Elsevier Editorial System(tm) for Journal of Sound and Vibration Manuscript Draft Manuscript Number: Title: Nonlinear Tracking Control of Vibration Amplitude for a Parametrically Excited Microcantilever Beam Article Type: Full Length Article Section/Category: A Active and adaptive control of sound and vibration Keywords: MEM cantilever; parametric resonance; amplitude control; nonlinear tracking control; singularities; Lyapunov method Corresponding Author: Dr. Quoc Chi Nguyen, Corresponding Author's Institution: Tel Aviv University First Author: Quoc Chi Nguyen Order of Authors: Quoc Chi Nguyen; Slava Krylov Abstract: In this paper, a feedback control algorithm to regulate oscillation amplitude of a microelectromechanical (MEM) cantilever beam operated at parametric resonances is developed. The control objective is to drive the oscillation amplitude of the micro beam, which amplified using parametric excitation, to the desired values. The principle of the control algorithm is to establish an output tracking control based on the nonlinear dynamic model of the micro beam, where the supply voltage is considered as a control input. The tracking control algorithm is designed to solve the singularities resulting from the zero-defection state of the micro beam. The Galerkin method is applied in order to reduce the partial differential equation describing the dynamics of the beam into a set of ordinary differential equations (ODEs). Uniformly ultimate boundedness stability of the control system is proved using Lyapunov method. The effectiveness of the proposed control algorithm is illustrated via numerical simulations. Suggested Reviewers: Mohammad Younis Ph.D. Professor, Mechanical Engineering, Binghamton University, State University of New York myounis@binghamton.edu Sungsu Park Ph.D. Professor, Department of Aerospace Engineering, Sejong University sungsu@sejong.ac.kr Qing Zheng Ph.D. Professor, Electrical & Computer Engineering, Gannon University zheng003@gannon.edu

October 3, 2013 Quoc Chi Nguyen, Ph. D.

Research Fellow

School of Mechanical Engineering

Tel Aviv University

Tel Aviv 69978, Israel

Tel.: +972-3-640-6246, Fax: +972-3-640-7617

Email: nqchi@tau.ac.il

Professor Matthew P. Cartmell, Ph. D.

Department of Mechanical Engineering

University of Sheffield

Sheffield, United Kingdom

Dear Professor Cartmell,

We would like to submit the paper entitled, “Nonlinear tracking control of vibration

amplitude for a parametrically excited microcantilever beam,” authored by myself and

Prof. Slava Krylov for possible publication in the Journal of Sound and Vibration.

This paper has not been published elsewhere nor has it been submitted for publication

elsewhere.

We have listed three prospective reviewers from the literature on the next page.

Sincerely,

Quoc Chi Nguyen

Cover Letter

(1) Professor Mohammad Younis

Department of Mechanical Engineering

Binghamton University, State University of New York (SUNY)

Binghamton, NY 13902-6000

Tel: (607)-777-4983, Fax: (607)-777-4620

Email: myounis@binghamton.edu

URL reference: http://ws.binghamton.edu/me/Younis/index.html

(2) Professor Sungsu Park

Department of Aerospace Engineering

Sejong University

98, Gunja-dong, Kwangjin-gu

Seoul, Korea

Tel.: +82-2-3408-3769 , Fax: +82-2-3408-3333

Email: sungsu@sejong.ac.kr

URL reference: http://fdcl.sejong.ac.kr/prof.html

(3) Professor Qing Zheng

Department of Electrical & Computer Engineering

Gannon University

109 University Square, Erie

Pennsylvania 16541, United States

Tel.: (814)-871-5617

Email: zheng003@gannon.edu

URL reference: http://www.gannon.edu/FacultyProfiles.aspx?profile=zheng003

Highlights

- Vibration amplitude control of a MEMS parametrically excited cantilever is

developed.

- Tracking control scheme is designed based on a nonlinear model of the cantilever

beam.

- The singularities resulting from the zero-defection of the cantilever beam is handled.

- Uniformly ultimately bounded stability of the control system is proved.

- The advantages of the proposed control method on performance and robustness are

shown.

Highlights (for review)

1

Nonlinear Tracking Control of Vibration Amplitude for a

Parametrically Excited Microcantilever Beam

Quoc Chi Nguyen† and Slava Krylov

School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv

69978, Israel,

Tel.:+972-3-640-6246,+972-3-640-5930,

Emails:nqchi@tau.ac.il, vadis@eng.tau.ac.il

† The author to whom all correspondence should be addressed

Abstract: In this paper, a feedback control algorithm to regulate oscillation amplitude of a

microelectromechanical (MEM) cantilever beam operated at parametric resonances is

developed. The control objective is to drive the oscillation amplitude of the micro beam,

which amplified using parametric excitation, to the desired values. The principle of the

control algorithm is to establish an output tracking control based on the nonlinear dynamic

model of the micro beam, where the supply voltage is considered as a control input. The

tracking control algorithm is designed to solve the singularities resulting from the zero-

defection state of the micro beam. The Galerkin method is applied in order to reduce the

partial differential equation describing the dynamics of the beam into a set of ordinary

differential equations (ODEs). Uniformly ultimate boundedness stability of the control system

is proved using Lyapunov method. The effectiveness of the proposed control algorithm is

illustrated via numerical simulations.

Keywords: MEM cantilever; Parametric resonance; Amplitude control; Nonlinear tracking

control; Singularities; Lyapunov method

ManuscriptClick here to download Manuscript: JSV 2013_Manuscript_Amplitude Vibration Control of PE Micro Beam.docxClick here to view linked References

2

1. Introduction

Microelectromechanical systems (MEMSs) have been developed to replace conventional

mechanical and electronic components such as actuators, transducers, and gears in microscale

devices [1-3]. The micro sensors and micro actuators have enabled development of smart

products that are able to sense and control the environment, applying to mechanical, thermal,

biological, chemical, optical, and magnetic systems [4-11]. MEMS cantilever is one of the

most popular MEMS structures widely used due to its ability to provide high sensitivity, high

selectivity, and flexibility of its working environment [7-11].

Large part of MEMS cantilever uses mechanical vibrating parts for sensing, where the

amplitude of the resonant peak and the bandwidth of resonance are important to achieve high

sensitivity. To increase the oscillation amplitude, design and fabrication techniques have been

developed in order to eliminate the effect of air damping [12,13]. In an active manner, the

oscillation amplitude of MEMS cantilevers can be increased externally using parametric

excitation [4,14-22], where the parametric pump voltage that yields electrostatic forces have

been employed. The periodic voltage signal provides a modulation of the stiffness of the

structure that and consequently, results in an increase of the oscillation amplitude

progressively at specific frequencies. The mechanical parametric excitation of MEMS was

introduced for the first time in [14]. Turner et al. [15] showed the measurement of the five

parametric resonances in MEMS and proposed the use of parametric excitation to improve the

quality of the MEMS sensing devices. Wang, Baskaran, & Turner [17] investigated the effect

of the nonlinearity to the parametrically excited resonant MEMS mass sensor. Baskaran &

Turner [18] provided an experimental result that shows parametric resonances of a torsional

MEMS resonator, considering two interacting mechanical modes of oscillation. Zhang, Meng,

& Wei [20] presented a study of nonlinear dynamics and chaos of a MEMS resonator under

two parametric and external excitations. Khirallah [22] proposed a parametric amplification of

3

a comb drive oscillator. Recently, Linzon et al. [4] introduced parametric excitation of silicon-

on-insulator microcantilever beams by fringing electrostatic fields.

The effectiveness of the parametric excitation to improve the performance of the MEMS

cantilevers has been verified. As shown in [4,16,21], with given amplitude, the specific

voltage and frequency of the pumping excitation can be determined through the resonance

curves. This can be considered as the open-loop control method for MEMS cantilever.

However, the dynamic nonlinearities affect significantly oscillations of the MEMS devices

and consequently, their sensitivity [23]. Besides, the imprecise fabrication and mechanical

and thermal noises limiting the performance of the MEMS devices are evident. Therefore,

reasonably prompt feedback control of oscillation amplitude for improvement of the

performance of a parametrically excited MEMS cantilever is desirable.

In recent years, there have been numerous papers published on control of MEMS [24–39].

The most focused control topic is to stabilize MEMS resonators [24-34]. Alsaleem & Younis

[31,32] used delay feedback controllers to stabilize MEMS resonators (especially near pull-in

point). The effects of the control gains in [31,32] were later investigated in [33]. Siewe &

Hegazy [34] indicated that reducing amplitude of parametric excitation can control chaotic

motion of a MEMS resonator. Seleim et al. [35] proposed a closed-loop control of a MEMS

resonator, providing optimal operating regions for the resonator.

There are a lot of researches investigating oscillation amplitude control of MEMS.

However, only linear resonant MEMS devices have been focused [26,36-39]. Leland [26]

designed two adaptive controllers for a MEMS gyroscope to regulate the amplitude of the

drive axis vibration. Batur, Sreeramreddy, & Khasawneh [36] used sliding mode control

algorithm to stabilize a MEMS gyroscope. Zheng et al. [37] proposed an active disturbance

rejection control strategy to regulate the output amplitude of the drive axis of a MEMS

gyroscope to a fixed level. Zhang, Meng, & Li [38] introduced an adaptive vibration control

4

of micro-cantilever beam to eliminate the useless high amplitude. Baghelani, Ghavifekr, &

Ebrahimi [39] used analytical formula to design an amplitude control for ring shape micro

disk resonator.

It is shown that, under parametric excitation, MEMS devices operated at nonlinear

parametric resonances possess much higher amplitude and consequently, higher sensitivity

than the ones operated at linear resonances. However, to the best of our knowledge, there has

not been a published research on control of vibration amplitude for parametric excited

MEMS. In this research, a control method to regulate vibration amplitude of a parametrically

excited microcantilever structure, which is introduced in [4], is developed. The distributed

electrostatic forces by tailored asymmetries in fringing fields provided by single co-planar

electrode located symmetrically around the actuated cantilever (see Fig. 1). The distributed

electrostatic forces depend on the supply voltage and the deflection of the microcantilever.

Under the effects of the AC supply voltages, the parametric excitation is achieved. The

control objective is to drive the vibration amplitude of the micro beam parametrically excited

to the desired values. The principle of control algorithm is to establish a tracking control

based on the nonlinear dynamic model of the micro beam, where the supply voltage is

considered as a control input. For the control design purpose, the reduced order model of the

micro beam obtained using the Galerkin method [40-43] is employed.

Contributions of this paper are the following. First, a nonlinear tracking control of

vibration amplitude for a parametrically excited micro cantilever beam is developed, where

the tracking through the singularities are handled. Second, uniformly ultimate boundedness

stability of the control system is proved mathematically using Lyapunov function. Third,

simulations results show the advantages of the proposed control method on performance and

robustness.

5

The rest of this paper is organized as follows. Section 2 presents the dynamic model of the

microcantilever beam system. Section 3 introduces the proposed control algorithm design. A

Lyapunov function-based stability analysis of the closed-loop system and the proof of

stability also are discussed. Section 4 includes numerical simulation results that illustrate the

effectiveness of the proposed control scheme. Finally, Section 5 draws conclusions.

2. Dynamic models of the micro beam

2.1. Continuous model

As shown in Fig.1, the microcantilever beam is designed such that it can freely move in the

out-of-plane direction (z-axis). The planar electrode placed in the area surrounding the micro

beam provides the fringing field, which becomes asymmetric when the micro beam deflects

from the equilibrium position in the x-y plane.

Under the assumption that the defections are small in comparison with the length of the

beam, the following governing equations of the micro cantilever beam are derived based on

Euler-Bernoulli theory [44].

2 42

2 4ˆˆ ( ) ( ) 0,

w w wA c EI V t f w

t t x

(1)

where A bh and 3 /12I bh . Since the width of the beam b is larger than its thickness h,

the effective modulus (plain strain) 2/ (1 )E E v is employed, where v is the Poisson’s

ratio. The beam is subject to the following homogenous boundary conditions (2) and the

nonzero initial conditions (3).

2 3

2 3

0

(0, ) 0, 0, 0, 0.x x l x l

w w ww t EI EI

x x x

(2)

0

0

(0, ) ( ), 0.t

ww x w x

t

(3)

6

The distributed electrostatic force ˆ ( )f w is approximated as follows [4].

2

( )ˆ ( )1 p

a wf w

w

, (4)

where a, , and p are fitting parameters. The distributed electrostatic forces act as restoring

forces that pull the micro beam back towards the equilibrium position. The relationship

between the electrostatic force f and the beam deflection w is plotted in [4, Fig.2, p.

163508-3]. It should be noted that the nonzero initial deflection of the beam (3) is necessary

to enable the actuation by the distributed electrostatic forces at the starting time.

Introduce the following nondimensional variables.

ˆˆ ˆ, , ,x t w

x t wL T h

(5)

where 4 /T AL EI is a time scale. Substituting Eq. (5) into Eqs. (1)-(3) and dropping the

hats, the nondimensional equations of motion, the boundary conditions, and the nonzero

initial condition that govern the transverse deflection of the micro beam are obtained as

follows.

2 42

2 4( ) ( ) 0,

w w wc V t f w

t t x

(6)

2 3

2 3

0 1 1

(0, ) 0, 0,x x x

w w ww t EI EI

x x x

(7)

0

0

( )(0, ) , 0.

t

w x ww x

h t

(8)

where

4

ˆL

c cEI A

, ˆ ( )

( )f w

f wa

,

4 2

0aL V

EIh , (9)

where V0 is the unit voltage.

7

2.2. Reduced order model

A reduced-order model is obtained through discretization of the partial differential

equation (6) with the boundary conditions (7) into a finite-degree-of-freedom system. Based

on the Galerkin procedure, the deflection of the beam is approximated by a combination of

smooth basis functions as

1

( , ) ( ) ( )n

i i

i

w x t q t x

, (10)

where ( )iq t is are the generalized displacement, and the set of the basis functions 1, ,

( )i i nx

is orthogonal, i.e.,1

0( ) ( )di j ijx x x , where ij is the Kronecker delta. In this paper, ( )i x

is the ith linear undamped mode shape of the straight micro beam given as

sinh sin

( ) sin sinh cos cosh ,cosh cos

i ii i i i i i

i i

x D x x x x

(11)

where iD is a constant, and i satisfies cos cosh 1i i , correspondingly giving infinite

number of frequencies

2

i i , (12)

where i is the ith natural frequency of the linearized straight micro beam. The finite

dimensional dynamic system will be obtained by carrying out the following procedure: (i)

multiplying Eq. (6) by a weighting function ( )j x , (ii) integrating the resultant equation

obtained in step (i) over the domain [0,1]x , (iii) utilizing the boundary conditions (7) and

substituting Eq. (10) into the resultant equation obtained in step (ii), (iv) collecting all terms

of the resultant equation obtained in step (iii) with respect to ( )iq t , ( )iq t , and ( )iq t . After step

(iv), Eq. (6) is rewritten into a set of ordinary differential equations (ODEs) as

2

0( ) ( ) ( ) ( ) 0, (0)t t t V Mq Cq Kq F q q = q , (13)

8

where T

1( ) ( ) ( )nt q t q tq is the time-dependent vector of generalized coordinates, and

the elements of M, C, and K are given as

1

0dij ij i jm x , (14)

1

2

0dij ij i i jk x , (15)

ij ijc cm . (16)

Note that M, C, and K are diagonal matrices. The vector 2 ( )V F q represents the distributed

electrostatic actuation, where T

1( ) ( ) ( )nf fF q q q with

1

1,...,0

( ) ( , ) ( )di i n if f x x q q . (17)

It should be noted that the parametric excitation is generated by applying the pumping voltage

( )V t .

2.3. State space model

For the control design purpose, it is convenient to rewrite the ODE (13) into a state space

model. Introduce the state vector T T T 2( ) ( , ) nt R z q q , and the control input

2( ) ( )u t V t ,

( ) 0u t . The reduced order model of the micro beam (13) can be converted to a state space

model:

T T T

1,..., 0 0( ) ( ), (0) ( , )i nz u t z Az B z q q , (18)

where

1 1

n n n n

0 IA

M K M C, (19)

1

1

1,...,( )

n

i nz

0B

M F. (20)

9

The control objective is to drive the deflection of the tip of the micro beam to a desired

oscillation. In the control design, the deflection of the tip of the micro beam is considered as a

control output, which is given as

,y Hz (21)

2 2 1(1) (1) (1)n n H 0 . (22)

3. Output tracking control problem

Consider the dynamic system (18)-(22) with the desired continuous oscillation ( )dy t . A

feedback control input ( )u t is designed to guarantee the tracking of the control output y(t),

that is, lim ( ) ( ) 0dt

y t y t

, and 1( )t L L z for the nonzero initial condition 0(0) z z .

Three assumptions about the desired output are made as follows: (i) The desired continuous

oscillation and its derivatives are sufficiently small; (ii) ( )

1,...,( )i

d i ny t

are bounded for all 0t ;

(iii) ( )

1,...,( )i

d i ny t

are piecewise continuous functions. Following the design procedure in [45],

the first step of the control design is to seek the nominal control input ( )du t and the desired

state ( )d tz satisfy the following statements: (i) ( )du t and ( )d tz satisfy the differential

equation

1,...,( ) ( )d d di n dz u t z Az B . (23)

(ii) The output tracking is achieved, i.e.,

d dy Hz . (24)

(iii) ( )du t and 1( )d t L L z .

Taking the first and second derivatives of the control input and using the dynamic equation

(18), the following equations is obtained.

y u Hz HAz HB HAz , (25a)

10

2y u HAz HA z HAB . (25b)

Let

2

1

( )( )

ni

i i i n

i i

Lt k z cz

m

HA z , (26)

1, ,

1

( ) ( ) ( )n

i i i i n

i

t L m f z

HAB . (27)

The Eq. (25) can be rewritten as

( ) ( ) ( )y t t u t . (28)

From Eq. (28), choose the nominal control input ( )du t as

1( ) ( )[ ( ) ( )]d du t t y t t . (29)

It should be noted that when 1,..., 0i nz (i.e., q 0 ), ( )t becomes zero, and consequently,

( )du t . Therefore, to avoid the unboundedness of the control input, for the neighborhood

of the singular point (.) ( ) ,nS t R r

q q q , the following nominal control input is

used.

1( ) ( )[ ( ) ( )]S

d r du t t y t t , (30)

where rt is the time when rq . Finally, the following control law is proposed.

( ) ( ) ( ( ) ( )) ( ( ) ( ))d p du t p t y t t t t k z z , (31)

where

sgn 1 1 sgn

( )2 ( ) 2 ( )r

r rp t

t t

q q

, (32)

and the control gain vector is given as

1 2p nk kk . (33)

11

Remark 1. The singularities of the system occur when the defection of micro beam is zero,

that is, the beam reaches the x-axis. At singular points, the electrostatic forces become zero,

and the system is temporarily uncontrollable.

Remark 2. The control law (31) includes the feedfoward term ( )( ( ) ( ))dp t y t t and

feedback term ( ) ( ( ) ( ))p dp t t tk z z . With the feedfoward term, the system is partially

linearized, as shown in Eq. (36)-(38) below. Meanwhile, the feedback term provides a

stabilizing effect by adjusting the control gains 1,...,2i i n

k

.

Remark 3. It should be noted that the control law (31) is a combination of the exact tracking

control law 1( ) ( ) ( ( ) ( )) ( ( ) ( ))d p du t t y t t t t k z z and the approximate tracking

control law 1( ) ( ) ( ( ) ( )) ( ( ) ( ))r d p du t t y t t t t k z z , which work in the regions

(.) ( ) ,nS t R r

q q q and (.) ( ) ,nS t R r

q q q , respectively.

Introduce the change of the state variables

1

1

2

2

1 3

2 1

2 2 2

2

0 0 1

0 0 1

n

n n

n

zy

zy

z

zz

z

H

HA

ξ Nz , (34)

which yields the inverse transformation

1z N ξ . (35)

Using the new state variables, the system dynamics becomes

1 2

2

1

( ) ( ) ( )

( , ) ( )

t t u t

u t

η s χ η A N ξ B

(36)

where

12

2 2 2 2

1 1

n n n n

0 IA

M K M C, (37)

2 1

1

1,...,( )

n

i n

0B

M F, (38)

and ( , )η s χ η is the internal dynamics the system (36).

Define the tracking error e as follows:

T

d dy y y y e (39)

Using the control law (31), the feedback control of the nonlinear system (36) is now

considered as follows.

1

1 1

(1 ) ( 1) ( ) ( )

( , ) ( ( ) ( )) ( )

e d e p d

d p d

p p y p

p y t t

e A e k k N ξ ξ

η s χ η A N ξ B k N ξ ξ (40)

where

0 1

, 0,e e

e e

kk k

A (41)

1 2( ) ( )e e e nk L k L k . (42)

The Jacobian matrix of the internal dynamics (0, )J η is obtained as

1 2

0

0(0, ) c c

η

sJ η J J J

η, (43)

where 0η is the equilibrium point of the internal dynamics (0, )η s η . The matrices J , 1c

J ,

and 2cJ are given in Appendix A. It is shown that the following Lipschitz condition holds for

all 0t .

0 0 1 2( , ) (0, ) ( , ) (0, )a a a b b b a b a b s χ η J η η s χ η J η η χ χ η η , (44)

where

1 2 1

1 max , , p

A N BHA k N , (45)

13

1 2 1

2 0max , , , (0, )p

A N BHA k N J η . (46)

Theorem 1. Consider the system (36)-(38). The control gain vector pk is chosen to satisfy the

following conditions:

(i) 0(0, )J η has no eigenvalues on the imaginary axis (the method to determine pk can be

refer to [46]), i.e., the origin of the system (0, )η s η is hyperbolic .

(ii) 0(0, )J η has no positive eigenvalue, i.e., the origin of the system (0, )η s η is

asymptotically stable.

Then, the control law (31) guarantees the uniformly ultimate boundedness of the tracking

error e .

Proof. Since the hyperbolic system 0(0, )η J η η and the Lipschitz condition (44) guarantee

that the approximation of ( , )s χ η by 0(0, )J η η satisfies the Condition 1 in [45, p. 932], for

given bounded dχ (i.e., the desired output dy and its first derivative dy ), it follows from the

Theorem 1 in [45, p. 932] that the unique bounded solution,

0( ) ( , ) (0, ) dd t

η ζ s χ η J η η , (47)

exists and satisfies lim 0dt

η , where ( )tζ is the solution of the following equation.

0(0, ) , ( ) X J η X X 0 . (48)

If the condition (ii) is satisfied, referring to [47,49], there exists a locally Lyapunov

function 1( )V t such that

2 2

1 1 2( )V t η η , (49)

21

3(0, )V

s η ηη

, (50)

14(0, )

V

s η η

η, (51)

14

where ( 1,..,4)i i are the positive constants, (see Theorem 4.16, p. 167 in [47]). From Eqs.

(49)-(51), the following inequality is obtained [48,49]:

213 4 5(0, )

dy

V

s η η η e

η, (52)

where 5 z e η and dd yy are utilized.

Consider the following Lyapunov function candidate:

T

1( ) ( )V t V t e Pe , (53)

where 0 , and the positive definite matrix P is the solution of the following equation.

e e TA P A P I . (54)

where I is identity matrix.

In the region (.) ( ) ,nS t R r

q q q , the error dynamics of the system (40) becomes

1( ) ( )e e p d

e A e k k N ξ ξ . (55)

The time derivative of the Lyapunov function (53) is obtained as

2 2T 1

3 4 5

2 2 2

3 3 4 5

22

24 5 3 4 53

3 3

22

4 54 53

3 3

( ) ( ) ( )

12 2

2 2

d

d

dd

e p d y

y

yy

V t

e e P k k N ξ ξ η η e

e e η η e

e η e

η

(56)

where

1

3 ( )e p k k N P . (57)

There exist pk , ek , and satisfying the following inequality.

2

4 5

3

3

1 02

. (58)

15

Then, one is obtained

2

4 5

3

( ) ( )2

dy

SV t V t

, (59)

where

2 3

3 4 5 3 1

1

min 1 / 2 ,2

s

. (60)

The inequality (59) implies that

2

4 5

3

( ) (0)2

dSyt

S

V t V e

(61)

From Eq. (61), it is concluded that e is bounded in the region S .

Let (.) ( ) ,nt R r

q q q be the set of singularity points, and 1,..., i i mT t be the

set of the time when the system is at the singularity points. It is assumed that there exist two

sets, 1,..., i i m

and 1,..., i i m

, such that ( )i it r

q and ( )i it r

q . In the region

(.) ( ) ,nS t R r

q q q , the time derivative of ( )V t is obtained as

2 1 T

2

3 4 5

2 2 2

4 3 3 4 5

2

3 4 4

3 4

24 3 4 4

3 4

( ) (1 ) ( 1) ( ) ( )

2 2 4

4 2 2

2 2

d

d

d

d

d d d

d e p d

y

y

y

y

y y y

V t p p y p

e k k N ξ ξ e P

η η e

e η e e e η η e

e

e η η

2

2 2 22244

3 4 3 4

2 2 2244

3 4 3 4

2

12 2

( )2 2

1

d

d

d

d d

d

d

d d

y

y

y p

y y

y

yS

y y

p

p

V t

p

k e

k e (62)

16

where

6 , z e η2 , HA P

dd yy , 4 dy P , (63)

are employed. pk , ek , and are selected from sets of the values of pk , ek , and

satisfying the inequality (58) such that the following inequalities hold.

3 42 2 0dy , (64)

3 4 0dy . (65)

Then, it is concluded that

2 2 22244

13 4 3 4

( ) ( ) d2 2

i id

di i

d d

m ty

y pS tiy y

V t V t

k e ,

(66)

where

8 6 3 1 6 3 1min (2 2 ) / 4 , / 2s , (67)

From Eq. (66), the boundedness of ( )V t is obtained as follows

2 2 2 2

7 6 3

8 6 3 5 6 3

2

1

1( ) (0)

2 2

d ,

S

d

i i

i i

t

y

S

m tp

tiS

V t V e

k

e

(68)

From the inequalities (61) and (68), the uniformly ultimate boundedness of the tracking error

e is concluded. Since e and η are bounded, the boundedness of the control law (31) is

guaranteed.

Remark 4. When the micro beam operates in the region S , it is proved that V(t) converges

to the ball of radius 2

4 5 32dy S

, that is, V(t) can be pushed in an arbitrarily small

boundedness region by setting a sufficiently small . Meanwhile, in the region S , the

17

convergence ball (see Eq. (68)) cannot easily be adjusted by changing . Therefore, to

improve the control performance, the region S should be decreased by choosing small r.

However, the small r makes the small ( )rt and consequently, the large control input ( )u t ,

which yields the difficulty in implementing the control scheme due to the supply voltage

saturation in practice. Therefore, the value of r yielding the best control performance is

determined by the maximum voltage, which could be applied.

4. Simulation results

To verify the effectiveness of the proposed control algorithm, numerical simulations were

carried out with the system parameters listed in Table 1. The single mode model was used to

describe the micro beam. Let the initial condition of the micro beam system (6)-(9) be

0 1( ) ( / )w x x L m. The control gain vector kp is selected as follows: 120 10p k .

The dynamic responses of the beam were simulated in two cases: open-loop and closed-

loop controls in which the closed-loop control is proposed in this research. In the case of open

loop control, the resonance curves (see Fig. 4 in [4]) for the micro beam that can predict

amplitude of an oscillation of the beam with respect to the excitation voltage is used to

determine the control input (i.e., 2 2/ 2 ( / 2)cos( )AC ACu V V t ), corresponding to the given

desired deflection. As shown in Figs. 3 & 4, the output tracking can be achieved with either

the open-loop or closed-loop controllers. However, the settling time and the robustness are

different.

It is believed that the closed-loop controller can drive the defection of the beam tip to the

desired oscillation faster than the use of the open loop controller (0.36 ms in the case of the

closed-loop vs. 1.08 ms in the case of the open-loop). To illustrate the robustness of the two

methods to the variation of the system parameters, the micro beam dynamics were simulated

with the assumption: both the open-loop and closed-loop control laws are obtained by using

18

the nominal Young modulus while the actual Young modulus in the beam system is different.

It should be noted that the difference of the Young modulus is chosen to represent typically

the variation of the beam stiffness due to the change in environmental thermal conditions,

materials, geometric shapes of the beam. With the deviation of Young modulus

10% 10% , the steady error tracking in the case of closed-loop control is less than 6%

(see Fig. 5), whereas the one in the case of open-loop control reaches -50% and 80% (see Fig.

6). It is shown that the closed-loop control is more robust than the open-loop control. The

feedback term in the control law (31) compensates the difference of the system parameters

and consequently, makes the control system more robust.

5. Conclusions

A control algorithm of vibration amplitude for a parametrically excited microcantilever

beam has been developed. The proposed control algorithm has been designed based on the

nonlinear model of the microcantilever beam. The tracking through singularities has been

solved. Uniformly ultimate boundedness stability of the control system has been proved using

Lyapunov method. Through the simulation results, the advantages of the closed-loop control

have been shown. The proposed control algorithm can drive the oscillation of the beam tip to

the desired one faster than the open-loop control. The use of the feedback control has

consolidated the robustness of the proposed control method. One of the main advantages of

the proposed closed-loop control is its ability to provide the desired vibrational amplitude

even in the case of wide variation of the system parameters.

19

Appendix A

2 2 2

1

3

3

0 00

0 0

0

00

n n n n

n

nn

c

m

k

m

ck

mm

0 I

J (A.1)

1

2 2 2

1

1 1 1

1 13 3 3 3

1 3 1

1 1

3 3 1

3 1

1 1 1 1

0 0

0 0

n n

n

n n

i i n i i

i i

n n

c i i n i i

i i

n n n

n n n n

n i i n n i i n i i n n i i

i i i i

cc

m m m m m mk k

m m m m m m

k k cc

m m m m m m m m m m m m

0

J

(A.2)

2

2

2 2 2

1

1

1

0

0

n

n

n n n n

p

n

i i

i

c

n n

p

n

n i i

i

k

m m

k

m m

0 0

J0

(A.3)

20

References

[1] J. Bryzek, Impact of MEMS technology on society, Sensors and Actuators A: Physical

56 (1-2) (1996) 1-9.

[2] B. Vigna, Future of MEMS: an industry point of view, Proceeding of the 7th.

International Conference on Thermal, Mechanical and Multiphysics Simulation and

Experiments in Micro-Electronics and Micro-Systems, EuroSimE 2006, Como, Italy,

2006.

[3] N. S. Barker, The future of N/MEMS, IEEE Microwave Magazine 8 (6) (2007) 52-55.

[4] Y. Linzon, B. Ilic, S. Lulinsky, S. Krylov, Efficient parametric excitation of silicon-on-

insulator microcantilever beams by fringing electrostatic fields, Journal of Applied

Physics 113 (16) (2013) 163508-1-163508-11.

[5] E. Buks , Mass detection with a nonlinear nanomechanical resonator, Physical Review E

74 (4) (2006) 046619-1-046619-9.

[6] R. Southworth, L. M. Bellan, Y. Linzon, H. G. Craighead, J. M. Parpia, Stress-based

vapor sensing using resonant microbridges, Applied Physics Letters 96 (16) (2010)

163503-1-163503-3.

[7] J. Fritz , Cantilever biosensors, The Analyst 133 (7) (2008) 855-863.

[8] G. Keskar, B. Elliott, J. Gaillard, M. J. Skove, A. M. Rao, Using electric actuation and

detection of oscillations in microcantilevers for pressure measurements, Sensors and

Actuators A: Physical 147 (1) (2008) 203-209.

[9] D. J. Joe, Y. Linzon, V. P. Adiga, R. A. Barton, M. Kim, B. Ilic, S. Krylov, J. M. Parpia,

H. G. Craighead, Stress-based resonant volatile gas microsensor operated near the

critically buckled state, Journal of Applied Physics 111 (10) (2012) 104517-1- 104517-

9.

[10] S. Boisseau, G. Despesse, B. Ahmed Seddik, Electrostatic Conversion for Vibration

Energy Harvesting, Small-Scale Energy Harvesting, Intech (2012), DOI: 10.5772/3078.

[11] H. S. Wasisto, S. Merzsch, A. Waag, E. Uhde, T. Salthammer, E. Peiner, Airborne

engineered nanoparticle mass sensor based on a silicon resonant cantilever, Sensors and

Actuators B: Chemical 180 (2013) 77-89.

[12] N. Yazdi, F. Ayazi, A. K. Najafi, Micromachined inertial sensors, The Proceedings of

IEEE, 86 (8) (1998), 1640-1659.

[13] B. L. Foulgoc, T. Bourouina, O. L. Traon, A. Bosseboeuf, F. Marty, C. Bréluzeau, J.-P.

Grandchamp, S. Masson, Highly decoupled single-crystal silicon resonators: an

approach for the intrinsic quality, Journal of Micromechanics and Microengineering 16

(6) 2006 S45-S53.

21

[14] D. Rugar, P. Grùtter, Mechanical parametric amplification and thermomechanical noise

squeezing, Physical Review Letters 67 (6) (1991) 699–702.

[15] K. L. Turner, S. A. Miller, P. G. Hartwell, N. C. MacDonald, S. H. Strogatz, S. G.

Adams, Five parametric resonances in a microelectromechanical system, Nature 396

(1998) 149-152.

[16] A. Dâna, F. Ho, Y. Yamamoto, Mechanical parametric amplification in piezoresistive

gallium arsenide microcantilevers, Applied Physics Letters 72 (10) (1998) 1152-1154.

[17] W. Zhang, R. Baskaran, K. L. Turner, Effect of cubic nonlinearity on auto-

parametrically amplified resonant MEMS mass sensor, Sensors and Actuators A 102 (1-

2) (2002) 139–150.

[18] R. Baskaran, K. L. Turner, Mechanical domain coupled mode parametric resonance and

amplification in a torsional mode micro electro mechanical oscillator, Journal of

Micromechanics Microengineering 13 (5) (2003) 701–707.

[19] S. Krylov, I. Harari, Y. Cohen, Stabilization of electrostatically actuated microstructures

using parametric excitation, Journal of Micromechanics Microengineering 15 (6) (2005)

1188–1204.

[20] W. M. Zhang, G. Meng, K. X. Wei, Dynamics of nonlinear coupled electrostatic

micromechanical resonators under two frequency parametric and external excitations,

Shock and Vibration 17 (6) (2010) 759–770.

[21] S. Krylov, Y. Gerson, T. Nachmias, U. Keren, Excitation of large-amplitude parametric

resonance by the mechanical stiffness modulation of a microstructure, Journal of

Micromechanicsl Microengineering 20 (1) (2010) 015041-1-015041-12.

[22] K. Khirallah, Parametric excitation, amplification, and tuning of MEMS folded-beam

comb drive oscillator, Journal of Microelectromechanical Systems 22 (2) (2013) 318-

330.

[23] A. Franks, Progress towards traceable nanometric surface metrology, Nanotechnology 4

(4) (1993) 200–205.

[24] P. K. C. Wang, Feedback control of vibrations in mircomachined cantilever beam with

electrostatic actuators, Journal of Sound and Vibration 213 (3) (1998) 537-550.

[25] L. Chen, C. H. Hanssen, Nonlinear control of a parametrically excited system subject to

actuator saturation, Proceedings of the 42nd IEEE Conference on Decision and Control,

Hawaii, USA, 2003.

[26] R. P. Leland, Adaptive control of a MEMS gyroscope using Lyapunov methods, IEEE

Transactions on Control Systems Technology 14 (2) (2006) 278-283.

[27] K-S. Shen, K.-S. Ou, Command-shaping techniques for electrostatic MEMS actuation:

analysis and simulation, Journal of Microelectromechanical systems 16 (3) 2007 537-

549 .

22

[28] S. Park, R. Horowitz, C.-W. Tan, Dynamics and control of a MEMS angle measuring

gyroscope, Sensors and Actutors A 144 (1) (2008) 56–63.

[29] K. Park, Q. Chen, Y.-C. Lai, Energy enhancement and chaos control in

microelectromechanical systems, Physical Review Letters E 77 (2) (2008) 026210-1-

026210-6.

[30] Y. Fu, J. Zhang, Active control of the nonlinear static and dynamic responses for

piezoelectric viscoelastic microplates, Smart Materials and Structures 18 (9) (2009) 1-

9.

[31] F. M. Alsaleem, M. I. Younis, Stabilization of electrostatic MEMS resonators using a

delayed feedback controller, Smart Materials and Structures 19 (3) (2010) 1-11.

[32] F. M. Alsaleem, M. I. Younis, Integrity analysis of electrically actuated resonators with

delayed feedback controller, Journal of Dynamics System, Measurement, and Control–

Transactions of the ASME 133 (3) (2011) 031011-1-051003-8.

[33] S. Shao, K. M. Masri, M. I. Younis, The effect of time-delayed feedback controller on

an electrically actuated resonator, Nonlinear Dynamics 74 (1-2) (2013) 257-270.

[34] M. S. Siewe, U. H. Hegazy, Homoclinic bifurcation and chaos control in MEMS

resonators, Applied Mathematical Modelling 35 (12) (2011) 5533–5552.

[35] A. Seleim, S. Towfighian, E. Delande, E. Abdel-Rahman, G. Heppler, Dynamics of a

close-loop controlled MEMS resonator, Nonlinear Dynamics 69 (1-2) (2012) 615–633.

[36] C. Batur, T. Sreeramreddy, Q. Khasawneh, Sliding mode control of a simulated MEMS

gyroscope, ISA Transactions 45 (1) (2006) 99–108.

[37] Q. Zheng, L. Dong, D. H. Lee, and Z. Gao, Active disturbance rejection control for

MEMS gyroscopes, IEEE Transactions on Control of Systems Technology 17 (6) (2009)

1432-1438.

[38] W. Zhang, G. Meng, H. Li, Adaptive vibration control of micro-cantilever beam with

piezoelectric actuator in MEMS, International Journal of Advanced Manufacturing

Technology 28 (3-4) (2006) 321-327.

[39] M. Baghelani, H. Ghavifekr, A. Ebrahimi, MEMS based oscillator for UHF applications

with automatic amplitude control, Microelectronics Journal 44 (4) (2013) 292–300.

[40] F. Pellicano, F. Vestroni, Complex dynamics of high-speed axially moving systems,

Journal of Sound and Vibration 258 (1) (2002) 31-44.

[41] K. Marynowski, T. Kapitaniak, Kelvin-Voigt versus Burgers internal damping in

modeling of axially moving viscoelastic web, International Journal of Non-Linear

Mechanics 37 (7) (2002) 1147-1161.

23

[42] Q. C. Nguyen, K.-S. Hong, Stabilization of an axially moving web via regulation of

axial velocity, Journal of Sound and Vibration 330 (20) (2011) 4676-4688.

[43] Q. C. Nguyen, K.-S. Hong, Transverse vibration control of axially moving membranes

by regulation of axial velocity, IEEE Transactions on Control Systems Technology 19

(4) (2012) 1124 – 1131.

[44] P. Villagio, Mathematical Models for Elastic Structures, Cambridge University Press,

Cambridge, UK, 1997.

[45] S. Devasia, D. Chen, & B. Paden, Nonlinear inversion-based out tracking, IEEE

Transactions on Automatic Control 41 (7) (1996) 930-942.

[46] N. K. Bose, On real eigenvalues of real nonsymmetric matrices, Proceedings of IEEE 56

(8) (1968) 1380.

[47] H. K. Khalil, Nonlinear systems, Prentice-Hall, New Jersey, USA, 2001.

[48] J. Hauser, S. Sastry, P. Kokotovic, Nonlinear control via approximate input-output

linearization: the ball and beam example, Proceedings of the 28th Conference on

Decision and Control, Tampa, Florida, USA December, 1989.

[49] J. Hauser, S. Sastry, P. Kokotovic, Nonlinear control via approximate input-output

linearization: the ball and beam example, IEEE Transactions on Automatic Control 31

(3) (1992) 392-398.

24

List of table

Table 1 System parameter values used in numerical simulations.

List of figures

Fig. 1. Micro cantilever beam with two symmetrically located electrodes.

Fig. 2. Control scheme of the micro beam system.

Fig. 3. Dynamic response of the micro beam in the case of the open-loop control.

Fig. 4. Dynamic response of the micro beam in the case of the closed-loop control (the

control law (31) used).

Fig. 5. The amplitude error vs. Young modulus difference in the case of the closed-

loop control.

Fig. 6. The amplitude error vs. Young modulus difference in the case of the open-loop

control.

25

Table 1 System parameter values used in numerical simulations.

Parameter Value

ρ

E

L

b

h

a

p

2330 kg/m3

169106 N/m

2

500 m

16 m

5m

1.310-6

N/mV2

1

1.45

26

Fig.1. Micro cantilever beam with two symmetrically located electrodes.

27

Micro beamcontrol input u

Actual state z + -

Stabilizing

compensation

k (z -z)d

+

p

Nonlinear

inversion

dReference state z

+d

u

d

Desired oscillation

of deflection y

Fig. 2. Control scheme of the micro beam system.

28

Fig. 3. Dynamic response of the micro beam in the case of the open-loop control.

29

Fig. 4. Dynamic response of the micro beam in the case of the closed-loop control (the control

law (31) used).

30

Fig. 5. The amplitude error vs. Yo ung modulus difference in the case of the closed-loop

control.

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 110

1

2

3

4

5

6

Young modulus difference (%)

Am

pli

tud

e err

or

(%)

31

Fig. 6. The amplitude error vs. Young modulus difference in the case of the open-loop

control.

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11-60

-40

-20

0

20

40

60

80

100

Young modulus difference (%)

Am

pli

tud

e err

or

(%)