## 3 APPSMC for MEMS Gyroscopes
Equivalent error \(\varepsilon_{i}\) can be derived by solving the inverse function of \(\theta_{i}(\varepsilon_{i})\), given in the form:  
\[\varepsilon_{i}=\frac{1}{\delta_{i}}\theta_{i}^{-1}(\frac{\varepsilon_{i}}{ \rho_{i}}),i=1,2 \tag{9}\]  
where \(\theta_{i}^{-1}(\star)\) represents the inverse function of \(\theta_{i}(\star)\). In the following parts, \(\theta_{i}(\varepsilon_{i})\), \(\rho_{i}(t)\), and \(e_{i}(t)\) are denoted as \(\theta_{i}\), \(\rho_{i}\), and \(e_{i}\) for simplicity. Consider (5) as the system dynamics, system tracking error is defined as:  
\[\mathbf{e}=\left[ \begin{array}{c}e_{1}\\ e_{2}\end{array} \right]=\mathbf{q_{d}}-\mathbf{q}, \tag{10}\]  
where \(\mathbf{q_{d}}\) is the reference trajectory, given in the form \(\mathbf{q_{d}}=\left[ \begin{array}{c}q_{dx}\\ q_{dy}\end{array} \right]\). The derivative of the tracking error is:  
\[\dot{\mathbf{e}}=\left[ \begin{array}{c}\dot{e}_{1}\\ \dot{e}_{2}\end{array} \right]=\mathbf{\dot{q}_{d}}-\dot{\mathbf{q}}. \tag{11}\]  
Conventionally, sliding mode control method uses system error and its derivative to form a sliding mode manifold. In APPSMC, the new derived error will be used in the design of the prescribed performance sliding mode manifold. Consider a linear sliding manifold given in the form:  
\[\mathbf{S}=\dot{\mathbf{\epsilon}}+\boldsymbol{\lambda}\mathbf{\varepsilon} \tag{12}\]  
where \(\boldsymbol{\lambda}=\left[ \begin{array}{cc}\lambda_{1}&0\\ 0&\lambda_{2}\end{array} \right]\) is an adjustable parameter of the manifold chosen to be a diagonal matrix with its main diagonal elements being positive constants. Differentiating (12) leads to:  
\[\dot{\mathbf{S}}=\ddot{\mathbf{\epsilon}}+\boldsymbol{\lambda}\dot{\mathbf{ \epsilon}} \tag{13}\]  
\(\varepsilon_{i}\) can be derived according to (9) and \(\varepsilon_{i}\) takes the form:  
\[ \begin{array}{l}\varepsilon_{i}=\frac{1}{2\delta_{i}}\ln\frac{1+\theta_{i}} {1-\theta_{i}}\\ =\frac{1}{2\delta_{i}}(\ln(1+\theta_{i})-\ln(1-\theta_{i})),i=1,2\end{array}  \tag{14}\]  
Differentiating (14) gives:  
\[\dot{\varepsilon}_{i}=\frac{d(\theta_{i}^{-1}(\frac{\varepsilon_{i}}{\rho_{i} }))}{d(\frac{\varepsilon_{i}}{\rho_{i}})}\bullet(\frac{\varepsilon_{i}}{\rho_ {i}})\prime=\frac{\delta_{i}\dot{\theta}_{i}}{1-\theta_{i}^{2}},i=1,2 \tag{15}\]  
Differentiating (15) leads to:  
\[ \begin{array}{l}\ddot{\varepsilon}_{i}=\frac{(1-\theta_{i}^{2})\ddot{\theta }_{i}+2\theta_{i}\dot{\theta}_{i}^{2}}{(1-\theta_{i}^{2})^{2}}\\ =\frac{\delta_{i}}{1-\theta_{i}^{2}}\ddot{\theta}_{i}+\frac{2\delta_{i}\dot{ \theta}_{i}\dot{\theta}_{i}^{2}}{(1-\theta_{i}^{2})^{2}}\\ =k_{1i}\theta(\varepsilon_{i})+k_{2i},i=1,2\end{array}  \tag{16}\]where \(k_{1i}=\frac{\delta_{i}}{1-\theta_{i}^{2}},k_{2i}=\frac{2\delta_{i}\theta_{i}\dot{ \theta}_{i}^{2}}{(1-\theta_{i}^{2})^{2}},i=1,2\)  
It can be proved that \(k_{1i}\) and \(k_{2i}\) are positive diagonal matrices for the fact that \(\delta_{i}\) and \(\theta_{i}\) are positive constants and \(-1<\theta_{i}<1,i=1,2\). For simplicity, equation (16) is rewritten in vector form:  
\[\bar{\epsilon}=\mathbf{k}_{1}\bar{\theta}(\epsilon)+\mathbf{k}_{2} \tag{17}\]  
where \(\theta=\left[ \begin{array}{c}\theta_{1}\\ \theta_{2}\end{array} \right]\), \(\mathbf{k}_{1}=\left[ \begin{array}{cc}k_{11}&0\\ 0&k_{12}\end{array} \right]\), \(\epsilon=\left[ \begin{array}{c}\varepsilon_{1}\\ \varepsilon_{2}\end{array} \right]\), and \(\mathbf{k}_{2}=\left[ \begin{array}{c}k_{21}\\ k_{22}\end{array} \right]\). Differentiating (8) with respect to time gives first and second order derivative of \(\theta_{i}\). \[\dot{\theta}_{i}=\frac{\dot{e}_{i}p_{i}+e_{i}\dot{\rho}_{i}}{\rho_{i}^{2}},i=1,2 \tag{18}\]  
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