## 3 APPSMC for MEMS Gyroscopes
\begin{array}{l}\dot{\mathbf{V}}\mathbf{=}\mathbf{S}^{T}(\mathbf{k_{1}r_{1 }}((\mathbf{\overline{D}}+2\mathbf{\overline{\Omega}})\mathbf{\hat{q}}+ \mathbf{\widetilde{K}q}+\mathbf{F})\\ \quad-\mathbf{k_{1}r_{1}}\mathbf{\gamma}\mathrm{sign}(\mathbf{S}))+tr(*)\end{array}  \tag{37}\]  
Figure 3: Structure of the adaptive prescribed performance sliding mode control (APPSMC) system. Recover \(tr(*)\) back to its original form, (37) can be rewritten as:  
\[ \begin{array}{l}\dot{\mathbf{V}}=\mathbf{S^{T}k_{1}r_{1}\widetilde{D}\dot{q}+2 \mathbf{S^{T}k_{1}r_{1}\widetilde{\Omega}\dot{q}}+\mathbf{S^{T}k_{1}r_{1} \widetilde{K}q}}\\ +\mathbf{S^{T}k_{1}r_{1}F-S^{T}k_{1}r_{1}\gamma sign(S)+tr(\eta_{1}\dot{ \widetilde{D}}^{T}\widetilde{\mathbf{D}})}\\ +tr(\eta_{2}\dot{\widetilde{K}}^{T}\widetilde{\mathbf{K}})+tr(\eta_{3}\dot{ \widetilde{\Omega}}^{T}\Omega)\end{array}  \tag{38}\]  
According to the property of matrix trace, (38) can be rewritten in the form:  
\[ \begin{array}{l}\dot{V}=tr(\dot{q}\mathbf{S^{T}R\widetilde{D}})+tr(2\dot{q} \mathbf{S^{T}R\widetilde{\Omega}})+tr(\mathbf{q}\mathbf{S^{T}R\widetilde{K}}) \\ +\mathbf{S^{T}k_{1}r_{1}F-S^{T}k_{1}r_{1}\gamma sign(S)+tr(\eta_{1}\dot{ \widetilde{D}}^{T}\widetilde{\mathbf{D}})}\\ +tr(\eta_{2}\dot{\widetilde{K}}^{T}\widetilde{\mathbf{K}})+tr(\eta_{3}\dot{ \widetilde{\Omega}}^{T}\Omega)\end{array}  \tag{39}\]  
Substituting adaptive laws (32), (33), (34) into (39) leads to:  
\[ \begin{array}{l}\dot{\mathbf{V}}\mathbf{=}\mathbf{S^{T}k_{1}r_{1}F-S^{T}k_ {1}r_{1}\gamma sign(S)}\\ \mathbf{=}\mathbf{S^{T}k_{1}r_{1}(-\gamma gsign(S)+F)}\\ \mathbf{\leq}\mathbf{S^{T}k_{1}r_{1}(-\gamma gsign(S)+|F|)}\\ \mathbf{\leq}\mathbf{S^{T}k_{1}r_{1}(-\gamma gsign(S)+F_{d})}\\ \mathbf{\leq}\mathbf{\Big{|}S^{T}\Big{|}k_{1}r_{1}(-\gamma+F_{d})}\end{array}  \tag{40}\]  
According to Assumption 1 together with the fact that \(\mathbf{k_{1}}\), \(\mathbf{r_{1}}\) are positive diagonal matrices, \(\dot{\mathbf{V}}\) can be proved to be semi positive definite with \(F_{di}\leq\gamma_{i},i=1,2\). \(\dot{\mathbf{V}}\) is negative definite i.e., \(\mathbf{V}(t)\leq\mathbf{V}(0)\) implies that \(\mathbf{S}\), \(\widetilde{\mathbf{D}}\), \(\widetilde{\mathbf{K}}\), and \(\widetilde{\mathbf{\Omega}}\) are all bounded. Integrating both sides of (40) leads to \(\int\limits_{0}^{t}\!\!\big{[}\mathbf{S^{T}}\big{]}dt\leq\left(\mathbf{k_{1}r _{1}(-\gamma+F_{d})}\right)^{-1}(\mathbf{V}(0)-\mathbf{V}(t))\). For the fact that \(\mathbf{V}(0)\) is bounded, \(\mathbf{V}(t)\) is bounded and non-increasing, we can obtain \(\lim\limits_{t\rightarrow\infty}\int\limits_{0}^{t}\!\!\big{[}S^{T}\big{]}dt<\infty\). Besides, since \(\dot{\mathbf{S}}\) is also bounded, therefore, we can prove that \(\mathbf{S}\) will asymptotically converge to zero, i.e., \(\lim\limits_{t\rightarrow\infty}\mathbf{S}=0\) by Barbalart lemma [14]. If the sliding manifold \(\mathbf{S}=0\), the equivalent error \(\varepsilon_{i}\) will converge to \(0\) with the sliding surface (12) being a Hurwitz polynomial. Furthermore, we can infer from Lemma 1 that the origin tracking error will converge to \(0\) with the equivalent error converging to \(0\). **Remark 1**. : _In this paper, a prescribed performance is proposed for the control of MEMS gyroscopes. For further improvement of gyroscope oscillation motion performance, design of the control of gyroscopes can be investigated using non-quadratic Lyapunov function [17; 18]._