Research ArticleAdaptive Fuzzy Sliding Mode Control of MEMS Gyroscope withFinite Time Convergence
Jianxin Ren Rui Zhang and Bin Xu
School of Automation Northwestern Polytechnical University Xirsquoan 710129 China
Correspondence should be addressed to Bin Xu smilefacebinxugmailcom
Received 25 April 2016 Accepted 18 July 2016
Academic Editor Jose A Somolinos
Copyright copy 2016 Jianxin Ren 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
This paper presents adaptive fuzzy finite time sliding mode control of microelectromechanical system gyroscope with uncertaintyand external disturbance Firstly fuzzy system is employed to approximate the uncertainty nonlinear dynamics Secondly nonlinearsliding mode hypersurface and double exponential reaching law are selected to design the finite time convergent sliding modecontroller Thirdly based on Lyapunov methods adaptive laws are presented to adjust the fuzzy weights and the system can beguaranteed to be stable Finally the effectiveness of the proposed method is verified with simulation
1 Introduction
MEMS gyroscopes have become themost growingmicrosen-sors in recent years due to the characteristics of compact sizelow cost andhigh sensitivityMostMEMSgyroscopes sales inthemarket are vibrating siliconmicromechanical gyroscopeswhose basic principle is to generate and detect Coriolis EffectAs depicted in Figure 1 under assumption that the proofmass119898 of gyroscope rotates around 119911-axis at a speed of Ω andmakes uniform motion along the 119909-axis at a speed of ^ aCoriolis force of = minus2119898Ω times ^ is produced along 119910-axis
In the last few years numerous advanced controlapproaches with intelligent design have been studied torealize the trajectory tracking in [1ndash4] and to handle thesystem parametric uncertainties and disturbances and theadaptive control can be found in [5ndash7] For control of MEMSgyroscope Park and Horowitz firstly applied adaptive statefeedback control method [8] Both drive shaft and sensitiveaxis were subjected to feedback control force in this controlmethod which administered two axial modal vibration trackspecified reference trajectories weakening the boundarybetween drive mode and test mode as well
Sliding mode control changes its structure to force thesystem in accordance with a predetermined trajectory Baturet al developed a sliding mode control for MEMS gyroscopesystem in [9] Since then adaptive sliding mode control
approach with the advantages of variable structure methodsand adaptive control strategies are presented to controlMEMS gyroscopes in [10 11]
Due to the necessity of ideal sliding mode good dynamicquality and high robustness several methods are extendedto improve the performance Yu and Man investigated anonlinear sliding mode hypersurface to ensure that systemsfrom any point of the sliding mode surface were able to reachthe balance point in a limited time in [12ndash16] Bartoszewicz[17] examined the reaching laws introduced byGao andHungin [18] and proposed an enhanced version of those reachinglaws which was more appropriate for systems subject toconstraints Recently Fallaha et al studied a novel approachwhich allowed chattering reduction on control input whilekeeping tracking performance in steady-state regime [19]This approach consisted of designing a nonlinear reachinglaw by using an exponential function that dynamicallyadapted to the variations of the controlled system Mei andWang in [20] proposed a nonlinear sliding mode surfacewhich converged to the equilibriumpoint with a higher speedthan both linear sliding mode surface and terminal slidingmode surface In addition a new two-power reaching lawwasproposed to make the system move toward the sliding modefaster
As a matter of fact the methods mentioned above arehighly dependent on the structure of the nonlinearity while
Hindawi Publishing CorporationJournal of SensorsVolume 2016 Article ID 1572303 7 pageshttpdxdoiorg10115520161572303
2 Journal of Sensors
z
y
m
x
F = minus2mΩ times
^
^
Ω
Figure 1 Coriolis Effect
currently accurate model is unavailable Thus fuzzy modelhas been widely used to approximate nonlinear objects in[21 22] Robust adaptive sliding mode control with on-lineidentification for the upper bounds of external disturbanceand estimator for the nonlinear dynamics of MEMS gyro-scope uncertainty parameters was proposed in [23]
In this paper an adaptive fuzzy sliding mode controlstrategy with nonlinear sliding mode hypersurface and dou-ble exponential reaching law is developed to track MEMSgyroscope Furthermore it converges faster compared withstrategies using conventional slidingmode surface in [23] andterminal sliding mode surface in [12ndash16]
The rest of this paper is organized as follows Thedynamics of MEMS gyroscope with parametric uncertaintiesand disturbances are given in Section 2 Controller designand stability analysis are discussed in Section 3 Numericalsimulations are conducted to verify the superiority of theproposed approach in Section 4 comparedwith conventionaladaptive fuzzy sliding mode control Conclusions are drawnin Section 5
2 Dynamics of MEMS Gyroscope
The basic principle of 119911-axis vibratory MEMS gyroscope isshown in Figure 2 which can be described as a quality-stiffness-damping system Owing to mechanical couplingcaused by fabrication imperfections the dynamics can bederived as
119898 + 119889119909119909 + (119889
119909119910minus 2119898Ω
lowast
119911) + (119896
119909119909minus 119898Ω
lowast2
119911) 119909
+ 119896119909119910119910 = 119906
lowast
119909
119898 + 119889119909119909 + (119889
119909119910+ 2119898Ω
lowast
119911) + (119896
119910119910minus 119898Ω
lowast2
119911) 119910
+ 119896119909119910119909 = 119906
lowast
119910
(1)
where 119898 is the mass of proof mass Ωlowast119911is the input angular
velocity 119909 119910 represent the system generalized coordinates119889119909119909 119889119910119910
represent damping terms 119889119909119910represents asymmetric
damping term 119896119909119909 119896119910119910
represent spring terms 119896119909119910represents
m
x
y
z
dxx
2
dxx
2
dyy
2
dyy
2
kxx2
kxx2
kyy
2
kyy
2
Ωlowastz
Figure 2 The basic principle diagram of 119911-axis vibratory MEMSgyroscope
asymmetric spring terms and 119906lowast
119909 119906lowast119910represent the control
forcesOn issues related to the study of mechanism the law
described by model is required to be independent of dimen-sions So it is necessary to establish nondimensional vectordynamics Because of the nondimensional time 119905
lowast
= 120596119900119905
both sides of (1) should be divided by reference frequency 1205962119900
reference length 119902119900 and referencemass119898Then the dynamics
can be rewritten in vector forms
119902lowast
119902119900
+
119863lowast
119898120596119900
119902lowast
119902119900
+ 2
119878lowast
120596119900
119902lowast
119902119900
minus
Ωlowast2
119911
1205962
119900
119902lowast
119902119900
+
119870lowast
1
1198981205962
119900
119902lowast
119902119900
=
119906lowast
1198981205962
119900119902119900
(2)
where 119902lowast = [119909
119910 ] 119906lowast = [
119906lowast
119909
119906lowast
119910
] 119863lowast = [
119889119909119909119889119909119910
119889119909119910119889119910119910
] 119878lowast = [0 minusΩ
lowast
119911
Ωlowast
1199110]
119870lowast
1= [
119896119909119909119896119909119910
119896119909119910119896119910119910
]New parameters are defined as follows
119902 =
119902lowast
119902119900
119906 =
119906lowast
1198981205962
119900119902119900
Ω119911=
Ωlowast
119911
120596119900
119863 =
119863lowast
119898120596119900
1198701=
119870lowast
1
1198981205962
119900
119878 = minus
119878lowast
120596119900
(3)
Journal of Sensors 3
Thus the final form of the nondimensional vector dynamicsis
= (2119878 minus 119863) + (Ω2
119911minus 1198701) 119902 + 119906 (4)
In presence of parametric uncertainties and externaldisturbance based on (4) state equation of dynamics isestablished as
= (119860 + Δ119860) + (119861 + Δ119861) 119902 + 119862119906 + 119889 (119905) (5)
where 119860 isin 1198772times2 119861 isin 119877
2times2 119862 isin 1198772times2 are system known
matrices Δ119860 Δ119861 are parametric uncertainties and 119889(119905) is anexternal disturbance Besides 119860 = 2119878 minus 119863 119861 = Ω
2
119911minus 1198701
If the system total interference (consisting of parametricuncertainties and external disturbance) is represented by119875(119905) we know
= 119860 + 119861119902 + 119862119906 + 119875 ( 119902 119905) (6)
where 119875( 119902 119905) = Δ119860 + Δ119861119902 + 119889(119905)It is vital that (6) must meet the following assumptions
Assumption 1 The total interference 119875( 119902 119905) le 119875119888 where
119875119888is an unknown positive vector
Assumption 2 The total interference 119875( 119902 119905) meets slidingmode matching conditions namely Δ119860 = 119862119867
1 Δ119861 = 119862119867
2
119889(119905) = 1198621198673 where 119867
1 1198672 1198673are unknown matrices with
appropriate dimensions
Assumption 3 119860 119861 are observability matrices
Based on the above assumptions the controller can bedesigned to compensate the total interference
3 Adaptive Fuzzy Finite Time SlidingMode Control
The fuzzy model of 119875(119905) could be composed of 119872 IF-THENrules and the 119894th rule has the form
119877119906119897119890 119894 IF 119894is 1198601119894and
119894is 1198602119894and 119909
119894is 1198603119894and 119910
119894is 1198604119894
THEN ( 119902 | 120579119875)119894is 119861119894 119894 = 1 2 119872
Based on singleton fuzzifier product inference and center-average defuzzifier its output can be expressed as
( 119902 | 120579119875) = 120579119879
119875120583 ( 119902) (7)
where 120583( 119902) = (1205781198601119894
times 1205781198602119894
times 1205781198603119894
times 1205781198604119894
)(sum119872
119895=11205781198601119895
times 1205781198602119895
times
1205781198603119895
times 1205781198604119895
) 1205781198601119894
1205781198602119894
1205781198603119894
1205781198604119894
are membership functionvalues of the fuzzy variables 119909 119910 with respect to fuzzysets 119860
1 1198602 1198603 1198604 respectively
The fuzzy sets of input variables are defined as 119873 119885 119875where 119873 is negative 119885 is zero and 119875 is positive Then
Table 1 Fuzzy control rules
119894
N Z P119894
N Z P119909119894
N Z P119910119894
N Z PN negative Z zero P positive
x
0 2 4 60
02
04
06
08
1
minus2minus4minus6
times10minus6
Mem
bers
hip
func
tion
degr
ee
Figure 3 The membership functions for 119894
membership functions of are selected as the followingtriangular functions
120578119873(119894) =
1 119909 le minus3
minus
1
3
119909 minus3 le 119909 le 0
120578119885(119894) =
1
3
119909 + 1 minus3 le 119909 le 0
minus
1
3
119909 + 1 0 le 119909 le 3
120578119875(119894) =
1 119909 ge 3
1
3
119909 0 le 119909 le 3
(8)
The corresponding membership functions of these fuzzysets labels are depicted in Figure 3 In addition the member-ship functions of
119894 119909119894 and 119910
119894are the same with
119894
Based on the aforementioned fuzzy sets and membershipfunctions the fuzzy rules are described in Table 1 Therefore81 fuzzy rules are chosen
The control target for MEMS gyroscope is to maintainthe proof mass oscillation at given frequency and amplitudesuch as 119909
119889= 119860119909sin(120596119909119905) 119910119889= 119860119910sin(120596119910119905) in the 119909 and 119910
directions respectively So reference model can be designedas
119889= 119860119889119902119889 (9)
where 119902119889= [119909119889
119910119889] 119860119889= [
minus1205962
1199090
0 minus1205962
119910
]And the tracking error is defined as
119890 = 119902 minus 119902119889 (10)
Nonlinear sliding mode hypersurface is chosen as
119904 = 119890 + 12057211989011989911198981+ 12057311989011989821198992 (11)
4 Journal of Sensors
where 120572 gt 0 120573 gt 01198981gt 1198991gt 0119898
2gt 1198992gt 0 what is more
1198981 11989911198982 1198992are odd
Then the reaching law is designed as the following doubleexponential function
119904 = minus1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904) (12)
where 1198961gt 0 119896
2gt 0 0 lt 119886 lt 1 119887 gt 1
It should be noted that the converge speed depends onparameters such as 120572 120573119898
1 11989911198982 1198992and 1198961 1198962 119886 119887
According to (6) equivalent control law is obtained as
119906eq = 119862minus1
( minus 119860 minus 119861119902 minus ( 119902 | 120579119875))
= 119862minus1
[(119889+ 119890) minus 119860 minus 119861119902 minus ( 119902 | 120579
119875)]
(13)
And the derivative of sliding surface (11) is
119904 = 119890 + 120572(1198991
1198981
) 11989011989911198981minus1
+ 120573(1198982
1198992
) 11989011989821198992minus1
(14)
Then substituting (12) into (14)
119890 = minus1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904) minus 120572(
1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
) 11989011989821198992minus1
(15)
And substituting (15) into (13)
119906eq = 119862minus1
[119889minus 1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904)
minus 120572(1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
) 11989011989821198992minus1
minus 119860 minus 119861119902
minus ( 119902 | 120579119875)]
(16)
Besides a robust item is designed to guarantee that the systemis asymptotically stable
119906119904= minus119862minus1
119870119904 (17)
Thus the adaptive fuzzy finite time sliding mode controller isobtained as
119906 = 119906eq + 119906119904 (18)
According to (10) we have
119890 = minus 119889= 119860 + 119861119902 + 119862119906 + 119875 ( 119902 119905) minus
119889 (19)
Substituting (18) into (19)
119890 = 119860 + 119861119902 + [119889minus 1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904)
minus 120572(1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
) 11989011989821198992minus1
] minus 119860
minus 119861119902 minus ( 119902 | 120579119875) minus 119870119904 + 119875 ( 119902 119905) minus
119889
= 119875 ( 119902 119905) minus ( 119902 | 120579119875) minus 119870119904 minus 119896
1|119904|119886 sgn (119904)
minus 1198962|119904|119887 sgn (119904) minus 120572(
1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
)
sdot 11989011989821198992minus1
(20)
Substituting (20) into (14)
119904 = 119875 ( 119902 119905) minus ( 119902 | 120579119875) minus 119870119904 minus 119896
1|119904|119886 sgn (119904)
minus 1198962|119904|119887 sgn (119904)
(21)
The optimal parameters are set as
120579lowast
119875= arg min [sup 1003816100381610038161003816
1003816 ( 119902 | 120579
119875) minus 119875 ( 119902 119905)
10038161003816100381610038161003816]
120579119875isin Ω119875 119902 isin 119877
2times2
(22)
whereΩ119875is a set of 120579
119875
And the minimum approximation errors are defined as
119908 = 119875 ( 119902 119905) minus ( 119902 | 120579119875) (23)
Substituting (23) into (21) we derive
119904 = ( 119902 | 120579lowast
119875) minus ( 119902 | 120579
119875) + 119908 minus 119870119904
minus 1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904)
(24)
Considering (7) (24) can be expressed as
119904 = 120593119879
119875120583 ( 119902) + 119908 minus 119870119904 minus 119896
1|119904|119886 sgn (119904)
minus 1198962|119904|119887 sgn (119904)
(25)
where 120593119875= 120579lowast
119875minus 120579119875
So adaptive law can be selected as
119875= minus119903119904
119879
120583 ( 119902) (26)
Namely
120579119875119909
= 119903119904 (1)120583119909( 119902)
120579119875119910
= 119903119904 (2)120583119910( 119902)
(27)
where 119875= minus
120579119875
Lyapunov function is defined as
119881 =
1
2
(119904119879
119904 +
1
119903
120593119879
119875120593119875) (28)
Journal of Sensors 5
Differentiate119881with respect to time yields and substitute (26)as
= 119904119879
119908 minus 119904119879
119870119904 minus 1198961119904119879
|119904|119886 sgn (119904)
minus 1198962119904119879
|119904|119887 sgn (119904)
(29)
Owing to the fuzzy approximation theory adaptive fuzzysystem can approximate nonlinear system closely Therefore le 0 namely the system is asymptotically stable
4 Simulation Study
In this section numerical simulations are investigated totrack the position and speed trajectories ofMEMS gyroscopecompensate parametric uncertainties and external distur-bances and verify the superiority of the proposed approachcompared with conventional adaptive fuzzy sliding modecontrol strategy using linear sliding mode surface Those twomethods are defined as follows
Method 1 Define the adaptive fuzzy sliding mode controlproposed in this paper as Method 1 whose sliding modesurface is shown in (11) and the reaching law is expressed in(12)
Method 2 Define the conventional adaptive fuzzy slidingmode control as Method 2 whose sliding mode surface is= 119890 + 120573119890 and the reaching law is 119904 = 0
Parameters of the MEMS gyroscope are as follows
119898 = 057 times 10minus8 kg
119889119909119909
= 0429 times 10minus6Nsm
119889119910119910
= 00429 times 10minus6Nsm
119889119909119910
= 00429 times 10minus6Nsm
119896119909119909
= 8098Nm
119896119910119910
= 7162Nm
119896119909119910
= 5Nm
Ω119911= 50 rads
(30)
Since the position of proof mass ranges within the scopeof submillimeter and the natural frequency is generally inthe range of kilohertz assume that reference length is 119902
119900=
10 times 10minus6m reference frequency is 120596
119900= 1 kHz and the
reference trajectories are 119909119889= sin(671119905) 119910
119889= 12 sin(511119905)
respectivelyThen set other simulation parameters as
119860 = [
minus0075 00025
minus00175 minus00075
]
119861 = [
minus14207 minus877
minus877 minus12564
]
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
minus202
minus202
Reference positionPosition tracking
x(120583
m)
y(120583
m)
Figure 4 Position tracking of Method 1
119862 = [
1 0
0 1
]
119870 = [
1000 0
0 1000
]
120572 = [
10 0
0 10
]
120573 = [
10 0
0 10
]
1198981= 3
1198991= 2
1198982= 3
1198992= 1
119875 (119905) = [
32 times 10minus6
5 times 10minus6
+ 5 times 10minus6 sin (511 (119905 + 03))
]
119903 = 001
119886 = 05
119887 = 10
1198961= 1000
1198962= 1000
(31)
And select the initial state values of the system as[08 0 1 0]
119879Then the position and speed trajectories of Method 1 are
shown in Figures 4 and 5 and those of Method 2 are depictedin Figures 6 and 7
The position tracking error and speed tracking error ofMethods 1 and 2 are shown in Figures 8ndash11 respectively
Through the tracking simulation of MEMS gyroscopethe proposed approach is with satisfying performance in
6 Journal of Sensors
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
minus100
10
minus20
0
20
Reference speedSpeed tracking
x(120583
ms
)y
(120583m
s)
Figure 5 Speed tracking of Method 1
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
Reference positionPosition tracking
minus202
minus202
x(120583
m)
y(120583
m)
Figure 6 Position tracking of Method 2
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
Reference speedSpeed tracking
minus100
10
minus10
0
10
x(120583
ms
)y
(120583m
s)
Figure 7 Speed tracking of Method 2
addition in comparison to Method 2 the convergence timeis shortened to 0310158401015840 from 0610158401015840
5 Conclusion and Future Work
An adaptive fuzzy finite time sliding mode control strategyusing nonlinear sliding mode hypersurface and double expo-nential reaching law is proposed to compensate parametricuncertainties and external disturbance of MEMS gyroscopein this paper Based on Lyapunov methods the stability ofsystem can be guaranteed Simulations verify that compared
Time (s)0 05 1 15 2 25 3
minus02
0
02
04
06
08
Method 1Method 2
e x(120583
m)
Figure 8 Position tracking error of gyroscope 119909
minus020
02040608
Time (s)0 05 1 15 2 25 3
Method 1Method 2
e y(120583
m)
Figure 9 Position tracking error of gyroscope 119910
minus15
minus10
minus5
0
5
Time (s)0 05 1 15 2 25 3
Method 1Method 2
e x(120583
ms
)
Figure 10 Speed tracking error of gyroscope 119909
Time (s)0 05 1 15 2 25 3
Method 1Method 2
minus20
minus15
minus10
minus5
0
5
e y(120583
ms
)
Figure 11 Speed tracking error of gyroscope 119910
Journal of Sensors 7
with conventional adaptive fuzzy linear sliding mode controlstrategy the convergence time of finite time convergent con-trol strategy proposed in this paper is shortened to 0310158401015840 from0610158401015840 namely convergence has been significantly improvedFor future work the novel adaptive online constructing fuzzyalgorithm [24] can be employed for more efficient learningwhile disturbance observer based design [25 26] can beconsidered to improve system performance
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by National Natural ScienceFoundation of China (61304098 60204005 and 60974109)Aeronautical Science Foundation of China (2015ZA53003)Natural Science Basic Research Plan in ShaanxiProvince (2014JQ8326 2015JM6272 and 2016KJXX-86)Fundamental Research Funds for the Central Universities(3102015AX001 3102015BJ(II)CG017) Fundamental ResearchFunds of Shenzhen Science and Technology Project(JCYJ20160229172341417) and International Science andTechnology Cooperation Program of China under Grant no2014DFA11580
References
[1] M Chen and S S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 no 12 pp 7706ndash7716 2015
[2] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015
[3] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 pp201ndash209 2015
[4] Q Yang S Jagannathan and Y Sun ldquoRobust integral of neuralnetwork and error sign control of MIMO nonlinear systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 12 pp 3278ndash3286 2015
[5] B Xu Q Zhang and Y Pan ldquoNeural network based dynamicsurface control of hypersonic flight dynamics using small-gaintheoremrdquo Neurocomputing vol 173 pp 690ndash699 2016
[6] Y-J Liu and S Tong ldquoAdaptive NN tracking control of uncer-tain nonlinear discrete-time systems with nonaffine dead-zoneinputrdquo IEEE Transactions on Cybernetics vol 45 no 3 pp 497ndash505 2015
[7] W He Y Chen and Z Yin ldquoAdaptive neural network controlof an uncertain robot with full-state constraintsrdquo IEEE Transac-tions on Cybernetics vol 46 no 3 pp 620ndash629 2016
[8] S Park and R Horowitz ldquoNew adaptive mode of operationfor MEMS gyroscopesrdquo ASME Journal of Dynamic SystemsMeasurement and Control vol 126 no 4 pp 800ndash810 2004
[9] C Batur T Sreeramreddy and Q Khasawneh ldquoSliding modecontrol of a simulatedMEMS gyroscoperdquo ISA Transactions vol45 no 1 pp 99ndash108 2006
[10] A Ebrahimi ldquoRegulated model-based and non-model-basedsliding mode control of a MEMS vibratory gyroscoperdquo Journalof Mechanical Science and Technology vol 28 no 6 pp 2343ndash2349 2014
[11] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009
[12] Z A K Michail ldquoTerminal attractors in neural networksrdquoNeural Networks vol 2 no 2 pp 259ndash274 1989
[13] X Yu and M Zhihong ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 49 no 2 pp 261ndash264 2002
[14] X Yu and Z Man ldquoTerminal sliding mode control of MIMOsystemsrdquo IEEE Trans on Circuits Systems I vol 44 no 11 pp1065ndash1070 1997
[15] AGhanbari andM RMoghanni-Bavil-Olyaei ldquoAdaptive fuzzyterminal sliding-mode control of MEMS z-axis gyroscope withextended Kalman filter observerrdquo Systems Science amp ControlEngineering vol 2 pp 183ndash191 2014
[16] W F Yan S X Hou Y M Fang and J Fei ldquoRobust adaptivenonsingular terminal slidingmode control ofMEMS gyroscopeusing fuzzy-neural-network compensatorrdquo International Jour-nal of Machine Learning and Cybernetics 2016
[17] A Bartoszewicz ldquoA new reaching law for sliding mode controlof continuous time systemswith constraintsrdquoTransactions of theInstitute of Measurement and Control vol 37 no 4 pp 515ndash5212015
[18] W Gao and J C Hung ldquoVariable structure control of nonlinearsystems a new approachrdquo IEEE Transactions on IndustrialElectronics vol 40 no 1 pp 45ndash55 1993
[19] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011
[20] H Mei and Y Wang ldquoFast convergent sliding mode variablestructure control of robotrdquo Information and Control vol 38 no5 pp 552ndash557 2009
[21] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992
[22] X Zeng andM G Singh ldquoSingh approximation theory of fuzzysystems-MIMO caserdquo IEEE Transactions on Fuzzy Systems vol3 no 2 pp 219ndash235 1995
[23] J Fei and S Wang ldquoRobust adaptive sliding mode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014
[24] NWang andM J Er ldquoDirect adaptive fuzzy tracking control ofmarine vehicles with fully unknown parametric dynamics anduncertaintiesrdquo IEEE Transactions on Control Systems Technol-ogy vol 24 no 5 pp 1845ndash1852 2016
[25] B Xu F Sun Y Pan and B Chen ldquoDisturbance observer basedcomposite learning fuzzy control of nonlinear systems withunknown dead zonerdquo IEEE Transactions on Systems Man andCybernetics Systems 2016
[26] B Xu ldquoDisturbance observer-based dynamic surface control oftransport aircraft with continuous heavy cargo airdroprdquo IEEETransactions on Systems Man and Cybernetics Systems 2016
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
2 Journal of Sensors
z
y
m
x
F = minus2mΩ times
^
^
Ω
Figure 1 Coriolis Effect
currently accurate model is unavailable Thus fuzzy modelhas been widely used to approximate nonlinear objects in[21 22] Robust adaptive sliding mode control with on-lineidentification for the upper bounds of external disturbanceand estimator for the nonlinear dynamics of MEMS gyro-scope uncertainty parameters was proposed in [23]
In this paper an adaptive fuzzy sliding mode controlstrategy with nonlinear sliding mode hypersurface and dou-ble exponential reaching law is developed to track MEMSgyroscope Furthermore it converges faster compared withstrategies using conventional slidingmode surface in [23] andterminal sliding mode surface in [12ndash16]
The rest of this paper is organized as follows Thedynamics of MEMS gyroscope with parametric uncertaintiesand disturbances are given in Section 2 Controller designand stability analysis are discussed in Section 3 Numericalsimulations are conducted to verify the superiority of theproposed approach in Section 4 comparedwith conventionaladaptive fuzzy sliding mode control Conclusions are drawnin Section 5
2 Dynamics of MEMS Gyroscope
The basic principle of 119911-axis vibratory MEMS gyroscope isshown in Figure 2 which can be described as a quality-stiffness-damping system Owing to mechanical couplingcaused by fabrication imperfections the dynamics can bederived as
119898 + 119889119909119909 + (119889
119909119910minus 2119898Ω
lowast
119911) + (119896
119909119909minus 119898Ω
lowast2
119911) 119909
+ 119896119909119910119910 = 119906
lowast
119909
119898 + 119889119909119909 + (119889
119909119910+ 2119898Ω
lowast
119911) + (119896
119910119910minus 119898Ω
lowast2
119911) 119910
+ 119896119909119910119909 = 119906
lowast
119910
(1)
where 119898 is the mass of proof mass Ωlowast119911is the input angular
velocity 119909 119910 represent the system generalized coordinates119889119909119909 119889119910119910
represent damping terms 119889119909119910represents asymmetric
damping term 119896119909119909 119896119910119910
represent spring terms 119896119909119910represents
m
x
y
z
dxx
2
dxx
2
dyy
2
dyy
2
kxx2
kxx2
kyy
2
kyy
2
Ωlowastz
Figure 2 The basic principle diagram of 119911-axis vibratory MEMSgyroscope
asymmetric spring terms and 119906lowast
119909 119906lowast119910represent the control
forcesOn issues related to the study of mechanism the law
described by model is required to be independent of dimen-sions So it is necessary to establish nondimensional vectordynamics Because of the nondimensional time 119905
lowast
= 120596119900119905
both sides of (1) should be divided by reference frequency 1205962119900
reference length 119902119900 and referencemass119898Then the dynamics
can be rewritten in vector forms
119902lowast
119902119900
+
119863lowast
119898120596119900
119902lowast
119902119900
+ 2
119878lowast
120596119900
119902lowast
119902119900
minus
Ωlowast2
119911
1205962
119900
119902lowast
119902119900
+
119870lowast
1
1198981205962
119900
119902lowast
119902119900
=
119906lowast
1198981205962
119900119902119900
(2)
where 119902lowast = [119909
119910 ] 119906lowast = [
119906lowast
119909
119906lowast
119910
] 119863lowast = [
119889119909119909119889119909119910
119889119909119910119889119910119910
] 119878lowast = [0 minusΩ
lowast
119911
Ωlowast
1199110]
119870lowast
1= [
119896119909119909119896119909119910
119896119909119910119896119910119910
]New parameters are defined as follows
119902 =
119902lowast
119902119900
119906 =
119906lowast
1198981205962
119900119902119900
Ω119911=
Ωlowast
119911
120596119900
119863 =
119863lowast
119898120596119900
1198701=
119870lowast
1
1198981205962
119900
119878 = minus
119878lowast
120596119900
(3)
Journal of Sensors 3
Thus the final form of the nondimensional vector dynamicsis
= (2119878 minus 119863) + (Ω2
119911minus 1198701) 119902 + 119906 (4)
In presence of parametric uncertainties and externaldisturbance based on (4) state equation of dynamics isestablished as
= (119860 + Δ119860) + (119861 + Δ119861) 119902 + 119862119906 + 119889 (119905) (5)
where 119860 isin 1198772times2 119861 isin 119877
2times2 119862 isin 1198772times2 are system known
matrices Δ119860 Δ119861 are parametric uncertainties and 119889(119905) is anexternal disturbance Besides 119860 = 2119878 minus 119863 119861 = Ω
2
119911minus 1198701
If the system total interference (consisting of parametricuncertainties and external disturbance) is represented by119875(119905) we know
= 119860 + 119861119902 + 119862119906 + 119875 ( 119902 119905) (6)
where 119875( 119902 119905) = Δ119860 + Δ119861119902 + 119889(119905)It is vital that (6) must meet the following assumptions
Assumption 1 The total interference 119875( 119902 119905) le 119875119888 where
119875119888is an unknown positive vector
Assumption 2 The total interference 119875( 119902 119905) meets slidingmode matching conditions namely Δ119860 = 119862119867
1 Δ119861 = 119862119867
2
119889(119905) = 1198621198673 where 119867
1 1198672 1198673are unknown matrices with
appropriate dimensions
Assumption 3 119860 119861 are observability matrices
Based on the above assumptions the controller can bedesigned to compensate the total interference
3 Adaptive Fuzzy Finite Time SlidingMode Control
The fuzzy model of 119875(119905) could be composed of 119872 IF-THENrules and the 119894th rule has the form
119877119906119897119890 119894 IF 119894is 1198601119894and
119894is 1198602119894and 119909
119894is 1198603119894and 119910
119894is 1198604119894
THEN ( 119902 | 120579119875)119894is 119861119894 119894 = 1 2 119872
Based on singleton fuzzifier product inference and center-average defuzzifier its output can be expressed as
( 119902 | 120579119875) = 120579119879
119875120583 ( 119902) (7)
where 120583( 119902) = (1205781198601119894
times 1205781198602119894
times 1205781198603119894
times 1205781198604119894
)(sum119872
119895=11205781198601119895
times 1205781198602119895
times
1205781198603119895
times 1205781198604119895
) 1205781198601119894
1205781198602119894
1205781198603119894
1205781198604119894
are membership functionvalues of the fuzzy variables 119909 119910 with respect to fuzzysets 119860
1 1198602 1198603 1198604 respectively
The fuzzy sets of input variables are defined as 119873 119885 119875where 119873 is negative 119885 is zero and 119875 is positive Then
Table 1 Fuzzy control rules
119894
N Z P119894
N Z P119909119894
N Z P119910119894
N Z PN negative Z zero P positive
x
0 2 4 60
02
04
06
08
1
minus2minus4minus6
times10minus6
Mem
bers
hip
func
tion
degr
ee
Figure 3 The membership functions for 119894
membership functions of are selected as the followingtriangular functions
120578119873(119894) =
1 119909 le minus3
minus
1
3
119909 minus3 le 119909 le 0
120578119885(119894) =
1
3
119909 + 1 minus3 le 119909 le 0
minus
1
3
119909 + 1 0 le 119909 le 3
120578119875(119894) =
1 119909 ge 3
1
3
119909 0 le 119909 le 3
(8)
The corresponding membership functions of these fuzzysets labels are depicted in Figure 3 In addition the member-ship functions of
119894 119909119894 and 119910
119894are the same with
119894
Based on the aforementioned fuzzy sets and membershipfunctions the fuzzy rules are described in Table 1 Therefore81 fuzzy rules are chosen
The control target for MEMS gyroscope is to maintainthe proof mass oscillation at given frequency and amplitudesuch as 119909
119889= 119860119909sin(120596119909119905) 119910119889= 119860119910sin(120596119910119905) in the 119909 and 119910
directions respectively So reference model can be designedas
119889= 119860119889119902119889 (9)
where 119902119889= [119909119889
119910119889] 119860119889= [
minus1205962
1199090
0 minus1205962
119910
]And the tracking error is defined as
119890 = 119902 minus 119902119889 (10)
Nonlinear sliding mode hypersurface is chosen as
119904 = 119890 + 12057211989011989911198981+ 12057311989011989821198992 (11)
4 Journal of Sensors
where 120572 gt 0 120573 gt 01198981gt 1198991gt 0119898
2gt 1198992gt 0 what is more
1198981 11989911198982 1198992are odd
Then the reaching law is designed as the following doubleexponential function
119904 = minus1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904) (12)
where 1198961gt 0 119896
2gt 0 0 lt 119886 lt 1 119887 gt 1
It should be noted that the converge speed depends onparameters such as 120572 120573119898
1 11989911198982 1198992and 1198961 1198962 119886 119887
According to (6) equivalent control law is obtained as
119906eq = 119862minus1
( minus 119860 minus 119861119902 minus ( 119902 | 120579119875))
= 119862minus1
[(119889+ 119890) minus 119860 minus 119861119902 minus ( 119902 | 120579
119875)]
(13)
And the derivative of sliding surface (11) is
119904 = 119890 + 120572(1198991
1198981
) 11989011989911198981minus1
+ 120573(1198982
1198992
) 11989011989821198992minus1
(14)
Then substituting (12) into (14)
119890 = minus1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904) minus 120572(
1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
) 11989011989821198992minus1
(15)
And substituting (15) into (13)
119906eq = 119862minus1
[119889minus 1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904)
minus 120572(1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
) 11989011989821198992minus1
minus 119860 minus 119861119902
minus ( 119902 | 120579119875)]
(16)
Besides a robust item is designed to guarantee that the systemis asymptotically stable
119906119904= minus119862minus1
119870119904 (17)
Thus the adaptive fuzzy finite time sliding mode controller isobtained as
119906 = 119906eq + 119906119904 (18)
According to (10) we have
119890 = minus 119889= 119860 + 119861119902 + 119862119906 + 119875 ( 119902 119905) minus
119889 (19)
Substituting (18) into (19)
119890 = 119860 + 119861119902 + [119889minus 1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904)
minus 120572(1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
) 11989011989821198992minus1
] minus 119860
minus 119861119902 minus ( 119902 | 120579119875) minus 119870119904 + 119875 ( 119902 119905) minus
119889
= 119875 ( 119902 119905) minus ( 119902 | 120579119875) minus 119870119904 minus 119896
1|119904|119886 sgn (119904)
minus 1198962|119904|119887 sgn (119904) minus 120572(
1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
)
sdot 11989011989821198992minus1
(20)
Substituting (20) into (14)
119904 = 119875 ( 119902 119905) minus ( 119902 | 120579119875) minus 119870119904 minus 119896
1|119904|119886 sgn (119904)
minus 1198962|119904|119887 sgn (119904)
(21)
The optimal parameters are set as
120579lowast
119875= arg min [sup 1003816100381610038161003816
1003816 ( 119902 | 120579
119875) minus 119875 ( 119902 119905)
10038161003816100381610038161003816]
120579119875isin Ω119875 119902 isin 119877
2times2
(22)
whereΩ119875is a set of 120579
119875
And the minimum approximation errors are defined as
119908 = 119875 ( 119902 119905) minus ( 119902 | 120579119875) (23)
Substituting (23) into (21) we derive
119904 = ( 119902 | 120579lowast
119875) minus ( 119902 | 120579
119875) + 119908 minus 119870119904
minus 1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904)
(24)
Considering (7) (24) can be expressed as
119904 = 120593119879
119875120583 ( 119902) + 119908 minus 119870119904 minus 119896
1|119904|119886 sgn (119904)
minus 1198962|119904|119887 sgn (119904)
(25)
where 120593119875= 120579lowast
119875minus 120579119875
So adaptive law can be selected as
119875= minus119903119904
119879
120583 ( 119902) (26)
Namely
120579119875119909
= 119903119904 (1)120583119909( 119902)
120579119875119910
= 119903119904 (2)120583119910( 119902)
(27)
where 119875= minus
120579119875
Lyapunov function is defined as
119881 =
1
2
(119904119879
119904 +
1
119903
120593119879
119875120593119875) (28)
Journal of Sensors 5
Differentiate119881with respect to time yields and substitute (26)as
= 119904119879
119908 minus 119904119879
119870119904 minus 1198961119904119879
|119904|119886 sgn (119904)
minus 1198962119904119879
|119904|119887 sgn (119904)
(29)
Owing to the fuzzy approximation theory adaptive fuzzysystem can approximate nonlinear system closely Therefore le 0 namely the system is asymptotically stable
4 Simulation Study
In this section numerical simulations are investigated totrack the position and speed trajectories ofMEMS gyroscopecompensate parametric uncertainties and external distur-bances and verify the superiority of the proposed approachcompared with conventional adaptive fuzzy sliding modecontrol strategy using linear sliding mode surface Those twomethods are defined as follows
Method 1 Define the adaptive fuzzy sliding mode controlproposed in this paper as Method 1 whose sliding modesurface is shown in (11) and the reaching law is expressed in(12)
Method 2 Define the conventional adaptive fuzzy slidingmode control as Method 2 whose sliding mode surface is= 119890 + 120573119890 and the reaching law is 119904 = 0
Parameters of the MEMS gyroscope are as follows
119898 = 057 times 10minus8 kg
119889119909119909
= 0429 times 10minus6Nsm
119889119910119910
= 00429 times 10minus6Nsm
119889119909119910
= 00429 times 10minus6Nsm
119896119909119909
= 8098Nm
119896119910119910
= 7162Nm
119896119909119910
= 5Nm
Ω119911= 50 rads
(30)
Since the position of proof mass ranges within the scopeof submillimeter and the natural frequency is generally inthe range of kilohertz assume that reference length is 119902
119900=
10 times 10minus6m reference frequency is 120596
119900= 1 kHz and the
reference trajectories are 119909119889= sin(671119905) 119910
119889= 12 sin(511119905)
respectivelyThen set other simulation parameters as
119860 = [
minus0075 00025
minus00175 minus00075
]
119861 = [
minus14207 minus877
minus877 minus12564
]
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
minus202
minus202
Reference positionPosition tracking
x(120583
m)
y(120583
m)
Figure 4 Position tracking of Method 1
119862 = [
1 0
0 1
]
119870 = [
1000 0
0 1000
]
120572 = [
10 0
0 10
]
120573 = [
10 0
0 10
]
1198981= 3
1198991= 2
1198982= 3
1198992= 1
119875 (119905) = [
32 times 10minus6
5 times 10minus6
+ 5 times 10minus6 sin (511 (119905 + 03))
]
119903 = 001
119886 = 05
119887 = 10
1198961= 1000
1198962= 1000
(31)
And select the initial state values of the system as[08 0 1 0]
119879Then the position and speed trajectories of Method 1 are
shown in Figures 4 and 5 and those of Method 2 are depictedin Figures 6 and 7
The position tracking error and speed tracking error ofMethods 1 and 2 are shown in Figures 8ndash11 respectively
Through the tracking simulation of MEMS gyroscopethe proposed approach is with satisfying performance in
6 Journal of Sensors
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
minus100
10
minus20
0
20
Reference speedSpeed tracking
x(120583
ms
)y
(120583m
s)
Figure 5 Speed tracking of Method 1
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
Reference positionPosition tracking
minus202
minus202
x(120583
m)
y(120583
m)
Figure 6 Position tracking of Method 2
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
Reference speedSpeed tracking
minus100
10
minus10
0
10
x(120583
ms
)y
(120583m
s)
Figure 7 Speed tracking of Method 2
addition in comparison to Method 2 the convergence timeis shortened to 0310158401015840 from 0610158401015840
5 Conclusion and Future Work
An adaptive fuzzy finite time sliding mode control strategyusing nonlinear sliding mode hypersurface and double expo-nential reaching law is proposed to compensate parametricuncertainties and external disturbance of MEMS gyroscopein this paper Based on Lyapunov methods the stability ofsystem can be guaranteed Simulations verify that compared
Time (s)0 05 1 15 2 25 3
minus02
0
02
04
06
08
Method 1Method 2
e x(120583
m)
Figure 8 Position tracking error of gyroscope 119909
minus020
02040608
Time (s)0 05 1 15 2 25 3
Method 1Method 2
e y(120583
m)
Figure 9 Position tracking error of gyroscope 119910
minus15
minus10
minus5
0
5
Time (s)0 05 1 15 2 25 3
Method 1Method 2
e x(120583
ms
)
Figure 10 Speed tracking error of gyroscope 119909
Time (s)0 05 1 15 2 25 3
Method 1Method 2
minus20
minus15
minus10
minus5
0
5
e y(120583
ms
)
Figure 11 Speed tracking error of gyroscope 119910
Journal of Sensors 7
with conventional adaptive fuzzy linear sliding mode controlstrategy the convergence time of finite time convergent con-trol strategy proposed in this paper is shortened to 0310158401015840 from0610158401015840 namely convergence has been significantly improvedFor future work the novel adaptive online constructing fuzzyalgorithm [24] can be employed for more efficient learningwhile disturbance observer based design [25 26] can beconsidered to improve system performance
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by National Natural ScienceFoundation of China (61304098 60204005 and 60974109)Aeronautical Science Foundation of China (2015ZA53003)Natural Science Basic Research Plan in ShaanxiProvince (2014JQ8326 2015JM6272 and 2016KJXX-86)Fundamental Research Funds for the Central Universities(3102015AX001 3102015BJ(II)CG017) Fundamental ResearchFunds of Shenzhen Science and Technology Project(JCYJ20160229172341417) and International Science andTechnology Cooperation Program of China under Grant no2014DFA11580
References
[1] M Chen and S S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 no 12 pp 7706ndash7716 2015
[2] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015
[3] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 pp201ndash209 2015
[4] Q Yang S Jagannathan and Y Sun ldquoRobust integral of neuralnetwork and error sign control of MIMO nonlinear systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 12 pp 3278ndash3286 2015
[5] B Xu Q Zhang and Y Pan ldquoNeural network based dynamicsurface control of hypersonic flight dynamics using small-gaintheoremrdquo Neurocomputing vol 173 pp 690ndash699 2016
[6] Y-J Liu and S Tong ldquoAdaptive NN tracking control of uncer-tain nonlinear discrete-time systems with nonaffine dead-zoneinputrdquo IEEE Transactions on Cybernetics vol 45 no 3 pp 497ndash505 2015
[7] W He Y Chen and Z Yin ldquoAdaptive neural network controlof an uncertain robot with full-state constraintsrdquo IEEE Transac-tions on Cybernetics vol 46 no 3 pp 620ndash629 2016
[8] S Park and R Horowitz ldquoNew adaptive mode of operationfor MEMS gyroscopesrdquo ASME Journal of Dynamic SystemsMeasurement and Control vol 126 no 4 pp 800ndash810 2004
[9] C Batur T Sreeramreddy and Q Khasawneh ldquoSliding modecontrol of a simulatedMEMS gyroscoperdquo ISA Transactions vol45 no 1 pp 99ndash108 2006
[10] A Ebrahimi ldquoRegulated model-based and non-model-basedsliding mode control of a MEMS vibratory gyroscoperdquo Journalof Mechanical Science and Technology vol 28 no 6 pp 2343ndash2349 2014
[11] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009
[12] Z A K Michail ldquoTerminal attractors in neural networksrdquoNeural Networks vol 2 no 2 pp 259ndash274 1989
[13] X Yu and M Zhihong ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 49 no 2 pp 261ndash264 2002
[14] X Yu and Z Man ldquoTerminal sliding mode control of MIMOsystemsrdquo IEEE Trans on Circuits Systems I vol 44 no 11 pp1065ndash1070 1997
[15] AGhanbari andM RMoghanni-Bavil-Olyaei ldquoAdaptive fuzzyterminal sliding-mode control of MEMS z-axis gyroscope withextended Kalman filter observerrdquo Systems Science amp ControlEngineering vol 2 pp 183ndash191 2014
[16] W F Yan S X Hou Y M Fang and J Fei ldquoRobust adaptivenonsingular terminal slidingmode control ofMEMS gyroscopeusing fuzzy-neural-network compensatorrdquo International Jour-nal of Machine Learning and Cybernetics 2016
[17] A Bartoszewicz ldquoA new reaching law for sliding mode controlof continuous time systemswith constraintsrdquoTransactions of theInstitute of Measurement and Control vol 37 no 4 pp 515ndash5212015
[18] W Gao and J C Hung ldquoVariable structure control of nonlinearsystems a new approachrdquo IEEE Transactions on IndustrialElectronics vol 40 no 1 pp 45ndash55 1993
[19] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011
[20] H Mei and Y Wang ldquoFast convergent sliding mode variablestructure control of robotrdquo Information and Control vol 38 no5 pp 552ndash557 2009
[21] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992
[22] X Zeng andM G Singh ldquoSingh approximation theory of fuzzysystems-MIMO caserdquo IEEE Transactions on Fuzzy Systems vol3 no 2 pp 219ndash235 1995
[23] J Fei and S Wang ldquoRobust adaptive sliding mode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014
[24] NWang andM J Er ldquoDirect adaptive fuzzy tracking control ofmarine vehicles with fully unknown parametric dynamics anduncertaintiesrdquo IEEE Transactions on Control Systems Technol-ogy vol 24 no 5 pp 1845ndash1852 2016
[25] B Xu F Sun Y Pan and B Chen ldquoDisturbance observer basedcomposite learning fuzzy control of nonlinear systems withunknown dead zonerdquo IEEE Transactions on Systems Man andCybernetics Systems 2016
[26] B Xu ldquoDisturbance observer-based dynamic surface control oftransport aircraft with continuous heavy cargo airdroprdquo IEEETransactions on Systems Man and Cybernetics Systems 2016
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Journal of Sensors 3
Thus the final form of the nondimensional vector dynamicsis
= (2119878 minus 119863) + (Ω2
119911minus 1198701) 119902 + 119906 (4)
In presence of parametric uncertainties and externaldisturbance based on (4) state equation of dynamics isestablished as
= (119860 + Δ119860) + (119861 + Δ119861) 119902 + 119862119906 + 119889 (119905) (5)
where 119860 isin 1198772times2 119861 isin 119877
2times2 119862 isin 1198772times2 are system known
matrices Δ119860 Δ119861 are parametric uncertainties and 119889(119905) is anexternal disturbance Besides 119860 = 2119878 minus 119863 119861 = Ω
2
119911minus 1198701
If the system total interference (consisting of parametricuncertainties and external disturbance) is represented by119875(119905) we know
= 119860 + 119861119902 + 119862119906 + 119875 ( 119902 119905) (6)
where 119875( 119902 119905) = Δ119860 + Δ119861119902 + 119889(119905)It is vital that (6) must meet the following assumptions
Assumption 1 The total interference 119875( 119902 119905) le 119875119888 where
119875119888is an unknown positive vector
Assumption 2 The total interference 119875( 119902 119905) meets slidingmode matching conditions namely Δ119860 = 119862119867
1 Δ119861 = 119862119867
2
119889(119905) = 1198621198673 where 119867
1 1198672 1198673are unknown matrices with
appropriate dimensions
Assumption 3 119860 119861 are observability matrices
Based on the above assumptions the controller can bedesigned to compensate the total interference
3 Adaptive Fuzzy Finite Time SlidingMode Control
The fuzzy model of 119875(119905) could be composed of 119872 IF-THENrules and the 119894th rule has the form
119877119906119897119890 119894 IF 119894is 1198601119894and
119894is 1198602119894and 119909
119894is 1198603119894and 119910
119894is 1198604119894
THEN ( 119902 | 120579119875)119894is 119861119894 119894 = 1 2 119872
Based on singleton fuzzifier product inference and center-average defuzzifier its output can be expressed as
( 119902 | 120579119875) = 120579119879
119875120583 ( 119902) (7)
where 120583( 119902) = (1205781198601119894
times 1205781198602119894
times 1205781198603119894
times 1205781198604119894
)(sum119872
119895=11205781198601119895
times 1205781198602119895
times
1205781198603119895
times 1205781198604119895
) 1205781198601119894
1205781198602119894
1205781198603119894
1205781198604119894
are membership functionvalues of the fuzzy variables 119909 119910 with respect to fuzzysets 119860
1 1198602 1198603 1198604 respectively
The fuzzy sets of input variables are defined as 119873 119885 119875where 119873 is negative 119885 is zero and 119875 is positive Then
Table 1 Fuzzy control rules
119894
N Z P119894
N Z P119909119894
N Z P119910119894
N Z PN negative Z zero P positive
x
0 2 4 60
02
04
06
08
1
minus2minus4minus6
times10minus6
Mem
bers
hip
func
tion
degr
ee
Figure 3 The membership functions for 119894
membership functions of are selected as the followingtriangular functions
120578119873(119894) =
1 119909 le minus3
minus
1
3
119909 minus3 le 119909 le 0
120578119885(119894) =
1
3
119909 + 1 minus3 le 119909 le 0
minus
1
3
119909 + 1 0 le 119909 le 3
120578119875(119894) =
1 119909 ge 3
1
3
119909 0 le 119909 le 3
(8)
The corresponding membership functions of these fuzzysets labels are depicted in Figure 3 In addition the member-ship functions of
119894 119909119894 and 119910
119894are the same with
119894
Based on the aforementioned fuzzy sets and membershipfunctions the fuzzy rules are described in Table 1 Therefore81 fuzzy rules are chosen
The control target for MEMS gyroscope is to maintainthe proof mass oscillation at given frequency and amplitudesuch as 119909
119889= 119860119909sin(120596119909119905) 119910119889= 119860119910sin(120596119910119905) in the 119909 and 119910
directions respectively So reference model can be designedas
119889= 119860119889119902119889 (9)
where 119902119889= [119909119889
119910119889] 119860119889= [
minus1205962
1199090
0 minus1205962
119910
]And the tracking error is defined as
119890 = 119902 minus 119902119889 (10)
Nonlinear sliding mode hypersurface is chosen as
119904 = 119890 + 12057211989011989911198981+ 12057311989011989821198992 (11)
4 Journal of Sensors
where 120572 gt 0 120573 gt 01198981gt 1198991gt 0119898
2gt 1198992gt 0 what is more
1198981 11989911198982 1198992are odd
Then the reaching law is designed as the following doubleexponential function
119904 = minus1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904) (12)
where 1198961gt 0 119896
2gt 0 0 lt 119886 lt 1 119887 gt 1
It should be noted that the converge speed depends onparameters such as 120572 120573119898
1 11989911198982 1198992and 1198961 1198962 119886 119887
According to (6) equivalent control law is obtained as
119906eq = 119862minus1
( minus 119860 minus 119861119902 minus ( 119902 | 120579119875))
= 119862minus1
[(119889+ 119890) minus 119860 minus 119861119902 minus ( 119902 | 120579
119875)]
(13)
And the derivative of sliding surface (11) is
119904 = 119890 + 120572(1198991
1198981
) 11989011989911198981minus1
+ 120573(1198982
1198992
) 11989011989821198992minus1
(14)
Then substituting (12) into (14)
119890 = minus1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904) minus 120572(
1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
) 11989011989821198992minus1
(15)
And substituting (15) into (13)
119906eq = 119862minus1
[119889minus 1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904)
minus 120572(1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
) 11989011989821198992minus1
minus 119860 minus 119861119902
minus ( 119902 | 120579119875)]
(16)
Besides a robust item is designed to guarantee that the systemis asymptotically stable
119906119904= minus119862minus1
119870119904 (17)
Thus the adaptive fuzzy finite time sliding mode controller isobtained as
119906 = 119906eq + 119906119904 (18)
According to (10) we have
119890 = minus 119889= 119860 + 119861119902 + 119862119906 + 119875 ( 119902 119905) minus
119889 (19)
Substituting (18) into (19)
119890 = 119860 + 119861119902 + [119889minus 1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904)
minus 120572(1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
) 11989011989821198992minus1
] minus 119860
minus 119861119902 minus ( 119902 | 120579119875) minus 119870119904 + 119875 ( 119902 119905) minus
119889
= 119875 ( 119902 119905) minus ( 119902 | 120579119875) minus 119870119904 minus 119896
1|119904|119886 sgn (119904)
minus 1198962|119904|119887 sgn (119904) minus 120572(
1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
)
sdot 11989011989821198992minus1
(20)
Substituting (20) into (14)
119904 = 119875 ( 119902 119905) minus ( 119902 | 120579119875) minus 119870119904 minus 119896
1|119904|119886 sgn (119904)
minus 1198962|119904|119887 sgn (119904)
(21)
The optimal parameters are set as
120579lowast
119875= arg min [sup 1003816100381610038161003816
1003816 ( 119902 | 120579
119875) minus 119875 ( 119902 119905)
10038161003816100381610038161003816]
120579119875isin Ω119875 119902 isin 119877
2times2
(22)
whereΩ119875is a set of 120579
119875
And the minimum approximation errors are defined as
119908 = 119875 ( 119902 119905) minus ( 119902 | 120579119875) (23)
Substituting (23) into (21) we derive
119904 = ( 119902 | 120579lowast
119875) minus ( 119902 | 120579
119875) + 119908 minus 119870119904
minus 1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904)
(24)
Considering (7) (24) can be expressed as
119904 = 120593119879
119875120583 ( 119902) + 119908 minus 119870119904 minus 119896
1|119904|119886 sgn (119904)
minus 1198962|119904|119887 sgn (119904)
(25)
where 120593119875= 120579lowast
119875minus 120579119875
So adaptive law can be selected as
119875= minus119903119904
119879
120583 ( 119902) (26)
Namely
120579119875119909
= 119903119904 (1)120583119909( 119902)
120579119875119910
= 119903119904 (2)120583119910( 119902)
(27)
where 119875= minus
120579119875
Lyapunov function is defined as
119881 =
1
2
(119904119879
119904 +
1
119903
120593119879
119875120593119875) (28)
Journal of Sensors 5
Differentiate119881with respect to time yields and substitute (26)as
= 119904119879
119908 minus 119904119879
119870119904 minus 1198961119904119879
|119904|119886 sgn (119904)
minus 1198962119904119879
|119904|119887 sgn (119904)
(29)
Owing to the fuzzy approximation theory adaptive fuzzysystem can approximate nonlinear system closely Therefore le 0 namely the system is asymptotically stable
4 Simulation Study
In this section numerical simulations are investigated totrack the position and speed trajectories ofMEMS gyroscopecompensate parametric uncertainties and external distur-bances and verify the superiority of the proposed approachcompared with conventional adaptive fuzzy sliding modecontrol strategy using linear sliding mode surface Those twomethods are defined as follows
Method 1 Define the adaptive fuzzy sliding mode controlproposed in this paper as Method 1 whose sliding modesurface is shown in (11) and the reaching law is expressed in(12)
Method 2 Define the conventional adaptive fuzzy slidingmode control as Method 2 whose sliding mode surface is= 119890 + 120573119890 and the reaching law is 119904 = 0
Parameters of the MEMS gyroscope are as follows
119898 = 057 times 10minus8 kg
119889119909119909
= 0429 times 10minus6Nsm
119889119910119910
= 00429 times 10minus6Nsm
119889119909119910
= 00429 times 10minus6Nsm
119896119909119909
= 8098Nm
119896119910119910
= 7162Nm
119896119909119910
= 5Nm
Ω119911= 50 rads
(30)
Since the position of proof mass ranges within the scopeof submillimeter and the natural frequency is generally inthe range of kilohertz assume that reference length is 119902
119900=
10 times 10minus6m reference frequency is 120596
119900= 1 kHz and the
reference trajectories are 119909119889= sin(671119905) 119910
119889= 12 sin(511119905)
respectivelyThen set other simulation parameters as
119860 = [
minus0075 00025
minus00175 minus00075
]
119861 = [
minus14207 minus877
minus877 minus12564
]
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
minus202
minus202
Reference positionPosition tracking
x(120583
m)
y(120583
m)
Figure 4 Position tracking of Method 1
119862 = [
1 0
0 1
]
119870 = [
1000 0
0 1000
]
120572 = [
10 0
0 10
]
120573 = [
10 0
0 10
]
1198981= 3
1198991= 2
1198982= 3
1198992= 1
119875 (119905) = [
32 times 10minus6
5 times 10minus6
+ 5 times 10minus6 sin (511 (119905 + 03))
]
119903 = 001
119886 = 05
119887 = 10
1198961= 1000
1198962= 1000
(31)
And select the initial state values of the system as[08 0 1 0]
119879Then the position and speed trajectories of Method 1 are
shown in Figures 4 and 5 and those of Method 2 are depictedin Figures 6 and 7
The position tracking error and speed tracking error ofMethods 1 and 2 are shown in Figures 8ndash11 respectively
Through the tracking simulation of MEMS gyroscopethe proposed approach is with satisfying performance in
6 Journal of Sensors
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
minus100
10
minus20
0
20
Reference speedSpeed tracking
x(120583
ms
)y
(120583m
s)
Figure 5 Speed tracking of Method 1
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
Reference positionPosition tracking
minus202
minus202
x(120583
m)
y(120583
m)
Figure 6 Position tracking of Method 2
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
Reference speedSpeed tracking
minus100
10
minus10
0
10
x(120583
ms
)y
(120583m
s)
Figure 7 Speed tracking of Method 2
addition in comparison to Method 2 the convergence timeis shortened to 0310158401015840 from 0610158401015840
5 Conclusion and Future Work
An adaptive fuzzy finite time sliding mode control strategyusing nonlinear sliding mode hypersurface and double expo-nential reaching law is proposed to compensate parametricuncertainties and external disturbance of MEMS gyroscopein this paper Based on Lyapunov methods the stability ofsystem can be guaranteed Simulations verify that compared
Time (s)0 05 1 15 2 25 3
minus02
0
02
04
06
08
Method 1Method 2
e x(120583
m)
Figure 8 Position tracking error of gyroscope 119909
minus020
02040608
Time (s)0 05 1 15 2 25 3
Method 1Method 2
e y(120583
m)
Figure 9 Position tracking error of gyroscope 119910
minus15
minus10
minus5
0
5
Time (s)0 05 1 15 2 25 3
Method 1Method 2
e x(120583
ms
)
Figure 10 Speed tracking error of gyroscope 119909
Time (s)0 05 1 15 2 25 3
Method 1Method 2
minus20
minus15
minus10
minus5
0
5
e y(120583
ms
)
Figure 11 Speed tracking error of gyroscope 119910
Journal of Sensors 7
with conventional adaptive fuzzy linear sliding mode controlstrategy the convergence time of finite time convergent con-trol strategy proposed in this paper is shortened to 0310158401015840 from0610158401015840 namely convergence has been significantly improvedFor future work the novel adaptive online constructing fuzzyalgorithm [24] can be employed for more efficient learningwhile disturbance observer based design [25 26] can beconsidered to improve system performance
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by National Natural ScienceFoundation of China (61304098 60204005 and 60974109)Aeronautical Science Foundation of China (2015ZA53003)Natural Science Basic Research Plan in ShaanxiProvince (2014JQ8326 2015JM6272 and 2016KJXX-86)Fundamental Research Funds for the Central Universities(3102015AX001 3102015BJ(II)CG017) Fundamental ResearchFunds of Shenzhen Science and Technology Project(JCYJ20160229172341417) and International Science andTechnology Cooperation Program of China under Grant no2014DFA11580
References
[1] M Chen and S S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 no 12 pp 7706ndash7716 2015
[2] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015
[3] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 pp201ndash209 2015
[4] Q Yang S Jagannathan and Y Sun ldquoRobust integral of neuralnetwork and error sign control of MIMO nonlinear systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 12 pp 3278ndash3286 2015
[5] B Xu Q Zhang and Y Pan ldquoNeural network based dynamicsurface control of hypersonic flight dynamics using small-gaintheoremrdquo Neurocomputing vol 173 pp 690ndash699 2016
[6] Y-J Liu and S Tong ldquoAdaptive NN tracking control of uncer-tain nonlinear discrete-time systems with nonaffine dead-zoneinputrdquo IEEE Transactions on Cybernetics vol 45 no 3 pp 497ndash505 2015
[7] W He Y Chen and Z Yin ldquoAdaptive neural network controlof an uncertain robot with full-state constraintsrdquo IEEE Transac-tions on Cybernetics vol 46 no 3 pp 620ndash629 2016
[8] S Park and R Horowitz ldquoNew adaptive mode of operationfor MEMS gyroscopesrdquo ASME Journal of Dynamic SystemsMeasurement and Control vol 126 no 4 pp 800ndash810 2004
[9] C Batur T Sreeramreddy and Q Khasawneh ldquoSliding modecontrol of a simulatedMEMS gyroscoperdquo ISA Transactions vol45 no 1 pp 99ndash108 2006
[10] A Ebrahimi ldquoRegulated model-based and non-model-basedsliding mode control of a MEMS vibratory gyroscoperdquo Journalof Mechanical Science and Technology vol 28 no 6 pp 2343ndash2349 2014
[11] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009
[12] Z A K Michail ldquoTerminal attractors in neural networksrdquoNeural Networks vol 2 no 2 pp 259ndash274 1989
[13] X Yu and M Zhihong ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 49 no 2 pp 261ndash264 2002
[14] X Yu and Z Man ldquoTerminal sliding mode control of MIMOsystemsrdquo IEEE Trans on Circuits Systems I vol 44 no 11 pp1065ndash1070 1997
[15] AGhanbari andM RMoghanni-Bavil-Olyaei ldquoAdaptive fuzzyterminal sliding-mode control of MEMS z-axis gyroscope withextended Kalman filter observerrdquo Systems Science amp ControlEngineering vol 2 pp 183ndash191 2014
[16] W F Yan S X Hou Y M Fang and J Fei ldquoRobust adaptivenonsingular terminal slidingmode control ofMEMS gyroscopeusing fuzzy-neural-network compensatorrdquo International Jour-nal of Machine Learning and Cybernetics 2016
[17] A Bartoszewicz ldquoA new reaching law for sliding mode controlof continuous time systemswith constraintsrdquoTransactions of theInstitute of Measurement and Control vol 37 no 4 pp 515ndash5212015
[18] W Gao and J C Hung ldquoVariable structure control of nonlinearsystems a new approachrdquo IEEE Transactions on IndustrialElectronics vol 40 no 1 pp 45ndash55 1993
[19] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011
[20] H Mei and Y Wang ldquoFast convergent sliding mode variablestructure control of robotrdquo Information and Control vol 38 no5 pp 552ndash557 2009
[21] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992
[22] X Zeng andM G Singh ldquoSingh approximation theory of fuzzysystems-MIMO caserdquo IEEE Transactions on Fuzzy Systems vol3 no 2 pp 219ndash235 1995
[23] J Fei and S Wang ldquoRobust adaptive sliding mode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014
[24] NWang andM J Er ldquoDirect adaptive fuzzy tracking control ofmarine vehicles with fully unknown parametric dynamics anduncertaintiesrdquo IEEE Transactions on Control Systems Technol-ogy vol 24 no 5 pp 1845ndash1852 2016
[25] B Xu F Sun Y Pan and B Chen ldquoDisturbance observer basedcomposite learning fuzzy control of nonlinear systems withunknown dead zonerdquo IEEE Transactions on Systems Man andCybernetics Systems 2016
[26] B Xu ldquoDisturbance observer-based dynamic surface control oftransport aircraft with continuous heavy cargo airdroprdquo IEEETransactions on Systems Man and Cybernetics Systems 2016
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 Journal of Sensors
where 120572 gt 0 120573 gt 01198981gt 1198991gt 0119898
2gt 1198992gt 0 what is more
1198981 11989911198982 1198992are odd
Then the reaching law is designed as the following doubleexponential function
119904 = minus1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904) (12)
where 1198961gt 0 119896
2gt 0 0 lt 119886 lt 1 119887 gt 1
It should be noted that the converge speed depends onparameters such as 120572 120573119898
1 11989911198982 1198992and 1198961 1198962 119886 119887
According to (6) equivalent control law is obtained as
119906eq = 119862minus1
( minus 119860 minus 119861119902 minus ( 119902 | 120579119875))
= 119862minus1
[(119889+ 119890) minus 119860 minus 119861119902 minus ( 119902 | 120579
119875)]
(13)
And the derivative of sliding surface (11) is
119904 = 119890 + 120572(1198991
1198981
) 11989011989911198981minus1
+ 120573(1198982
1198992
) 11989011989821198992minus1
(14)
Then substituting (12) into (14)
119890 = minus1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904) minus 120572(
1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
) 11989011989821198992minus1
(15)
And substituting (15) into (13)
119906eq = 119862minus1
[119889minus 1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904)
minus 120572(1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
) 11989011989821198992minus1
minus 119860 minus 119861119902
minus ( 119902 | 120579119875)]
(16)
Besides a robust item is designed to guarantee that the systemis asymptotically stable
119906119904= minus119862minus1
119870119904 (17)
Thus the adaptive fuzzy finite time sliding mode controller isobtained as
119906 = 119906eq + 119906119904 (18)
According to (10) we have
119890 = minus 119889= 119860 + 119861119902 + 119862119906 + 119875 ( 119902 119905) minus
119889 (19)
Substituting (18) into (19)
119890 = 119860 + 119861119902 + [119889minus 1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904)
minus 120572(1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
) 11989011989821198992minus1
] minus 119860
minus 119861119902 minus ( 119902 | 120579119875) minus 119870119904 + 119875 ( 119902 119905) minus
119889
= 119875 ( 119902 119905) minus ( 119902 | 120579119875) minus 119870119904 minus 119896
1|119904|119886 sgn (119904)
minus 1198962|119904|119887 sgn (119904) minus 120572(
1198991
1198981
) 11989011989911198981minus1
minus 120573(1198982
1198992
)
sdot 11989011989821198992minus1
(20)
Substituting (20) into (14)
119904 = 119875 ( 119902 119905) minus ( 119902 | 120579119875) minus 119870119904 minus 119896
1|119904|119886 sgn (119904)
minus 1198962|119904|119887 sgn (119904)
(21)
The optimal parameters are set as
120579lowast
119875= arg min [sup 1003816100381610038161003816
1003816 ( 119902 | 120579
119875) minus 119875 ( 119902 119905)
10038161003816100381610038161003816]
120579119875isin Ω119875 119902 isin 119877
2times2
(22)
whereΩ119875is a set of 120579
119875
And the minimum approximation errors are defined as
119908 = 119875 ( 119902 119905) minus ( 119902 | 120579119875) (23)
Substituting (23) into (21) we derive
119904 = ( 119902 | 120579lowast
119875) minus ( 119902 | 120579
119875) + 119908 minus 119870119904
minus 1198961|119904|119886 sgn (119904) minus 119896
2|119904|119887 sgn (119904)
(24)
Considering (7) (24) can be expressed as
119904 = 120593119879
119875120583 ( 119902) + 119908 minus 119870119904 minus 119896
1|119904|119886 sgn (119904)
minus 1198962|119904|119887 sgn (119904)
(25)
where 120593119875= 120579lowast
119875minus 120579119875
So adaptive law can be selected as
119875= minus119903119904
119879
120583 ( 119902) (26)
Namely
120579119875119909
= 119903119904 (1)120583119909( 119902)
120579119875119910
= 119903119904 (2)120583119910( 119902)
(27)
where 119875= minus
120579119875
Lyapunov function is defined as
119881 =
1
2
(119904119879
119904 +
1
119903
120593119879
119875120593119875) (28)
Journal of Sensors 5
Differentiate119881with respect to time yields and substitute (26)as
= 119904119879
119908 minus 119904119879
119870119904 minus 1198961119904119879
|119904|119886 sgn (119904)
minus 1198962119904119879
|119904|119887 sgn (119904)
(29)
Owing to the fuzzy approximation theory adaptive fuzzysystem can approximate nonlinear system closely Therefore le 0 namely the system is asymptotically stable
4 Simulation Study
In this section numerical simulations are investigated totrack the position and speed trajectories ofMEMS gyroscopecompensate parametric uncertainties and external distur-bances and verify the superiority of the proposed approachcompared with conventional adaptive fuzzy sliding modecontrol strategy using linear sliding mode surface Those twomethods are defined as follows
Method 1 Define the adaptive fuzzy sliding mode controlproposed in this paper as Method 1 whose sliding modesurface is shown in (11) and the reaching law is expressed in(12)
Method 2 Define the conventional adaptive fuzzy slidingmode control as Method 2 whose sliding mode surface is= 119890 + 120573119890 and the reaching law is 119904 = 0
Parameters of the MEMS gyroscope are as follows
119898 = 057 times 10minus8 kg
119889119909119909
= 0429 times 10minus6Nsm
119889119910119910
= 00429 times 10minus6Nsm
119889119909119910
= 00429 times 10minus6Nsm
119896119909119909
= 8098Nm
119896119910119910
= 7162Nm
119896119909119910
= 5Nm
Ω119911= 50 rads
(30)
Since the position of proof mass ranges within the scopeof submillimeter and the natural frequency is generally inthe range of kilohertz assume that reference length is 119902
119900=
10 times 10minus6m reference frequency is 120596
119900= 1 kHz and the
reference trajectories are 119909119889= sin(671119905) 119910
119889= 12 sin(511119905)
respectivelyThen set other simulation parameters as
119860 = [
minus0075 00025
minus00175 minus00075
]
119861 = [
minus14207 minus877
minus877 minus12564
]
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
minus202
minus202
Reference positionPosition tracking
x(120583
m)
y(120583
m)
Figure 4 Position tracking of Method 1
119862 = [
1 0
0 1
]
119870 = [
1000 0
0 1000
]
120572 = [
10 0
0 10
]
120573 = [
10 0
0 10
]
1198981= 3
1198991= 2
1198982= 3
1198992= 1
119875 (119905) = [
32 times 10minus6
5 times 10minus6
+ 5 times 10minus6 sin (511 (119905 + 03))
]
119903 = 001
119886 = 05
119887 = 10
1198961= 1000
1198962= 1000
(31)
And select the initial state values of the system as[08 0 1 0]
119879Then the position and speed trajectories of Method 1 are
shown in Figures 4 and 5 and those of Method 2 are depictedin Figures 6 and 7
The position tracking error and speed tracking error ofMethods 1 and 2 are shown in Figures 8ndash11 respectively
Through the tracking simulation of MEMS gyroscopethe proposed approach is with satisfying performance in
6 Journal of Sensors
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
minus100
10
minus20
0
20
Reference speedSpeed tracking
x(120583
ms
)y
(120583m
s)
Figure 5 Speed tracking of Method 1
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
Reference positionPosition tracking
minus202
minus202
x(120583
m)
y(120583
m)
Figure 6 Position tracking of Method 2
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
Reference speedSpeed tracking
minus100
10
minus10
0
10
x(120583
ms
)y
(120583m
s)
Figure 7 Speed tracking of Method 2
addition in comparison to Method 2 the convergence timeis shortened to 0310158401015840 from 0610158401015840
5 Conclusion and Future Work
An adaptive fuzzy finite time sliding mode control strategyusing nonlinear sliding mode hypersurface and double expo-nential reaching law is proposed to compensate parametricuncertainties and external disturbance of MEMS gyroscopein this paper Based on Lyapunov methods the stability ofsystem can be guaranteed Simulations verify that compared
Time (s)0 05 1 15 2 25 3
minus02
0
02
04
06
08
Method 1Method 2
e x(120583
m)
Figure 8 Position tracking error of gyroscope 119909
minus020
02040608
Time (s)0 05 1 15 2 25 3
Method 1Method 2
e y(120583
m)
Figure 9 Position tracking error of gyroscope 119910
minus15
minus10
minus5
0
5
Time (s)0 05 1 15 2 25 3
Method 1Method 2
e x(120583
ms
)
Figure 10 Speed tracking error of gyroscope 119909
Time (s)0 05 1 15 2 25 3
Method 1Method 2
minus20
minus15
minus10
minus5
0
5
e y(120583
ms
)
Figure 11 Speed tracking error of gyroscope 119910
Journal of Sensors 7
with conventional adaptive fuzzy linear sliding mode controlstrategy the convergence time of finite time convergent con-trol strategy proposed in this paper is shortened to 0310158401015840 from0610158401015840 namely convergence has been significantly improvedFor future work the novel adaptive online constructing fuzzyalgorithm [24] can be employed for more efficient learningwhile disturbance observer based design [25 26] can beconsidered to improve system performance
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by National Natural ScienceFoundation of China (61304098 60204005 and 60974109)Aeronautical Science Foundation of China (2015ZA53003)Natural Science Basic Research Plan in ShaanxiProvince (2014JQ8326 2015JM6272 and 2016KJXX-86)Fundamental Research Funds for the Central Universities(3102015AX001 3102015BJ(II)CG017) Fundamental ResearchFunds of Shenzhen Science and Technology Project(JCYJ20160229172341417) and International Science andTechnology Cooperation Program of China under Grant no2014DFA11580
References
[1] M Chen and S S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 no 12 pp 7706ndash7716 2015
[2] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015
[3] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 pp201ndash209 2015
[4] Q Yang S Jagannathan and Y Sun ldquoRobust integral of neuralnetwork and error sign control of MIMO nonlinear systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 12 pp 3278ndash3286 2015
[5] B Xu Q Zhang and Y Pan ldquoNeural network based dynamicsurface control of hypersonic flight dynamics using small-gaintheoremrdquo Neurocomputing vol 173 pp 690ndash699 2016
[6] Y-J Liu and S Tong ldquoAdaptive NN tracking control of uncer-tain nonlinear discrete-time systems with nonaffine dead-zoneinputrdquo IEEE Transactions on Cybernetics vol 45 no 3 pp 497ndash505 2015
[7] W He Y Chen and Z Yin ldquoAdaptive neural network controlof an uncertain robot with full-state constraintsrdquo IEEE Transac-tions on Cybernetics vol 46 no 3 pp 620ndash629 2016
[8] S Park and R Horowitz ldquoNew adaptive mode of operationfor MEMS gyroscopesrdquo ASME Journal of Dynamic SystemsMeasurement and Control vol 126 no 4 pp 800ndash810 2004
[9] C Batur T Sreeramreddy and Q Khasawneh ldquoSliding modecontrol of a simulatedMEMS gyroscoperdquo ISA Transactions vol45 no 1 pp 99ndash108 2006
[10] A Ebrahimi ldquoRegulated model-based and non-model-basedsliding mode control of a MEMS vibratory gyroscoperdquo Journalof Mechanical Science and Technology vol 28 no 6 pp 2343ndash2349 2014
[11] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009
[12] Z A K Michail ldquoTerminal attractors in neural networksrdquoNeural Networks vol 2 no 2 pp 259ndash274 1989
[13] X Yu and M Zhihong ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 49 no 2 pp 261ndash264 2002
[14] X Yu and Z Man ldquoTerminal sliding mode control of MIMOsystemsrdquo IEEE Trans on Circuits Systems I vol 44 no 11 pp1065ndash1070 1997
[15] AGhanbari andM RMoghanni-Bavil-Olyaei ldquoAdaptive fuzzyterminal sliding-mode control of MEMS z-axis gyroscope withextended Kalman filter observerrdquo Systems Science amp ControlEngineering vol 2 pp 183ndash191 2014
[16] W F Yan S X Hou Y M Fang and J Fei ldquoRobust adaptivenonsingular terminal slidingmode control ofMEMS gyroscopeusing fuzzy-neural-network compensatorrdquo International Jour-nal of Machine Learning and Cybernetics 2016
[17] A Bartoszewicz ldquoA new reaching law for sliding mode controlof continuous time systemswith constraintsrdquoTransactions of theInstitute of Measurement and Control vol 37 no 4 pp 515ndash5212015
[18] W Gao and J C Hung ldquoVariable structure control of nonlinearsystems a new approachrdquo IEEE Transactions on IndustrialElectronics vol 40 no 1 pp 45ndash55 1993
[19] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011
[20] H Mei and Y Wang ldquoFast convergent sliding mode variablestructure control of robotrdquo Information and Control vol 38 no5 pp 552ndash557 2009
[21] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992
[22] X Zeng andM G Singh ldquoSingh approximation theory of fuzzysystems-MIMO caserdquo IEEE Transactions on Fuzzy Systems vol3 no 2 pp 219ndash235 1995
[23] J Fei and S Wang ldquoRobust adaptive sliding mode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014
[24] NWang andM J Er ldquoDirect adaptive fuzzy tracking control ofmarine vehicles with fully unknown parametric dynamics anduncertaintiesrdquo IEEE Transactions on Control Systems Technol-ogy vol 24 no 5 pp 1845ndash1852 2016
[25] B Xu F Sun Y Pan and B Chen ldquoDisturbance observer basedcomposite learning fuzzy control of nonlinear systems withunknown dead zonerdquo IEEE Transactions on Systems Man andCybernetics Systems 2016
[26] B Xu ldquoDisturbance observer-based dynamic surface control oftransport aircraft with continuous heavy cargo airdroprdquo IEEETransactions on Systems Man and Cybernetics Systems 2016
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Journal of Sensors 5
Differentiate119881with respect to time yields and substitute (26)as
= 119904119879
119908 minus 119904119879
119870119904 minus 1198961119904119879
|119904|119886 sgn (119904)
minus 1198962119904119879
|119904|119887 sgn (119904)
(29)
Owing to the fuzzy approximation theory adaptive fuzzysystem can approximate nonlinear system closely Therefore le 0 namely the system is asymptotically stable
4 Simulation Study
In this section numerical simulations are investigated totrack the position and speed trajectories ofMEMS gyroscopecompensate parametric uncertainties and external distur-bances and verify the superiority of the proposed approachcompared with conventional adaptive fuzzy sliding modecontrol strategy using linear sliding mode surface Those twomethods are defined as follows
Method 1 Define the adaptive fuzzy sliding mode controlproposed in this paper as Method 1 whose sliding modesurface is shown in (11) and the reaching law is expressed in(12)
Method 2 Define the conventional adaptive fuzzy slidingmode control as Method 2 whose sliding mode surface is= 119890 + 120573119890 and the reaching law is 119904 = 0
Parameters of the MEMS gyroscope are as follows
119898 = 057 times 10minus8 kg
119889119909119909
= 0429 times 10minus6Nsm
119889119910119910
= 00429 times 10minus6Nsm
119889119909119910
= 00429 times 10minus6Nsm
119896119909119909
= 8098Nm
119896119910119910
= 7162Nm
119896119909119910
= 5Nm
Ω119911= 50 rads
(30)
Since the position of proof mass ranges within the scopeof submillimeter and the natural frequency is generally inthe range of kilohertz assume that reference length is 119902
119900=
10 times 10minus6m reference frequency is 120596
119900= 1 kHz and the
reference trajectories are 119909119889= sin(671119905) 119910
119889= 12 sin(511119905)
respectivelyThen set other simulation parameters as
119860 = [
minus0075 00025
minus00175 minus00075
]
119861 = [
minus14207 minus877
minus877 minus12564
]
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
minus202
minus202
Reference positionPosition tracking
x(120583
m)
y(120583
m)
Figure 4 Position tracking of Method 1
119862 = [
1 0
0 1
]
119870 = [
1000 0
0 1000
]
120572 = [
10 0
0 10
]
120573 = [
10 0
0 10
]
1198981= 3
1198991= 2
1198982= 3
1198992= 1
119875 (119905) = [
32 times 10minus6
5 times 10minus6
+ 5 times 10minus6 sin (511 (119905 + 03))
]
119903 = 001
119886 = 05
119887 = 10
1198961= 1000
1198962= 1000
(31)
And select the initial state values of the system as[08 0 1 0]
119879Then the position and speed trajectories of Method 1 are
shown in Figures 4 and 5 and those of Method 2 are depictedin Figures 6 and 7
The position tracking error and speed tracking error ofMethods 1 and 2 are shown in Figures 8ndash11 respectively
Through the tracking simulation of MEMS gyroscopethe proposed approach is with satisfying performance in
6 Journal of Sensors
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
minus100
10
minus20
0
20
Reference speedSpeed tracking
x(120583
ms
)y
(120583m
s)
Figure 5 Speed tracking of Method 1
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
Reference positionPosition tracking
minus202
minus202
x(120583
m)
y(120583
m)
Figure 6 Position tracking of Method 2
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
Reference speedSpeed tracking
minus100
10
minus10
0
10
x(120583
ms
)y
(120583m
s)
Figure 7 Speed tracking of Method 2
addition in comparison to Method 2 the convergence timeis shortened to 0310158401015840 from 0610158401015840
5 Conclusion and Future Work
An adaptive fuzzy finite time sliding mode control strategyusing nonlinear sliding mode hypersurface and double expo-nential reaching law is proposed to compensate parametricuncertainties and external disturbance of MEMS gyroscopein this paper Based on Lyapunov methods the stability ofsystem can be guaranteed Simulations verify that compared
Time (s)0 05 1 15 2 25 3
minus02
0
02
04
06
08
Method 1Method 2
e x(120583
m)
Figure 8 Position tracking error of gyroscope 119909
minus020
02040608
Time (s)0 05 1 15 2 25 3
Method 1Method 2
e y(120583
m)
Figure 9 Position tracking error of gyroscope 119910
minus15
minus10
minus5
0
5
Time (s)0 05 1 15 2 25 3
Method 1Method 2
e x(120583
ms
)
Figure 10 Speed tracking error of gyroscope 119909
Time (s)0 05 1 15 2 25 3
Method 1Method 2
minus20
minus15
minus10
minus5
0
5
e y(120583
ms
)
Figure 11 Speed tracking error of gyroscope 119910
Journal of Sensors 7
with conventional adaptive fuzzy linear sliding mode controlstrategy the convergence time of finite time convergent con-trol strategy proposed in this paper is shortened to 0310158401015840 from0610158401015840 namely convergence has been significantly improvedFor future work the novel adaptive online constructing fuzzyalgorithm [24] can be employed for more efficient learningwhile disturbance observer based design [25 26] can beconsidered to improve system performance
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by National Natural ScienceFoundation of China (61304098 60204005 and 60974109)Aeronautical Science Foundation of China (2015ZA53003)Natural Science Basic Research Plan in ShaanxiProvince (2014JQ8326 2015JM6272 and 2016KJXX-86)Fundamental Research Funds for the Central Universities(3102015AX001 3102015BJ(II)CG017) Fundamental ResearchFunds of Shenzhen Science and Technology Project(JCYJ20160229172341417) and International Science andTechnology Cooperation Program of China under Grant no2014DFA11580
References
[1] M Chen and S S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 no 12 pp 7706ndash7716 2015
[2] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015
[3] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 pp201ndash209 2015
[4] Q Yang S Jagannathan and Y Sun ldquoRobust integral of neuralnetwork and error sign control of MIMO nonlinear systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 12 pp 3278ndash3286 2015
[5] B Xu Q Zhang and Y Pan ldquoNeural network based dynamicsurface control of hypersonic flight dynamics using small-gaintheoremrdquo Neurocomputing vol 173 pp 690ndash699 2016
[6] Y-J Liu and S Tong ldquoAdaptive NN tracking control of uncer-tain nonlinear discrete-time systems with nonaffine dead-zoneinputrdquo IEEE Transactions on Cybernetics vol 45 no 3 pp 497ndash505 2015
[7] W He Y Chen and Z Yin ldquoAdaptive neural network controlof an uncertain robot with full-state constraintsrdquo IEEE Transac-tions on Cybernetics vol 46 no 3 pp 620ndash629 2016
[8] S Park and R Horowitz ldquoNew adaptive mode of operationfor MEMS gyroscopesrdquo ASME Journal of Dynamic SystemsMeasurement and Control vol 126 no 4 pp 800ndash810 2004
[9] C Batur T Sreeramreddy and Q Khasawneh ldquoSliding modecontrol of a simulatedMEMS gyroscoperdquo ISA Transactions vol45 no 1 pp 99ndash108 2006
[10] A Ebrahimi ldquoRegulated model-based and non-model-basedsliding mode control of a MEMS vibratory gyroscoperdquo Journalof Mechanical Science and Technology vol 28 no 6 pp 2343ndash2349 2014
[11] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009
[12] Z A K Michail ldquoTerminal attractors in neural networksrdquoNeural Networks vol 2 no 2 pp 259ndash274 1989
[13] X Yu and M Zhihong ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 49 no 2 pp 261ndash264 2002
[14] X Yu and Z Man ldquoTerminal sliding mode control of MIMOsystemsrdquo IEEE Trans on Circuits Systems I vol 44 no 11 pp1065ndash1070 1997
[15] AGhanbari andM RMoghanni-Bavil-Olyaei ldquoAdaptive fuzzyterminal sliding-mode control of MEMS z-axis gyroscope withextended Kalman filter observerrdquo Systems Science amp ControlEngineering vol 2 pp 183ndash191 2014
[16] W F Yan S X Hou Y M Fang and J Fei ldquoRobust adaptivenonsingular terminal slidingmode control ofMEMS gyroscopeusing fuzzy-neural-network compensatorrdquo International Jour-nal of Machine Learning and Cybernetics 2016
[17] A Bartoszewicz ldquoA new reaching law for sliding mode controlof continuous time systemswith constraintsrdquoTransactions of theInstitute of Measurement and Control vol 37 no 4 pp 515ndash5212015
[18] W Gao and J C Hung ldquoVariable structure control of nonlinearsystems a new approachrdquo IEEE Transactions on IndustrialElectronics vol 40 no 1 pp 45ndash55 1993
[19] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011
[20] H Mei and Y Wang ldquoFast convergent sliding mode variablestructure control of robotrdquo Information and Control vol 38 no5 pp 552ndash557 2009
[21] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992
[22] X Zeng andM G Singh ldquoSingh approximation theory of fuzzysystems-MIMO caserdquo IEEE Transactions on Fuzzy Systems vol3 no 2 pp 219ndash235 1995
[23] J Fei and S Wang ldquoRobust adaptive sliding mode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014
[24] NWang andM J Er ldquoDirect adaptive fuzzy tracking control ofmarine vehicles with fully unknown parametric dynamics anduncertaintiesrdquo IEEE Transactions on Control Systems Technol-ogy vol 24 no 5 pp 1845ndash1852 2016
[25] B Xu F Sun Y Pan and B Chen ldquoDisturbance observer basedcomposite learning fuzzy control of nonlinear systems withunknown dead zonerdquo IEEE Transactions on Systems Man andCybernetics Systems 2016
[26] B Xu ldquoDisturbance observer-based dynamic surface control oftransport aircraft with continuous heavy cargo airdroprdquo IEEETransactions on Systems Man and Cybernetics Systems 2016
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 Journal of Sensors
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
minus100
10
minus20
0
20
Reference speedSpeed tracking
x(120583
ms
)y
(120583m
s)
Figure 5 Speed tracking of Method 1
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
Reference positionPosition tracking
minus202
minus202
x(120583
m)
y(120583
m)
Figure 6 Position tracking of Method 2
Time (s)0 05 1 15 2 25 3
Time (s)0 05 1 15 2 25 3
Reference speedSpeed tracking
minus100
10
minus10
0
10
x(120583
ms
)y
(120583m
s)
Figure 7 Speed tracking of Method 2
addition in comparison to Method 2 the convergence timeis shortened to 0310158401015840 from 0610158401015840
5 Conclusion and Future Work
An adaptive fuzzy finite time sliding mode control strategyusing nonlinear sliding mode hypersurface and double expo-nential reaching law is proposed to compensate parametricuncertainties and external disturbance of MEMS gyroscopein this paper Based on Lyapunov methods the stability ofsystem can be guaranteed Simulations verify that compared
Time (s)0 05 1 15 2 25 3
minus02
0
02
04
06
08
Method 1Method 2
e x(120583
m)
Figure 8 Position tracking error of gyroscope 119909
minus020
02040608
Time (s)0 05 1 15 2 25 3
Method 1Method 2
e y(120583
m)
Figure 9 Position tracking error of gyroscope 119910
minus15
minus10
minus5
0
5
Time (s)0 05 1 15 2 25 3
Method 1Method 2
e x(120583
ms
)
Figure 10 Speed tracking error of gyroscope 119909
Time (s)0 05 1 15 2 25 3
Method 1Method 2
minus20
minus15
minus10
minus5
0
5
e y(120583
ms
)
Figure 11 Speed tracking error of gyroscope 119910
Journal of Sensors 7
with conventional adaptive fuzzy linear sliding mode controlstrategy the convergence time of finite time convergent con-trol strategy proposed in this paper is shortened to 0310158401015840 from0610158401015840 namely convergence has been significantly improvedFor future work the novel adaptive online constructing fuzzyalgorithm [24] can be employed for more efficient learningwhile disturbance observer based design [25 26] can beconsidered to improve system performance
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by National Natural ScienceFoundation of China (61304098 60204005 and 60974109)Aeronautical Science Foundation of China (2015ZA53003)Natural Science Basic Research Plan in ShaanxiProvince (2014JQ8326 2015JM6272 and 2016KJXX-86)Fundamental Research Funds for the Central Universities(3102015AX001 3102015BJ(II)CG017) Fundamental ResearchFunds of Shenzhen Science and Technology Project(JCYJ20160229172341417) and International Science andTechnology Cooperation Program of China under Grant no2014DFA11580
References
[1] M Chen and S S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 no 12 pp 7706ndash7716 2015
[2] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015
[3] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 pp201ndash209 2015
[4] Q Yang S Jagannathan and Y Sun ldquoRobust integral of neuralnetwork and error sign control of MIMO nonlinear systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 12 pp 3278ndash3286 2015
[5] B Xu Q Zhang and Y Pan ldquoNeural network based dynamicsurface control of hypersonic flight dynamics using small-gaintheoremrdquo Neurocomputing vol 173 pp 690ndash699 2016
[6] Y-J Liu and S Tong ldquoAdaptive NN tracking control of uncer-tain nonlinear discrete-time systems with nonaffine dead-zoneinputrdquo IEEE Transactions on Cybernetics vol 45 no 3 pp 497ndash505 2015
[7] W He Y Chen and Z Yin ldquoAdaptive neural network controlof an uncertain robot with full-state constraintsrdquo IEEE Transac-tions on Cybernetics vol 46 no 3 pp 620ndash629 2016
[8] S Park and R Horowitz ldquoNew adaptive mode of operationfor MEMS gyroscopesrdquo ASME Journal of Dynamic SystemsMeasurement and Control vol 126 no 4 pp 800ndash810 2004
[9] C Batur T Sreeramreddy and Q Khasawneh ldquoSliding modecontrol of a simulatedMEMS gyroscoperdquo ISA Transactions vol45 no 1 pp 99ndash108 2006
[10] A Ebrahimi ldquoRegulated model-based and non-model-basedsliding mode control of a MEMS vibratory gyroscoperdquo Journalof Mechanical Science and Technology vol 28 no 6 pp 2343ndash2349 2014
[11] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009
[12] Z A K Michail ldquoTerminal attractors in neural networksrdquoNeural Networks vol 2 no 2 pp 259ndash274 1989
[13] X Yu and M Zhihong ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 49 no 2 pp 261ndash264 2002
[14] X Yu and Z Man ldquoTerminal sliding mode control of MIMOsystemsrdquo IEEE Trans on Circuits Systems I vol 44 no 11 pp1065ndash1070 1997
[15] AGhanbari andM RMoghanni-Bavil-Olyaei ldquoAdaptive fuzzyterminal sliding-mode control of MEMS z-axis gyroscope withextended Kalman filter observerrdquo Systems Science amp ControlEngineering vol 2 pp 183ndash191 2014
[16] W F Yan S X Hou Y M Fang and J Fei ldquoRobust adaptivenonsingular terminal slidingmode control ofMEMS gyroscopeusing fuzzy-neural-network compensatorrdquo International Jour-nal of Machine Learning and Cybernetics 2016
[17] A Bartoszewicz ldquoA new reaching law for sliding mode controlof continuous time systemswith constraintsrdquoTransactions of theInstitute of Measurement and Control vol 37 no 4 pp 515ndash5212015
[18] W Gao and J C Hung ldquoVariable structure control of nonlinearsystems a new approachrdquo IEEE Transactions on IndustrialElectronics vol 40 no 1 pp 45ndash55 1993
[19] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011
[20] H Mei and Y Wang ldquoFast convergent sliding mode variablestructure control of robotrdquo Information and Control vol 38 no5 pp 552ndash557 2009
[21] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992
[22] X Zeng andM G Singh ldquoSingh approximation theory of fuzzysystems-MIMO caserdquo IEEE Transactions on Fuzzy Systems vol3 no 2 pp 219ndash235 1995
[23] J Fei and S Wang ldquoRobust adaptive sliding mode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014
[24] NWang andM J Er ldquoDirect adaptive fuzzy tracking control ofmarine vehicles with fully unknown parametric dynamics anduncertaintiesrdquo IEEE Transactions on Control Systems Technol-ogy vol 24 no 5 pp 1845ndash1852 2016
[25] B Xu F Sun Y Pan and B Chen ldquoDisturbance observer basedcomposite learning fuzzy control of nonlinear systems withunknown dead zonerdquo IEEE Transactions on Systems Man andCybernetics Systems 2016
[26] B Xu ldquoDisturbance observer-based dynamic surface control oftransport aircraft with continuous heavy cargo airdroprdquo IEEETransactions on Systems Man and Cybernetics Systems 2016
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Journal of Sensors 7
with conventional adaptive fuzzy linear sliding mode controlstrategy the convergence time of finite time convergent con-trol strategy proposed in this paper is shortened to 0310158401015840 from0610158401015840 namely convergence has been significantly improvedFor future work the novel adaptive online constructing fuzzyalgorithm [24] can be employed for more efficient learningwhile disturbance observer based design [25 26] can beconsidered to improve system performance
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work was supported by National Natural ScienceFoundation of China (61304098 60204005 and 60974109)Aeronautical Science Foundation of China (2015ZA53003)Natural Science Basic Research Plan in ShaanxiProvince (2014JQ8326 2015JM6272 and 2016KJXX-86)Fundamental Research Funds for the Central Universities(3102015AX001 3102015BJ(II)CG017) Fundamental ResearchFunds of Shenzhen Science and Technology Project(JCYJ20160229172341417) and International Science andTechnology Cooperation Program of China under Grant no2014DFA11580
References
[1] M Chen and S S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics vol 62 no 12 pp 7706ndash7716 2015
[2] B Xu C Yang and Y Pan ldquoGlobal neural dynamic surfacetracking control of strict-feedback systems with applicationto hypersonic flight vehiclerdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 10 pp 2563ndash25752015
[3] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 pp201ndash209 2015
[4] Q Yang S Jagannathan and Y Sun ldquoRobust integral of neuralnetwork and error sign control of MIMO nonlinear systemsrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 26 no 12 pp 3278ndash3286 2015
[5] B Xu Q Zhang and Y Pan ldquoNeural network based dynamicsurface control of hypersonic flight dynamics using small-gaintheoremrdquo Neurocomputing vol 173 pp 690ndash699 2016
[6] Y-J Liu and S Tong ldquoAdaptive NN tracking control of uncer-tain nonlinear discrete-time systems with nonaffine dead-zoneinputrdquo IEEE Transactions on Cybernetics vol 45 no 3 pp 497ndash505 2015
[7] W He Y Chen and Z Yin ldquoAdaptive neural network controlof an uncertain robot with full-state constraintsrdquo IEEE Transac-tions on Cybernetics vol 46 no 3 pp 620ndash629 2016
[8] S Park and R Horowitz ldquoNew adaptive mode of operationfor MEMS gyroscopesrdquo ASME Journal of Dynamic SystemsMeasurement and Control vol 126 no 4 pp 800ndash810 2004
[9] C Batur T Sreeramreddy and Q Khasawneh ldquoSliding modecontrol of a simulatedMEMS gyroscoperdquo ISA Transactions vol45 no 1 pp 99ndash108 2006
[10] A Ebrahimi ldquoRegulated model-based and non-model-basedsliding mode control of a MEMS vibratory gyroscoperdquo Journalof Mechanical Science and Technology vol 28 no 6 pp 2343ndash2349 2014
[11] J Fei and C Batur ldquoA novel adaptive sliding mode control withapplication to MEMS gyroscoperdquo ISA Transactions vol 48 no1 pp 73ndash78 2009
[12] Z A K Michail ldquoTerminal attractors in neural networksrdquoNeural Networks vol 2 no 2 pp 259ndash274 1989
[13] X Yu and M Zhihong ldquoFast terminal sliding-mode controldesign for nonlinear dynamical systemsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 49 no 2 pp 261ndash264 2002
[14] X Yu and Z Man ldquoTerminal sliding mode control of MIMOsystemsrdquo IEEE Trans on Circuits Systems I vol 44 no 11 pp1065ndash1070 1997
[15] AGhanbari andM RMoghanni-Bavil-Olyaei ldquoAdaptive fuzzyterminal sliding-mode control of MEMS z-axis gyroscope withextended Kalman filter observerrdquo Systems Science amp ControlEngineering vol 2 pp 183ndash191 2014
[16] W F Yan S X Hou Y M Fang and J Fei ldquoRobust adaptivenonsingular terminal slidingmode control ofMEMS gyroscopeusing fuzzy-neural-network compensatorrdquo International Jour-nal of Machine Learning and Cybernetics 2016
[17] A Bartoszewicz ldquoA new reaching law for sliding mode controlof continuous time systemswith constraintsrdquoTransactions of theInstitute of Measurement and Control vol 37 no 4 pp 515ndash5212015
[18] W Gao and J C Hung ldquoVariable structure control of nonlinearsystems a new approachrdquo IEEE Transactions on IndustrialElectronics vol 40 no 1 pp 45ndash55 1993
[19] C J Fallaha M Saad H Y Kanaan and K Al-HaddadldquoSliding-mode robot control with exponential reaching lawrdquoIEEE Transactions on Industrial Electronics vol 58 no 2 pp600ndash610 2011
[20] H Mei and Y Wang ldquoFast convergent sliding mode variablestructure control of robotrdquo Information and Control vol 38 no5 pp 552ndash557 2009
[21] L-X Wang and J M Mendel ldquoFuzzy basis functions universalapproximation and orthogonal least-squares learningrdquo IEEETransactions on Neural Networks vol 3 no 5 pp 807ndash814 1992
[22] X Zeng andM G Singh ldquoSingh approximation theory of fuzzysystems-MIMO caserdquo IEEE Transactions on Fuzzy Systems vol3 no 2 pp 219ndash235 1995
[23] J Fei and S Wang ldquoRobust adaptive sliding mode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014
[24] NWang andM J Er ldquoDirect adaptive fuzzy tracking control ofmarine vehicles with fully unknown parametric dynamics anduncertaintiesrdquo IEEE Transactions on Control Systems Technol-ogy vol 24 no 5 pp 1845ndash1852 2016
[25] B Xu F Sun Y Pan and B Chen ldquoDisturbance observer basedcomposite learning fuzzy control of nonlinear systems withunknown dead zonerdquo IEEE Transactions on Systems Man andCybernetics Systems 2016
[26] B Xu ldquoDisturbance observer-based dynamic surface control oftransport aircraft with continuous heavy cargo airdroprdquo IEEETransactions on Systems Man and Cybernetics Systems 2016
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of