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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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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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
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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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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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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
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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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International 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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International 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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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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
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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
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