## 2 Generalized Super-Twisting Algorithm-Based Controller and Observer Design
### Generalized Super-Twisting Algorithm Controller Design  
Consider the following first-order single-input single-output (SISO) nonlinear system  
\[\dot{\theta}=bu+f(\theta,t) \tag{4}\]  
where \(\theta\) denotes the sliding variable, \(f(\theta,t)\) represents an unknown perturbation that is bounded by \(|f(\theta,t)|\leq\delta\) with \(\delta\) as an unknown positive constant, \(b\) is the control gain, and \(u\) is the control input. The generalized STA (GSTA) [41] is a modified version of the conventional STA by simply adding some extra correction terms, which was originally introduced by [PERSON]. Since the added linear terms are stronger than the nonlinear ones when the system trajectories are far from the origin, the robustness and the convergence speed of the closed-loop system are improved accordingly. The GSTA-based controller of the system (4) is given by  
\[ \begin{cases}u=\frac{1}{b}(-k_{1}\overline{\omega}_{1}(\theta)+v)\\ \dot{v}=-k_{2}\overline{\omega}_{2}(\theta)\end{cases}  \tag{5}\]  
where \(k_{1}\) and \(k_{2}\) are positive constants to be designed, and the two functions \(\overline{\omega}_{1}(\theta)\) and \(\overline{\omega}_{2}(\theta)\) are given by  
\[ \begin{split}\overline{\omega}_{1}(\theta)&=\mu_{1}| \theta|^{1/2}\text{sign}(\theta)+\mu_{2}\theta\\ \overline{\omega}_{2}(\theta)&=\frac{\mu_{1}^{2}}{2} \text{sign}(\theta)+\frac{3}{2}\mu_{1}\mu_{2}|\theta|^{1/2}\text{sign}(\theta )+\mu_{2}^{2}\theta\end{split}  \tag{6}\]  
where \(\mu_{1}>0\), \(\mu_{2}>0\), and \(\text{sign}(\bullet)\) is the standard signum function, which is given by  
\[\text{sign}(\vartheta)=\left\{ \begin{array}{lcl}1,&\text{if}&\vartheta>0\\ 0,&\text{if}&\vartheta=0\\ -1,&\text{otherwise}\end{array} \right. \tag{7}\]  
Based on the definition of the functions \(\overline{\omega}_{1}(\theta)\) and \(\overline{\omega}_{2}(\theta)\) in (6), we have  
\[ \begin{split}\overline{\omega}^{\prime}{}_{1}(\theta)&= \frac{1}{2}\mu_{1}|\theta|^{-1/2}+\mu_{2}\\ \overline{\omega}_{2}(\theta)&=\overline{\omega}_{1}(\theta) \hat{\omega}^{\prime}_{1}(\theta)\end{split}  \tag{8}\]  
where \(\overline{\omega}^{\prime}{}_{1}(\theta)\) denotes the partial derivative with respect to \(\theta\). According to the control laws (5) and the system dynamics (4), the closed-loop system dynamics become  
\[ \begin{split}\dot{\theta}&=-k_{1}\overline{\omega}_{1}( \theta)+v+f(\theta,t)\\ \dot{v}&=-k_{2}\overline{\omega}_{2}(\theta)\end{split}  \tag{9}\]  
**Lemma 2**. : _Consider the system dynamics (4), by using the control laws given in (5), the closed-loop system is finite-time stable in the presence of the perturbation, which is bounded by an unknown positive constant._  
Proof of Lemma 2.: Define the augmented state vector as \(\mathbf{\chi}=[\overline{\omega}_{1}(\theta),v]^{T}\). Taking the time derivative, one obtains  
\[ \begin{split}\dot{\mathbf{\chi}}&=\left[\begin{array}{ cc}\bar{\omega}^{\prime}_{1}(\theta)\dot{\theta}\\ -k_{2}\overline{\omega}_{2}(\theta)\end{array}\right]\\ &=\overline{\omega}^{\prime}{}_{1}(\theta)\left[\begin{array}{ cc}-k_{1}&1\\ -k_{2}&0\end{array}\right]\left[\begin{array}{cc}\overline{\omega}_{1}(\theta )\\ v\end{array}\right]+\left[\begin{array}{cc}1\\ 0\end{array}\right]\overline{\omega}^{\prime}{}_{1}(\theta)f(\theta,t)\\ &=\overline{\omega}^{\prime}{}_{1}(\theta)\mathbf{A}_{\mathbf{\chi}}\mathbf{\chi}+ \mathbf{B}_{\mathbf{\chi}}\overline{\omega}^{\prime}{}_{1}(\theta)f(\theta,t) \end{split}  \tag{10}\]where the matrices \(\mathbf{A}_{\chi}\) and \(\mathbf{B}_{\chi}\) are given by  
\[\mathbf{A}_{\chi}=\left[ \begin{array}{cc}-k_{1}&1\\ -k_{2}&0\end{array} \right];\mathbf{B}_{\chi}=\left[ \begin{array}{c}1\\ 0\end{array}