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

Yunmei Fang1 Shitao Wang2 Juntao Fei2 and Mingang Hua2

1College of Mechanical and Electrical Engineering Hohai University Changzhou 213022 China2College of IOT Engineering Hohai University Changzhou 213022 China

Correspondence should be addressed to Juntao Fei jtfeiyahoocom

Received 23 November 2014 Revised 20 January 2015 Accepted 22 January 2015

Academic Editor Manuel De la Sen

Copyright copy 2015 Yunmei Fang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 Introduction

Gyroscopes are commonly used sensors in navigation auto-mobile and so forth The performance of the MEMS gyro-scope is often deteriorated by the effects of time varyingparameters quadrature errors and external disturbancesAdvanced control such as adaptive control and intelligentcontrol are necessary to be adopted to control the MEMSgyroscope In the last few years various control approacheshave been proposed to control theMEMSgyroscope Increas-ing attention has been given to the tracking control ofMEMSgyroscope Leland [1] derived two adaptive controllersfor a MEMS gyroscope that tune the drive axis naturalfrequency to a preselected frequency and drive the sense axisvibration to zero for force-to-rebalance operation Sung etal [2] proposed a phase-domain design approach to studythe mode-matched control of gyroscope Antonello et al [3]derived extremum-seeking control to automatically matchthe vibrationmode inMEMS vibrating gyroscopes Park et al[4] presented an adaptive controller for a MEMS gyroscopewhich drives both axes of vibration and controls the entireoperation of the gyroscope John and Vinay [5] developeda novel concept for an adaptively controlled triaxial angu-lar velocity sensor device Sliding mode control has been

incorporated into adaptive controller to control the MEMSgyroscopes [6 7] Model uncertainties and disturbances areinevitable in actual engineering and require the controllerto be either adaptive or robust to these model uncertaintiesAdaptive fuzzy sliding mode controller can be utilized tocompensate the model uncertainties and disturbances sinceit combines the merits of the sliding mode control the fuzzyinference mechanism and the adaptive algorithm Wang[8] demonstrated that an arbitrary function of a certain setof functions can be approximated with arbitrary accuracyusing fuzzy system on a compact domain using universalapproximation theorem Guo and Woo [9] derived adaptivefuzzy sliding mode controller for robot manipulator Yoo andHam [10] developed adaptive controller for robot manipula-tor using fuzzy compensator Wai [11] proposed fuzzy slidingmode controller using adaptive tuning technique Chang et al[12] designed and implemented fuzzy slidingmode formationcontrol for multirobot systems Chien et al [13] developedrobust adaptive controller design for a class of uncertainnonlinear systems using online T-S fuzzy-neural modelingapproach Stabilizing controller design for uncertain nonlin-ear systems using fuzzy models was investigated in [14] Parkand Cho [15] designed T-S model based indirect adaptivefuzzy controller using online parameter estimation

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 419643 10 pageshttpdxdoiorg1011552015419643

2 Discrete Dynamics in Nature and Society

The system nonlinearities in MEMS gyroscope modelhave been described in [16ndash18] Since there are systemnonlinearities in MEMS gyroscope it is necessary to userobust adaptive fuzzy controller to MEMS gyroscope andutilize T-S fuzzymodel to represent the systemnonlinearities

Systematic stability analysis and controller design of theadaptive fuzzy controller using T-S fuzzy model for MEMSgyroscope have not been investigated before The advantageof the proposed adaptive T-S fuzzy controller over othercontrollers is that it can represent system nonlinearities usingT-S fuzzy model in the aspect of robust design Thus itcan overcome the impacts on output tracking error that arecaused bymodel uncertainties and external disturbancesTheproposed T-S modeling method can provide a possibilityfor developing a systematic analysis and design methodfor complex nonlinear control systems thus improving thetracking and compensation performanceThus it is necessaryto adopt the adaptive fuzzy control scheme to approximatethe nonlinear system and compensate model uncertaintiesand external disturbances in the control of MEMS gyroscopeusing T-S fuzzy model

In this paper the Lyapunov-based robust adaptive fuzzycontrol strategy is applied to the tracking control of MEMSgyroscope using T-S fuzzy model The paper integratesadaptive control and the nonlinear approximation of fuzzycontrol with T-S fuzzy model The proposed adaptive fuzzyT-S controller can guarantee the asymptotic stability of theclosed loop system and improve the robustness of controlsystem in the presence of model uncertainties and externaldisturbances The proposed controller has the followingcharacteristics and contributions

(1) Since the reference model is marginally stable thereference model is required to be revised so that thelocal linear state feedback controller can be designedfor each T-S fuzzy submodel based on PDC Thepartial parameters of the reference control matrixare changed and then the reference input is changedaccordingly to guarantee that the changed referencemodel is equivalent to previous one

(2) To improve the performance of the control systemthe corresponding improvement is made to the adap-tive law The new adaptive law can be chosen tobe the previous adaptive law plus a robust termto guarantee that the derivative of the Lyapunovfunction is negative rather than nonpositive

(3) The proposed estimator for the existing fuzzy statefeedback controller can achieve a good robust per-formance against parameter uncertainties The con-troller output is implemented in the T-S fuzzy modeland nonlinear model respectively to verify the valid-ity of the adaptive fuzzy control with parameterestimation scheme and prove the correctness of the T-S model and the feasibility of the proposed controlleron the nonlinear model of gyroscope

kxx2

2

kxx2

2

kyy

2

2

kyy

2

2

dyy

2

Ωz

y

x

dxx2

dyy

2

dxx2

m

kx3 kx3

ky3

ky3

Figure 1 119885-axis vibratory MEMS gyroscope with nonlinear effec-tive spring

2 Dynamics of MEMS Gyroscope

Assume that the gyroscope is moving with a constant linearspeed the gyroscope is rotating at a constant angular velocitythe centrifugal forces are assumed negligible the gyroscopeundergoes rotations along 119911-axis as shown in Figure 1Referring to [4 16 17] the dynamics equations of gyroscopesystem are as follows

119898 + 119889119909119909 + (119889

119909119910minus 2119898Ω

lowast

119911) 119910

+ (119896119909119909

minus 119898Ωlowast

119911

2

) 119909 + 119896119909119910119910 + 11989611990931199093

= 119906119909

119898 119910 + 119889119910119910

119910 + (119889119909119910

+ 2119898Ωlowast

119911)

+ (119896119910119910

minus 119898Ωlowast

119911

2

) 119910 + 119896119909119910119909 + 11989611991031199103

= 119906119910

(1)

where119898 is the mass of proof mass Fabrication imperfectionscontribute mainly to the asymmetric spring term 119889

119909119910 and

asymmetric damping terms 119896119909119910 119889119909119909 and 119889

119910119910are damping

terms 119896119909119909 119896119910119910

are linear spring terms 1198961199093 1198961199103 are nonlinear

spring terms Ωlowast119911is angular velocity 119906

119909 119906119910are the control

forcesDividing the equation by the reference mass and because

of the nondimensional time 119905lowast = 1205960119905 dividing both sides of

equation by reference frequency 12059620and reference length 119902

0

and rewriting the dynamics in vector forms result in

119902lowast

1

1199020

+119863lowast

1198981205960

119902lowast

1

1199020

+ 2119878lowast

1205960

119902lowast

1

1199020

minusΩlowast

119911

2

1198981205960

119902lowast

1

1199020

+119870lowast

1

1198981205962

0

119902lowast

1

1199020

+119870lowast

3

1198981205962

0

119902lowast

1

3

1199020

=119906lowast

1198981205962

01199020

(2)

Discrete Dynamics in Nature and Society 3

where

119902lowast

1= [

119909

119910] 119906

lowast

= [

119906119909

119906119910

] 119863lowast

= [

119889119909119909

119889119909119910

119889119909119910

119889119910119910

]

119878lowast

= [

0 minusΩlowast

119911

Ωlowast

1199110

] 119870lowast

1= [

119896119909119909

119896119909119910

119896119909119910

119896119910119910

] 119870lowast

3= [

1198961199093 0

0 1198961199103

]

(3)

Define new parameters as follows

1199021=119902lowast

1

1199020

119906 =119906lowast

1198981205962

01199020

Ω119911=Ωlowast

119911

1205960

119863 =119863lowast

1198981205960

119878 =119878lowast

1205960

1198701=

119870lowast

1

1198981205962

0

1198703=119870lowast

31199022

0

1198981205962

0

(4)

For the convenience of notation ignoring the superscriptyields the final form of the nondimensional equation ofmotion for the 119911-axis gyroscope

1199021= (2119878 minus 119863) 119902

1+ (Ω2

119911minus 1198701) 1199021minus 11987031199023

1+ 119906 (5)

3 Adaptive Fuzzy Control Based on PDC

TheMIMO T-S fuzzy model of the MEMS gyroscope whichis composed of 119894 rules is established based on the nonlinearmodel of MEMS gyroscope Use the fuzzy implications andthe fuzzy reasoningmethods suggested by Takagi and Sugeno[19] to express a real plant model the T-S fuzzy model usesfuzzy implication of the following form

Rule 119894 IF 119909 is about 1198721198941and 119910 is about 119872

1198942and

is about 1198721198943and 119910 is about 119872

1198944

THEN 119902 = 119860119894119902 + 119861119894119906 119894 = 1 2 9

(6)

The defuzzification fuzzy dynamic system model can beexpressed using center-average defuzzifier product inferenceand singleton fuzzifier

119902 =sum9

119894=1120583119894(120578) [119860

119894119902 + 119861119894119906]

sum9

119894=1120583119894(120578)

(7)

where

119860119894=

[[[[[[

[

119886119894

11119886119894

12119886119894

13119886119894

14

119886119894

21119886119894

22119886119894

23119886119894

24

1 0 0 0

0 1 0 0

]]]]]]

]

119861119894=

[[[[[

[

1 0

0 1

0 0

0 0

]]]]]

]

119902 =

[[[[[

[

119910

119910

]]]]]

]

119902 =

[[[[[

[

119910

119909

119910

]]]]]

]

120583119894(120578) = 119872

1198941(119909)119872

1198942(119910)119872

1198943()119872

1198944( 119910)

(8)

1198721198941(119909)119872

1198942(119910)119872

1198943() and119872

1198944( 119910) aremembership function

values of the fuzzy variable 119909119910 and 119910with respect to fuzzyset119872

119894111987211989421198721198943 and119872

1198944 respectively

Since the control target for MEMS gyroscope is tomaintain the proof mass to oscillate in the 119909 and 119910 directionat given frequency and amplitude 119909 = 119860

119909sin(120596119909119905) 119910 =

119860119910sin(120596119910119905) generally the reference model can be defined as

119902119889= 119860119889119902119889 where

119902119889=

[[[[[

[

119910

119910

]]]]]

]

=

[[[[[[

[

1198601199091205962

119909sin (120596

119909119905)

1198601199101205962

119910sin (120596

119910119905)

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)

]]]]]]

]

119902119889=

[[[[[

[

119910

119909

119910

]]]]]

]

=

[[[[[[

[

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)

119860119909sin (120596

119909119905)

119860119910sin (120596

119910119905)

]]]]]]

]

119860119889=

[[[[[

[

0 0 minus1205962

1199090

0 0 0 minus1205962

119910

1 0 0 0

0 1 0 0

]]]]]

]

(9)

Since the reference model is nonasymptotically stable thenonlinear T-S fuzzy model cannot be asymptotically stableIn order to design local linear state feedback controller foreach T-S fuzzy submodel based on PDC the reference modelis adjusted to be 119902

119898= 119860119898119902119898+ 119861119898119903 where

119860119898=

[[[[[[

[

119886119898

11119886119898

12119886119898

13119886119898

14

119886119898

21119886119898

22119886119898

23119886119898

24

1 0 0 0

0 1 0 0

]]]]]]

]

=

[[[[[

[

minus1 0 minus1205962

1199090

0 minus1 0 minus1205962

119910

1 0 0 0

0 1 0 0

]]]]]

]

119861119898=

[[[[[

[

1 0

0 1

0 0

0 0

]]]]]

]

119902119898=

[[[[[

[

119910

119910

]]]]]

]

=

[[[[[[

[

1198601199091205962

119909sin (120596

119909119905)

1198601199101205962

119910sin (120596

119910119905)

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)

]]]]]]

]

4 Discrete Dynamics in Nature and Society

119902119898=

[[[[[

[

119910

119909

119910

]]]]]

]

=

[[[[[[

[

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)

119860119909sin (120596

119909119905)

119860119910sin (120596

119910119905)

]]]]]]

]

119903 = [

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)]

(10)

Remark 1 Substituting 119860119898 119861119898 119902119898 119903 into 119902

119898yields 119902

119889 we

can get that the changed reference model 119902119898= 119860119898119902119898+ 119861119898119903

is equivalent to previous one 119902119889= 119860119889119902119889

According to PDC [14] we design local state linearfeedback controllers for each linear submodel The fuzzycontrol rules are as the following forms

Rule 119894 IF 119909 is about 1198721198941and 119910 is about 119872

1198942and

is about 1198721198943and 119910 is about 119872

1198944

THEN 119906 (119905) = minus119870119894(119905) 119909 (119905) + 119903 (119905) 119894 = 1 2 9

(11)

The controller output can be expressed using center-averagedefuzzifier product inference and singleton fuzzifier

119906 (119905) =sum9

119894=1120583119894(120578) [minus119870

119894119902 (119905) + 119903 (119905)]

sum9

119894=1120583119894(120578)

(12)

where

119870119894= 119860119894minus 119860119898

=

[[[[[[

[

119886119894

11minus 119886119898

11119886119894

12minus 119886119898

12119886119894

13minus 119886119898

13119886119894

14minus 119886119898

14

119886119894

21minus 119886119898

21119886119894

22minus 119886119898

22119886119894

23minus 119886119898

23119886119894

24minus 119886119898

24

1 0 0 0

0 1 0 0

]]]]]]

]

119894 = 1 2 9

(13)

Substituting the controller force (12) into the fuzzy system (7)yields the desired model

119902119898= 119860119898119902119898+ 119861119898119903 (14)

4 Parameter Estimation

The block diagram of the adaptive fuzzy control using T-Sfuzzy model is shown as in Figure 2 Taking the parameteruncertainty into account then the parameter estimation isapplied to the adaptive fuzzy control The adaptive law canguarantee the asymptotic convergence of the error betweenthe plant model state and the estimation model state andmake the closed loop system of fuzzy controller and estima-tion model be a desired linear model and consequently theplant model state can follow the state of the desired linearmodelThe controller output is implemented in the nonlinearmodel also to prove the correctness of the T-S model and thefeasibility of the proposed control scheme on the nonlinearmodel of MEMS gyroscope

With unknown parameters matrix 119860119894 the controller

changes to

119906 (119905) =

sum9

119894=1120583119894(120578) [minus

119894119902 (119905) + 119903 (119905)]

sum9

119894=1120583119894(120578)

(15)

where

119894= 119860119894minus 119860119898

=

[[[[[[

[

119886119894

11minus 119886119898

11119886119894

12minus 119886119898

12119886119894

13minus 119886119898

13119886119894

14minus 119886119898

14

119886119894

21minus 119886119898

21119886119894

22minus 119886119898

22119886119894

23minus 119886119898

23119886119894

24minus 119886119898

24

1 0 0 0

0 1 0 0

]]]]]]

]

119860119894=

[[[[[[

[

119886119894

11119886119894

12119886119894

13119886119894

14

119886119894

21119886119894

22119886119894

23119886119894

24

1 0 0 0

0 1 0 0

]]]]]]

]

119894 = 1 2 9

(16)

is the estimate of 119860119894

By considering the plant parameterization (7) can berewritten as

119902 (119905) = 119860119904119902 (119905) +

sum9

119894=1120583119894(120578) [(119860

119894minus 119860119904) 119902 (119905) + 119861

119894119906 (119905)]

sum9

119894=1120583119894(120578)

(17)

where 119860119904is an arbitrary stable matrix

Define the estimation mode as the following formula

119902 (119905) = 119860119904119902 (119905) +

sum9

119894=1120583119894(120578) [(119860

119894minus 119860119904) 119902 (119905) + 119861

119894119906 (119905)]

sum9

119894=1120583119894(120578)

(18)

The estimationmodel (18) can be the same one as by adoptingthe control input (15)

119902119898= 119860119898119902119898+ 119861119898119903 (19)

The estimation error 119890(119905) = 119902(119905) minus 119902119898(119905)meets (19) and (20)

119890 (119905) = 119860119904119890 (119905) +

sum9

119894=1120583119894(120578) (119860

119894minus 119860119894) 119902 (119905)

sum9

119894=1120583119894(120578)

= 119860119904119890 (119905) minus

sum9

119894=1120583119894(120578) [

11198942119894

0 0]119879

119902 (119905)

sum9

119894=1120583119894(120578)

(20)

119890119879

(119905) = 119890 (119905) 119860119879

119904minussum9

119894=1120583119894(120578) 119902119879

(119905) [1119894

2119894

0 0]

sum9

119894=1120583119894(120578)

(21)

Discrete Dynamics in Nature and Society 5

Σ

Σ

+

minusqnon

qd

eT-S

+

minus

q

q

Desired trajectory

Reference input

Fuzzy controller

Adaptive law

U

Plant model

Estimation model

Nonlinear model

Ki(t)120583i(120578)

eNON

qdx = Axsin 120596xt)

qdy = Aysin 120596yt)

rx = Ax120596xcosry = Ay120596ycos

u(t) =sum9

i=1120583i(120578)

sum9i=1120583i(120578)

[minus Kiq(t) + r(t)]

aTi = 120574i

120583i(120578)

sum9i=1120583i(120578)

e(t)qT(t) minus 120588sgn(ai)[PT1 PT

2 ]

1 = (2S minus D) 1 + (Ω2z minus K1)q1 minus K3q

31 + u

(t) = Amq(t) +sum9

i=1120583i(120578)[(Ai minus Am)q(t) + Biu(t)]

Ni=1120583i(120578)sum

(t) = Amq(t) +sum9

i=1120583i(120578)[(Ai minus Am)q(t) + Biu(t)]

Ni=1120583i(120578)sum

q

)

120596xt))

)

120596yt))

q

q

q

Figure 2 The block diagram of the adaptive fuzzy control using T-S fuzzy model

where 119890(119905) actually is the tracking error between theplant model (7) and the estimation model (18)

119894(119905) =

[1119894(119905) 2119894(119905) 0 0]

1119894(119905) =

1119894(119905) minus 120572

1119894 1119894(119905) =

[119886119894

11(119905) 119886

119894

12(119905) 119886

119894

13(119905) 119886

119894

14(119905)] 120572

1119894= [119886

119894

11119886119894

12119886119894

13119886119894

14]

2119894(119905) =

2119894(119905) minus 120572

2119894 2119894

= [119886119894

21(119905) 119886

119894

22(119905) 119886

119894

23(119905) 119886

119894

24(119905)]

1205722119894= [119886119894

21119886119894

22119886119894

23119886119894

24]

Remark 2 The estimation error 119890(119905) can be thought of asthe tracking error 119890T-S between the plant model (7) and thedesired linear model (14) because 119890(119905) = 119902(119905) minus 119902

119898(119905) and

119890T-S(119905) = 119902(119905) minus 119902119898(119905) have the same control matrix while the

tracking error 119890NON between the referencemodel (14) and thenonlinear model (5) is used to prove the correctness of the T-S model and the feasibility of the proposed controller on thenonlinear model of gyroscope

Define a Lyapunov function candidate as

119881 (119905) = 119881 (119890 1198861119894 1198862119894) = 119890119879

(119905) 119875119890 (119905)

+

9

sum

119894=1

119886119879

1119894(119905) 1198861119894(119905)

1205741119894

+

9

sum

119894=1

119886119879

2119894(119905) 1198862119894(119905)

1205742119894

(22)

where 119875meet 119860119879119904119875 + 119875119860

119904= minus119868

The derivative of 119881 with respect to time becomes

(119905) = 119890119879

(119905) 119875119890 (119905) + 119890119879

(119905) 119875 119890 (119905)

+

9

sum

119894=1

2

119886119879

1119894(119905) 1198861119894(119905)

1205741119894

+

9

sum

119894=1

2

119886119879

2119894(119905) 1198862119894(119905)

1205742119894

(23)

Substituting (20) (21) into the derivative of 119881 yields

(119905) = minus 119890119879

(119905) (119860119879

119904119875 + 119875119860

119904) 119890 (119905)

minus 2sum9

119894=1120583119894(120578) 119875119879

1119890 (119905) 119902

119879

(119905) 1198862119894(119905)

sum9

119894=1120583119894(120578)

minus 2sum9

119894=1120583119894(120578) 119875119879

2119890 (119905) 119902

119879

(119905) 1198862119894(119905)

sum9

119894=1120583119894(120578)

+

9

sum

119894=1

2

119886119879

1119894(119905) 1198861119894(119905)

1205741119894

+

9

sum

119894=1

2

119886119879

2119894(119905) 1198862119894(119905)

1205742119894

(24)

where 119875 = [1198751

1198752

1198753

1198754]

The adaptive law can be chosen as

119886119879

119894= 120574119894

120583119894(120578)

sum9

119894=1120583119894(120578)

[119875119879

1119875119879

2] 119890 (119905) 119902

119879

(119905)

minus 120588 sgn (119886119894(119905)) 120588 gt 0

(25)

where the sgn function can be defined as

sgn (119886119894) =

1 119886119894gt 0

0 119886119894= 0

minus1 119886119894lt 0

(26)

6 Discrete Dynamics in Nature and Society

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 10

05

1

0

05

1

x

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

y

120583(x)

0

05

1

120583(y)

minus6 minus4 minus2 0 2 4 60

05

1

minus6 minus4 minus2 0 2 4 6

120583(d

x)

120583(d

y)

dx dy

M71M81M91

M11

M21M31

M41M51M61

M73M83M93

M32M62M92

M12M42M72

M22M52M82

M34M64M94

M14M44M74

M24M54M84

M13M23M33

M43M53M63

Figure 3 Membership functions

Substituting (25) into (24) yields

(119905) = minus 119890119879

(119905) 119890 (119905) minus 2120588

9

sum

119894=1

sgn (1198861119894(119905)) 1198861119894(119905)

1205741119894

minus 2120588

9

sum

119894=1

sgn (1198862119894(119905)) 1198862119894(119905)

1205742119894

= minus 119890119879

(119905) 119890 (119905) minus 2120588

9

sum

119894=1

10038161003816100381610038161198861119894 (119905)1003816100381610038161003816

1205741119894

minus 2120588

9

sum

119894=1

10038161003816100381610038161198862119894 (119905)1003816100381610038161003816

1205742119894

le minus 119890119879

(119905) 119890 (119905) le minus 119890 (119905)2

le 0

(27)

119881(119905) ge 0 and (119905) le 0 imply the boundedness of 119890(119905) and119894(119905) the control is bounded for all time (21) implies 119890(119905) is

also bounded and inequality (27) implies 119890(119905) isin 1198712 Then

with 119890(119905) isin 1198712

cap 119871infin and 119890(119905) isin 119871

infin according to Barbalatrsquoslemma [20] the tracking error 119890(119905) asymptotically convergesto zero

5 Simulation Analysis

In this section we will evaluate the proposed adaptive fuzzycontrol using T-S model approach on the lumped MEMSgyroscope sensormodel Parameters of theMEMS gyroscopesensor are as follows

119898 = 057119890 minus 8 kg 1205960= 1 kHz 119902

0= 10119890 minus 6m

119889119909119909

= 0429119890 minus 6Nsm 119889119910119910

= 00429119890 minus 6Nsm

119889119909119910

= 00429119890 minus 6Nsm 119896119909119909

= 8098Nm

119896119910119910

= 7162Nm 119896119909119910

= 5Nm

1198961199093 = 35611989012Nm

1198961199103 = 35611989012Nm Ω

119911= 50 rads

(28)

Since the general displacement range of the MEMSgyroscope sensor in each axis is submicrometer level itis reasonable to choose 1 120583m as the reference length 119902

0

Given that the usual natural frequency of each axle of avibratory MEMS gyroscope sensor is in the kHz range 120596

0is

chosen as 1 kHz The unknown angular velocity is assumedΩ119911

= 50 rads The desired motion trajectories are 119909119898

=

119860119909sin(120596119909119905) 119910119898

= 119860119910sin(120596119910119905) where 119860

119909= 1 119860

119910= 12

120596119909= 671 kHz and 120596

119910= 511 kHz

The plant parameters are adjusted online by adaptivelaw (24) where the adaptive gain 120574

119894= 1 and the adaptive

parameter 120588 = 1 The initial values of T-S model are[02 024 0 0] initial values of the estimation model are[0 0 0 0] and initial values of the nonlinear model are[02 024 0 0] The external disturbance is 119889 = [

119889119909

119889119910

] =

[10 sin(2120587119905)10 sin(2120587119905) ]119860119898 = [

minus1 0 minus1205962

1199090

0 minus1 0 minus1205962

119910

1 0 0 0

0 1 0 0

] and the eigenvalues of119860119898

are minus05plusmn66916119894 minus05plusmn50855119894 Although119860119904canmeet the

requirement of 119860119904in 119860119879

119904119875 + 119875119860

119904= minus119868 the values of first and

second column in 119875 which are [051 0

0 052

001 0

0 002

] cannot adjust each

parameter since partial parameters are zero which is used as

adaptive gain in (24) so 119860119904is chosen as 119860

119904= [

minus1 minus1 minus1 minus1

0 minus1 0 0

0 0 minus1 0

0 0 0 minus1

]

Discrete Dynamics in Nature and Society 7

0 1 2 3minus008

minus0075

minus007

minus0065

Time (s)0 1 2 3

22

23

24

25

Time (s)0 1 2 3

minus145

minus14

minus135

minus13

minus125

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)

00 1 2 3minus0018

minus0017

minus0016

minus0015

Time (s)

1 2 3minus013

minus012

minus011

minus01

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)0 1 2 3

minus13

minus125

minus12

minus11

minus115

Time (s)

A2

[11

]

A2

[12

]

A2

[13

]

A2

[14

]

A2

[21

]

A2

[24

]

A2

[22

]

A2

[23

]

times104

times104

times10minus3

Figure 4 The change of plant parameters in 1198602

10 2 385

90

95

100

Time (s)10 2 3

80

85

90

95

Time (s)0 1 2 3

minus17

minus16

minus15

minus14

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)

0 1 2 3150

160

170

180

Time (s)0 1 2 3

Time (s)

0 1 2 3

Time (s)0 1 2 3

Time (s)

130

140

150

160

minus900

minus850

minus800

minus750

minus155

minus15

minus145

minus14

minus135

A5

[11

]

A5

[12

]

A5

[13

]

A5

[14

]

A5

[21

]

A5

[24

]

A5

[22

]

A5

[23

]

times104

times104

Figure 5 The change of plant parameters in 1198605

to ensure that each parameter of first and second column in

119875 is nonzero values as [05 minus025

minus025 075

minus025 025

minus025 025

]

The fuzzy rules for T-S fuzzy model for the system can beobtained by linearizing the nonlinear model (5) at the points119909 isin minus1 0 1 isin minus119860

119909120596119909

0 119860119909120596119909 119910 isin minus12 0 12

and 119910 isin minus119860119910120596119910

0 119860119910120596119910 and then the membership

functions of states 119909 119910 and 119910with respect to fuzzy set1198721198941

11987211989421198721198943 and119872

1198944used in the T-S fuzzy model are shown as

in Figure 3In this simulation it is assumed that the physical parame-

ters in the T-S fuzzy model are not known exactly Hence the

8 Discrete Dynamics in Nature and Society

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

22

23

24

25

65

7

75

minus008

minus0075

minus007

minus0065

minus16

minus155

minus15

minus145

minus14

minus270

minus260

minus250

minus240

minus230

minus145

minus14

minus135

minus13

minus125

minus013

minus012

minus011

minus01

minus0018

minus0017

minus0016

minus0015

A7

[11

]

A7

[12

]

A7

[13

]

A7

[14

]

A7

[21

]

A7

[22

]

A7

[23

]

A7

[24

]

times104

times104

times10minus3

Figure 6 The change of plant parameters in 1198607

parameters are tuned by the proposed adaptive law The truevalues of 119860lowast

119894of fuzzy T-S model are as shown in Table 1 The

initial value 119860119894= 09 lowast 119860

lowast

119894 The variations of each parameter

in 1198602 1198605 and 119860

7are shown in Figures 4 5 and 6 Here

only parts of parameters are shown in table for examples InFigures 4 5 and 6 we can see each parameter converges to itstrue value asymptotically

Figure 7 depicts the tracking error of 119909- and 119910-axisbetween the reference model and T-S model It can beobserved from Figure 7 that the position of 119909 and 119910 inthe T-S model can track the position of reference modelin very short time and tracking errors converge to zeroasymptotically The tracking error 119890T-S converges to zeroasymptotically rather than shocks near zero because thenew adaptive law is chosen to be the previous adaptivelaw 120574119894(120583119894(120578)sum

9

119894=1120583119894(120578)) [119875

119879

1119875119879

2] 119890(119905)119902

119879

(119905) plus a robust term120588 sgn(

119894) to guarantee that the derivative of the Lyapunov

function is negative rather than nonpositive Figure 8 plotsthe tracking error of 119909- and 119910-axis between the referencemodel and the nonlinear model We can notice that theperformance of the tracking errors in Figure 8 is worse thanthat in Figure 7

The tracking error 119890T-S shocks near zero because thecontroller is designed based on the T-S model rather thantheNONmodel FromFigures 7 and 8 we can see the validityof the adaptive fuzzy control with parameter estimationscheme on the T-S model and the feasibility of the proposedcontroller on the nonlinear model of gyroscope Then itcan be concluded that the MEMS gyroscope can maintain

0 5 10 15 20 25 30minus1

minus1

minus2

minus05

0

05

1

Time (s)

0 5 10 15 20 25 30

Time (s)

012

e xts

e yts

Figure 7 The tracking error 119890T-S between the reference model andT-S model

the proof mass oscillating in the 119909 and 119910 direction atgiven frequency and amplitude by using adaptive T-S fuzzycontroller The proposed T-S modeling method can providea possibility for developing a systematic analysis and designmethod for complex nonlinear control systems which canapproximate the nonlinear system and compensate modeluncertainties and external disturbances It can be observedfrom Figure 9 that the controller input of the adaptive fuzzycontroller is stable

Discrete Dynamics in Nature and Society 9

Table 1 Parameters of T-S fuzzy model

119860lowast

1=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

2=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

3=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

4=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

5=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

6=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

7=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

8=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

9=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

0 5 10 15 20 25 30minus1

minus05

05

1

Time (s)

0

minus1

minus20 5 10 15 20 25 30

Time (s)

012

e xno

ne y

non

Figure 8The tracking error 119890NON between the reference model andnonlinear model

6 Conclusions

A T-S fuzzy model based adaptive controller for angularvelocity sensor is presented in this paper An adaptiveT-S fuzzy compensator is used to approximate the modeluncertainties and external disturbances The output of thefuzzy controller is used as compensator to reduce the effectsof the system nonlinearities The proposed new adaptive lawcan guarantee that the derivative of the Lyapunov functionis negative rather than nonpositive The proposed estimatorfor the existing fuzzy state feedback controller can achievea good robust performance against parameter uncertaintiesSimulation studies are implemented to verify the effectivenessof the proposed adaptive fuzzy control and the correctness ofthe T-S fuzzy model However experimental demonstrationshould be investigated to verify the validity of the proposedapproach in the next research step

minus1

minus20 5 10 15 20 25 30

Time (s)

012

minus1

minus20 5 10 15 20 25 30

Time (s)

012

times104

times104

uy

ux

Figure 9 Control input with the proposed controller

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the anonymous reviewers for usefulcomments that improved the quality of the paperThiswork ispartially supported by National Science Foundation of Chinaunder Grant no 61374100 and Natural Science Foundation ofJiangsu Province under Grant no BK20131136

References

[1] R P Leland ldquoAdaptive control of a MEMS gyroscope usinglyapunov methodsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 2 pp 278ndash283 2006

10 Discrete Dynamics in Nature and Society

[2] S Sung W-T Sung C Kim S Yun and Y J Lee ldquoOnthe mode-matched control of MEMS vibratory gyroscope viaphase-domain analysis and designrdquo IEEEASME Transactionson Mechatronics vol 14 no 4 pp 446ndash455 2009

[3] R Antonello R Oboe L Prandi and F Biganzoli ldquoAuto-matic mode matching in MEMS vibrating gyroscopes usingextremum-seeking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 10 pp 3880ndash3891 2009

[4] S Park R Horowitz S K Hong and Y Nam ldquoTrajectory-switching algorithm for aMEMS gyroscoperdquo IEEE Transactionson Instrumentation and Measurement vol 56 no 6 pp 2561ndash2569 2007

[5] J D John and T Vinay ldquoNovel concept of a single-massadaptively controlled triaxial angular rate sensorrdquo IEEE SensorsJournal vol 6 no 3 pp 588ndash595 2006

[6] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009

[7] J Fei and F Chowdhury ldquoRobust adaptive controller for triaxialangular velocity sensorrdquo International Journal of InnovativeComputing Information and Control vol 7 no 6 pp 2439ndash2448 2010

[8] L Wang Adaptive Fuzzy Systems and Control-Design andStability Analysis Prentice Hall Upper Saddle River NJ USA1994

[9] Y Guo and P-Y Woo ldquoAn adaptive fuzzy sliding mode con-troller for robotic manipulatorsrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 33 no2 pp 149ndash159 2003

[10] B K Yoo and W C Ham ldquoAdaptive control of robot manip-ulator using fuzzy compensatorrdquo IEEE Transactions on FuzzySystems vol 8 no 2 pp 186ndash199 2000

[11] R-J Wai ldquoFuzzy sliding-mode control using adaptive tuningtechniquerdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 586ndash594 2007

[12] Y-H Chang C-W Chang C-L Chen and C-W Tao ldquoFuzzysliding-mode formation control for multirobot systems designand implementationrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 42 no 2 pp 444ndash457 2012

[13] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinearsystems using online T-S fuzzy-neural modeling approachrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 41 no 2 pp 542ndash552 2011

[14] M C M Teixeira and S H Zak ldquoStabilizing controller designfor uncertain nonlinear systems using fuzzy modelsrdquo IEEETransactions on Fuzzy Systems vol 7 no 2 pp 133ndash142 1999

[15] C-W Park and Y-W Cho ldquoT-S model based indirect adaptivefuzzy control using online parameter estimationrdquo IEEE Trans-actions on Systems Man and Cybernetics Part B Cyberneticsvol 34 no 6 pp 2293ndash2302 2004

[16] S F Asokanthan and T Wang ldquoNonlinear instabilities in avibratory gyroscope subjected to angular speed fluctuationsrdquoNonlinear Dynamics vol 54 no 1-2 pp 69ndash78 2008

[17] T Wang Nonlinear and stochastic dynamics of MEMS-basedangular rate sensing and switching systems [PhD thesis] TheUniversity of Western of Ontario Ontario Canada 2009

[18] Z Chi M Jia and Q Xu ldquoFuzzy PID feedback control ofpiezoelectric actuator with feedforward compensationrdquo Math-ematical Problems in Engineering vol 2014 Article ID 10718414 pages 2014

[19] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics Part B vol 15 no 1 pp 116ndash1321985

[20] P Ioannou and J Sun Robust Adaptive Control Prentice HallUpper Saddle River NJ USA 1996

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Differential EquationsInternational Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

The system nonlinearities in MEMS gyroscope modelhave been described in [16ndash18] Since there are systemnonlinearities in MEMS gyroscope it is necessary to userobust adaptive fuzzy controller to MEMS gyroscope andutilize T-S fuzzymodel to represent the systemnonlinearities

Systematic stability analysis and controller design of theadaptive fuzzy controller using T-S fuzzy model for MEMSgyroscope have not been investigated before The advantageof the proposed adaptive T-S fuzzy controller over othercontrollers is that it can represent system nonlinearities usingT-S fuzzy model in the aspect of robust design Thus itcan overcome the impacts on output tracking error that arecaused bymodel uncertainties and external disturbancesTheproposed T-S modeling method can provide a possibilityfor developing a systematic analysis and design methodfor complex nonlinear control systems thus improving thetracking and compensation performanceThus it is necessaryto adopt the adaptive fuzzy control scheme to approximatethe nonlinear system and compensate model uncertaintiesand external disturbances in the control of MEMS gyroscopeusing T-S fuzzy model

In this paper the Lyapunov-based robust adaptive fuzzycontrol strategy is applied to the tracking control of MEMSgyroscope using T-S fuzzy model The paper integratesadaptive control and the nonlinear approximation of fuzzycontrol with T-S fuzzy model The proposed adaptive fuzzyT-S controller can guarantee the asymptotic stability of theclosed loop system and improve the robustness of controlsystem in the presence of model uncertainties and externaldisturbances The proposed controller has the followingcharacteristics and contributions

(1) Since the reference model is marginally stable thereference model is required to be revised so that thelocal linear state feedback controller can be designedfor each T-S fuzzy submodel based on PDC Thepartial parameters of the reference control matrixare changed and then the reference input is changedaccordingly to guarantee that the changed referencemodel is equivalent to previous one

(2) To improve the performance of the control systemthe corresponding improvement is made to the adap-tive law The new adaptive law can be chosen tobe the previous adaptive law plus a robust termto guarantee that the derivative of the Lyapunovfunction is negative rather than nonpositive

(3) The proposed estimator for the existing fuzzy statefeedback controller can achieve a good robust per-formance against parameter uncertainties The con-troller output is implemented in the T-S fuzzy modeland nonlinear model respectively to verify the valid-ity of the adaptive fuzzy control with parameterestimation scheme and prove the correctness of the T-S model and the feasibility of the proposed controlleron the nonlinear model of gyroscope

kxx2

2

kxx2

2

kyy

2

2

kyy

2

2

dyy

2

Ωz

y

x

dxx2

dyy

2

dxx2

m

kx3 kx3

ky3

ky3

Figure 1 119885-axis vibratory MEMS gyroscope with nonlinear effec-tive spring

2 Dynamics of MEMS Gyroscope

Assume that the gyroscope is moving with a constant linearspeed the gyroscope is rotating at a constant angular velocitythe centrifugal forces are assumed negligible the gyroscopeundergoes rotations along 119911-axis as shown in Figure 1Referring to [4 16 17] the dynamics equations of gyroscopesystem are as follows

119898 + 119889119909119909 + (119889

119909119910minus 2119898Ω

lowast

119911) 119910

+ (119896119909119909

minus 119898Ωlowast

119911

2

) 119909 + 119896119909119910119910 + 11989611990931199093

= 119906119909

119898 119910 + 119889119910119910

119910 + (119889119909119910

+ 2119898Ωlowast

119911)

+ (119896119910119910

minus 119898Ωlowast

119911

2

) 119910 + 119896119909119910119909 + 11989611991031199103

= 119906119910

(1)

where119898 is the mass of proof mass Fabrication imperfectionscontribute mainly to the asymmetric spring term 119889

119909119910 and

asymmetric damping terms 119896119909119910 119889119909119909 and 119889

119910119910are damping

terms 119896119909119909 119896119910119910

are linear spring terms 1198961199093 1198961199103 are nonlinear

spring terms Ωlowast119911is angular velocity 119906

119909 119906119910are the control

forcesDividing the equation by the reference mass and because

of the nondimensional time 119905lowast = 1205960119905 dividing both sides of

equation by reference frequency 12059620and reference length 119902

0

and rewriting the dynamics in vector forms result in

119902lowast

1

1199020

+119863lowast

1198981205960

119902lowast

1

1199020

+ 2119878lowast

1205960

119902lowast

1

1199020

minusΩlowast

119911

2

1198981205960

119902lowast

1

1199020

+119870lowast

1

1198981205962

0

119902lowast

1

1199020

+119870lowast

3

1198981205962

0

119902lowast

1

3

1199020

=119906lowast

1198981205962

01199020

(2)

Discrete Dynamics in Nature and Society 3

where

119902lowast

1= [

119909

119910] 119906

lowast

= [

119906119909

119906119910

] 119863lowast

= [

119889119909119909

119889119909119910

119889119909119910

119889119910119910

]

119878lowast

= [

0 minusΩlowast

119911

Ωlowast

1199110

] 119870lowast

1= [

119896119909119909

119896119909119910

119896119909119910

119896119910119910

] 119870lowast

3= [

1198961199093 0

0 1198961199103

]

(3)

Define new parameters as follows

1199021=119902lowast

1

1199020

119906 =119906lowast

1198981205962

01199020

Ω119911=Ωlowast

119911

1205960

119863 =119863lowast

1198981205960

119878 =119878lowast

1205960

1198701=

119870lowast

1

1198981205962

0

1198703=119870lowast

31199022

0

1198981205962

0

(4)

For the convenience of notation ignoring the superscriptyields the final form of the nondimensional equation ofmotion for the 119911-axis gyroscope

1199021= (2119878 minus 119863) 119902

1+ (Ω2

119911minus 1198701) 1199021minus 11987031199023

1+ 119906 (5)

3 Adaptive Fuzzy Control Based on PDC

TheMIMO T-S fuzzy model of the MEMS gyroscope whichis composed of 119894 rules is established based on the nonlinearmodel of MEMS gyroscope Use the fuzzy implications andthe fuzzy reasoningmethods suggested by Takagi and Sugeno[19] to express a real plant model the T-S fuzzy model usesfuzzy implication of the following form

Rule 119894 IF 119909 is about 1198721198941and 119910 is about 119872

1198942and

is about 1198721198943and 119910 is about 119872

1198944

THEN 119902 = 119860119894119902 + 119861119894119906 119894 = 1 2 9

(6)

The defuzzification fuzzy dynamic system model can beexpressed using center-average defuzzifier product inferenceand singleton fuzzifier

119902 =sum9

119894=1120583119894(120578) [119860

119894119902 + 119861119894119906]

sum9

119894=1120583119894(120578)

(7)

where

119860119894=

[[[[[[

[

119886119894

11119886119894

12119886119894

13119886119894

14

119886119894

21119886119894

22119886119894

23119886119894

24

1 0 0 0

0 1 0 0

]]]]]]

]

119861119894=

[[[[[

[

1 0

0 1

0 0

0 0

]]]]]

]

119902 =

[[[[[

[

119910

119910

]]]]]

]

119902 =

[[[[[

[

119910

119909

119910

]]]]]

]

120583119894(120578) = 119872

1198941(119909)119872

1198942(119910)119872

1198943()119872

1198944( 119910)

(8)

1198721198941(119909)119872

1198942(119910)119872

1198943() and119872

1198944( 119910) aremembership function

values of the fuzzy variable 119909119910 and 119910with respect to fuzzyset119872

119894111987211989421198721198943 and119872

1198944 respectively

Since the control target for MEMS gyroscope is tomaintain the proof mass to oscillate in the 119909 and 119910 directionat given frequency and amplitude 119909 = 119860

119909sin(120596119909119905) 119910 =

119860119910sin(120596119910119905) generally the reference model can be defined as

119902119889= 119860119889119902119889 where

119902119889=

[[[[[

[

119910

119910

]]]]]

]

=

[[[[[[

[

1198601199091205962

119909sin (120596

119909119905)

1198601199101205962

119910sin (120596

119910119905)

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)

]]]]]]

]

119902119889=

[[[[[

[

119910

119909

119910

]]]]]

]

=

[[[[[[

[

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)

119860119909sin (120596

119909119905)

119860119910sin (120596

119910119905)

]]]]]]

]

119860119889=

[[[[[

[

0 0 minus1205962

1199090

0 0 0 minus1205962

119910

1 0 0 0

0 1 0 0

]]]]]

]

(9)

Since the reference model is nonasymptotically stable thenonlinear T-S fuzzy model cannot be asymptotically stableIn order to design local linear state feedback controller foreach T-S fuzzy submodel based on PDC the reference modelis adjusted to be 119902

119898= 119860119898119902119898+ 119861119898119903 where

119860119898=

[[[[[[

[

119886119898

11119886119898

12119886119898

13119886119898

14

119886119898

21119886119898

22119886119898

23119886119898

24

1 0 0 0

0 1 0 0

]]]]]]

]

=

[[[[[

[

minus1 0 minus1205962

1199090

0 minus1 0 minus1205962

119910

1 0 0 0

0 1 0 0

]]]]]

]

119861119898=

[[[[[

[

1 0

0 1

0 0

0 0

]]]]]

]

119902119898=

[[[[[

[

119910

119910

]]]]]

]

=

[[[[[[

[

1198601199091205962

119909sin (120596

119909119905)

1198601199101205962

119910sin (120596

119910119905)

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)

]]]]]]

]

4 Discrete Dynamics in Nature and Society

119902119898=

[[[[[

[

119910

119909

119910

]]]]]

]

=

[[[[[[

[

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)

119860119909sin (120596

119909119905)

119860119910sin (120596

119910119905)

]]]]]]

]

119903 = [

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)]

(10)

Remark 1 Substituting 119860119898 119861119898 119902119898 119903 into 119902

119898yields 119902

119889 we

can get that the changed reference model 119902119898= 119860119898119902119898+ 119861119898119903

is equivalent to previous one 119902119889= 119860119889119902119889

According to PDC [14] we design local state linearfeedback controllers for each linear submodel The fuzzycontrol rules are as the following forms

Rule 119894 IF 119909 is about 1198721198941and 119910 is about 119872

1198942and

is about 1198721198943and 119910 is about 119872

1198944

THEN 119906 (119905) = minus119870119894(119905) 119909 (119905) + 119903 (119905) 119894 = 1 2 9

(11)

The controller output can be expressed using center-averagedefuzzifier product inference and singleton fuzzifier

119906 (119905) =sum9

119894=1120583119894(120578) [minus119870

119894119902 (119905) + 119903 (119905)]

sum9

119894=1120583119894(120578)

(12)

where

119870119894= 119860119894minus 119860119898

=

[[[[[[

[

119886119894

11minus 119886119898

11119886119894

12minus 119886119898

12119886119894

13minus 119886119898

13119886119894

14minus 119886119898

14

119886119894

21minus 119886119898

21119886119894

22minus 119886119898

22119886119894

23minus 119886119898

23119886119894

24minus 119886119898

24

1 0 0 0

0 1 0 0

]]]]]]

]

119894 = 1 2 9

(13)

Substituting the controller force (12) into the fuzzy system (7)yields the desired model

119902119898= 119860119898119902119898+ 119861119898119903 (14)

4 Parameter Estimation

The block diagram of the adaptive fuzzy control using T-Sfuzzy model is shown as in Figure 2 Taking the parameteruncertainty into account then the parameter estimation isapplied to the adaptive fuzzy control The adaptive law canguarantee the asymptotic convergence of the error betweenthe plant model state and the estimation model state andmake the closed loop system of fuzzy controller and estima-tion model be a desired linear model and consequently theplant model state can follow the state of the desired linearmodelThe controller output is implemented in the nonlinearmodel also to prove the correctness of the T-S model and thefeasibility of the proposed control scheme on the nonlinearmodel of MEMS gyroscope

With unknown parameters matrix 119860119894 the controller

changes to

119906 (119905) =

sum9

119894=1120583119894(120578) [minus

119894119902 (119905) + 119903 (119905)]

sum9

119894=1120583119894(120578)

(15)

where

119894= 119860119894minus 119860119898

=

[[[[[[

[

119886119894

11minus 119886119898

11119886119894

12minus 119886119898

12119886119894

13minus 119886119898

13119886119894

14minus 119886119898

14

119886119894

21minus 119886119898

21119886119894

22minus 119886119898

22119886119894

23minus 119886119898

23119886119894

24minus 119886119898

24

1 0 0 0

0 1 0 0

]]]]]]

]

119860119894=

[[[[[[

[

119886119894

11119886119894

12119886119894

13119886119894

14

119886119894

21119886119894

22119886119894

23119886119894

24

1 0 0 0

0 1 0 0

]]]]]]

]

119894 = 1 2 9

(16)

is the estimate of 119860119894

By considering the plant parameterization (7) can berewritten as

119902 (119905) = 119860119904119902 (119905) +

sum9

119894=1120583119894(120578) [(119860

119894minus 119860119904) 119902 (119905) + 119861

119894119906 (119905)]

sum9

119894=1120583119894(120578)

(17)

where 119860119904is an arbitrary stable matrix

Define the estimation mode as the following formula

119902 (119905) = 119860119904119902 (119905) +

sum9

119894=1120583119894(120578) [(119860

119894minus 119860119904) 119902 (119905) + 119861

119894119906 (119905)]

sum9

119894=1120583119894(120578)

(18)

The estimationmodel (18) can be the same one as by adoptingthe control input (15)

119902119898= 119860119898119902119898+ 119861119898119903 (19)

The estimation error 119890(119905) = 119902(119905) minus 119902119898(119905)meets (19) and (20)

119890 (119905) = 119860119904119890 (119905) +

sum9

119894=1120583119894(120578) (119860

119894minus 119860119894) 119902 (119905)

sum9

119894=1120583119894(120578)

= 119860119904119890 (119905) minus

sum9

119894=1120583119894(120578) [

11198942119894

0 0]119879

119902 (119905)

sum9

119894=1120583119894(120578)

(20)

119890119879

(119905) = 119890 (119905) 119860119879

119904minussum9

119894=1120583119894(120578) 119902119879

(119905) [1119894

2119894

0 0]

sum9

119894=1120583119894(120578)

(21)

Discrete Dynamics in Nature and Society 5

Σ

Σ

+

minusqnon

qd

eT-S

+

minus

q

q

Desired trajectory

Reference input

Fuzzy controller

Adaptive law

U

Plant model

Estimation model

Nonlinear model

Ki(t)120583i(120578)

eNON

qdx = Axsin 120596xt)

qdy = Aysin 120596yt)

rx = Ax120596xcosry = Ay120596ycos

u(t) =sum9

i=1120583i(120578)

sum9i=1120583i(120578)

[minus Kiq(t) + r(t)]

aTi = 120574i

120583i(120578)

sum9i=1120583i(120578)

e(t)qT(t) minus 120588sgn(ai)[PT1 PT

2 ]

1 = (2S minus D) 1 + (Ω2z minus K1)q1 minus K3q

31 + u

(t) = Amq(t) +sum9

i=1120583i(120578)[(Ai minus Am)q(t) + Biu(t)]

Ni=1120583i(120578)sum

(t) = Amq(t) +sum9

i=1120583i(120578)[(Ai minus Am)q(t) + Biu(t)]

Ni=1120583i(120578)sum

q

)

120596xt))

)

120596yt))

q

q

q

Figure 2 The block diagram of the adaptive fuzzy control using T-S fuzzy model

where 119890(119905) actually is the tracking error between theplant model (7) and the estimation model (18)

119894(119905) =

[1119894(119905) 2119894(119905) 0 0]

1119894(119905) =

1119894(119905) minus 120572

1119894 1119894(119905) =

[119886119894

11(119905) 119886

119894

12(119905) 119886

119894

13(119905) 119886

119894

14(119905)] 120572

1119894= [119886

119894

11119886119894

12119886119894

13119886119894

14]

2119894(119905) =

2119894(119905) minus 120572

2119894 2119894

= [119886119894

21(119905) 119886

119894

22(119905) 119886

119894

23(119905) 119886

119894

24(119905)]

1205722119894= [119886119894

21119886119894

22119886119894

23119886119894

24]

Remark 2 The estimation error 119890(119905) can be thought of asthe tracking error 119890T-S between the plant model (7) and thedesired linear model (14) because 119890(119905) = 119902(119905) minus 119902

119898(119905) and

119890T-S(119905) = 119902(119905) minus 119902119898(119905) have the same control matrix while the

tracking error 119890NON between the referencemodel (14) and thenonlinear model (5) is used to prove the correctness of the T-S model and the feasibility of the proposed controller on thenonlinear model of gyroscope

Define a Lyapunov function candidate as

119881 (119905) = 119881 (119890 1198861119894 1198862119894) = 119890119879

(119905) 119875119890 (119905)

+

9

sum

119894=1

119886119879

1119894(119905) 1198861119894(119905)

1205741119894

+

9

sum

119894=1

119886119879

2119894(119905) 1198862119894(119905)

1205742119894

(22)

where 119875meet 119860119879119904119875 + 119875119860

119904= minus119868

The derivative of 119881 with respect to time becomes

(119905) = 119890119879

(119905) 119875119890 (119905) + 119890119879

(119905) 119875 119890 (119905)

+

9

sum

119894=1

2

119886119879

1119894(119905) 1198861119894(119905)

1205741119894

+

9

sum

119894=1

2

119886119879

2119894(119905) 1198862119894(119905)

1205742119894

(23)

Substituting (20) (21) into the derivative of 119881 yields

(119905) = minus 119890119879

(119905) (119860119879

119904119875 + 119875119860

119904) 119890 (119905)

minus 2sum9

119894=1120583119894(120578) 119875119879

1119890 (119905) 119902

119879

(119905) 1198862119894(119905)

sum9

119894=1120583119894(120578)

minus 2sum9

119894=1120583119894(120578) 119875119879

2119890 (119905) 119902

119879

(119905) 1198862119894(119905)

sum9

119894=1120583119894(120578)

+

9

sum

119894=1

2

119886119879

1119894(119905) 1198861119894(119905)

1205741119894

+

9

sum

119894=1

2

119886119879

2119894(119905) 1198862119894(119905)

1205742119894

(24)

where 119875 = [1198751

1198752

1198753

1198754]

The adaptive law can be chosen as

119886119879

119894= 120574119894

120583119894(120578)

sum9

119894=1120583119894(120578)

[119875119879

1119875119879

2] 119890 (119905) 119902

119879

(119905)

minus 120588 sgn (119886119894(119905)) 120588 gt 0

(25)

where the sgn function can be defined as

sgn (119886119894) =

1 119886119894gt 0

0 119886119894= 0

minus1 119886119894lt 0

(26)

6 Discrete Dynamics in Nature and Society

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 10

05

1

0

05

1

x

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

y

120583(x)

0

05

1

120583(y)

minus6 minus4 minus2 0 2 4 60

05

1

minus6 minus4 minus2 0 2 4 6

120583(d

x)

120583(d

y)

dx dy

M71M81M91

M11

M21M31

M41M51M61

M73M83M93

M32M62M92

M12M42M72

M22M52M82

M34M64M94

M14M44M74

M24M54M84

M13M23M33

M43M53M63

Figure 3 Membership functions

Substituting (25) into (24) yields

(119905) = minus 119890119879

(119905) 119890 (119905) minus 2120588

9

sum

119894=1

sgn (1198861119894(119905)) 1198861119894(119905)

1205741119894

minus 2120588

9

sum

119894=1

sgn (1198862119894(119905)) 1198862119894(119905)

1205742119894

= minus 119890119879

(119905) 119890 (119905) minus 2120588

9

sum

119894=1

10038161003816100381610038161198861119894 (119905)1003816100381610038161003816

1205741119894

minus 2120588

9

sum

119894=1

10038161003816100381610038161198862119894 (119905)1003816100381610038161003816

1205742119894

le minus 119890119879

(119905) 119890 (119905) le minus 119890 (119905)2

le 0

(27)

119881(119905) ge 0 and (119905) le 0 imply the boundedness of 119890(119905) and119894(119905) the control is bounded for all time (21) implies 119890(119905) is

also bounded and inequality (27) implies 119890(119905) isin 1198712 Then

with 119890(119905) isin 1198712

cap 119871infin and 119890(119905) isin 119871

infin according to Barbalatrsquoslemma [20] the tracking error 119890(119905) asymptotically convergesto zero

5 Simulation Analysis

In this section we will evaluate the proposed adaptive fuzzycontrol using T-S model approach on the lumped MEMSgyroscope sensormodel Parameters of theMEMS gyroscopesensor are as follows

119898 = 057119890 minus 8 kg 1205960= 1 kHz 119902

0= 10119890 minus 6m

119889119909119909

= 0429119890 minus 6Nsm 119889119910119910

= 00429119890 minus 6Nsm

119889119909119910

= 00429119890 minus 6Nsm 119896119909119909

= 8098Nm

119896119910119910

= 7162Nm 119896119909119910

= 5Nm

1198961199093 = 35611989012Nm

1198961199103 = 35611989012Nm Ω

119911= 50 rads

(28)

Since the general displacement range of the MEMSgyroscope sensor in each axis is submicrometer level itis reasonable to choose 1 120583m as the reference length 119902

0

Given that the usual natural frequency of each axle of avibratory MEMS gyroscope sensor is in the kHz range 120596

0is

chosen as 1 kHz The unknown angular velocity is assumedΩ119911

= 50 rads The desired motion trajectories are 119909119898

=

119860119909sin(120596119909119905) 119910119898

= 119860119910sin(120596119910119905) where 119860

119909= 1 119860

119910= 12

120596119909= 671 kHz and 120596

119910= 511 kHz

The plant parameters are adjusted online by adaptivelaw (24) where the adaptive gain 120574

119894= 1 and the adaptive

parameter 120588 = 1 The initial values of T-S model are[02 024 0 0] initial values of the estimation model are[0 0 0 0] and initial values of the nonlinear model are[02 024 0 0] The external disturbance is 119889 = [

119889119909

119889119910

] =

[10 sin(2120587119905)10 sin(2120587119905) ]119860119898 = [

minus1 0 minus1205962

1199090

0 minus1 0 minus1205962

119910

1 0 0 0

0 1 0 0

] and the eigenvalues of119860119898

are minus05plusmn66916119894 minus05plusmn50855119894 Although119860119904canmeet the

requirement of 119860119904in 119860119879

119904119875 + 119875119860

119904= minus119868 the values of first and

second column in 119875 which are [051 0

0 052

001 0

0 002

] cannot adjust each

parameter since partial parameters are zero which is used as

adaptive gain in (24) so 119860119904is chosen as 119860

119904= [

minus1 minus1 minus1 minus1

0 minus1 0 0

0 0 minus1 0

0 0 0 minus1

]

Discrete Dynamics in Nature and Society 7

0 1 2 3minus008

minus0075

minus007

minus0065

Time (s)0 1 2 3

22

23

24

25

Time (s)0 1 2 3

minus145

minus14

minus135

minus13

minus125

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)

00 1 2 3minus0018

minus0017

minus0016

minus0015

Time (s)

1 2 3minus013

minus012

minus011

minus01

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)0 1 2 3

minus13

minus125

minus12

minus11

minus115

Time (s)

A2

[11

]

A2

[12

]

A2

[13

]

A2

[14

]

A2

[21

]

A2

[24

]

A2

[22

]

A2

[23

]

times104

times104

times10minus3

Figure 4 The change of plant parameters in 1198602

10 2 385

90

95

100

Time (s)10 2 3

80

85

90

95

Time (s)0 1 2 3

minus17

minus16

minus15

minus14

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)

0 1 2 3150

160

170

180

Time (s)0 1 2 3

Time (s)

0 1 2 3

Time (s)0 1 2 3

Time (s)

130

140

150

160

minus900

minus850

minus800

minus750

minus155

minus15

minus145

minus14

minus135

A5

[11

]

A5

[12

]

A5

[13

]

A5

[14

]

A5

[21

]

A5

[24

]

A5

[22

]

A5

[23

]

times104

times104

Figure 5 The change of plant parameters in 1198605

to ensure that each parameter of first and second column in

119875 is nonzero values as [05 minus025

minus025 075

minus025 025

minus025 025

]

The fuzzy rules for T-S fuzzy model for the system can beobtained by linearizing the nonlinear model (5) at the points119909 isin minus1 0 1 isin minus119860

119909120596119909

0 119860119909120596119909 119910 isin minus12 0 12

and 119910 isin minus119860119910120596119910

0 119860119910120596119910 and then the membership

functions of states 119909 119910 and 119910with respect to fuzzy set1198721198941

11987211989421198721198943 and119872

1198944used in the T-S fuzzy model are shown as

in Figure 3In this simulation it is assumed that the physical parame-

ters in the T-S fuzzy model are not known exactly Hence the

8 Discrete Dynamics in Nature and Society

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

22

23

24

25

65

7

75

minus008

minus0075

minus007

minus0065

minus16

minus155

minus15

minus145

minus14

minus270

minus260

minus250

minus240

minus230

minus145

minus14

minus135

minus13

minus125

minus013

minus012

minus011

minus01

minus0018

minus0017

minus0016

minus0015

A7

[11

]

A7

[12

]

A7

[13

]

A7

[14

]

A7

[21

]

A7

[22

]

A7

[23

]

A7

[24

]

times104

times104

times10minus3

Figure 6 The change of plant parameters in 1198607

parameters are tuned by the proposed adaptive law The truevalues of 119860lowast

119894of fuzzy T-S model are as shown in Table 1 The

initial value 119860119894= 09 lowast 119860

lowast

119894 The variations of each parameter

in 1198602 1198605 and 119860

7are shown in Figures 4 5 and 6 Here

only parts of parameters are shown in table for examples InFigures 4 5 and 6 we can see each parameter converges to itstrue value asymptotically

Figure 7 depicts the tracking error of 119909- and 119910-axisbetween the reference model and T-S model It can beobserved from Figure 7 that the position of 119909 and 119910 inthe T-S model can track the position of reference modelin very short time and tracking errors converge to zeroasymptotically The tracking error 119890T-S converges to zeroasymptotically rather than shocks near zero because thenew adaptive law is chosen to be the previous adaptivelaw 120574119894(120583119894(120578)sum

9

119894=1120583119894(120578)) [119875

119879

1119875119879

2] 119890(119905)119902

119879

(119905) plus a robust term120588 sgn(

119894) to guarantee that the derivative of the Lyapunov

function is negative rather than nonpositive Figure 8 plotsthe tracking error of 119909- and 119910-axis between the referencemodel and the nonlinear model We can notice that theperformance of the tracking errors in Figure 8 is worse thanthat in Figure 7

The tracking error 119890T-S shocks near zero because thecontroller is designed based on the T-S model rather thantheNONmodel FromFigures 7 and 8 we can see the validityof the adaptive fuzzy control with parameter estimationscheme on the T-S model and the feasibility of the proposedcontroller on the nonlinear model of gyroscope Then itcan be concluded that the MEMS gyroscope can maintain

0 5 10 15 20 25 30minus1

minus1

minus2

minus05

0

05

1

Time (s)

0 5 10 15 20 25 30

Time (s)

012

e xts

e yts

Figure 7 The tracking error 119890T-S between the reference model andT-S model

the proof mass oscillating in the 119909 and 119910 direction atgiven frequency and amplitude by using adaptive T-S fuzzycontroller The proposed T-S modeling method can providea possibility for developing a systematic analysis and designmethod for complex nonlinear control systems which canapproximate the nonlinear system and compensate modeluncertainties and external disturbances It can be observedfrom Figure 9 that the controller input of the adaptive fuzzycontroller is stable

Discrete Dynamics in Nature and Society 9

Table 1 Parameters of T-S fuzzy model

119860lowast

1=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

2=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

3=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

4=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

5=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

6=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

7=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

8=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

9=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

0 5 10 15 20 25 30minus1

minus05

05

1

Time (s)

0

minus1

minus20 5 10 15 20 25 30

Time (s)

012

e xno

ne y

non

Figure 8The tracking error 119890NON between the reference model andnonlinear model

6 Conclusions

A T-S fuzzy model based adaptive controller for angularvelocity sensor is presented in this paper An adaptiveT-S fuzzy compensator is used to approximate the modeluncertainties and external disturbances The output of thefuzzy controller is used as compensator to reduce the effectsof the system nonlinearities The proposed new adaptive lawcan guarantee that the derivative of the Lyapunov functionis negative rather than nonpositive The proposed estimatorfor the existing fuzzy state feedback controller can achievea good robust performance against parameter uncertaintiesSimulation studies are implemented to verify the effectivenessof the proposed adaptive fuzzy control and the correctness ofthe T-S fuzzy model However experimental demonstrationshould be investigated to verify the validity of the proposedapproach in the next research step

minus1

minus20 5 10 15 20 25 30

Time (s)

012

minus1

minus20 5 10 15 20 25 30

Time (s)

012

times104

times104

uy

ux

Figure 9 Control input with the proposed controller

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the anonymous reviewers for usefulcomments that improved the quality of the paperThiswork ispartially supported by National Science Foundation of Chinaunder Grant no 61374100 and Natural Science Foundation ofJiangsu Province under Grant no BK20131136

References

[1] R P Leland ldquoAdaptive control of a MEMS gyroscope usinglyapunov methodsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 2 pp 278ndash283 2006

10 Discrete Dynamics in Nature and Society

[2] S Sung W-T Sung C Kim S Yun and Y J Lee ldquoOnthe mode-matched control of MEMS vibratory gyroscope viaphase-domain analysis and designrdquo IEEEASME Transactionson Mechatronics vol 14 no 4 pp 446ndash455 2009

[3] R Antonello R Oboe L Prandi and F Biganzoli ldquoAuto-matic mode matching in MEMS vibrating gyroscopes usingextremum-seeking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 10 pp 3880ndash3891 2009

[4] S Park R Horowitz S K Hong and Y Nam ldquoTrajectory-switching algorithm for aMEMS gyroscoperdquo IEEE Transactionson Instrumentation and Measurement vol 56 no 6 pp 2561ndash2569 2007

[5] J D John and T Vinay ldquoNovel concept of a single-massadaptively controlled triaxial angular rate sensorrdquo IEEE SensorsJournal vol 6 no 3 pp 588ndash595 2006

[6] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009

[7] J Fei and F Chowdhury ldquoRobust adaptive controller for triaxialangular velocity sensorrdquo International Journal of InnovativeComputing Information and Control vol 7 no 6 pp 2439ndash2448 2010

[8] L Wang Adaptive Fuzzy Systems and Control-Design andStability Analysis Prentice Hall Upper Saddle River NJ USA1994

[9] Y Guo and P-Y Woo ldquoAn adaptive fuzzy sliding mode con-troller for robotic manipulatorsrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 33 no2 pp 149ndash159 2003

[10] B K Yoo and W C Ham ldquoAdaptive control of robot manip-ulator using fuzzy compensatorrdquo IEEE Transactions on FuzzySystems vol 8 no 2 pp 186ndash199 2000

[11] R-J Wai ldquoFuzzy sliding-mode control using adaptive tuningtechniquerdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 586ndash594 2007

[12] Y-H Chang C-W Chang C-L Chen and C-W Tao ldquoFuzzysliding-mode formation control for multirobot systems designand implementationrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 42 no 2 pp 444ndash457 2012

[13] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinearsystems using online T-S fuzzy-neural modeling approachrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 41 no 2 pp 542ndash552 2011

[14] M C M Teixeira and S H Zak ldquoStabilizing controller designfor uncertain nonlinear systems using fuzzy modelsrdquo IEEETransactions on Fuzzy Systems vol 7 no 2 pp 133ndash142 1999

[15] C-W Park and Y-W Cho ldquoT-S model based indirect adaptivefuzzy control using online parameter estimationrdquo IEEE Trans-actions on Systems Man and Cybernetics Part B Cyberneticsvol 34 no 6 pp 2293ndash2302 2004

[16] S F Asokanthan and T Wang ldquoNonlinear instabilities in avibratory gyroscope subjected to angular speed fluctuationsrdquoNonlinear Dynamics vol 54 no 1-2 pp 69ndash78 2008

[17] T Wang Nonlinear and stochastic dynamics of MEMS-basedangular rate sensing and switching systems [PhD thesis] TheUniversity of Western of Ontario Ontario Canada 2009

[18] Z Chi M Jia and Q Xu ldquoFuzzy PID feedback control ofpiezoelectric actuator with feedforward compensationrdquo Math-ematical Problems in Engineering vol 2014 Article ID 10718414 pages 2014

[19] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics Part B vol 15 no 1 pp 116ndash1321985

[20] P Ioannou and J Sun Robust Adaptive Control Prentice HallUpper Saddle River NJ USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

where

119902lowast

1= [

119909

119910] 119906

lowast

= [

119906119909

119906119910

] 119863lowast

= [

119889119909119909

119889119909119910

119889119909119910

119889119910119910

]

119878lowast

= [

0 minusΩlowast

119911

Ωlowast

1199110

] 119870lowast

1= [

119896119909119909

119896119909119910

119896119909119910

119896119910119910

] 119870lowast

3= [

1198961199093 0

0 1198961199103

]

(3)

Define new parameters as follows

1199021=119902lowast

1

1199020

119906 =119906lowast

1198981205962

01199020

Ω119911=Ωlowast

119911

1205960

119863 =119863lowast

1198981205960

119878 =119878lowast

1205960

1198701=

119870lowast

1

1198981205962

0

1198703=119870lowast

31199022

0

1198981205962

0

(4)

For the convenience of notation ignoring the superscriptyields the final form of the nondimensional equation ofmotion for the 119911-axis gyroscope

1199021= (2119878 minus 119863) 119902

1+ (Ω2

119911minus 1198701) 1199021minus 11987031199023

1+ 119906 (5)

3 Adaptive Fuzzy Control Based on PDC

TheMIMO T-S fuzzy model of the MEMS gyroscope whichis composed of 119894 rules is established based on the nonlinearmodel of MEMS gyroscope Use the fuzzy implications andthe fuzzy reasoningmethods suggested by Takagi and Sugeno[19] to express a real plant model the T-S fuzzy model usesfuzzy implication of the following form

Rule 119894 IF 119909 is about 1198721198941and 119910 is about 119872

1198942and

is about 1198721198943and 119910 is about 119872

1198944

THEN 119902 = 119860119894119902 + 119861119894119906 119894 = 1 2 9

(6)

The defuzzification fuzzy dynamic system model can beexpressed using center-average defuzzifier product inferenceand singleton fuzzifier

119902 =sum9

119894=1120583119894(120578) [119860

119894119902 + 119861119894119906]

sum9

119894=1120583119894(120578)

(7)

where

119860119894=

[[[[[[

[

119886119894

11119886119894

12119886119894

13119886119894

14

119886119894

21119886119894

22119886119894

23119886119894

24

1 0 0 0

0 1 0 0

]]]]]]

]

119861119894=

[[[[[

[

1 0

0 1

0 0

0 0

]]]]]

]

119902 =

[[[[[

[

119910

119910

]]]]]

]

119902 =

[[[[[

[

119910

119909

119910

]]]]]

]

120583119894(120578) = 119872

1198941(119909)119872

1198942(119910)119872

1198943()119872

1198944( 119910)

(8)

1198721198941(119909)119872

1198942(119910)119872

1198943() and119872

1198944( 119910) aremembership function

values of the fuzzy variable 119909119910 and 119910with respect to fuzzyset119872

119894111987211989421198721198943 and119872

1198944 respectively

Since the control target for MEMS gyroscope is tomaintain the proof mass to oscillate in the 119909 and 119910 directionat given frequency and amplitude 119909 = 119860

119909sin(120596119909119905) 119910 =

119860119910sin(120596119910119905) generally the reference model can be defined as

119902119889= 119860119889119902119889 where

119902119889=

[[[[[

[

119910

119910

]]]]]

]

=

[[[[[[

[

1198601199091205962

119909sin (120596

119909119905)

1198601199101205962

119910sin (120596

119910119905)

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)

]]]]]]

]

119902119889=

[[[[[

[

119910

119909

119910

]]]]]

]

=

[[[[[[

[

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)

119860119909sin (120596

119909119905)

119860119910sin (120596

119910119905)

]]]]]]

]

119860119889=

[[[[[

[

0 0 minus1205962

1199090

0 0 0 minus1205962

119910

1 0 0 0

0 1 0 0

]]]]]

]

(9)

Since the reference model is nonasymptotically stable thenonlinear T-S fuzzy model cannot be asymptotically stableIn order to design local linear state feedback controller foreach T-S fuzzy submodel based on PDC the reference modelis adjusted to be 119902

119898= 119860119898119902119898+ 119861119898119903 where

119860119898=

[[[[[[

[

119886119898

11119886119898

12119886119898

13119886119898

14

119886119898

21119886119898

22119886119898

23119886119898

24

1 0 0 0

0 1 0 0

]]]]]]

]

=

[[[[[

[

minus1 0 minus1205962

1199090

0 minus1 0 minus1205962

119910

1 0 0 0

0 1 0 0

]]]]]

]

119861119898=

[[[[[

[

1 0

0 1

0 0

0 0

]]]]]

]

119902119898=

[[[[[

[

119910

119910

]]]]]

]

=

[[[[[[

[

1198601199091205962

119909sin (120596

119909119905)

1198601199101205962

119910sin (120596

119910119905)

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)

]]]]]]

]

4 Discrete Dynamics in Nature and Society

119902119898=

[[[[[

[

119910

119909

119910

]]]]]

]

=

[[[[[[

[

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)

119860119909sin (120596

119909119905)

119860119910sin (120596

119910119905)

]]]]]]

]

119903 = [

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)]

(10)

Remark 1 Substituting 119860119898 119861119898 119902119898 119903 into 119902

119898yields 119902

119889 we

can get that the changed reference model 119902119898= 119860119898119902119898+ 119861119898119903

is equivalent to previous one 119902119889= 119860119889119902119889

According to PDC [14] we design local state linearfeedback controllers for each linear submodel The fuzzycontrol rules are as the following forms

Rule 119894 IF 119909 is about 1198721198941and 119910 is about 119872

1198942and

is about 1198721198943and 119910 is about 119872

1198944

THEN 119906 (119905) = minus119870119894(119905) 119909 (119905) + 119903 (119905) 119894 = 1 2 9

(11)

The controller output can be expressed using center-averagedefuzzifier product inference and singleton fuzzifier

119906 (119905) =sum9

119894=1120583119894(120578) [minus119870

119894119902 (119905) + 119903 (119905)]

sum9

119894=1120583119894(120578)

(12)

where

119870119894= 119860119894minus 119860119898

=

[[[[[[

[

119886119894

11minus 119886119898

11119886119894

12minus 119886119898

12119886119894

13minus 119886119898

13119886119894

14minus 119886119898

14

119886119894

21minus 119886119898

21119886119894

22minus 119886119898

22119886119894

23minus 119886119898

23119886119894

24minus 119886119898

24

1 0 0 0

0 1 0 0

]]]]]]

]

119894 = 1 2 9

(13)

Substituting the controller force (12) into the fuzzy system (7)yields the desired model

119902119898= 119860119898119902119898+ 119861119898119903 (14)

4 Parameter Estimation

The block diagram of the adaptive fuzzy control using T-Sfuzzy model is shown as in Figure 2 Taking the parameteruncertainty into account then the parameter estimation isapplied to the adaptive fuzzy control The adaptive law canguarantee the asymptotic convergence of the error betweenthe plant model state and the estimation model state andmake the closed loop system of fuzzy controller and estima-tion model be a desired linear model and consequently theplant model state can follow the state of the desired linearmodelThe controller output is implemented in the nonlinearmodel also to prove the correctness of the T-S model and thefeasibility of the proposed control scheme on the nonlinearmodel of MEMS gyroscope

With unknown parameters matrix 119860119894 the controller

changes to

119906 (119905) =

sum9

119894=1120583119894(120578) [minus

119894119902 (119905) + 119903 (119905)]

sum9

119894=1120583119894(120578)

(15)

where

119894= 119860119894minus 119860119898

=

[[[[[[

[

119886119894

11minus 119886119898

11119886119894

12minus 119886119898

12119886119894

13minus 119886119898

13119886119894

14minus 119886119898

14

119886119894

21minus 119886119898

21119886119894

22minus 119886119898

22119886119894

23minus 119886119898

23119886119894

24minus 119886119898

24

1 0 0 0

0 1 0 0

]]]]]]

]

119860119894=

[[[[[[

[

119886119894

11119886119894

12119886119894

13119886119894

14

119886119894

21119886119894

22119886119894

23119886119894

24

1 0 0 0

0 1 0 0

]]]]]]

]

119894 = 1 2 9

(16)

is the estimate of 119860119894

By considering the plant parameterization (7) can berewritten as

119902 (119905) = 119860119904119902 (119905) +

sum9

119894=1120583119894(120578) [(119860

119894minus 119860119904) 119902 (119905) + 119861

119894119906 (119905)]

sum9

119894=1120583119894(120578)

(17)

where 119860119904is an arbitrary stable matrix

Define the estimation mode as the following formula

119902 (119905) = 119860119904119902 (119905) +

sum9

119894=1120583119894(120578) [(119860

119894minus 119860119904) 119902 (119905) + 119861

119894119906 (119905)]

sum9

119894=1120583119894(120578)

(18)

The estimationmodel (18) can be the same one as by adoptingthe control input (15)

119902119898= 119860119898119902119898+ 119861119898119903 (19)

The estimation error 119890(119905) = 119902(119905) minus 119902119898(119905)meets (19) and (20)

119890 (119905) = 119860119904119890 (119905) +

sum9

119894=1120583119894(120578) (119860

119894minus 119860119894) 119902 (119905)

sum9

119894=1120583119894(120578)

= 119860119904119890 (119905) minus

sum9

119894=1120583119894(120578) [

11198942119894

0 0]119879

119902 (119905)

sum9

119894=1120583119894(120578)

(20)

119890119879

(119905) = 119890 (119905) 119860119879

119904minussum9

119894=1120583119894(120578) 119902119879

(119905) [1119894

2119894

0 0]

sum9

119894=1120583119894(120578)

(21)

Discrete Dynamics in Nature and Society 5

Σ

Σ

+

minusqnon

qd

eT-S

+

minus

q

q

Desired trajectory

Reference input

Fuzzy controller

Adaptive law

U

Plant model

Estimation model

Nonlinear model

Ki(t)120583i(120578)

eNON

qdx = Axsin 120596xt)

qdy = Aysin 120596yt)

rx = Ax120596xcosry = Ay120596ycos

u(t) =sum9

i=1120583i(120578)

sum9i=1120583i(120578)

[minus Kiq(t) + r(t)]

aTi = 120574i

120583i(120578)

sum9i=1120583i(120578)

e(t)qT(t) minus 120588sgn(ai)[PT1 PT

2 ]

1 = (2S minus D) 1 + (Ω2z minus K1)q1 minus K3q

31 + u

(t) = Amq(t) +sum9

i=1120583i(120578)[(Ai minus Am)q(t) + Biu(t)]

Ni=1120583i(120578)sum

(t) = Amq(t) +sum9

i=1120583i(120578)[(Ai minus Am)q(t) + Biu(t)]

Ni=1120583i(120578)sum

q

)

120596xt))

)

120596yt))

q

q

q

Figure 2 The block diagram of the adaptive fuzzy control using T-S fuzzy model

where 119890(119905) actually is the tracking error between theplant model (7) and the estimation model (18)

119894(119905) =

[1119894(119905) 2119894(119905) 0 0]

1119894(119905) =

1119894(119905) minus 120572

1119894 1119894(119905) =

[119886119894

11(119905) 119886

119894

12(119905) 119886

119894

13(119905) 119886

119894

14(119905)] 120572

1119894= [119886

119894

11119886119894

12119886119894

13119886119894

14]

2119894(119905) =

2119894(119905) minus 120572

2119894 2119894

= [119886119894

21(119905) 119886

119894

22(119905) 119886

119894

23(119905) 119886

119894

24(119905)]

1205722119894= [119886119894

21119886119894

22119886119894

23119886119894

24]

Remark 2 The estimation error 119890(119905) can be thought of asthe tracking error 119890T-S between the plant model (7) and thedesired linear model (14) because 119890(119905) = 119902(119905) minus 119902

119898(119905) and

119890T-S(119905) = 119902(119905) minus 119902119898(119905) have the same control matrix while the

tracking error 119890NON between the referencemodel (14) and thenonlinear model (5) is used to prove the correctness of the T-S model and the feasibility of the proposed controller on thenonlinear model of gyroscope

Define a Lyapunov function candidate as

119881 (119905) = 119881 (119890 1198861119894 1198862119894) = 119890119879

(119905) 119875119890 (119905)

+

9

sum

119894=1

119886119879

1119894(119905) 1198861119894(119905)

1205741119894

+

9

sum

119894=1

119886119879

2119894(119905) 1198862119894(119905)

1205742119894

(22)

where 119875meet 119860119879119904119875 + 119875119860

119904= minus119868

The derivative of 119881 with respect to time becomes

(119905) = 119890119879

(119905) 119875119890 (119905) + 119890119879

(119905) 119875 119890 (119905)

+

9

sum

119894=1

2

119886119879

1119894(119905) 1198861119894(119905)

1205741119894

+

9

sum

119894=1

2

119886119879

2119894(119905) 1198862119894(119905)

1205742119894

(23)

Substituting (20) (21) into the derivative of 119881 yields

(119905) = minus 119890119879

(119905) (119860119879

119904119875 + 119875119860

119904) 119890 (119905)

minus 2sum9

119894=1120583119894(120578) 119875119879

1119890 (119905) 119902

119879

(119905) 1198862119894(119905)

sum9

119894=1120583119894(120578)

minus 2sum9

119894=1120583119894(120578) 119875119879

2119890 (119905) 119902

119879

(119905) 1198862119894(119905)

sum9

119894=1120583119894(120578)

+

9

sum

119894=1

2

119886119879

1119894(119905) 1198861119894(119905)

1205741119894

+

9

sum

119894=1

2

119886119879

2119894(119905) 1198862119894(119905)

1205742119894

(24)

where 119875 = [1198751

1198752

1198753

1198754]

The adaptive law can be chosen as

119886119879

119894= 120574119894

120583119894(120578)

sum9

119894=1120583119894(120578)

[119875119879

1119875119879

2] 119890 (119905) 119902

119879

(119905)

minus 120588 sgn (119886119894(119905)) 120588 gt 0

(25)

where the sgn function can be defined as

sgn (119886119894) =

1 119886119894gt 0

0 119886119894= 0

minus1 119886119894lt 0

(26)

6 Discrete Dynamics in Nature and Society

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 10

05

1

0

05

1

x

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

y

120583(x)

0

05

1

120583(y)

minus6 minus4 minus2 0 2 4 60

05

1

minus6 minus4 minus2 0 2 4 6

120583(d

x)

120583(d

y)

dx dy

M71M81M91

M11

M21M31

M41M51M61

M73M83M93

M32M62M92

M12M42M72

M22M52M82

M34M64M94

M14M44M74

M24M54M84

M13M23M33

M43M53M63

Figure 3 Membership functions

Substituting (25) into (24) yields

(119905) = minus 119890119879

(119905) 119890 (119905) minus 2120588

9

sum

119894=1

sgn (1198861119894(119905)) 1198861119894(119905)

1205741119894

minus 2120588

9

sum

119894=1

sgn (1198862119894(119905)) 1198862119894(119905)

1205742119894

= minus 119890119879

(119905) 119890 (119905) minus 2120588

9

sum

119894=1

10038161003816100381610038161198861119894 (119905)1003816100381610038161003816

1205741119894

minus 2120588

9

sum

119894=1

10038161003816100381610038161198862119894 (119905)1003816100381610038161003816

1205742119894

le minus 119890119879

(119905) 119890 (119905) le minus 119890 (119905)2

le 0

(27)

119881(119905) ge 0 and (119905) le 0 imply the boundedness of 119890(119905) and119894(119905) the control is bounded for all time (21) implies 119890(119905) is

also bounded and inequality (27) implies 119890(119905) isin 1198712 Then

with 119890(119905) isin 1198712

cap 119871infin and 119890(119905) isin 119871

infin according to Barbalatrsquoslemma [20] the tracking error 119890(119905) asymptotically convergesto zero

5 Simulation Analysis

In this section we will evaluate the proposed adaptive fuzzycontrol using T-S model approach on the lumped MEMSgyroscope sensormodel Parameters of theMEMS gyroscopesensor are as follows

119898 = 057119890 minus 8 kg 1205960= 1 kHz 119902

0= 10119890 minus 6m

119889119909119909

= 0429119890 minus 6Nsm 119889119910119910

= 00429119890 minus 6Nsm

119889119909119910

= 00429119890 minus 6Nsm 119896119909119909

= 8098Nm

119896119910119910

= 7162Nm 119896119909119910

= 5Nm

1198961199093 = 35611989012Nm

1198961199103 = 35611989012Nm Ω

119911= 50 rads

(28)

Since the general displacement range of the MEMSgyroscope sensor in each axis is submicrometer level itis reasonable to choose 1 120583m as the reference length 119902

0

Given that the usual natural frequency of each axle of avibratory MEMS gyroscope sensor is in the kHz range 120596

0is

chosen as 1 kHz The unknown angular velocity is assumedΩ119911

= 50 rads The desired motion trajectories are 119909119898

=

119860119909sin(120596119909119905) 119910119898

= 119860119910sin(120596119910119905) where 119860

119909= 1 119860

119910= 12

120596119909= 671 kHz and 120596

119910= 511 kHz

The plant parameters are adjusted online by adaptivelaw (24) where the adaptive gain 120574

119894= 1 and the adaptive

parameter 120588 = 1 The initial values of T-S model are[02 024 0 0] initial values of the estimation model are[0 0 0 0] and initial values of the nonlinear model are[02 024 0 0] The external disturbance is 119889 = [

119889119909

119889119910

] =

[10 sin(2120587119905)10 sin(2120587119905) ]119860119898 = [

minus1 0 minus1205962

1199090

0 minus1 0 minus1205962

119910

1 0 0 0

0 1 0 0

] and the eigenvalues of119860119898

are minus05plusmn66916119894 minus05plusmn50855119894 Although119860119904canmeet the

requirement of 119860119904in 119860119879

119904119875 + 119875119860

119904= minus119868 the values of first and

second column in 119875 which are [051 0

0 052

001 0

0 002

] cannot adjust each

parameter since partial parameters are zero which is used as

adaptive gain in (24) so 119860119904is chosen as 119860

119904= [

minus1 minus1 minus1 minus1

0 minus1 0 0

0 0 minus1 0

0 0 0 minus1

]

Discrete Dynamics in Nature and Society 7

0 1 2 3minus008

minus0075

minus007

minus0065

Time (s)0 1 2 3

22

23

24

25

Time (s)0 1 2 3

minus145

minus14

minus135

minus13

minus125

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)

00 1 2 3minus0018

minus0017

minus0016

minus0015

Time (s)

1 2 3minus013

minus012

minus011

minus01

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)0 1 2 3

minus13

minus125

minus12

minus11

minus115

Time (s)

A2

[11

]

A2

[12

]

A2

[13

]

A2

[14

]

A2

[21

]

A2

[24

]

A2

[22

]

A2

[23

]

times104

times104

times10minus3

Figure 4 The change of plant parameters in 1198602

10 2 385

90

95

100

Time (s)10 2 3

80

85

90

95

Time (s)0 1 2 3

minus17

minus16

minus15

minus14

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)

0 1 2 3150

160

170

180

Time (s)0 1 2 3

Time (s)

0 1 2 3

Time (s)0 1 2 3

Time (s)

130

140

150

160

minus900

minus850

minus800

minus750

minus155

minus15

minus145

minus14

minus135

A5

[11

]

A5

[12

]

A5

[13

]

A5

[14

]

A5

[21

]

A5

[24

]

A5

[22

]

A5

[23

]

times104

times104

Figure 5 The change of plant parameters in 1198605

to ensure that each parameter of first and second column in

119875 is nonzero values as [05 minus025

minus025 075

minus025 025

minus025 025

]

The fuzzy rules for T-S fuzzy model for the system can beobtained by linearizing the nonlinear model (5) at the points119909 isin minus1 0 1 isin minus119860

119909120596119909

0 119860119909120596119909 119910 isin minus12 0 12

and 119910 isin minus119860119910120596119910

0 119860119910120596119910 and then the membership

functions of states 119909 119910 and 119910with respect to fuzzy set1198721198941

11987211989421198721198943 and119872

1198944used in the T-S fuzzy model are shown as

in Figure 3In this simulation it is assumed that the physical parame-

ters in the T-S fuzzy model are not known exactly Hence the

8 Discrete Dynamics in Nature and Society

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

22

23

24

25

65

7

75

minus008

minus0075

minus007

minus0065

minus16

minus155

minus15

minus145

minus14

minus270

minus260

minus250

minus240

minus230

minus145

minus14

minus135

minus13

minus125

minus013

minus012

minus011

minus01

minus0018

minus0017

minus0016

minus0015

A7

[11

]

A7

[12

]

A7

[13

]

A7

[14

]

A7

[21

]

A7

[22

]

A7

[23

]

A7

[24

]

times104

times104

times10minus3

Figure 6 The change of plant parameters in 1198607

parameters are tuned by the proposed adaptive law The truevalues of 119860lowast

119894of fuzzy T-S model are as shown in Table 1 The

initial value 119860119894= 09 lowast 119860

lowast

119894 The variations of each parameter

in 1198602 1198605 and 119860

7are shown in Figures 4 5 and 6 Here

only parts of parameters are shown in table for examples InFigures 4 5 and 6 we can see each parameter converges to itstrue value asymptotically

Figure 7 depicts the tracking error of 119909- and 119910-axisbetween the reference model and T-S model It can beobserved from Figure 7 that the position of 119909 and 119910 inthe T-S model can track the position of reference modelin very short time and tracking errors converge to zeroasymptotically The tracking error 119890T-S converges to zeroasymptotically rather than shocks near zero because thenew adaptive law is chosen to be the previous adaptivelaw 120574119894(120583119894(120578)sum

9

119894=1120583119894(120578)) [119875

119879

1119875119879

2] 119890(119905)119902

119879

(119905) plus a robust term120588 sgn(

119894) to guarantee that the derivative of the Lyapunov

function is negative rather than nonpositive Figure 8 plotsthe tracking error of 119909- and 119910-axis between the referencemodel and the nonlinear model We can notice that theperformance of the tracking errors in Figure 8 is worse thanthat in Figure 7

The tracking error 119890T-S shocks near zero because thecontroller is designed based on the T-S model rather thantheNONmodel FromFigures 7 and 8 we can see the validityof the adaptive fuzzy control with parameter estimationscheme on the T-S model and the feasibility of the proposedcontroller on the nonlinear model of gyroscope Then itcan be concluded that the MEMS gyroscope can maintain

0 5 10 15 20 25 30minus1

minus1

minus2

minus05

0

05

1

Time (s)

0 5 10 15 20 25 30

Time (s)

012

e xts

e yts

Figure 7 The tracking error 119890T-S between the reference model andT-S model

the proof mass oscillating in the 119909 and 119910 direction atgiven frequency and amplitude by using adaptive T-S fuzzycontroller The proposed T-S modeling method can providea possibility for developing a systematic analysis and designmethod for complex nonlinear control systems which canapproximate the nonlinear system and compensate modeluncertainties and external disturbances It can be observedfrom Figure 9 that the controller input of the adaptive fuzzycontroller is stable

Discrete Dynamics in Nature and Society 9

Table 1 Parameters of T-S fuzzy model

119860lowast

1=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

2=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

3=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

4=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

5=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

6=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

7=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

8=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

9=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

0 5 10 15 20 25 30minus1

minus05

05

1

Time (s)

0

minus1

minus20 5 10 15 20 25 30

Time (s)

012

e xno

ne y

non

Figure 8The tracking error 119890NON between the reference model andnonlinear model

6 Conclusions

A T-S fuzzy model based adaptive controller for angularvelocity sensor is presented in this paper An adaptiveT-S fuzzy compensator is used to approximate the modeluncertainties and external disturbances The output of thefuzzy controller is used as compensator to reduce the effectsof the system nonlinearities The proposed new adaptive lawcan guarantee that the derivative of the Lyapunov functionis negative rather than nonpositive The proposed estimatorfor the existing fuzzy state feedback controller can achievea good robust performance against parameter uncertaintiesSimulation studies are implemented to verify the effectivenessof the proposed adaptive fuzzy control and the correctness ofthe T-S fuzzy model However experimental demonstrationshould be investigated to verify the validity of the proposedapproach in the next research step

minus1

minus20 5 10 15 20 25 30

Time (s)

012

minus1

minus20 5 10 15 20 25 30

Time (s)

012

times104

times104

uy

ux

Figure 9 Control input with the proposed controller

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the anonymous reviewers for usefulcomments that improved the quality of the paperThiswork ispartially supported by National Science Foundation of Chinaunder Grant no 61374100 and Natural Science Foundation ofJiangsu Province under Grant no BK20131136

References

[1] R P Leland ldquoAdaptive control of a MEMS gyroscope usinglyapunov methodsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 2 pp 278ndash283 2006

10 Discrete Dynamics in Nature and Society

[2] S Sung W-T Sung C Kim S Yun and Y J Lee ldquoOnthe mode-matched control of MEMS vibratory gyroscope viaphase-domain analysis and designrdquo IEEEASME Transactionson Mechatronics vol 14 no 4 pp 446ndash455 2009

[3] R Antonello R Oboe L Prandi and F Biganzoli ldquoAuto-matic mode matching in MEMS vibrating gyroscopes usingextremum-seeking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 10 pp 3880ndash3891 2009

[4] S Park R Horowitz S K Hong and Y Nam ldquoTrajectory-switching algorithm for aMEMS gyroscoperdquo IEEE Transactionson Instrumentation and Measurement vol 56 no 6 pp 2561ndash2569 2007

[5] J D John and T Vinay ldquoNovel concept of a single-massadaptively controlled triaxial angular rate sensorrdquo IEEE SensorsJournal vol 6 no 3 pp 588ndash595 2006

[6] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009

[7] J Fei and F Chowdhury ldquoRobust adaptive controller for triaxialangular velocity sensorrdquo International Journal of InnovativeComputing Information and Control vol 7 no 6 pp 2439ndash2448 2010

[8] L Wang Adaptive Fuzzy Systems and Control-Design andStability Analysis Prentice Hall Upper Saddle River NJ USA1994

[9] Y Guo and P-Y Woo ldquoAn adaptive fuzzy sliding mode con-troller for robotic manipulatorsrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 33 no2 pp 149ndash159 2003

[10] B K Yoo and W C Ham ldquoAdaptive control of robot manip-ulator using fuzzy compensatorrdquo IEEE Transactions on FuzzySystems vol 8 no 2 pp 186ndash199 2000

[11] R-J Wai ldquoFuzzy sliding-mode control using adaptive tuningtechniquerdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 586ndash594 2007

[12] Y-H Chang C-W Chang C-L Chen and C-W Tao ldquoFuzzysliding-mode formation control for multirobot systems designand implementationrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 42 no 2 pp 444ndash457 2012

[13] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinearsystems using online T-S fuzzy-neural modeling approachrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 41 no 2 pp 542ndash552 2011

[14] M C M Teixeira and S H Zak ldquoStabilizing controller designfor uncertain nonlinear systems using fuzzy modelsrdquo IEEETransactions on Fuzzy Systems vol 7 no 2 pp 133ndash142 1999

[15] C-W Park and Y-W Cho ldquoT-S model based indirect adaptivefuzzy control using online parameter estimationrdquo IEEE Trans-actions on Systems Man and Cybernetics Part B Cyberneticsvol 34 no 6 pp 2293ndash2302 2004

[16] S F Asokanthan and T Wang ldquoNonlinear instabilities in avibratory gyroscope subjected to angular speed fluctuationsrdquoNonlinear Dynamics vol 54 no 1-2 pp 69ndash78 2008

[17] T Wang Nonlinear and stochastic dynamics of MEMS-basedangular rate sensing and switching systems [PhD thesis] TheUniversity of Western of Ontario Ontario Canada 2009

[18] Z Chi M Jia and Q Xu ldquoFuzzy PID feedback control ofpiezoelectric actuator with feedforward compensationrdquo Math-ematical Problems in Engineering vol 2014 Article ID 10718414 pages 2014

[19] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics Part B vol 15 no 1 pp 116ndash1321985

[20] P Ioannou and J Sun Robust Adaptive Control Prentice HallUpper Saddle River NJ USA 1996

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

Volume 2014

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Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

119902119898=

[[[[[

[

119910

119909

119910

]]]]]

]

=

[[[[[[

[

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)

119860119909sin (120596

119909119905)

119860119910sin (120596

119910119905)

]]]]]]

]

119903 = [

119860119909120596119909cos (120596

119909119905)

119860119910120596119910cos (120596

119910119905)]

(10)

Remark 1 Substituting 119860119898 119861119898 119902119898 119903 into 119902

119898yields 119902

119889 we

can get that the changed reference model 119902119898= 119860119898119902119898+ 119861119898119903

is equivalent to previous one 119902119889= 119860119889119902119889

According to PDC [14] we design local state linearfeedback controllers for each linear submodel The fuzzycontrol rules are as the following forms

Rule 119894 IF 119909 is about 1198721198941and 119910 is about 119872

1198942and

is about 1198721198943and 119910 is about 119872

1198944

THEN 119906 (119905) = minus119870119894(119905) 119909 (119905) + 119903 (119905) 119894 = 1 2 9

(11)

The controller output can be expressed using center-averagedefuzzifier product inference and singleton fuzzifier

119906 (119905) =sum9

119894=1120583119894(120578) [minus119870

119894119902 (119905) + 119903 (119905)]

sum9

119894=1120583119894(120578)

(12)

where

119870119894= 119860119894minus 119860119898

=

[[[[[[

[

119886119894

11minus 119886119898

11119886119894

12minus 119886119898

12119886119894

13minus 119886119898

13119886119894

14minus 119886119898

14

119886119894

21minus 119886119898

21119886119894

22minus 119886119898

22119886119894

23minus 119886119898

23119886119894

24minus 119886119898

24

1 0 0 0

0 1 0 0

]]]]]]

]

119894 = 1 2 9

(13)

Substituting the controller force (12) into the fuzzy system (7)yields the desired model

119902119898= 119860119898119902119898+ 119861119898119903 (14)

4 Parameter Estimation

The block diagram of the adaptive fuzzy control using T-Sfuzzy model is shown as in Figure 2 Taking the parameteruncertainty into account then the parameter estimation isapplied to the adaptive fuzzy control The adaptive law canguarantee the asymptotic convergence of the error betweenthe plant model state and the estimation model state andmake the closed loop system of fuzzy controller and estima-tion model be a desired linear model and consequently theplant model state can follow the state of the desired linearmodelThe controller output is implemented in the nonlinearmodel also to prove the correctness of the T-S model and thefeasibility of the proposed control scheme on the nonlinearmodel of MEMS gyroscope

With unknown parameters matrix 119860119894 the controller

changes to

119906 (119905) =

sum9

119894=1120583119894(120578) [minus

119894119902 (119905) + 119903 (119905)]

sum9

119894=1120583119894(120578)

(15)

where

119894= 119860119894minus 119860119898

=

[[[[[[

[

119886119894

11minus 119886119898

11119886119894

12minus 119886119898

12119886119894

13minus 119886119898

13119886119894

14minus 119886119898

14

119886119894

21minus 119886119898

21119886119894

22minus 119886119898

22119886119894

23minus 119886119898

23119886119894

24minus 119886119898

24

1 0 0 0

0 1 0 0

]]]]]]

]

119860119894=

[[[[[[

[

119886119894

11119886119894

12119886119894

13119886119894

14

119886119894

21119886119894

22119886119894

23119886119894

24

1 0 0 0

0 1 0 0

]]]]]]

]

119894 = 1 2 9

(16)

is the estimate of 119860119894

By considering the plant parameterization (7) can berewritten as

119902 (119905) = 119860119904119902 (119905) +

sum9

119894=1120583119894(120578) [(119860

119894minus 119860119904) 119902 (119905) + 119861

119894119906 (119905)]

sum9

119894=1120583119894(120578)

(17)

where 119860119904is an arbitrary stable matrix

Define the estimation mode as the following formula

119902 (119905) = 119860119904119902 (119905) +

sum9

119894=1120583119894(120578) [(119860

119894minus 119860119904) 119902 (119905) + 119861

119894119906 (119905)]

sum9

119894=1120583119894(120578)

(18)

The estimationmodel (18) can be the same one as by adoptingthe control input (15)

119902119898= 119860119898119902119898+ 119861119898119903 (19)

The estimation error 119890(119905) = 119902(119905) minus 119902119898(119905)meets (19) and (20)

119890 (119905) = 119860119904119890 (119905) +

sum9

119894=1120583119894(120578) (119860

119894minus 119860119894) 119902 (119905)

sum9

119894=1120583119894(120578)

= 119860119904119890 (119905) minus

sum9

119894=1120583119894(120578) [

11198942119894

0 0]119879

119902 (119905)

sum9

119894=1120583119894(120578)

(20)

119890119879

(119905) = 119890 (119905) 119860119879

119904minussum9

119894=1120583119894(120578) 119902119879

(119905) [1119894

2119894

0 0]

sum9

119894=1120583119894(120578)

(21)

Discrete Dynamics in Nature and Society 5

Σ

Σ

+

minusqnon

qd

eT-S

+

minus

q

q

Desired trajectory

Reference input

Fuzzy controller

Adaptive law

U

Plant model

Estimation model

Nonlinear model

Ki(t)120583i(120578)

eNON

qdx = Axsin 120596xt)

qdy = Aysin 120596yt)

rx = Ax120596xcosry = Ay120596ycos

u(t) =sum9

i=1120583i(120578)

sum9i=1120583i(120578)

[minus Kiq(t) + r(t)]

aTi = 120574i

120583i(120578)

sum9i=1120583i(120578)

e(t)qT(t) minus 120588sgn(ai)[PT1 PT

2 ]

1 = (2S minus D) 1 + (Ω2z minus K1)q1 minus K3q

31 + u

(t) = Amq(t) +sum9

i=1120583i(120578)[(Ai minus Am)q(t) + Biu(t)]

Ni=1120583i(120578)sum

(t) = Amq(t) +sum9

i=1120583i(120578)[(Ai minus Am)q(t) + Biu(t)]

Ni=1120583i(120578)sum

q

)

120596xt))

)

120596yt))

q

q

q

Figure 2 The block diagram of the adaptive fuzzy control using T-S fuzzy model

where 119890(119905) actually is the tracking error between theplant model (7) and the estimation model (18)

119894(119905) =

[1119894(119905) 2119894(119905) 0 0]

1119894(119905) =

1119894(119905) minus 120572

1119894 1119894(119905) =

[119886119894

11(119905) 119886

119894

12(119905) 119886

119894

13(119905) 119886

119894

14(119905)] 120572

1119894= [119886

119894

11119886119894

12119886119894

13119886119894

14]

2119894(119905) =

2119894(119905) minus 120572

2119894 2119894

= [119886119894

21(119905) 119886

119894

22(119905) 119886

119894

23(119905) 119886

119894

24(119905)]

1205722119894= [119886119894

21119886119894

22119886119894

23119886119894

24]

Remark 2 The estimation error 119890(119905) can be thought of asthe tracking error 119890T-S between the plant model (7) and thedesired linear model (14) because 119890(119905) = 119902(119905) minus 119902

119898(119905) and

119890T-S(119905) = 119902(119905) minus 119902119898(119905) have the same control matrix while the

tracking error 119890NON between the referencemodel (14) and thenonlinear model (5) is used to prove the correctness of the T-S model and the feasibility of the proposed controller on thenonlinear model of gyroscope

Define a Lyapunov function candidate as

119881 (119905) = 119881 (119890 1198861119894 1198862119894) = 119890119879

(119905) 119875119890 (119905)

+

9

sum

119894=1

119886119879

1119894(119905) 1198861119894(119905)

1205741119894

+

9

sum

119894=1

119886119879

2119894(119905) 1198862119894(119905)

1205742119894

(22)

where 119875meet 119860119879119904119875 + 119875119860

119904= minus119868

The derivative of 119881 with respect to time becomes

(119905) = 119890119879

(119905) 119875119890 (119905) + 119890119879

(119905) 119875 119890 (119905)

+

9

sum

119894=1

2

119886119879

1119894(119905) 1198861119894(119905)

1205741119894

+

9

sum

119894=1

2

119886119879

2119894(119905) 1198862119894(119905)

1205742119894

(23)

Substituting (20) (21) into the derivative of 119881 yields

(119905) = minus 119890119879

(119905) (119860119879

119904119875 + 119875119860

119904) 119890 (119905)

minus 2sum9

119894=1120583119894(120578) 119875119879

1119890 (119905) 119902

119879

(119905) 1198862119894(119905)

sum9

119894=1120583119894(120578)

minus 2sum9

119894=1120583119894(120578) 119875119879

2119890 (119905) 119902

119879

(119905) 1198862119894(119905)

sum9

119894=1120583119894(120578)

+

9

sum

119894=1

2

119886119879

1119894(119905) 1198861119894(119905)

1205741119894

+

9

sum

119894=1

2

119886119879

2119894(119905) 1198862119894(119905)

1205742119894

(24)

where 119875 = [1198751

1198752

1198753

1198754]

The adaptive law can be chosen as

119886119879

119894= 120574119894

120583119894(120578)

sum9

119894=1120583119894(120578)

[119875119879

1119875119879

2] 119890 (119905) 119902

119879

(119905)

minus 120588 sgn (119886119894(119905)) 120588 gt 0

(25)

where the sgn function can be defined as

sgn (119886119894) =

1 119886119894gt 0

0 119886119894= 0

minus1 119886119894lt 0

(26)

6 Discrete Dynamics in Nature and Society

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 10

05

1

0

05

1

x

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

y

120583(x)

0

05

1

120583(y)

minus6 minus4 minus2 0 2 4 60

05

1

minus6 minus4 minus2 0 2 4 6

120583(d

x)

120583(d

y)

dx dy

M71M81M91

M11

M21M31

M41M51M61

M73M83M93

M32M62M92

M12M42M72

M22M52M82

M34M64M94

M14M44M74

M24M54M84

M13M23M33

M43M53M63

Figure 3 Membership functions

Substituting (25) into (24) yields

(119905) = minus 119890119879

(119905) 119890 (119905) minus 2120588

9

sum

119894=1

sgn (1198861119894(119905)) 1198861119894(119905)

1205741119894

minus 2120588

9

sum

119894=1

sgn (1198862119894(119905)) 1198862119894(119905)

1205742119894

= minus 119890119879

(119905) 119890 (119905) minus 2120588

9

sum

119894=1

10038161003816100381610038161198861119894 (119905)1003816100381610038161003816

1205741119894

minus 2120588

9

sum

119894=1

10038161003816100381610038161198862119894 (119905)1003816100381610038161003816

1205742119894

le minus 119890119879

(119905) 119890 (119905) le minus 119890 (119905)2

le 0

(27)

119881(119905) ge 0 and (119905) le 0 imply the boundedness of 119890(119905) and119894(119905) the control is bounded for all time (21) implies 119890(119905) is

also bounded and inequality (27) implies 119890(119905) isin 1198712 Then

with 119890(119905) isin 1198712

cap 119871infin and 119890(119905) isin 119871

infin according to Barbalatrsquoslemma [20] the tracking error 119890(119905) asymptotically convergesto zero

5 Simulation Analysis

In this section we will evaluate the proposed adaptive fuzzycontrol using T-S model approach on the lumped MEMSgyroscope sensormodel Parameters of theMEMS gyroscopesensor are as follows

119898 = 057119890 minus 8 kg 1205960= 1 kHz 119902

0= 10119890 minus 6m

119889119909119909

= 0429119890 minus 6Nsm 119889119910119910

= 00429119890 minus 6Nsm

119889119909119910

= 00429119890 minus 6Nsm 119896119909119909

= 8098Nm

119896119910119910

= 7162Nm 119896119909119910

= 5Nm

1198961199093 = 35611989012Nm

1198961199103 = 35611989012Nm Ω

119911= 50 rads

(28)

Since the general displacement range of the MEMSgyroscope sensor in each axis is submicrometer level itis reasonable to choose 1 120583m as the reference length 119902

0

Given that the usual natural frequency of each axle of avibratory MEMS gyroscope sensor is in the kHz range 120596

0is

chosen as 1 kHz The unknown angular velocity is assumedΩ119911

= 50 rads The desired motion trajectories are 119909119898

=

119860119909sin(120596119909119905) 119910119898

= 119860119910sin(120596119910119905) where 119860

119909= 1 119860

119910= 12

120596119909= 671 kHz and 120596

119910= 511 kHz

The plant parameters are adjusted online by adaptivelaw (24) where the adaptive gain 120574

119894= 1 and the adaptive

parameter 120588 = 1 The initial values of T-S model are[02 024 0 0] initial values of the estimation model are[0 0 0 0] and initial values of the nonlinear model are[02 024 0 0] The external disturbance is 119889 = [

119889119909

119889119910

] =

[10 sin(2120587119905)10 sin(2120587119905) ]119860119898 = [

minus1 0 minus1205962

1199090

0 minus1 0 minus1205962

119910

1 0 0 0

0 1 0 0

] and the eigenvalues of119860119898

are minus05plusmn66916119894 minus05plusmn50855119894 Although119860119904canmeet the

requirement of 119860119904in 119860119879

119904119875 + 119875119860

119904= minus119868 the values of first and

second column in 119875 which are [051 0

0 052

001 0

0 002

] cannot adjust each

parameter since partial parameters are zero which is used as

adaptive gain in (24) so 119860119904is chosen as 119860

119904= [

minus1 minus1 minus1 minus1

0 minus1 0 0

0 0 minus1 0

0 0 0 minus1

]

Discrete Dynamics in Nature and Society 7

0 1 2 3minus008

minus0075

minus007

minus0065

Time (s)0 1 2 3

22

23

24

25

Time (s)0 1 2 3

minus145

minus14

minus135

minus13

minus125

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)

00 1 2 3minus0018

minus0017

minus0016

minus0015

Time (s)

1 2 3minus013

minus012

minus011

minus01

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)0 1 2 3

minus13

minus125

minus12

minus11

minus115

Time (s)

A2

[11

]

A2

[12

]

A2

[13

]

A2

[14

]

A2

[21

]

A2

[24

]

A2

[22

]

A2

[23

]

times104

times104

times10minus3

Figure 4 The change of plant parameters in 1198602

10 2 385

90

95

100

Time (s)10 2 3

80

85

90

95

Time (s)0 1 2 3

minus17

minus16

minus15

minus14

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)

0 1 2 3150

160

170

180

Time (s)0 1 2 3

Time (s)

0 1 2 3

Time (s)0 1 2 3

Time (s)

130

140

150

160

minus900

minus850

minus800

minus750

minus155

minus15

minus145

minus14

minus135

A5

[11

]

A5

[12

]

A5

[13

]

A5

[14

]

A5

[21

]

A5

[24

]

A5

[22

]

A5

[23

]

times104

times104

Figure 5 The change of plant parameters in 1198605

to ensure that each parameter of first and second column in

119875 is nonzero values as [05 minus025

minus025 075

minus025 025

minus025 025

]

The fuzzy rules for T-S fuzzy model for the system can beobtained by linearizing the nonlinear model (5) at the points119909 isin minus1 0 1 isin minus119860

119909120596119909

0 119860119909120596119909 119910 isin minus12 0 12

and 119910 isin minus119860119910120596119910

0 119860119910120596119910 and then the membership

functions of states 119909 119910 and 119910with respect to fuzzy set1198721198941

11987211989421198721198943 and119872

1198944used in the T-S fuzzy model are shown as

in Figure 3In this simulation it is assumed that the physical parame-

ters in the T-S fuzzy model are not known exactly Hence the

8 Discrete Dynamics in Nature and Society

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

22

23

24

25

65

7

75

minus008

minus0075

minus007

minus0065

minus16

minus155

minus15

minus145

minus14

minus270

minus260

minus250

minus240

minus230

minus145

minus14

minus135

minus13

minus125

minus013

minus012

minus011

minus01

minus0018

minus0017

minus0016

minus0015

A7

[11

]

A7

[12

]

A7

[13

]

A7

[14

]

A7

[21

]

A7

[22

]

A7

[23

]

A7

[24

]

times104

times104

times10minus3

Figure 6 The change of plant parameters in 1198607

parameters are tuned by the proposed adaptive law The truevalues of 119860lowast

119894of fuzzy T-S model are as shown in Table 1 The

initial value 119860119894= 09 lowast 119860

lowast

119894 The variations of each parameter

in 1198602 1198605 and 119860

7are shown in Figures 4 5 and 6 Here

only parts of parameters are shown in table for examples InFigures 4 5 and 6 we can see each parameter converges to itstrue value asymptotically

Figure 7 depicts the tracking error of 119909- and 119910-axisbetween the reference model and T-S model It can beobserved from Figure 7 that the position of 119909 and 119910 inthe T-S model can track the position of reference modelin very short time and tracking errors converge to zeroasymptotically The tracking error 119890T-S converges to zeroasymptotically rather than shocks near zero because thenew adaptive law is chosen to be the previous adaptivelaw 120574119894(120583119894(120578)sum

9

119894=1120583119894(120578)) [119875

119879

1119875119879

2] 119890(119905)119902

119879

(119905) plus a robust term120588 sgn(

119894) to guarantee that the derivative of the Lyapunov

function is negative rather than nonpositive Figure 8 plotsthe tracking error of 119909- and 119910-axis between the referencemodel and the nonlinear model We can notice that theperformance of the tracking errors in Figure 8 is worse thanthat in Figure 7

The tracking error 119890T-S shocks near zero because thecontroller is designed based on the T-S model rather thantheNONmodel FromFigures 7 and 8 we can see the validityof the adaptive fuzzy control with parameter estimationscheme on the T-S model and the feasibility of the proposedcontroller on the nonlinear model of gyroscope Then itcan be concluded that the MEMS gyroscope can maintain

0 5 10 15 20 25 30minus1

minus1

minus2

minus05

0

05

1

Time (s)

0 5 10 15 20 25 30

Time (s)

012

e xts

e yts

Figure 7 The tracking error 119890T-S between the reference model andT-S model

the proof mass oscillating in the 119909 and 119910 direction atgiven frequency and amplitude by using adaptive T-S fuzzycontroller The proposed T-S modeling method can providea possibility for developing a systematic analysis and designmethod for complex nonlinear control systems which canapproximate the nonlinear system and compensate modeluncertainties and external disturbances It can be observedfrom Figure 9 that the controller input of the adaptive fuzzycontroller is stable

Discrete Dynamics in Nature and Society 9

Table 1 Parameters of T-S fuzzy model

119860lowast

1=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

2=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

3=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

4=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

5=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

6=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

7=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

8=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

9=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

0 5 10 15 20 25 30minus1

minus05

05

1

Time (s)

0

minus1

minus20 5 10 15 20 25 30

Time (s)

012

e xno

ne y

non

Figure 8The tracking error 119890NON between the reference model andnonlinear model

6 Conclusions

A T-S fuzzy model based adaptive controller for angularvelocity sensor is presented in this paper An adaptiveT-S fuzzy compensator is used to approximate the modeluncertainties and external disturbances The output of thefuzzy controller is used as compensator to reduce the effectsof the system nonlinearities The proposed new adaptive lawcan guarantee that the derivative of the Lyapunov functionis negative rather than nonpositive The proposed estimatorfor the existing fuzzy state feedback controller can achievea good robust performance against parameter uncertaintiesSimulation studies are implemented to verify the effectivenessof the proposed adaptive fuzzy control and the correctness ofthe T-S fuzzy model However experimental demonstrationshould be investigated to verify the validity of the proposedapproach in the next research step

minus1

minus20 5 10 15 20 25 30

Time (s)

012

minus1

minus20 5 10 15 20 25 30

Time (s)

012

times104

times104

uy

ux

Figure 9 Control input with the proposed controller

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the anonymous reviewers for usefulcomments that improved the quality of the paperThiswork ispartially supported by National Science Foundation of Chinaunder Grant no 61374100 and Natural Science Foundation ofJiangsu Province under Grant no BK20131136

References

[1] R P Leland ldquoAdaptive control of a MEMS gyroscope usinglyapunov methodsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 2 pp 278ndash283 2006

10 Discrete Dynamics in Nature and Society

[2] S Sung W-T Sung C Kim S Yun and Y J Lee ldquoOnthe mode-matched control of MEMS vibratory gyroscope viaphase-domain analysis and designrdquo IEEEASME Transactionson Mechatronics vol 14 no 4 pp 446ndash455 2009

[3] R Antonello R Oboe L Prandi and F Biganzoli ldquoAuto-matic mode matching in MEMS vibrating gyroscopes usingextremum-seeking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 10 pp 3880ndash3891 2009

[4] S Park R Horowitz S K Hong and Y Nam ldquoTrajectory-switching algorithm for aMEMS gyroscoperdquo IEEE Transactionson Instrumentation and Measurement vol 56 no 6 pp 2561ndash2569 2007

[5] J D John and T Vinay ldquoNovel concept of a single-massadaptively controlled triaxial angular rate sensorrdquo IEEE SensorsJournal vol 6 no 3 pp 588ndash595 2006

[6] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009

[7] J Fei and F Chowdhury ldquoRobust adaptive controller for triaxialangular velocity sensorrdquo International Journal of InnovativeComputing Information and Control vol 7 no 6 pp 2439ndash2448 2010

[8] L Wang Adaptive Fuzzy Systems and Control-Design andStability Analysis Prentice Hall Upper Saddle River NJ USA1994

[9] Y Guo and P-Y Woo ldquoAn adaptive fuzzy sliding mode con-troller for robotic manipulatorsrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 33 no2 pp 149ndash159 2003

[10] B K Yoo and W C Ham ldquoAdaptive control of robot manip-ulator using fuzzy compensatorrdquo IEEE Transactions on FuzzySystems vol 8 no 2 pp 186ndash199 2000

[11] R-J Wai ldquoFuzzy sliding-mode control using adaptive tuningtechniquerdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 586ndash594 2007

[12] Y-H Chang C-W Chang C-L Chen and C-W Tao ldquoFuzzysliding-mode formation control for multirobot systems designand implementationrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 42 no 2 pp 444ndash457 2012

[13] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinearsystems using online T-S fuzzy-neural modeling approachrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 41 no 2 pp 542ndash552 2011

[14] M C M Teixeira and S H Zak ldquoStabilizing controller designfor uncertain nonlinear systems using fuzzy modelsrdquo IEEETransactions on Fuzzy Systems vol 7 no 2 pp 133ndash142 1999

[15] C-W Park and Y-W Cho ldquoT-S model based indirect adaptivefuzzy control using online parameter estimationrdquo IEEE Trans-actions on Systems Man and Cybernetics Part B Cyberneticsvol 34 no 6 pp 2293ndash2302 2004

[16] S F Asokanthan and T Wang ldquoNonlinear instabilities in avibratory gyroscope subjected to angular speed fluctuationsrdquoNonlinear Dynamics vol 54 no 1-2 pp 69ndash78 2008

[17] T Wang Nonlinear and stochastic dynamics of MEMS-basedangular rate sensing and switching systems [PhD thesis] TheUniversity of Western of Ontario Ontario Canada 2009

[18] Z Chi M Jia and Q Xu ldquoFuzzy PID feedback control ofpiezoelectric actuator with feedforward compensationrdquo Math-ematical Problems in Engineering vol 2014 Article ID 10718414 pages 2014

[19] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics Part B vol 15 no 1 pp 116ndash1321985

[20] P Ioannou and J Sun Robust Adaptive Control Prentice HallUpper Saddle River NJ USA 1996

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

Σ

Σ

+

minusqnon

qd

eT-S

+

minus

q

q

Desired trajectory

Reference input

Fuzzy controller

Adaptive law

U

Plant model

Estimation model

Nonlinear model

Ki(t)120583i(120578)

eNON

qdx = Axsin 120596xt)

qdy = Aysin 120596yt)

rx = Ax120596xcosry = Ay120596ycos

u(t) =sum9

i=1120583i(120578)

sum9i=1120583i(120578)

[minus Kiq(t) + r(t)]

aTi = 120574i

120583i(120578)

sum9i=1120583i(120578)

e(t)qT(t) minus 120588sgn(ai)[PT1 PT

2 ]

1 = (2S minus D) 1 + (Ω2z minus K1)q1 minus K3q

31 + u

(t) = Amq(t) +sum9

i=1120583i(120578)[(Ai minus Am)q(t) + Biu(t)]

Ni=1120583i(120578)sum

(t) = Amq(t) +sum9

i=1120583i(120578)[(Ai minus Am)q(t) + Biu(t)]

Ni=1120583i(120578)sum

q

)

120596xt))

)

120596yt))

q

q

q

Figure 2 The block diagram of the adaptive fuzzy control using T-S fuzzy model

where 119890(119905) actually is the tracking error between theplant model (7) and the estimation model (18)

119894(119905) =

[1119894(119905) 2119894(119905) 0 0]

1119894(119905) =

1119894(119905) minus 120572

1119894 1119894(119905) =

[119886119894

11(119905) 119886

119894

12(119905) 119886

119894

13(119905) 119886

119894

14(119905)] 120572

1119894= [119886

119894

11119886119894

12119886119894

13119886119894

14]

2119894(119905) =

2119894(119905) minus 120572

2119894 2119894

= [119886119894

21(119905) 119886

119894

22(119905) 119886

119894

23(119905) 119886

119894

24(119905)]

1205722119894= [119886119894

21119886119894

22119886119894

23119886119894

24]

Remark 2 The estimation error 119890(119905) can be thought of asthe tracking error 119890T-S between the plant model (7) and thedesired linear model (14) because 119890(119905) = 119902(119905) minus 119902

119898(119905) and

119890T-S(119905) = 119902(119905) minus 119902119898(119905) have the same control matrix while the

tracking error 119890NON between the referencemodel (14) and thenonlinear model (5) is used to prove the correctness of the T-S model and the feasibility of the proposed controller on thenonlinear model of gyroscope

Define a Lyapunov function candidate as

119881 (119905) = 119881 (119890 1198861119894 1198862119894) = 119890119879

(119905) 119875119890 (119905)

+

9

sum

119894=1

119886119879

1119894(119905) 1198861119894(119905)

1205741119894

+

9

sum

119894=1

119886119879

2119894(119905) 1198862119894(119905)

1205742119894

(22)

where 119875meet 119860119879119904119875 + 119875119860

119904= minus119868

The derivative of 119881 with respect to time becomes

(119905) = 119890119879

(119905) 119875119890 (119905) + 119890119879

(119905) 119875 119890 (119905)

+

9

sum

119894=1

2

119886119879

1119894(119905) 1198861119894(119905)

1205741119894

+

9

sum

119894=1

2

119886119879

2119894(119905) 1198862119894(119905)

1205742119894

(23)

Substituting (20) (21) into the derivative of 119881 yields

(119905) = minus 119890119879

(119905) (119860119879

119904119875 + 119875119860

119904) 119890 (119905)

minus 2sum9

119894=1120583119894(120578) 119875119879

1119890 (119905) 119902

119879

(119905) 1198862119894(119905)

sum9

119894=1120583119894(120578)

minus 2sum9

119894=1120583119894(120578) 119875119879

2119890 (119905) 119902

119879

(119905) 1198862119894(119905)

sum9

119894=1120583119894(120578)

+

9

sum

119894=1

2

119886119879

1119894(119905) 1198861119894(119905)

1205741119894

+

9

sum

119894=1

2

119886119879

2119894(119905) 1198862119894(119905)

1205742119894

(24)

where 119875 = [1198751

1198752

1198753

1198754]

The adaptive law can be chosen as

119886119879

119894= 120574119894

120583119894(120578)

sum9

119894=1120583119894(120578)

[119875119879

1119875119879

2] 119890 (119905) 119902

119879

(119905)

minus 120588 sgn (119886119894(119905)) 120588 gt 0

(25)

where the sgn function can be defined as

sgn (119886119894) =

1 119886119894gt 0

0 119886119894= 0

minus1 119886119894lt 0

(26)

6 Discrete Dynamics in Nature and Society

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 10

05

1

0

05

1

x

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

y

120583(x)

0

05

1

120583(y)

minus6 minus4 minus2 0 2 4 60

05

1

minus6 minus4 minus2 0 2 4 6

120583(d

x)

120583(d

y)

dx dy

M71M81M91

M11

M21M31

M41M51M61

M73M83M93

M32M62M92

M12M42M72

M22M52M82

M34M64M94

M14M44M74

M24M54M84

M13M23M33

M43M53M63

Figure 3 Membership functions

Substituting (25) into (24) yields

(119905) = minus 119890119879

(119905) 119890 (119905) minus 2120588

9

sum

119894=1

sgn (1198861119894(119905)) 1198861119894(119905)

1205741119894

minus 2120588

9

sum

119894=1

sgn (1198862119894(119905)) 1198862119894(119905)

1205742119894

= minus 119890119879

(119905) 119890 (119905) minus 2120588

9

sum

119894=1

10038161003816100381610038161198861119894 (119905)1003816100381610038161003816

1205741119894

minus 2120588

9

sum

119894=1

10038161003816100381610038161198862119894 (119905)1003816100381610038161003816

1205742119894

le minus 119890119879

(119905) 119890 (119905) le minus 119890 (119905)2

le 0

(27)

119881(119905) ge 0 and (119905) le 0 imply the boundedness of 119890(119905) and119894(119905) the control is bounded for all time (21) implies 119890(119905) is

also bounded and inequality (27) implies 119890(119905) isin 1198712 Then

with 119890(119905) isin 1198712

cap 119871infin and 119890(119905) isin 119871

infin according to Barbalatrsquoslemma [20] the tracking error 119890(119905) asymptotically convergesto zero

5 Simulation Analysis

In this section we will evaluate the proposed adaptive fuzzycontrol using T-S model approach on the lumped MEMSgyroscope sensormodel Parameters of theMEMS gyroscopesensor are as follows

119898 = 057119890 minus 8 kg 1205960= 1 kHz 119902

0= 10119890 minus 6m

119889119909119909

= 0429119890 minus 6Nsm 119889119910119910

= 00429119890 minus 6Nsm

119889119909119910

= 00429119890 minus 6Nsm 119896119909119909

= 8098Nm

119896119910119910

= 7162Nm 119896119909119910

= 5Nm

1198961199093 = 35611989012Nm

1198961199103 = 35611989012Nm Ω

119911= 50 rads

(28)

Since the general displacement range of the MEMSgyroscope sensor in each axis is submicrometer level itis reasonable to choose 1 120583m as the reference length 119902

0

Given that the usual natural frequency of each axle of avibratory MEMS gyroscope sensor is in the kHz range 120596

0is

chosen as 1 kHz The unknown angular velocity is assumedΩ119911

= 50 rads The desired motion trajectories are 119909119898

=

119860119909sin(120596119909119905) 119910119898

= 119860119910sin(120596119910119905) where 119860

119909= 1 119860

119910= 12

120596119909= 671 kHz and 120596

119910= 511 kHz

The plant parameters are adjusted online by adaptivelaw (24) where the adaptive gain 120574

119894= 1 and the adaptive

parameter 120588 = 1 The initial values of T-S model are[02 024 0 0] initial values of the estimation model are[0 0 0 0] and initial values of the nonlinear model are[02 024 0 0] The external disturbance is 119889 = [

119889119909

119889119910

] =

[10 sin(2120587119905)10 sin(2120587119905) ]119860119898 = [

minus1 0 minus1205962

1199090

0 minus1 0 minus1205962

119910

1 0 0 0

0 1 0 0

] and the eigenvalues of119860119898

are minus05plusmn66916119894 minus05plusmn50855119894 Although119860119904canmeet the

requirement of 119860119904in 119860119879

119904119875 + 119875119860

119904= minus119868 the values of first and

second column in 119875 which are [051 0

0 052

001 0

0 002

] cannot adjust each

parameter since partial parameters are zero which is used as

adaptive gain in (24) so 119860119904is chosen as 119860

119904= [

minus1 minus1 minus1 minus1

0 minus1 0 0

0 0 minus1 0

0 0 0 minus1

]

Discrete Dynamics in Nature and Society 7

0 1 2 3minus008

minus0075

minus007

minus0065

Time (s)0 1 2 3

22

23

24

25

Time (s)0 1 2 3

minus145

minus14

minus135

minus13

minus125

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)

00 1 2 3minus0018

minus0017

minus0016

minus0015

Time (s)

1 2 3minus013

minus012

minus011

minus01

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)0 1 2 3

minus13

minus125

minus12

minus11

minus115

Time (s)

A2

[11

]

A2

[12

]

A2

[13

]

A2

[14

]

A2

[21

]

A2

[24

]

A2

[22

]

A2

[23

]

times104

times104

times10minus3

Figure 4 The change of plant parameters in 1198602

10 2 385

90

95

100

Time (s)10 2 3

80

85

90

95

Time (s)0 1 2 3

minus17

minus16

minus15

minus14

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)

0 1 2 3150

160

170

180

Time (s)0 1 2 3

Time (s)

0 1 2 3

Time (s)0 1 2 3

Time (s)

130

140

150

160

minus900

minus850

minus800

minus750

minus155

minus15

minus145

minus14

minus135

A5

[11

]

A5

[12

]

A5

[13

]

A5

[14

]

A5

[21

]

A5

[24

]

A5

[22

]

A5

[23

]

times104

times104

Figure 5 The change of plant parameters in 1198605

to ensure that each parameter of first and second column in

119875 is nonzero values as [05 minus025

minus025 075

minus025 025

minus025 025

]

The fuzzy rules for T-S fuzzy model for the system can beobtained by linearizing the nonlinear model (5) at the points119909 isin minus1 0 1 isin minus119860

119909120596119909

0 119860119909120596119909 119910 isin minus12 0 12

and 119910 isin minus119860119910120596119910

0 119860119910120596119910 and then the membership

functions of states 119909 119910 and 119910with respect to fuzzy set1198721198941

11987211989421198721198943 and119872

1198944used in the T-S fuzzy model are shown as

in Figure 3In this simulation it is assumed that the physical parame-

ters in the T-S fuzzy model are not known exactly Hence the

8 Discrete Dynamics in Nature and Society

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

22

23

24

25

65

7

75

minus008

minus0075

minus007

minus0065

minus16

minus155

minus15

minus145

minus14

minus270

minus260

minus250

minus240

minus230

minus145

minus14

minus135

minus13

minus125

minus013

minus012

minus011

minus01

minus0018

minus0017

minus0016

minus0015

A7

[11

]

A7

[12

]

A7

[13

]

A7

[14

]

A7

[21

]

A7

[22

]

A7

[23

]

A7

[24

]

times104

times104

times10minus3

Figure 6 The change of plant parameters in 1198607

parameters are tuned by the proposed adaptive law The truevalues of 119860lowast

119894of fuzzy T-S model are as shown in Table 1 The

initial value 119860119894= 09 lowast 119860

lowast

119894 The variations of each parameter

in 1198602 1198605 and 119860

7are shown in Figures 4 5 and 6 Here

only parts of parameters are shown in table for examples InFigures 4 5 and 6 we can see each parameter converges to itstrue value asymptotically

Figure 7 depicts the tracking error of 119909- and 119910-axisbetween the reference model and T-S model It can beobserved from Figure 7 that the position of 119909 and 119910 inthe T-S model can track the position of reference modelin very short time and tracking errors converge to zeroasymptotically The tracking error 119890T-S converges to zeroasymptotically rather than shocks near zero because thenew adaptive law is chosen to be the previous adaptivelaw 120574119894(120583119894(120578)sum

9

119894=1120583119894(120578)) [119875

119879

1119875119879

2] 119890(119905)119902

119879

(119905) plus a robust term120588 sgn(

119894) to guarantee that the derivative of the Lyapunov

function is negative rather than nonpositive Figure 8 plotsthe tracking error of 119909- and 119910-axis between the referencemodel and the nonlinear model We can notice that theperformance of the tracking errors in Figure 8 is worse thanthat in Figure 7

The tracking error 119890T-S shocks near zero because thecontroller is designed based on the T-S model rather thantheNONmodel FromFigures 7 and 8 we can see the validityof the adaptive fuzzy control with parameter estimationscheme on the T-S model and the feasibility of the proposedcontroller on the nonlinear model of gyroscope Then itcan be concluded that the MEMS gyroscope can maintain

0 5 10 15 20 25 30minus1

minus1

minus2

minus05

0

05

1

Time (s)

0 5 10 15 20 25 30

Time (s)

012

e xts

e yts

Figure 7 The tracking error 119890T-S between the reference model andT-S model

the proof mass oscillating in the 119909 and 119910 direction atgiven frequency and amplitude by using adaptive T-S fuzzycontroller The proposed T-S modeling method can providea possibility for developing a systematic analysis and designmethod for complex nonlinear control systems which canapproximate the nonlinear system and compensate modeluncertainties and external disturbances It can be observedfrom Figure 9 that the controller input of the adaptive fuzzycontroller is stable

Discrete Dynamics in Nature and Society 9

Table 1 Parameters of T-S fuzzy model

119860lowast

1=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

2=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

3=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

4=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

5=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

6=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

7=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

8=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

9=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

0 5 10 15 20 25 30minus1

minus05

05

1

Time (s)

0

minus1

minus20 5 10 15 20 25 30

Time (s)

012

e xno

ne y

non

Figure 8The tracking error 119890NON between the reference model andnonlinear model

6 Conclusions

A T-S fuzzy model based adaptive controller for angularvelocity sensor is presented in this paper An adaptiveT-S fuzzy compensator is used to approximate the modeluncertainties and external disturbances The output of thefuzzy controller is used as compensator to reduce the effectsof the system nonlinearities The proposed new adaptive lawcan guarantee that the derivative of the Lyapunov functionis negative rather than nonpositive The proposed estimatorfor the existing fuzzy state feedback controller can achievea good robust performance against parameter uncertaintiesSimulation studies are implemented to verify the effectivenessof the proposed adaptive fuzzy control and the correctness ofthe T-S fuzzy model However experimental demonstrationshould be investigated to verify the validity of the proposedapproach in the next research step

minus1

minus20 5 10 15 20 25 30

Time (s)

012

minus1

minus20 5 10 15 20 25 30

Time (s)

012

times104

times104

uy

ux

Figure 9 Control input with the proposed controller

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the anonymous reviewers for usefulcomments that improved the quality of the paperThiswork ispartially supported by National Science Foundation of Chinaunder Grant no 61374100 and Natural Science Foundation ofJiangsu Province under Grant no BK20131136

References

[1] R P Leland ldquoAdaptive control of a MEMS gyroscope usinglyapunov methodsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 2 pp 278ndash283 2006

10 Discrete Dynamics in Nature and Society

[2] S Sung W-T Sung C Kim S Yun and Y J Lee ldquoOnthe mode-matched control of MEMS vibratory gyroscope viaphase-domain analysis and designrdquo IEEEASME Transactionson Mechatronics vol 14 no 4 pp 446ndash455 2009

[3] R Antonello R Oboe L Prandi and F Biganzoli ldquoAuto-matic mode matching in MEMS vibrating gyroscopes usingextremum-seeking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 10 pp 3880ndash3891 2009

[4] S Park R Horowitz S K Hong and Y Nam ldquoTrajectory-switching algorithm for aMEMS gyroscoperdquo IEEE Transactionson Instrumentation and Measurement vol 56 no 6 pp 2561ndash2569 2007

[5] J D John and T Vinay ldquoNovel concept of a single-massadaptively controlled triaxial angular rate sensorrdquo IEEE SensorsJournal vol 6 no 3 pp 588ndash595 2006

[6] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009

[7] J Fei and F Chowdhury ldquoRobust adaptive controller for triaxialangular velocity sensorrdquo International Journal of InnovativeComputing Information and Control vol 7 no 6 pp 2439ndash2448 2010

[8] L Wang Adaptive Fuzzy Systems and Control-Design andStability Analysis Prentice Hall Upper Saddle River NJ USA1994

[9] Y Guo and P-Y Woo ldquoAn adaptive fuzzy sliding mode con-troller for robotic manipulatorsrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 33 no2 pp 149ndash159 2003

[10] B K Yoo and W C Ham ldquoAdaptive control of robot manip-ulator using fuzzy compensatorrdquo IEEE Transactions on FuzzySystems vol 8 no 2 pp 186ndash199 2000

[11] R-J Wai ldquoFuzzy sliding-mode control using adaptive tuningtechniquerdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 586ndash594 2007

[12] Y-H Chang C-W Chang C-L Chen and C-W Tao ldquoFuzzysliding-mode formation control for multirobot systems designand implementationrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 42 no 2 pp 444ndash457 2012

[13] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinearsystems using online T-S fuzzy-neural modeling approachrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 41 no 2 pp 542ndash552 2011

[14] M C M Teixeira and S H Zak ldquoStabilizing controller designfor uncertain nonlinear systems using fuzzy modelsrdquo IEEETransactions on Fuzzy Systems vol 7 no 2 pp 133ndash142 1999

[15] C-W Park and Y-W Cho ldquoT-S model based indirect adaptivefuzzy control using online parameter estimationrdquo IEEE Trans-actions on Systems Man and Cybernetics Part B Cyberneticsvol 34 no 6 pp 2293ndash2302 2004

[16] S F Asokanthan and T Wang ldquoNonlinear instabilities in avibratory gyroscope subjected to angular speed fluctuationsrdquoNonlinear Dynamics vol 54 no 1-2 pp 69ndash78 2008

[17] T Wang Nonlinear and stochastic dynamics of MEMS-basedangular rate sensing and switching systems [PhD thesis] TheUniversity of Western of Ontario Ontario Canada 2009

[18] Z Chi M Jia and Q Xu ldquoFuzzy PID feedback control ofpiezoelectric actuator with feedforward compensationrdquo Math-ematical Problems in Engineering vol 2014 Article ID 10718414 pages 2014

[19] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics Part B vol 15 no 1 pp 116ndash1321985

[20] P Ioannou and J Sun Robust Adaptive Control Prentice HallUpper Saddle River NJ USA 1996

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 10

05

1

0

05

1

x

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

y

120583(x)

0

05

1

120583(y)

minus6 minus4 minus2 0 2 4 60

05

1

minus6 minus4 minus2 0 2 4 6

120583(d

x)

120583(d

y)

dx dy

M71M81M91

M11

M21M31

M41M51M61

M73M83M93

M32M62M92

M12M42M72

M22M52M82

M34M64M94

M14M44M74

M24M54M84

M13M23M33

M43M53M63

Figure 3 Membership functions

Substituting (25) into (24) yields

(119905) = minus 119890119879

(119905) 119890 (119905) minus 2120588

9

sum

119894=1

sgn (1198861119894(119905)) 1198861119894(119905)

1205741119894

minus 2120588

9

sum

119894=1

sgn (1198862119894(119905)) 1198862119894(119905)

1205742119894

= minus 119890119879

(119905) 119890 (119905) minus 2120588

9

sum

119894=1

10038161003816100381610038161198861119894 (119905)1003816100381610038161003816

1205741119894

minus 2120588

9

sum

119894=1

10038161003816100381610038161198862119894 (119905)1003816100381610038161003816

1205742119894

le minus 119890119879

(119905) 119890 (119905) le minus 119890 (119905)2

le 0

(27)

119881(119905) ge 0 and (119905) le 0 imply the boundedness of 119890(119905) and119894(119905) the control is bounded for all time (21) implies 119890(119905) is

also bounded and inequality (27) implies 119890(119905) isin 1198712 Then

with 119890(119905) isin 1198712

cap 119871infin and 119890(119905) isin 119871

infin according to Barbalatrsquoslemma [20] the tracking error 119890(119905) asymptotically convergesto zero

5 Simulation Analysis

In this section we will evaluate the proposed adaptive fuzzycontrol using T-S model approach on the lumped MEMSgyroscope sensormodel Parameters of theMEMS gyroscopesensor are as follows

119898 = 057119890 minus 8 kg 1205960= 1 kHz 119902

0= 10119890 minus 6m

119889119909119909

= 0429119890 minus 6Nsm 119889119910119910

= 00429119890 minus 6Nsm

119889119909119910

= 00429119890 minus 6Nsm 119896119909119909

= 8098Nm

119896119910119910

= 7162Nm 119896119909119910

= 5Nm

1198961199093 = 35611989012Nm

1198961199103 = 35611989012Nm Ω

119911= 50 rads

(28)

Since the general displacement range of the MEMSgyroscope sensor in each axis is submicrometer level itis reasonable to choose 1 120583m as the reference length 119902

0

Given that the usual natural frequency of each axle of avibratory MEMS gyroscope sensor is in the kHz range 120596

0is

chosen as 1 kHz The unknown angular velocity is assumedΩ119911

= 50 rads The desired motion trajectories are 119909119898

=

119860119909sin(120596119909119905) 119910119898

= 119860119910sin(120596119910119905) where 119860

119909= 1 119860

119910= 12

120596119909= 671 kHz and 120596

119910= 511 kHz

The plant parameters are adjusted online by adaptivelaw (24) where the adaptive gain 120574

119894= 1 and the adaptive

parameter 120588 = 1 The initial values of T-S model are[02 024 0 0] initial values of the estimation model are[0 0 0 0] and initial values of the nonlinear model are[02 024 0 0] The external disturbance is 119889 = [

119889119909

119889119910

] =

[10 sin(2120587119905)10 sin(2120587119905) ]119860119898 = [

minus1 0 minus1205962

1199090

0 minus1 0 minus1205962

119910

1 0 0 0

0 1 0 0

] and the eigenvalues of119860119898

are minus05plusmn66916119894 minus05plusmn50855119894 Although119860119904canmeet the

requirement of 119860119904in 119860119879

119904119875 + 119875119860

119904= minus119868 the values of first and

second column in 119875 which are [051 0

0 052

001 0

0 002

] cannot adjust each

parameter since partial parameters are zero which is used as

adaptive gain in (24) so 119860119904is chosen as 119860

119904= [

minus1 minus1 minus1 minus1

0 minus1 0 0

0 0 minus1 0

0 0 0 minus1

]

Discrete Dynamics in Nature and Society 7

0 1 2 3minus008

minus0075

minus007

minus0065

Time (s)0 1 2 3

22

23

24

25

Time (s)0 1 2 3

minus145

minus14

minus135

minus13

minus125

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)

00 1 2 3minus0018

minus0017

minus0016

minus0015

Time (s)

1 2 3minus013

minus012

minus011

minus01

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)0 1 2 3

minus13

minus125

minus12

minus11

minus115

Time (s)

A2

[11

]

A2

[12

]

A2

[13

]

A2

[14

]

A2

[21

]

A2

[24

]

A2

[22

]

A2

[23

]

times104

times104

times10minus3

Figure 4 The change of plant parameters in 1198602

10 2 385

90

95

100

Time (s)10 2 3

80

85

90

95

Time (s)0 1 2 3

minus17

minus16

minus15

minus14

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)

0 1 2 3150

160

170

180

Time (s)0 1 2 3

Time (s)

0 1 2 3

Time (s)0 1 2 3

Time (s)

130

140

150

160

minus900

minus850

minus800

minus750

minus155

minus15

minus145

minus14

minus135

A5

[11

]

A5

[12

]

A5

[13

]

A5

[14

]

A5

[21

]

A5

[24

]

A5

[22

]

A5

[23

]

times104

times104

Figure 5 The change of plant parameters in 1198605

to ensure that each parameter of first and second column in

119875 is nonzero values as [05 minus025

minus025 075

minus025 025

minus025 025

]

The fuzzy rules for T-S fuzzy model for the system can beobtained by linearizing the nonlinear model (5) at the points119909 isin minus1 0 1 isin minus119860

119909120596119909

0 119860119909120596119909 119910 isin minus12 0 12

and 119910 isin minus119860119910120596119910

0 119860119910120596119910 and then the membership

functions of states 119909 119910 and 119910with respect to fuzzy set1198721198941

11987211989421198721198943 and119872

1198944used in the T-S fuzzy model are shown as

in Figure 3In this simulation it is assumed that the physical parame-

ters in the T-S fuzzy model are not known exactly Hence the

8 Discrete Dynamics in Nature and Society

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

22

23

24

25

65

7

75

minus008

minus0075

minus007

minus0065

minus16

minus155

minus15

minus145

minus14

minus270

minus260

minus250

minus240

minus230

minus145

minus14

minus135

minus13

minus125

minus013

minus012

minus011

minus01

minus0018

minus0017

minus0016

minus0015

A7

[11

]

A7

[12

]

A7

[13

]

A7

[14

]

A7

[21

]

A7

[22

]

A7

[23

]

A7

[24

]

times104

times104

times10minus3

Figure 6 The change of plant parameters in 1198607

parameters are tuned by the proposed adaptive law The truevalues of 119860lowast

119894of fuzzy T-S model are as shown in Table 1 The

initial value 119860119894= 09 lowast 119860

lowast

119894 The variations of each parameter

in 1198602 1198605 and 119860

7are shown in Figures 4 5 and 6 Here

only parts of parameters are shown in table for examples InFigures 4 5 and 6 we can see each parameter converges to itstrue value asymptotically

Figure 7 depicts the tracking error of 119909- and 119910-axisbetween the reference model and T-S model It can beobserved from Figure 7 that the position of 119909 and 119910 inthe T-S model can track the position of reference modelin very short time and tracking errors converge to zeroasymptotically The tracking error 119890T-S converges to zeroasymptotically rather than shocks near zero because thenew adaptive law is chosen to be the previous adaptivelaw 120574119894(120583119894(120578)sum

9

119894=1120583119894(120578)) [119875

119879

1119875119879

2] 119890(119905)119902

119879

(119905) plus a robust term120588 sgn(

119894) to guarantee that the derivative of the Lyapunov

function is negative rather than nonpositive Figure 8 plotsthe tracking error of 119909- and 119910-axis between the referencemodel and the nonlinear model We can notice that theperformance of the tracking errors in Figure 8 is worse thanthat in Figure 7

The tracking error 119890T-S shocks near zero because thecontroller is designed based on the T-S model rather thantheNONmodel FromFigures 7 and 8 we can see the validityof the adaptive fuzzy control with parameter estimationscheme on the T-S model and the feasibility of the proposedcontroller on the nonlinear model of gyroscope Then itcan be concluded that the MEMS gyroscope can maintain

0 5 10 15 20 25 30minus1

minus1

minus2

minus05

0

05

1

Time (s)

0 5 10 15 20 25 30

Time (s)

012

e xts

e yts

Figure 7 The tracking error 119890T-S between the reference model andT-S model

the proof mass oscillating in the 119909 and 119910 direction atgiven frequency and amplitude by using adaptive T-S fuzzycontroller The proposed T-S modeling method can providea possibility for developing a systematic analysis and designmethod for complex nonlinear control systems which canapproximate the nonlinear system and compensate modeluncertainties and external disturbances It can be observedfrom Figure 9 that the controller input of the adaptive fuzzycontroller is stable

Discrete Dynamics in Nature and Society 9

Table 1 Parameters of T-S fuzzy model

119860lowast

1=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

2=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

3=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

4=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

5=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

6=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

7=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

8=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

9=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

0 5 10 15 20 25 30minus1

minus05

05

1

Time (s)

0

minus1

minus20 5 10 15 20 25 30

Time (s)

012

e xno

ne y

non

Figure 8The tracking error 119890NON between the reference model andnonlinear model

6 Conclusions

A T-S fuzzy model based adaptive controller for angularvelocity sensor is presented in this paper An adaptiveT-S fuzzy compensator is used to approximate the modeluncertainties and external disturbances The output of thefuzzy controller is used as compensator to reduce the effectsof the system nonlinearities The proposed new adaptive lawcan guarantee that the derivative of the Lyapunov functionis negative rather than nonpositive The proposed estimatorfor the existing fuzzy state feedback controller can achievea good robust performance against parameter uncertaintiesSimulation studies are implemented to verify the effectivenessof the proposed adaptive fuzzy control and the correctness ofthe T-S fuzzy model However experimental demonstrationshould be investigated to verify the validity of the proposedapproach in the next research step

minus1

minus20 5 10 15 20 25 30

Time (s)

012

minus1

minus20 5 10 15 20 25 30

Time (s)

012

times104

times104

uy

ux

Figure 9 Control input with the proposed controller

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the anonymous reviewers for usefulcomments that improved the quality of the paperThiswork ispartially supported by National Science Foundation of Chinaunder Grant no 61374100 and Natural Science Foundation ofJiangsu Province under Grant no BK20131136

References

[1] R P Leland ldquoAdaptive control of a MEMS gyroscope usinglyapunov methodsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 2 pp 278ndash283 2006

10 Discrete Dynamics in Nature and Society

[2] S Sung W-T Sung C Kim S Yun and Y J Lee ldquoOnthe mode-matched control of MEMS vibratory gyroscope viaphase-domain analysis and designrdquo IEEEASME Transactionson Mechatronics vol 14 no 4 pp 446ndash455 2009

[3] R Antonello R Oboe L Prandi and F Biganzoli ldquoAuto-matic mode matching in MEMS vibrating gyroscopes usingextremum-seeking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 10 pp 3880ndash3891 2009

[4] S Park R Horowitz S K Hong and Y Nam ldquoTrajectory-switching algorithm for aMEMS gyroscoperdquo IEEE Transactionson Instrumentation and Measurement vol 56 no 6 pp 2561ndash2569 2007

[5] J D John and T Vinay ldquoNovel concept of a single-massadaptively controlled triaxial angular rate sensorrdquo IEEE SensorsJournal vol 6 no 3 pp 588ndash595 2006

[6] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009

[7] J Fei and F Chowdhury ldquoRobust adaptive controller for triaxialangular velocity sensorrdquo International Journal of InnovativeComputing Information and Control vol 7 no 6 pp 2439ndash2448 2010

[8] L Wang Adaptive Fuzzy Systems and Control-Design andStability Analysis Prentice Hall Upper Saddle River NJ USA1994

[9] Y Guo and P-Y Woo ldquoAn adaptive fuzzy sliding mode con-troller for robotic manipulatorsrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 33 no2 pp 149ndash159 2003

[10] B K Yoo and W C Ham ldquoAdaptive control of robot manip-ulator using fuzzy compensatorrdquo IEEE Transactions on FuzzySystems vol 8 no 2 pp 186ndash199 2000

[11] R-J Wai ldquoFuzzy sliding-mode control using adaptive tuningtechniquerdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 586ndash594 2007

[12] Y-H Chang C-W Chang C-L Chen and C-W Tao ldquoFuzzysliding-mode formation control for multirobot systems designand implementationrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 42 no 2 pp 444ndash457 2012

[13] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinearsystems using online T-S fuzzy-neural modeling approachrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 41 no 2 pp 542ndash552 2011

[14] M C M Teixeira and S H Zak ldquoStabilizing controller designfor uncertain nonlinear systems using fuzzy modelsrdquo IEEETransactions on Fuzzy Systems vol 7 no 2 pp 133ndash142 1999

[15] C-W Park and Y-W Cho ldquoT-S model based indirect adaptivefuzzy control using online parameter estimationrdquo IEEE Trans-actions on Systems Man and Cybernetics Part B Cyberneticsvol 34 no 6 pp 2293ndash2302 2004

[16] S F Asokanthan and T Wang ldquoNonlinear instabilities in avibratory gyroscope subjected to angular speed fluctuationsrdquoNonlinear Dynamics vol 54 no 1-2 pp 69ndash78 2008

[17] T Wang Nonlinear and stochastic dynamics of MEMS-basedangular rate sensing and switching systems [PhD thesis] TheUniversity of Western of Ontario Ontario Canada 2009

[18] Z Chi M Jia and Q Xu ldquoFuzzy PID feedback control ofpiezoelectric actuator with feedforward compensationrdquo Math-ematical Problems in Engineering vol 2014 Article ID 10718414 pages 2014

[19] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics Part B vol 15 no 1 pp 116ndash1321985

[20] P Ioannou and J Sun Robust Adaptive Control Prentice HallUpper Saddle River NJ USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

0 1 2 3minus008

minus0075

minus007

minus0065

Time (s)0 1 2 3

22

23

24

25

Time (s)0 1 2 3

minus145

minus14

minus135

minus13

minus125

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)

00 1 2 3minus0018

minus0017

minus0016

minus0015

Time (s)

1 2 3minus013

minus012

minus011

minus01

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)0 1 2 3

minus13

minus125

minus12

minus11

minus115

Time (s)

A2

[11

]

A2

[12

]

A2

[13

]

A2

[14

]

A2

[21

]

A2

[24

]

A2

[22

]

A2

[23

]

times104

times104

times10minus3

Figure 4 The change of plant parameters in 1198602

10 2 385

90

95

100

Time (s)10 2 3

80

85

90

95

Time (s)0 1 2 3

minus17

minus16

minus15

minus14

Time (s)0 1 2 3

minus900

minus850

minus800

minus750

Time (s)

0 1 2 3150

160

170

180

Time (s)0 1 2 3

Time (s)

0 1 2 3

Time (s)0 1 2 3

Time (s)

130

140

150

160

minus900

minus850

minus800

minus750

minus155

minus15

minus145

minus14

minus135

A5

[11

]

A5

[12

]

A5

[13

]

A5

[14

]

A5

[21

]

A5

[24

]

A5

[22

]

A5

[23

]

times104

times104

Figure 5 The change of plant parameters in 1198605

to ensure that each parameter of first and second column in

119875 is nonzero values as [05 minus025

minus025 075

minus025 025

minus025 025

]

The fuzzy rules for T-S fuzzy model for the system can beobtained by linearizing the nonlinear model (5) at the points119909 isin minus1 0 1 isin minus119860

119909120596119909

0 119860119909120596119909 119910 isin minus12 0 12

and 119910 isin minus119860119910120596119910

0 119860119910120596119910 and then the membership

functions of states 119909 119910 and 119910with respect to fuzzy set1198721198941

11987211989421198721198943 and119872

1198944used in the T-S fuzzy model are shown as

in Figure 3In this simulation it is assumed that the physical parame-

ters in the T-S fuzzy model are not known exactly Hence the

8 Discrete Dynamics in Nature and Society

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

22

23

24

25

65

7

75

minus008

minus0075

minus007

minus0065

minus16

minus155

minus15

minus145

minus14

minus270

minus260

minus250

minus240

minus230

minus145

minus14

minus135

minus13

minus125

minus013

minus012

minus011

minus01

minus0018

minus0017

minus0016

minus0015

A7

[11

]

A7

[12

]

A7

[13

]

A7

[14

]

A7

[21

]

A7

[22

]

A7

[23

]

A7

[24

]

times104

times104

times10minus3

Figure 6 The change of plant parameters in 1198607

parameters are tuned by the proposed adaptive law The truevalues of 119860lowast

119894of fuzzy T-S model are as shown in Table 1 The

initial value 119860119894= 09 lowast 119860

lowast

119894 The variations of each parameter

in 1198602 1198605 and 119860

7are shown in Figures 4 5 and 6 Here

only parts of parameters are shown in table for examples InFigures 4 5 and 6 we can see each parameter converges to itstrue value asymptotically

Figure 7 depicts the tracking error of 119909- and 119910-axisbetween the reference model and T-S model It can beobserved from Figure 7 that the position of 119909 and 119910 inthe T-S model can track the position of reference modelin very short time and tracking errors converge to zeroasymptotically The tracking error 119890T-S converges to zeroasymptotically rather than shocks near zero because thenew adaptive law is chosen to be the previous adaptivelaw 120574119894(120583119894(120578)sum

9

119894=1120583119894(120578)) [119875

119879

1119875119879

2] 119890(119905)119902

119879

(119905) plus a robust term120588 sgn(

119894) to guarantee that the derivative of the Lyapunov

function is negative rather than nonpositive Figure 8 plotsthe tracking error of 119909- and 119910-axis between the referencemodel and the nonlinear model We can notice that theperformance of the tracking errors in Figure 8 is worse thanthat in Figure 7

The tracking error 119890T-S shocks near zero because thecontroller is designed based on the T-S model rather thantheNONmodel FromFigures 7 and 8 we can see the validityof the adaptive fuzzy control with parameter estimationscheme on the T-S model and the feasibility of the proposedcontroller on the nonlinear model of gyroscope Then itcan be concluded that the MEMS gyroscope can maintain

0 5 10 15 20 25 30minus1

minus1

minus2

minus05

0

05

1

Time (s)

0 5 10 15 20 25 30

Time (s)

012

e xts

e yts

Figure 7 The tracking error 119890T-S between the reference model andT-S model

the proof mass oscillating in the 119909 and 119910 direction atgiven frequency and amplitude by using adaptive T-S fuzzycontroller The proposed T-S modeling method can providea possibility for developing a systematic analysis and designmethod for complex nonlinear control systems which canapproximate the nonlinear system and compensate modeluncertainties and external disturbances It can be observedfrom Figure 9 that the controller input of the adaptive fuzzycontroller is stable

Discrete Dynamics in Nature and Society 9

Table 1 Parameters of T-S fuzzy model

119860lowast

1=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

2=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

3=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

4=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

5=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

6=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

7=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

8=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

9=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

0 5 10 15 20 25 30minus1

minus05

05

1

Time (s)

0

minus1

minus20 5 10 15 20 25 30

Time (s)

012

e xno

ne y

non

Figure 8The tracking error 119890NON between the reference model andnonlinear model

6 Conclusions

A T-S fuzzy model based adaptive controller for angularvelocity sensor is presented in this paper An adaptiveT-S fuzzy compensator is used to approximate the modeluncertainties and external disturbances The output of thefuzzy controller is used as compensator to reduce the effectsof the system nonlinearities The proposed new adaptive lawcan guarantee that the derivative of the Lyapunov functionis negative rather than nonpositive The proposed estimatorfor the existing fuzzy state feedback controller can achievea good robust performance against parameter uncertaintiesSimulation studies are implemented to verify the effectivenessof the proposed adaptive fuzzy control and the correctness ofthe T-S fuzzy model However experimental demonstrationshould be investigated to verify the validity of the proposedapproach in the next research step

minus1

minus20 5 10 15 20 25 30

Time (s)

012

minus1

minus20 5 10 15 20 25 30

Time (s)

012

times104

times104

uy

ux

Figure 9 Control input with the proposed controller

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the anonymous reviewers for usefulcomments that improved the quality of the paperThiswork ispartially supported by National Science Foundation of Chinaunder Grant no 61374100 and Natural Science Foundation ofJiangsu Province under Grant no BK20131136

References

[1] R P Leland ldquoAdaptive control of a MEMS gyroscope usinglyapunov methodsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 2 pp 278ndash283 2006

10 Discrete Dynamics in Nature and Society

[2] S Sung W-T Sung C Kim S Yun and Y J Lee ldquoOnthe mode-matched control of MEMS vibratory gyroscope viaphase-domain analysis and designrdquo IEEEASME Transactionson Mechatronics vol 14 no 4 pp 446ndash455 2009

[3] R Antonello R Oboe L Prandi and F Biganzoli ldquoAuto-matic mode matching in MEMS vibrating gyroscopes usingextremum-seeking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 10 pp 3880ndash3891 2009

[4] S Park R Horowitz S K Hong and Y Nam ldquoTrajectory-switching algorithm for aMEMS gyroscoperdquo IEEE Transactionson Instrumentation and Measurement vol 56 no 6 pp 2561ndash2569 2007

[5] J D John and T Vinay ldquoNovel concept of a single-massadaptively controlled triaxial angular rate sensorrdquo IEEE SensorsJournal vol 6 no 3 pp 588ndash595 2006

[6] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009

[7] J Fei and F Chowdhury ldquoRobust adaptive controller for triaxialangular velocity sensorrdquo International Journal of InnovativeComputing Information and Control vol 7 no 6 pp 2439ndash2448 2010

[8] L Wang Adaptive Fuzzy Systems and Control-Design andStability Analysis Prentice Hall Upper Saddle River NJ USA1994

[9] Y Guo and P-Y Woo ldquoAn adaptive fuzzy sliding mode con-troller for robotic manipulatorsrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 33 no2 pp 149ndash159 2003

[10] B K Yoo and W C Ham ldquoAdaptive control of robot manip-ulator using fuzzy compensatorrdquo IEEE Transactions on FuzzySystems vol 8 no 2 pp 186ndash199 2000

[11] R-J Wai ldquoFuzzy sliding-mode control using adaptive tuningtechniquerdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 586ndash594 2007

[12] Y-H Chang C-W Chang C-L Chen and C-W Tao ldquoFuzzysliding-mode formation control for multirobot systems designand implementationrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 42 no 2 pp 444ndash457 2012

[13] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinearsystems using online T-S fuzzy-neural modeling approachrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 41 no 2 pp 542ndash552 2011

[14] M C M Teixeira and S H Zak ldquoStabilizing controller designfor uncertain nonlinear systems using fuzzy modelsrdquo IEEETransactions on Fuzzy Systems vol 7 no 2 pp 133ndash142 1999

[15] C-W Park and Y-W Cho ldquoT-S model based indirect adaptivefuzzy control using online parameter estimationrdquo IEEE Trans-actions on Systems Man and Cybernetics Part B Cyberneticsvol 34 no 6 pp 2293ndash2302 2004

[16] S F Asokanthan and T Wang ldquoNonlinear instabilities in avibratory gyroscope subjected to angular speed fluctuationsrdquoNonlinear Dynamics vol 54 no 1-2 pp 69ndash78 2008

[17] T Wang Nonlinear and stochastic dynamics of MEMS-basedangular rate sensing and switching systems [PhD thesis] TheUniversity of Western of Ontario Ontario Canada 2009

[18] Z Chi M Jia and Q Xu ldquoFuzzy PID feedback control ofpiezoelectric actuator with feedforward compensationrdquo Math-ematical Problems in Engineering vol 2014 Article ID 10718414 pages 2014

[19] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics Part B vol 15 no 1 pp 116ndash1321985

[20] P Ioannou and J Sun Robust Adaptive Control Prentice HallUpper Saddle River NJ USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

0 1 2 3

Time (s)

22

23

24

25

65

7

75

minus008

minus0075

minus007

minus0065

minus16

minus155

minus15

minus145

minus14

minus270

minus260

minus250

minus240

minus230

minus145

minus14

minus135

minus13

minus125

minus013

minus012

minus011

minus01

minus0018

minus0017

minus0016

minus0015

A7

[11

]

A7

[12

]

A7

[13

]

A7

[14

]

A7

[21

]

A7

[22

]

A7

[23

]

A7

[24

]

times104

times104

times10minus3

Figure 6 The change of plant parameters in 1198607

parameters are tuned by the proposed adaptive law The truevalues of 119860lowast

119894of fuzzy T-S model are as shown in Table 1 The

initial value 119860119894= 09 lowast 119860

lowast

119894 The variations of each parameter

in 1198602 1198605 and 119860

7are shown in Figures 4 5 and 6 Here

only parts of parameters are shown in table for examples InFigures 4 5 and 6 we can see each parameter converges to itstrue value asymptotically

Figure 7 depicts the tracking error of 119909- and 119910-axisbetween the reference model and T-S model It can beobserved from Figure 7 that the position of 119909 and 119910 inthe T-S model can track the position of reference modelin very short time and tracking errors converge to zeroasymptotically The tracking error 119890T-S converges to zeroasymptotically rather than shocks near zero because thenew adaptive law is chosen to be the previous adaptivelaw 120574119894(120583119894(120578)sum

9

119894=1120583119894(120578)) [119875

119879

1119875119879

2] 119890(119905)119902

119879

(119905) plus a robust term120588 sgn(

119894) to guarantee that the derivative of the Lyapunov

function is negative rather than nonpositive Figure 8 plotsthe tracking error of 119909- and 119910-axis between the referencemodel and the nonlinear model We can notice that theperformance of the tracking errors in Figure 8 is worse thanthat in Figure 7

The tracking error 119890T-S shocks near zero because thecontroller is designed based on the T-S model rather thantheNONmodel FromFigures 7 and 8 we can see the validityof the adaptive fuzzy control with parameter estimationscheme on the T-S model and the feasibility of the proposedcontroller on the nonlinear model of gyroscope Then itcan be concluded that the MEMS gyroscope can maintain

0 5 10 15 20 25 30minus1

minus1

minus2

minus05

0

05

1

Time (s)

0 5 10 15 20 25 30

Time (s)

012

e xts

e yts

Figure 7 The tracking error 119890T-S between the reference model andT-S model

the proof mass oscillating in the 119909 and 119910 direction atgiven frequency and amplitude by using adaptive T-S fuzzycontroller The proposed T-S modeling method can providea possibility for developing a systematic analysis and designmethod for complex nonlinear control systems which canapproximate the nonlinear system and compensate modeluncertainties and external disturbances It can be observedfrom Figure 9 that the controller input of the adaptive fuzzycontroller is stable

Discrete Dynamics in Nature and Society 9

Table 1 Parameters of T-S fuzzy model

119860lowast

1=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

2=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

3=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

4=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

5=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

6=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

7=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

8=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

9=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

0 5 10 15 20 25 30minus1

minus05

05

1

Time (s)

0

minus1

minus20 5 10 15 20 25 30

Time (s)

012

e xno

ne y

non

Figure 8The tracking error 119890NON between the reference model andnonlinear model

6 Conclusions

A T-S fuzzy model based adaptive controller for angularvelocity sensor is presented in this paper An adaptiveT-S fuzzy compensator is used to approximate the modeluncertainties and external disturbances The output of thefuzzy controller is used as compensator to reduce the effectsof the system nonlinearities The proposed new adaptive lawcan guarantee that the derivative of the Lyapunov functionis negative rather than nonpositive The proposed estimatorfor the existing fuzzy state feedback controller can achievea good robust performance against parameter uncertaintiesSimulation studies are implemented to verify the effectivenessof the proposed adaptive fuzzy control and the correctness ofthe T-S fuzzy model However experimental demonstrationshould be investigated to verify the validity of the proposedapproach in the next research step

minus1

minus20 5 10 15 20 25 30

Time (s)

012

minus1

minus20 5 10 15 20 25 30

Time (s)

012

times104

times104

uy

ux

Figure 9 Control input with the proposed controller

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the anonymous reviewers for usefulcomments that improved the quality of the paperThiswork ispartially supported by National Science Foundation of Chinaunder Grant no 61374100 and Natural Science Foundation ofJiangsu Province under Grant no BK20131136

References

[1] R P Leland ldquoAdaptive control of a MEMS gyroscope usinglyapunov methodsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 2 pp 278ndash283 2006

10 Discrete Dynamics in Nature and Society

[2] S Sung W-T Sung C Kim S Yun and Y J Lee ldquoOnthe mode-matched control of MEMS vibratory gyroscope viaphase-domain analysis and designrdquo IEEEASME Transactionson Mechatronics vol 14 no 4 pp 446ndash455 2009

[3] R Antonello R Oboe L Prandi and F Biganzoli ldquoAuto-matic mode matching in MEMS vibrating gyroscopes usingextremum-seeking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 10 pp 3880ndash3891 2009

[4] S Park R Horowitz S K Hong and Y Nam ldquoTrajectory-switching algorithm for aMEMS gyroscoperdquo IEEE Transactionson Instrumentation and Measurement vol 56 no 6 pp 2561ndash2569 2007

[5] J D John and T Vinay ldquoNovel concept of a single-massadaptively controlled triaxial angular rate sensorrdquo IEEE SensorsJournal vol 6 no 3 pp 588ndash595 2006

[6] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009

[7] J Fei and F Chowdhury ldquoRobust adaptive controller for triaxialangular velocity sensorrdquo International Journal of InnovativeComputing Information and Control vol 7 no 6 pp 2439ndash2448 2010

[8] L Wang Adaptive Fuzzy Systems and Control-Design andStability Analysis Prentice Hall Upper Saddle River NJ USA1994

[9] Y Guo and P-Y Woo ldquoAn adaptive fuzzy sliding mode con-troller for robotic manipulatorsrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 33 no2 pp 149ndash159 2003

[10] B K Yoo and W C Ham ldquoAdaptive control of robot manip-ulator using fuzzy compensatorrdquo IEEE Transactions on FuzzySystems vol 8 no 2 pp 186ndash199 2000

[11] R-J Wai ldquoFuzzy sliding-mode control using adaptive tuningtechniquerdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 586ndash594 2007

[12] Y-H Chang C-W Chang C-L Chen and C-W Tao ldquoFuzzysliding-mode formation control for multirobot systems designand implementationrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 42 no 2 pp 444ndash457 2012

[13] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinearsystems using online T-S fuzzy-neural modeling approachrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 41 no 2 pp 542ndash552 2011

[14] M C M Teixeira and S H Zak ldquoStabilizing controller designfor uncertain nonlinear systems using fuzzy modelsrdquo IEEETransactions on Fuzzy Systems vol 7 no 2 pp 133ndash142 1999

[15] C-W Park and Y-W Cho ldquoT-S model based indirect adaptivefuzzy control using online parameter estimationrdquo IEEE Trans-actions on Systems Man and Cybernetics Part B Cyberneticsvol 34 no 6 pp 2293ndash2302 2004

[16] S F Asokanthan and T Wang ldquoNonlinear instabilities in avibratory gyroscope subjected to angular speed fluctuationsrdquoNonlinear Dynamics vol 54 no 1-2 pp 69ndash78 2008

[17] T Wang Nonlinear and stochastic dynamics of MEMS-basedangular rate sensing and switching systems [PhD thesis] TheUniversity of Western of Ontario Ontario Canada 2009

[18] Z Chi M Jia and Q Xu ldquoFuzzy PID feedback control ofpiezoelectric actuator with feedforward compensationrdquo Math-ematical Problems in Engineering vol 2014 Article ID 10718414 pages 2014

[19] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics Part B vol 15 no 1 pp 116ndash1321985

[20] P Ioannou and J Sun Robust Adaptive Control Prentice HallUpper Saddle River NJ USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

Table 1 Parameters of T-S fuzzy model

119860lowast

1=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

2=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

3=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus14207 minus877 0 0

minus877 minus12564 0 0

]]]]]]]]

]

119879

119860lowast

4=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

5=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

6=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

7=

[[[[[[[[

[

minus00753 minus00175 1 0

00025 minus01205 0 1

minus15568 7432 0 0

minus262 minus14201 0 0

]]]]]]]]

]

119879

119860lowast

8=

[[[[[[[[

[

minus98 minus170 1 0

minus90 minus155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

119860lowast

9=

[[[[[[[[

[

98 170 1 0

90 155 0 1

minus16066 minus851 0 0

minus859 minus15232 0 0

]]]]]]]]

]

119879

0 5 10 15 20 25 30minus1

minus05

05

1

Time (s)

0

minus1

minus20 5 10 15 20 25 30

Time (s)

012

e xno

ne y

non

Figure 8The tracking error 119890NON between the reference model andnonlinear model

6 Conclusions

A T-S fuzzy model based adaptive controller for angularvelocity sensor is presented in this paper An adaptiveT-S fuzzy compensator is used to approximate the modeluncertainties and external disturbances The output of thefuzzy controller is used as compensator to reduce the effectsof the system nonlinearities The proposed new adaptive lawcan guarantee that the derivative of the Lyapunov functionis negative rather than nonpositive The proposed estimatorfor the existing fuzzy state feedback controller can achievea good robust performance against parameter uncertaintiesSimulation studies are implemented to verify the effectivenessof the proposed adaptive fuzzy control and the correctness ofthe T-S fuzzy model However experimental demonstrationshould be investigated to verify the validity of the proposedapproach in the next research step

minus1

minus20 5 10 15 20 25 30

Time (s)

012

minus1

minus20 5 10 15 20 25 30

Time (s)

012

times104

times104

uy

ux

Figure 9 Control input with the proposed controller

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank the anonymous reviewers for usefulcomments that improved the quality of the paperThiswork ispartially supported by National Science Foundation of Chinaunder Grant no 61374100 and Natural Science Foundation ofJiangsu Province under Grant no BK20131136

References

[1] R P Leland ldquoAdaptive control of a MEMS gyroscope usinglyapunov methodsrdquo IEEE Transactions on Control SystemsTechnology vol 14 no 2 pp 278ndash283 2006

10 Discrete Dynamics in Nature and Society

[2] S Sung W-T Sung C Kim S Yun and Y J Lee ldquoOnthe mode-matched control of MEMS vibratory gyroscope viaphase-domain analysis and designrdquo IEEEASME Transactionson Mechatronics vol 14 no 4 pp 446ndash455 2009

[3] R Antonello R Oboe L Prandi and F Biganzoli ldquoAuto-matic mode matching in MEMS vibrating gyroscopes usingextremum-seeking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 10 pp 3880ndash3891 2009

[4] S Park R Horowitz S K Hong and Y Nam ldquoTrajectory-switching algorithm for aMEMS gyroscoperdquo IEEE Transactionson Instrumentation and Measurement vol 56 no 6 pp 2561ndash2569 2007

[5] J D John and T Vinay ldquoNovel concept of a single-massadaptively controlled triaxial angular rate sensorrdquo IEEE SensorsJournal vol 6 no 3 pp 588ndash595 2006

[6] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009

[7] J Fei and F Chowdhury ldquoRobust adaptive controller for triaxialangular velocity sensorrdquo International Journal of InnovativeComputing Information and Control vol 7 no 6 pp 2439ndash2448 2010

[8] L Wang Adaptive Fuzzy Systems and Control-Design andStability Analysis Prentice Hall Upper Saddle River NJ USA1994

[9] Y Guo and P-Y Woo ldquoAn adaptive fuzzy sliding mode con-troller for robotic manipulatorsrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 33 no2 pp 149ndash159 2003

[10] B K Yoo and W C Ham ldquoAdaptive control of robot manip-ulator using fuzzy compensatorrdquo IEEE Transactions on FuzzySystems vol 8 no 2 pp 186ndash199 2000

[11] R-J Wai ldquoFuzzy sliding-mode control using adaptive tuningtechniquerdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 586ndash594 2007

[12] Y-H Chang C-W Chang C-L Chen and C-W Tao ldquoFuzzysliding-mode formation control for multirobot systems designand implementationrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 42 no 2 pp 444ndash457 2012

[13] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinearsystems using online T-S fuzzy-neural modeling approachrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 41 no 2 pp 542ndash552 2011

[14] M C M Teixeira and S H Zak ldquoStabilizing controller designfor uncertain nonlinear systems using fuzzy modelsrdquo IEEETransactions on Fuzzy Systems vol 7 no 2 pp 133ndash142 1999

[15] C-W Park and Y-W Cho ldquoT-S model based indirect adaptivefuzzy control using online parameter estimationrdquo IEEE Trans-actions on Systems Man and Cybernetics Part B Cyberneticsvol 34 no 6 pp 2293ndash2302 2004

[16] S F Asokanthan and T Wang ldquoNonlinear instabilities in avibratory gyroscope subjected to angular speed fluctuationsrdquoNonlinear Dynamics vol 54 no 1-2 pp 69ndash78 2008

[17] T Wang Nonlinear and stochastic dynamics of MEMS-basedangular rate sensing and switching systems [PhD thesis] TheUniversity of Western of Ontario Ontario Canada 2009

[18] Z Chi M Jia and Q Xu ldquoFuzzy PID feedback control ofpiezoelectric actuator with feedforward compensationrdquo Math-ematical Problems in Engineering vol 2014 Article ID 10718414 pages 2014

[19] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics Part B vol 15 no 1 pp 116ndash1321985

[20] P Ioannou and J Sun Robust Adaptive Control Prentice HallUpper Saddle River NJ USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

[2] S Sung W-T Sung C Kim S Yun and Y J Lee ldquoOnthe mode-matched control of MEMS vibratory gyroscope viaphase-domain analysis and designrdquo IEEEASME Transactionson Mechatronics vol 14 no 4 pp 446ndash455 2009

[3] R Antonello R Oboe L Prandi and F Biganzoli ldquoAuto-matic mode matching in MEMS vibrating gyroscopes usingextremum-seeking controlrdquo IEEE Transactions on IndustrialElectronics vol 56 no 10 pp 3880ndash3891 2009

[4] S Park R Horowitz S K Hong and Y Nam ldquoTrajectory-switching algorithm for aMEMS gyroscoperdquo IEEE Transactionson Instrumentation and Measurement vol 56 no 6 pp 2561ndash2569 2007

[5] J D John and T Vinay ldquoNovel concept of a single-massadaptively controlled triaxial angular rate sensorrdquo IEEE SensorsJournal vol 6 no 3 pp 588ndash595 2006

[6] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009

[7] J Fei and F Chowdhury ldquoRobust adaptive controller for triaxialangular velocity sensorrdquo International Journal of InnovativeComputing Information and Control vol 7 no 6 pp 2439ndash2448 2010

[8] L Wang Adaptive Fuzzy Systems and Control-Design andStability Analysis Prentice Hall Upper Saddle River NJ USA1994

[9] Y Guo and P-Y Woo ldquoAn adaptive fuzzy sliding mode con-troller for robotic manipulatorsrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 33 no2 pp 149ndash159 2003

[10] B K Yoo and W C Ham ldquoAdaptive control of robot manip-ulator using fuzzy compensatorrdquo IEEE Transactions on FuzzySystems vol 8 no 2 pp 186ndash199 2000

[11] R-J Wai ldquoFuzzy sliding-mode control using adaptive tuningtechniquerdquo IEEE Transactions on Industrial Electronics vol 54no 1 pp 586ndash594 2007

[12] Y-H Chang C-W Chang C-L Chen and C-W Tao ldquoFuzzysliding-mode formation control for multirobot systems designand implementationrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 42 no 2 pp 444ndash457 2012

[13] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinearsystems using online T-S fuzzy-neural modeling approachrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 41 no 2 pp 542ndash552 2011

[14] M C M Teixeira and S H Zak ldquoStabilizing controller designfor uncertain nonlinear systems using fuzzy modelsrdquo IEEETransactions on Fuzzy Systems vol 7 no 2 pp 133ndash142 1999

[15] C-W Park and Y-W Cho ldquoT-S model based indirect adaptivefuzzy control using online parameter estimationrdquo IEEE Trans-actions on Systems Man and Cybernetics Part B Cyberneticsvol 34 no 6 pp 2293ndash2302 2004

[16] S F Asokanthan and T Wang ldquoNonlinear instabilities in avibratory gyroscope subjected to angular speed fluctuationsrdquoNonlinear Dynamics vol 54 no 1-2 pp 69ndash78 2008

[17] T Wang Nonlinear and stochastic dynamics of MEMS-basedangular rate sensing and switching systems [PhD thesis] TheUniversity of Western of Ontario Ontario Canada 2009

[18] Z Chi M Jia and Q Xu ldquoFuzzy PID feedback control ofpiezoelectric actuator with feedforward compensationrdquo Math-ematical Problems in Engineering vol 2014 Article ID 10718414 pages 2014

[19] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics Part B vol 15 no 1 pp 116ndash1321985

[20] P Ioannou and J Sun Robust Adaptive Control Prentice HallUpper Saddle River NJ USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of