## 3 APPSMC for MEMS Gyroscopes  
An APPSMC is introduced to accomplish trajectory tracking where several adaptive laws are derived to adjust controller parameters. The stability of the control system is guaranteed according to Lyapunov stability theory. In the design procedure of the PPC method in this section, an error transformation function is introduced to transform the origin tracking error into an equivalent unconstrained one and the transformed error is used in the design of adaptive sliding mode controller with prescribe performance. According to the theory of PPC scheme in [20; 21], error dynamics shall always stay in some specified error bound described by a PB function and the error convergence rate is no less than the performance function. Generally speaking, any smooth decreasing function can be chosen as an error bound function. As can be seen from [20; 21] that the performance bound function is chosen as a decreasing function. Prescribed performance can be achieved by transforming the original tracking error into an unconstrained equivalent error with appropriate original error bound and steady state error bound. In the paper, we adopted the prescribed performance control method in the control of gyroscopes to provide transient tracking performance of the gyroscopes. Relative articles have been cited in the reference. In this section, a decreasing function described by Equation (6) is adopted as an error bound function. \[\rho_{i}(t)=(\rho_{0i}-\rho_{\infty i})e^{-l_{i}t}+\rho_{\infty i},i=1,2 \tag{6}\]  
According to the theory of prescribed performance control (PPC) [20; 21], \(\rho_{0i}\) is a non-zero positive constant and choices for \(\rho_{0i}\) shall comply with the principle that the initial tracking error is less than \(\rho_{0i}\). \(\rho_{\infty i}\) is also a positive constant and it is the value of the PB imposed on error dynamics at the steady state. \(e\) is the Euler number and the term \(e^{-l_{i}t}\) determines the convergence speed of PB function \(\rho_{i}(t)\). It can be found from \(e^{-l_{i}t}\) that \(l_{i}\) is the dominant parameter which is chosen as a positive constant. Different choices of \(l_{i}\) can result in different PB function with different convergence speed. If gyroscope tracking errors satisfy \(-\rho_{i}(t)<e_{i}(t)<\rho_{i}(t),i=1,2\), prescribed performance is guaranteed. A new performance index is introduced construct connection between the origin tracking error and the PB, given in the form:  
\[\theta_{i}(\varepsilon_{i})=\frac{e_{i}(t)}{\rho_{i}(t)},i=1,2 \tag{7}\]  
We can easily get the range of (7) such that \(-1<\theta_{i}(\varepsilon_{i})<1\) if prescribed performance is achieved. A hyperbolic tangent function is chosen to transform the constrained error index \(\frac{\varepsilon_{i}(t)}{\rho_{i}(t)}\) into an equivalent unconstrained one. The error transformation equation is expressed as:  
\[\theta_{i}(\varepsilon_{i})=\frac{e_{i}(t)}{\rho_{i}(t)}=\frac{e^{\delta_{i} \varepsilon_{i}}-e^{-\delta_{i}\varepsilon_{i}}}{e^{\delta_{i}\varepsilon_{i} }+e^{-\delta_{i}\varepsilon_{i}}},i=1,2 \tag{8}\]  
where \(\delta_{i}\) is the adjust parameter for the transformation function chosen to be a positive constant. \(\varepsilon_{i}\) is the derived \"equivalent\" error which is used to compose the sliding manifold afterwards. Figure 2 depicts the hyperbolic tangent function. Figure 2: Graphical illustration of error transformation function. **Lemma 1**. : _If the equivalent error \(\varepsilon_{i}\) in (8) converges 0, the origin error in (8) will also converge to 0._  
Proof. : From the definition of the hyperbolic tangent function defined in (8) and the graphic depicting the hyperbolic tangent function in (8), we can easily get the conclusion that \(\theta_{i}(\varepsilon_{i})\) converges to 0 with \(\varepsilon_{i}\) converging to 0.