# NESO-Based Path Following Control for
## 3 Control Strategy Design
### The NESO for Hovercrafts Subject to Unknown Nonlinear Time-Varying Uncertainties  
In this subsection, the total uncertainties including the unknown internal unmodeled dynamics and the unknown external ocean disturbances are needed to be estimated. For the nonlinear system (9) with unknown nonlinear time-varying uncertainties, a homogenous nonlinear extended state observer is constructed as  
\[\left\{ \begin{array}{l}\dot{\hat{\mathbf{\eta}}}=\mathbf{\mathrm{R}}(\psi)\hat{\mathbf{ \upsilon}}-\beta_{1}\mathbf{\mathrm{R}}(\psi)\big{[}\mathbf{\mathrm{R}}^{T}(\psi)( \hat{\mathbf{\eta}}-\mathbf{\mathrm{\eta}})\big{]}^{\alpha}\\ \dot{\hat{\mathbf{\upsilon}}}=\hat{\mathbf{d}}-\beta_{2}\big{[}\mathbf{\mathrm{R}}^{T}(\psi )(\hat{\mathbf{\eta}}-\mathbf{\mathrm{\eta}})\big{]}^{2\alpha-1}+\mathbf{\mathrm{M}}^{-1} \tau\\ \dot{\hat{\mathbf{\mathrm{d}}}}=-\beta_{3}\big{[}\mathbf{\mathrm{R}}^{T}(\psi)(\hat{ \mathbf{\eta}}-\mathbf{\mathrm{\eta}})\big{]}^{3\alpha-2}\end{array} \right., \tag{16}\]  
where \([\mathbf{\mathrm{x}}]^{\alpha}=[sign(x_{1})|x_{1}|^{\alpha}\), \(sign(x_{2})|x_{2}|^{\alpha}\), \(sign(x_{3})|x_{3}|^{\alpha}]^{T}\in\mathfrak{R}^{3}\), \(\mathbf{\mathrm{x}}=[x_{1},x_{2},x_{3}]^{T}\in\mathfrak{R}^{3}\), \(\alpha\in(2/3,1)\), \(\beta_{i}(i=1,2,3)\in\mathfrak{R}\) are the designed positive parameters, and \(\hat{\mathbf{\mathrm{\eta}}}=\big{[}\hat{x},\hat{y},\hat{\psi}\big{]}^{T}\in \mathfrak{R}^{3}\), \(\hat{\mathbf{\upsilon}}=[\hat{u},\hat{\mathbf{\upsilon}},\hat{\mathbf{\upsilon}}]^{T}\in \mathfrak{R}^{3}\), \(\hat{\mathbf{\mathrm{d}}}=\big{[}\hat{d}_{u},\hat{d}_{v},\hat{d}_{t}\big{]}^{T}\in \mathfrak{R}^{3}\) with \(\hat{x},\hat{y},\hat{\psi},\hat{u},\hat{\mathbf{\upsilon}},\hat{\mathbf{\upsilon}}, \hat{\mathbf{\upsilon}},\hat{\mathbf{\upsilon}},\hat{\mathbf{\upsilon}},\hat{d}_{u},\hat{d }_{v}\), and \(\hat{d}_{r}\) are the estimations of \(x,y,\psi,u,v,r,d_{u},d_{v}\) and \(d_{r}\), respectively. Denoting the estimation error vectors as  
\[\left\{ \begin{array}{l}\overline{\mathbf{\eta}}=\hat{\mathbf{\mathrm{\eta}}}-\mathbf{ \mathrm{\eta}}\\ \overline{\mathbf{\upsilon}}=\hat{\mathbf{\upsilon}}-\mathbf{\mathrm{\upsilon}}\\ \overline{\mathbf{\mathrm{d}}}=\hat{\mathbf{\mathrm{d}}}-\mathbf{\mathrm{d}}\end{array} \right. \tag{17}\]  
and  
\[\left\{ \begin{array}{l}\mathbf{\mathrm{E}}_{1}(t)=\mathbf{\mathrm{R}}^{T}(\psi) \overline{\mathbf{\mathrm{\eta}}}(t)\\ \mathbf{\mathrm{E}}_{2}(t)=\overline{\mathbf{\upsilon}}(t)\\ \mathbf{\mathrm{E}}_{3}(t)=\overline{\mathbf{\mathrm{d}}}(t)\end{array} \right. \tag{18}\]  
then, it is follows (17) and (18) that the error dynamics of the NESO can be described as  
\[\left\{ \begin{array}{l}\dot{\mathbf{\mathrm{E}}}_{1}(t)=\mathbf{\mathrm{E}}_{2}(t)- \beta_{1}[\mathbf{\mathrm{E}}_{1}]^{\alpha}\\ \dot{\mathbf{\mathrm{E}}}_{2}(t)=\mathbf{\mathrm{E}}_{3}(t)-\beta_{2}[\mathbf{\mathrm{E}}_{1 }]^{2\alpha-1}\\ \dot{\mathbf{\mathrm{E}}}_{3}(t)=-\beta_{3}[\mathbf{\mathrm{E}}_{1}]^{3\alpha-2}-\mathbf{ \mathrm{d}}(t)\end{array} \right. \tag{19}\]To facilitate stability analysis of observer error dynamics (19), define observer error vector \(\mathbf{E}_{o}=\left[\mathbf{E}_{1}^{T},\mathbf{E}_{2}^{T},\mathbf{E}_{3}^{T} \right]^{T}\in\Re^{9}\), then the observer error dynamics can be rewritten as  
\[\dot{\mathbf{E}}_{o}=\mathbf{F}(\mathbf{E}_{o})+\mathbf{B}\dot{\mathbf{d}}(t) \tag{20}\]  
where \(\mathbf{F}(\mathbf{E}_{o})=\left[-\beta_{1}[\mathbf{E}_{1}(t)]^{\alpha}+ \mathbf{E}_{2}(t),-\beta_{2}[\mathbf{E}_{1}(t)]^{2\alpha-1}+\mathbf{E}_{3}(t),- \beta_{3}[\mathbf{E}_{1}(t)]^{3\alpha-2}\right]^{T}\in\Re^{9}\) and \(\mathbf{B}=[0_{3\times 3},0_{3\times 3},-\mathbf{I}_{3\times 3}]^{T}\in\Re^{9 \times 3}\). **Theorem 1**.