当前位置：文档库 › Disturbance-Observer-Based Hysteresis Compensation for Piezoelectric Actuators

Disturbance-Observer-Based Hysteresis Compensation for Piezoelectric Actuators

Disturbance-Observer-Based Hysteresis Compensation for Piezoelectric Actuators Jingang Yi,Senior Member,IEEE,Steven Chang,and Yantao Shen,Member,IEEE

Abstract—We present a novel hysteresis compensation method for piezoelectric actuators.We consider the hysteresis nonlinearity of the actuator as a disturbance over a linear system.A disturbance observer(DOB)is then utilized to estimate and compensate for the hysteresis nonlinearity.In contrast to the existing inverse-model-based approach,the DOB-based hysteresis compensation does not rely on any particular hysteresis model,and therefore provides a simple and effective compensation mechanism.We design and fab-ricate a lead magnesium niobate-lead titanate(PMN-PT)piezo-electric actuator for microscale tip-based power sintering process. Experimental validation of the proposed hysteresis compensation is performed on the PMN-PT cantilever piezoelectric actuator.The experimental results demonstrate the effectiveness and ef?ciency of the approach.

Index Terms—Compensation,disturbance observer(DOB), hysteresis,piezoelectric actuator,smart materials.

I.I NTRODUCTION

D URING the last decade,micro/nanomanipulation have

been extensively studied in robotics and control commu-nities[1],[2].For manipulation platforms,for example,such as that used in atomic force microscopy(AFM),piezoelectric can-tilever beams and piezoelectric tubes are the most widely used actuation mechanisms.However,one of the potential issues of using these actuators is poor performance and limited band-width due to the nonlinear hysteresis effect of the piezoelectric materials[3].

To compensate for hysteresis nonlinearity,various control strategies have been proposed in the past decade.A review of the hysteresis modeling and compensation methods for mi-cro/nanoapplications can be found in recent two special issues in control engineering community[3],[4].Two control struc-tures are mainly used for hysteresis compensation[5],[6]:1) an inverse-model feedforward hysteresis compensation and2) a closed-loop feedback hysteresis compensation.The inverse-model feedforward hysteresis compensation mechanism uses an

Manuscript received November2,2008;revised March24,2009.First pub-lished June19,2009;current version published August14,2009.Recommended by Guest Editor P.X.Liu.The work of J.Yi was supported in part by the San Diego State Research Foundation and in part by the National Science Foundation under Grant CMMI-0826532.This work was presented at the2009American Control Conference,St.Louis,MO,June10–12,2009.

J.Yi is with the Department of Mechanical and Aerospace Engineering,Rut-gers University,Piscataway,NJ08854USA(e-mail:jgyi@https://www.wendangku.net/doc/215361053.html,).

S.Chang is with the Department of Mechanical Engineering,San Diego State University,San Diego,CA92182USA(e-mail:changs@https://www.wendangku.net/doc/215361053.html,). Y.Shen is with the Department of Electrical and Biomedical Engineering, University of Nevada,Reno,NV89557USA(e-mail:ytshen@https://www.wendangku.net/doc/215361053.html,). Color versions of one or more of the?gures in this paper are available online at https://www.wendangku.net/doc/215361053.html,.

Digital Object Identi?er10.1109/TMECH.2009.2023986inverse hysteresis model in the feedforward loop to cancel the hysteresis nonlinearity.While this implementation does not re-quire displacement or force sensors,an accurate mathematical model for hysteresis is essential to such an approach.Addition-ally,the hysteresis model is typically complicated[7],[8],and the method is prone to robustness issues due to disturbances.On the other hand,although the feedback hysteresis compensation approach requires external displacement or force sensors,it pro-vides an effective and robust means to suppress the hysteresis nonlinearity.Moreover,the feedback approach does not rely on a precise mathematical model of the hysteresis.

We present a novel hysteresis compensation mechanism for a piezoelectric cantilever actuator used for micromanipulation applications.Our approach is based on the closed-loop com-pensation mechanism.Instead of precisely modeling the hys-teresis,we treat the hysteresis as an external disturbance added to the linear dynamic behavior of the cantilever actuator.Then, we use a disturbance observer(DOB)to estimate and com-pensate for the hysteresis nonlinearity.The signi?cance of the proposed DOB-based hysteresis compensation is its simplic-ity in implementation and robustness due to its independence from any hysteresis models.We have also experimentally val-idated the proposed compensation method on a piezoelectric cantilever actuator.The focus of this paper is to demonstrate the effectiveness,robustness,and simple implementation of the DOB-based compensation scheme.We leave the comparison of the proposed compensation with other existing model-based feedback/feedforward methods to ongoing future work.

Our approach is inspired by several related work[9],[11].

A DOB-based nonlinearity cancellation is proposed in[9]for a large class of single-input single-output(SISO)nonlinear sys-tems.The output of the nonlinear SISO system is assumed to be the sum of the outputs of a stable SISO linear-time-invariant system and a bounded function of time.The nonlinearity in the system,for example,could include dead zone,backlash, and hysteresis.Our work is an extension of the approach in[9] and[10].We use a piezoelectric cantilever actuator as an ex-ample to show that hysteresis nonlinearity can be decomposed into a bounded hysteresis operator and an approximated linear dynamic system.We also provide experimental validation.

A linear active disturbance rejection control(LADRC)is pro-posed in[11]for hysteresis compensation.Hysteresis nonlinear-ity is considered as a disturbance and a linear observer is pro-posed as a disturbance estimator.However,it is not clear why the hysteresis-induced disturbance can be treated as a linear system. Moreover,no experimental validation has been presented for the proposed approach.We relax the linear disturbance estimation

1083-4435/$26.00?2009IEEE

Fig.1.Schematic of decomposition of a nonlinear system with hysteresis nonlinearity H(·)as an integrated part of the system.

assumption and provide a more comprehensive treatment for the hysteresis nonlinearity.

The remainder of the paper is organized as follows.We present the DOB-based hysteresis compensation design in Section II.In Section III,we discuss the fabrication and mod-eling of a piezoelectric cantilever actuator.The experimental implementation of the DOB-based hysteresis compensator for the piezoelectric cantilever actuator is presented in Section IV. Finally,we conclude the paper in Section V.

II.DOB-B ASED H YSTERESIS C OMPENSATION

In this section,we?rst discuss the decomposition of a class of nonlinear dynamic systems with hysteresis relationship as a component of the system.Then,we present a DOB-based hysteresis compensation mechanism.

A.Decomposition of a Class of Nonlinear Dynamic Systems With Hysteresis

We consider here a class of nonlinear dynamic systems in which the hysteresis nonlinear relationship is one part of the dynamics.For the piezoelectric cantilever actuator that we will discuss in the next section,the applied voltage input v(t)pro-duces a charge q(t)between the electrodes.This charge then produces a deformation in the piezoelectric material and the accompanying tip de?ection y(t).The left sub?gure in Fig.1 shows such a nonlinear relationship.It is well-known that a hys-teresis relationship exists between the input voltage v(t)and the charge q(t)for piezoelectric materials[5].We denote the hysteresis relationship between v(t)and q(t)as

q(t)=(H(v))(t).(1) The dynamic relationship between q(t)and displacement y(t) at the tip of the cantilever is given by the linear time-invariant system P(s).We will discuss the development of P(s)for the cantilever piezoelectric actuator example in the next section. Let N(H,P)denote the nonlinear dynamic system between input v(t)and output y(t).We propose to decompose N(H,P) into a linear system L(K,P,d),as shown in the right sub?gure in Fig.1.Here,K(s)denotes a linear time-invariant relation-ship between input variable v(t)and intermediate variable q(t). Also,the disturbance d(t)is a bounded nonlinear function of input v(t).We assume that the intermediate variable q(t)is not measurable.

To further elucidate the earlier decomposition,we consider a Duhem model to capture the rate-independent hysteresis re-lationship between the input v(t)and intermediate

variable Fig.2.Schematic of the linear representation for the hysteresis operator q(t)=(H(v))(t)=αv(t)+d(t).

q(t)[8],[12].The Duhem model represents H(·)by a?rst-order nonlinear differential equation as

˙q(t)=α|˙v(t)|(av(t)?q(t))+b˙v(t)(2) whereα>0and a>b≥1/2a are model constants that de-pend on the shape and area of the hysteresis curves.In(2), q(t)is considered as the state variable of the differential equa-tion and depends on the values of both v(t)and˙v(t).Such a mathematical relationship(2)can reproduce the hysteresis phe-nomena that we observe in experiments.Readers can refer to[8] and[12]for more details on how to estimate these hysteresis model parameters using experimental data.

We consider the solution properties of the Duhem dynamic model(2).Following a similar derivation in[13],we solve(2) explicitly as follows:

q(t)=αv(t)+d(t)(3) where

d(t):=d(v(t))=(q0?av0)e?α(v?v0)sgn(˙v)

+e?αv sgn(˙v)

(b?a)eατsgn(˙v)dτ(4)

q0:=q(0),v0:=v(0),and function sgn(x)=1if x≥0and sgn(x)=?1if x<0.It is straightforward to check that for ˙v>0or˙v<0,the previous solution satis?es

lim

v→+∞

d(t)=?

a?b

or lim

v→?∞

d(t)=

a?b

(5) respectively.Therefore,d(t)is bounded,that is,|d(t)|≤d m:= sup t≥0|d(t)|.

Fig.2shows the linear representation of the nonlinear hys-teresis operator H(·)by(3)with the bounded unknown distur-bance d(t).Note that for the Duhem model

K(s)=α

for the linear system L(K,P,d)shown in Fig.1.

Fig.3.Schematic of the DOB-based control design.

Although the decomposition analysis is proved using the Duhem hysteresis model,it is possible to show a similar treat-ment for other hysteresis models.With the earlier decompo-sition,we are now ready to apply the DOB scheme for the nonlinear hysteretic dynamic system(1).

B.DOB-Based Hysteresis Compensation

The general idea of DOB-based compensation is to estimate any unknown disturbances introduced into a system[14],[15]. The design of the DOB is accomplished through augmenting autonomous dynamical systems to the controlled plant.Because the DOB-based compensation is mainly utilized for disturbance rejection,additional feedback or feedforward controllers may be included to achieve the desired tracking performance if needed [16].

Fig.3shows the block diagram of the DOB-based hysteresis compensation mechanism[9].Here,C(s)denotes the controller for the plant L(K,P,d)andξ(t)denotes the measurement noise. K N(s)and P N(s)denote the nominal transfer functions for K(s)and P(s),respectively.It is straightforward to calculate that the estimated disturbance?d(t)will be close to d(t)if the measurement noiseξ(t)is negligible,and the nominal plants K N(s)and P N(s)are close to K(s)and P(s),respectively.

The?lter Q(s)is used to make Q(s)K?1

N (s)P?1

(s)real-

izable since K?1

N (s)and P?1

(s)are typically noncasual.The

relative degree of Q(s)must be greater than or equal to that of the nominal plant K N(s)P N(s)[15].Moreover,all unstable zeros of K N(s)P N(s)must be zeros of Q(s)[17].Typically, Q(s)can be chosen as[15]

Q(s):=Q(s;τ)=N Q(s;τ)

D Q(s;τ)

N?r

k=1

βk(τs)k+1

k=1

βk(τs)k+1

where r d is the relative degree of Q(s)andτis the time constant of the?lter.

The stability of the DOB-based control design has been re-cently presented in[16]and[18].It has been shown in[16]that for certain disturbances,the DOB design is equivalent to the in-ternal model principle.Particularly,for robot control,the DOB-based control is the same as the passivity-based approaches discussed in the aforementioned papers.Also,because of

these Fig.4.PMN-PT/PDMS cantilever actuator.

equivalences,the stability and robustness of the DOB design can be obtained using the same methods.In the following,we brie?y present a recent result in robust stability[18]that will be used in our hysteresis compensation design.

Consider the plant P(s)∈P,where the set P is de?ned as a collection of transfer functions for which the leading coef?cients of the numerator and denominator do not change sign.We de?ne the robust internal stability for each P(s)∈P,as the closed-loop systems from the input set[r(t)d(t)ξ(t)]T to the output set[e(t)v(t)ˉy(t)]T(Fig.3)that are stable under the DOB-based control systems design.We have the following stability results.

Proposition1([18]):For P(s)∈P,there exists a constant τ?>0such that,for all0<τ<τ?,the closed-loop system under the DOB-based control shown in Fig.3is robust internally stable if the following conditions hold:1)P N(s)C(s)/(1+ P N(s)C(s))is stable,2)P(s)is minimum phase,and3)P f(s) is Hurwitz,where

P f(s):=D Q(s;1)+

lim

s→∞

P(s)

P N(s)

N Q(s;1).(6)

Note that it is possible to design the?lter Q(s)to meet the third condition in Proposition1[18].

III.A CTUATOR M ODELING AND S TABILITY OF THE

DOB-B ASED C ONTROL D ESIGN

In this section,we?rst describe the fabrication of the piezo-electric cantilever actuator that is used as the testing device for the proposed hysteresis compensation.We then present a dynamic model of the actuator.Finally,we brie?y check the robust stability conditions presented in Proposition1.

A.Actuator Fabrication

Fig.4shows a prototype of the piezoelectric cantilever ac-tuator.We use lead magnesium niobate-lead titanate(PMN-PT),a single crystal relaxor ferroelectric material,as the ac-tuator material.The PMN-PT piezoelectric actuator was orig-inally developed for microscale tip-based nanopowder sinter-ing processes at San Diego State University.We also use an

interdigitated electrode(IDE)design for better actuation per-formance.To reduce the stress concentration at the support, and therefore,increase the reliability of the actuator for cyclic loading,we use polydimethylsiloxane(PDMS)to form a coat-ing layer on the PMN-PT cantilever.A proof mass at the can-tilever tip is also formed by PDMS to emulate any attached end-effector,such as a conductive tip,for micro/nanomanipulation.

A similar cantilever PMN-PT/PDMS design is also used as an energy harvester[19].

To fabricate the actuator prototype,we use 001 -factory-oriented PMN-PT single crystal plate with a thickness of 110μm.The PMN-PT plate is polished and a thin layer(50 nm)of gold or Cr is deposited on the top and the bottom sur-faces to form?ne primary electrode layers.The IDE patterns are custom fabricated by photolithography.The PMN-PT can-tilever is then attached to a PCB board(8mm×8mm×1 mm),as shown in Fig.4.A PDMS epoxy coat is applied to the cantilever.Any air cavities are carefully removed.The PMN-PT/PDMS cantilever is then heated to cure the PDMS epoxy. Once the PDMS has cured and hardened,it is cooled down to room temperature.A similar process is used to add the proof mass.Finally,the PMN-PT/PDMS cantilever undergoes poling.

B.Dynamic Modeling

For the purpose of modeling,the actuator prototype is con-sidered as a composite cantilever with a point mass on its tip (see Fig.5).Let l and b denote length and width of the PMN-PT/PDMS composite cantilever,respectively.The mass of the proof mass is denoted by M p.The heights of the PMN-PT and PDMS layers are denoted by h p and h b,respectively.The mass density per unit length for the PMN-PT and the base beams are denoted byρp andρb,respectively.

We consider the composite layer of the PMN-PT and PDMS as a single beam with?exural rigidity EI eq[19]. The mass density per unit length for the composite beam is m:=(ρp h p+ρb h b)/(h p+h b).The vertical de?ection of the beam is denoted by w(x,t)at location x and time t in the coor-dinate system xoz that is located on the base(see Fig.5).The kinetic and potential energy of the system are

T=1

m˙w(x,t)2dx+

M p˙w(l,t)2

V=1

EI eq(w )2dx

where˙w(x,t):=?w(x,t)/?t and w (x,t):=?2w(x,t)/?x2. Let c denote the viscous damping coef?cient for the beam.The virtual work done by the damping force is

δW c=

(?c˙w(x,t)+M n(x,t))δwdx

whereδw is the virtual displacement.The bending moment M n(x,t)due to the electric?eld by the input charge q B(t) (bottom electrode)and q T(t)(top electrode)is

approximated Fig.5.(a)Operating diagram of the PMN-PT/PDMS actuator with IDEs.

(b)1-D schematic of the composite cantilever.

[20]as

M n(x,t)=?

Ebd33C E h2

12h F

q d(t)=K e q d(t)

where q d(t):=q T(t)?q B(t),K e:=?Ebd33C E h2/12h F, C E is the capacitance of the IDE,h F is the interelectrode dis-tance,and d33is the piezoelectric coef?cient.We assume that the electrode starts at location x s≥0and ends at location x e≤l. Therefore,the spatial derivative M n(x,t)can be represented as M n(x,t)=K e[δ (x?x s)?δ (x?x e)]q d(t)

=K a q d(t)(7) where K a=K e[δ (x?x s)?δ (x?x e)],andδ (·),the spa-tial derivative of the Dirac delta function,represents the unit dipole function[21].

Using the extended Hamilton’s principles,we obtain the equa-tions of motion and boundary conditions

M p

?2w(l,t)

?t2

?2w

?t2

+EI eq

?4w

?x4

=M n(x,t)(8a) M p

?2w(l,t)

?t2

?EI eq?

3w(l,t)

?x3

=0(8b) w(0,t)=w (0,t)=w (l,t)=0.(8c) We de?neξ:=x/l,β:=M p/ml,andλ4:=ml4ω2/EI eq as the dimensionless length of the beam,mass ratio of the proof mass and the composite beam,and the natural frequency param-eters of the beam–mass system,respectively.The characteristic equation of the dynamic systems given in(8)is then obtained as

1+cosλcoshλ+λβ(cosλsinhλ?sinλcoshλ)=0.(9)

The natural frequency decreases asβ(or as the tip mass)

increases[19].

By variable separation,we write w(x,t)= ∞

i=1

φi(x)p i(t),

whereφi(x)is the i th modal shape and p i(t)is the i th gener-alized coordinate for the system.For i th mode,the equation of motion for the beam-mass system in terms of p i(t)is obtained as m ei¨p i(t)+c ei˙p i(t)+k ei p i(t)=f ei q d(t)(10)

where the generalized mass,damping,and stiffness coef?cients are

m ei=

mφ2i(x)dx+M pφ2i(l)

c ei=

cφ2i(x)dx

k ei=

EI eq(φ i(x))2dx

respectively.The generalized force coef?cient is given by

f ei=

0K aφi(x)dx=K e

dφi

(x s)?

dφi

(x e)

Let w p(t):=w(l,t)= ∞

i=1

φi(l)p i(t)denote the measur-

able tip displacement.Taking Laplace transformation,from (10),we?nd the transfer function between the excitation charge q d(t)and the tip displacement w p(t)as

P(s):=W p(s)

Q d(s)

∞

i=1

P i(s)=

∞

i=1

A v iω2ni

s2+2ζiωni s+ω2

(11)

whereω2ni=k ei/m ei,2ζiωni=c ei/m ei,and A v i=φi(l)f ei/k ei.

Remark1:In the previous analysis,we neglect the nonlin-ear effects of the deformation of the composite beam and the electromechanical characteristic of the PMN-PT piezoelectric material.The treatment of these nonlinear properties on dynam-ical modeling can be found in[22].The damping effect due to the electromechanical coupling of the piezoelectric material is also neglected in the models.

C.Robust Stability of the DOB-Based Control Design

Since the transfer function(11)shows the system to exhibit vibrational behavior,it is appropriate to use the lowest mode to approximate the dynamics for applications utilizing a low frequency range.Therefore,we take the?rst mode as the ap-proximation for the plant dynamics in the DOB-based hysteresis control systems design,namely,

P N(s)=P1(s)=

A v1ω2n1

s+2ζ1ωn1s+ω2n1

.(12)

Since the plant dynamics P(s)in(11)is stable and minimum-phase with relative degree of2,we choose a second-order But-terworth?lter for Q(s)

Q(s)=Q(s;τ)=

(τs)2

√

2τs+1

(13)

Fig.6.Schematic of the testing systems.

TABLE I

E STIMATED AND M EASURED

F IRST AND S ECOND N ATURAL F REQUENCIES AND

THE D AMPING

C OEFFICIENTζ1OF THE A CTUATOR

whereτ=1/ωc andωc is the cutoff frequency of the?lter Q(s).

Note that the relative degree of Q(s)is the same as that of the

plant P(s).Q(s)is chosen as a low-pass?lter that admits the

lower frequencies used in disturbance rejection while rejecting

higher frequencies associated with sensor noises[16].

To check the stability conditions given in Proposition1,the

?rst condition can easily be satis?ed by designing the feed-

back controller C(s)such that the closed-loop systems with the

nominal plant P N(s)is stable.For the second condition,P(s)is

minimum phase and stable given the fact that the gain of the?rst

mode is relatively large and the cantilever piezoelectric beam is

a stable system.To check the third condition,we note that

lim

s→∞

P(s)

P N(s)

?1=lim

s→∞

∞

i=2

P i(s)

P1(s)

∞

i=2

A v iω2ni

A v1ω2n1

=:R p

and therefore

P f(s)=D Q(s;1)=s2+

√

2s+(1?R p).

Since typically A v i A v1from vibration theory,we assume

without loss of generality1that R p<1.Therefore,P f(s)is

Hurwitz and the robust internal stability can be obtained by

Proposition1.

IV.E XPERIMENTS

A.Experimental Setup

A PMN-PT cantilever actuator prototype(shown in Fig.4)

is fabricated with a dimension of7.4mm×2mm×110μm.

To test the cantilever actuator,we set up a testbed,as shown in

Fig.6.The cantilever actuator is mounted on a?xed base.A

high-precision?ber optic displacement sensor system(model

1If R p≥1,we can always redesign the constant term of D Q(s;1)to make

it larger than R p.A similar explanation is also presented in[18].

Fig.7.Bode plots of the cantilever actuator.

D11,Philtec,Inc.)is used to measure the tip displacement of the cantilever.The output sensitivity of the?ber optic displace-ment sensor is12.3nm/mV.The cantilever actuator is driven by an ampli?er(model790A,PCB Piezotronics,Inc.).A real-time control system(model ACE1104,dSPACE,Inc.)is used to control the motion of the actuator and also to collect the displacement data.

B.Estimation of Model Parameters

The estimation of the actuator parameters,such as geom-etry and mechanical properties,can be found in[19].With these parameters,we can analytically estimate the?rst and sec-ond natural frequencies of the actuator using(9),as shown in Table I.The Bode plot of the transfer function P(s)is shown in Fig.7.We obtain the frequency response by impulse excitation input.The estimated?rst and second natural frequencies are also listed in Table I.The estimates of these natural frequencies from the Bode plots match the analysis closely.Note that the Bode plot of the nominal plant P N(s)shown in Fig.7matches the frequency responses of P(s)at low frequencies.We also obtained an estimate of the damping coef?cientζ1through the step response of the system[19].The estimated value is listed in Table I.

For our experiments,we only excite the electrodes on the top surface of the actuator.The bottom electrodes are connected to ground.Because we cannot measure the charge signal q(t)in experiments,we cannot directly determine the parameter A v in the plant model(11).However,from an input/output perspec-tive,we can treat A v=1nm/C and let the parameterαin the hysteresis operator(3)determine the dc gain of the system. Fig.8(a)and(c)show the actuator response under a si-nusoidal input with a frequency of10and100Hz,re-spectively.Clearly,the hysteresis nonlinearity in actuator re-sponse is independent of input frequency.We estimate the value ofαby taking an average of the estimates from various input frequencies,and the following estimate is obtained:

K(s)=K N(s)=α=76.4C/V.

C.Hysteresis Compensation Performance

To demonstrate the effectiveness of the DOB-based hystere-sis compensation,we show several experimental examples un-der various input excitations.Fig.8(b)and(d)show the rela-tionship of the tip displacement versus the input voltage under DOB compensation without the use of additional feedback con-trollers C(s).In these experiments,we design the cutoff fre-quencyωc of the DOB?lter Q(s)to be500Hz since we only conduct low frequency excitations.Although the effectiveness of the proposed compensation is only shown under two input frequencies,our testing results have demonstrated effectiveness of the compensation up to150Hz.For these particular experi-ments,the DOB-based compensation was effective in removing all hysteresis detectable given the limitations of the sensors used.Fig.9shows a comparison of the tracking performance of the actuator under sinusoidal input with various frequencies. The desired tracking trajectory is a sinusoidal signal with an

Fig.8.Tip displacement versus the input voltage.(a)Without DOB under a 10-Hz excitation.(b)With DOB compensation under a 10-Hz excitation.(c)Without DOB under a 100-Hz excitation.(d)With DOB compensation under a 100-Hz

excitation.

Fig.9.(a)Comparison results of the tracking performance with and without DOB hysteresis compensation under a 100-Hz sinusoidal input.(b)Tracking errors e p (t )under sinusoidal inputs with various frequencies.

amplitude of 500nm and frequencies of 1,10,and 100Hz.For each case,we only compare the compensation performance with and without DOB.In Fig.9(a),we illustrate the trajectory tracking performance with and without DOB compensation at 100-Hz input excitation,while Fig.9(b)shows the tracking er-rors e p (t ):=r (t )?y (t )under 1-,10-,and 100-Hz frequencies inputs.We clearly see that without the DOB compensation,the tip displacement displays a “phase lag”with respect to the refer-ence trajectory.With the DOB-based hysteresis compensation,the tip displacement follows the reference closely.It is also noted

Fig.10.(a)Comparison results of the tracking performance with and without DOB hysteresis compensation under a 50-Hz triangular-shape input.(b)Tracking errors e p (t )with various input

frequencies.

https://www.wendangku.net/doc/215361053.html,parison results of the step responses with and without DOB hysteresis compensation.

that with increased input frequencies,the DOB-based tracking errors increase as well.

In Fig.10,we demonstrate the tracking performance for pe-riodic triangular input trajectories at various frequencies.Such input trajectories have been used for micropositioning systems such as tip motion under the scanning mode of AFM.Similar to the previous example,Fig.10(a)shows the tracking perfor-mance under an input frequency of 50Hz,while Fig.10(b)illustrates the tracking errors e p (t )at 1,25,50,and 100Hz.These results clearly demonstrate that under only DOB com-pensation,the output displacements track the input trajectory closely.

As a last example,Fig.11shows the tracking performance of the system to a step input with and without the DOB-based hysteresis compensation.For these experiments,we introduce a PID controller C (s )to track the reference signal r (t ).For the PID controller,we used the gains K p =10?4,K i =125,and K d =10?10.These gains were selected to offer the best

performance under a step input without DOB compensation.We also include the tracking performance of the system un-der open-loop as well as under only DOB compensation.We observe that in the open-loop case,the output exhibits oscilla-tions.Furthermore,the output converges slowly with a steady state error of roughly 80nm.Under only PID control,the out-put does not exhibit oscillations;however,convergence is also slow due to the presence of hysteresis.In the case with only DOB compensation,the output converges quickly,but with considerable oscillations.Finally,with a combination of DOB and PID control,the output converges quickly and exhibits no oscillations.

V .C ONCLUSION

In this paper,we demonstrated a novel hysteresis compen-sation method for piezoelectric actuators.The compensation mechanism treats the hysteresis nonlinearity as an unknown disturbance added to a linear system.As a result,the proposed hysteresis compensation method does not rely on any mathe-matical model of hysteresis.Simplicity in implementation and robustness of performance are the main advantages of the pro-posed DOB-based hysteresis compensation approach.We have demonstrated the effectiveness and ef?ciency of the method as applied to a PMN-PT cantilever piezoelectric actuator for micromanipulation applications.The experimental results con-?rmed that the proposed DOB-based compensation scheme suc-cessfully suppressed the hysteresis of the piezoelectric actu-ators to a satisfactory level under an input frequencies up to 150Hz.

In the future,we plan to further improve the DOB design and compare its performance with other feedback/feedforward hysteresis compensation methods.We are also interested in enhancing the DOB-based design for high-bandwidth hystere-sis compensation that can be applied to micro/nanopositioning systems.

A CKNOWLEDGMENT

The authors thank Dr.S.M.Shahruz at Berkeley Engineering Research Institute,Prof.K.S.Moon,Prof.K.Morsi,and Prof. S.Kassegne at San Diego State University for their helpful discussions.They are also grateful to A.Mathers for his help in fabricating the PMN-PT/PDMS cantilever.

R EFERENCES

[1]T.Fukuda,F.Arai,and L.Dong,“Assembly of nanodevices with carbon

nanotubes through nanorobotic manipulations,”Proc.IEEE,vol.91, no.11,pp.1803–1818,Nov.2003.

[2]N.Xi and W.Li,“Recent development in nanoscale manipulation and

assembly,”IEEE Trans.Autom.Sci.Eng.,vol.3,no.3,pp.194–198,Jul.

2006.

[3]S.Devasia,E.Eleftheriou,and S.Moheimani,“A survey of control issues

in nanopositioning,”IEEE Trans.Control Syst.Technol.,vol.15,no.5, pp.802–823,Sep.2007.

[4]X.Tan and R.Iyer,“Modeling and control of hysteresis,”IEEE Control

Syst.Mag.,vol.29,no.1,pp.26–28,Feb.2009.

[5] A.J.Fleming and S.O.R.Moheimani,“A grounded-load ampli?er for

reducing hysteresis in piezoelectric tube scanners,”Rev.Sci.Instrum., vol.76,pp.073707-1–073707-5,2005.

[6]H.Janocha,D.Pesotski,and K.Kuhnen,“FPGA-based compensator

of hysteretic actuator nonlinearities for highly dynamic applications,”

IEEE/ASME Trans.Mechatronics,vol.13,no.1,pp.112–116,Feb.

2008.

[7]J.Macki,P.Nistri,and P.Zecca,“Mathematical models for hysteresis,”

SIAM Rev.,vol.35,no.1,pp.94–123,1993.

[8] A.Visintin,“Mathematical models of hysteresis,”in The Science of

Hysteresis,G.Bertotti and I.Mayergoyz,Eds.London,U.K.:Academic, 2006,pp.1–123.

[9]S.Shahruz,“Performance enhancement of a class of nonlinear systems by

disturbance observers,”IEEE/ASME Trans.Mechatronics,vol.5,no.3, pp.319–323,Sep.2000.

[10]S.Chang,J.Yi,and Y.Shen,“Disturbance observer-based hysteresis

compensation for piezoelectric micro-actuators,”in Proc.Amer.Control Conf.,St.Louis,MO,2009,pp.4196–4201.

[11] F.Goforth and Z.Gao,“An active disturbance rejection control solution

for hysteresis compensation,”in Proc.Amer.Control Conf.,Seattle,WA, 2008,pp.2202–2208.

[12]H.Adriaens,W.de Koning,and R.Banning,“Modeling piezoelectric

actuators,”IEEE/ASME Trans.Mechatronics,vol.5,no.4,pp.331–341, Dec.2000.

[13] C.-Y.Su,Y.Stepanenko,J.Svoboda,and T.Leung,“Robust adpative

control of a class of nonlinear systems with unknown backlash-like hys-teresis,”IEEE Trans.Autom.Control,vol.45,no.12,pp.2427–2432, Dec.2000.

[14]K.Ohishi,M.Nakao,K.Ohnishi,and K.Miyachi,“Microprocessor-

controlled DC motor for load-insensitive position servo system,”IEEE Trans.Ind.Electron.,vol.34,no.1,pp.44–49,Feb.1987.

[15]T.Umeno and Y.Hori,“Robust speed control of DC servomotors us-

ing modern two-degree-of-freedom controller design,”IEEE Trans.Ind.

Electron.,vol.38,no.5,pp.363–368,Oct.1991.

[16] E.Schrijver and J.van Dijk,“Disturbance observers for rigid mechanical

system:Equivalence,stability,and design,”Trans.ASME,J.Dyn.Syst.

Meas.Control,vol.124,no.1,pp.539–548,2002.

[17] A.Tesfaye,H.Lee,and M.Tomizuka,“A sensitivity optimization ap-

proach to design of a disturbance observer in digital motion control sys-tems,”IEEE/ASME Trans.Mechatronics,vol.5,no.1,pp.32–38,Mar.

2000.

[18]H.Shim and N.Jo,“An almost necessary and suf?cient condition for

robust stability of closed-loop systems with disturbance observer,”Auto-matica,vol.45,pp.296–299,2009.

[19] A.Mathers,K.S.Moon,and J.Yi,“A vibration-based PMN-PT energy

harvester,”IEEE Sens.J.,vol.9,no.7,pp.731–739,Jul.2009.

[20]J.Yi,S.Chang,K.Moon,and Y.Shi,“Dynamic modeling of an L-shape

PMN-PT piezo-based manipulator,”in Proc.Amer.Control Conf.,Seattle, WA,2008,pp.3755–3760.

[21]H.R.Pota and T.E.Alberts,“Multivariable transfer functions for a slewing

piezoelectric laminate beam,”Trans.ASME,J.Dyn.Syst.,Meas.,Control, vol.117,no.1,pp.352–359,1995.[22]S.Mahmoodi,N.Jalili,and M.Daqaq,“Modeling,nonlinear dynamics,

and identi?cation of a piezoelectrically actuated microcantilever sensor,”

IEEE/ASME Trans.Mechatronics,vol.13,no.1,pp.58–65,Feb.

2008.

Jingang Yi(S’99–M’02–SM’07)received the B.S.

degree in electrical engineering from Zhejiang Uni-

versity,Hangzhou,China,in1993,the M.Eng.de-

gree in precision instruments from Tsinghua Univer-

sity,Beijing,China,in1996,and the M.A.degree

in mathematics and the Ph.D.degree in mechani-

cal engineering from the University of California,

Berkeley,in2001and2002,respectively.

From May2002to January2005,he was with Lam

Research Corporation,Fremont,CA,as a Member

of Technical Staff.From January2005to December 2006,he was with the Department of Mechanical Engineering,Texas A&M University,as a Visiting Assistant Professor.Prior to joining Rutgers University in August2008,he was an Assistant Professor of mechanical engineering at San Diego State University beginning in January2007.He is currently an As-sistant Professor of mechanical engineering at Rutgers University,Piscataway, NJ.His current research interests include autonomous robotic systems,dynamic systems and control,intelligent sensing and actuation systems,mechatronics, and automation science and engineering,with applications to semiconductor manufacturing,intelligent transportation,and biomedical systems.

Dr.Yi is a member of the American Society of Mechanical Engineers (ASME).He has coauthored papers that have been awarded the Best Student Paper Award Finalist of the2008ASME Dynamic Systems and Control Con-ference,the Best Conference Paper Award Finalists of the2007and2008 IEEE International Conference on Automation Science and Engineering,and the Kayamori Best Paper Award of the2005IEEE International Conference on Robotics and Automation.He is currently an Associate Editor of the ASME Dynamic Systems and Control Division and the IEEE Robotics and Automation Society Conference Editorial Boards.He has also been a Guest Editor of the IEEE T RANSACTIONS ON A UTOMATION S CIENCE AND E NGINEERING

Steven Chang received the B.S.degree in mechan-

ical engineering from the University of California,

Berkeley,in2003.He is currently working toward

the M.S.degree in mechanical engineering with em-

phases in controls and mechatronics at San Diego

State University,San Diego,CA.

He was a Technical Assistant with the Institute of

Transportation Studies Pavement Research Center.

From October2004to August2007,he was a Prod-

uct Quality Engineer before enrolling at San Diego

State

University.

Yantao Shen(M’02)received the B.Eng.degree

in mechanical and electronic engineering and the

M.Eng.degree in mechatronic control and automa-

tion,both from Beijing Institute of Technology,

Beijing,China,in1994and1997,respectively,and

the Ph.D.degree in sensor-based robotic systems

from the Chinese University of Hong Kong,Hong

Kong,in2002.

From2002to2007,he was a Research Asso-

ciate in the Department of Electrical and Computer

Engineering,Michigan State University.In January 2008,he joined the University of Nevada,Reno,where he is currently an As-sistant Professor in the Department of Electrical and Biomedical Engineering. He has authored or coauthored more than60papers in professional journals and conference proceedings.His current research interests include the areas of bioinstrumentation,smart sensors and actuators for biosystems,sensorized micro/nanosystems,visual servo systems,and haptic/tactile interfaces.

Dr.Shen is a member of Sigma Xi.He is currently an Associate Editor for several conferences of the IEEE Robotics and Automation Society.He was also a?nalist for the Best Vision Paper Award at the IEEE International Conference on Robotics and Automation in2001.