System Dynamics and Adaptive Control for MEMS Gyroscope Sensor

System Dynamics and Adaptive Control for MEMS Gyroscope Sensor Juntao Fei1,2 and Hongfei Ding2 1 2

Jiangsu Key Laboratory of Power Transmission and Distribution Equipment Technology College of Computer and Information, Hohai University, Changzhou, 213022, P. R. China

Abstract: This paper presents an adaptive control approach for Micro‐Electro‐Mechanical Systems (MEMS) z‐axis gyroscope sensor. The dynamical model of MEMS gyroscope sensor is derived and adaptive state tracking control for MEMS gyroscope is developed. The proposed adaptive control approaches can estimate the angular velocity and the damping and stiffness coefficients including the coupling terms due to the fabrication imperfection. The stability of the closed‐loop systems is established with the proposed adaptive control strategy. Numerical simulation is investigated to verify the effectiveness of the proposed control scheme. Keywords: Adaptive control, MEMS gyroscope, angular velocity 1. Introduction Gyroscopes are commonly used sensors for measuring angular velocity in many areas of applications such as navigation, homing, and control stabilization. Vibratory gyroscopes are the devices that transfer energy from one axis to other axis through Coriolis forces. The conventional mode of operation drives one of the modes of the gyroscope into a known oscillatory motion and then detects the Coriolis acceleration coupling along the sense mode of vibration, which is orthogonal to the driven mode. The response of the sense mode provides information about the applied angular velocity. Fabrication imperfections result in some cross stiffness and cross damping effects that may hinder the measurement of angular velocity of MEMS gyroscope. Therefore the angular velocity measurement and minimization of the cross coupling between two axes are challenging problems in gyroscopes that need to be solved using advanced control methods. Adaptive control is an effective approach to handle parameter variations. In the presence of model uncertainties and external disturbances, sliding mode control is necessary to be incorporated into the adaptive control to improve the robust performance of control system. Sliding mode control is a robust control technique which has many attractive features such as robustness to parameter variations and insensitivity to disturbances. Adaptive sliding mode control has the advantages of combining the robustness of variable structure methods with the tracking capability of adaptive control. In the last few years, many applications have been developed using sliding mode control and adaptive control. Utkin [1] showed that variable structure control is insensitive to parameters perturbations and external disturbances. Ioannou and Sun [2] described the model reference adaptive control. Chou et al. [3] proposed an integral sliding surface and derived an adaptive law to estimate

International Journal of Advanced Robotic Systems, Vol. 7, No. 4 (2010) ISSN 1729‐8806, pp. 81‐86

the upper bound of uncertainties. Some control algorithms have been proposed to control the MEMS gyroscope. Batur et.al. [4] developed a sliding mode control for a MEMS gyroscope system. Leland [5] presented an adaptive force balanced controller for tuning the natural frequency of the drive axis of a vibratory gyroscope. Novel robust adaptive controllers are proposed in [6‐7] to control the vibration of MEMS gyroscope. Sun et al. [8] developed a phase‐domain design approach to study the mode‐matched control of MEMS vibratory gyroscope. Antonelli et al. [9] used extremum‐seeking control to automatically match the vibration mode in MEMS vibrating gyroscopes. Feng et al. [10] presented an adaptive estimator‐based technique to estimate the angular motion by providing the Coriolis force as the input to the adaptive estimator and to improve the bandwidth of microgyroscope. Tasi et al. [11] proposed integrated model reference adaptive control and time‐varying angular rate estimation algorithm for micro‐machined gyroscopes. Raman et al. [12] developed a closed‐loop digitally controlled MEMS gyroscope using unconstrained sigma‐delta force balanced feedback control. Some robust adaptive controllers for vibratory gyroscope system have been proposed in [13‐14]. Park et al. [15] presented an adaptive controller for a MEMS gyroscope which drives both axes of vibration and controls the entire operation of the gyroscope. In this paper, the proposed adaptive control is different from the adaptive controller [15] in that an addition controller is incorporated into the state feedback controller to give more freedom in designing the adaptive controller. Therefore the error dynamics is determined by the reference model dynamics and addition control. This paper investigates the adaptive control approach to identify the angular velocity of MEMS gyroscope using state tracking controllers. The contribution of this paper is that novel adaptive control is proposed to control the MEMS gyroscope and to estimate the angular velocity

77

International Journal of Advanced Robotic Systems, Vol. 7, No. 4 (2010)

and all unknown gyroscope parameters. The paper is organized as follows. In section II, the dynamics of MEMS gyroscope is described. The adaptive controller is derived in section III. Simulation results are presented in section IV. The Conclusion is provided in section V. 2. Dynamics of Mems Vibrational Gyroscope

velocity vector of an arbitrary point A as measured against the axes of the rotating system. The velocity of A relative to B is therefore made up of two terms ‐ the velocity measured against the rotating axes and a component that results from the rotation of the axes, and is thus invisible to the observer in the rotating frame.

The inertial frame and rotating frame are shown in Fig. 1. The typical MEMS vibratory gyroscope includes a proof mass suspended by springs, an electrostatic actuation and sensing mechanisms for forcing an oscillatory motion and sensing the position and velocity of the proof mass.

The property

y

Y

 rA

 rA

A as measured against the axes of the inertial frame. rA is an arbitrary point A relative to the origin of the B

rotating axes , rB is position vector of the origin of the rotating frame relative to the origin of the inertial frame B . Their relation can be expressed as

rA  rB  rA

B

 rB  x i  y j  z k .

(1)

The velocity vector of an arbitrary point A as measured against the axes of the inertial frame can be derived as

dx dy dz di dj dk . (2) i j k x  y z dt dt dt dt dt dt

,



d z  rA   z  v A   z   z  rA B B B dt

d z  rA   z  ( v A   z  rA ) B B B dt (4)

dt d z   rA   z  v A   z   z  rA . B B B dt

dv A

X

di  z  i dt



Fig. 1. Inertial frame and rotating frame In the Fig. 1, rA is position vector of an arbitrary point

Since

B

B

O

v A  vB 

d (  z  rA )

x

B

B

dt

can be proved as

The property

 rB

d(  z  rA )

dj   z j dt

,

dk  z  k dt

,

dt

B

 a A   z  v A can be proved as B

B

dy dz   dx d i  j  k dv A dt dt dt   B  dt dt d 2x d2y d 2z dx di dy dj dz dk  2 i 2 j 2 k      dt dt dt dt dt dt dt dt dt (5) d 2x d2y d 2z dx dy dz  2 i 2 j 2 k ( z  i )  ( z  j )  ( z  k ) dt dt dt dt dt dt 2 2 2 d x d y d z dx dy dz  2 i  2 j  2 k  z  i  z  j  z  k dt dt dt dt dt dt  aA  z  vA . B

B

Differentiating (3) and using (4) and (5) yields

a A  aB  a A  2  z  v A  B

B

d z  rA   z  (  z  rA ) (6) B B dt

where a A and a B are the accelerations with respect to

d2x d2y d2z i  2 j  2 k is the 2 B dt dt dt acceleration vector of an arbitrary point A as measured the inertial frame, a A 

against the axes of the rotating system. The acceleration of A relative to B is therefore made up of four terms ‐ the acceleration measured against the rotating axes and three components that result from the rotation of the axes, and are thus invisible to the observer in the rotating frame. These are the Euler (tangential), Centripetal and Coriolis accelerations respectively. Multiplying (6) by mass m gives

 d z   rA   m z  (  z  rA ) ma A  ma B  ma A  2m  z  v A  m B B B B  dt

substituting these properties into (2) yields

(7)

dx dy dz v A  vB  i  j  k  x(  z  i )  y(  z  j )  k(  z  k ) (3) dt dt dt  v B  v A   z  rA

where v A and v B are the velocities with respect to the

78

B



dx dy dz i j  k is the relative dt dt dt

2m  z v A

B

is the Coriolis force and

m z  (  z  rA ) is the Centrifugal force. B

B

B

inertial frame, v A

where

The Coriolis force acting on the proof mass along x direction is derived as

Fcoriolis x  2m z k  y j  2m z y i (8)

Juntao Fei and Hongfei Ding: System Dynamics and Adaptive Control for MEMS Gyroscope Sensor

The Coriolis force acting on the proof mass along y direction is derived as

Fcoriolis y  2m z k  xi  2m z xj . (9)

Dividing (12) and (13) by the reference mass and rewriting the gyroscope dynamics in vector forms result in

q 

By using the property of k  (k  i )   i , the Centripetal force acting on the proof mass along x direction can be derived as

Fcentripetal  x  m z k   z k  xi    m z xi (10). 2

By using the property of k  ( k  j )   j , the Centripetal forces acting on the proof mass along y direction can be derived as

Fcentripetal  y  m z k   z k  yj    m z yj . (11) 2

(14)

 0 z  ux  x Ω  u    , q    ,  , z 0  y uy  d xx d xy  k xx k xy  D , K a     . d xy d yy  k xy k yy   Using non‐dimensional time t  w0t , and dividing both where

2 sides of (14) by w0 and reference length q0 give the

final form of the non‐dimensional equation of motion as

q D q K q u Ω q . (15)    2 q0 mw0 q0 mw02 q0 mw02 q0 w0 q0

Defining a set of new parameters as follows:

2

2

We assume that the table where the proof mass is mounted is moving with a constant velocity; the gyroscope is rotating at a constant angular velocity  z over a sufficiently long time interval; the centripetal

K D u q  a q   2 Ωq m m m

forces m z x and m z y are assumed to be

D  q  , D  ,  z  z ， (16) q0 w0 mw0

q 

negligible; gyroscope undergoes rotation about the z axis only, and thereby Coriolis force is generated in a direction perpendicular to the drive and rotational axes. A z‐axis MEMS gyroscope is depicted in Fig. 2. With the assumptions the dynamics of gyroscope become

u 

u 2

mw0 q 0

, wx 

wxy 

k xx , wy  2 mw0 k xy mw0

2

k yy mw0

2

,

. (17)

mx  d xx x  d xy y  k xx x  k xy y  u x  2m z y (12)

my  d xy x  d yy y  k xy x  k yy y  u y  2m z x . (13)

Ignoring the superscript (*) for notational clarity, the nondimensional representation of (12) and (13) is

Fabrication imperfections contribute mainly to the symmetric spring and damping terms, k xy and d xy .

q  Dq  K b q  u  2Ωq (18)

 wx 2

wxy  2  . wy 

The x and y axes spring and damping terms k xx , k yy , d xx and d yy are mostly known, but have

where K b  

small unknown variations from their nominal values. The mass of proof mass can be determined very accurately, and u x , u y are the control forces in the x and y

3. Adaptive Control Design

direction.

y k xx

z

k yy

d yy

In this section an adaptive controller is proposed to identify the angular velocity and all unknown gyroscope parameters. The block diagram is shown in Fig. 3. In the adaptive control design, we consider the equation (9) as the system model. Disturbance

k xx

m

Reference Model

Proof Mass d xx

d xx k yy

 wxy

Adaptive Controller

MEMS Gyroscope

+

Adaptive Law

d yy

Fig. 2. A simple model of a MEMS Z axis gyroscope

-

X

x

e

Xm

Estimation of Angular rate

Fig. 3. Block diagram of adaptive control for MEMS gyroscope 79


Rewriting the gyroscope model (18) in state space form as

yields

~ X (t)  Am X (t)  BK T (t) X (t) (26)

X  AX  Bu (19)

Where

Then, we have the tracking error equation

 0  w 2 x A  0    wxy

1

0

 d xx

 wxy

0

0

 ( d xy  2 z )  w y

2

 x  x  u x  u    , X    . (20)  y u y     y 

T

0 1 0 0  B  , 0 0 0 1 

  ( d xy  2 z )  1   d yy 

0

~ e(t)  (Am  BK f )e  BK T (t) X (t) (27)

Define a Lyapunov function





1 T 1 ~ ~ e Pe  tr KM 1 K T (28) 2 2

V

where P and M  diagm1 m2  are positive definite matrix. Differentiating V with respect to time yields

~ ~ V  e T Pe  tr  KM 1 K T    (29) ~ ~ 1 ~ T  T T T   e Qe  e PBK X  tr KM K qm  K m qm  0 (21)  

The reference model x m  A1 sin( w1t ) , y m  A2 sin( w2 t ) is defined as



where K m  diag w1

2

2



where P(Am  BK f )  (Am  BK f )T P  Q , Q is

w2 .

Similar to (19), the reference model can be written as

 0  w 2 X m   1  0   0

1

0

0

0

0

0

0  w2

2

0 0  X m  Am X m (22) 1  0

positive definite matrix. To make V  0 , we choose the adaptive law as

~ K T (t)  K T (t)   MB T P T eX T (t) (30)

with K (0) being arbitrary. This adaptive law yields

V  e T Qe  min ( Q ) e  0 (31)

where Am is a known constant matrix. We make the following assumptions.

*

Assumption. There exists a constant matrix K such that the following matching condition A  BK

*T

 Am

can always be satisfied. The control target for MEMS gyroscope is (i) to design an adaptive controller so that the trajectory of X can track the state of reference model X m ; (ii) to estimate the angular velocity of MEMS gyroscope and all unknown gyroscope parameters. The tracking error and its derivative are

The inequality V    min ( Q ) e implies that e is integrable as

t



0

e dt 

1

min ( Q )

V ( 0 )  V ( t )

. Since

V ( 0 ) is bounded and V ( t ) is nonincreasing and bounded, it can be concluded that lim

t

 e dt

t  0

bounded. Since lim

is

t

 e dt is bounded and e is also

t  0

bounded, according to Barbalat’s lemma, e will asymptotically converge to zero, lim e( t )  0 . t 

e(t)  X (t)  X m (t) (23) It can be shown that if f the persistent excitation can be satisfied, ie. w1  w2 controller parameter converges to ~ e  Am e  (A  Am )X  Bu . (24) its true values, K  0 . In other words, excitation of proof mass should be persistently exciting. ~ The adaptive controller is proposed as Since K  0 , then the unknown angular velocity as well

as all other unknown parameters can be determined from

u(t)  K T (t) X (t)  K f e (25) T A  BK  Am and we obtain  z  0.25( k 22  k 41 ) .

where K ( t ) is an estimate of K * , the constant matrix K f satisfies the condition that ( Am  BK Hurwitz.

~

f

) is

* We define the estimation error as K (t)  K (t)  K and

substitute this estimation error and (25) into (19) 80

Conclusion: if persistently exciting drive signals, xm  A1 sin(w1t ) and y m  A2 sin( w2t ) are used, then

~ K and e( t ) all converge to zero asymptotically.

Consequently the unknown angular velocity can be

Juntao Fei and Hongfei Ding: System Dynamics and Adaptive Control for MEMS Gyroscope Sensor

determined as lim t   z (t )   z . However it is difficult to establish the convergence rate. 4. Simulation Example

including the angular velocity converge to their true values, and tracking error is going to zero asymptotically as time go on. -3

2

We will evaluate the proposed adaptive control on the lumped MEMS gyroscope model [7] using MATLAB/SIMULINK. The control objective is to design adaptive state tracking controller so that a consistent estimate of z can be obtained.

e1

0 -2 -4

In the simulation, we allowed  5% parameter variations for the spring and damping coefficients with respect to their nominal values. We further assumed  5% magnitude changes in the coupling terms i.e. d xy and  xy , again with respect to their nominal

values. The external disturbance is a random variable signal with zero mean and unity variance. Parameters of the MEMS gyroscope are as follows: m  0.57e  8 kg, d xx  0.429e  6 N s/m,

k xy  5

N/m,

6

e3

-2

80

90

100

10

20

30

40 50 60 Time(Second)

70

80

90

100

x 10

0

8 6

Angular Rate

K matrix

is K ( 0 )  0.95 K . The desired motion trajectories are *

0 -2

y m  1.2 sin( w2 t ) , where w1  6.17 kHz and w2  5.11kHz. The adaptive gain of and

-4 -6

20 . The K f in (29) is chosen as

-8

0

10

20

30


70

80

90

100

Fig. 5. Adaptation of angular velocity using adaptive control

k11

1540 1535 1530

0

10

20

30


70

80

90

100

0

10

20

30


70

80

90

100

0

10

20

30


70

80

90

100

0

10

20

30


70

80

90

100

k12

50 0 -50

k13

100 90 80

20 k14

Fig. 4 depicts the tracking errors. It is observed that the tracking errors converge to zero asymptotically. Figs. 5 and 6 draw the adaptation of the angular velocity and controller parameters. It is shown that the estimates of angular velocity and controller parameters converge to their values. Fig.7 plots the control input using adaptive control . The estimate of angular velocity using adaptive control has larger overshoot at the beginning but much smaller rise time. The model uncertainties and external disturbances are difficult to compensate in the adaptive controller, because there is no disturbance term in the derivation whereas the disturbance term can be dealt with well in the adaptive sliding mode control, therefore, adaptive sliding mode control is better than adaptive control in the presence of model uncertainties and external disturbance. Simulations demonstrate that with the control laws (29), and the parameter adaptation laws (34), if the gyroscope is controlled to follow the mode‐unmatched reference model, the persistent excitation condition is satisfied, i.e. w1  w2 , and all unknown gyroscope parameters,

70

2

gain of (41) is M  diag20 20


Fig. 4. The tracking error using adaptive control

k yy  71.62

10000 10000 1000 20000 Kf   . The adaptive  1000 1000 1000 1000

30

0

The unknown angular velocity is assumed  z  5.0

(34) is M  diag 20

20

2

4

xm  sin(w1t )

10

4

N/m, w0  1kHz , q 0  10 6 m . rad/s and the initial condition on

0 -3

d xy  0.0429e  6 N s/m, d yy  0.687e  36 N s/m, k xx  80.98 N/m,

x 10

0 -20

81


7. Reference

k21

100 90 80

0

10

20

30


70

80

90

100

0

10

20

30


70

80

90

100

0

10

20

30


70

80

90

100

0

10

20

30


70

80

90

100

V. I. Utkin, Variable structure systems with sliding modes, IEEE Trans. on Automatic Control, 22, pp. 212‐222, 1977. P. Ioannou, J. Sun, Robust Adaptive Control. Prentice‐Hall, 1996. C. Chou, C. Cheng , A decentralized model reference adaptive variable structure controller for large‐scale time‐varying delay systems, IEEE Transactions on Automatic Control, 48(7), pp. 1213‐1217, 2003. C. Batur, T. Sreeramreddy, Sliding mode control of a

k22

50 0 -50

k23

1380 1360 1340

k24

100 0 -100

simulated MEMS gyroscope, ISA Transaction, 45(1),

Fig. 6. Adaptation of control parameters using adaptive control 4

Control Signal u1

2

x 10

0 -2 -4 -6

0

10

20

30


70

80

90

100

0

10

20

30


70

80

90

100

Control Signal u2

15000 10000 5000 0 -5000

Fig. 7. Adaptive control input 5. Conclusion

This paper investigates the design of adaptive control for MEMS gyroscope. The dynamics model of the MEMS gyroscope is developed and nondimensionized. Novel adaptive approach is proposed and stability condition is established. Simulation results demonstrate that the effectiveness of the proposed adaptive control techniques in identifying the gyroscope parameters and angular velocity. 6. Acknowledgment The author thanks to the anonymous reviewers for useful comments that improved the quality of the manuscript. This work was supported by National Science Foundation of China under grant No. 61074056, The Natural Science Foundation of Jiangsu Province under grant No. BK2010201.

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pp.99‐108 , 2006 . R. Leland, Adaptive control of a MEMS gyroscope using Lyapunov methods. IEEE Transactions on Control Systems Technology, 14, pp.278–283, 2006. J. Fei, C. Batur, Robust adaptive control for a MEMS vibratory gyroscope, International Journal of Advanced Manufacturing Technology, 42(3), pp. 293‐300, 2009. J. Fei, C. Batur，A novel adaptive sliding mode control for MEMS gyroscope ， ISA Transactions ， 48(1), pp. 73‐78, 2009. S.Sung ,W. Sung,C. Kim, S.Yun, Y. Lee ， On the mode‐matched control of MEMS vibratory gyroscope via phase‐domain analysis and design, IEEE/ASME Transactions on Mechatronics, 14(4), pp.446‐455, 2009. R. Antonello, R. Oboe, L. Prandi, F. Biganzoli, Automatic mode matching in MEMS vibrating gyroscopes using extremum‐seeking control, IEEE Transactions on Industrial Electronics, 56(10), pp.3880‐3891, 2009. Z. Feng, M. Fan, Adaptive input estimation methods for improving the bandwidth of microgyroscopes, IEEE Sensors Journal, 7(4), pp.562‐567, 2007. Tsai, Nan‐Chyuan, Sue, Chung‐Yang, Integrated model reference adaptive control and time‐varying angular rate estimation for micro‐machined gyroscopes, International Journal of Control, 83(2), pp. 246‐256, 2010. Raman, J. Cretu, E.; Rombouts, P.; Weyten, L. A closed‐loop digitally controlled MEMS gyroscope with unconstrained sigma‐delta force‐feedback, IEEE Sensors Journal, 9(3), pp. 297‐305, 2009. J. Fei, F. Chowdhury, Robust adaptive controller for triaxial angular velocity sensor, International Journal of Innovative Computing, Information and Control, Vol. 7, No. 6, pp. 2439‐2448, 2010. J. Fei, C. Batur, A novel adaptive sliding mode control with application to MEMS gyroscope, ISA Transactions, Vol. 48, No 1, pp. 73‐78, 2009. S.

Park, R. Horowitz, S.Hong, Y. Nam, Trajectory‐switching algorithm for a MEMS gyroscope, IEEE Transactions on Instrumentation and Measurement, 56(60), pp.2561‐2569, 2007.