## 1 Introduction
However, for the fact that MEMS gyroscopes are very small while its structure is quite complex and precise, control of dynamics of gyroscope oscillation are of great importance and deserve more focus on the design of corresponding control strategies. For example, big overshoot or undershot may cause detecting capacity combs to collide resulting in damage to measure chips. At the same time, error convergence rates are also very important in measurement meters, if it takes a very long time for the tracking errors to converge (in other words a very long settling time), it may lead to a very long measuring time and it is not likely to choose measuring meters which need a long time for measure values to stabilize in practical situations. However, very little attention is paid to these important performances. To improve the control performance of the systems, terminal sliding mode control, non-quadratic Lyapunov function [17; 18] can be utilized in the design of control systems. Meanwhile, prescribed performance control (PPC) method [19; 20; 21] is an advanced control method which ensures that system tracking error will not go beyond a predefined error bound and the error dynamics converge faster than the error bound. In [20], [PERSON] and [PERSON] introduced a kind of PPC method for nonlinear MIMO systems where the \"constrained\" error dynamic system is stabilized using the proposed control method with the \"unconstrained\" error signals. For the characteristic of PPC that the method can impose some specified error bound on system error dynamics to achieve a favorable performance. PPC technique has been applied in diverse occasions including space vehicle altitude control [19; 22; 23] and so on [24; 25]. In [26], [PERSON] and [PERSON] adopted PPC method for MEMS gyroscope control, and the method can successfully drive the detecting mass into a desired oscillation. However, the predefinederror bound is treated as a slow time varying function and it shall be further investigated how to accomplish the trajectory tracking objective without the assumption [27]. Motivated by the above mentioned work, this paper aims to systematically study the MEMS gyroscope trajectory tracking via adaptive PPC approach. The main contribution of this paper can be concluded as follows:  
The proposed adaptive prescribed performance sliding mode control (APPSMC) utilizes a transformed error in the design of sliding mode control (SMC) surface where a performance bound (PB) function regarding system tracking error is proposed together with an error transformation function to connect the origin system tracking error with the transformed equivalent error. The MEMS gyroscope tracking error can converge to the 0 and remaining in a predefined PB (both transient state and steady state) under the proposed APPSMC. It is worth pointing out that above mentioned gyroscope control papers cannot guarantee tracking error to be within a specific bound while the gyroscope tracking error in this paper will not go beyond the proposed error bound. The propose control method can accomplish trajectory tracking with information of the nominal values of the system where some adaptive laws are derived in the Lyapunov stability framework where gains for adaptive laws can be calculated according to the calculation method introduced in the paper. This paper is organized as follows: the gyroscope dynamics are introduced in Section 2. In Section 3, design and stability proof for the Z-axis gyroscope using APPSMC is investigated. Section 4 provides simulation results and Section 5 gives the conclusions.