## 2 Generalized Super-Twisting Algorithm-Based Controller and Observer Design
### Generalized Super-Twisting Algorithm Controller Design
\right]\]  
Based on the definition of vector \(\chi\), an important property can be derived as  
\[|\theta|^{-1/2}\geq\frac{\mu_{1}}{\|\chi\|};|\theta|^{-1}\geq\frac{\mu_{2}}{ \|\chi\|} \tag{11}\]  
Since the matrix \(\mathbf{A}_{\chi}\) is Hurwitz, there exists a symmetric positive definite \(\mathbf{P}_{\chi}\) satisfying the following Lyapunov equation  
\[\mathbf{P}_{\chi}\mathbf{A}_{\chi}+\mathbf{A}_{\chi}^{\mathrm{T}}\mathbf{P}_ {\chi}=-\mathbf{Q}_{\chi} \tag{12}\]  
where \(\mathbf{Q}_{\chi}\) is a symmetric positive definite matrix. Consider the following Lyapunov function  
\[V_{\chi}=\boldsymbol{\chi}^{\mathrm{T}}\mathbf{P}_{\chi}\boldsymbol{\chi} \tag{13}\]  
From (13), we have  
\[\lambda_{\min}(\mathbf{P}_{\chi})||\boldsymbol{\chi}||^{2}\leq V_{\chi}\leq \lambda_{\max}(\mathbf{P}_{\chi})||\boldsymbol{\chi}||^{2} \tag{14}\]  
Taking the derivative of (13) and then combining with (12), one obtains  
\[\dot{V}_{\chi} =\boldsymbol{\chi}^{\mathrm{T}}\mathbf{P}_{\chi}\boldsymbol{\chi }+\boldsymbol{\chi}^{\mathrm{T}}\mathbf{P}_{\chi}\boldsymbol{\hat{x}} \tag{15}\] \[=-\overline{\omega}^{\prime}{}_{1}(\theta)\boldsymbol{\chi}^{ \mathrm{T}}\mathbf{Q}_{\chi}\boldsymbol{\chi}+2\boldsymbol{\partial}^{T} \mathbf{P}_{\chi}\mathbf{B}_{\chi}\overline{\omega}^{\prime}{}_{1}(\theta)f( \theta,t)\]  
Substituting (8) and (11) into (15), we have  
\[\dot{V}_{\chi}= -\bigg{(}\frac{1}{2}\mu_{1}|\theta|^{-1/2}+\mu_{2}\bigg{)} \boldsymbol{\chi}^{\mathrm{T}}\mathbf{Q}_{\chi}\boldsymbol{\chi}+2 \boldsymbol{\chi}^{\mathrm{T}}\mathbf{P}_{\chi}\mathbf{B}_{\chi}\bigg{(}\frac{ 1}{2}\mu_{1}|\theta|^{-1/2}+\mu_{2}\bigg{)}f(\theta,t) \tag{16}\] \[\leq -\frac{1}{2}\mu_{1}^{2}\lambda_{\min}(\mathbf{Q}_{\chi})\| \boldsymbol{\chi}\|-\mu_{2}\lambda_{\min}(\mathbf{Q}_{\chi})\|\boldsymbol{ \chi}\|^{2}+\mu_{1}|\theta|^{-1/2}\boldsymbol{\chi}^{\mathrm{T}}\mathbf{P}_{ \chi}\mathbf{B}_{\chi}\overline{\omega}^{\prime}{}_{1}(\theta,t)\] \[+2\mu_{2}\boldsymbol{\chi}^{\mathrm{T}}\mathbf{P}_{\chi}\mathbf{B }_{\chi}f(\theta,t)\]  
Applying the [PERSON]'s inequality and applying (11) leads to  
\[\dot{V}_{\chi}\leq -\frac{1}{2}\mu_{1}^{2}\lambda_{\min}(\mathbf{Q}_{\chi})\| \boldsymbol{\chi}\|-\mu_{2}\lambda_{\min}(\mathbf{Q}_{\chi})\|\boldsymbol{\chi }\|^{2}+\frac{1}{2}\mu_{1}^{2}\mu_{2}\|\boldsymbol{\chi}\|+\frac{1}{2}\mathbf{ B}_{\chi}^{\mathrm{T}}\mathbf{P}_{\chi}^{\mathrm{T}}\mathbf{P}_{\chi} \mathbf{B}_{\chi}\delta^{2} \tag{17}\] \[\mathbf{P}_{\chi}\mathbf{A}_{\chi}+\mathbf{A}_{\chi}^{\mathrm{T}}\mathbf{P}_ {\chi}=-\mathbf{Q}_{\chi} \tag{12}\]0 \[\mathbf{P}_{\chi}\mathbf{A}_{\chi}+\mathbf{A}_{\chi}^{\mathrm{T}}\mathbf{P}_ {\chi}=-\mathbf{Q}_{\chi} \tag{12}\]1 \[\mathbf{P}_{\chi}\mathbf{A}_{\chi}+\mathbf{A}_{\chi}^{\mathrm{T}}\mathbf{P}_ {\chi}=-\mathbf{Q}_{\chi} \tag{12}\]2  
where  
\[\mathbf{P}_{\chi}\mathbf{A}_{\chi}+\mathbf{A}_{\chi}^{\mathrm{T}}\mathbf{P}_ {\chi}=-\mathbf{Q}_{\chi} \tag{12}\]3  
Based on Lemma 1, the function \(V_{\chi}\) given by (13) reaches a region whose bound is determined by \(k_{1}\), \(k_{2}\), \(\mu_{1}\), and \(\mu_{2}\) in finite time. Hence, according to the definition of the vector \(\boldsymbol{\chi}\), the sliding variable \(\theta\) is restricted by an arbitrarily small value as a result. This completes the Proof of Lemma 2.