DSpace at VNU: Nonlinear tracking control of vibration amplitude for a parametrically excited microcantilever beam

The control objective is to drive the oscillation amplitude of the micro-beam, which is amplified using parametric excitation, to the desired values.. The principle of the control algori

Nonlinear tracking control of vibration amplitude

for a parametrically excited microcantilever beam

Quoc Chi Nguyena,n, Slava Krylovb

a

Department of Mechatronics, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet, Ho Chi Minh City, Vietnam

b

School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel

a r t i c l e i n f o

Article history:

Received 3 October 2013

Received in revised form

7 September 2014

Accepted 17 October 2014

Handling Editor: D.J Wagg

a b s t r a c t

In this paper, a feedback control algorithm to regulate oscillation amplitude of a microelec-tromechanical (MEMS) cantilever beam operated at parametric resonances is developed The control objective is to drive the oscillation amplitude of the micro-beam, which is 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-deflection 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

& 2014 Elsevier Ltd All rights reserved

1 Introduction

Microelectromechanical systems (MEMS) 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 [4–12] 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[7–12]

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[13,14] In an active manner, the oscillation amplitude of MEMS cantilevers can be increased externally using parametric excitation[4,5,15–23], 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[15] Turner et al.[16]showed the measurement of the five parametric resonances in MEMS and proposed the use of parametric

Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/jsvi

Journal of Sound and Vibration

http://dx.doi.org/10.1016/j.jsv.2014.10.029

0022-460X/& 2014 Elsevier Ltd All rights reserved.

n Corresponding author Tel.: þ84 8 38688611.

E-mail addresses: nqchi@hcmut.edu.vn (Q.C Nguyen), vadis@eng.tau.ac.il (S Krylov).

excitation to improve the quality of the MEMS sensing devices Zhang et al.[18]investigated the effect of the nonlinearity in the parametrically excited resonant MEMS mass sensor Baskaran and Turner[19]provided an experimental result that shows parametric resonances of a torsional MEMS resonator, considering two interacting mechanical modes of oscillation Zhang

et al.[21] presented a study of nonlinear dynamics and chaos of a MEMS resonator under two parametric and external excitations Khirallah[23]proposed a parametric amplification of a comb drive oscillator Recently, Krylov et al.[4]and Linzon

et al.[5]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[4,5,17,22], 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[24] 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[25–40] The most focused control topic

is to stabilize MEMS resonators[25–35] Alsaleem and Younis[32,33]used delay feedback controllers to stabilize MEMS resonators (especially near pull-in point) The effects of the control gains in[32,33]were later investigated in[34] Siewe and Hegazy [35] indicated that reducing amplitude of parametric excitation can control chaotic motion of a MEMS resonator Seleim et al.[36]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 considered[27,37–40] Leland[27]designed two adaptive controllers for a MEMS gyroscope to regulate the amplitude of the drive axis vibration Batur et al [37] used sliding mode control algorithm to stabilize a MEMS gyroscope Zheng et al.[38]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 et al.[39]introduced an adaptive vibration control of micro-cantilever beam to eliminate the useless high amplitude Baghelani et al.[40]used analytical formula to design an amplitude control for ring shape micro-disk resonator

It is shown that, under parametric excitation, MEMS devices possess much higher amplitude and consequently, higher sensitivity than the ones operated at linear resonances However, the high amplitude is insufficient to obtain the high quality of the MEMS devices In many applications, the amplitude should be controlled For example, in atomic force microscopy (AFM) scanners, the distance between the tip of the AFM beam and the scanned surface should be controlled

In optical scanners, the optical scanned angles should be constant under the effects of temperature and voltage fluctuation Actually, even in the MEMS mass sensors, to be able to measure with high precision, the amplitude should be usually known Since the uncertainties such as temperature, voltage fluctuation, uncertain damping, etc usually affect to the MEMS devices, the amplitude need to be controlled In the present work, a control method to regulate vibration amplitude of a parametrically excited microcantilever structure, which is introduced in[4,5], is developed The distributed electrostatic forces are generated 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

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

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[41–44]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

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

system and the proof of stability also are discussed Section 4includes numerical simulation results that illustrate the effectiveness of the proposed control scheme Finally,Section 5draws conclusions

2 Dynamic models of the micro-beam

2.1 Continuous model

As shown inFig 1, the rectangular cross-section 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 deflections 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[45]:

∂t2þ^c∂w

∂tþ ~EI∂

∂x4þ ^V2ðtÞ^f ¼ 0; (1) where A ¼ bh and I ¼ bh3=12 Since the width of the beam b is larger than its thickness h, the effective (plane strain) modulus ~E ¼ E=ð1v2Þ is employed, where v is Poisson's ratio The supply voltage is denoted as ^VðtÞ The beam is subject to the following homogenous boundary conditions(2)and the nonzero initial conditions(3):

wð0; tÞ ¼ 0; ∂w∂x

x ¼ 0¼ 0; ~EI∂2w

∂x2

x ¼ l¼ 0; ~EI∂3w

∂x3

wð0; xÞ ¼ w0ðxÞ; ∂w∂t

The distributed electrostatic force ^f, which is generated due to asymmetry of fringing fields between the beam and the electrode, act as restoring forces that pull the micro-beam back towards the equilibrium position Since the distributed electrostatic force ^f cannot be calculated analytically in a closed form, in this paper, the distributed electrostatic force is approximated as follows[5]:

^fð ^wÞ ¼ aðσ^wÞ

where ^w ¼ w=h, and a,σ, and p are fitting parameters The unit of the value of a is N/mV2, and the parameters,σand p, are non-dimensional The relationship between the electrostatic force ^f and the non-dimensional beam deflection ^w is plotted

in [5, Fig 2, p 163508-3] It should be noted that the nonzero initial deflection of beam(3)is necessary to enable the actuation by the distributed electrostatic forces at the starting time

Remark 1 In this paper, it should be noted that since the contribution of the electrostatic nonlinearity associated with the actuation deflection-dependent force is dominant, the influence of the nonlinear curvature and rotational inertia[46–48], which have only minor effects, is neglected Moreover, since the force function(4)decreases with the large amplitude, it may result in the limitation of the amplitude as well The experiment and simulation results in [4,5] show that the parametric amplitude is limited by the electrostatic nonlinearity

Introduce the following non-dimensional variables:

^x ¼x

L; ^t ¼Tt; ^w ¼w

where T ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ρAL4= ~EI

q

is a time scale Substituting Eq.(5) into Eqs (1)–(3) and dropping the hats, the non-dimensional equations of motion, the boundary conditions, and the nonzero initial condition that govern the transverse deflection of the micro-beam are obtained as follows:

∂2w

∂t2þc∂w

∂tþ∂

∂x4þβV2ðtÞf ðwÞ ¼ 0; (6)

wð0; tÞ ¼∂w

∂x

x ¼ 0

¼ 0; ~EI∂2w

∂x2

x ¼ 1

¼ ~EI∂3w

∂x3

x ¼ 1

wð0; xÞ ¼w0ðxÞ

h ; ∂w∂t

where

c ¼^c

ffiffiffiffiffiffiffiffiffiffiffi

L4 eEIρA

s

; f ðwÞ ¼^fðwÞ

a ; β¼aL

4

V2

where V0is the unit voltage Hereafter, for the sake of convenience, V0¼1 is adopted As a result, the numerical values of the non-dimensional voltage V are equal to the actual applied voltages values

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

wðx; tÞ ¼ ∑n

i ¼ 1

where qiðtÞ is are the generalized displacement, and the set of the basis functions ϕiðxÞ

R1

0ϕiðxÞϕjðxÞdx ¼δij, whereδijis the Kronecker delta In this paper,ϕiðxÞ is the ith linear undamped mode shape of the straight micro-beam given as

ϕiðxÞ ¼ Di sinλix  sinhλix þsinhλiþ sinλi

coshλiþ cos λi

 cosλix þ coshλix

where Diis a constant, andλisatisfies cos λicoshλi¼ 1, correspondingly giving infinite number of frequencies

whereωiis 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 functionϕjðxÞ, (ii) integrating the resultant equation obtained in step (i) over the domain x ¼ ½0; 1, (iii) utilizing the boundary conditions(7)and substituting

Eq.(10)into the resultant equation obtained in step (ii), and (iv) collecting all terms of the resultant equation obtained in step (iii) with respect to€qiðtÞ,_qiðtÞ, and qiðtÞ After step (iv), Eq.(6)is rewritten into a set of ordinary differential equations (ODEs) as

M€qðtÞþC _qðtÞþKqðtÞþβV2FðqÞ ¼ 0; qð0Þ ¼ q0; (13) where qðtÞ ¼ ½ q1ðtÞ … qnðtÞ Tis the time-dependent vector of generalized coordinates, and the elements of M, C, and K are given as

mij¼δij

Z 1

kij¼δij

Z1

iϕ″

Note that M, C, and K are diagonal matrices The vectorβV2FðqÞ represents the distributed electrostatic actuation, where FðqÞ ¼ ½ f1ðqÞ … fnðqÞ T

with

fiðqÞ ¼

Z 1 0

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

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 zðtÞ ¼ ðqT; _qT

ÞT

AR2n

, and the control input uðtÞ ¼ V2ðtÞ, uðtÞZ0 The reduced-order model of the micro-beam(13)can

be converted to a state space model:

_z ¼ AzþBðzi ¼ 1;:::;nÞuðtÞ; zð0Þ ¼ ðqT; _qTÞT

where

A ¼ 0nn Inn

M 1K M 1C

B ¼ βM 10n1Fðzi ¼ 1 ;:::;nÞ

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

H ¼ ½ϕ1ð1Þ ϕ2ð1Þ … ϕnð1Þ 01n: (22)

3 Output tracking control problem

Consider the dynamic system(18)–(22) with the desired continuous oscillation ydðtÞ A feedback control input uðtÞ is designed to guarantee the tracking of the control output y(t), that is, lim

t-1yðtÞydðtÞ

¼ 0, and zðtÞAL1\ L1for the nonzero initial condition zð0Þ ¼ z0 Three assumptions about the desired output are made as follows: (i) the desired oscillation and its derivatives are continuous; (ii) fyðiÞdðtÞgi ¼ 1;…;n are bounded for all tZ0; and (iii) fyðiÞdðtÞgi ¼ 1;…;n are piecewise continuous functions

Following the design procedure in[49], the first step of the control design is to seek the nominal control input udðtÞ and the desired state zdðtÞ satisfy the following statements:

(i) udðtÞ and zdðtÞ satisfy the differential equation

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

yd¼ Hzd: (24) (iii) udðtÞ and zdðtÞAL1\ L1

Taking the first and second derivatives of the control input and using the dynamic equation(18), the following equations

is obtained:

_y ¼ H _z ¼ HAzþHBu ¼ HAz; (25a)

€y ¼ H A_z ¼ HA2

It should be noted that the expression HB ¼ 0, which obtained from Eqs.(20) and (22), is employed to derive Eq.(25a) Let

θðtÞ ¼ HA2

z ¼ ∑n

i ¼ 1

ϕið1Þ

mi

kiziþczi þ n

ψðtÞ ¼ HAB ¼  ∑n

Eq.(25)can be rewritten as

€y ¼θðtÞþψðtÞuðtÞ: (28) From Eq.(28), choose the nominal control input udðtÞ as

udðtÞ ¼ψ 1ðtÞ½€ydðtÞθðtÞ: (29)

It should be noted that when zi ¼ 1 ;:::;n¼ 0 (i.e., q ¼ 0),ψðtÞ becomes zero, and consequently, udðtÞ-1 Therefore, to avoid the unboundedness of the control input, for the neighborhood of the singular point S9 qð:ÞjqðtÞARn

; jjqjj1rr

, the

following nominal control input is used:

uS

dðtÞ ¼ψ 1ðtrÞ½€ydðtÞθðtÞ; (30) where tr is the time when jjqjj1¼ r Finally, the following control law is proposed:

uðtÞ ¼ pðtÞfð€ydðtÞθðtÞÞkpðzðtÞzdðtÞÞg; (31) where

pðtÞ ¼sgn jjqjjð 1rÞþ1

2ψðtÞ þ

1  sgn jjqjjð 1rÞ

and the control gain vector is given as

kp¼ k1 … k2n

Remark 2 The singularities of the system occur when the deflection 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 3 The control law(31)includes the feedfoward term pðtÞð€ydðtÞθðtÞÞ and feedback term pðtÞkpðzðtÞzdðtÞÞ With the feedfoward term, the system is partially linearized, as shown in Eqs.(36)–(38)below Meanwhile, the feedback term provides a stabilizing effect by adjusting the control gains ki

 

i ¼ 1 ;…;2n Remark 4 It should be noted that the control law(31)is a combination of the exact tracking control law uðtÞ ¼ψ 1ðtÞ fð€ydðtÞθðtÞÞkpðzðtÞzdðtÞÞg and the approximate tracking control law uðtÞ ¼ψ 1ðtrÞfð€ydðtÞθðtÞÞkpðzðtÞzdðtÞÞg, which work in the regions S9 qð:ÞjqðtÞARn

; jjqjj14r

and S9 qð:ÞjqðtÞARn

; jjqjj1rr

, respectively

Remark 5 From the control law(31), it is shown that the large amplitude and the high frequency of the desired oscillation result in the large€ydðtÞ and consequently, the large control input Since uðtÞ ¼ V2ðtÞ, the control input has a limit determined

by the maximum voltage, which can be applied to the micro-beam The amplitude and the frequency of the desired oscillation are limited by the maximum voltage

Introduce the change of the state variables

⋮

η2n  2

2 6 6 6 4

3 7 7 7 5

¼

y _y

z3

⋮

z2n

2 6 6 6 4

3 7 7 7 5

¼

H HA

0 … … … 0 1

2 6 6 6 6 4

3 7 7 7 7 5

z1

z2

⋮

⋮

z2n  1

z2n

2 6 6 6 6 6

3 7 7 7 7 7

Since HA ¼ ½ 01n ϕ1ð1Þ ϕ2ð1Þ … ϕnð1Þ , it is concluded that the matrix N has full rank and consequently, is invertible Therefore, Eq.(34)yields the following inverse transformation:

Using the new state variables, the system dynamics becomes

_

_

χ2¼θðtÞþψðtÞuðtÞ _

η¼ sðχ;ηÞ ¼ AηN 1ξþBηuðtÞ

8

>

where

Aη¼ 0n  2n þ 2 In  2n  2

M 1K M 1C

Bη¼ βM0 1n  21Fðξi ¼ 1 ;:::;nÞ

andη_¼ sðχ;ηÞ is the internal dynamics of system(36)

Define the tracking error e as follows:

e ¼ ½ y yd _y _ydT (39)

Using control law(31), the feedback control of the nonlinear system(36)is now considered as follows:

_e ¼ Aee þð1 ψpÞθþðψp  1Þ€ydþðkeψpkpÞN 1ðξξdÞ _

η¼ sðχ;ηÞ ¼ AηN 1ξþBηpfð€ydðtÞθðtÞÞkpN 1ðξξdÞg

8

<

where

Ae¼ 0 1

ke ke

" #

ke¼ ½ keϕ1ðLÞ … keϕ2nðLÞ : (42) The Jacobian matrix of the internal dynamics Jð0;ηÞ is obtained as

Jð0;η0Þ ¼ ∂s

∂η

η

0

¼ JηþJc1þJc2; (43) whereη0is the equilibrium point of the internal dynamicsη_¼ sð0;ηÞ The matrices Jη, Jc1, and Jc2are given inAppendix A

It is shown that the following Lipschitz condition holds for all tZ0:

jjfsðχa;ηaÞJð0;η0Þηagfsðχb;ηbÞJð0;η0Þηbgjjrμ1jjχaχbjjþμ2jjηaηbjj; (44) where

μ1Z max jjAηN 1jj1; jjBHA2jj1; jjkpN 1jj1

μ2Z max jjAn ηN 1jj1; jjBHA2jj1; jjkpN 1jj1; Jð0;η0Þo

(46)

Theorem 1 Consider systems(36)–(38) The control gain vector kpis chosen to satisfy the following conditions:

(i) Jð0;η0Þ has no eigenvalues on the imaginary axis (the method to determine kpcan be refer to[50]), i.e., the origin of the systemη_¼ sð0;ηÞ is hyperbolic

(ii) Jð0;η0Þ has no positive eigenvalue, i.e., the origin of the systemη_¼ sð0;ηÞ is asymptotically stable

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

Proof Since the hyperbolic system η_¼ Jð0;η0Þηand the Lipschitz condition (44)guarantee that the approximation of

sðχ;ηÞ by Jð0;η0Þηsatisfies Condition 1 in[49, p 932], for given boundedχd(i.e., the desired output ydand its first derivative _yd), it follows fromTheorem 1in[49, p 932] that the unique bounded solution,

Z þ 1

 1 ζðt τÞ sð χ;ηÞJð0;η0Þηdτ; (47) exists and satisfies lim

_X ¼ Jð0;η0ÞX; Xð71Þ ¼ 0: (48)

If condition (ii) is satisfied, referring to[51,52], there exists a locally Lyapunov function W1ðtÞ such that

rW1ðtÞrκ2jjηjj2

∂W1

∂ηsð0;ηÞr κ3jjηjj2

∂W1

∂ηsð0;ηÞr κ4jjηjj; (51) where κiði ¼ 1; …; 4Þ are the positive constants (see Theorem 4.16, p 167 in [51]) From Eqs (49)–(51), the following inequality is obtained[52,53]:

∂W1

∂ηsð0;ηÞr κ3jjηjj2þκ4κ5jjηjjðjjejjþμydÞ; (52) where jjzjjrκ5jjejjþjjηjj

and jydjrμyd are utilized

Consider the following Lyapunov function candidate:

WðtÞ ¼ eTPeþγW1ðtÞ; (53)

whereγ40, and the positive definite matrix P is the solution of the following equation:

AeP þATeP ¼ I: (54) where I is identity matrix

In the region S ¼ qð:ÞjqðtÞARn

; jjqjj14r

, the error dynamics of system(40)becomes _e ¼ Aee þðkekpÞN 1ðξξdÞ: (55) The time derivative of the Lyapunov function(53)is obtained as

_ WðtÞ ¼  jjejj2þeTPðkekpÞN 1ðξξdÞþγ κ3jjηjj2þκ4κ5jjηjj jjejjþμyd

r jjejj2þμ3jjejj2þγ κ3jjηjj2þκ4κ5jjηjj jjejjþμyd

r  1μ3γ κð 4κ5Þ2

2κ3

! jjejj2γκ3

2 jjηjjκ4κ5

jjejj

γκ3

2 jjηjjκ4κ5μyd

þγ κ4κ5μyd

 2

2κ3

(56)

where

μ3¼ jjðkekpÞN 1jj jjPjj: (57) There exist kp, ke, andγsatisfying the following inequality:

1 μ3γ κð 4κ5Þ2

Then, one is obtained

_ WðtÞo μSWðtÞþγðκ4κ5μydÞ2

where

μs¼ min 1μ3γ κð 4κ5Þ2=2κ3κ1;γκ3

2κ1

Inequality(59)implies that

WðtÞrWð0ÞeμStþγðκ4κ5μydÞ2

2κ3μS

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

Let Ω9 qð:ÞjqðtÞARn

; jjqjj1¼ r

be the set of singularity points, and T ¼ ftigi ¼ 1 ;…;m be the set of the time when the system is at the singularity points It is assumed that there exist two sets, fε

jjqðtiε

i Þjj1¼ r and jjqðtiþε

i Þjj1¼ r In the region S ¼ qð:ÞjqðtÞAR n; jjqjj1rr, the time derivative of WðtÞ is obtained as _

WðtÞ ¼  jjejj2þ ð1n ψpÞθþðψp  1Þ€ydþðkeψpkpÞN 1ðξξdÞo

eTP

þγ κ3jjηjj2þκ4κ5jjηjj jjejjþμyd

r jjejj2þμθjjηjjjjejjþμ4jjejjþμ3jjejj2þγ κ3jjηjj2þκ4κ5jjηjj jjejjþμyd

r 2  2μ3μθγκ4μyd

4 jjejj 4μ4

2 2μ3μθγκ4μyd

!2

ðμθþγκ4μydÞ

2 jjejjjjηjj2

κ3γμθγκ4μyd

2 jjηjj γκ4μyd

!2

2 2μ3μθγκ4μyd

þγμydþ γ2κ2μ2

d

þ 1 ψp

jjkpjjjjejj2

r μSWðtÞþ μ2

2  2μ3μθγκ4μyd

þ γ2κ2μ2

d

þγμyd

þ 1 ψp

jjkpjj jjejj2; (62) where

jjzjjrκ6jjejjþjjηjj

; μθ¼κjjHA2jj jjPjj; jydjrμ€y d; μ4¼μ€y djjPjj (63)

are employed kp, ke, andγ are selected from sets of the values of kp, ke, andγ satisfying inequality(58)such that the following inequalities hold:

2  2μ3μθγκ4μyd40; (64)

Then, it is concluded that

_

WðtÞr μSWðtÞþ μ2

2  2μ3μθγκ4μyd

þ γ2κ2μ2

d

þγμydþjjkpjj ∑m

i ¼ 1

Z t i þε i

t i εþ i

jjejj2dτ; (66) where

μs¼ min ð22μ8μθγμ6μ3Þ=4κ1; μθþγμ6μ3

=2κ1

From Eq.(66), the boundedness of WðtÞ is obtained as follows:

WðtÞrWð0ÞeμS tþ1

2  2μ8μθγμ6μ3

þ γ2μ2μ2

þγμyd

þjjkpjj

i ¼ 1

Z tiþε i

t i εþ i

From inequalities(61) and (68), the uniformly ultimate boundedness of the tracking error e is concluded Since e andηare bounded, the boundedness of the control law(31)is guaranteed □

Remark 6 When the micro-beam operates in the region S, it is proved that W(t) converges to the ball of radius

γðκ4κ5μydÞ2=2κ3μS, that is, W(t) can be pushed in an arbitrarily small boundedness region by setting a sufficiently smallγ Meanwhile, in the region S, the convergence ball (see Eq.(68)) cannot easily be adjusted by changing γ Therefore, to improve the control performance, the region S should be decreased by choosing small r However, the small r makes the smallψðtrÞ and consequently, the large control input uðtÞ, 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 (Fig 2)

4 Simulation results

To verify the effectiveness of the proposed control algorithm, numerical simulations were carried out with the system parameters listed inTable 1 The single mode model was used to describe the micro-beam Let the initial condition of the micro-beam system(6)–(9) be w0ðxÞ ¼ϕ1ðx=LÞμm The control gain vector kpis selected as follows: kp¼ 120 10

The dynamical 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 (seeFig 4in[5]) 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., u ¼ ^V2AC=2þð ^V2AC=2Þ cos ðωtÞ), corresponding to the given desired deflection As shown in

time and the robustness are different

Micro beam control inputu

Actual statez

+

-Stabilizing compensation

k (z -z)d

+

p

Nonlinear inversion

d

Reference statez

+

d

u

d

Desired oscillation

of deflectiony

Fig 2 Control scheme of the micro-beam system.

It is believed that the closed-loop controller can drive the deflection 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, as shown

micro-beam dynamics was simulated with the assumption: both the open-loop and closed-loop control laws are obtained

by using 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 ½  10% 10% , the steady tracking error in the case of closed-loop control is less than 6 percent (see Fig 5), whereas the one in the case of open-loop control reaches 50 and 80 percent (seeFig 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

closed-loop and open-loop controls, in which the results were collected from the numerical simulations illustrating the robustness of the closed-loop and open-loop controls With the deviation of Young modulus [  10% 10%], the settling time in the case of closed-loop control (seeFig 7) varies from 0.27 ms (a decrease of  23 percent, in comparison to the case of no

Fig 3 Dynamic response of the micro-beam in the case of the closed-loop control (the control law (31) used).

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

Table 1 System parameter values used in numerical simulations.

N/m 2

... 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 The control law(31)includes... _ydT (39)

Using control law(31), the feedback control of the nonlinear. .. In the case of open loop control, the resonance curves (seeFig 4in[5]) for the micro -beam that can predict amplitude of an oscillation of the beam with respect to the excitation voltage is used