# NESO-Based Path Following Control for
## 3 Control Strategy Design
### The NESO for Hovercrafts Subject to Unknown Nonlinear Time-Varying Uncertainties
: _Consider Assumption 1 and Assumption 2, the homogenous nonlinear extended state observer (16), nonlinear system (9) with the unknown nonlinear time-varying uncertainties, the estimation error dynamic (20) is convergent for any initial values \(\mathbf{\eta}(0),\mathbf{v}(0),\mathbf{\hat{\eta}}(0),\mathbf{\hat{v}}(0)\) and the bounded total uncertainties \(\mathbf{d}(t)\)_  
**Proof. ** It follows Definition 2 that the vector field \(\mathbf{F}(\mathbf{E}_{o})\) is homogenous of degree \(\alpha\)\(-\)\(1\) with respect to weights {1,\(\alpha\),2\(\alpha\)\(-\)\(1\)}, then it follows Lemma 1 that there exists a positive definite radial unbounded function \(V_{o}(E_{o}):\Re^{9}\rightarrow\Re\) satisfying that \(V_{o}(E_{o})\) is homogeneous of degree \(\gamma\) in weights {1,\(\alpha,2\alpha-1\)}, and \(\frac{\partial}{\partial\mathbf{E}_{1}}\)\(V_{o}(\mathbf{E}_{o})\left(\mathbf{E}_{2}(t)-\beta_{1}[\mathbf{E}_{1}]^{ \alpha}\right)+\frac{\partial}{\partial\mathbf{E}_{2}}\)\(V_{o}(\mathbf{E}_{0})\left(\mathbf{E}_{3}(t)-\beta_{2}[\mathbf{E}_{1}]^{2\alpha-1} \right)+\frac{\partial}{\partial\mathbf{E}_{3}}\)\(V_{o}(\mathbf{E}_{o})\left(-\beta_{3}[\mathbf{E}_{1}]^{3\alpha-2}\right)\) is homogeneous of degree \(\gamma+\alpha-1\) and negative definite. Then, there exists those positive constants \(c_{i}(i=1,\cdots,4)>0\) according to Lemma 2 such that  
\[ \begin{array}{l}-c_{1}V_{o}(\mathbf{E}_{o})^{\frac{\gamma+\alpha-1}{\gamma}} \leq\frac{\partial}{\partial\mathbf{E}_{1}}\)\(V_{o}(\mathbf{E}_{0})\left(\mathbf{E}_{2}(t)-\beta_{1}[\mathbf{E}_{1}]^{ \alpha}\right)+\frac{\partial}{\partial\mathbf{E}_{2}}\)\(V_{o}(\mathbf{E}_{0})\left(\mathbf{E}_{3}(t)-\beta_{2}[\mathbf{E}_{1}]^{2 \alpha-1}\right)\\ +\frac{\partial}{\partial\mathbf{E}_{3}}\)\(V_{o}(\mathbf{E}_{o})\left(-\beta_{3}[\mathbf{E}_{1}]^{3\alpha-2}\right)\leq-c_{2}V_{o}( \mathbf{E}_{0})^{\frac{\gamma+\alpha-1}{\gamma}}\end{array}  \tag{21}\]  
and  
\[\left|\frac{\partial}{\partial\mathbf{E}_{3}}\)\(V_{o}(\mathbf{E}_{o})\right|\leq c_{3}V_{o}(\mathbf{E}_{o})^{\frac{ \gamma-2\alpha+1}{\gamma}} \tag{22}\]  
Next, taking the time derivative of \(V_{o}(\mathbf{E}_{o})\), we have  
\[ \begin{array}{l}\dot{V}_{o}(\mathbf{E}_{o})=\frac{\partial}{\partial\mathbf{E }_{1}}\)\(V_{o}(\mathbf{E}_{o})\left(\mathbf{E}_{2}(t)-\beta_{1}[\mathbf{E}_{1}]^{ \alpha}\right)+\frac{\partial}{\partial\mathbf{E}_{2}}\)\(V_{o}(\mathbf{E}_{o})\left(\mathbf{E}_{3}(t)-\beta_{2}[\mathbf{E}_{1}]^{2 \alpha-1}\right)\\ +\frac{\partial}{\partial\mathbf{E}_{3}}\)\(V_{o}(\mathbf{E}_{o})\left(-\beta_{3}[\mathbf{E}_{1}]^{3\alpha-2}\right)+\frac{ \partial}{\partial\mathbf{E}_{3}}\)\(V_{o}(\mathbf{E}_{o})\dot{\mathbf{d}}(t)\leq-c_{2}V_{o}( \mathbf{E}_{o})^{\frac{\gamma+\alpha-1}{\gamma}}+\left|\frac{\partial}{\partial \mathbf{E}_{3}}\)\(V_{o}(\mathbf{E}_{o})\left\|\dot{\mathbf{d}}(t)\right\|\end{array}  \tag{23}\]  
Let \(Q(\mathbf{E}_{o})=c_{4}V_{o}(\mathbf{E}_{o})^{\frac{\gamma+\alpha-1}{\gamma}}\). Since \(V_{o}(\mathbf{E}_{o})\) is a radially unbounded and positive definite function, thus that \(\{\mathbf{E}_{o}\}[V_{o}(\mathbf{E}_{o})\leq d_{0}]\) for any \(d_{0}>0\) is bounded and \(\lim\limits_{\|\mathbf{E}_{o}\|\rightarrow\infty}V(\mathbf{E}_{o})\rightarrow\infty\). It follows (22) and \(\alpha\in(2/3,1)\) that \(\lim\limits_{\|\mathbf{E}_{o}\|\rightarrow\infty}Q(\mathbf{E}_{o})/\left| \partial V_{o}(\mathbf{E}_{o})/\partial\mathbf{E}_{3}\right|\rightarrow\infty\). Thus, there exists a positive constant \(\mathcal{B}_{3}\) such that \(\left|\partial V_{o}(\mathbf{E}_{o})/\partial\mathbf{E}_{3}\right|\leq Q( \mathbf{E}_{o})\) for all \(\|\mathbf{E}\|\geq\mathcal{B}_{3}\), yields  
\[\dot{V}_{o}(\mathbf{E}_{o})\leq-c_{2}V_{o}(\mathbf{E}_{o})^{\frac{\gamma+\alpha-1 }{\gamma}}+c_{4}V_{o}(\mathbf{E}_{o})^{\frac{\gamma+\alpha-1}{\gamma}}\dot{ \mathbf{d}}(t)=-\left(c_{2}-c_{4}\|\dot{\mathbf{d}}(t)\|\right)V_{o}(\mathbf{E }_{o})^{\frac{\gamma+\alpha-1}{\gamma}}\leq-\kappa V_{o}(\mathbf{E}_{o})^{ \frac{\gamma+\alpha-1}{\gamma}}\leq 0 \tag{24}\]  
where \(\kappa=c_{2}-c_{4}\mathcal{B}_{2}>0\).