An adaptive sliding mode control using robust feedback compensator is presented for a MEMS gyroscope in the presence of external disturbances and parameter uncertainties. An adaptive controller with a robust term is used to improve the robustness of the control system and compensate the system nonlinearities. The proposed robust adaptive control can estimate the angular velocity and all the system parameters including damping and stiffness coefficients in the Lyapunov framework. In addition, standard adaptive control scheme without robust algorithm is compared with the proposed robust adaptive scheme in the aspect of numerical simulation and algorithm derivation. Numerical simulations show that the robust adaptive control has better robustness in the presence of external disturbances than the standard adaptive control.

In recent years, MEMS gyroscopes are commonly used microsensors for measuring angular velocity because of their compact size, low cost, and high sensitivity. Most MEMS gyroscopes utilize electrostatic force and capacitive detection. However, there are some damping effects and cross-stiffness in the presence of fabrication imperfections, and the performance of MEMS deteriorates due to the effects of time-varying parameters, external disturbances, and quadrature errors. Therefore, the measurement of angular velocity and minimization of the cross-coupling in the MEMS gyroscope should be investigated and studied. In recent years, the applications of adaptive control and robust control have achieved great developments. To control the MEMS gyroscopes, people proposed advanced control methods such as adaptive control, which can modify its characteristics with the varying parameters. Arancibia et al. [

In this paper, robust adaptive control scheme which has the advantage of combining the tracking capability of adaptive control with the robustness of robust control is proposed for the control of MEMS gyroscope. In addition, in the presence of external disturbance and model uncertainty, robust control is incorporated into the adaptive control to improve the dynamic characteristics and the robustness of the control system. The robust term can maintain some characteristics of the control system in the presence of parameter perturbations.

The robust adaptive control scheme with an additional robust term is firstly applied to the MEMS gyroscopes to improve the transient performance. Moreover, a standard adaptive control without feedback robust algorithm is proposed to estimate angular velocity and all unknown parameters for the purpose of comparison. The novelty in the paper is that a feedback algorithm is incorporated into the robust adaptive controller to improve the robustness of the control system and minimize the cross-coupling between two axes. The stability and convergence of the closed-loop control system can be guaranteed.

This paper focuses on the design and stability analysis of the robust adaptive control for MEMS gyroscopes. Meanwhile, in order to discuss the effect of the proposed control scheme, a comparative study between the standard adaptive control and the robust adaptive control is conducted, thus evaluating the performance index such as the convergence of angular velocity, tracking error, and the robustness of the control system. The advantages of the proposed control scheme are summarized compared to existing ones.

A feedback algorithm is incorporated into the robust adaptive controller to deal with nonlinearities of a system. The robust adaptive controller can improve the robustness of the MEMS and estimate angular velocity and all unknown system parameters correctly. Meanwhile, cross-coupling between two axes is minimized in the presence of external disturbances. A novel robust adaptive control scheme is proposed and used to estimate angular velocity and all uncertain parameters with improved transient performance.

Using Lyapunov analysis, the stability and convergence of the control system can be guaranteed. The tracking errors all converge to zero, and all the unknown parameters including angular velocity converge to their true values. Incorporating feedback algorithm into a robust adaptive controller, the control system tracks the desired reference trajectory much more quickly compared with other ones.

This paper is organized as follows. In Section

The dynamics of MEMS gyroscope are described in this section. Dynamics of a MEMS gyroscope are derived from Newton’s law in the rotating frame.

In a

Simplified model of a

Taking fabrication imperfections, which cause extra coupling between

In (

Dividing both sides of (

The vector form of MEMS gyroscope dynamics model can be written as

We make the following assumption: the lumped uncertainty and disturbance

In this section, a robust adaptive control scheme with application to a MEMS gyroscope is given. The control objective is to improve the robustness of the control system and minimize system nonlinearities; thus all unknown system parameters and angular velocity can be estimated correctly. The block diagram of a robust adaptive control for a MEMS gyroscope is shown in Figure

Block diagram of a robust adaptive control for a MEMS gyroscope.

Suppose that a reference trajectory is generated by an ideal oscillator and the control target is to make the trajectory of the gyroscopes follow that of the reference model. The reference model

The sliding surface is designed as

The derivative of the sliding surface is

Then, (

Setting

The adaptive controller

Substituting (

Consider the following Lyapunov function:

The derivative of

Taking the feedback component into account,

A continuous function

If V is decrescent and there exist

For the purpose of comparison, a standard adaptive control is also proposed for a MEMS gyroscope. The difference between robust adaptive control and standard adaptive control schemes is that a feedback controller is not incorporated into the adaptive controller which is proposed as

In order to eliminate the control discontinuities, a smooth sliding mode control that can reduce chattering problem is proposed as

In this section, the robust adaptive sliding control and the adaptive sliding control are both evaluated on the MEMS gyroscope model for the purpose of comparison. The control objective is to estimate angular velocity and all the unknown parameters and make the control system track the reference trajectory. Random signal with zero mean and unity variance is considered as external disturbance. The parameters of the MEMS gyroscope are listed as follows:

Figures

Tracking error using robust adaptive control.

Tracking error using standard adaptive control.

Adaptation of angular velocity using robust control.

Adaptation of angular velocity using standard adaptive control.

Adaptation of control parameters using robust adaptive control.

Adaptation of control parameters using standard adaptive control.

Comparison of system parameters between robust adaptive control and adaptive control.

In summary, with the control law (

In the paper, a novel robust adaptive control scheme is proposed for MEMS gyroscopes. For the purpose of comparison, a robust adaptive control and a standard adaptive control without feedback algorithm are investigated. Numerical simulations show that if the persistent excitation can be satisfied, all unknown parameters, including the angular velocity, converge to their true values, and tracking error converges to zero asymptotically, using these two adaptive approaches. But in the presence of external disturbance and model uncertainties, the robust adaptive control has better robustness and dynamic characteristics than standard adaptive control.

The authors thank the anonymous reviewers for useful comments that improved the quality of the paper. This work is partially supported by National Science Foundation of China under Grant no. 61374100; Natural Science Foundation of Jiangsu Province under Grant no. BK20131136; the Fundamental Research Funds for the Central Universities under Grant no. 2012B06714.