## 3 APPSMC for MEMS Gyroscopes
\begin{array}{l}\ddot{\theta}_{i}=\frac{\rho_{i}^{3}\ddot{\epsilon}_{i}-\rho_ {i}^{2}\ddot{\rho}_{i}\dot{\epsilon}-2\rho_{i}^{2}\dot{\rho}_{i}\dot{\epsilon }+2\epsilon_{i}\dot{\rho}_{i}\dot{\rho}_{i}^{2}\dot{\epsilon}_{i}}{\rho_{i}^{ 3}}\\ =\frac{1}{\rho_{i}}\ddot{\epsilon}_{i}+\frac{-\rho_{i}\ddot{\rho}_{i}\dot{\epsilon }-2\rho_{i}\dot{\rho}_{i}\dot{\epsilon}+2\epsilon_{i}\dot{\rho}_{i}^{2}\dot{ \epsilon}_{i}}{\rho_{i}^{3}},i=1,2\\ =r_{1i}\ddot{\epsilon}_{i}+r_{2i},i=1,2\end{array}  \tag{19}\]  
where:  
\(r_{1i}=\frac{1}{\rho_{i}},r_{2i}=\frac{-\rho_{i}\ddot{\rho}_{i}\dot{\epsilon }-2\rho_{i}\dot{\rho}_{i}\dot{\epsilon}+2\dot{\rho}_{i}\dot{\rho}_{i}^{2}\dot {\epsilon}_{i}}{\rho_{i}^{3}},i=1,2\)  
Equation (19) is rewritten in vector form as:  
\[\ddot{\bar{\theta}}=\mathbf{r_{1}}\bar{\mathbf{e}}+\mathbf{r_{2}} \tag{20}\]  
where \(\theta=\left[ \begin{array}{c}\theta_{1}\\ \theta_{2}\end{array} \right]\), \(\mathbf{r_{1}}=\left[ \begin{array}{cc}r_{11}&0\\ 0&r_{12}\end{array} \right]\), \(\mathbf{e}=\left[ \begin{array}{c}e_{1}\\ e_{2}\end{array} \right]\) and \(\mathbf{r_{2}}=\left[ \begin{array}{c}r_{21}\\ r_{22}\end{array} \right]\)  
Substitute (20) into (17) gives:  
\[\bar{\epsilon}=\mathbf{k_{1}r_{1}}\bar{\mathbf{e}}+\mathbf{k_{1}r_{2}}+ \mathbf{k_{2}} \tag{21}\]  
Substitute (21) into (13) leads to:  
\[ \begin{array}{l}\dot{\mathbf{S}}=\bar{\epsilon}+\lambda\dot{\epsilon}\\ =\mathbf{k_{1}r_{1}}\bar{\mathbf{e}}+\mathbf{k_{1}r_{2}}+\mathbf{k_{2}}+ \lambda\dot{\epsilon}\end{array}  \tag{22}\]  
Substitute system dynamics in (4) into (22), it reveals that:  
\[ \begin{array}{l}\dot{\mathbf{S}}=\mathbf{k_{1}r_{1}}(\ddot{\bar{\mathbf{q}}} _{\mathbf{d}}-(\mathbf{U}-(\mathbf{D}+2\mathbf{\Omega})\dot{\mathbf{q}}_{1}- \mathbf{Kq_{1}}+\mathbf{F}))\\ +\mathbf{k_{1}r_{2}}+\mathbf{k_{2}}+\lambda\dot{\epsilon}\end{array}  \tag{23}\]  
Equivalent control force can be derived by ignoring the lumped error \(F\) and setting \(\dot{S}=0\):  
\[ \begin{array}{l}\mathbf{u_{eq}}=\ddot{\mathbf{q}}_{\mathbf{d}}+(\mathbf{D}+2 \mathbf{\Omega})\dot{\mathbf{q}}+\mathbf{Kq}\\ +(\mathbf{k_{1}r_{1}})^{-1}(\mathbf{k_{1}r_{2}}+\mathbf{k_{2}}+\lambda\dot{ \epsilon})\end{array}  \tag{24}\]  
The actual sliding mode control force is designed in the form:  
\[\mathbf{U}=\mathbf{u_{eq}}+\mathbf{u_{sw}} \tag{25}\]  
where the switching control force \(\mathbf{u_{sw}}\) is designed in the form:  
\[\mathbf{u_{sw}}=\mathbf{\gamma}\text{sign}(\mathbf{S}) \tag{26}\]where \(\mathbf{\gamma}=\left[ \begin{array}{cc}\gamma_{1}&0\\ 0&\gamma_{2}\end{array} \right]\) is the gain of the switching term. **Theorem 1**. : _If the control force in (25) is applied to the MEMS system with robust gain in (26) larger than the lumped disturbance bound, system tracking error will asymptotically converge to 0._  
Proof. : Choose a Lyapunov candidate in the form:  
\[\mathbf{V}=\frac{1}{2}\mathbf{S}^{\mathsf{T}}\mathbf{S} \tag{27}\]  
Differentiating (27) with respect to time leads to:  
\[ \begin{array}{l}\dot{\mathbf{V}}=\mathbf{S}^{\mathsf{T}}\dot{\mathbf{S}}\\ =\mathbf{S}^{\mathsf{T}}(\mathbf{k}_{1}\mathbf{r}_{1}(\mathbf{\tilde{q}_{d}}-( \mathbf{U}-(\mathbf{D}+2\mathbf{\Omega})\dot{\mathbf{q}}-\mathbf{Kq}+\mathbf{F})) )\\ +\mathbf{k}_{1}\mathbf{k}_{2}+\mathbf{k}_{2}+\lambda\dot{\mathbf{\varepsilon}} )\end{array}  \tag{28}\]  
Substitute the control force in (25) into (28) leads to:  
\[ \begin{array}{l}\dot{\mathbf{V}}\leq\mathbf{S}^{\mathsf{T}}(\mathbf{k}_{1} \mathbf{r}_{1}(-\mathbf{\gamma}\cdot\text{sign}(\mathbf{S})+\mathbf{F}))\\ \leq\mathbf{S}^{\mathsf{T}}(\mathbf{k}_{1}\mathbf{r}_{1}(-\mathbf{\gamma}\cdot \text{sign}(\mathbf{S})+\mathbf{F}_{\mathbf{d}}))\\ \leq\left|\mathbf{S}^{\mathsf{T}}\right|(\mathbf{k}_{1}\mathbf{r}_{1}(-\mathbf{ \gamma}+\mathbf{F}_{\mathbf{d}}))\end{array}