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

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1 Discrete Dynamics in Nature and Society Volume 5 Article ID 4643 pages Research Article Adaptive Control of MEMS Gyroscope Based on -S Fuzzy Model Yunmei Fang Shitao Wang Juntao Fei and Mingang Hua College of Mechanical and Electrical Engineering Hohai University Changzhou 3 China College of IO Engineering Hohai University Changzhou 3 China Correspondence should be addressed to Juntao Fei; jtfei@yahoo.com Received 3 November 4; Revised January 5; Accepted January 5 Academic Editor: Manuel De la Sen Copyright 5 Yunmei Fang et al. his is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. A multi-input multioutput (MIMO) akagi-sugeno (-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 -S fuzzy submodel based on a parallel distributed compensation (PDC) method. A parameter estimation scheme for updating the parameters of the -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. he controller output is implemented on the nonlinear model and -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 -S fuzzy model.. Introduction Gyroscopes are commonly used sensors in navigation automobile and so forth. he 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 derived two adaptive controllers for a MEMS gyroscope that tune the drive axis natural frequencytoapreselectedfrequencyanddrivethesenseaxis vibration to zero for force-to-rebalance operation. Sung et al. 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 derived adaptive fuzzy sliding mode controller for robot manipulator. Yoo and Ham developed adaptive controller for robot manipulator using fuzzy compensator. Wai proposed fuzzy sliding mode controller using adaptive tuning technique. Chang et al. designed and implemented fuzzy sliding mode formation control for multirobot systems. Chien et al. 3 developed robust adaptive controller design for a class of uncertain nonlinear systems using online -S fuzzy-neural modeling approach. Stabilizing controller design for uncertain nonlinear systems using fuzzy models was investigated in 4. Park and Cho 5 designed -S model based indirect adaptive fuzzy controller using online parameter estimation.

2 Discrete Dynamics in Nature and Society he system nonlinearities in MEMS gyroscope model have been described in 6 8. Since there are system nonlinearities in MEMS gyroscope it is necessary to use robust adaptive fuzzy controller to MEMS gyroscope and utilize -S fuzzy model to represent the system nonlinearities. Systematic stability analysis and controller design of the adaptive fuzzy controller using -S fuzzy model for MEMS gyroscopehavenotbeeninvestigatedbefore.headvantage oftheproposedadaptive-sfuzzycontrolleroverother controllers is that it can represent system nonlinearities using -S fuzzy model in the aspect of robust design. hus it can overcome the impacts on output tracking error that are caused by model uncertainties and external disturbances. he proposed -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. hus it is necessary to adopt the adaptive fuzzy control scheme to approximate thenonlinearsystemandcompensatemodeluncertainties and external disturbances in the control of MEMS gyroscope using -S fuzzy model. In this paper the Lyapunov-based robust adaptive fuzzy control strategy is applied to the tracking control of MEMS gyroscope using -S fuzzy model. he paper integrates adaptive control and the nonlinear approximation of fuzzy control with -S fuzzy model. he proposed adaptive fuzzy -S controller can guarantee the asymptotic stability of the closedloopsystemandimprovetherobustnessofcontrol system in the presence of model uncertainties and external disturbances. he proposed controller has the following characteristics and contributions. () 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 foreach-sfuzzysubmodelbasedonpdc.he 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. () o improve the performance of the control system the corresponding improvement is made to the adaptive law. he 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) he proposed estimator for the existing fuzzy state feedback controller can achieve a good robust performance against parameter uncertainties. he controller output is implemented in the -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 - S model and the feasibility of the proposed controller on the nonlinear model of gyroscope. Ω z d k yy yy k ky 3 xx kx 3 k xx kx 3 d xx k yy ky 3 m Figure : Z-axis vibratory MEMS gyroscope with nonlinear effective spring. d yy d xx. 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 z-axis as shown in Figure. Referring to the dynamics equations of gyroscope system are as follows: mxd xx x(d xy mω z ) y (k xx mω z )xkxy yk x 3x 3 u x myd yy y(d xy mω z ) x (k yy mω z )ykxy xk y 3y 3 u y where m is the mass of proof mass. Fabrication imperfections contribute mainly to the asymmetric spring term d xy and asymmetric damping terms k xy d xx andd yy are damping terms; k xx k yy are linear spring terms; k x 3 k y 3 are nonlinear spring terms Ω z is angular velocity; u x u y are the control forces. Dividingtheequationbythereferencemassandbecause of the nondimensional time t ω t dividing both sides of equation by reference frequency ω and reference length q andrewritingthedynamicsinvectorformsresultin q q D mω q q S ω q Ω z q mω q K q mω K 3 q 3 q mω q q u mω q y x () ()

3 Discrete Dynamics in Nature and Society 3 where q x y u u x u y D d xx d xy d xy d yy S Ω z Ω z K k xx k xy K 3 k xy k k x3. yy k y 3 (3) x x y y B i q x q x y y μ M i (x) M i (y) M i3 ( x) M i4 ( y). (8) Define new parameters as follows: D D mω q q q u u mω q Ω z Ω z ω S S K ω K mω K 3 K 3 q mω. (4) For the convenience of notation ignoring the superscript yields the final form of the nondimensional equation of motion for the z-axis gyroscope q (S D) q (Ω z K )q K 3 q 3 u. (5) 3. Adaptive Fuzzy Control Based on PDC he MIMO -S fuzzy model of the MEMS gyroscope which is composed of i rules is established based on the nonlinear model of MEMS gyroscope. Use the fuzzy implications and the fuzzy reasoning methods suggested by akagi and Sugeno to express a real plant model; the -S fuzzy model uses fuzzy implication of the following form: Rule i: IFx is about M i and y is about M i and is about M i3 and HEN qa i qb i u y is about M i4... he defuzzification fuzzy dynamic system model can be expressed using center-average defuzzifier product inference and singleton fuzzifier x (6) M i (x) M i (y) M i3 ( x)andm i4 ( y) are membership function values of the fuzzy variable x y xand y with respect to fuzzy set M i M i M i3 andm i4 respectively. Since the control target for MEMS gyroscope is to maintain the proof mass to oscillate in the x and y direction at given frequency and amplitude x A x sin(ω x t) y A y sin(ω y t) generally the reference model can be defined as q d A d q d where q d x A x ω x sin (ω xt) y A y ω y x sin (ω yt) A x ω x cos (ω x t) y A y ω y cos (ω y t) x A x ω x cos (ω x t) y q d A y ω y cos (ω y t) x A x sin (ω x t) y A y sin (ω y t) ω x ω A d y. Since the reference model is nonasymptotically stable the nonlinear -S fuzzy model cannot be asymptotically stable. In order to design local linear state feedback controller for each -S fuzzy submodel based on PDC the reference model is adjusted to be q m A m q m B m rwhere () where q μ A i qb i u μ (7) a i ai ai 3 ai 4 A i a i ai ai 3 ai 4 a m am am 3 am 4 ω x a m A m am am 3 am 4 ω y x A x ω x sin (ω xt) y B m q m A y ω y x sin (ω yt) A x ω x cos (ω x t) y A y ω y cos (ω y t)

4 4 Discrete Dynamics in Nature and Society x A x ω x cos (ω x t) y q m A y ω y cos (ω y t) x r A xω x cos (ω x t) A x sin (ω x t) A y ω y cos (ω y t). y A y sin (ω y t) () Remark. Substituting A m B m q m r into q m yields q d ;we can get that the changed reference model q m A m q m B m r is equivalent to previous one q d A d q d. According to PDC 4 we design local state linear feedback controllers for each linear submodel. he fuzzy control rules are as the following forms: Rule i: IFx is about M i and y is about M i and is about M i3 and y is about M i4 HEN u (t) K i (t) x (t) r(t) x... () he controller output can be expressed using center-average defuzzifier product inference and singleton fuzzifier where K i A i A m u (t) μ K i q (t) r(t) μ () a i am ai am ai 3 am 3 ai 4 am 4 a i am ai am ai 3 am 3 ai 4 am 4... (3) Substituting the controller force () into the fuzzy system (7) yields the desired model 4. Parameter Estimation q m A m q m B m r. (4) heblockdiagramoftheadaptivefuzzycontrolusing-s fuzzy model is shown as in Figure. aking the parameter uncertainty into account then the parameter estimation is applied to the adaptive fuzzy control. he 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. he controller output is implemented in the nonlinear model also to prove the correctness of the -S model and the feasibilityoftheproposedcontrolschemeonthenonlinear model of MEMS gyroscope. With unknown parameters matrix A i the controller changes to where K i A i A m u (t) μ K i q (t) r(t) μ (5) a i am a i am a i 3 am 3 a i 4 am 4 a i am a i am a i 3 am 3 a i 4 am 4 a i a i a i 3 a i 4 A i a i a i a i 3 a i 4... (6) istheestimateofa i. By considering the plant parameterization (7) can be rewritten as q (t) A s q (t) μ (A i A s )q(t) B i u (t) μ (7) where A s is an arbitrary stable matrix. Define the estimation mode as the following formula: q (t) A s q (t) μ ( A i A s )q(t) B i u (t) μ. (8) he estimation model (8) canbethesameoneasbyadopting the control input (5) q m A m q m B m r. () he estimation error e(t) q(t) q m (t) meets () and () e (t) A s e (t) μ (A i A i )q(t) μ A s e (t) μ α i α i q (t) μ () e (t) e(t) A s μ q (t) α i α i μ ()

5 Discrete Dynamics in Nature and Society 5 Desired trajectory Reference input q dx A x sin ω x t) q dy A y sin ω y t) r x A x ω x cos ω x t) r y A y ω y cos ω y t) ) ) ) ).. (S D). q q (Ω z K )q K 3 q 3 u Nonlinear model q d q non Σ e NON μ K i q(t) r(t) u(t) μ U Fuzzy controller K i (t)μ. ã μ i γ i μ P e(t)q P (t) ρsgn(ã i ) Adaptive law. μ (A i A m )q(t) B i u(t) q (t) A m q(t) N μ Estimation model μ (A i A m )q(t) B i u(t). q(t) A m q(t) N μ Plant model e -S Σ q q Figure : he block diagram of the adaptive fuzzy control using -S fuzzy model. where e(t) actually is the tracking error between the plant model (7) and the estimation model (8) α i (t) α i (t) α i (t) α i (t) α i (t) α i α i (t) a i (t) a i (t) a i 3 (t) a i 4 (t) α i a i ai ai 3 ai 4 α i (t) α i (t) α i α i a i (t) a i (t) a i 3 (t) a i 4 (t) α i a i ai ai 3 ai 4. Remark. he estimation error e(t) can be thought of as the tracking error e -S between the plant model (7) and the desired linear model (4) because e(t) q(t) q m (t) and e -S (t) q(t) q m (t) havethesamecontrolmatrixwhilethe tracking error e NON between the reference model (4) and the nonlinear model (5) is used to prove the correctness of the - S model and the feasibility of the proposed controller on the nonlinear model of gyroscope. Define a Lyapunov function candidate as V (t) V (e a i a i )e (t) Pe (t) a i (t) a i (t) γ i a i (t) a i (t) γ i where P meet A s PPA s I. he derivative of V with respect to time becomes () Substituting () () into the derivative of V yields V (t) e (t) (A s PPA s)e(t) μ P e (t) q (t) a i(t) μ μ P e (t) q (t) a i (t) μ a i (t) a i (t) γ i where PP P P 3 P 4. he adaptive law can be chosen as a μ i γ i μ P ρsgn ( a i (t)) ρ > where the sgn function can be defined as a i (t) a i (t) γ i P e(t) q (t) (4) (5) V (t) e (t) Pe (t) e (t) Pe (t) a i (t) a i (t) γ i a i (t) a i (t). γ i (3) a i > { sgn ( a i ) a i { { a i <. (6)

6 6 Discrete Dynamics in Nature and Society M 7 M 8 M M M M 3 M 4 M 5 M 73 M 83 M 6 M 3 M 3 M 3 M 33 M 43 M 53 M 63 μ(x).5 μ(y) x y M 3 M 6 M M M 4 M 7 M M 34 M 5 M 64 M 8 M 4 M 4 M 44 M 74 M 4 M 54 M 84 μ(d x ).5 μ(d y ) d x d y Figure 3: Membership functions. Substituting (5) into (4) yields k yy 7.6 N/m k xy 5N/m V (t) e (t) e (t) ρ ρ e (t) e (t) ρ sgn ( a i (t)) a i (t) γ i sgn ( a i (t)) a i (t) γ i a i (t) γ i e (t) e (t) e (t). ρ a i (t) γ i (7) V(t) and V(t) imply the boundedness of e(t) and α i (t) the control is bounded for all time () implies e(t) is also bounded and inequality (7) implies e(t) L.hen with e(t) L L and e(t) L according to Barbalat s lemma the tracking error e(t) asymptotically converges to zero. 5. Simulation Analysis In this section we will evaluate the proposed adaptive fuzzy control using -S model approach on the lumped MEMS gyroscope sensor model. Parameters of the MEMS gyroscope sensor are as follows: m.57e 8 kg ω khz q e 6 m d xx.4e 6 Ns/m d xy.4e 6 Ns/m d yy.4e 6 Ns/m k xx 8.8 N/m k x e N/m k y e N/m Ω z 5. rad/s. (8) Since the general displacement range of the MEMS gyroscope sensor in each axis is submicrometer level it is reasonable to choose μm as the reference length q. Given that the usual natural frequency of each axle of a vibratory MEMS gyroscope sensor is in the khz range ω is chosenaskhz.heunknownangularvelocityisassumed Ω z 5. rad/s. he desired motion trajectories are x m A x sin(ω x t) y m A y sin(ω y t) wherea x A y. ω x 6.7 khz and ω y 5. khz. he plant parameters are adjusted online by adaptive law (4) where the adaptive gain γ i and the adaptive parameter ρ. he initial values of -S model are..4 initial values of the estimation model are and initial values of the nonlinear model are..4. he external disturbance is d d x d y ω sin(πt) sin(πt). A x m ω y and the eigenvalues of A m are {.5±6.66i.5±5.855i}.AlthoughA s can meet the requirement of A s in A s PPA s Ithevaluesoffirstand second column in P which are.5.5 cannot adjust each.. parameter since partial parameters are zero which is used as adaptive gain in (4) soa s is chosen as A s

7 Discrete Dynamics in Nature and Society A.7.75 A.4.3 A A A.6.7 A.. A A Figure 4: he change of plant parameters in A A 5 5 A 5 85 A A A A A A Figure 5: he change of plant parameters in A 5. to ensure that each parameter of first and second column in P is nonzero values as he fuzzy rules for -S fuzzy model for the system can be obtained by linearizing the nonlinear model (5) at the points x { } x { A x ω x A x ω x } y {..}. and y { A y ω y A y ω y } and then the membership functions of states x y xand y with respect to fuzzy set M i M i M i3 andm i4 used in the -S fuzzy model are shown as in Figure 3. In this simulation it is assumed that the physical parameters in the -S fuzzy model are not known exactly. Hence the

8 8 Discrete Dynamics in Nature and Society A A A A A A 7.. A A Figure 6: he change of plant parameters in A 7. parametersaretunedbytheproposedadaptivelaw.hetrue values of A i of fuzzy -S model are as shown in able.he initial value A i. A i. he variations of each parameter in A A 5 anda 7 are shown in Figures 4 5and6. Here only parts of parameters are shown in table for examples. In Figures 4 5and6we can see each parameter converges to its true value asymptotically. Figure 7 depicts the tracking error of x- andy-axis between the reference model and -S model. It can be observed from Figure 7 that the position of x and y in the-smodelcantrackthepositionofreferencemodel in very short time and tracking errors converge to zero asymptotically. he tracking error e -S converges to zero asymptotically rather than shocks near zero because the new adaptive law is chosen to be the previous adaptive law γ i (μ / μ i(η)) P P e(t)q (t) plus a robust term ρ sgn( α i ) to guarantee that the derivative of the Lyapunov function is negative rather than nonpositive. Figure 8 plots the tracking error of x- andy-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. he tracking error e -S shocks near zero because the controller is designed based on the -S model rather than the NON model. From Figures 7 and 8wecanseethevalidity of the adaptive fuzzy control with parameter estimation scheme on the -S model and the feasibility of the proposed controller on the nonlinear model of gyroscope. hen it can be concluded that the MEMS gyroscope can maintain e xts e yts Figure 7: he tracking error e -S between the reference model and -S model. the proof mass oscillating in the x and y direction at givenfrequencyandamplitudebyusingadaptive-sfuzzy controller. he proposed -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 that the controller input of the adaptive fuzzy controller is stable.

9 Discrete Dynamics in Nature and Society able : Parameters of -S fuzzy model A A A A A A A A A e xnon e ynon Figure 8: he tracking error e NON between the reference model and nonlinear model. u y u x Figure : Control input with the proposed controller. 6. Conclusions A -S fuzzy model based adaptive controller for angular velocity sensor is presented in this paper. An adaptive -S fuzzy compensator is used to approximate the model uncertainties and external disturbances. he output of the fuzzy controller is used as compensator to reduce the effects of the system nonlinearities. he proposed new adaptive law can guarantee that the derivative of the Lyapunov function is negative rather than nonpositive. he 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 -S fuzzy model. However experimental demonstration should be investigated to verify the validity of the proposed approach in the next research step. Conflict of Interests he authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments he authors thank the anonymous reviewers for useful comments that improved the quality of the paper. his work is partially supported by National Science Foundation of China under Grant no and Natural Science Foundation of Jiangsu Province under Grant no. BK336. References R. P. Leland Adaptive control of a MEMS gyroscope using lyapunov methods IEEE ransactions on Control Systems echnologyvol.4no.pp

10 Discrete Dynamics in Nature and Society S. Sung W.-. Sung C. Kim S. Yun and Y. J. Lee On the mode-matched control of MEMS vibratory gyroscope via phase-domain analysis and design IEEE/ASME ransactions on Mechatronicsvol.4no.4pp R.AntonelloR.OboeL.PrandiandF.Biganzoli Automatic mode matching in MEMS vibrating gyroscopes using extremum-seeking control IEEE ransactions on Industrial Electronics vol. 56 no. pp S.ParkR.HorowitzS.K.HongandY.Nam rajectoryswitching algorithm for a MEMS gyroscope IEEE ransactions on Instrumentation and Measurement vol.56no.6pp J. D. John and. Vinay Novel concept of a single-mass adaptively controlled triaxial angular rate sensor IEEE Sensors Journalvol.6no.3pp J. Fei and C. Batur A novel adaptive sliding mode control with application to MEMS gyroscope ISA ransactionsvol.48no. pp J. Fei and F. Chowdhury Robust adaptive controller for triaxial angular velocity sensor International Innovative Computing Information and Control vol.7no.6pp L. Wang Adaptive Fuzzy Systems and Control-Design and Stability Analysis Prentice Hall Upper Saddle River NJ USA 4. Y. Guo and P.-Y. Woo An adaptive fuzzy sliding mode controller for robotic manipulators IEEE ransactions on Systems Man and Cybernetics Part A: Systems and Humans.vol.33no. pp B. K. Yoo and W. C. Ham Adaptive control of robot manipulator using fuzzy compensator IEEE ransactions on Fuzzy Systemsvol.8no.pp.86. R.-J. Wai Fuzzy sliding-mode control using adaptive tuning technique IEEE ransactions on Industrial Electronicsvol.54 no. pp Y.-H. Chang C.-W. Chang C.-L. Chen and C.-W. ao Fuzzy sliding-mode formation control for multirobot systems: design and implementation IEEE ransactions on Systems Man and Cybernetics Part B: Cyberneticsvol.4no.pp Y.-H. Chien W.-Y. Wang Y.-G. Leu and.-. Lee Robust adaptive controller design for a class of uncertain nonlinear systems using online -S fuzzy-neural modeling approach IEEE ransactions on Systems Man and Cybernetics Part B: Cybernetics vol. 4 no. pp M. C. M. eixeira and S. H. Zak Stabilizing controller design for uncertain nonlinear systems using fuzzy models IEEE ransactions on Fuzzy Systemsvol.7no.pp C.-W. Park and Y.-W. Cho -S model based indirect adaptive fuzzy control using online parameter estimation IEEE ransactions on Systems Man and Cybernetics Part B: Cybernetics vol. 34 no. 6 pp S. F. Asokanthan and. Wang Nonlinear instabilities in a vibratory gyroscope subjected to angular speed fluctuations Nonlinear Dynamicsvol.54no.-pp Wang Nonlinear and stochastic dynamics of MEMS-based angular rate sensing and switching systems Ph.D. thesis he University of Western of Ontario Ontario Canada. 8 Z. Chi M. Jia and Q. Xu Fuzzy PID feedback control of piezoelectric actuator with feedforward compensation Mathematical Problems in Engineering vol. 4 Article ID pages 4.. akagi and M. Sugeno Fuzzy identification of systems and its applications to modeling and control IEEE ransactions on Systems Man and Cybernetics Part Bvol.5no.pp P. Ioannou and J. Sun Robust Adaptive Control Prentice Hall Upper Saddle River NJ USA 6.

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