research article adaptive fuzzy sliding mode control of mems...

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)


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)


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)


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


119889119909119910


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


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


minus202

minus202

x(120583

m)

y(120583

m)


Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3


minus100

10

minus10

0

10

x(120583

ms

)y

(120583m

s)


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)


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

)



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


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



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)



= (2119878 minus 119863) + (Ω2

119911minus 1198701) 119902 + 119906 (4)


= (119860 + Δ119860) + (119861 + Δ119861) 119902 + 119862119906 + 119889 (119905) (5)




2

119911minus 1198701


= 119860 + 119861119902 + 119862119906 + 119875 ( 119902 119905) (6)





1 Δ119861 = 119862119867

2

119889(119905) = 1198621198673 where 119867







119877119906119897119890 119894 IF 119894is 1198601119894and

119894is 1198602119894and 119909

119894is 1198603119894and 119910

119894is 1198604119894

THEN ( 119902 | 120579119875)119894is 119861119894 119894 = 1 2 119872


( 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


1 1198602 1198603 1198604 respectively



119894

N Z P119894

N Z P119909119894

N Z P119910119894


x

0 2 4 60

02

04

06

08

1

minus2minus4minus6

times10minus6

Mem

bers

hip

func

tion

degr

ee



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)


119894 119909119894 and 119910


119894



119889= 119860119909sin(120596119909119905) 119910119889= 119860119910sin(120596119910119905) in the 119909 and 119910


119889= 119860119889119902119889 (9)

where 119902119889= [119909119889

119910119889] 119860119889= [

minus1205962

1199090

0 minus1205962

119910


119890 = 119902 minus 119902119889 (10)


119904 = 119890 + 12057211989011989911198981+ 12057311989011989821198992 (11)


where 120572 gt 0 120573 gt 01198981gt 1198991gt 0119898


1198981 11989911198982 1198992are odd


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


1 11989911198982 1198992and 1198961 1198962 119886 119887


119906eq = 119862minus1


= 119862minus1


119875)]

(13)


119904 = 119890 + 120572(1198991

1198981

) 11989011989911198981minus1

+ 120573(1198982

1198992

) 11989011989821198992minus1

(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)


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)


119906119904= minus119862minus1

119870119904 (17)


119906 = 119906eq + 119906119904 (18)


119890 = minus 119889= 119860 + 119861119902 + 119862119906 + 119875 ( 119902 119905) minus

119889 (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


119889


1|119904|119886 sgn (119904)

minus 1198962|119904|119887 sgn (119904) minus 120572(

1198991

1198981

) 11989011989911198981minus1

minus 120573(1198982

1198992

)

sdot 11989011989821198992minus1

(20)



1|119904|119886 sgn (119904)

minus 1198962|119904|119887 sgn (119904)

(21)


120579lowast

119875= arg min [sup 1003816100381610038161003816

1003816 ( 119902 | 120579

119875) minus 119875 ( 119902 119905)

10038161003816100381610038161003816]

120579119875isin Ω119875 119902 isin 119877

2times2

(22)


119875


119908 = 119875 ( 119902 119905) minus ( 119902 | 120579119875) (23)


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)


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


119875= minus119903119904

119879

120583 ( 119902) (26)

Namely

120579119875119909

= 119903119904 (1)120583119909( 119902)

120579119875119910

= 119903119904 (2)120583119910( 119902)

(27)

where 119875= minus

120579119875


119881 =

1

2

(119904119879

119904 +

1

119903

120593119879

119875120593119875) (28)



= 119904119879

119908 minus 119904119879

119870119904 minus 1198961119904119879

|119904|119886 sgn (119904)

minus 1198962119904119879

|119904|119887 sgn (119904)

(29)


4 Simulation Study






119889119909119909


119889119910119910


119889119909119910


119896119909119909

= 8098Nm

119896119910119910

= 7162Nm

119896119909119910

= 5Nm

Ω119911= 50 rads

(30)


119900=




119889= 12 sin(511119905)


119860 = [

minus0075 00025


]

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


x(120583

m)

y(120583

m)


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)







Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3

minus100

10

minus20

0

20


x(120583

ms

)y

(120583m

s)


Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3


minus202

minus202

x(120583

m)

y(120583

m)


Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3


minus100

10

minus10

0

10

x(120583

ms

)y

(120583m

s)





Time (s)0 05 1 15 2 25 3

minus02

0

02

04

06

08

Method 1Method 2

e x(120583

m)


minus020

02040608

Time (s)0 05 1 15 2 25 3

Method 1Method 2

e y(120583

m)


minus15

minus10

minus5

0

5

Time (s)0 05 1 15 2 25 3

Method 1Method 2

e x(120583

ms

)


Time (s)0 05 1 15 2 25 3

Method 1Method 2

minus20

minus15

minus10

minus5

0

5

e y(120583

ms

)




Competing Interests


Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











= (2119878 minus 119863) + (Ω2

119911minus 1198701) 119902 + 119906 (4)


= (119860 + Δ119860) + (119861 + Δ119861) 119902 + 119862119906 + 119889 (119905) (5)




2

119911minus 1198701


= 119860 + 119861119902 + 119862119906 + 119875 ( 119902 119905) (6)





1 Δ119861 = 119862119867

2

119889(119905) = 1198621198673 where 119867







119877119906119897119890 119894 IF 119894is 1198601119894and

119894is 1198602119894and 119909

119894is 1198603119894and 119910

119894is 1198604119894

THEN ( 119902 | 120579119875)119894is 119861119894 119894 = 1 2 119872


( 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


1 1198602 1198603 1198604 respectively



119894

N Z P119894

N Z P119909119894

N Z P119910119894


x

0 2 4 60

02

04

06

08

1

minus2minus4minus6

times10minus6

Mem

bers

hip

func

tion

degr

ee



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)


119894 119909119894 and 119910


119894



119889= 119860119909sin(120596119909119905) 119910119889= 119860119910sin(120596119910119905) in the 119909 and 119910


119889= 119860119889119902119889 (9)

where 119902119889= [119909119889

119910119889] 119860119889= [

minus1205962

1199090

0 minus1205962

119910


119890 = 119902 minus 119902119889 (10)


119904 = 119890 + 12057211989011989911198981+ 12057311989011989821198992 (11)


where 120572 gt 0 120573 gt 01198981gt 1198991gt 0119898


1198981 11989911198982 1198992are odd


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


1 11989911198982 1198992and 1198961 1198962 119886 119887


119906eq = 119862minus1


= 119862minus1


119875)]

(13)


119904 = 119890 + 120572(1198991

1198981

) 11989011989911198981minus1

+ 120573(1198982

1198992

) 11989011989821198992minus1

(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)


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)


119906119904= minus119862minus1

119870119904 (17)


119906 = 119906eq + 119906119904 (18)


119890 = minus 119889= 119860 + 119861119902 + 119862119906 + 119875 ( 119902 119905) minus

119889 (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


119889


1|119904|119886 sgn (119904)

minus 1198962|119904|119887 sgn (119904) minus 120572(

1198991

1198981

) 11989011989911198981minus1

minus 120573(1198982

1198992

)

sdot 11989011989821198992minus1

(20)



1|119904|119886 sgn (119904)

minus 1198962|119904|119887 sgn (119904)

(21)


120579lowast

119875= arg min [sup 1003816100381610038161003816

1003816 ( 119902 | 120579

119875) minus 119875 ( 119902 119905)

10038161003816100381610038161003816]

120579119875isin Ω119875 119902 isin 119877

2times2

(22)


119875


119908 = 119875 ( 119902 119905) minus ( 119902 | 120579119875) (23)


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)


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


119875= minus119903119904

119879

120583 ( 119902) (26)

Namely

120579119875119909

= 119903119904 (1)120583119909( 119902)

120579119875119910

= 119903119904 (2)120583119910( 119902)

(27)

where 119875= minus

120579119875


119881 =

1

2

(119904119879

119904 +

1

119903

120593119879

119875120593119875) (28)



= 119904119879

119908 minus 119904119879

119870119904 minus 1198961119904119879

|119904|119886 sgn (119904)

minus 1198962119904119879

|119904|119887 sgn (119904)

(29)


4 Simulation Study






119889119909119909


119889119910119910


119889119909119910


119896119909119909

= 8098Nm

119896119910119910

= 7162Nm

119896119909119910

= 5Nm

Ω119911= 50 rads

(30)


119900=




119889= 12 sin(511119905)


119860 = [

minus0075 00025


]

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


x(120583

m)

y(120583

m)


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)







Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3

minus100

10

minus20

0

20


x(120583

ms

)y

(120583m

s)


Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3


minus202

minus202

x(120583

m)

y(120583

m)


Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3


minus100

10

minus10

0

10

x(120583

ms

)y

(120583m

s)





Time (s)0 05 1 15 2 25 3

minus02

0

02

04

06

08

Method 1Method 2

e x(120583

m)


minus020

02040608

Time (s)0 05 1 15 2 25 3

Method 1Method 2

e y(120583

m)


minus15

minus10

minus5

0

5

Time (s)0 05 1 15 2 25 3

Method 1Method 2

e x(120583

ms

)


Time (s)0 05 1 15 2 25 3

Method 1Method 2

minus20

minus15

minus10

minus5

0

5

e y(120583

ms

)




Competing Interests


Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where 120572 gt 0 120573 gt 01198981gt 1198991gt 0119898


1198981 11989911198982 1198992are odd


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


1 11989911198982 1198992and 1198961 1198962 119886 119887


119906eq = 119862minus1


= 119862minus1


119875)]

(13)


119904 = 119890 + 120572(1198991

1198981

) 11989011989911198981minus1

+ 120573(1198982

1198992

) 11989011989821198992minus1

(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)


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)


119906119904= minus119862minus1

119870119904 (17)


119906 = 119906eq + 119906119904 (18)


119890 = minus 119889= 119860 + 119861119902 + 119862119906 + 119875 ( 119902 119905) minus

119889 (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


119889


1|119904|119886 sgn (119904)

minus 1198962|119904|119887 sgn (119904) minus 120572(

1198991

1198981

) 11989011989911198981minus1

minus 120573(1198982

1198992

)

sdot 11989011989821198992minus1

(20)



1|119904|119886 sgn (119904)

minus 1198962|119904|119887 sgn (119904)

(21)


120579lowast

119875= arg min [sup 1003816100381610038161003816

1003816 ( 119902 | 120579

119875) minus 119875 ( 119902 119905)

10038161003816100381610038161003816]

120579119875isin Ω119875 119902 isin 119877

2times2

(22)


119875


119908 = 119875 ( 119902 119905) minus ( 119902 | 120579119875) (23)


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)


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


119875= minus119903119904

119879

120583 ( 119902) (26)

Namely

120579119875119909

= 119903119904 (1)120583119909( 119902)

120579119875119910

= 119903119904 (2)120583119910( 119902)

(27)

where 119875= minus

120579119875


119881 =

1

2

(119904119879

119904 +

1

119903

120593119879

119875120593119875) (28)



= 119904119879

119908 minus 119904119879

119870119904 minus 1198961119904119879

|119904|119886 sgn (119904)

minus 1198962119904119879

|119904|119887 sgn (119904)

(29)


4 Simulation Study






119889119909119909


119889119910119910


119889119909119910


119896119909119909

= 8098Nm

119896119910119910

= 7162Nm

119896119909119910

= 5Nm

Ω119911= 50 rads

(30)


119900=




119889= 12 sin(511119905)


119860 = [

minus0075 00025


]

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


x(120583

m)

y(120583

m)


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)







Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3

minus100

10

minus20

0

20


x(120583

ms

)y

(120583m

s)


Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3


minus202

minus202

x(120583

m)

y(120583

m)


Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3


minus100

10

minus10

0

10

x(120583

ms

)y

(120583m

s)





Time (s)0 05 1 15 2 25 3

minus02

0

02

04

06

08

Method 1Method 2

e x(120583

m)


minus020

02040608

Time (s)0 05 1 15 2 25 3

Method 1Method 2

e y(120583

m)


minus15

minus10

minus5

0

5

Time (s)0 05 1 15 2 25 3

Method 1Method 2

e x(120583

ms

)


Time (s)0 05 1 15 2 25 3

Method 1Method 2

minus20

minus15

minus10

minus5

0

5

e y(120583

ms

)




Competing Interests


Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











= 119904119879

119908 minus 119904119879

119870119904 minus 1198961119904119879

|119904|119886 sgn (119904)

minus 1198962119904119879

|119904|119887 sgn (119904)

(29)


4 Simulation Study






119889119909119909


119889119910119910


119889119909119910


119896119909119909

= 8098Nm

119896119910119910

= 7162Nm

119896119909119910

= 5Nm

Ω119911= 50 rads

(30)


119900=




119889= 12 sin(511119905)


119860 = [

minus0075 00025


]

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


x(120583

m)

y(120583

m)


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)







Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3

minus100

10

minus20

0

20


x(120583

ms

)y

(120583m

s)


Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3


minus202

minus202

x(120583

m)

y(120583

m)


Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3


minus100

10

minus10

0

10

x(120583

ms

)y

(120583m

s)





Time (s)0 05 1 15 2 25 3

minus02

0

02

04

06

08

Method 1Method 2

e x(120583

m)


minus020

02040608

Time (s)0 05 1 15 2 25 3

Method 1Method 2

e y(120583

m)


minus15

minus10

minus5

0

5

Time (s)0 05 1 15 2 25 3

Method 1Method 2

e x(120583

ms

)


Time (s)0 05 1 15 2 25 3

Method 1Method 2

minus20

minus15

minus10

minus5

0

5

e y(120583

ms

)




Competing Interests


Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3

minus100

10

minus20

0

20


x(120583

ms

)y

(120583m

s)


Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3


minus202

minus202

x(120583

m)

y(120583

m)


Time (s)0 05 1 15 2 25 3

Time (s)0 05 1 15 2 25 3


minus100

10

minus10

0

10

x(120583

ms

)y

(120583m

s)





Time (s)0 05 1 15 2 25 3

minus02

0

02

04

06

08

Method 1Method 2

e x(120583

m)


minus020

02040608

Time (s)0 05 1 15 2 25 3

Method 1Method 2

e y(120583

m)


minus15

minus10

minus5

0

5

Time (s)0 05 1 15 2 25 3

Method 1Method 2

e x(120583

ms

)


Time (s)0 05 1 15 2 25 3

Method 1Method 2

minus20

minus15

minus10

minus5

0

5

e y(120583

ms

)




Competing Interests


Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Competing Interests


Acknowledgments


References





























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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