## 3 APPSMC for MEMS Gyroscopes
 \tag{29}\]  
It can be inferred from Assumption 1 that if the robust gain \(\mathbf{\gamma}\) is set such that \(F_{di}\leq\gamma_{i},i=1,2\), together with the fact that \(\mathbf{k}_{1}\) and \(\mathbf{r}_{1}\) are positive diagonal matrices, the derivative of the Lyapunov function is non-positive definite in the following form. \[\dot{\mathbf{V}}\leq 0 \tag{30}\]  
It can be seen from the designed control force (24) that the control force contains three different system parameter matrices which are actually unknown. Thus, we actually cannot carried out the control force directly. AC can be used in the paper to tune control force parameters and the control force in (25) is modified in the following form where three adaptive laws are given in (32)-(34). \[ \begin{array}{l}\mathbf{U}=\hat{\mathbf{u}}_{\mathbf{eq}}+\mathbf{u}_{ \mathbf{sw}}\\ =\ddot{\mathbf{q}}_{\mathbf{d}}+(\mathbf{\dot{D}}+2\mathbf{\hat{\Omega}})\dot{ \mathbf{q}}\\ +\dot{\mathbf{Kq}}+(\mathbf{k}_{1}\mathbf{r}_{1})^{-1}\Big{(}\mathbf{k}_{1} \mathbf{r}_{2}\mathbf{+k}_{2}+\lambda\dot{\mathbf{\varepsilon}}\Big{)}+ \mathbf{u}_{\mathbf{sw}}\end{array}  \tag{31}\]  
where \(\mathbf{\dot{D}}\), \(\mathbf{\hat{\Omega}}\), and \(\mathbf{\dot{K}}\) are set to replace the corresponding parameters and they are the estimates for \(\mathbf{D}\), \(\mathbf{K}\), and \(\mathbf{\Omega}\),respectively. Adaptive laws for \(\mathbf{\dot{D}}\), \(\mathbf{\hat{\Omega}}\), and \(\mathbf{\dot{K}}\) are set in the following form. \[\dot{\mathbf{\dot{D}}}^{\mathsf{T}}=\frac{\dot{\mathbf{q}}\mathbf{S}^{\mathsf{ T}}\mathbf{k}_{1}\mathbf{r}_{1}}{\eta_{1}} \tag{32}\]  
\[\dot{\mathbf{\dot{\Omega}}}^{\mathsf{T}}=2\frac{\dot{\mathbf{q}}\mathbf{S}^{ \mathsf{T}}\mathbf{k}_{1}\mathbf{r}_{1}}{\eta_{2}} \tag{33}\]  
\[\dot{\mathbf{\dot{K}}}^{\mathsf{T}}=\frac{\dot{\mathbf{q}}\mathbf{S}^{ \mathsf{T}}\mathbf{k}_{1}\mathbf{r}_{1}}{\eta_{3}} \tag{34}\]  
**Theorem 2**. : _if the control force in (31) in applied to the MEMS system, adaptive laws (32)-(34) are incorporated to online tune control force parameters, system tracking error will asymptotically converge to 0. Besides, according to parameter identification theory [30], all system parameters can be correctly estimated if the persistent excitation condition is satisfied._  
Figure 3 shows the block diagram of the entire APPSMC system. **Proof. ** Select a Lyapunov candidate in the form:  
\[V=\frac{1}{2}\mathbf{S}^{T}\mathbf{S}+\frac{1}{2}tr(\eta_{1}\mathbf{\widetilde{ D}}^{T}\mathbf{\widetilde{D}})+\frac{1}{2}tr(\eta_{2}\mathbf{\widetilde{K}}^{T} \mathbf{\widetilde{K}})+\frac{1}{2}tr(\eta_{3}\mathbf{\Omega}^{T}\mathbf{ \Omega}) \tag{35}\]  
where the function contains sliding surface and parameter estimation error. Parameter estimation errors are more specifically described in the following form:  
\[\mathbf{\widetilde{D}}=\mathbf{D}-\mathbf{\hat{D}},\mathbf{\widetilde{K}}= \mathbf{K}-\mathbf{\hat{K}},\mathbf{\widetilde{\Omega}}=\mathbf{\Omega}- \mathbf{\hat{\Omega}}\]  
\(\eta_{1}\), \(\eta_{2}\), and \(\eta_{3}\) are positive constants which can be viewed as fix gains for adaptive laws. Differentiating right hand and left hand of (35) and substituting system dynamic in (4) into it yields:  
\[ \begin{array}{l}\dot{V}=\mathbf{S}^{T}\mathbf{\hat{S}}+tr(\eta_{1}\mathbf{ \hat{\widetilde{D}}}^{T}\mathbf{\widetilde{D}})+tr(\eta_{2}\mathbf{\hat{ \widetilde{K}}}^{T}\mathbf{\widetilde{K}})+tr(\eta_{3}\mathbf{\hat{\Omega}}^{ T}\mathbf{\Omega})\\ \quad=\mathbf{S}^{T}(\mathbf{k_{1}r_{1}}(\mathbf{\tilde{q}_{d}}-(\mathbf{U}-( \mathbf{D}+2\mathbf{\Omega})\mathbf{\hat{q}}-\mathbf{Kq}+\mathbf{F})))\\ \quad+\mathbf{k_{1}k_{2}+k_{2}+\lambda\dot{\varepsilon}})+tr(\eta_{1}\mathbf{ \hat{\widetilde{D}}}^{T}\mathbf{\widetilde{D}})+tr(\eta_{2}\mathbf{\hat{ \widetilde{K}}}^{T}\mathbf{\widetilde{K}})\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad+tr(\eta_{3}\mathbf{\hat{\Omega}}^{ T}\mathbf{\Omega})\end{array}  \tag{36}\]  
The last three terms in (36) are denoted as \(tr(*)\) for clarity. Substituting control force in (31) into (36) yields:  
\[