(t)-x_{i}(t)\big{|}^{2/3} \operatorname{sgn}\big{(}\hat{x}_{i}(t)-x_{i}(t)\big{)}\]4 \[\mu_{i1}(t)= -\lambda_{0}L_{i}^{1/3}\big{|}\dot{x}_{i}(t)-x_{i}(t)\big{|}^{2/3} \operatorname{sgn}\big{(}\hat{x}_{i}(t)-x_{i}(t)\big{)}\]5 \[\mu_{i1}(t)= -\lambda_{0}L_{i}^{1/3}\big{|}\dot{x}_{i}(t)-x_{i}(t)\big{|}^{2/3} \operatorname{sgn}\big{(}\hat{x}_{i}(t)-x_{i}(t)\big{)}\]6 \[\mu_{i1}(t)= -\lambda_{0}L_{i}^{1/3}\big{|}\dot{x}_{i}(t)-x_{i}(t)\big{|}^{2/3} \operatorname{sgn}\big{(}\hat{x}_{i}(t)-x_{i}(t)\big{)}\]7By (4) and (12), the derivative of \(\hat{\varpi}_{i1}(t)\) is  
\[ \begin{split}\hat{\varpi}_{i1}(t)=&\frac{\dot{\hat{ \varpi}}_{i1}(t)-\hat{\varpi}_{i1}(t)}{L_{i}}\\ =&\frac{1}{L_{i}}(-\lambda_{1}L_{i}^{1/2}|\hat{ \varpi}_{i1}(t)-\mu_{i1}(t)|^{1/2}\operatorname{sgn}(\hat{\varpi}_{i1}(t)-\mu_ {i1}(t))\\ &-\varrho_{1}(\hat{\varpi}_{i1}(t)-\mu_{i1}(t))+\hat{\varpi}_{i}( t)-\hat{\varpi}_{i1}(t))\end{split}  \tag{14}\]  
Using (12), we can obtain  
\[