Gyroscopes are commonly used sensors for measuring angular velocity in many areas of application, such as navigation, homing and control stabilisation. The performance of the MEMS gyroscope often deteriorates due to the effects of time-varying parameters as well as noise sources, such as mechanical and circuitry noise, quadrature errors, parameter variations and external disturbances.
It is necessary to use an advanced control, such as adaptive control or intelligent control, to control MEMS gyroscopes and in the last few years, various control approaches have been developed. Increasing attention has been given to the tracking control of MEMS gyroscopes. Leland  derived two adaptive controllers for a vibrational MEMS gyroscope which tune the drive axis’ natural frequency and the sense axis’ vibration to zero by a force-to-rebalance operation. Batur et al.  developed a sliding mode control for a MEMS gyroscope. Sun et al.  derived a phase-domain design approach in order to study the mode-matched control of gyroscopes. Antonello et al.  used an extremum-seeking control to automatically match the vibration mode in MEMS vibrating gyroscopes. Park et al.  presented an adaptive controller for a MEMS gyroscope which drives both of the axes of vibration and controls the entire operation of the gyroscope. Some adaptive sliding mode controllers have been developed to control MEMS gyroscopes [6-7].
System nonlinearities are inevitable in actual engineering and they require the controller to be either adaptive or robust. Intelligent control approaches such as neural networks and fuzzy control do not require mathematical models and have the ability to approximate nonlinear systems. Neural network technologies have been applied to nonlinear control systems [8-9]. Wang  proposed a universal approximation theorem and demonstrated that an arbitrary function of a certain set of functions can be approximated with arbitrary accuracy using a fuzzy system over a compact domain. Therefore a fuzzy logic system for approximating arbitrary nonlinear functions makes it a useful tool for adaptive applications. The key idea behind adaptive fuzzy logic systems is that a wide class of nonlinear systems can be approximated to arbitrary closeness by them. An adaptive fuzzy sliding mode controller combines the merits of a sliding mode control, a fuzzy inference mechanism and an adaptive algorithm. Guo et al.  proposed an adaptive fuzzy sliding mode controller for a robot manipulator. Tong et al.  designed a fuzzy indirect and direct adaptive control for a nonlinear system. Yoo et al.  developed adaptive controller for a robot manipulator using a fuzzy compensator. Wai et al.  presented an adaptive fuzzy neural network control design via a T–S fuzzy model for a robot manipulator which included actuator dynamics. Wai et al.  investigated an adaptive sliding mode control system for an indirect field-oriented induction motor drive so as to track periodic commands. Lee  proposed a robust adaptive fuzzy control by back-stepping for a class of MIMO nonlinear systems. Islam et al.  used a robust adaptive fuzzy output feedback control system for robot manipulators. Zhou et al.  developed an adaptive output-feedback fuzzy tracking control for a class of nonlinear systems. However, the application of fuzzy control to MEMS gyroscopes has never been performed in the literature and, therefore, it is necessary to adopt an adaptive fuzzy control that can adjust the parameter vector of the member functions on-line using an adaptive law for the vibration control of MEMS gyroscopes.
This paper focuses on the design of a direct adaptive fuzzy control for a MEMS gyroscope using a supervisory compensator. A direct adaptive fuzzy controller use fuzzy logic systems as controllers; therefore linguistic fuzzy control rules can be directly incorporated into the controller. The contribution of this paper lies is the integration of an adaptive control, a nonlinear approximation of a fuzzy control and a supervisory compensator. The study of the adaptive fuzzy controller is conducted on a MEMS gyroscope and it is applied to control the gyroscope and guarantee that the closed loop system is globally stable and tracking errors are as small as possible. In order to guarantee the robustness of the adaptive fuzzy controller, a supervisory compensator is incorporated into the adaptive fuzzy control scheme in the Lyapunov framework.
This paper is organised as follows. In section 2, the dynamics of the MEMS gyroscope are introduced and a non-dimensional procedure is described. In section 3, an adaptive fuzzy control is derived and Lyapunov analysis is implemented to guarantee the asymptotic stability of the closed-loop system. Simulation results are presented in section 4 to verify the effectiveness of the proposed adaptive fuzzy control. Conclusions are provided in section 5.
2. Dynamics of the Mems gyroscope
The dynamics of the MEMS gyroscope are described in Fig. 1. A typical MEMS gyroscope configuration includes a proof mass suspended by spring beams, electrostatic actuations and sensing mechanisms for forcing an oscillatory motion and sensing the position and velocity of the proof mass, as well as a rigid frame which is rotated along the rotation axis. The dynamics of a MEMS gyroscope are derived by Newton’s Law in the rotating frame.
In a z-axis gyroscope, by supposing the stiffness of spring in direction z to be much larger than that in directions x, y, the motion of the poof mass is constrained to just along the x-y plane. Assume that the measured angular velocity is almost constant over a long enough time interval. Considering fabrication imperfections which cause extra coupling between the x and y axes, and ignoring centrifugal forces, the governing equation for a z-axis MEMS gyroscope is simplified as:
where x and y are the coordinates of the proof mass with respect to the gyro frame in a Cartesian coordinate system. In equation (1) and are damping coefficients; and are spring coefficients; and , quadrature errors, all of which are coupled damping and spring terms, respectively, mainly due to the asymmetries in suspension structure and the misalignment of sensors and actuators. The coupled spring and damping terms are unknown, but can be assumed to be small. The nominal values of the x and y axes’ spring and damping terms are known, but there are small unknown variations. The proof mass can be determined accurately.
Dividing both sides of equation (1) by , , , which are a reference proof mass of a gyroscope, the length and natural resonance frequency respectively, yields the non-dimensional motion equation as:
where , , , , , .
The vector form of the MEMS gyroscope dynamic model can be written as:
where , , , , .
If , and are unknown, (3) can be rewritten as:
where is an unknown dynamic function . Fuzzy control will be investigated in order to approximate the unknown function in the next section.
3. Adaptive Fuzzy Controller Design
Fuzzy control is very robust and capable of handling nonlinear systems, providing a methodology for representing, manipulating and implementing a human’s heuristic knowledge about how to control a system. A fuzzy controller is composed of the following four elements: fuzzier, some fuzzy IF-THEN rules, a fuzzy inference engine and a defuzzifier. The fuzzy inference engine uses the fuzzy IF-THEN rules to perform a mapping from an input linguistic vector to an output variable . The i-th fuzzy rule can be expressed as:
where ,…, are fuzzy variables and is a singleton number.
The output of the fuzzy system can be expressed using a centre-average defuzzifier, product inference and a singleton fuzzifier:
where is the membership function value of the fuzzy variable , is the number of fuzzy rules and is an adaptive parameter vector.
is fuzzy basis function vector and
is i-th column vector of fuzzy basis function vector.
The control target for the MEMS gyroscope is to maintain the proof mass so as to oscillate in the and directions at a given frequency and amplitude:
The closed loop system must be globally stable in the sense that all variables and must be uniformly stable. The tracking error should be as small as possible.
The procedure of the proposed adaptive fuzzy control as applied to the MEMS gyroscope is described in this section. The proposed adaptive fuzzy control scheme is shown by Fig. 2.
Rewriting (3) as:
The reference model is defined as:
Rewriting (8) as:
The tracking error is defined as:
Suppose that the controller is composed of a fuzzy controller and a supervisory compensator :
where the fuzzy controller , is a fuzzy basis function, the supervisory compensator is:
where , .
Choosing the control force:
the controller will force the tracking error to converge to zero, where and are chosen such that all roots of are located on the left side of the plane.
The dynamic of tracking error can be derived as:
where . Since is a stable matrix, there exists a unique positive definite and symmetrical matrix , satisfying the Lyapunov equation:
where , Q is an arbitrary positive definite matrix.
We define the optimal parameter vector thus:
The fuzzy approximation error vector is defined as:
As such, (16) becomes:
We define the Lyapunov function candidate:
where is a positive constant.
Suppose , then (20) becomes .
Differentiating with respect to time yields:
If we choose the adaptive law:
Choosing , (24) becomes:
Here, we use the inequality:
where is the eigenvalue of the matrix with a minimum real part.
, becomes negative semi-definite and ensures that , and are all bounded. LaSalle’s invariant set theorem can be used to prove that . implies that and that there is no other solution but . According to LaSalle’s invariant set theorem, defining then if contains no trajectories other than , the origin zero is asymptotically stable. Therefore, is an invariant set which implies that any trajectory starting from an initial condition within that set remains in the set constantly, that is will asymptotically converge on zero.
4. Simulation Study
In this section, we will evaluate the proposed adaptive fuzzy control approach on the lumped MEMS gyroscope sensor model [1, 5] by using MATLAB/SIMULINK. The control objective is to maintain the system to track the desired reference trajectory and adopt the adaptive fuzzy controller for the gyroscope model. The parameters of the MEMS gyroscope sensor are shown as follows:
The desired motion trajectories are where . The following six membership functions are selected in the region [-3,3] respectively, as shown by Fig. 3.
In the simulation, we allowed 10% parameter variations for the spring and damping coefficients with respect to their nominal values. We further assumed a 10% magnitude change in the coupling terms. The external disturbance and . The initial states of the system are [1,0,0,0], the initial values of are , where is i-th column vector of the estimated parameter matrix, adaptive gain is chosen as in (23), is chosen as in (16) to satisfy that all of the roots of are located on the left side of the plane, Q is an arbitrary positive definite matrix, chosen as =[50 0;0 50] in (17) and should be a positive constant since in (12).
In order to eliminate chattering, the discontinuous control component in (11) can be replaced by a smooth-sliding mode component to yield:
, where .
First, we simulate the system response of MEMS gyroscope with and respectively. The tracking responses for these two different fuzzy controllers in the presence of the disturbance will be simulated as shown by Figs. 4-9.
Figs. 4 and 5 show that the position-tracking trajectory and tracking errors of x, y only using . It can be observed the tracking error is relatively large which is around the value of 0.2. Figs. 6 and 7 depict the position-tracking trajectory and tracking errors of x, y using . It can be seen here that the tracking error is almost zero. Figs. 4-7 compare the tracking error between the system using and . It can be seen that the system using a supervisory compensator has better trajectory-tracking performance than that which uses only a fuzzy controller , and the position of x, y can track the position of the reference model in a short time with the supervisory compensator. Therefore, it can be seen that the adaptive fuzzy tracking performance is satisfactory, the MEMS gyroscope can maintain the proof mass to oscillate in the x, y direction at a given frequency and amplitude with the proposed adaptive fuzzy control.
Fig. 8 plots the adaptive fuzzy control inputs using . It can be seen that the chattering is diminished because of the use of a smooth supervisory sliding controller which can create a small boundary layer around the switching surface in which the trajectory will remain. Fig. 9 illustrates the property of the sliding surface using , showing that it converges to zero asymptotically.
In order to show the robustness of the performance of the MEMS gyroscope, the disturbance with a large magnitude and a high frequency is considered in the simulation. Comparing Figs. 10 and 11, we can see that in the case of a high-frequency sinusoidal disturbance signal the system using also has better performance than one using only ; the tracking error can still approach zero in a short time, showing that the robustness of the system using (11) is better than that using the fuzzy controller in the presence of external disturbances.
A fuzzy logic-based adaptive control for an angular velocity sensor is presented in this paper. An adaptive fuzzy controller that can incorporate fuzzy control rules directly into itself is used to adaptively adjust the parameters of member functions and make the tracking error as small as possible. Moreover, a supervisory compensator is added to the adaptive fuzzy control system in order to guarantee the asymptotically stability of the control system. A simulation study is implemented to verify the effectiveness of the proposed adaptive fuzzy control in relation to external disturbances. It can be concluded that the dynamic gyroscope system’s responses are as expected, and adaptive fuzzy control with a supervisory compensator has better tracking performance and tracking errors asymptotically converge on zero.