Abstract

A multi-input multioutput (MIMO) Takagi-Sugeno (T-S) fuzzy model is built on the basis of a nonlinear model of MEMS gyroscope. A reference model is adjusted so that a local linear state feedback controller could be designed for each T-S fuzzy submodel based on a parallel distributed compensation (PDC) method. A parameter estimation scheme for updating the parameters of the T-S fuzzy models is designed and analyzed based on the Lyapunov theory. A new adaptive law can be selected to be the former adaptive law plus a nonnegative in variable to guarantee that the derivative of the Lyapunov function is smaller than zero. The controller output is implemented on the nonlinear model and T-S fuzzy model, respectively, for the purpose of comparison. Numerical simulations are investigated to verify the effectiveness of the proposed control scheme and the correctness of the T-S fuzzy model.

1. Introduction

Gyroscopes are commonly used sensors in navigation, automobile, and so forth. The performance of the MEMS gyroscope is often deteriorated by the effects of time varying parameters, quadrature errors, and external disturbances. Advanced control such as adaptive control and intelligent control are necessary to be adopted to control the MEMS gyroscope. In the last few years, various control approaches have been proposed to control the MEMS gyroscope. Increasing attention has been given to the tracking control of MEMS gyroscope. Leland [1] derived two adaptive controllers for a MEMS gyroscope that tune the drive axis natural frequency to a preselected frequency and drive the sense axis vibration to zero for force-to-rebalance operation. Sung et al. [2] proposed a phase-domain design approach to study the mode-matched control of gyroscope. Antonello et al. [3] derived extremum-seeking control to automatically match the vibration mode in MEMS vibrating gyroscopes. Park et al. [4] presented an adaptive controller for a MEMS gyroscope which drives both axes of vibration and controls the entire operation of the gyroscope. John and Vinay [5] developed a novel concept for an adaptively controlled triaxial angular velocity sensor device. Sliding mode control has been incorporated into adaptive controller to control the MEMS gyroscopes [6, 7]. Model uncertainties and disturbances are inevitable in actual engineering and require the controller to be either adaptive or robust to these model uncertainties. Adaptive fuzzy sliding mode controller can be utilized to compensate the model uncertainties and disturbances since it combines the merits of the sliding mode control, the fuzzy inference mechanism, and the adaptive algorithm. Wang [8] demonstrated that an arbitrary function of a certain set of functions can be approximated with arbitrary accuracy using fuzzy system on a compact domain using universal approximation theorem. Guo and Woo [9] derived adaptive fuzzy sliding mode controller for robot manipulator. Yoo and Ham [10] developed adaptive controller for robot manipulator using fuzzy compensator. Wai [11] proposed fuzzy sliding mode controller using adaptive tuning technique. Chang et al. [12] designed and implemented fuzzy sliding mode formation control for multirobot systems. Chien et al. [13] developed robust adaptive controller design for a class of uncertain nonlinear systems using online T-S fuzzy-neural modeling approach. Stabilizing controller design for uncertain nonlinear systems using fuzzy models was investigated in [14]. Park and Cho [15] designed T-S model based indirect adaptive fuzzy controller using online parameter estimation.

The system nonlinearities in MEMS gyroscope model have been described in [16–18]. Since there are system nonlinearities in MEMS gyroscope, it is necessary to use robust adaptive fuzzy controller to MEMS gyroscope and utilize T-S fuzzy model to represent the system nonlinearities.

Systematic stability analysis and controller design of the adaptive fuzzy controller using T-S fuzzy model for MEMS gyroscope have not been investigated before. The advantage of the proposed adaptive T-S fuzzy controller over other controllers is that it can represent system nonlinearities using T-S fuzzy model in the aspect of robust design. Thus, it can overcome the impacts on output tracking error that are caused by model uncertainties and external disturbances. The proposed T-S modeling method can provide a possibility for developing a systematic analysis and design method for complex nonlinear control systems, thus improving the tracking and compensation performance. Thus it is necessary to adopt the adaptive fuzzy control scheme to approximate the nonlinear system and compensate model uncertainties and external disturbances in the control of MEMS gyroscope using T-S fuzzy model.

In this paper, the Lyapunov-based robust adaptive fuzzy control strategy is applied to the tracking control of MEMS gyroscope using T-S fuzzy model. The paper integrates adaptive control and the nonlinear approximation of fuzzy control with T-S fuzzy model. The proposed adaptive fuzzy T-S controller can guarantee the asymptotic stability of the closed loop system and improve the robustness of control system in the presence of model uncertainties and external disturbances. The proposed controller has the following characteristics and contributions.(1)Since the reference model is marginally stable, the reference model is required to be revised so that the local linear state feedback controller can be designed for each T-S fuzzy submodel based on PDC. The partial parameters of the reference control matrix are changed and then the reference input is changed accordingly to guarantee that the changed reference model is equivalent to previous one.(2)To improve the performance of the control system, the corresponding improvement is made to the adaptive law. The new adaptive law can be chosen to be the previous adaptive law plus a robust term to guarantee that the derivative of the Lyapunov function is negative rather than nonpositive.(3)The proposed estimator for the existing fuzzy state feedback controller can achieve a good robust performance against parameter uncertainties. The controller output is implemented in the T-S fuzzy model and nonlinear model, respectively, to verify the validity of the adaptive fuzzy control with parameter estimation scheme and prove the correctness of the T-S model and the feasibility of the proposed controller on the nonlinear model of gyroscope.

2. Dynamics of MEMS Gyroscope

Assume that the gyroscope is moving with a constant linear speed; the gyroscope is rotating at a constant angular velocity; the centrifugal forces are assumed negligible; the gyroscope undergoes rotations along -axis as shown in Figure 1. Referring to [4, 16, 17], the dynamics equations of gyroscope system are as follows: where is the mass of proof mass. Fabrication imperfections contribute mainly to the asymmetric spring term , and asymmetric damping terms , , and are damping terms; , are linear spring terms; , are nonlinear spring terms, is angular velocity; , are the control forces.

Figure 1 

axis vibratory MEMS gyroscope with nonlinear effective spring.

Dividing the equation by the reference mass, and because of the nondimensional time , dividing both sides of equation by reference frequency and reference length , and rewriting the dynamics in vector forms result in where Define new parameters as follows: For the convenience of notation, ignoring the superscript yields the final form of the nondimensional equation of motion for the -axis gyroscope

3. Adaptive Fuzzy Control Based on PDC

The MIMO T-S fuzzy model of the MEMS gyroscope, which is composed of rules, is established based on the nonlinear model of MEMS gyroscope. Use the fuzzy implications and the fuzzy reasoning methods suggested by Takagi and Sugeno [19] to express a real plant model; the T-S fuzzy model uses fuzzy implication of the following form: The defuzzification fuzzy dynamic system model can be expressed using center-average defuzzifier, product inference, and singleton fuzzifier where , , , and are membership function values of the fuzzy variable , , , and with respect to fuzzy set , , , and , respectively.

Since the control target for MEMS gyroscope is to maintain the proof mass to oscillate in the and direction at given frequency and amplitude, , , generally the reference model can be defined as , where Since the reference model is nonasymptotically stable, the nonlinear T-S fuzzy model cannot be asymptotically stable. In order to design local linear state feedback controller for each T-S fuzzy submodel based on PDC, the reference model is adjusted to be , where

Remark 1. Substituting into yields ; we can get that the changed reference model is equivalent to previous one .
According to PDC [14], we design local state linear feedback controllers for each linear submodel. The fuzzy control rules are as the following forms: The controller output can be expressed using center-average defuzzifier, product inference, and singleton fuzzifier where Substituting the controller force (12) into the fuzzy system (7) yields the desired model

4. Parameter Estimation

The block diagram of the adaptive fuzzy control using T-S fuzzy model is shown as in Figure 2. Taking the parameter uncertainty into account, then the parameter estimation is applied to the adaptive fuzzy control. The adaptive law can guarantee the asymptotic convergence of the error between the plant model state and the estimation model state and make the closed loop system of fuzzy controller and estimation model be a desired linear model and, consequently, the plant model state can follow the state of the desired linear model. The controller output is implemented in the nonlinear model also to prove the correctness of the T-S model and the feasibility of the proposed control scheme on the nonlinear model of MEMS gyroscope.

Figure 2 

The block diagram of the adaptive fuzzy control using T-S fuzzy model.

With unknown parameters matrix , the controller changes to where is the estimate of .

By considering the plant parameterization, (7) can be rewritten as where is an arbitrary stable matrix.

Define the estimation mode as the following formula: The estimation model (18) can be the same one as by adopting the control input (15) The estimation error meets (19) and (20) where actually is the tracking error between the plant model (7) and the estimation model (18), , , , , , , .

Remark 2. The estimation error can be thought of as the tracking error between the plant model (7) and the desired linear model (14) because and have the same control matrix, while the tracking error between the reference model (14) and the nonlinear model (5) is used to prove the correctness of the T-S model and the feasibility of the proposed controller on the nonlinear model of gyroscope.
Define a Lyapunov function candidate as where meet .
The derivative of with respect to time becomes Substituting (20), (21) into the derivative of yields where .
The adaptive law can be chosen as where the sgn function can be defined as Substituting (25) into (24) yields and imply the boundedness of and , the control is bounded for all time, (21) implies is also bounded, and inequality (27) implies . Then, with and , according to Barbalat’s lemma [20], the tracking error asymptotically converges to zero.

5. Simulation Analysis

In this section, we will evaluate the proposed adaptive fuzzy control using T-S model approach on the lumped MEMS gyroscope sensor model. Parameters of the MEMS gyroscope sensor are as follows:

Since the general displacement range of the MEMS gyroscope sensor in each axis is submicrometer level, it is reasonable to choose 1 μm as the reference length . Given that the usual natural frequency of each axle of a vibratory MEMS gyroscope sensor is in the kHz range, is chosen as 1 kHz. The unknown angular velocity is assumed  rad/s. The desired motion trajectories are , , where , ,  kHz, and  kHz.

The plant parameters are adjusted online by adaptive law (24) where the adaptive gain and the adaptive parameter . The initial values of T-S model are , initial values of the estimation model are , and initial values of the nonlinear model are . The external disturbance is . and the eigenvalues of are . Although can meet the requirement of in , the values of first and second column in which are cannot adjust each parameter since partial parameters are zero which is used as adaptive gain in (24), so is chosen as to ensure that each parameter of first and second column in is nonzero values as .

The fuzzy rules for T-S fuzzy model for the system can be obtained by linearizing the nonlinear model (5) at the points , , , and and then the membership functions of states , , , and with respect to fuzzy set , , , and used in the T-S fuzzy model are shown as in Figure 3.

Figure 3 

Membership functions.

In this simulation, it is assumed that the physical parameters in the T-S fuzzy model are not known exactly. Hence, the parameters are tuned by the proposed adaptive law. The true values of of fuzzy T-S model are as shown in Table 1. The initial value . The variations of each parameter in , , and are shown in Figures 4, 5, and 6. Here only parts of parameters are shown in table for examples. In Figures 4, 5, and 6 we can see each parameter converges to its true value asymptotically.

Figure 4 

The change of plant parameters in .

Figure 5 

The change of plant parameters in .