## 1 Introduction  
As one kind of inertial sensors, gyroscopes are used in application areas where there exist particular requirements for angular velocities measurement and MEMS gyroscopes have attracted increasing attention for the advantages of low cost, easy installation and so on. Due to above mentioned prominent features, MEMS gyroscopes have been widely installed in cell phones, navigation devices, self-balance platforms and other application aspects. Generally speaking, the principle of vibrating gyroscopes is based on the Coriolis phenomenon where a detecting mass is driven into a stable oscillatory motion along the driving axis, Coriolis force occurs along the sense axis when some angular velocity is applied to the gyroscope. However, gyroscope performances are often degraded by manufacturing errors (mainly contribute to aspects of parameter deviation and asymmetric structure) and external disturbances, which exist in manufacturing and measuring environments. As a result, aiming at improving gyroscope oscillation performance as well as robustness to disturbances, advanced control methods such as sliding mode control (SMC), adaptive control (AC) are practical and feasible choices in the control of gyroscopes. It is known that there are many factors that may hinder the angular velocity measurement such as asymmetrical structure, temperature [1; 2; 3; 4], humidity in packing environment, and other disturbances. Great efforts have been devoted to improve system performances and robustness by suppressing parameter uncertainties and external disturbances [5; 6; 7; 8; 9]. Various control methods are adopted to stabilize gyroscope systems and improve system performances. In [5], [PERSON] used an extended state observer and presented a disturbance rejection-based controller for MEMS gyroscopes. As one kind of advanced control scheme with extraordinary strong robustness [10; 11; 12; 13], SMC can also be used to control gyroscope dynamics and the method has been adopted in various implementation occasions for its outstanding features such as easy implementation, insensitive to parameter variation, strong robustness and so on. In [12], [PERSON] introduced the concept of sliding mode control method where design and analysis for sliding mode systems are investigated. Although SMC is a good tool for control problems in the presence of disturbance and uncertain parameters, SMC can achieve excellent control performance on the basis that there exist a switching term varying with error dynamics which may cause high frequency chattering in control forces. To overcome the drawback of SMC, great efforts have been devoted to the investigation of SMC method, ASMC, dynamic sliding mode control, super-twisting algorithm are gradually introduced and adopted in many areas. AC technique is one effective method to solve system parameter variation problems and it is often integrated with SMC to alleviate chattering in the face of different kinds of disturbances. The basic principle of AC strategy is that controller parameters can vary automatically according to different system parameters and different system dynamics. As a result, it is natural to combine AC with SMC method for system performance improvement in the presence of parameter deviation. In [14], [PERSON] and [PERSON] proposed a robust sliding mode controller for gyroscopes where all parameters can be estimated by adaptive laws. [PERSON] and [PERSON] [15; 16] further proposed terminal sliding mode controller for gyroscopes where finite time error convergence can be achieved. Although the control methods [14; 15; 16] mentioned above can accomplish trajectory tracking under parameter variation and other disturbances, transient performance indexes especially maximum overshot and undershot [15] and error convergence rate are not fully studied.