## 3 Controller and Observer
### Objective Sliding Mode Control (OSMC)
 \tag{17}\]  
Substituting \(\dot{\textbf{s}}\) = 0 into (17), Equation (17) can be expressed as  
\[\ddot{\textbf{e}}=-\frac{\alpha mf(\textbf{e})^{m-1}f(\textbf{e})^{\prime}\dot {\textbf{e}}+\textbf{e}f(\textbf{e})^{m}+\textbf{e}mf(\textbf{e})^{m-1}f( \textbf{e})^{\prime}\dot{\textbf{e}}+\dot{\textbf{e}}f\big{(}\dot{\textbf{e }}\big{)}^{n}}{\Big{(}1+\beta nf\big{(}\dot{\textbf{e}}\big{)}^{n-1}f\big{(} \dot{\textbf{e  
\[\dot{V}_{1}= \quad\mathbf{e}^{\mathrm{T}}\dot{\mathbf{e}} \tag{25}\] \[=\mathbf{e}^{\mathrm{T}}\Bigl{(}-(\alpha+\mathbf{e})f(\mathbf{e})^{m}-\bigl{(} \beta+\dot{\mathbf{e}}\bigr{)}f\bigl{(}\dot{\mathbf{e}}\bigr{)}^{n}\Bigr{)}\] \[=-(\alpha+\mathbf{e})\mathbf{e}^{\mathrm{T}}\mathbf{e}\biggl{(}1+\frac{a}{ \exp\bigl{(}\left\lVert\dot{\mathbf{e}}\right\rVert^{2}\bigr{)}}\biggr{)}^{m}- \bigl{(}\beta+\dot{\mathbf{e}}\bigr{)}\mathbf{e}^{\mathrm{T}}\mathbf{e}\biggl{(}1+\frac{a}{ \exp\bigl{(}\left\lVert\dot{\mathbf{e}}\right\rVert^{2}\bigr{)}}\biggr{)}^{n}\] \[\leq-\lVert\dot{\mathbf{e}}\rVert\biggl{(}(\alpha+\mathbf{e})\biggl{(}1+ \frac{a}{\exp\bigl{(}\left\lVert\dot{\mathbf{e}}\right\rVert^{2}\bigr{)}}\biggr{)} ^{m}+\bigl{(}\beta+\dot{\mathbf{e}}\bigr{)}\biggl{(}1+\frac{a}{\exp\bigl{(}\left \lVert\dot{\mathbf{e}}\right\rVert^{2}\bigr{)}}\biggr{)}^{n}\biggr{)}\frac{\sqrt{ \beta}\left\lVert\dot{\mathbf{e}}\right\rVert}{\sqrt{2}}\] \[=-\sqrt{2}\lVert\dot{\mathbf{e}}\rVert\biggl{(}(\alpha+\mathbf{e})\biggl{(} 1+\frac{a}{\exp\bigl{(}\left\lVert\dot{\mathbf{e}}\right\rVert^{2}\bigr{)}} \biggr{)}^{m}+\bigl{(}\beta+\dot{\mathbf{e}}\bigr{)}\biggl{(}1+\frac{a}{\exp\bigl{(} \left\lVert\dot{\mathbf{e}}\right\rVert^{2}\bigr{)}}\biggr{)}^{n}\biggr{)}V_{1}^{ \frac{1}{2}}\]  
According to Lemma 1, \(t_{\varepsilon}\) satisfies  
\[t_{\varepsilon}\leq\frac{\sqrt{2}V_{10}^{\frac{1}{2}}}{\lVert\mathbf{e}_{0} \rVert\biggl{(}(\alpha+\mathbf{e})\biggl{(}1+\frac{a}{\exp\bigl{(}\left\lVert\dot {\mathbf{e}}\right\rVert^{2}\bigr{)}}\biggr{)}^{m}+\bigl{(}\beta+\dot{\mathbf{e}} \bigr{)}\biggl{(}1+\frac{a}{\exp\bigl{(}\left\lVert\dot{\mathbf{e}}\right\rVert^ {2}\bigr{)}}\biggr{)}^{n}\biggr{)}} \tag{26}\]  
In (26), \(V_{10}\) is the initial value of \(V_{1}\). In order to make the system closer to the actual situation, the control strategy can be divided into two parts (switching control and equivalent control), which can be expressed as  
\[\mathbf{\tau}=\mathbf{\tau}_{eq}+\mathbf{\tau}_{sw} \tag{27}\]  
And in order to make the controller performance achieve the expected effect, the form of the function is designed as:  
\[\mathrm{dst}(x)=\xi\frac{\mathbf{e}^{\lvert x\rvert}+c_{1}}{\mathbf{e}^{\lvert x\rvert }+c_{2}} \tag{28}\]  
Now, we will prove that the function \(\mathrm{dst}(x)\) is bounded. Proof. : Function \(\mathrm{dst}(x)\) can be expressed in the following form  
\[\mathrm{dst}(x)=\xi\biggl{(}1+\frac{C_{1}-C_{2}}{e^{\lvert x\rvert}+C_{2}} \biggr{)} \tag{29}\]  
where \(\xi\), \(C_{1}\) and \(C_{2}\) are all specified small positive numbers, and \(e^{\lvert x\rvert}\geq 1\). Thus, we can conclude that \(C_{1}-C_{2}\) and \(e^{\lvert x\rvert}+C_{2}\) are bounded. So, \(1+\frac{C_{1}-C_{2}}{e^{\lvert x\rvert}+C_{2}}\) is bounded, and the function \(\mathrm{dst}(x)\) is bounded. \(\square\)  
The switching control \(\mathbf{\tau}_{sw}\) is  
\[\mathbf{\tau}_{sw}=\mathrm{dst}\Bigl{(}\delta_{1}^{-1}\mathrm{s}\Bigr{)}+c_{3}M_{ 0}\mathrm{dst}(\mathbf{s}) \tag{30}\]  
In (30), \(\delta_{1}\) is the switching coefficient matrix. \[=\mathbf{e}^{\mathrm{T}}\Bigl{(}-(\alpha+\mathbf{e})f(\mathbf{e})^{m}-\bigl{(} \beta+\dot{\mathbf{e}}\bigr{)}f\bigl{(}\dot{\mathbf{e}}\bigr{)}^{n}\Bigr{)}\]0  
The total control law of OSMC is \[