## 2 Generalized Super-Twisting Algorithm-Based Controller and Observer Design
### Nonlinear Continuous ESO Design  
Consider the following system  
\[\dot{x}=\psi u+\hbar(x,t) \tag{18}\]  
where \(\hbar(x,t)\) is the lumped disturbance term, \(u\) is the input of the system, \(\psi\) denotes the known control gain, and \(x\) represents the system state that is measurable. It is assumed that the lumped disturbances \(\hbar(x,t)\) are differential, and the first-order time derivative of it is bounded by an unknown constant, i.e., \(\left|\hbar(x,t)\right|\leq\Delta\). Defining \(x_{e}=\hbar(x,t)\) as an extended system state, the system (18) can be transformed into the following augmented system  
\[ \begin{split}\dot{x}&=\psi u+x_{e}\\ \dot{x}_{e}&=g(x,t)\end{split}  \tag{19}\]  
where \(g(x,t)\) is the first-order derivative of \(\hbar(x,t)\). The nonlinear continuous ESO [42] based on GSTA is constructed to estimate the lumped disturbance term as follows:  
\[ \begin{cases}\dot{x}=\psi u+\hat{x}_{e}+\tau_{1}\phi_{1}(\hat{x})\\ \dot{\hat{x}}_{e}=\pi_{2}\phi_{2}(\hat{x})\end{cases}  \tag{20}\]  
where \(\hat{x}\) and \(\hat{x}_{e}\) are estimates of \(x\) and \(x_{e}\), respectively; \(\tilde{x}=x-\hat{x}\) is the correction term; \(\tau_{1}>0\) and \(\pi_{2}>0\) represent the observer gains; and the two functions \(\phi_{1}(\tilde{x})\) and \(\phi_{2}(\tilde{x})\) are given by  
\[ \begin{split}\phi_{1}(\tilde{x})&=\gamma_{1}|\tilde{ x}|^{1/2}\text{sign}(\tilde{x})+\gamma_{2}\tilde{x}\\ \phi_{2}(\tilde{x})&=\frac{1}{2}\gamma_{1}^{2} \text{sign}(\tilde{x})+\frac{3}{2}\gamma_{1}\gamma_{2}|\tilde{x}|^{1/2}\text{ sign}(\tilde{x})+\gamma_{2}\tilde{x}\end{split}  \tag{21}\]  
where \(\gamma_{1}>0\) and \(\gamma_{2}>0\). According to (19) and (20), the error dynamics of the observer are obtained as  
\[ \begin{split}\dot{\tilde{x}}&=\tilde{x}_{e}-\tau_ {1}\phi_{1}(\tilde{x})\\ \dot{\hat{x}}_{e}&=-\tau_{2}\phi_{2}(\tilde{x})+g(t) \end{split}  \tag{22}\]  
**Lemma 3**. : _Consider the system dynamics (19), the GSTA-based nonlinear continuous ESO (20) guarantees that the estimation errors converge to an arbitrarily small region of the origin in finite time whose bounds depend on the selection of the observer gains \(\tau_{1}\) and \(\pi_{2}\), and the tuning parameters \(\gamma_{1}\) and \(\gamma_{2}\)._  
Proof of Lemma 3.: Define the new state vector \(\boldsymbol{\zeta}=\left[\xi_{1},\xi_{2}\right]^{T}=\left[\phi_{1}(\tilde{x}),\tilde{x}_{e}\right]^{T}\). Based on this, we have  
\[ \begin{split}\boldsymbol{\dot{\zeta}}&=\left[ \begin{array}{c}\phi^{\prime}{}_{1}(\tilde{x})\hat{x}\\ -\tau_{2}\phi_{2}(\tilde{x})+g(t)\end{array}\right]\\ &=\phi^{\prime}{}_{1}(\tilde{x})\mathbf{A}_{\tilde{\zeta}}\boldsymbol{ \zeta}+\mathbf{B}_{\xi}g(t)\end{split}  \tag{23}\]  
where the matrices \(\mathbf{A}_{\tilde{\zeta}}\) and \(\mathbf{B}_{\tilde{\zeta}}\) are given by  
\[\mathbf{A}_{\tilde{\zeta}}=\left[ \begin{array}{cc}-\tau_{1}&1\\ -\tau_{2}&0\end{array} \right];\mathbf{B}_{\tilde{\zeta}}=\left[ \begin{array}{ cc}0\\ 1\end{array} \right]\]  
Since the matrix \(\mathbf{A}_{\tilde{\zeta}}\) is negative definite, there exists a positive symmetric matrix \(\mathbf{P}_{\tilde{\zeta}}\) satisfying the following Lyapunov equation\[\mathbf{P}_{\zeta}\mathbf{A}_{\zeta}+\mathbf{A}_{\zeta}^{T}\mathbf{P}_{\zeta}=- \mathbf{Q}_{\zeta} \tag{24}\]  
A candidate Lyapunov function is chosen as  
\[V_{\zeta}=\boldsymbol{\zeta}^{T}\mathbf{P}_{\zeta}\boldsymbol{\zeta} \tag{25}\]  
Based on this, we have  
\[\lambda_{\min}\big{(}\mathbf{P}_{\zeta}\big{)}\|\boldsymbol{\xi}\|^{2}\leq \boldsymbol{\zeta}^{T}\mathbf{P}_{\zeta}\boldsymbol{\xi}\leq\lambda_{\max} \big{(}\mathbf{P}_{\zeta}\big{)}\|\boldsymbol{\xi}\|^{2} \tag{26}\]  
Moreover, according to the definition of \(\boldsymbol{\zeta}\), one obtains  
\[|\boldsymbol{\xi}|^{-1/2}\geq\gamma_{1}\sqrt{\lambda_{\min}(\mathbf{P}_{ \zeta})}V_{\zeta}^{-1/2} \tag{27}\]  
Taking the derivative of (25) yields  
\[