# Adaptive Control of MEMS Gyroscope Based on T-S Fuzzy Model

Authors: Yunmei Fang; Shitao Wang; Juntao Fei; Mingang Hua
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## 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.


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## Body

## 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 alongz-axis as shown in Figure 1. Referring to [4, 16, 17], the dynamics equations of gyroscope system are as follows:
(1)mx¨+dxxx˙+dxy-2mΩz*y˙+kxx-mΩz*2x+kxyy+kx3x3=ux,my¨+dyyy˙+dxy+2mΩz*x˙+kyy-mΩz*2y+kxyx+ky3y3=uy,
where m is the mass of proof mass. Fabrication imperfections contribute mainly to the asymmetric spring term dxy, and asymmetric damping terms kxy, dxx, and dyy are damping terms; kxx, kyy are linear spring terms; kx3, ky3 are nonlinear spring terms, Ωz* is angular velocity; ux, uy are the control forces.Figure 1
Z
-axis vibratory MEMS gyroscope with nonlinear effective spring.Dividing the equation by the reference mass, and because of the nondimensional timet*=ω0t, dividing both sides of equation by reference frequency ω02 and reference length q0, and rewriting the dynamics in vector forms result in
(2)q¨1*q0+D*mω0q˙1*q0+2S*ω0q˙1*q0-Ωz*2mω0q1*q0+K1*mω02q1*q0+K3*mω02q1*3q0=u*mω02q0,
where
(3)q1*=xy,u*=uxuy,D*=dxxdxydxydyy,S*=0-Ωz*Ωz*0,K1*=kxxkxykxykyy,K3*=kx300ky3.
Define new parameters as follows:
(4)q1=q1*q0,u=u*mω02q0,Ωz=Ωz*ω0,D=D*mω0,S=S*ω0,K1=K1*mω02,K3=K3*q02mω02.
For the convenience of notation, ignoring the superscript yields the final form of the nondimensional equation of motion for the z-axis gyroscope
(5)q¨1=2S-Dq˙1+Ωz2-K1q1-K3q13+u.


## 3. Adaptive Fuzzy Control Based on PDC

The MIMO T-S fuzzy model of the MEMS gyroscope, which is composed ofi 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:
(6)Rulei:IFxisaboutMi1andyisaboutMi2andx˙isaboutMi3andy˙isaboutMi4,THENq˙=Aiq+Biu,i=1,2,…,9.
The defuzzification fuzzy dynamic system model can be expressed using center-average defuzzifier, product inference, and singleton fuzzifier
(7)q˙=∑i=19μiηAiq+Biu∑i=19μiη,
where
(8)Ai=a11ia12ia13ia14ia21ia22ia23ia24i10000100,Bi=10010000,q˙=x¨y¨x˙y˙,q=x˙y˙xy,μiη=Mi1xMi2yMi3x˙Mi4y˙.Mi1x, Mi2y, Mi3x˙, and Mi4y˙ are membership function values of the fuzzy variable x, y, x˙, and y˙ with respect to fuzzy set Mi1, Mi2, Mi3, and Mi4, respectively.Since the control target for MEMS gyroscope is to maintain the proof mass to oscillate in thex and y direction at given frequency and amplitude, x=Axsinωxt, y=Aysinωyt, generally the reference model can be defined as q˙d=Adqd, where
(9)q˙d=x¨y¨x˙y˙=Axωx2sin(ωxt)Ayωy2sin(ωyt)Axωxcos⁡(ωxt)Ayωycos⁡(ωyt),qd=x˙y˙xy=Axωxcos⁡(ωxt)Ayωycos⁡(ωyt)Axsin(ωxt)Aysin(ωyt),Ad=00-ωx20000-ωy210000100.
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 q˙m=Amqm+Bmr, where
(10)Am=a11ma12ma13ma14ma21ma22ma23ma24m10000100=-10-ωx200-10-ωy210000100,Bm=10010000,q˙m=x¨y¨x˙y˙=Axωx2sin(ωxt)Ayωy2sin(ωyt)Axωxcos⁡(ωxt)Ayωycos⁡(ωyt),qm=x˙y˙xy=Axωxcos⁡(ωxt)Ayωycos⁡(ωyt)Axsin(ωxt)Aysin(ωyt),r=Axωxcos⁡(ωxt)Ayωycos⁡(ωyt).Remark 1.
SubstitutingAm,Bm,qm,r into q˙m yields q˙d; we can get that the changed reference model q˙m=Amqm+Bmr is equivalent to previous one q˙d=Adqd.
According to PDC [14], we design local state linear feedback controllers for each linear submodel. The fuzzy control rules are as the following forms:
(11)Rulei:IFxisaboutMi1andyisaboutMi2andx˙isaboutMi3andy˙isaboutMi4,THENut=-Kitxt+rt,i=1,2,…,9.
The controller output can be expressed using center-average defuzzifier, product inference, and singleton fuzzifier
(12)ut=∑i=19μiη-Kiqt+rt∑i=19μiη,
where
(13)Ki=Ai-Am=a11i-a11ma12i-a12ma13i-a13ma14i-a14ma21i-a21ma22i-a22ma23i-a23ma24i-a24m10000100,aaaaaaaaaaaaaaaaaaaaaaaaai=1,2,…,9.
Substituting the controller force (12) into the fuzzy system (7) yields the desired model
(14)q˙m=Amqm+Bmr.


## 4. Parameter Estimation

The block diagram of the adaptive fuzzy control using T-S fuzzy model is shown as in Figure2. 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 matrixAi, the controller changes to
(15)ut=∑i=19μiη-K^iqt+rt∑i=19μiη,
where
(16)K^i=A^i-Am=a^11i-a11ma^12i-a12ma^13i-a13ma^14i-a14ma^21i-a21ma^22i-a22ma^23i-a23ma^24i-a24m10000100,A^i=a^11ia^12ia^13ia^14ia^21ia^22ia^23ia^24i10000100aaaaaaaaaaaaaaaaaaaaaaaaaaaai=1,2,…,9
is the estimate of Ai.By considering the plant parameterization, (7) can be rewritten as
(17)q˙t=Asqt+∑i=19μiηAi-Asqt+Biut∑i=19μiη,
where As is an arbitrary stable matrix.Define the estimation mode as the following formula:(18)q^˙t=Asq^t+∑i=19μiηA^i-Asqt+Biut∑i=19μiη.
The estimation model (18) can be the same one as by adopting the control input (15)(19)q˙^m=Amq^m+Bmr.
The estimation error et=qt-q^mt meets (19) and (20)(20)e˙t=Aset+∑i=19μiηAi-A^iqt∑i=19μiη=Aset-∑i=19μiηα~1iα~2i00Tqt∑i=19μiη,(21)e˙Tt=etAsT-∑i=19μiηqTtα~1iα~2i00∑i=19μiη,
where et actually is the tracking error between the plant model (7) and the estimation model (18), α~i(t)=α~1i(t)α~2i(t)00, α~1i(t)=α^1i(t)-α1i, α^1i(t)=a^11i(t)a^12i(t)a^13i(t)a^14i(t), α1i=a11ia12ia13ia14i, α~2i(t)=α^2i(t)-α2i, α^2i=a^21i(t)a^22i(t)a^23i(t)a^24i(t), α2i=a21ia22ia23ia24i.Remark 2.
The estimation erroret can be thought of as the tracking error eT-S between the plant model (7) and the desired linear model (14) because et=qt-q^mt and eT-St=qt-qmt have the same control matrix, while the tracking error eNON 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(22)Vt=Ve,a~1i,a~2i=eTtPet+∑i=19a~1iTta~1itγ1i+∑i=19a~2iTta~2itγ2i,
where P meet AsTP+PAs=-I.
The derivative ofV with respect to time becomes
(23)V˙t=e˙TtPet+eTtPe˙t+∑i=192a~˙1iTta~1itγ1i+∑i=192a~˙2iTta~2itγ2i.
Substituting (20), (21) into the derivative of V yields
(24)V˙t=-eTtAsTP+PAset-2∑i=19μiηP1TetqTta~2it∑i=19μiη-2∑i=19μiηP2TetqTta~2i(t)∑i=19μiη+∑i=192a~˙1iTta~1itγ1i+∑i=192a~˙2iTta~2itγ2i,
where P=P1P2P3P4.
The adaptive law can be chosen as(25)a~˙iT=γiμiη∑i=19μiηP1TP2TetqTt-ρsgn(a~i(t))ρ>0,
where the sgn function can be defined as
(26)sgn(a~i)=1a~i>00a~i=0-1a~i<0.
Substituting (25) into (24) yields
(27)V˙t=-eTtet-2ρ∑i=19sgna~1ita~1itγ1i-2ρ∑i=19sgna~2ita~2itγ2i=-eTtet-2ρ∑i=19a~1itγ1i-2ρ∑i=19a~2i(t)γ2i≤-eTtet≤-et2≤0.V(t)≥0 and V˙(t)≤0 imply the boundedness of e(t) and α~i(t), the control is bounded for all time, (21) implies e˙(t) is also bounded, and inequality (27) implies e(t)∈L2. Then, with e(t)∈L2∩L∞ and e˙(t)∈L∞, according to Barbalat’s lemma [20], the tracking error e(t) 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:(28)m=0.57e-8kg,ω0=1kHz,q0=10e-6m,dxx=0.429e-6Ns/m,dyy=0.0429e-6Ns/m,dxy=0.0429e-6Ns/m,kxx=80.98N/m,kyy=71.62N/m,kxy=5N/m,kx3=3.56e12N/m,ky3=3.56e12N/m,Ωz=5.0rad/s.Since the general displacement range of the MEMS gyroscope sensor in each axis is submicrometer level, it is reasonable to choose 1μm as the reference length q0. Given that the usual natural frequency of each axle of a vibratory MEMS gyroscope sensor is in the kHz range, ω0 is chosen as 1 kHz. The unknown angular velocity is assumed Ωz=5.0 rad/s. The desired motion trajectories are xm=Axsinωxt, ym=Aysinωyt, where Ax=1, Ay=1.2, ωx=6.71 kHz, and ωy=5.11 kHz.The plant parameters are adjusted online by adaptive law (24) where the adaptive gain γi=1 and the adaptive parameter ρ=1. The initial values of T-S model are [0.2,0.24,0,0], initial values of the estimation model are [0,0,0,0], and initial values of the nonlinear model are [0.2,0.24,0,0]. The external disturbance is d=dxdy=10sin2πt10sin2πt. Am=-10-ωx200-10-ωy210000100 and the eigenvalues of Am are -0.5±6.6916i,-0.5±5.0855i. Although As can meet the requirement of As in AsTP+PAs=-I, the values of first and second column in P which are 0.51000.520.01000.02 cannot adjust each parameter since partial parameters are zero which is used as adaptive gain in (24), so As is chosen as As=-1-1-1-10-10000-10000-1 to ensure that each parameter of first and second column in P is nonzero values as 0.5-0.25-0.250.75-0.250.25-0.250.25.The fuzzy rules for T-S fuzzy model for the system can be obtained by linearizing the nonlinear model (5) at the points x∈-101, x˙∈-Axωx0Axωx, y∈-1.201.2, and y˙∈-Ayωy0Ayωy and then the membership functions of states x, y, x˙, and y˙ with respect to fuzzy set Mi1, Mi2, Mi3, and Mi4 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 ofAi* of fuzzy T-S model are as shown in Table 1. The initial value Ai=0.9*Ai*. The variations of each parameter in A2, A5, and A7 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.Table 1
Parameters of T-S fuzzy model.
A
1
*
=
-
0.0753
-
0.0175
1
0
0.0025
-
0.1205
0
1
-
14207
-
877
0
0
-
877
-
12564
0
0
T
A
2
*
=
-
0.0753
-
0.0175
1
0
0.0025
-
0.1205
0
1
-
14207
-
877
0
0
-
877
-
12564
0
0
T
A
3
*
=
-
0.0753
-
0.0175
1
0
0.0025
-
0.1205
0
1
-
14207
-
877
0
0
-
877
-
12564
0
0
T
A
4
*
=
-
0.0753
-
0.0175
1
0
0.0025
-
0.1205
0
1
-
15568
7.432
0
0
-
262
-
14201
0
0
T
A
5
*
=
98
170
1
0
90
155
0
1
-
16066
-
851
0
0
-
859
-
15232
0
0
T
A
6
*
=
-
98
-
170
1
0
-
90
-
155
0
1
-
16066
-
851
0
0
-
859
-
15232
0
0
T
A
7
*
=
-
0.0753
-
0.0175
1
0
0.0025
-
0.1205
0
1
-
15568
7.432
0
0
-
262
-
14201
0
0
T
A
8
*
=
-
98
-
170
1
0
-
90
-
155
0
1
-
16066
-
851
0
0
-
859
-
15232
0
0
T
A
9
*
=
98
170
1
0
90
155
0
1
-
16066
-
851
0
0
-
859
-
15232
0
0
TFigure 4
The change of plant parameters inA2.Figure 5
The change of plant parameters inA5.Figure 6
The change of plant parameters inA7.Figure7 depicts the tracking error of x- and y-axis between the reference model and T-S model. It can be observed from Figure 7 that the position of x and y in the T-S model can track the position of reference model in very short time and tracking errors converge to zero asymptotically. The tracking error eT-S converges to zero asymptotically rather than shocks near zero because the new adaptive law is chosen to be the previous adaptive law γi(μiη/∑i=19μiη)P1TP2TetqTt plus a robust term ρsgn(α~i) to guarantee that the derivative of the Lyapunov function is negative rather than nonpositive. Figure 8 plots the tracking error of x- and y-axis between the reference model and the nonlinear model. We can notice that the performance of the tracking errors in Figure 8 is worse than that in Figure 7.Figure 7
The tracking erroreT-S between the reference model and T-S model.Figure 8
The tracking erroreNON between the reference model and nonlinear model.The tracking erroreT-S shocks near zero because the controller is designed based on the T-S model rather than the NON model. From Figures 7 and 8, we can see the validity of the adaptive fuzzy control with parameter estimation scheme on the T-S model and the feasibility of the proposed controller on the nonlinear model of gyroscope. Then it can be concluded that the MEMS gyroscope can maintain the proof mass oscillating in the x and y direction at given frequency and amplitude by using adaptive T-S fuzzy controller. The proposed T-S modeling method can provide a possibility for developing a systematic analysis and design method for complex nonlinear control systems, which can approximate the nonlinear system and compensate model uncertainties and external disturbances. It can be observed from Figure 9 that the controller input of the adaptive fuzzy controller is stable.Figure 9
Control input with the proposed controller.


## 6. Conclusions

A T-S fuzzy model based adaptive controller for angular velocity sensor is presented in this paper. An adaptive T-S fuzzy compensator is used to approximate the model uncertainties and external disturbances. The output of the fuzzy controller is used as compensator to reduce the effects of the system nonlinearities. The proposed new adaptive law can guarantee that the derivative of the Lyapunov function is negative rather than nonpositive. The proposed estimator for the existing fuzzy state feedback controller can achieve a good robust performance against parameter uncertainties. Simulation studies are implemented to verify the effectiveness of the proposed adaptive fuzzy control and the correctness of the T-S fuzzy model. However, experimental demonstration should be investigated to verify the validity of the proposed approach in the next research step.



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