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Research ArticleMinimal-Learning-Parameter Technique Based Adaptive NeuralSliding Mode Control of MEMS Gyroscope

Bin Xu and Pengchao Zhang

Shaanxi Provincial Key Laboratory of Industrial Automation Shaanxi University of Technology HanzhongShaanxi 723000 China

Correspondence should be addressed to Bin Xu smilefacebinxugmailcom

Received 15 May 2017 Accepted 27 June 2017 Published 26 July 2017

Academic Editor Yanan Li

Copyright copy 2017 Bin Xu and Pengchao ZhangThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper investigates an adaptive neural sliding mode controller for MEMS gyroscopes with minimal-learning-parametertechnique Considering the system uncertainty in dynamics neural network is employed for approximation Minimal-learning-parameter technique is constructed to decrease the number of update parameters and in this way the computation burdenis greatly reduced Sliding mode control is designed to cancel the effect of time-varying disturbance The closed-loop stabilityanalysis is established via Lyapunov approach Simulation results are presented to demonstrate the effectiveness of themethod

1 Introduction

Recently MEMS gyroscopes have been drawing growingattention because they intend to employ advanced controlapproaches to realize trajectories tracking and to handle sys-tem parametric uncertainties and disturbances These intel-ligent control methods improve the performance of gyro-scopes so that the applications of MEMS gyroscopes areexpanded With the vigorous development of nonlinear sys-tem control methods [1ndash8] a variety of gyroscopic modalcontrol methods emerged

In [9 10] one adaptive operation strategy is presentedto control the MEMS 119911-axis gyroscope which offers a largeroperational bandwidth absence of zero-rate output self-cal-ibration and large robustness to parameter variations causedby fabrication defects and ambient conditions In [11 12]the sliding mode control is proposed to handle the vibratingproofmass which achieves better estimation of the unknownangular velocity than conventional model reference adaptivefeedback controller Since then in the presence of significantuncertainties the regulated model-based and non-model-based sliding model control approaches are presented toimprove tracking control of the drive and sense modes ofthe vibratory gyroscope in [13] In [14] an adaptive tracking

controller with a proportional and integral sliding surface isproposed

Neural network has an inherent ability to learn andapproximate nonlinear functions [15 16] which can be uti-lized for unstructured uncertainties Thus an adaptive con-trol strategy using radial basis function (RBF) networkFuzzyLogic System compensator is presented for robust trackingof MEMS gyroscope in the presence of model uncertaintiesand external disturbances to compensate such system non-linearities and improve the tracking performance in [17ndash19]However in practical application large amount of updateparameters results in the computation burden of onlinelearning In [20] the minimal-learning-parameter techniqueis further incorporated into the high gain observer to greatlyreduce the online computation burden

Inspired by the above-mentioned discussions on design-ing intelligent controllers and reducing the number of onlineparameters this paper will focus on constructing the newcontrol scheme for MEMS gyroscopes to suppress the systemparametric uncertainties and disturbances The main contri-bution of this paper is that a single parameter is employedto replace weight matrix which significantly reduces thecomputation burden

HindawiComplexityVolume 2017 Article ID 6019175 8 pageshttpsdoiorg10115520176019175

2 Complexity

dyy

2

dyy

2

kyy

2

kyy

2

dxx

2

dxx

2

kxx2

kxx2

m

y

x

z

Ωlowastz

Figure 1 The basic principle diagram of 119911-axis MEMS gyroscope

The structure of this paper is organized as followsSection 2 formulates the dynamics of MEMS gyroscopeSection 3 studies the neural network of lumped parametricuncertainties In Section 4 an adaptive neural sliding modecontrol strategy using the minimal-learning-parameter tech-nique is designed and stability analysis is discussed Numer-ical simulations are investigated to verify the superiority ofthe proposed approach in Section 5 Conclusions are given inSection 6

2 Problem Formulation

21 Dynamics of MEMS Gyroscope As Figure 1 shows theidealMEMS gyroscope is a quality-stiffness-damping systemConsidering the mechanical coupling caused by manufac-turing defects the dynamics of MEMS gyroscopes can beexpressed as

119898 + 119889119909119909 + (119889119909119910 minus 2119898Ωlowast119911) 119910 + (119896119909119909 minus 119898Ωlowast2119911 ) 119909+ 119896119909119910119910 = 119906lowast119909

119898 119910 + 119889119909119909 119910 + (119889119909119910 + 2119898Ωlowast119911) + (119896119910119910 minus 119898Ωlowast2119911 ) 119910+ 119896119909119910119909 = 119906lowast119910

(1)

where119898 represents themass of proofmassΩlowast119911 represents theinput angular velocity 119909 and 119910 represent the system general-ized coordinates 119889119909119909 and 119889119910119910 represent damping terms 119889119909119910represents asymmetric damping term 119896119909119909 and 119896119910119910 representspring terms 119896119909119910 represents asymmetric spring terms and 119906lowast119909and 119906lowast119910 represent the control forces

In (1) the damping terms are affected by atmosphericpressure and spring terms are affected by ambient tempera-ture That means

119889119909119909 = 1198891199090 + Δ119889119909119909119889119910119910 = 1198891199100 + Δ119889119910119910

119896119909119909 = 1198961199090 + Δ119896119909119909119896119910119910 = 1198961199100 + Δ119896119910119910119896119909119910 = 1198961199091199100 + Δ119896119909119910

(2)

where 1198891199090 1198891199100 and 1198891199091199100 are the damping terms of MEMSgyroscope in normal atmospheric pressure 1198961199090 1198961199100 and1198961199091199100 are the spring terms of MEMS gyroscope in roomtemperature environment Δ119889119909119909 Δ119889119910119910 and Δ119889119909119910 are thedeviation in damping coefficients due to the changes ofatmospheric pressure and Δ119896119909119909 Δ119896119910119910 and Δ119896119909119910 are thedeviation of damping coefficients because of the changes ofambient temperature

Dividing both sides of (1) by reference mass119898 referencefrequency 1205962119900 and reference length 119902119900 the dynamics can bederived as

qlowast

119902119900 +Dlowast

119898120596119900qlowast

119902119900 + 2Slowast

120596119900qlowast

119902119900 minusΩlowast21199111205962119900

qlowast

119902119900 +Klowast

1198981205962119900qlowast

119902119900= ulowast

1198981205962119900119902119900 (3)

where qlowast = [ 119909119910 ] ulowast = [ 119906lowast119909119906lowast119910 ] Dlowast = [ 119889119909119909 119889119909119910119889119909119910 119889119910119910] Slowast = [ 0 minusΩlowast119911

Ωlowast119911 0]

and Klowast = [ 119896119909119909 119896119909119910119896119909119910 119896119910119910]

Define new parameters as q = qlowast119902119900 u = ulowast1198981205962119900119902119900Ω119911 = Ωlowast119911 120596119900D = Dlowast119898120596119900 K = Klowast1198981205962119900 and S = minusSlowast120596119900Then (3) has the following form

q = (2S minusD) q + (Ω2119911 minus K) q + u (4)

where D = D0 + ΔD K = K0 + ΔK and D0 =[ 1198891199090119898120596119900 11988911990911991001198981205961199001198891199091199100119898120596119900 1198891199100119898120596119900

] K0 = [ 11989611990901198981205962119900 119896119909119910011989812059621199001198961199091199100119898120596

2119900 1198961199100119898120596

2119900

] ΔD =[ Δ119889119909119909119898120596119900 Δ119889119909119910119898120596119900Δ119889119909119910119898120596119900 Δ119889119910119910119898120596119900

] and ΔK = [ Δ1198961199091199091198981205962119900 Δ1198961199091199101198981205962119900Δ119896119909119910119898120596

2119900 Δ119896119910119910119898120596

2119900

]With the external disturbances caused by compound

maneuvers (4) is replaced by

q = (2S minusD0 minus ΔD) q + (Ω2z minus K0 minus ΔK) q + Cu

+ d (t) (5)

where d(t) is the external disturbance and dm = sup|d(t)|Define lumped parametric uncertainties as

P (q q) = minusΔDq minus ΔKq (6)

Equation (5) can be written as

q = Aq + Bq + Cu + P (q q) + d (t) (7)

where A = 2S minusD0 and B = Ω2z minus K0

Complexity 3

Remark 1 In order to make the uncertainty of MEMS gyro-scope depicted in (6) controllable there must be unknownmatrices of appropriate dimensions G H and p(t) such thatΔA = CG ΔB = CH and d(t) = Cp(t) And C is selected asC = [ 1 00 1 ]22 Control Goal The control goal of this paper is to design acontroller to steer the position 119902 and the speed 119902 to the desiredtrajectories qm(119905) = [119909119898 119910119898]119879 and qm(119905) = [119898 119910119898]119879Besides the minimal-learning-parameter technique is fur-ther incorporated into the estimation of lumped parametricuncertainties P(q q)3 Brief Description of RBF Neural Network

A neural network is established to approximate the lumpedparametric uncertainties P(q q) which can be expressed as

P (q q | 120579) = T120583 (q q) (8)

where sube Rn is the adjustable parameter matrix 120583(q q) is anonlinear vector function of the inputs and the RBF has theform

120583i = exp(minus1003817100381710038171003817q minus qmi100381710038171003817100381721205901198942 ) 119894 = 1 2 119899 (9)

where qmi is an 119899-dimensional vector representing the centerof the 119894th basis function and 120590119894 is the variance representingthe spread of the basis function

Suppose that 120579lowast are the optimal weight parametersparametric uncertainties could be reexpressed as

P (q q) = 120579lowastT120583 (q q) + 120576 (10)

where 120576 is the optimal estimation error of RBFneural networkand 120576119899 = sup|120576|

Thus the estimation error can be written as

P minus P = 120579lowastT120583 (q q) + 120576 minus T120583 (q q)= minusT120583 (q q) + 120576

(11)

where = minus 120579lowast4 Adaptive Sliding Mode Control withMinimal-Learning-Parameter Technique

Define the system tracking error as

e (t) = q (t) minus qm (t) (12)

Select the sliding mode function as

s (t) = e (t) + 120573e (t) (13)

where 120573 is satisfied with Hurwitz conditionThe derivative of s(t) iss (t) = e (t) + 120573e (t) = [q (t) minus qm (t)] + 120573e (t)

= Aq + Bq + Cu + P (q q) + d (t) minus qm (t)+ 120573e (t)

(14)

Define 120601 = 120579lowast2 and the estimation error is 120601 = 120601minus120601 where120601 is the estimation of 120601

Assume that s = 0 according to (14) controller could bedesigned as

u = Cminus1 [qm minus Aq minus Bq minus 12 s120601120583T120583 minus m lowast sgn (s)

minus 120573e minus Ks] (15)

where 120578m = sup|120578| 120578 = 120576 + d(t) m lowast sgn(s) is Hadamardproduct item and minusCminus1Ks is a robust item

Remark 2 Compared with traditional results of MEMS con-trol [15ndash17] in this paper minimal-learning-parameter tech-nique is employed for controller design to reduce computa-tion burden

Remark 3 When strong time-varying disturbances exist theboundary layer is bigger than before and the estimation errorsare increased

The adaptive law of single parameter 120601 can be designed as120601 = 120574

2 sTs120583119879120583 minus 120581120574120601 (16)

where 120574 gt 0 and 120581 gt 0An adaptive item 119898 is employed to estimate 120578119898 and the

estimated error is 119898 = 119898 minus 120578119898 The adaptive law of 119898 isselected as

120578119898 = 120591 (|s| minus 120572119898) (17)

where 120591 gt 0 and 120572 gt 0Substitute (15) into (14)

s (t) = [qm minus Aq minus Bq minus 12 s120601120583T120583 minus m lowast sgn (s)

minus 120573e minus Ks] + Aq + Bq + P (q q) + d (t) minus qm (t)+ 120573e (t) = minus12 s120601120583T120583 minus m lowast sgn (s) + [120579lowastT120583 (q q)+ 120576] + d (t) minus Ks

(18)

4 Complexity

Theorem 4 Considering that the nonlinear system (7) iswith parametric uncertainties and disturbances if controller(15) and updating laws (16) and (17) are designed then theboundedness of all the closed-loop system signals included in(19) can be guaranteed

Proof Lyapunov function is selected as

119871 = 12 sTs +

121205741206012 +

12120591 Tmm (19)

The derivative of Lyapunov function is

= sT s + 1120574120601 120601 + 1

120591 Tm 120578m = sT [minus12 s120601120583119879120583 minus mlowast sgn (s) + [120579lowastT120583 (q q) + 120576] + d (119905) minus Ks] + 1

120574120601 120601+ 1120591 Tm 120578m = minus12 sTs120601120583119879120583 + sT120579lowastT120583 (q q) + sT [120576

+ d (119905) minus m lowast sgn (s)] minus sTKs + 1120574120601 120601 + 1

120591 Tm 120578mle minus12 sTs120601120583119879120583 +

12 sTs120601120583119879120583 +

12 + sT [120578 minus m

lowast sgn (s)] minus sTKs + 1120574120601 120601 + 1

120591 Tm 120578m le minus12sdot sTs120601120583119879120583 + 1

2 + sT [120578 minus 120578m lowast sgn (s) + 120578mlowast sgn (s) minus m lowast sgn (s)] minus sTKs + 1

120574120601 120601 + 1120591 Tm 120578m

le 120601(minus12 sTs120583119879120583 +1120574 120601) + 1

2 + sT [(120578m minus m)

lowast sgn (s)] minus sTKs + 1120591 Tm 120578m

(20)

Substituting (16) and (17) into (20) the following inequalityis obtained

le minus120581120601120601 + 12 minus sTKs minus sT [m lowast sgn (s)] + Tm |s|

minus 120572Tmm le minus1205812 (1206012 minus 1206012) +12 minus sTKs minus 120572Tmm

= minus12058121206012 minus sTKs + (12058121206012 +12) minus 120572Tmm

(21)

where 120581 = 2120582120574 and 120582 = min 1205821 1205822 1205821 1205822 are the eigen-values of matrix K Furthermore we have

le minus12058121206012 minus sTKs + (12058121206012 +12)


= minus2120582( 121205741206012 +

12 sTs) + (

12058121206012 +

12)

= minus2120582 (119871 minus 12120591 Tmm) + (

12058121206012 +

12) = minus2120582119871 + 119876

(22)

where 119876 = (120582120591)Tmm + (1205812)1206012 + 12The solution of (22) is

119871 le 1198762120582 + (119871 (0) minus 119876

2120582) 119890minus2120582119905 (23)

Then all the signals included in the Lyapunov function arebounded This concludes the proof

Remark 5 In practical application the high-frequencyswitching control signals of MEMS gyroscopes result inserious chatteringTherefore the saturation function sat(119904) isused to replace the sign function sgn(s) in (15)The saturationfunction sat(119909) has the form

sat (119909) =

1 119909 gt 119886119909119886 |119909| le 119886minus1 119909 lt minus119886

(24)

where 119886 is a positive constant5 Numerical Simulation

In this section the aforementioned control scheme of MEMSgyroscope is simulated the controller of which is designed as(15) and the adaptive laws are proposed as (16) and (17)

Parameters of the MEMS gyroscope are as follows

119898 = 057 times 10minus8 kg119889119909119909 = 0429 times 10minus6Nsm119889119910119910 = 00429 times 10minus6Nsm119889119909119910 = 00429 times 10minus6Nsm119896119909119909 = 8098Nm119896119910119910 = 7162Nm119896119909119910 = 5NmΩ119911 = 50 rads

(25)

Since the position of proof mass ranges within the scopeof submillimeter and the natural frequency is generally inthe range of kilohertz the reference length is assumed as119902119900 = 10 times 10minus6m and reference frequency is assumed as120596119900 = 1 kHz

Suppose that the reference trajectories are 119909119898 =sin(171119905) 119898 = 171 cos(171119905) 119910119898 = 12 sin(111119905) and119910119898 = 12 times 111 cos(111119905) respectively

Complexity 5

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus5

05

1015

minus50

050

100

minus50

050

100

x y

Φ

Figure 2 Adaptive signals

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus2

minus1

0

1

2

minus2

minus1

0

1

2

times104

times104

ux

uy

Figure 3 Control inputs

Then set other simulation parameters as

A = [ minus0075 00025minus00175 minus00075]

B = [minus14207 minus877minus877 minus12564]

C = [1 00 1]

K = [8 00 4]

120573 = [20 00 15]

120574 = 80120581 = 001120591 = 80120572 = 001

(26)

And select the initial state values of the system as[08 0 1 0]119879 The centers of basis function for networkare uniformly valued in [minus1 1] and the spreads of thebasis function are 120590119894 = 1 The number of neural networknodes is chosen as 256 The adaptive signals are presentedin Figure 2 and the control inputs are shown in Figure 3 As

6 Complexity

Reference positionPosition tracking


Reference speedSpeed tracking


˙ 2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

minus1

minus05

0

05

1

15

minus1

minus15

minus05

0

05

1

15

minus4

minus3

minus2

minus1

0

1

2

minus5

minus4

minus3

minus2

minus1

0

1

2

x y

x

y

Figure 4 Position and speed trajectories

depicted in Figure 4 the system using adaptive neural net-work slidingmode control withminimal-learning-parametertechnique could track the reference signals very well Theposition tracking errors and speed tracking errors are shownin Figure 5 Through the tracking simulations of MEMSgyroscope the proposed approach has satisfying perform-ance

6 Conclusions

In this paper an adaptive neural network sliding modecontrol strategy is proposed to compensate parametric uncer-tainties and external disturbances of MEMS gyroscopesBased on Lyapunov criterion system stability is guaranteedWith minimal-learning-parameter technique the online

computation burden is significantly reduced Numericalsimulations verify that the novel control scheme couldtrack reference trajectories very well which is similar toconventional adaptive neural network sliding mode controlschemeTherefore the control scheme proposed in this papercould force the mass moves along reference trajectoriesso that the performances of MEMS gyroscopes are im-proved

For future work more efficient learning methods [21ndash25]will be tested on the dynamics while the implementation forreal systems will be analyzed

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 7

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

minus02

0

02

04

06

08

minus02

0

02

04

06

08

1

12

minus5

minus4

minus3

minus2

minus1

0

1

minus6

minus5

minus4

minus3

minus2

minus1

0

1

e x

e y

e x

e y

Figure 5 Position and speed tracking errors

Acknowledgments

This work was supported by Industrial Science and Tech-nology Key Project of Shaanxi Province (Grant no 2016GY-070) and Key Project of Shaanxi Provincersquos Department ofEducation (Grant no 2016JS017)

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 December 2015

[2] B Xu D Wang Y Zhang and Z Shi ldquoDOB based neuralcontrol of flexible hypersonic flight vehicle considering windeffectsrdquo IEEE Transactions on Industrial Electronics vol PP no99 p 1 2017

[3] B Xu ldquoComposite learning finite-time control with applicationto quadrotorsrdquo IEEE Transactions on Systems Man and Cyber-netics Systems vol PP no 99 pp 1ndash10 2017

[4] G-X Wen C L Chen Y-J Liu and Z Liu ldquoNeural-network-based adaptive leader-following consensus control for second-order non-linear multi-agent systemsrdquo IET Control Theory andApplications vol 9 no 13 pp 1927ndash1934 2015

[5] C Yang XWang Z Li Y Li and C Su ldquoTeleoperation controlbased on combination of wave variable and neural networksrdquoIEEE Transactions on Systems Man and Cybernetics Systemsvol PP no 99 pp 1ndash12 2016

[6] C Yang Y Jiang Z Li W He and C-Y Su ldquoNeural controlof bimanual robots with guaranteed global stability and motionprecisionrdquo IEEE Transactions on Industrial Informatics vol 13no 3 pp 1162ndash1171 2016

[7] C Yang X Wang L Cheng and H Ma ldquoNeural-learning-based telerobot control with guaranteed performancerdquo IEEETransactions on Cybernetics vol PP no 99 pp 1ndash12 2016

[8] 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

[9] S Park and R Horowitz ldquoNew adaptive mode of operation forMEMS gyroscopesrdquo Journal of Dynamic Systems Measurementand Control Transactions of the ASME vol 126 no 4 pp 800ndash810 2004

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

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

8 Complexity

[12] C Batur T Sreeramreddy and Q Khasawneh ldquoSliding modecontrol of a simulatedMEMS gyroscoperdquo ISA Transactions vol45 no 1 pp 99ndash108 2006

[13] 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

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

[15] F-J Lin S-Y Chen and K-K Shyu ldquoRobust dynamic sliding-mode control using adaptive RENN for magnetic levitationsystemrdquo IEEE Transactions on Neural Networks vol 20 no 6pp 938ndash951 2009

[16] L Y Wang T Y Chai and L F Zhai ldquoNeural-network-based terminal sliding-mode control of robotic manipulatorsincluding actuator dynamicsrdquo IEEE Transactions on IndustrialElectronics vol 56 no 9 pp 3296ndash3304 2009

[17] J Fei and Y Yang ldquoAdaptive neural compensation schemefor robust tracking of MEMS gyroscoperdquo in Proceedings ofthe 2012 IEEE International Conference on Systems Man andCybernetics SMC 2012 pp 1546ndash1551 October 2012

[18] Y P Asad A Shamsi and J Tavoosi ldquoBackstepping-basedrecurrent type-2 fuzzy sliding mode control for MIMO systems(MEMS triaxial gyroscope case study)rdquo International Journalof Uncertainty Fuzziness and Knowledge-Based Systems vol 25no 2 pp 213ndash233 2017

[19] J X Ren R Zhang and B Xu ldquoAdaptive fuzzy sliding modecontrol of MEMS gyroscope with finite time convergencerdquoJournal of Sensors vol 2016 Article ID 1572303 7 pages 2016

[20] B Xu Y Fan and S Zhang ldquoMinimal-learning-parametertechnique based adaptive neural control of hypersonic flightdynamicswithout back-steppingrdquoNeurocomputing vol 164 no1-2 pp 201ndash209 2015

[21] J Yang S Li and X Yu ldquoSliding-mode control for systems withmismatched uncertainties via a disturbance observerrdquo IEEETransactions on Industrial Electronics vol 60 no 1 pp 160ndash1692013

[22] B Xu and F Sun ldquoComposite intelligent learning control ofstrict-feedback systemswith disturbancerdquo IEEETransactions onCybernetics vol PP no 99 pp 1ndash12 2017

[23] M Wang Y Zhang and H Ye ldquoDynamic learning fromadaptive neural control of uncertain robots with guaranteedfull-state tracking precisionrdquo Complexity In press

[24] B Luo T Huang H-N Wu and X Yang ldquoData-driven Hinfincontrol for nonlinear distributed parameter systemsrdquo IEEETransactions on Neural Networks and Learning Systems vol 26no 11 pp 2949ndash2961 2015

[25] B Xu and P Zhang ldquoComposite learning sliding mode controlof flexible-link manipulatorrdquo Complexity In press

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Complexity

dyy

2

dyy

2

kyy

2

kyy

2

dxx

2

dxx

2

kxx2

kxx2

m

y

x

z

Ωlowastz

Figure 1 The basic principle diagram of 119911-axis MEMS gyroscope

The structure of this paper is organized as followsSection 2 formulates the dynamics of MEMS gyroscopeSection 3 studies the neural network of lumped parametricuncertainties In Section 4 an adaptive neural sliding modecontrol strategy using the minimal-learning-parameter tech-nique is designed and stability analysis is discussed Numer-ical simulations are investigated to verify the superiority ofthe proposed approach in Section 5 Conclusions are given inSection 6

2 Problem Formulation

21 Dynamics of MEMS Gyroscope As Figure 1 shows theidealMEMS gyroscope is a quality-stiffness-damping systemConsidering the mechanical coupling caused by manufac-turing defects the dynamics of MEMS gyroscopes can beexpressed as

119898 + 119889119909119909 + (119889119909119910 minus 2119898Ωlowast119911) 119910 + (119896119909119909 minus 119898Ωlowast2119911 ) 119909+ 119896119909119910119910 = 119906lowast119909

119898 119910 + 119889119909119909 119910 + (119889119909119910 + 2119898Ωlowast119911) + (119896119910119910 minus 119898Ωlowast2119911 ) 119910+ 119896119909119910119909 = 119906lowast119910

(1)

where119898 represents themass of proofmassΩlowast119911 represents theinput angular velocity 119909 and 119910 represent the system general-ized coordinates 119889119909119909 and 119889119910119910 represent damping terms 119889119909119910represents asymmetric damping term 119896119909119909 and 119896119910119910 representspring terms 119896119909119910 represents asymmetric spring terms and 119906lowast119909and 119906lowast119910 represent the control forces

In (1) the damping terms are affected by atmosphericpressure and spring terms are affected by ambient tempera-ture That means

119889119909119909 = 1198891199090 + Δ119889119909119909119889119910119910 = 1198891199100 + Δ119889119910119910

119896119909119909 = 1198961199090 + Δ119896119909119909119896119910119910 = 1198961199100 + Δ119896119910119910119896119909119910 = 1198961199091199100 + Δ119896119909119910

(2)

where 1198891199090 1198891199100 and 1198891199091199100 are the damping terms of MEMSgyroscope in normal atmospheric pressure 1198961199090 1198961199100 and1198961199091199100 are the spring terms of MEMS gyroscope in roomtemperature environment Δ119889119909119909 Δ119889119910119910 and Δ119889119909119910 are thedeviation in damping coefficients due to the changes ofatmospheric pressure and Δ119896119909119909 Δ119896119910119910 and Δ119896119909119910 are thedeviation of damping coefficients because of the changes ofambient temperature

Dividing both sides of (1) by reference mass119898 referencefrequency 1205962119900 and reference length 119902119900 the dynamics can bederived as

qlowast

119902119900 +Dlowast

119898120596119900qlowast

119902119900 + 2Slowast

120596119900qlowast

119902119900 minusΩlowast21199111205962119900

qlowast

119902119900 +Klowast

1198981205962119900qlowast

119902119900= ulowast

1198981205962119900119902119900 (3)

where qlowast = [ 119909119910 ] ulowast = [ 119906lowast119909119906lowast119910 ] Dlowast = [ 119889119909119909 119889119909119910119889119909119910 119889119910119910] Slowast = [ 0 minusΩlowast119911

Ωlowast119911 0]

and Klowast = [ 119896119909119909 119896119909119910119896119909119910 119896119910119910]

Define new parameters as q = qlowast119902119900 u = ulowast1198981205962119900119902119900Ω119911 = Ωlowast119911 120596119900D = Dlowast119898120596119900 K = Klowast1198981205962119900 and S = minusSlowast120596119900Then (3) has the following form

q = (2S minusD) q + (Ω2119911 minus K) q + u (4)

where D = D0 + ΔD K = K0 + ΔK and D0 =[ 1198891199090119898120596119900 11988911990911991001198981205961199001198891199091199100119898120596119900 1198891199100119898120596119900

] K0 = [ 11989611990901198981205962119900 119896119909119910011989812059621199001198961199091199100119898120596

2119900 1198961199100119898120596

2119900

] ΔD =[ Δ119889119909119909119898120596119900 Δ119889119909119910119898120596119900Δ119889119909119910119898120596119900 Δ119889119910119910119898120596119900

] and ΔK = [ Δ1198961199091199091198981205962119900 Δ1198961199091199101198981205962119900Δ119896119909119910119898120596

2119900 Δ119896119910119910119898120596

2119900

]With the external disturbances caused by compound

maneuvers (4) is replaced by

q = (2S minusD0 minus ΔD) q + (Ω2z minus K0 minus ΔK) q + Cu

+ d (t) (5)

where d(t) is the external disturbance and dm = sup|d(t)|Define lumped parametric uncertainties as

P (q q) = minusΔDq minus ΔKq (6)

Equation (5) can be written as

q = Aq + Bq + Cu + P (q q) + d (t) (7)

where A = 2S minusD0 and B = Ω2z minus K0

Complexity 3



P (q q | 120579) = T120583 (q q) (8)





P (q q) = 120579lowastT120583 (q q) + 120576 (10)




(11)



e (t) = q (t) minus qm (t) (12)


s (t) = e (t) + 120573e (t) (13)



(14)









2 sTs120583119879120583 minus 120581120574120601 (16)



120578119898 = 120591 (|s| minus 120572119898) (17)




(18)

4 Complexity



119871 = 12 sTs +

121205741206012 +

12120591 Tmm (19)


= sT s + 1120574120601 120601 + 1




120591 Tm 120578mle minus12 sTs120601120583119879120583 +

12 sTs120601120583119879120583 +

12 + sT [120578 minus m




120574120601 120601 + 1120591 Tm 120578m

le 120601(minus12 sTs120583119879120583 +1120574 120601) + 1

2 + sT [(120578m minus m)


(20)





(21)




= minus2120582( 121205741206012 +

12 sTs) + (

12058121206012 +

12)

= minus2120582 (119871 minus 12120591 Tmm) + (

12058121206012 +

12) = minus2120582119871 + 119876

(22)


119871 le 1198762120582 + (119871 (0) minus 119876

2120582) 119890minus2120582119905 (23)



sat (119909) =


(24)





(25)



Complexity 5

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus5

05

1015

minus50

050

100

minus50

050

100

x y

Φ


1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus2

minus1

0

1

2

minus2

minus1

0

1

2

times104

times104

ux

uy





C = [1 00 1]

K = [8 00 4]

120573 = [20 00 15]

120574 = 80120581 = 001120591 = 80120572 = 001

(26)


6 Complexity





˙ 2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

minus1

minus05

0

05

1

15

minus1

minus15

minus05

0

05

1

15

minus4

minus3

minus2

minus1

0

1

2

minus5

minus4

minus3

minus2

minus1

0

1

2

x y

x

y



6 Conclusions






Complexity 7

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

minus02

0

02

04

06

08

minus02

0

02

04

06

08

1

12

minus5

minus4

minus3

minus2

minus1

0

1

minus6

minus5

minus4

minus3

minus2

minus1

0

1

e x

e y

e x

e y


Acknowledgments


References












8 Complexity






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 3



P (q q | 120579) = T120583 (q q) (8)





P (q q) = 120579lowastT120583 (q q) + 120576 (10)




(11)



e (t) = q (t) minus qm (t) (12)


s (t) = e (t) + 120573e (t) (13)



(14)









2 sTs120583119879120583 minus 120581120574120601 (16)



120578119898 = 120591 (|s| minus 120572119898) (17)




(18)

4 Complexity



119871 = 12 sTs +

121205741206012 +

12120591 Tmm (19)


= sT s + 1120574120601 120601 + 1




120591 Tm 120578mle minus12 sTs120601120583119879120583 +

12 sTs120601120583119879120583 +

12 + sT [120578 minus m




120574120601 120601 + 1120591 Tm 120578m

le 120601(minus12 sTs120583119879120583 +1120574 120601) + 1

2 + sT [(120578m minus m)


(20)





(21)




= minus2120582( 121205741206012 +

12 sTs) + (

12058121206012 +

12)

= minus2120582 (119871 minus 12120591 Tmm) + (

12058121206012 +

12) = minus2120582119871 + 119876

(22)


119871 le 1198762120582 + (119871 (0) minus 119876

2120582) 119890minus2120582119905 (23)



sat (119909) =


(24)





(25)



Complexity 5

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus5

05

1015

minus50

050

100

minus50

050

100

x y

Φ


1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus2

minus1

0

1

2

minus2

minus1

0

1

2

times104

times104

ux

uy





C = [1 00 1]

K = [8 00 4]

120573 = [20 00 15]

120574 = 80120581 = 001120591 = 80120572 = 001

(26)


6 Complexity





˙ 2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

minus1

minus05

0

05

1

15

minus1

minus15

minus05

0

05

1

15

minus4

minus3

minus2

minus1

0

1

2

minus5

minus4

minus3

minus2

minus1

0

1

2

x y

x

y



6 Conclusions






Complexity 7

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

minus02

0

02

04

06

08

minus02

0

02

04

06

08

1

12

minus5

minus4

minus3

minus2

minus1

0

1

minus6

minus5

minus4

minus3

minus2

minus1

0

1

e x

e y

e x

e y


Acknowledgments


References












8 Complexity






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




4 Complexity



119871 = 12 sTs +

121205741206012 +

12120591 Tmm (19)


= sT s + 1120574120601 120601 + 1




120591 Tm 120578mle minus12 sTs120601120583119879120583 +

12 sTs120601120583119879120583 +

12 + sT [120578 minus m




120574120601 120601 + 1120591 Tm 120578m

le 120601(minus12 sTs120583119879120583 +1120574 120601) + 1

2 + sT [(120578m minus m)


(20)





(21)




= minus2120582( 121205741206012 +

12 sTs) + (

12058121206012 +

12)

= minus2120582 (119871 minus 12120591 Tmm) + (

12058121206012 +

12) = minus2120582119871 + 119876

(22)


119871 le 1198762120582 + (119871 (0) minus 119876

2120582) 119890minus2120582119905 (23)



sat (119909) =


(24)





(25)



Complexity 5

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus5

05

1015

minus50

050

100

minus50

050

100

x y

Φ


1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus2

minus1

0

1

2

minus2

minus1

0

1

2

times104

times104

ux

uy





C = [1 00 1]

K = [8 00 4]

120573 = [20 00 15]

120574 = 80120581 = 001120591 = 80120572 = 001

(26)


6 Complexity





˙ 2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

minus1

minus05

0

05

1

15

minus1

minus15

minus05

0

05

1

15

minus4

minus3

minus2

minus1

0

1

2

minus5

minus4

minus3

minus2

minus1

0

1

2

x y

x

y



6 Conclusions






Complexity 7

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

minus02

0

02

04

06

08

minus02

0

02

04

06

08

1

12

minus5

minus4

minus3

minus2

minus1

0

1

minus6

minus5

minus4

minus3

minus2

minus1

0

1

e x

e y

e x

e y


Acknowledgments


References












8 Complexity






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 5

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus5

05

1015

minus50

050

100

minus50

050

100

x y

Φ


1 2 3 4 5 6 7 8 9 100Time (s)

1 2 3 4 5 6 7 8 9 100Time (s)

minus2

minus1

0

1

2

minus2

minus1

0

1

2

times104

times104

ux

uy





C = [1 00 1]

K = [8 00 4]

120573 = [20 00 15]

120574 = 80120581 = 001120591 = 80120572 = 001

(26)


6 Complexity





˙ 2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

minus1

minus05

0

05

1

15

minus1

minus15

minus05

0

05

1

15

minus4

minus3

minus2

minus1

0

1

2

minus5

minus4

minus3

minus2

minus1

0

1

2

x y

x

y



6 Conclusions






Complexity 7

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

minus02

0

02

04

06

08

minus02

0

02

04

06

08

1

12

minus5

minus4

minus3

minus2

minus1

0

1

minus6

minus5

minus4

minus3

minus2

minus1

0

1

e x

e y

e x

e y


Acknowledgments


References












8 Complexity






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




6 Complexity





˙ 2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

minus1

minus05

0

05

1

15

minus1

minus15

minus05

0

05

1

15

minus4

minus3

minus2

minus1

0

1

2

minus5

minus4

minus3

minus2

minus1

0

1

2

x y

x

y



6 Conclusions






Complexity 7

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

minus02

0

02

04

06

08

minus02

0

02

04

06

08

1

12

minus5

minus4

minus3

minus2

minus1

0

1

minus6

minus5

minus4

minus3

minus2

minus1

0

1

e x

e y

e x

e y


Acknowledgments


References












8 Complexity






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 7

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

2 4 6 8 100Time (s)

minus02

0

02

04

06

08

minus02

0

02

04

06

08

1

12

minus5

minus4

minus3

minus2

minus1

0

1

minus6

minus5

minus4

minus3

minus2

minus1

0

1

e x

e y

e x

e y


Acknowledgments


References












8 Complexity






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




8 Complexity






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




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