Journal of Vibration and Control-2015-Xin-808-17

Article

Adaptive vibration control for MEMS

vibratory gyroscope using backstepping

sliding mode control

Mingyuan Xin and Juntao Fei

Abstract

In this paper,an adaptive backstepping sliding mode control approach is proposed to control the MEMS triaxial gyro-scope.An adaptive backstepping controller that can compensate the system nonlinearities is incorporated into the sliding mode control scheme in the Lyapunov framework.First,a robust backstepping control system incorporated with sliding mode control technique is designed.Then,an adaptive law is derived to estimate the upper bound value of the lumped uncertainty in the backstepping sliding mode control.With the adaptive backstepping sliding mode controller,the chattering in control efforts can be obviously reduced.Numerical simulation is provided to verify the effectiveness of the proposed scheme.

Keywords

Adaptive backstepping,chattering,MEMS gyroscope,sliding mode control,upper bound

1.Introduction

Gyroscopes are commonly used sensors for measuring angular velocity in many areas of applications such as, for example,navigation.The performance of the MEMS gyroscope often deteriorates because of param-eter variations,quadrature errors,and external disturb-ances,etc.It is feasible to use controls such as adaptive control,sliding mode control and intelligent control to control the MEMS gyroscope.

Over the last few years,some control approaches have been proposed to control the MEMS gyroscope. Leland(2006)proposed an adaptive control of a MEMS gyroscope using Lyapunov methods where both controllers tune the drive axis natural frequency, and drive the sense axis vibration to zero for a force-to-rebalance operation.Batur and Sreeramreddy(2006) developed a sliding mode control for a MEMS gyro-scope.Sung and Lee(2009)derived a phase-domain design approach to study the mode-matched control of a gyroscope.Park et al.(2007)presented an adaptive controller of a MEMS gyroscope.Antonello et al. (2009)presented an extremum-seeking controller to automatically match mode in MEMS gyroscope.Ma et al.(2010)proposed a compact H1robust rebalance loop control for MEMS gyroscopes.Adaptive sliding mode controllers have been developed to control the vibration of MEMS gyroscope(Fei and Batur,2009; Fei,2010).System nonlinearities are inevitable in actual engineering and require the controller to be either adap-tive or robust to these model uncertainties.Guo and Woo(2004)proposed an adaptive fuzzy sliding mode controller for a robot manipulator.Wai(2007)pre-sented adaptive fuzzy sliding-mode control with appli-cation to an electrical servo drive.In the nonlinear control area,backstepping is a powerful method for the design and analysis of adaptive controllers for some special classes of uncertain nonlinear systems with its constructive Lyapunov design procedure. Adaptive backstepping is a system and recursive design methodology for nonlinear feedback control. Krstic et al.(1995)proposed an adaptive backstepping nonlinear observer for nonlinear systems.Lin et al. (2002)derived adaptive backstepping sliding mode College of Computer and Information,Hohai University,China

Corresponding author:

Juntao Fei,College of Computer and Information,Hohai University, Changzhou,213022,China.

Email:jtfei@https://www.sodocs.net/doc/c68093386.html,

Received:6November2012;accepted:11April2013

Journal of Vibration and Control

2015,Vol.21(4)808–817

!The Author(s)2013

Reprints and permissions:

https://www.sodocs.net/doc/c68093386.html,/journalsPermissions.nav

DOI:10.1177/1077546313492363

https://www.sodocs.net/doc/c68093386.html,

control for a linear induction motor drive.Sun et al. (2011)investigated adaptive backstepping sliding mode control of a static var compensator.Wang and Lin (2012)developed multivariable adaptive backstepping control using a norm estimation approach.Shieh and Hsu(2007)presented an adaptive backstepping control scheme for precise trajectory tracking of a piezoactua-tor-driven stage.Khot et al.(2012)using proportional-integral-derivative(PID)-based output feedback controller for the active vibration control of a cantilever beam.Hacioglu and Yagiz(2012)developed adaptive backstepping control for the vibration isolation of build-ings.Hu and Xiao(2011)proposed a robust adaptive backstepping control for the attitude stabilization and vibration reduction of?exible spacecraft subject to actu-ator saturation.However,adaptive backstepping con-trollers have not previously been investigated in the control of a MEMS gyroscope.In this paper,a back-stepping control technique is utilized to compensate the model uncertainties in the MEMS gyroscope because this provides a systematic procedure for stabilizing the controller,following a step-by-step algorithm,Another advantage of backstepping control is that it has the?exi-bility to avoid cancellations of useful nonlinearities and achieve stabilization and tracking.

In the control of a MEMS gyroscope,the parameter uncertainties and external disturbances are lumped,and the bound of the lumped uncertainty is necessary in the design of the backstepping sliding mode controller. However,the bound of the lumped uncertainty is di?-cult to obtain in advance in practical applications. Therefore,an adaptive law is derived to adapt the value of the lumped uncertainty in real-time,and an adaptive backstepping sliding mode control law is pro-posed to reduce the chattering.This is the most import-ant feature of the proposed control as compared with the existing work.The contribution of this paper can be emphasized as:

1.The novelty about the controller design part is that

adaptive backstepping control methodology has been integrated with a sliding mode control technique to achieve robust backstepping control.The adaptive backstepping control is proposed to adjust the con-trol system to attenuate the tracking error.The pro-posed controller is divided into two parts,one is a backstepping controller and another is a robust com-pensator that is designed to attenuate the e?ect of system nonlinearities.

2.This paper comprehensively combines adaptive con-

trol,sliding mode control and backstepping control.

The proposed adaptive backstepping sliding mode controller can guarantee the stability of the closed loop system and improve the robustness for external disturbances and model uncertainties.Moreover,a

simple adaptive algorithm is investigated to estimate the bound of system nonlinearities.

The paper is organized as follows.In Section2,the dynamics of the MEMS gyroscope is described through non-dimensional transformation.In Sections3and4, backstepping sliding mode control and adaptive back-stepping sliding mode control are investigated in the Lyapunov framework.Simulation results are presented in Section5to verify the e?ectiveness of the proposed approach.Section6gives the conclusions.

2.Dynamics of MEMS gyroscope

We assume that the gyroscope is moving with a con-stant linear speed and is rotating at a constant angular velocity.The centrifugal forces are assumed negligible because of small displacements.The gyroscope under-goes rotations along x,y and z axis.Then the dynamics of the triaxial gyroscope becomes

m€xtd xx_xtd xy_ytd xz_ztk xx xtk xy y

tk xz z?u xt2m z_yà2m y_z

m€ytd xy_xtd yy_ytd yz_ztk xy xtk yy y

tk yz z?u yà2m z_xt2m x_z

m€ztd xz_xtd yz_ytd zz_ztk xz xtk yz y

tk zz z?u zt2m y_xà2m x_y

e1T

where m is the mass of proof mass.Fabrication imper-fections contribute mainly to the asymmetric spring terms k xy,k xz and k yz and asymmetric damping terms d xy,d xz and d yz.The spring terms are k xx,k yy and k zz, respectively.The damping terms are d xx,d yy and d zz. x, y and z are angular velocities.u x,u y and u z are the control forces.

Dividing equation(1)by the reference mass and rewriting the dynamics in vector forms result in

€qt

D

m

_qt

K a

m

q?

u

m

à2 _qe2Twhere

q?

x

y

z

2

64

3

75u?

u x

u y

u z

2

64

3

75D?

d xx d xy d xz

d xy d yy d yz

d xz d yz d zz

2

64

3

75

K a?

k xx k xy k xz

k xy k yy k yz

k xz k yz k zz

2

64

3

75 ?

0à z y

z0à x

à y x0

2

64

3

75

Using non-dimensional time t??w0t,and dividing

both sides of the equation by reference length w2

and

Xin and Fei809

the reference length q0,and then de?ning new param-eters as

q??q

q0

,D??

D

mw0

, ??

w0

,u??

u

mw

q0

w x?

?????????

k xx

mw

s

,w y?

?????????

k yy

mw

s

,w z?

?????????

k zz

mw

s

w xy?k xy

mw

,w yz?

k yz

mw

,w xz?

k xz

mw

Therefore,after ignoring the superstar for the con-venience of notation,the?nal non-dimensional equa-tion of(1)and(2)can be expressed as

€qtD_qtK b q?uà2 _qe3Twhere

K b?

w2

x

w xy w xz

w xy w2

y

w yz

w xz w yz w2

z

2

4

3

5

Remark1The linearized model(4)can be derived under some assumptions.The damping term in equa-tion(3)originates from the Coriolis forces.The linear-ized model(4)with the lumped parameter uncertainties and external disturbances can be expressed as

€qtD_qtK b q?uà2 _qtd fe4Twhere d f is external disturbance and modeling error including the ignored centrifugal forces and other non-linear terms in the linearization procedure,mechanical–thermal noises and other unknown external disturbance can be regarded as nonlinear terms.The objective of the control system is to maintain the proof mass to oscillate in the x,y and z direction as:x m?A1sine!1tT,y m?A2sine!2tTand z m?A3sine!3tTThe reference model can be expressed as:€q mtK m q m?0,

where K m?diag f!2

1!2

2

!2

3

g.

In other words,the control objective is to force q to follow a given bounded reference signal q m.

3.Backstepping sliding mode

controller

In this section,the state variables are de?ned as X1?q and X2?_q,then the system model(4)can be rewritten in state-space form

_X 1?X2

_X 2?àeDt2:TX2àK b X1tu

&

e5T

And the output equation can be expressed as

Y?X1e6T

Considering the system with the parameter uncertain-ties and external disturbances,equation(5)can be writ-ten as

_X

2

??àeDt2:TtáA1 X2teàK btáA2TX1

teItáBTutfetT

?àeDt2:TX2àK b X1tutFetTe7Twhere FetTrepresents the lumped uncertainty,which contains the parameter uncertainties and external dis-turbances,given by

FetT?áA1X2táA2X1táButfetTe8T

The design procedure of a backstepping sliding mode controller has two steps.First,a virtual control function will be constructed via a Lyapunov function V1.Then,a real control law is constructed.We will investigate the two design steps in detail.

De?ne Y d?q m.

Step1Assume the reference trajectory is Y d.De?ne the tracking error as

e1?YàY de9T

and

e2?X2àa1e10TThen the derivative of_e1becomes

_e1?X2à_Y de11TSelect the virtual control as

a1?àc1e1t_Y de12Twhere c1is a non-zero positive constant.

The?rst Lyapunov function is chosen as

V1?

1

2

e T

1

e1e13Tand the derivative of V1is

_V

1

?

1

2

_e T

1

e1t

1

2

e T

1

_e1?e T

1

_e1?e T

1

eX2à_Y dT

?e T

1

ee2ta1à_Y dT?e T

1

ee2àc1e1t_Y dà_Y dT

?àc1e T

1

e1te T

1

e2?àc1

X3

i?1

e2

1i

te T

1

e2e14T

810Journal of Vibration and Control21(4)

If e 2?0,then

_V

1?àc 1X 3i ?1

e 21i 0

e15T

Step 2Di?erentiating equation (10)with respect to

time yields and using equation (7),we obtain

_e

2?_X 2à_a 1?àeD t2:TX 2àK b X 1tu tF et Tà_a 1e16TDe?ne the sliding surface as

S ?k 1e 1te 2

e17T

where k 1is a non-zero positive constant.De?ne the second Lyapunov function as

V 2?V 1t1

2

S T S

e18T

and the derivative of V 2is _V

2?_V 1tS T _S ?àc 1X 3i ?1

e 21i te T 1e 2tS T

ek 1_e

1t_e 2Te19T

Substituting equations(11)(16)and (17)into (19)

gives _V

2?àc 1X 3i ?1

e 21i te T 1e 2tS T ?k 1ee 2ta 1à_Y d T

àeD t2:TX 2àK b X 1tu tF et Tà_a

1 ?àc 1X 3i ?1e 21i à1k 1e T 2

e 2tS T "

1k 1e 2tk 1ee 2àc 1e 1TàeD t2:Tee 2ta 1TàK b ee 1tY d Ttu tF et Tà_a

1#

e20T

According to equation (20),the backstepping sliding mode control law is designed as

u ?à1

k 1

e 2àk 1ee 2àc 1e 1TteD t2:Tee 2ta 1TtK b ee 1tY d Tt_a

1àF max sgn eS Te21T

where F max ?F et T

t , 40Substituting (21)into (20)gives _V

2?àc 1X 3i ?1

e 21i

à1k 1

e T 2e 2tS T

?F et TàF max sgn eS T e22T

then

_V

2?àc 1X 3i ?1

e 21i

à1k 1X 3

i ?1

e 22i tS T ?F et TàF max sgn eS T àc 1X 3i ?1

e 21i

à1k 1X 3i ?1e 22i

t

S T F et T àS T F max ?àc 1X 3i ?1e 21i à1k 1X 3i ?1

e 22i

àS T ?àc 1

X 3i ?1

e 21i

à1k 1X 3i ?1e 2

2i àX 3i ?1

i s i j j 0e23T

4.Adaptive backstepping sliding mode controller

In the design of the backstepping sliding mode control-ler,the sliding mode control requires the upper bound of uncertainties and disturbances to specify the sliding mode gain to satisfy the requirement of stability and robustness.However,the upper bound of the lumped uncertainty F et Tis di?cult to measure in practical appli-cations.Therefore an adaptive law is incorporated into the backstepping control system to adapt the value of the upper bound F max of lumped uncertainty F et T.

Replacing F max by ^F

in equation (21),the adaptive backstepping sliding mode controller is designed as u ?à

1

k 1e 2àk 1ee 2àc 1e 1TteD t2:Tee 2ta 1TtK b ee 1tY d Tt_a

1à^F sgn eS Te24T

De?ne a Lyapunov function as

V 3?V 2t

12r 1

~F T ~

F e25T

where ~F

?F max à^F ,^F is the estimate of F max ,r 1is a non-zero positive constant.The derivative of V 3is _V

3?_V 2t1r 1

~F T _~F ?àc 1X 3i ?1

e 21i à1k 1e T 2

e 2tS T "

1k 1e 2tk 1ee 2àc 1e 1TàeD t2:Tee 2ta 1T

àK b ee 1tY d Ttu tF et Tà_a 1#

t11

~F T _~F

e26T

Xin and Fei 811

Substituting(24)into(26)gives

_V 3?àc1

X3

i?1

e2

1i

à

1

k1

X3

i?1

e2

2i

tS T?FetTà^F sgneST

t

1

r1

~F T_~F

?àc1

X3

i?1

e2

1i

à

1

k1

X3

i?1

e2

2i

tS T FetTàF max sgneST

?

t~F sgneST

?

t

1

r1

~F T_~F

?àc1

X3

i?1

e2

1i

à

1

k1

X3

i?1

e2

2i

tS T?FetTàF max sgneST

tS T

~

Ft

1

r1

~F T_~F

àc1

X3

i?1

e2

1i

à

1

k1

X3

i?1

e2

2i

tS T

FetT

àS T

F max

t

X3

i?1

~F

i

s i j jt

1

r1

X3

i?1

~F

i

_~F

i

?àc1

X3

i?1

e2

1i

à

1

k1

X3

i?1

e2

2i

àS T

t

1

r1

X3

i?1

~F

i

er1s i j jt_~F iT

e27T

To make_V30,the adaptive law is designed as

_^

F i?à_~F i?r1s i j j,i?1,2,3e28TThen

_V 3?àc1

X3

i?1

e2

1i

à

1

k1

X3

i?1

e2

2i

àS T

?àc1

X3

i?1

e2

1i

à

1

k1

X3

i?1

e2

2i

à

X3

i?1

i s i j j0e29T

This implies that_V3is a negative semi-de?nite func-tion._V3becomes negative semi-de?nite implying that the trajectory reaches the sliding surface in?nite time and remains on the sliding surface and e1,S,~F i are all bounded.Furthermore,we have

Z t 0_V

3

e Td ?V3etTàV3e0T

à

Z t

0c1

X3

i?1

e2

1i

t

1

k1

X3

i?1

e2

2i

t

X3

i?1

i s i j j

!

d ,

that is

V3etTt

Z t

0c1

X3

i?1

e2

1i

t

1

k1

X3

i?1

e2

2i

t

X3

i?1

i s i j j

!

d V3e0T:

Since V3e0Tis bounded and V3etTis non-increasing

bounded function,therefore

lim t!1

Z t

c1

X3

i?1

e2

1i

t

1

k1

X3

i?1

e2

2i

t

X3

i?1

i s i j j

!

d 51:

According to Barbalat Lemma,it can be concluded that

lim

t!1

c1

X3

i?1

e2

1i

t

1

k1

X3

i?1

e2

2i

t

X3

i?1

i s i j j

!

?0,

which means e1i,e2i,s i will converge to zero as

t!1.Consequently S also converges to zero

asymptotically.

5.Simulation results

The proposed adaptive backstepping sliding mode con-

trol with application to the lumped MEMS gyroscope

model is investigated.The parameters of the MEMS

gyroscope used in this paper are chosen as follows:

m?0:57?10à8kg,q0?1m m,!0?3kHz,

d xx?0:429?10à6Nás=m,d yy?0:687?10à6Nás=m

d zz?0:895?10à6Nás=m,d xy?0:0429?10à6Nás=m

d xz?0:0687?10à6Nás=m,d yz?0:0895?10à6Nás=m

k xx?80:98N=m,k yy?71:62N=m,k zz?60:97N=m

k xy?5N=m,k xz?6N=m,k yz?7N=m

The unknown angular velocity is assumed as

x?3rad=s, y?2rad=s, z?5rad=s

The reference input signals are

x m?sine6:71tT,y m?1:2sine5:11tT,z m?1:5sine4:17tT

In this system,the initial state condition is chosen as

qe0T??0:50:50:5 T,and the parameters of back-

stepping sliding mode controller are c1?70,k1?50.

The?xed gain is chosen as F max?150.

5.1.Backstepping sliding mode control

The simulated results of the backstepping sliding mode

control system are depicted in Figures1–4.Assuming

the external disturbances exist,the disturbances

are expressed by fetT?50sine6:71tT50cose5:11tT

?

50cose4:17tT T.

Actually,we can choose any random disturbances.

But,in order to verify the e?ectiveness of the proposed

812Journal of Vibration and Control21(4)

method,the resonance frequency and higher-power are chosen to view the track performance and control responses.The resonance frequency will cause the res-onance phenomenon.If the results with resonance fre-quency are good,other frequencies will also have a good tracking performance.Besides,for the 50times disturbances that are chosen in this section ,the ?xed gain should be chosen more than 50,so we chose F max ?150.The position tracking response with disturbances is shown in Figure 1.It can be seen that the position can e?ectively follow the desired reference model in the presence of the external disturbances.Figure 2plots the tracking error of the system.Figure 3shows that the sliding surface converges to zero asymptotically in a short time,demonstrating the control system will get into the sliding surface and remain alongside it.Figure 4depicts the control input,and it is

obvious

Figure 1.The position tracking response of X,Y ,Z axis with

disturbances.

Figure 2.T racking error of X,Y ,Z axis with disturbances.

Xin and Fei 813

that the undesirable chattering phenomena are serious because of the excess selection of ?xed gain F max .

5.2.Adaptive backstepping sliding mode control

In the simulation,the adaptive backstepping sliding mode control system is simulated under the same con-ditions,and the parameter of adaptive law is chosen as r 1?30.The results are shown in Figures 5–8.

Figure 5shows the position tracking response and Figure 6plots the tracking https://www.sodocs.net/doc/c68093386.html,pared with the simulation results in Figure 1and Figure 5,we can see that the adaptive backstepping sliding mode control also has good tracking performance.Figure 7depicts the control e?https://www.sodocs.net/doc/c68093386.html,pared with Figure 4,it is obvi-ous that the chattering has been obviously reduced in Figure 7due to the upper bound of lumped uncertainty.

Figure 8shows the adaptation of ^F

i .It can be seen

that Figure 3.Convergence of the sliding surface

s.

Figure 4.The control efforts with fixed gain F max .

814Journal of Vibration and Control 21(4)

the adjustable parameter ^F

i converges to constant values.

In summary,we compare the proposed method with backstepping sliding mode control.From the simula-tion results,we can see the control e?orts using the proposed method are better,and the chattering is obvi-ously reduced.Also,the proposed method has good tracking performance.It just takes a little time to track the given trajectory due to the adaptive law.

6.Conclusion

In this paper,both backstepping sliding mode control and adaptive backstepping sliding mode control approaches are developed to control the MEMS tri-axial gyroscope respectively.In order to relax the requirement for the bound of lumped uncertainty,an adaptive backstepping sliding mode control method is proposed for the MEMS triaxial gyroscope system

to

Figure 5.The position tracking response of X,Y ,Z

axis.

Figure 6.T racking error of X,Y ,Z axis.

Xin and Fei 815

estimate online the value of the upper bound of lumped

uncertainty and reduce the chattering phenomena.The proposed adaptive backstepping controller can guaran-tee the stability of the closed loop system and improve the robustness for external disturbances and model uncertainties.Simulations are implemented to verify the e?ectiveness of the proposed adaptive backstepping sliding mode control and demonstrate that the pro-posed method has good tracking performance and superior control responses.In the next step,we can utilize other intelligent control methods and combine two or more of them in the control MEMS gyroscope.For example,fuzzy control can be incorporated into

the adaptive backstepping sliding mode control by using the fuzzy control to approximate the upper bound of the lumped uncertainties.Acknowledgments

The authors thank to the anonymous reviewers for useful comments that improved the quality of the manuscript.

Funding

This work was partially supported by National Science Foundation of China (grant number 61074056),and the Fundamental Research Funds for the Central Universities (grant number

2012B06714).

Figure 8.The adaptation of ^F

i

.Figure 7.The control efforts.

816Journal of Vibration and Control 21(4)

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Xin and Fei817