research article adaptive control of mems gyroscope based on global terminal sliding...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 797626 9 pageshttpdxdoiorg1011552013797626

Research ArticleAdaptive Control of MEMS Gyroscope Based onGlobal Terminal Sliding Mode Controller

Weifeng Yan and Juntao Fei

Jiangsu Key Lab of Power Transmission and Distribution Equipment Technology College of Computer and InformationHohai University Changzhou 213022 China

Correspondence should be addressed to Juntao Fei jtfeiyahoocom

Received 25 June 2013 Accepted 31 October 2013

Academic Editor Shihua Li

Copyright copy 2013 W Yan and J Fei 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

An adaptive global fast terminal sliding mode control (GFTSM) is proposed for tracking control of Micro-Electro-MechanicalSystems (MEMS) vibratory gyroscopes under unknownmodel uncertainties and external disturbances To improve the convergencerate of reaching the sliding surface a global fast terminal sliding surface is employed which can integrate the advantages oftraditional sliding mode control and terminal sliding mode control It can be guaranteed that sliding surface and equilibrium pointcan be reached in a shorter finite time from any initial state In the presence of unknown upper bound of system nonlinearitiesan adaptive global fast terminal sliding mode controller is derived to estimate this unknown upper bound Simulation resultsdemonstrate that the tracking error can be attenuated efficiently and robustness of the control system can be improved with theproposed adaptive global fast terminal sliding mode control

1 Introduction

MEMS gyroscopes are the basic measuring elements ofinertial navigation and inertial guidance system which arecommonly used for measuring angular velocity Becauseof their enormous advantages in size and cost MEMSgyroscopes have a wide range of applications in the areaof navigation platform stabilization control and trafficBut due to the manufacturing error and the temperaturedisturbance which can cause deviations from the desiredqualities and degrade the performance of a gyroscopeBesides MEMS gyroscopes belong to MIMO systems whichhave parameter uncertainties and are vulnerable to externalenvironment influence Therefore it is necessary to solvethese challenges by utilizing advanced control methods Inthe last few years many advanced control approaches havebeen proposed to enhance performance and robustness ofMEMS gyroscope Batur et al [1] developed a sliding modecontrol for MEMS gyroscope system Leland [2] presentedan adaptive controller for tuning the natural frequency of thedrive axis of a vibratory gyroscope Park et al [3 4] presentedan adaptive controller for a MEMS gyroscope which drivesboth axes of vibration and controls the entire operation

of the gyroscope Adaptive controller has been developedusing sliding mode control to control the vibration of MEMSgyroscope [5] Raman et al [6] proposed a closed-loopdigitally controlled MEMS gyroscope using unconstrainedsigma-delta force balanced feedback control Tsai and Sue [7]developed integrated model reference adaptive control andtime-varying angular rate estimator for microgyroscopes

As we know sliding mode control has many attractivefeatures such as robustness to parameter variations andinsensitivity to disturbance The basic idea of the slidingmode control is to drive and maintain the system trajec-tory on a sliding surface designed a priori in the statespace However one of the representative characteristics ofconventional sliding mode control is that the convergenceof the system states to the equilibrium point is usuallyasymptotical but not in finite time Recently a new type ofsliding mode control technique called terminal sliding modecontrol is developed Instead of using linear hyperplanesas the sliding surfaces the terminal sliding mode controladopts nonlinear sliding surfaces With the nonlinear slidingsurfaces the terminal sliding mode controller can make thesystem states reach the equilibrium point in a finite time

2 Mathematical Problems in Engineering

period Saif et al [8] presented a novel methodology forapproximation of the unknown time-varying rotation rate byusing a sliding mode observer along with a robust controlsystem based on terminal slidingmode control for improvingthe performance of the MEMS gyroscope Zhu and Chai[9] presented a global finite-time tracking controller basedon terminal sliding mode control method for the trajectorytracking control problem of biped multilinked robots withuncertain disturbances In order to solve the optimal problemof convergence time and estimate the unknown upper boundof system uncertainties much research has been done toapply intelligent control approaches such as the adaptiveglobal fast terminal sliding mode control Feng et al [10]and Keleher and Stonier [11] proposed an adaptive fastterminal sliding mode tracking control scheme for roboticmanipulators to estimate the upper bound of uncertaintiesTao et al [12] developed a new adaptive fuzzy terminalsliding mode controller for linear systems with mismatchedtime varying uncertainties Zhang et al [13] proposed a newapproach of adaptive fuzzy global identical terminal slidingmode control for cross-beam system Neila and Tarak [14]designed adaptive terminal sliding mode control for rigidrobotic manipulators Feng et al [15] proposed a nonsingularterminal sliding mode controller for rigid manipulators Inthis paper robust adaptive global fast terminal sliding modecontrol strategy for trajectory tracking control of MEMSvibratory gyroscopes is proposed The main motivations inthe paper are highlighted as follows

(1) A global fast terminal sliding mode surface is de-signed to improve the convergence rate of reachingthe sliding surface It integrates the advantages of tra-ditional sliding and terminal sliding control andguarantees that the tracking error can converge tozero in a shorter finite time Adaptive control andterminal sliding control are successfully integratedwith MEMS gyroscope

(2) An adaptive global fast terminal sliding mode con-trol is incorporated into the MEMS gyroscope toimprove the tracking performance and strengthen therobustness of the control system With this controlthe upper bound of model uncertainties and externaldisturbances are not needed because the proposedcontroller can estimate the upper bound of theseuncertainties using an adaptive mechanism Alsothe proposed controller eliminates the chatteringeffect without losing the robustness property and theprecision

This paper is organized as follows In Section 2 thedynamics of MEMS gyroscope are described In Section 3global fast terminal sliding mode control is introduced InSection 4 the proposed adaptive global fast terminal slidingmode controller is derived and the stability of this controlsystem is proved by Lyapunovrsquos stability theory Simulationresults are presented in Section 5 Finally conclusions areprovided in Section 6

proof massm

kxx

kyy

kyy

kxx

dxxdxx

dyy

dyy

x

x

y

y

Ωz

Figure 1 A simplified model of a 119911-axis MEMS Gyroscope

2 Dynamics of MEMS Gyroscope

A 119911-axis MEMS gyroscope is depicted in Figure 1 Generallya typical MEMS vibratory gyroscope includes a proof masssuspended by spring beams electrostatic actuations andsensing mechanisms for forcing an oscillatory motion andsensing the position and velocity of the proofmass Dynamicsof a MEMS gyroscope are derived from Newtonrsquos law in therotating frame

As we know Newtonrsquos law in the rotating frame becomes

119865119903= 119865phy + 119865centri + 119865Coriolis + 119865Euler = 119898119886

119903 (1)

In (1) 119865119903is the total applied force to the proof mass in

the gyro frame 119865phy is the total physical force to the proofmass in the inertial frame 119865centri is the centrifugal force119865Coriolis is the Coriolis force 119865Euler is the Euler force and 119886

119903

is the acceleration of the proof mass with respect to the gyroframe119865phy119865Coriolis and119865Euler are inertial forces caused by therotation of the gyro frame

With the definition of 119903119903and V

119903as the position and

velocity vectors relative to the rotating gyroscope frame andΩ as the angular velocity vector of the gyroscope frame theexpressions for the inertial forces are reduced to

119865Coriolis = minus2119898Ω times V119903 119865centri = minus119898Ω times (Ω times 119903

119903)

119865Euler = minus119898

119889Ω

119889119905

times 119903119903

(2)

then

119898119886119903+ 119898Ω times (Ω times 119903

119903) + 2119898Ω times V

119903+ 119898Ω times 119903

119903= 119865phy (3)

where 119865phy contains spring damping and control forcesapplied to the proof mass

In a 119911-axis MEMS gyroscope by supposing the stiffnessof spring in 119911 direction much larger than that in 119909 and 119910

directions motion of proof mass is constrained to only alongthe 119909 minus 119910 plane Referring to [16] with assuming that thegyroscope is almost rotating at a constant angular velocity

Mathematical Problems in Engineering 3

Ω119911over a sufficiently long time interval the dynamics of

gyroscope based on (3) are simplified as follows

119898 + 119889119909 + [119896

119909minus 119898(Ω

2

119910+ Ω2

119911)] 119909 + 119898Ω

119909Ω119910119910

= 119906119909+ 2119898Ω

119911119910

119898 119910 + 119889119910

119910 + [119896119910minus 119898(Ω

2

119909+ Ω2

119911)] 119910 + 119898Ω

119909Ω119910119909

= 119906119910minus 2119898Ω

119911

(4)

where 119909 and 119910 are the coordinates of the proof mass withrespect to the gyro frame in a Cartesian coordinate system119889119909119910

and 119896119909119910

are damping and spring coefficients Ω119909119910119911

arethe angular rate components along each axis of the gyroframe and 119906

119909119910are the control forces in 119909 and 119910 directions

In this design the control forces are the electrostatic forces inparallel plate actuator which can be expressed as the gradientof the potential energy stored on the capacitor The last twoterms in (4) 2119898Ω

119911119910 and 2119898Ω

119911 are the Coriolis forces used

to reconstruct the unknown input angular rate Ω119911 Under

typical assumptions Ω119909

asymp Ω119910

asymp 0 only the component ofthe angular rate Ω

119911causes a dynamic coupling between 119909-

and 119910-axesTaking fabrication imperfections into account which

cause extra coupling between 119909- and 119910-axes the lumpedparameter mathematical model for an actual 119911-axis MEMSgyroscope becomes

119898 + 119889119909119909 + 119889119909119910

119910 + 119896119909119909119909 + 119896119909119910119910 = 119906

119909+ 2119898Ω

119911119910

119898 119910 + 119889119909119910 + 119889119910119910

119910 + 119896119909119910119909 + 119896119910119910119910 = 119906

119910minus 2119898Ω

119911

(5)

In (5) 119889119909119909

and 119889119910119910

are damping coefficients 119896119909119909

and119896119910119910

are spring coefficients 119889119909119910

and 119896119909119910 called quadrature

errors are coupled damping and spring terms respectivelymainly due to the asymmetries in suspension structure andmisalignment of sensors and actuators The coupled springand damping terms are unknown but can be assumed tobe small The nominal values of the 119909- and 119910-axes springand damping terms are known but there are small unknownvariations The proof mass can be determined accurately

Dividing both sides of (5) by 119898 1199020 and 120596

2

0 which are

a reference mass length and natural resonance frequencyrespectively we get the form of the nondimensional equationof motion as

+ 119889119909119909 + 119889119909119910

119910 + 1205962

119909119909 + 120596119909119910119910 = 119906

119909+ 2Ω119911

119910

119910 + 119889119909119910 + 119889119910119910

119910 + 120596119909119910119909 + 1205962

119910119910 = 119906

119910minus 2Ω119911

(6)

where 1198891199091199091198981205960rarr 119889119909119909 1198891199091199101198981205960rarr 119889119909119910 1198891199101199101198981205960rarr 119889119910119910

radic1198961199091199091198981205962

0rarr 120596

119909 1198961199091199101198981205962

0rarr 120596

119909119910 radic1198961199101199101198981205962

0rarr 120596

119910

Ω1199111205960rarr Ω119911

The vector form of MEMS gyroscope dynamics modelcan be written as

119902 + 119863 119902 + 119870119902 = 119906 minus 2Ω 119902 (7)

where 119902 = [119909

119910 ] 119863 = [119889119909119909

119889119909119910

119889119909119910

119889119910119910

] 119870 = [1205962

119909

120596119909119910

120596119909119910

1205962

119910

] 119906 = [119906119909

119906119910]

Ω = [0

Ω119911

minusΩ119911

0]

Considering the system with parametric uncertaintiesand external disturbance the dynamics of MEMS gyroscope(7) can be expressed as

119902 + (119863 + 2Ω + Δ119863) 119902 + (119870 + Δ119870) 119902 = 119906 + 119889 (8)

where Δ119863 is the unknown parameter uncertainties of thematrix119863+2ΩΔ119870 is the unknownparameter uncertainties ofthe matrix119870 and 119889 is an uncertain extraneous disturbance

The control target for MEMS gyroscope is to find acontrol law so that the proof mass position q(119905) can track thedesired trajectory q

119903(119905) In addition in the presence of model

uncertainties and external disturbances a robust controldesign is required to improve the robustness and trackingresolution of the control system In this paper adaptive globalfast terminal sliding mode control is designed to make thesystem track the desired trajectory and improve the systemrobustness in the unknownmodel uncertainties and externaldisturbances

3 Global Fast Terminal Sliding Mode Control

The traditional fast terminal sliding mode form can bedefined as

119904119886= 1+ 12057301199091

11990121199011

= 0 (9)

where 1199091isin 1198771 is state variable 120573

0gt 0 is constant and 119901

1

and 1199012are odd positive integers satisfying 119901

1gt 1199012 Given an

initial state 1199091(0) = 0 the system will reach the equilibrium

point 1199091= 0 along the sliding mode (9) in finite time 119905

1198881 The

time to reach 1199091= 0 from 119909

1(0) = 0 is determined by

1199051198881

=

1199011

1205730(1199011minus 1199012)

10038161003816100381610038161199091(0)

1003816100381610038161003816

(1199011minus1199012)1199011

(10)

The global fast terminal sliding surface can be defined as

119904119887= 1+ 12057201199091+ 120573011990911990121199011

1= 0 (11)

where 1205720gt 0 is constant Given an initial state 119909

1(0) = 0 the

system will reach the equilibrium point 1199091

= 0 along thesliding mode (11) in finite time 119905

1198882 The time to reach 119909

1= 0

from 1199091(0) = 0 is determined by

1199051198882

=

1199011

1205720(1199011minus 1199012)

ln12057201199091(0)(1199011minus1199012)1199011+ 1205730

1205730

(12)

Referring to [17] we can demonstrate that 1199051198882

le 1199051198881

Comparing (9) and (11) it can be seen that the convergencetimes of the two types of terminal sliding modes are alldetermined by the fast terminal attractor

1= minus120573

011990911990121199011

1

when it is far from the equilibrium point At this time theirconvergence rates are basically the same But when close tothe equilibrium point the convergence time of global fastterminal sliding mode is mainly determined by the linearslidingmode

1= minus12057201199091and the convergence rate accelerates

exponentially At this time its convergence rate ismuch betterthan the traditional fast terminal sliding mode


Expectationmodel

GFSTMcontroller

MEMSgyroscope

Adaptivelaw

Estimatedparameters

Input

Disturbance

qr

minus

+

e u q

b

Figure 2 Block diagram of an adaptive global fast terminal sliding mode control for a MEMS gyroscope

Remark 1 The global fast terminal sliding mode not onlyintroduces the fast terminal attractormaking the system stateconverge in finite time but also keeps the rapidity of the linearsliding mode when it is close to the equilibrium state so as torealize the goal of that the system state will converge to theequilibrium state quickly and accurately

4 Adaptive Global Fast TerminalSliding Mode Control

The global fast terminal sliding mode control that is putforward has solved the optimal problem of convergencetime It integrates the advantages of traditional sliding modecontrol and terminal sliding mode control in the designof sliding mode and uses the concept of fast access to theequilibrium point at the arriving stage to ensure that thetracking error converges to zero in a shorter finite timeHowever as a result of the design principle that is similarto the traditional sliding mode control there still existschattering in the global fast terminal sliding mode controlsignals Also the uncertainty boundary needs to be knownin advance and it is hard to accomplish that in actual MEMSgyroscopes In allusion to the trajectory tracking control ofMEMS gyroscopes with unknown model uncertainties andexternal disturbances a new adaptive global fast terminalsliding mode control algorithm is synthesized in this sectionwhich uses the respective advantages of adaptive and slidingmode control and the estimates of unknown uncertaintywhich can be obtained by adaptive control

The block diagram of an adaptive global fast terminalsliding mode control for a MEMS gyroscope is shown inFigure 2 the tracking error between expectation state andgyroscope state comes to the global fast terminal slidingmode controller The adaptive global fast terminal slidingmode controller is proposed to control theMEMS gyroscopeThe unknown upper bound of system uncertainties can beidentified by adaptive estimator

41 Design of Global Fast Terminal Sliding Mode Surface Thetracking control problem of a MEMS gyroscope is what thispaper considers so the control target is to design a suitable

control law to make the system output 119902 track the state of thedesired trajectory 119902

119903completely in finite time

For the trajectory tracking control of aMEMS gyroscopetwo input and two output systems the global fast terminalsliding surface is chosen as

119904119888= [1199041198881 1199041198882]119879

= 119890 + 120572119890 + 12057311989011990121199011

(13)

where 120572 = diag (1205721 1205722) and 120573 = diag (120573

1 1205732) are sliding

mode surface constants 1205721 1205722 1205731 1205732are constants 119890 = 119902 minus

119902119903= [1199021minus 1199021199031 1199022minus 1199021199032]119879 is the system tracking error and 119902

119903

is the desired trajectory In the sliding mode 119904119888= [1199041198881 1199041198882]119879

the system reaches the equilibrium state 119902 = 0 from anyinitial state 119902(0) = 0 in finite time 119905

119904= [1199051199041 1199051199042]119879 The time 119905

119904is

determined by

119905119904= [

1199011

1205721(1199011minus 1199012)

ln12057211199021(0)(1199011minus1199012)1199011+ 1205731

1205731

1199011

1205722(1199011minus 1199012)

ln12057221199022(0)(1199011minus1199012)1199011+ 1205732

1205732

]

(14)

We can make the system reach equilibrium state in finitetime 119905

119904through setting the suitable parameters 120572 120573 119901

1 and

1199012 After determining the sliding surface the next step is to

design a slidingmode controller to ensure the existence of thesliding mode stage

42 Design of Global Fast Terminal Sliding Mode ControllerIn the sliding mode control the control input should be ableto force all the system state trajectories to converge the slidingmode surface 119904 = 0 so as to ensure the existence of the slidingmode stageThe behavior of the sliding mode is equivalent tothat of the stability of the state trajectory in the sliding surfaceIn other words the control law should be able to ensure thatthe tracking error converges to zero from any state

Furthermore rewriting (8) as

119902 + (119863 + 2Ω) 119902 + 119870119902 = 119906 + 119891 (15)

where 119891 represents the matched lumped uncertainty anddisturbance which is given by

119891 = 119889 minus Δ119863 119902 minus Δ119870119902 (16)


We make the following assumption prior to furtherdiscussion

Assumption 2 The lumped uncertainty and disturbance 119891

is input related and if the control input does not containthe acceleration signal the system uncertainties will bebounded by a positive function of the position and velocitymeasurements in the following form

10038171003817100381710038171198911003817100381710038171003817lt 1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

(17)

where 1198870 1198871 and 119887

2are unknown positive constants

Remark 3 If lumped uncertainty and disturbance 119891 is notrelated to with state position and velocity signals the systemuncertainty will be only bounded by a positive constant 119887

0

Using the sliding surface (13) the global fast slidingmodecontroller is proposed as follows

119906 = 1199060+ 1199061+ 1199062 (18)

where 1199060is defined in (19) 119906

1is defined in (20) and 119906

2is

defined in (21) Consider

1199060= 119902119903+ (119863 + 2Ω) 119902 + 119870119902 (19)

1199061= minus120572 119890 minus

1199012

1199011

120573 diag (11989011990121199011minus11

11989011990121199011minus1

2) 119890 (20)

1199062= minus120588

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(21)

And 120588 has the following expression

120588 = 120585 + 1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817 (22)

where 120585 is a positive constantAccording to [18] it is known that the control system is

stable in this control law and reaches the stable point in finitetime But the control law requires that 120588 should be knownWe usually choose the larger 120588 conservatively to compensatethe system model uncertainties and the upper bound of thedisturbances fully but this often enhances the chattering

In order to overcome this problem we introduce a properadaptation law combining with global fast terminal slidingmode control to realize the adaptive global fast terminalsliding mode control in the next part This method achievesthe goal of tracking system model errors and uncertaindisturbances automatically weakening the chattering andensuring the stability of the control system

43 Design of Adaptive Global Fast Terminal Sliding ModeController Supposing that 119887

0 1198871 and

1198872are the estimates

of 1198870 1198871 and 119887

2 respectively the following simple adaptive

control laws can realize the estimation of the unknownparameters 119887

0 1198871 and 119887

2 Consider

1198870= 1198980

1003817100381710038171003817119904119888

1003817100381710038171003817

1198871= 1198981

1003817100381710038171003817119904119888

1003817100381710038171003817

10038171003817100381710038171199021003817100381710038171003817

1198872= 1198982

1003817100381710038171003817119904119888

1003817100381710038171003817

1003817100381710038171003817

1199021003817100381710038171003817

2

(23)

where 1198980 1198981 and 119898

2are arbitrary positive constants

representing the adaptive gainsBased on the given adaptation law for the MEMS gyro-

scope system with the dynamics (15) the adaptive global fastterminal sliding mode controller is designed synthetically asfollows which uses the sliding surface (13)

119906 = 1199060+ 1199061+ 1199062 (24)



2is

defined in (27)

1199060= 119886 + (119863 + 2Ω) V + 119870119902 (25)

1199061= minus119882

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(26)

1199062= minus

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) (27)

where 119882 = diag (1199081 1199082) 119908119894gt 0 119894 = 1 2 is a designed

parameter matrix and

V = 119902119903minus 120572119890 minus 120573119890

11990121199011

119886 = V = 119902119903minus 120572 119890 minus

1199012

1199011

120573 diag (11989011990121199011minus11

11989011990121199011minus1

2) 119890

(28)

Theorem 4 If the control law (24) with the sliding surface(13) and the adaptation law (23) is applied to the nonlinearuncertain system defined by (15) then the systemrsquos trackingerror can converge to zero in finite time

Proof Let us consider the following positive definite functionas a Lyapunov function candidate

119881 =

1

2

119904119879

119888119904119888+

1

2

2

sum

119894=0

119898minus1

119894

1198872

119894 (29)

where 119887119894= 119887119894minus119887119894and 119894 = 0 1 2 are the estimation errors

between actual values and their estimated values respectivelySince 119887

0 1198871 and 119887

2are constants we have

119887119894= minus

119887119894

Substituting the control 119906 in (24) into the dynamic ofMEMS gyroscope (15) yields

119902 + (119863 + 2Ω + Δ119863) 119902 + (119870 + Δ119870) 119902

= 119886 + (119863 + 2Ω) V + 119870119902 + 1199061+ 1199062+ 119889

(30)

Rewriting (30) yields

( 119902 minus 119886) + (119863 + 2Ω) ( 119902 minus V) + Δ119863 119902 + Δ119870119902

= 1199061+ 1199062+ 119889

(31)

Substituting (28) into (31) generates

119904119888+ (119863 + 2Ω) 119904

119888+ Δ119863 119902 + Δ119870119902 = 119906

1+ 1199062+ 119889 (32)

Equation (32) can be expressed as

119904119888= minus (119863 + 2Ω) 119904

119888minus Δ119863 119902 minus Δ119870119902 + 119906

1+ 1199062+ 119889 (33)


from the definition of 119891 in (16) we have

119904119888= minus (119863 + 2Ω) 119904

119888+ 1199061+ 1199062+ 119891 (34)

Differentiating 119881 with respect to time yields we have

= 119904119879

119888119904119888minus

2

sum

119894=0

119898minus1

119894

119887119894

119887119894

= 119904119879

119888[minus (119863 + 2Ω) 119904

119888+ 1199061+ 1199062+ 119891] minus 119898

minus1

0

1198870

1198870

minus 119898minus1

1

1198871

1198871minus 119898minus1

2

1198872

1198872

= 119904119879

119888(1199061+ 1199062+ 119891) + 119904

119879

119888[minus (119863 + 2Ω)] 119904

119888

minus 119898minus1

0

1198870


1

1198871


2

1198872

1198872

le 119904119879

119888(1199061+ 1199062+ 119891) minus 119898

minus1

0

1198870


1

1198871


2

1198872

1198872

le minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888(1199062+ 119891) minus 119898

minus1

0

1198870

1198870

minus 119898minus1

1

1198871


2

1198872

1198872= minus120582min (119882)

1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888119891

minus1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

)

minus1198870

1003817100381710038171003817119904119888

1003817100381710038171003817minus1198871

1003817100381710038171003817119904119888

1003817100381710038171003817

10038171003817100381710038171199021003817100381710038171003817minus1198872

1003817100381710038171003817119904119888

1003817100381710038171003817

1003817100381710038171003817

1199021003817100381710038171003817

2

= minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888119891 minus

1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

)

le minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+1003817100381710038171003817119904119888

1003817100381710038171003817119891

minus1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) le minus1205781003817100381710038171003817119904119888

1003817100381710038171003817lt 0

(35)

where 120578 = 120582min(119882) minus 119891 + (1198870+ 1198871119902 + 119887

2 1199022

) gt 0 and119904119888 = 0 [15]Therefore according to Lyapunovrsquos second method the

designed controller can guarantee the globally asymptoticalstability of the system and make the output tracking errorconverge to zero in finite time On the other hand in theglobal fast terminal sliding mode 119904

119888= 119890 + 120572119890 + 120573119890

11990121199011= 0

the output tracking error 119890 = 119902minus119902119903converges to zero in finite

time

Remark 5 In the proof of Theorem 4 we used the Rayleighprinciple namely

120582min (119882) 1199042

le 119904119879

119882119904 le 120582max (119882) 1199042

(36)

where 120582min(119882) and 120582max(119882) are the minimum and maxi-mum eigenvalues of 119882 respectively Since 119882 is a diagnosematrix that is119882 = diag [119908

1 119908

119899] 120582min(119882) and 120582max(119882)

have the simple types described as follows

120582min (119882) = min119894=1119899

119908119894

120582max (119882) = max119894=1119899

119908119894

(37)

Remark 6 Since there are two switching functions sgn inadaptive terminal sliding mode control (26) and (27) there

exists chattering in the global fast terminal sliding modecontrol signals In order to reduce the chattering we canuse adaptive global fast terminal quasi-sliding mode controlinstead of adaptive global fast terminal sliding mode controlThe so-called adaptive global fast terminal quasi-slidingmode control is a control method that the system trajectoryis confined to a neighborhood 120575 within the ideal slidingmode The quasi-sliding mode control adopts normal slidingmode control in the boundary layer outside and continuousstate feedback control in the boundary layer so as to avoidor weaken the chattering Therefore the controller (24) isimproved as follows

1199061=

minus119882

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

if 1003817100381710038171003817119904119888

1003817100381710038171003817gt 120575

minus119882

119904119888

120575

if 1003817100381710038171003817119904119888

1003817100381710038171003817le 120575

1199062=

minus

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) if 1003817100381710038171003817119904119888

1003817100381710038171003817gt 120575

minus

119904119888

120575

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) if 1003817100381710038171003817119904119888

1003817100381710038171003817le 120575

(38)

where 120575 is a designed positive parameter then the chatteringwill be eliminated

5 Simulation Example

In this section we will evaluate the proposed adaptive globalfast terminal sliding mode control scheme on the lumpedMEMS gyroscope sensor model by using MatlabSimulinkThe control objective is to design an adaptive global fastterminal sliding mode controller so that the trajectory 119902 (119905)

can track the state of the desired model 119902119903(119905) completely in

finite time In addition the unpredictably upper bound ofsystem uncertainties can be estimated by adaptive estimatorThe parameters of the MEMS gyroscope are as follows

119898 = 18 times 10minus7 kg 119896

119909119909= 63955Nm

119896119910119910

= 9592Nm 119896119909119910

= 12779Nm

119889119909119909

= 18 times 10minus6Nsm 119889

119910119910= 18 times 10

minus6Nsm

119889119909119910

= 36 times 10minus7Nsm

(39)

Since the general displacement range of the MEMS gyro-scope in each axis is sub-micrometer level it is reasonable tochoose 1 120583m as the reference length 119902

0 Given that the usual

natural frequency of each axis of a MEMS gyroscope is inthe KHz range so choose the 120596

0as 1 KHz The unknown

angular velocity is assumed Ω119911

= 100 rads Then thenondimensional values of the MEMS gyroscope parametersare listed as follows

1205962

119909= 3553 120596

2

119910= 5329

120596119909119910

= 7099 119889119909119909

= 001

119889119910119910

= 001 119889119909119910

= 0002 Ω119911= 01

(40)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

15

Time (s)

Posit

ion

trac

king

of x

-axi

s

Desired trajectory of x-axisActual trajectory of x-axis

minus05

minus1

Figure 3 Position tracking of 119909-axis

0 1 2 3 4 5 6 7 8 9 10Time (s)

Desired trajectory of y-axisActual trajectory of y-axis

0

05

1

15

Posit

ion

trac

king

of y

-axi

s

minus05

minus15

minus1


In this simulation example the initial values of thesystem are selected as 119902

1(0) = 10 119902

1(0) = 0 119902

2(0) =

05 and 1199022(0) = 0 The desired motion trajectories are

described by 1199021199031

= sin(120587119905) and 1199021199032

= cos(120587119905) The lumpedparametric uncertainties and external disturbances given by119891 = 119889 minus Δ119863 119902 minus Δ119870119902 are composed of model uncertaintiesand external disturbances As for model uncertainties thereare plusmn10 parameter variations for the spring and dampingcoefficients with respect to their nominal values and plusmn10magnitude changes in the coupling terms that is 119889

119909119910and

120596119909119910

with respect to their nominal values Random signal119889 = [05 lowast randn(1 1) 05 lowast randn(1 1)] is considered asexternal disturbances The chosen sliding surface parametersare 1199011= 5 119901

2= 3 120572

1= 1205722= 025 and 120573

1= 1205732= 05 and

the initial conditions of the upper bound of the uncertaintyare 11988700

= 01 11988710

= 02 and 11988720

= 03 The adaptive gainsare chosen as 119898

0= 10 119898

1= 16 and 119898

2= 16 The sliding

mode controller parameter of (26) is 119882 = diag (2 4)(1199081=

2 1199082= 4) In order to have a small boundary layer around

the sliding surface and reduce the chattering we have chosenthe designed positive parameter 120575 equal to 006 and used theadaptive global fast terminal quasi-slidingmode control (36)The simulation results are shown in Figures 3ndash9

0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

SMCAGFTSMC

minus02

minus04

Posit

ion

trac

king

erro

r of x

-axi

s

Figure 5 Position tracking error of 119909-axis comparison betweenAGFTSMC and SMC

0 1 2 3 4 5 6 7 8 9 10

0

01

02

Time (s)

minus01

minus02

minus03

minus04

minus05

Posit

ion

trac

king

erro

r of y

-axi

s

SMCAGFTSMC


Figures 3 and 4 depict the MEMS gyroscope along 119909-axis and 119910-axis tracking trajectories using the adaptive globalfast terminal sliding mode control law respectively FromFigures 3 and 4 it can be observed intuitively that after about 3seconds the actualmotion trajectory of theMEMSgyroscopeis consistent with the desired reference trajectory makingthe system state track the state of the desired trajectorycompletely in a shorter finite time

Subsequently in order to demonstrate the advantages ofthe paper a comparable investigation on MEMS gyroscopemodel (15) is accomplished between the proposed adaptiveglobal fast terminal sliding mode control and conventionalsliding mode control and the results are shown in Figures5 and 6 where the tracking errors of 119909-axis and 119910-axiscorresponding to adaptive global fast terminal sliding modecontrol more severely decrease in contrast with the conven-tional sliding mode control due to the proposed finite timeconvergence algorithm Besides from the Figures 5 and 6 itcan be also seen that the tracking errors of adaptive global


0 1 2 3 4 5 6 7 8 9 10

005

Time (s)

002

minus25minus2

minus15minus1

minus05

minus02

minus04

minus06

Term

inal

slid

ing

Term

inal

slid

ing

0 1 2 3 4 5 6 7 8 9 10Time (s)

surfa

ces 1

surfa

ces 2

Figure 7 Convergence of the terminal sliding surface 119904(119905)

0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux

Figure 8 Control input signal of 119909-axis

fast terminal slidingmode control converge to zero in about 3seconds which improves the dynamic behavior of theMEMSgyroscope and verify that the adaptive global fast terminalsliding mode control can ensure the global stability of thecontrol system

Moreover we can set the appropriate slidingmode surfaceparameters to adjust the convergence rate so as to improvethe work efficiency Figure 7 describes the convergence of theglobal fast terminal sliding mode surface 119904(119905) It is shownthat the sliding mode surface converges to zero in a veryshort time The global fast terminal sliding mode control hassolved the optimal problem of convergence time comparedwith the traditional terminal slidingmode so that it canmakethe system state converge to the equilibrium state quicklyand accurately It should be noted that we can also chooseappropriate reaching law to improve the convergence ratefurther such as ideal reaching law and adaptive reachinglaw which have been proposed in recent years Sliding modecontrol system utilized these two types of reaching laws thatnot only can reach the region of the switching surface infinite time but also enhance the dynamic performance of thecontrol system

Figures 8 and 9 depict the smooth sliding mode controlforces of the MEMS gyroscope along 119909-axis and 119910-axis withthe adaptive global fast terminal sliding mode controllerrespectively It is clearly observed from Figures 8 and 9 thatthere is no chattering using the proposed controller Figure 10

0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux


0

05

1

005

115

0123

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Estim

ates

ofb

2Es

timat

es o

fb1

Estim

ates

ofb

0

Figure 10 Adaptively estimated parameters 11988701198871 and

1198872

presents the adaptively estimated parameters 1198870 1198871 and

1198872

These results show a stable convergence of the upper boundparameters

In the presence of large uncertainties the global fastterminal sliding mode control can give large tracking errorwhich is not the case in adaptive global fast terminal slidingmode controlTherefore the introduction of adaptive controltechnique can adapt to the changes of external environmentand control structure parameters which maintains the opti-mal performance of the control system In addition adaptivetechnology can effectively solve the robustness problem in thepresence of unknown model parameters and external distur-bances so as to improve theMEMS gyroscopersquos reliability anddisturbance rejection ability

6 Conclusion

In this paper we proposed the design of a new adaptive globalfast terminal sliding mode controller for trajectory trackingcontrol of MEMS vibratory gyroscopes which enables therobustness in the presence of parameter uncertainties andthe external disturbances The main feature of this designis that it combines the global fast terminal sliding modecontrol with a boundary layer and the adaptive approach


Based on the Lyapunovrsquos stability theory the system statewith the designed global fast terminal slidingmode controlleris able to converge to equilibrium point in a shorter finitetime In order to reduce the chattering we present a softeningcontrol approach Considering the unknown upper boundof system uncertainties an adaptive global fast terminalsliding mode controller with unpredictably upper bound ofparameter estimation is derived The adaptive algorithm isused to estimate the bounds of uncertainties and externaldisturbances Simulation results are provided to demonstratethe validity and reliability of the proposed control scheme

Acknowledgments

The authors thank the anonymous reviewers for their usefulcomments that improved the quality of the paper This workis partially supported by the National Science Foundation ofChina under Grant no 61374100 the Natural Science Foun-dation of Jiangsu Province under Grant no BK20131136 andthe Fundamental Research Funds for the Central Universitiesunder Grant no 2012B06714

References

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

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

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

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

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

[6] J Raman E Cretu P Rombouts and L Weyten ldquoA closed-loop digitally controlled MEMS gyroscope with unconstrainedsigma-delta force-feedbackrdquo IEEE Sensors Journal vol 9 no 3pp 297ndash305 2009

[7] N-C Tsai and C-Y Sue ldquoIntegrated model reference adaptivecontrol and time-varying angular rate estimation for micro-machined gyroscopesrdquo International Journal of Control vol 83no 2 pp 246ndash256 2010

[8] M Saif B Ebrahimi and M Vali ldquoTerminal sliding modecontrol of Z-axis MEMS gyroscope with observer based rota-tion rate estimationrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3483ndash3489 July 2011

[9] D Zhu and Y Chai ldquoTracking control of multi-linked robotsbased on terminal sliding moderdquo in Proceedings of the 30thChinese Control Conference (CCC rsquo11) pp 2539ndash2543 July 2011

[10] Y Feng X Yu and ZMan ldquoAdaptive fast terminal slidingmodetracking control of robotic manipulatorrdquo in Proceedings of the40th IEEE Conference on Decision and Control (CDC rsquo01) pp4021ndash4026 December 2001

[11] P G Keleher and R J Stonier ldquoAdaptive terminal sliding modecontrol of rigid robotic manipulator with uncertain dynamicincorporating constraint inequalitiesrdquo ANZIAM Journal vol43 pp 102ndash153 2001

[12] C W Tao J S Taur and M-L Chan ldquoAdaptive fuzzy terminalsliding mode controller for linear systems with mismatchedtime-varying uncertaintiesrdquo IEEE Transactions on SystemsMan and Cybernetics Part B vol 34 no 1 pp 255ndash262 2004

[13] J Y Zhang W J Gu Z D Liu and R P Chen ldquoAdaptive fuzzyglobal identical terminal sliding mode control for cross-beamsystemrdquo in International Conference on Systems and Informaticspp 51ndash55 2012

[14] M B R Neila and D Tarak ldquoAdaptive terminal sliding modecontrol for rigid robotic manipulatorsrdquo International Journal ofAutomation and Computing vol 8 no 2 pp 215ndash220 2011

[15] Y Feng X Yu and Z Man ldquoNon-singular terminal slidingmode control of rigid manipulatorsrdquo Automatica vol 38 no12 pp 2159ndash2167 2002

[16] S Park Adaptive control strategies for MEMS gyroscope [PhDdissertation] University of California Berkeley Calif USA2000

[17] K-B Park and T Tsuji ldquoTerminal sliding mode control ofsecond-order nonlinear uncertain systemsrdquo International Jour-nal of Robust and Nonlinear Control vol 9 no 11 pp 769ndash7801999

[18] S Yu X Yu and Z Man ldquoRobust global terminal sliding modecontrol of SISO nonlinear uncertain systemsrdquo in Proceedings ofthe 39th IEEEConfernce onDecision andControl pp 2198ndash2203December 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


period Saif et al [8] presented a novel methodology forapproximation of the unknown time-varying rotation rate byusing a sliding mode observer along with a robust controlsystem based on terminal slidingmode control for improvingthe performance of the MEMS gyroscope Zhu and Chai[9] presented a global finite-time tracking controller basedon terminal sliding mode control method for the trajectorytracking control problem of biped multilinked robots withuncertain disturbances In order to solve the optimal problemof convergence time and estimate the unknown upper boundof system uncertainties much research has been done toapply intelligent control approaches such as the adaptiveglobal fast terminal sliding mode control Feng et al [10]and Keleher and Stonier [11] proposed an adaptive fastterminal sliding mode tracking control scheme for roboticmanipulators to estimate the upper bound of uncertaintiesTao et al [12] developed a new adaptive fuzzy terminalsliding mode controller for linear systems with mismatchedtime varying uncertainties Zhang et al [13] proposed a newapproach of adaptive fuzzy global identical terminal slidingmode control for cross-beam system Neila and Tarak [14]designed adaptive terminal sliding mode control for rigidrobotic manipulators Feng et al [15] proposed a nonsingularterminal sliding mode controller for rigid manipulators Inthis paper robust adaptive global fast terminal sliding modecontrol strategy for trajectory tracking control of MEMSvibratory gyroscopes is proposed The main motivations inthe paper are highlighted as follows

(1) A global fast terminal sliding mode surface is de-signed to improve the convergence rate of reachingthe sliding surface It integrates the advantages of tra-ditional sliding and terminal sliding control andguarantees that the tracking error can converge tozero in a shorter finite time Adaptive control andterminal sliding control are successfully integratedwith MEMS gyroscope

(2) An adaptive global fast terminal sliding mode con-trol is incorporated into the MEMS gyroscope toimprove the tracking performance and strengthen therobustness of the control system With this controlthe upper bound of model uncertainties and externaldisturbances are not needed because the proposedcontroller can estimate the upper bound of theseuncertainties using an adaptive mechanism Alsothe proposed controller eliminates the chatteringeffect without losing the robustness property and theprecision

This paper is organized as follows In Section 2 thedynamics of MEMS gyroscope are described In Section 3global fast terminal sliding mode control is introduced InSection 4 the proposed adaptive global fast terminal slidingmode controller is derived and the stability of this controlsystem is proved by Lyapunovrsquos stability theory Simulationresults are presented in Section 5 Finally conclusions areprovided in Section 6

proof massm

kxx

kyy

kyy

kxx

dxxdxx

dyy

dyy

x

x

y

y

Ωz

Figure 1 A simplified model of a 119911-axis MEMS Gyroscope

2 Dynamics of MEMS Gyroscope

A 119911-axis MEMS gyroscope is depicted in Figure 1 Generallya typical MEMS vibratory gyroscope includes a proof masssuspended by spring beams electrostatic actuations andsensing mechanisms for forcing an oscillatory motion andsensing the position and velocity of the proofmass Dynamicsof a MEMS gyroscope are derived from Newtonrsquos law in therotating frame

As we know Newtonrsquos law in the rotating frame becomes

119865119903= 119865phy + 119865centri + 119865Coriolis + 119865Euler = 119898119886

119903 (1)

In (1) 119865119903is the total applied force to the proof mass in

the gyro frame 119865phy is the total physical force to the proofmass in the inertial frame 119865centri is the centrifugal force119865Coriolis is the Coriolis force 119865Euler is the Euler force and 119886

119903

is the acceleration of the proof mass with respect to the gyroframe119865phy119865Coriolis and119865Euler are inertial forces caused by therotation of the gyro frame

With the definition of 119903119903and V

119903as the position and

velocity vectors relative to the rotating gyroscope frame andΩ as the angular velocity vector of the gyroscope frame theexpressions for the inertial forces are reduced to

119865Coriolis = minus2119898Ω times V119903 119865centri = minus119898Ω times (Ω times 119903

119903)

119865Euler = minus119898

119889Ω

119889119905

times 119903119903

(2)

then

119898119886119903+ 119898Ω times (Ω times 119903

119903) + 2119898Ω times V

119903+ 119898Ω times 119903

119903= 119865phy (3)

where 119865phy contains spring damping and control forcesapplied to the proof mass

In a 119911-axis MEMS gyroscope by supposing the stiffnessof spring in 119911 direction much larger than that in 119909 and 119910

directions motion of proof mass is constrained to only alongthe 119909 minus 119910 plane Referring to [16] with assuming that thegyroscope is almost rotating at a constant angular velocity




119898 + 119889119909 + [119896

119909minus 119898(Ω

2

119910+ Ω2

119911)] 119909 + 119898Ω

119909Ω119910119910

= 119906119909+ 2119898Ω

119911119910

119898 119910 + 119889119910

119910 + [119896119910minus 119898(Ω

2

119909+ Ω2

119911)] 119910 + 119898Ω

119909Ω119910119909

= 119906119910minus 2119898Ω

119911

(4)


and 119896119909119910





119911119910 and 2119898Ω




asymp Ω119910





119898 + 119889119909119909 + 119889119909119910

119910 + 119896119909119909119909 + 119896119909119910119910 = 119906

119909+ 2119898Ω

119911119910

119898 119910 + 119889119909119910 + 119889119910119910

119910 + 119896119909119910119909 + 119896119910119910119910 = 119906

119910minus 2119898Ω

119911

(5)

In (5) 119889119909119909

and 119889119910119910


and119896119910119910





2

0 which are


+ 119889119909119909 + 119889119909119910

119910 + 1205962

119909119909 + 120596119909119910119910 = 119906

119909+ 2Ω119911

119910

119910 + 119889119909119910 + 119889119910119910

119910 + 120596119909119910119909 + 1205962

119910119910 = 119906

119910minus 2Ω119911

(6)


radic1198961199091199091198981205962

0rarr 120596

119909 1198961199091199101198981205962

0rarr 120596

119909119910 radic1198961199101199101198981205962

0rarr 120596

119910

Ω1199111205960rarr Ω119911


119902 + 119863 119902 + 119870119902 = 119906 minus 2Ω 119902 (7)

where 119902 = [119909

119910 ] 119863 = [119889119909119909

119889119909119910

119889119909119910

119889119910119910

] 119870 = [1205962

119909

120596119909119910

120596119909119910

1205962

119910

] 119906 = [119906119909

119906119910]

Ω = [0

Ω119911

minusΩ119911

0]


119902 + (119863 + 2Ω + Δ119863) 119902 + (119870 + Δ119870) 119902 = 119906 + 119889 (8)







119904119886= 1+ 12057301199091

11990121199011

= 0 (9)



1





1198881 The



1199051198881

=

1199011

1205730(1199011minus 1199012)

10038161003816100381610038161199091(0)

1003816100381610038161003816

(1199011minus1199012)1199011

(10)


119904119887= 1+ 12057201199091+ 120573011990911990121199011

1= 0 (11)


1(0) = 0 the




1= 0


1199051198882

=

1199011

1205720(1199011minus 1199012)

ln12057201199091(0)(1199011minus1199012)1199011+ 1205730

1205730

(12)


le 1199051198881


1= minus120573

011990911990121199011

1





Expectationmodel

GFSTMcontroller

MEMSgyroscope

Adaptivelaw

Estimatedparameters

Input

Disturbance

qr

minus

+

e u q

b










119904119888= [1199041198881 1199041198882]119879

= 119890 + 120572119890 + 12057311989011990121199011

(13)





119903



119904= [1199051199041 1199051199042]119879 The time 119905

119904is

determined by

119905119904= [

1199011

1205721(1199011minus 1199012)

ln12057211199021(0)(1199011minus1199012)1199011+ 1205731

1205731

1199011

1205722(1199011minus 1199012)

ln12057221199022(0)(1199011minus1199012)1199011+ 1205732

1205732

]

(14)



1 and





119902 + (119863 + 2Ω) 119902 + 119870119902 = 119906 + 119891 (15)


119891 = 119889 minus Δ119863 119902 minus Δ119870119902 (16)





10038171003817100381710038171198911003817100381710038171003817lt 1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

(17)

where 1198870 1198871 and 119887



0


119906 = 1199060+ 1199061+ 1199062 (18)



2is


1199060= 119902119903+ (119863 + 2Ω) 119902 + 119870119902 (19)

1199061= minus120572 119890 minus

1199012

1199011

120573 diag (11989011990121199011minus11

11989011990121199011minus1

2) 119890 (20)

1199062= minus120588

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(21)


120588 = 120585 + 1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817 (22)





0 1198871 and


of 1198870 1198871 and 119887



0 1198871 and 119887

2 Consider

1198870= 1198980

1003817100381710038171003817119904119888

1003817100381710038171003817

1198871= 1198981

1003817100381710038171003817119904119888

1003817100381710038171003817

10038171003817100381710038171199021003817100381710038171003817

1198872= 1198982

1003817100381710038171003817119904119888

1003817100381710038171003817

1003817100381710038171003817

1199021003817100381710038171003817

2

(23)

where 1198980 1198981 and 119898




119906 = 1199060+ 1199061+ 1199062 (24)



2is

defined in (27)

1199060= 119886 + (119863 + 2Ω) V + 119870119902 (25)

1199061= minus119882

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(26)

1199062= minus

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) (27)



V = 119902119903minus 120572119890 minus 120573119890

11990121199011

119886 = V = 119902119903minus 120572 119890 minus

1199012

1199011

120573 diag (11989011990121199011minus11

11989011990121199011minus1

2) 119890

(28)



119881 =

1

2

119904119879

119888119904119888+

1

2

2

sum

119894=0

119898minus1

119894

1198872

119894 (29)



0 1198871 and 119887


119887119894= minus

119887119894


119902 + (119863 + 2Ω + Δ119863) 119902 + (119870 + Δ119870) 119902

= 119886 + (119863 + 2Ω) V + 119870119902 + 1199061+ 1199062+ 119889

(30)


( 119902 minus 119886) + (119863 + 2Ω) ( 119902 minus V) + Δ119863 119902 + Δ119870119902

= 1199061+ 1199062+ 119889

(31)


119904119888+ (119863 + 2Ω) 119904

119888+ Δ119863 119902 + Δ119870119902 = 119906

1+ 1199062+ 119889 (32)


119904119888= minus (119863 + 2Ω) 119904

119888minus Δ119863 119902 minus Δ119870119902 + 119906

1+ 1199062+ 119889 (33)



119904119888= minus (119863 + 2Ω) 119904

119888+ 1199061+ 1199062+ 119891 (34)


= 119904119879

119888119904119888minus

2

sum

119894=0

119898minus1

119894

119887119894

119887119894

= 119904119879

119888[minus (119863 + 2Ω) 119904

119888+ 1199061+ 1199062+ 119891] minus 119898

minus1

0

1198870

1198870

minus 119898minus1

1

1198871


2

1198872

1198872

= 119904119879

119888(1199061+ 1199062+ 119891) + 119904

119879

119888[minus (119863 + 2Ω)] 119904

119888

minus 119898minus1

0

1198870


1

1198871


2

1198872

1198872

le 119904119879

119888(1199061+ 1199062+ 119891) minus 119898

minus1

0

1198870


1

1198871


2

1198872

1198872

le minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888(1199062+ 119891) minus 119898

minus1

0

1198870

1198870

minus 119898minus1

1

1198871


2

1198872

1198872= minus120582min (119882)

1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888119891

minus1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

)

minus1198870

1003817100381710038171003817119904119888

1003817100381710038171003817minus1198871

1003817100381710038171003817119904119888

1003817100381710038171003817

10038171003817100381710038171199021003817100381710038171003817minus1198872

1003817100381710038171003817119904119888

1003817100381710038171003817

1003817100381710038171003817

1199021003817100381710038171003817

2

= minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888119891 minus

1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

)

le minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+1003817100381710038171003817119904119888

1003817100381710038171003817119891

minus1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) le minus1205781003817100381710038171003817119904119888

1003817100381710038171003817lt 0

(35)

where 120578 = 120582min(119882) minus 119891 + (1198870+ 1198871119902 + 119887

2 1199022



119888= 119890 + 120572119890 + 120573119890

11990121199011= 0


time


120582min (119882) 1199042

le 119904119879

119882119904 le 120582max (119882) 1199042

(36)


1 119908

119899] 120582min(119882) and 120582max(119882)


120582min (119882) = min119894=1119899

119908119894

120582max (119882) = max119894=1119899

119908119894

(37)



1199061=

minus119882

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

if 1003817100381710038171003817119904119888

1003817100381710038171003817gt 120575

minus119882

119904119888

120575

if 1003817100381710038171003817119904119888

1003817100381710038171003817le 120575

1199062=

minus

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) if 1003817100381710038171003817119904119888

1003817100381710038171003817gt 120575

minus

119904119888

120575

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) if 1003817100381710038171003817119904119888

1003817100381710038171003817le 120575

(38)






119898 = 18 times 10minus7 kg 119896

119909119909= 63955Nm

119896119910119910

= 9592Nm 119896119909119910

= 12779Nm

119889119909119909


119910119910= 18 times 10

minus6Nsm

119889119909119910


(39)







1205962

119909= 3553 120596

2

119910= 5329

120596119909119910

= 7099 119889119909119909

= 001

119889119910119910

= 001 119889119909119910

= 0002 Ω119911= 01

(40)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

15

Time (s)

Posit

ion

trac

king

of x

-axi

s


minus05

minus1


0 1 2 3 4 5 6 7 8 9 10Time (s)


0

05

1

15

Posit

ion

trac

king

of y

-axi

s

minus05

minus15

minus1



1(0) = 10 119902

1(0) = 0 119902

2(0) =



= sin(120587119905) and 1199021199032


119909119910and

120596119909119910


2= 3 120572

1= 1205722= 025 and 120573

1= 1205732= 05 and


= 01 11988710

= 02 and 11988720


0= 10 119898

1= 16 and 119898

2= 16 The sliding




0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

SMCAGFTSMC

minus02

minus04

Posit

ion

trac

king

erro

r of x

-axi

s


0 1 2 3 4 5 6 7 8 9 10

0

01

02

Time (s)

minus01

minus02

minus03

minus04

minus05

Posit

ion

trac

king

erro

r of y

-axi

s

SMCAGFTSMC





0 1 2 3 4 5 6 7 8 9 10

005

Time (s)

002

minus25minus2

minus15minus1

minus05

minus02

minus04

minus06

Term

inal

slid

ing

Term

inal

slid

ing

0 1 2 3 4 5 6 7 8 9 10Time (s)

surfa

ces 1

surfa

ces 2


0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux





0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux


0

05

1

005

115

0123

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Estim

ates

ofb

2Es

timat

es o

fb1

Estim

ates

ofb

0


1198872


1198872



6 Conclusion




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119898 + 119889119909 + [119896

119909minus 119898(Ω

2

119910+ Ω2

119911)] 119909 + 119898Ω

119909Ω119910119910

= 119906119909+ 2119898Ω

119911119910

119898 119910 + 119889119910

119910 + [119896119910minus 119898(Ω

2

119909+ Ω2

119911)] 119910 + 119898Ω

119909Ω119910119909

= 119906119910minus 2119898Ω

119911

(4)


and 119896119909119910





119911119910 and 2119898Ω




asymp Ω119910





119898 + 119889119909119909 + 119889119909119910

119910 + 119896119909119909119909 + 119896119909119910119910 = 119906

119909+ 2119898Ω

119911119910

119898 119910 + 119889119909119910 + 119889119910119910

119910 + 119896119909119910119909 + 119896119910119910119910 = 119906

119910minus 2119898Ω

119911

(5)

In (5) 119889119909119909

and 119889119910119910


and119896119910119910





2

0 which are


+ 119889119909119909 + 119889119909119910

119910 + 1205962

119909119909 + 120596119909119910119910 = 119906

119909+ 2Ω119911

119910

119910 + 119889119909119910 + 119889119910119910

119910 + 120596119909119910119909 + 1205962

119910119910 = 119906

119910minus 2Ω119911

(6)


radic1198961199091199091198981205962

0rarr 120596

119909 1198961199091199101198981205962

0rarr 120596

119909119910 radic1198961199101199101198981205962

0rarr 120596

119910

Ω1199111205960rarr Ω119911


119902 + 119863 119902 + 119870119902 = 119906 minus 2Ω 119902 (7)

where 119902 = [119909

119910 ] 119863 = [119889119909119909

119889119909119910

119889119909119910

119889119910119910

] 119870 = [1205962

119909

120596119909119910

120596119909119910

1205962

119910

] 119906 = [119906119909

119906119910]

Ω = [0

Ω119911

minusΩ119911

0]


119902 + (119863 + 2Ω + Δ119863) 119902 + (119870 + Δ119870) 119902 = 119906 + 119889 (8)







119904119886= 1+ 12057301199091

11990121199011

= 0 (9)



1





1198881 The



1199051198881

=

1199011

1205730(1199011minus 1199012)

10038161003816100381610038161199091(0)

1003816100381610038161003816

(1199011minus1199012)1199011

(10)


119904119887= 1+ 12057201199091+ 120573011990911990121199011

1= 0 (11)


1(0) = 0 the




1= 0


1199051198882

=

1199011

1205720(1199011minus 1199012)

ln12057201199091(0)(1199011minus1199012)1199011+ 1205730

1205730

(12)


le 1199051198881


1= minus120573

011990911990121199011

1





Expectationmodel

GFSTMcontroller

MEMSgyroscope

Adaptivelaw

Estimatedparameters

Input

Disturbance

qr

minus

+

e u q

b










119904119888= [1199041198881 1199041198882]119879

= 119890 + 120572119890 + 12057311989011990121199011

(13)





119903



119904= [1199051199041 1199051199042]119879 The time 119905

119904is

determined by

119905119904= [

1199011

1205721(1199011minus 1199012)

ln12057211199021(0)(1199011minus1199012)1199011+ 1205731

1205731

1199011

1205722(1199011minus 1199012)

ln12057221199022(0)(1199011minus1199012)1199011+ 1205732

1205732

]

(14)



1 and





119902 + (119863 + 2Ω) 119902 + 119870119902 = 119906 + 119891 (15)


119891 = 119889 minus Δ119863 119902 minus Δ119870119902 (16)





10038171003817100381710038171198911003817100381710038171003817lt 1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

(17)

where 1198870 1198871 and 119887



0


119906 = 1199060+ 1199061+ 1199062 (18)



2is


1199060= 119902119903+ (119863 + 2Ω) 119902 + 119870119902 (19)

1199061= minus120572 119890 minus

1199012

1199011

120573 diag (11989011990121199011minus11

11989011990121199011minus1

2) 119890 (20)

1199062= minus120588

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(21)


120588 = 120585 + 1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817 (22)





0 1198871 and


of 1198870 1198871 and 119887



0 1198871 and 119887

2 Consider

1198870= 1198980

1003817100381710038171003817119904119888

1003817100381710038171003817

1198871= 1198981

1003817100381710038171003817119904119888

1003817100381710038171003817

10038171003817100381710038171199021003817100381710038171003817

1198872= 1198982

1003817100381710038171003817119904119888

1003817100381710038171003817

1003817100381710038171003817

1199021003817100381710038171003817

2

(23)

where 1198980 1198981 and 119898




119906 = 1199060+ 1199061+ 1199062 (24)



2is

defined in (27)

1199060= 119886 + (119863 + 2Ω) V + 119870119902 (25)

1199061= minus119882

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(26)

1199062= minus

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) (27)



V = 119902119903minus 120572119890 minus 120573119890

11990121199011

119886 = V = 119902119903minus 120572 119890 minus

1199012

1199011

120573 diag (11989011990121199011minus11

11989011990121199011minus1

2) 119890

(28)



119881 =

1

2

119904119879

119888119904119888+

1

2

2

sum

119894=0

119898minus1

119894

1198872

119894 (29)



0 1198871 and 119887


119887119894= minus

119887119894


119902 + (119863 + 2Ω + Δ119863) 119902 + (119870 + Δ119870) 119902

= 119886 + (119863 + 2Ω) V + 119870119902 + 1199061+ 1199062+ 119889

(30)


( 119902 minus 119886) + (119863 + 2Ω) ( 119902 minus V) + Δ119863 119902 + Δ119870119902

= 1199061+ 1199062+ 119889

(31)


119904119888+ (119863 + 2Ω) 119904

119888+ Δ119863 119902 + Δ119870119902 = 119906

1+ 1199062+ 119889 (32)


119904119888= minus (119863 + 2Ω) 119904

119888minus Δ119863 119902 minus Δ119870119902 + 119906

1+ 1199062+ 119889 (33)



119904119888= minus (119863 + 2Ω) 119904

119888+ 1199061+ 1199062+ 119891 (34)


= 119904119879

119888119904119888minus

2

sum

119894=0

119898minus1

119894

119887119894

119887119894

= 119904119879

119888[minus (119863 + 2Ω) 119904

119888+ 1199061+ 1199062+ 119891] minus 119898

minus1

0

1198870

1198870

minus 119898minus1

1

1198871


2

1198872

1198872

= 119904119879

119888(1199061+ 1199062+ 119891) + 119904

119879

119888[minus (119863 + 2Ω)] 119904

119888

minus 119898minus1

0

1198870


1

1198871


2

1198872

1198872

le 119904119879

119888(1199061+ 1199062+ 119891) minus 119898

minus1

0

1198870


1

1198871


2

1198872

1198872

le minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888(1199062+ 119891) minus 119898

minus1

0

1198870

1198870

minus 119898minus1

1

1198871


2

1198872

1198872= minus120582min (119882)

1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888119891

minus1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

)

minus1198870

1003817100381710038171003817119904119888

1003817100381710038171003817minus1198871

1003817100381710038171003817119904119888

1003817100381710038171003817

10038171003817100381710038171199021003817100381710038171003817minus1198872

1003817100381710038171003817119904119888

1003817100381710038171003817

1003817100381710038171003817

1199021003817100381710038171003817

2

= minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888119891 minus

1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

)

le minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+1003817100381710038171003817119904119888

1003817100381710038171003817119891

minus1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) le minus1205781003817100381710038171003817119904119888

1003817100381710038171003817lt 0

(35)

where 120578 = 120582min(119882) minus 119891 + (1198870+ 1198871119902 + 119887

2 1199022



119888= 119890 + 120572119890 + 120573119890

11990121199011= 0


time


120582min (119882) 1199042

le 119904119879

119882119904 le 120582max (119882) 1199042

(36)


1 119908

119899] 120582min(119882) and 120582max(119882)


120582min (119882) = min119894=1119899

119908119894

120582max (119882) = max119894=1119899

119908119894

(37)



1199061=

minus119882

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

if 1003817100381710038171003817119904119888

1003817100381710038171003817gt 120575

minus119882

119904119888

120575

if 1003817100381710038171003817119904119888

1003817100381710038171003817le 120575

1199062=

minus

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) if 1003817100381710038171003817119904119888

1003817100381710038171003817gt 120575

minus

119904119888

120575

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) if 1003817100381710038171003817119904119888

1003817100381710038171003817le 120575

(38)






119898 = 18 times 10minus7 kg 119896

119909119909= 63955Nm

119896119910119910

= 9592Nm 119896119909119910

= 12779Nm

119889119909119909


119910119910= 18 times 10

minus6Nsm

119889119909119910


(39)







1205962

119909= 3553 120596

2

119910= 5329

120596119909119910

= 7099 119889119909119909

= 001

119889119910119910

= 001 119889119909119910

= 0002 Ω119911= 01

(40)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

15

Time (s)

Posit

ion

trac

king

of x

-axi

s


minus05

minus1


0 1 2 3 4 5 6 7 8 9 10Time (s)


0

05

1

15

Posit

ion

trac

king

of y

-axi

s

minus05

minus15

minus1



1(0) = 10 119902

1(0) = 0 119902

2(0) =



= sin(120587119905) and 1199021199032


119909119910and

120596119909119910


2= 3 120572

1= 1205722= 025 and 120573

1= 1205732= 05 and


= 01 11988710

= 02 and 11988720


0= 10 119898

1= 16 and 119898

2= 16 The sliding




0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

SMCAGFTSMC

minus02

minus04

Posit

ion

trac

king

erro

r of x

-axi

s


0 1 2 3 4 5 6 7 8 9 10

0

01

02

Time (s)

minus01

minus02

minus03

minus04

minus05

Posit

ion

trac

king

erro

r of y

-axi

s

SMCAGFTSMC





0 1 2 3 4 5 6 7 8 9 10

005

Time (s)

002

minus25minus2

minus15minus1

minus05

minus02

minus04

minus06

Term

inal

slid

ing

Term

inal

slid

ing

0 1 2 3 4 5 6 7 8 9 10Time (s)

surfa

ces 1

surfa

ces 2


0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux





0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux


0

05

1

005

115

0123

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Estim

ates

ofb

2Es

timat

es o

fb1

Estim

ates

ofb

0


1198872


1198872



6 Conclusion




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Expectationmodel

GFSTMcontroller

MEMSgyroscope

Adaptivelaw

Estimatedparameters

Input

Disturbance

qr

minus

+

e u q

b










119904119888= [1199041198881 1199041198882]119879

= 119890 + 120572119890 + 12057311989011990121199011

(13)





119903



119904= [1199051199041 1199051199042]119879 The time 119905

119904is

determined by

119905119904= [

1199011

1205721(1199011minus 1199012)

ln12057211199021(0)(1199011minus1199012)1199011+ 1205731

1205731

1199011

1205722(1199011minus 1199012)

ln12057221199022(0)(1199011minus1199012)1199011+ 1205732

1205732

]

(14)



1 and





119902 + (119863 + 2Ω) 119902 + 119870119902 = 119906 + 119891 (15)


119891 = 119889 minus Δ119863 119902 minus Δ119870119902 (16)





10038171003817100381710038171198911003817100381710038171003817lt 1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

(17)

where 1198870 1198871 and 119887



0


119906 = 1199060+ 1199061+ 1199062 (18)



2is


1199060= 119902119903+ (119863 + 2Ω) 119902 + 119870119902 (19)

1199061= minus120572 119890 minus

1199012

1199011

120573 diag (11989011990121199011minus11

11989011990121199011minus1

2) 119890 (20)

1199062= minus120588

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(21)


120588 = 120585 + 1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817 (22)





0 1198871 and


of 1198870 1198871 and 119887



0 1198871 and 119887

2 Consider

1198870= 1198980

1003817100381710038171003817119904119888

1003817100381710038171003817

1198871= 1198981

1003817100381710038171003817119904119888

1003817100381710038171003817

10038171003817100381710038171199021003817100381710038171003817

1198872= 1198982

1003817100381710038171003817119904119888

1003817100381710038171003817

1003817100381710038171003817

1199021003817100381710038171003817

2

(23)

where 1198980 1198981 and 119898




119906 = 1199060+ 1199061+ 1199062 (24)



2is

defined in (27)

1199060= 119886 + (119863 + 2Ω) V + 119870119902 (25)

1199061= minus119882

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(26)

1199062= minus

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) (27)



V = 119902119903minus 120572119890 minus 120573119890

11990121199011

119886 = V = 119902119903minus 120572 119890 minus

1199012

1199011

120573 diag (11989011990121199011minus11

11989011990121199011minus1

2) 119890

(28)



119881 =

1

2

119904119879

119888119904119888+

1

2

2

sum

119894=0

119898minus1

119894

1198872

119894 (29)



0 1198871 and 119887


119887119894= minus

119887119894


119902 + (119863 + 2Ω + Δ119863) 119902 + (119870 + Δ119870) 119902

= 119886 + (119863 + 2Ω) V + 119870119902 + 1199061+ 1199062+ 119889

(30)


( 119902 minus 119886) + (119863 + 2Ω) ( 119902 minus V) + Δ119863 119902 + Δ119870119902

= 1199061+ 1199062+ 119889

(31)


119904119888+ (119863 + 2Ω) 119904

119888+ Δ119863 119902 + Δ119870119902 = 119906

1+ 1199062+ 119889 (32)


119904119888= minus (119863 + 2Ω) 119904

119888minus Δ119863 119902 minus Δ119870119902 + 119906

1+ 1199062+ 119889 (33)



119904119888= minus (119863 + 2Ω) 119904

119888+ 1199061+ 1199062+ 119891 (34)


= 119904119879

119888119904119888minus

2

sum

119894=0

119898minus1

119894

119887119894

119887119894

= 119904119879

119888[minus (119863 + 2Ω) 119904

119888+ 1199061+ 1199062+ 119891] minus 119898

minus1

0

1198870

1198870

minus 119898minus1

1

1198871


2

1198872

1198872

= 119904119879

119888(1199061+ 1199062+ 119891) + 119904

119879

119888[minus (119863 + 2Ω)] 119904

119888

minus 119898minus1

0

1198870


1

1198871


2

1198872

1198872

le 119904119879

119888(1199061+ 1199062+ 119891) minus 119898

minus1

0

1198870


1

1198871


2

1198872

1198872

le minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888(1199062+ 119891) minus 119898

minus1

0

1198870

1198870

minus 119898minus1

1

1198871


2

1198872

1198872= minus120582min (119882)

1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888119891

minus1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

)

minus1198870

1003817100381710038171003817119904119888

1003817100381710038171003817minus1198871

1003817100381710038171003817119904119888

1003817100381710038171003817

10038171003817100381710038171199021003817100381710038171003817minus1198872

1003817100381710038171003817119904119888

1003817100381710038171003817

1003817100381710038171003817

1199021003817100381710038171003817

2

= minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888119891 minus

1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

)

le minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+1003817100381710038171003817119904119888

1003817100381710038171003817119891

minus1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) le minus1205781003817100381710038171003817119904119888

1003817100381710038171003817lt 0

(35)

where 120578 = 120582min(119882) minus 119891 + (1198870+ 1198871119902 + 119887

2 1199022



119888= 119890 + 120572119890 + 120573119890

11990121199011= 0


time


120582min (119882) 1199042

le 119904119879

119882119904 le 120582max (119882) 1199042

(36)


1 119908

119899] 120582min(119882) and 120582max(119882)


120582min (119882) = min119894=1119899

119908119894

120582max (119882) = max119894=1119899

119908119894

(37)



1199061=

minus119882

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

if 1003817100381710038171003817119904119888

1003817100381710038171003817gt 120575

minus119882

119904119888

120575

if 1003817100381710038171003817119904119888

1003817100381710038171003817le 120575

1199062=

minus

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) if 1003817100381710038171003817119904119888

1003817100381710038171003817gt 120575

minus

119904119888

120575

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) if 1003817100381710038171003817119904119888

1003817100381710038171003817le 120575

(38)






119898 = 18 times 10minus7 kg 119896

119909119909= 63955Nm

119896119910119910

= 9592Nm 119896119909119910

= 12779Nm

119889119909119909


119910119910= 18 times 10

minus6Nsm

119889119909119910


(39)







1205962

119909= 3553 120596

2

119910= 5329

120596119909119910

= 7099 119889119909119909

= 001

119889119910119910

= 001 119889119909119910

= 0002 Ω119911= 01

(40)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

15

Time (s)

Posit

ion

trac

king

of x

-axi

s


minus05

minus1


0 1 2 3 4 5 6 7 8 9 10Time (s)


0

05

1

15

Posit

ion

trac

king

of y

-axi

s

minus05

minus15

minus1



1(0) = 10 119902

1(0) = 0 119902

2(0) =



= sin(120587119905) and 1199021199032


119909119910and

120596119909119910


2= 3 120572

1= 1205722= 025 and 120573

1= 1205732= 05 and


= 01 11988710

= 02 and 11988720


0= 10 119898

1= 16 and 119898

2= 16 The sliding




0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

SMCAGFTSMC

minus02

minus04

Posit

ion

trac

king

erro

r of x

-axi

s


0 1 2 3 4 5 6 7 8 9 10

0

01

02

Time (s)

minus01

minus02

minus03

minus04

minus05

Posit

ion

trac

king

erro

r of y

-axi

s

SMCAGFTSMC





0 1 2 3 4 5 6 7 8 9 10

005

Time (s)

002

minus25minus2

minus15minus1

minus05

minus02

minus04

minus06

Term

inal

slid

ing

Term

inal

slid

ing

0 1 2 3 4 5 6 7 8 9 10Time (s)

surfa

ces 1

surfa

ces 2


0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux





0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux


0

05

1

005

115

0123

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Estim

ates

ofb

2Es

timat

es o

fb1

Estim

ates

ofb

0


1198872


1198872



6 Conclusion




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













10038171003817100381710038171198911003817100381710038171003817lt 1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

(17)

where 1198870 1198871 and 119887



0


119906 = 1199060+ 1199061+ 1199062 (18)



2is


1199060= 119902119903+ (119863 + 2Ω) 119902 + 119870119902 (19)

1199061= minus120572 119890 minus

1199012

1199011

120573 diag (11989011990121199011minus11

11989011990121199011minus1

2) 119890 (20)

1199062= minus120588

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(21)


120588 = 120585 + 1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817 (22)





0 1198871 and


of 1198870 1198871 and 119887



0 1198871 and 119887

2 Consider

1198870= 1198980

1003817100381710038171003817119904119888

1003817100381710038171003817

1198871= 1198981

1003817100381710038171003817119904119888

1003817100381710038171003817

10038171003817100381710038171199021003817100381710038171003817

1198872= 1198982

1003817100381710038171003817119904119888

1003817100381710038171003817

1003817100381710038171003817

1199021003817100381710038171003817

2

(23)

where 1198980 1198981 and 119898




119906 = 1199060+ 1199061+ 1199062 (24)



2is

defined in (27)

1199060= 119886 + (119863 + 2Ω) V + 119870119902 (25)

1199061= minus119882

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(26)

1199062= minus

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) (27)



V = 119902119903minus 120572119890 minus 120573119890

11990121199011

119886 = V = 119902119903minus 120572 119890 minus

1199012

1199011

120573 diag (11989011990121199011minus11

11989011990121199011minus1

2) 119890

(28)



119881 =

1

2

119904119879

119888119904119888+

1

2

2

sum

119894=0

119898minus1

119894

1198872

119894 (29)



0 1198871 and 119887


119887119894= minus

119887119894


119902 + (119863 + 2Ω + Δ119863) 119902 + (119870 + Δ119870) 119902

= 119886 + (119863 + 2Ω) V + 119870119902 + 1199061+ 1199062+ 119889

(30)


( 119902 minus 119886) + (119863 + 2Ω) ( 119902 minus V) + Δ119863 119902 + Δ119870119902

= 1199061+ 1199062+ 119889

(31)


119904119888+ (119863 + 2Ω) 119904

119888+ Δ119863 119902 + Δ119870119902 = 119906

1+ 1199062+ 119889 (32)


119904119888= minus (119863 + 2Ω) 119904

119888minus Δ119863 119902 minus Δ119870119902 + 119906

1+ 1199062+ 119889 (33)



119904119888= minus (119863 + 2Ω) 119904

119888+ 1199061+ 1199062+ 119891 (34)


= 119904119879

119888119904119888minus

2

sum

119894=0

119898minus1

119894

119887119894

119887119894

= 119904119879

119888[minus (119863 + 2Ω) 119904

119888+ 1199061+ 1199062+ 119891] minus 119898

minus1

0

1198870

1198870

minus 119898minus1

1

1198871


2

1198872

1198872

= 119904119879

119888(1199061+ 1199062+ 119891) + 119904

119879

119888[minus (119863 + 2Ω)] 119904

119888

minus 119898minus1

0

1198870


1

1198871


2

1198872

1198872

le 119904119879

119888(1199061+ 1199062+ 119891) minus 119898

minus1

0

1198870


1

1198871


2

1198872

1198872

le minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888(1199062+ 119891) minus 119898

minus1

0

1198870

1198870

minus 119898minus1

1

1198871


2

1198872

1198872= minus120582min (119882)

1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888119891

minus1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

)

minus1198870

1003817100381710038171003817119904119888

1003817100381710038171003817minus1198871

1003817100381710038171003817119904119888

1003817100381710038171003817

10038171003817100381710038171199021003817100381710038171003817minus1198872

1003817100381710038171003817119904119888

1003817100381710038171003817

1003817100381710038171003817

1199021003817100381710038171003817

2

= minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888119891 minus

1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

)

le minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+1003817100381710038171003817119904119888

1003817100381710038171003817119891

minus1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) le minus1205781003817100381710038171003817119904119888

1003817100381710038171003817lt 0

(35)

where 120578 = 120582min(119882) minus 119891 + (1198870+ 1198871119902 + 119887

2 1199022



119888= 119890 + 120572119890 + 120573119890

11990121199011= 0


time


120582min (119882) 1199042

le 119904119879

119882119904 le 120582max (119882) 1199042

(36)


1 119908

119899] 120582min(119882) and 120582max(119882)


120582min (119882) = min119894=1119899

119908119894

120582max (119882) = max119894=1119899

119908119894

(37)



1199061=

minus119882

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

if 1003817100381710038171003817119904119888

1003817100381710038171003817gt 120575

minus119882

119904119888

120575

if 1003817100381710038171003817119904119888

1003817100381710038171003817le 120575

1199062=

minus

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) if 1003817100381710038171003817119904119888

1003817100381710038171003817gt 120575

minus

119904119888

120575

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) if 1003817100381710038171003817119904119888

1003817100381710038171003817le 120575

(38)






119898 = 18 times 10minus7 kg 119896

119909119909= 63955Nm

119896119910119910

= 9592Nm 119896119909119910

= 12779Nm

119889119909119909


119910119910= 18 times 10

minus6Nsm

119889119909119910


(39)







1205962

119909= 3553 120596

2

119910= 5329

120596119909119910

= 7099 119889119909119909

= 001

119889119910119910

= 001 119889119909119910

= 0002 Ω119911= 01

(40)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

15

Time (s)

Posit

ion

trac

king

of x

-axi

s


minus05

minus1


0 1 2 3 4 5 6 7 8 9 10Time (s)


0

05

1

15

Posit

ion

trac

king

of y

-axi

s

minus05

minus15

minus1



1(0) = 10 119902

1(0) = 0 119902

2(0) =



= sin(120587119905) and 1199021199032


119909119910and

120596119909119910


2= 3 120572

1= 1205722= 025 and 120573

1= 1205732= 05 and


= 01 11988710

= 02 and 11988720


0= 10 119898

1= 16 and 119898

2= 16 The sliding




0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

SMCAGFTSMC

minus02

minus04

Posit

ion

trac

king

erro

r of x

-axi

s


0 1 2 3 4 5 6 7 8 9 10

0

01

02

Time (s)

minus01

minus02

minus03

minus04

minus05

Posit

ion

trac

king

erro

r of y

-axi

s

SMCAGFTSMC





0 1 2 3 4 5 6 7 8 9 10

005

Time (s)

002

minus25minus2

minus15minus1

minus05

minus02

minus04

minus06

Term

inal

slid

ing

Term

inal

slid

ing

0 1 2 3 4 5 6 7 8 9 10Time (s)

surfa

ces 1

surfa

ces 2


0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux





0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux


0

05

1

005

115

0123

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Estim

ates

ofb

2Es

timat

es o

fb1

Estim

ates

ofb

0


1198872


1198872



6 Conclusion




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119904119888= minus (119863 + 2Ω) 119904

119888+ 1199061+ 1199062+ 119891 (34)


= 119904119879

119888119904119888minus

2

sum

119894=0

119898minus1

119894

119887119894

119887119894

= 119904119879

119888[minus (119863 + 2Ω) 119904

119888+ 1199061+ 1199062+ 119891] minus 119898

minus1

0

1198870

1198870

minus 119898minus1

1

1198871


2

1198872

1198872

= 119904119879

119888(1199061+ 1199062+ 119891) + 119904

119879

119888[minus (119863 + 2Ω)] 119904

119888

minus 119898minus1

0

1198870


1

1198871


2

1198872

1198872

le 119904119879

119888(1199061+ 1199062+ 119891) minus 119898

minus1

0

1198870


1

1198871


2

1198872

1198872

le minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888(1199062+ 119891) minus 119898

minus1

0

1198870

1198870

minus 119898minus1

1

1198871


2

1198872

1198872= minus120582min (119882)

1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888119891

minus1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

)

minus1198870

1003817100381710038171003817119904119888

1003817100381710038171003817minus1198871

1003817100381710038171003817119904119888

1003817100381710038171003817

10038171003817100381710038171199021003817100381710038171003817minus1198872

1003817100381710038171003817119904119888

1003817100381710038171003817

1003817100381710038171003817

1199021003817100381710038171003817

2

= minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+ 119904119879

119888119891 minus

1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

)

le minus120582min (119882)1003817100381710038171003817119904119888

1003817100381710038171003817+1003817100381710038171003817119904119888

1003817100381710038171003817119891

minus1003817100381710038171003817119904119888

1003817100381710038171003817(1198870+ 1198871

10038171003817100381710038171199021003817100381710038171003817+ 1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) le minus1205781003817100381710038171003817119904119888

1003817100381710038171003817lt 0

(35)

where 120578 = 120582min(119882) minus 119891 + (1198870+ 1198871119902 + 119887

2 1199022



119888= 119890 + 120572119890 + 120573119890

11990121199011= 0


time


120582min (119882) 1199042

le 119904119879

119882119904 le 120582max (119882) 1199042

(36)


1 119908

119899] 120582min(119882) and 120582max(119882)


120582min (119882) = min119894=1119899

119908119894

120582max (119882) = max119894=1119899

119908119894

(37)



1199061=

minus119882

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

if 1003817100381710038171003817119904119888

1003817100381710038171003817gt 120575

minus119882

119904119888

120575

if 1003817100381710038171003817119904119888

1003817100381710038171003817le 120575

1199062=

minus

119904119888

1003817100381710038171003817119904119888

1003817100381710038171003817

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) if 1003817100381710038171003817119904119888

1003817100381710038171003817gt 120575

minus

119904119888

120575

(1198870+1198871

10038171003817100381710038171199021003817100381710038171003817+1198872

1003817100381710038171003817

1199021003817100381710038171003817

2

) if 1003817100381710038171003817119904119888

1003817100381710038171003817le 120575

(38)






119898 = 18 times 10minus7 kg 119896

119909119909= 63955Nm

119896119910119910

= 9592Nm 119896119909119910

= 12779Nm

119889119909119909


119910119910= 18 times 10

minus6Nsm

119889119909119910


(39)







1205962

119909= 3553 120596

2

119910= 5329

120596119909119910

= 7099 119889119909119909

= 001

119889119910119910

= 001 119889119909119910

= 0002 Ω119911= 01

(40)


0 1 2 3 4 5 6 7 8 9 10

0

05

1

15

Time (s)

Posit

ion

trac

king

of x

-axi

s


minus05

minus1


0 1 2 3 4 5 6 7 8 9 10Time (s)


0

05

1

15

Posit

ion

trac

king

of y

-axi

s

minus05

minus15

minus1



1(0) = 10 119902

1(0) = 0 119902

2(0) =



= sin(120587119905) and 1199021199032


119909119910and

120596119909119910


2= 3 120572

1= 1205722= 025 and 120573

1= 1205732= 05 and


= 01 11988710

= 02 and 11988720


0= 10 119898

1= 16 and 119898

2= 16 The sliding




0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

SMCAGFTSMC

minus02

minus04

Posit

ion

trac

king

erro

r of x

-axi

s


0 1 2 3 4 5 6 7 8 9 10

0

01

02

Time (s)

minus01

minus02

minus03

minus04

minus05

Posit

ion

trac

king

erro

r of y

-axi

s

SMCAGFTSMC





0 1 2 3 4 5 6 7 8 9 10

005

Time (s)

002

minus25minus2

minus15minus1

minus05

minus02

minus04

minus06

Term

inal

slid

ing

Term

inal

slid

ing

0 1 2 3 4 5 6 7 8 9 10Time (s)

surfa

ces 1

surfa

ces 2


0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux





0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux


0

05

1

005

115

0123

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Estim

ates

ofb

2Es

timat

es o

fb1

Estim

ates

ofb

0


1198872


1198872



6 Conclusion




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 10

0

05

1

15

Time (s)

Posit

ion

trac

king

of x

-axi

s


minus05

minus1


0 1 2 3 4 5 6 7 8 9 10Time (s)


0

05

1

15

Posit

ion

trac

king

of y

-axi

s

minus05

minus15

minus1



1(0) = 10 119902

1(0) = 0 119902

2(0) =



= sin(120587119905) and 1199021199032


119909119910and

120596119909119910


2= 3 120572

1= 1205722= 025 and 120573

1= 1205732= 05 and


= 01 11988710

= 02 and 11988720


0= 10 119898

1= 16 and 119898

2= 16 The sliding




0 1 2 3 4 5 6 7 8 9 10

0

02

04

06

08

1

Time (s)

SMCAGFTSMC

minus02

minus04

Posit

ion

trac

king

erro

r of x

-axi

s


0 1 2 3 4 5 6 7 8 9 10

0

01

02

Time (s)

minus01

minus02

minus03

minus04

minus05

Posit

ion

trac

king

erro

r of y

-axi

s

SMCAGFTSMC





0 1 2 3 4 5 6 7 8 9 10

005

Time (s)

002

minus25minus2

minus15minus1

minus05

minus02

minus04

minus06

Term

inal

slid

ing

Term

inal

slid

ing

0 1 2 3 4 5 6 7 8 9 10Time (s)

surfa

ces 1

surfa

ces 2


0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux





0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux


0

05

1

005

115

0123

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Estim

ates

ofb

2Es

timat

es o

fb1

Estim

ates

ofb

0


1198872


1198872



6 Conclusion




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 10

005

Time (s)

002

minus25minus2

minus15minus1

minus05

minus02

minus04

minus06

Term

inal

slid

ing

Term

inal

slid

ing

0 1 2 3 4 5 6 7 8 9 10Time (s)

surfa

ces 1

surfa

ces 2


0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux





0100200300400500

minus100

minus200

minus300

minus4000 1 2 3 4 5 6 7 8 9 10

Time (s)

Con

trol i

nput

Ux


0

05

1

005

115

0123

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

0 1 2 3 4 5 6 7 8 9 10Time (s)

Estim

ates

ofb

2Es

timat

es o

fb1

Estim

ates

ofb

0


1198872


1198872



6 Conclusion




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article adaptive control of mems gyroscope based on global terminal sliding...

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