Modeling and H

∞

Composite Control of the Coupled Hysteretic

Dynamics in Piezoelectric Micro-displacement Systems

Liang Tang

, Lei Liu

and Xin Guan

Beijing Institute of Control Engineering, National Laboratory of Space Intelligent Control, 100080, Beijing, China

State Key Laboratory of Structural Analysis for Industrial Equipment, School of Aeronautics and Astronautics,

Dalian University of Technology, Dalian 116024, China

Keywords:

Hysteretic Dynamics, Piezoelectric Actuator, Broadband Control, Jitter Control.

Abstract:

This paper investigates the modeling and H

∞

composite control of the coupled hysteretic dynamics in a piezo-

electric micro-displacement system (PMS). First, the coupled multi-ﬁeld hysteretic dynamics with physical

meanings is presented for PMS. Next, the composite control analysis of the hysteretic dynamics is proposed.

Then, a H

∞

synthesis controller is designed by using the simpliﬁed hysteretic dynamics. To enhance the H

∞

performance, the inversion-based feedforward compensation is augmented. The proposed H

∞

feedback con-

trol and the inversion-based feedforward can be designed separately. Finally, the experimental studies are

provided to demonstrate the proposed H

∞

composite control approach.

1 INTRODUCTION

Piezoelectric micro-displacement systems (PMSs) are

widely investigated to suppress jitters and micro vi-

brations produced by reaction ﬂywheels, control mo-

mentum gyroscopes, Stirling coolers and step motors

of precision spacecrafts, such as inter-satellite laser

communication, space telescope and missile warn-

ing satellite with staring camera (Kamesha and Ghos-

alb, 2010; Maillarda and LeLettya, 2009; Nagashima

and Agrawal, 2014; Dewella and Blaurockb, 2005;

Maillarda and LeLettya, 2009; Laneand and Lacy,

2008; Neat and Goullioud, 1998; McMickell and

Hansen, 2007). For instance, piezoelectric fast steer-

ing mirrors are employed to suppress jitters of line-of-

sight(LOS) in inter-satellite laser communication and

space telescopes. To reject jitters and micro vibra-

tions, broadband control of PMS is increasingly ap-

pealing, but most of the operating bandwidth of PMS

is still insufﬁcient.

To enhance the bandwidth and performance of

PMS, various controllers were designed (Devasia and

Moheimani, 2007). If tracking signals are at low fre-

quencies, proportional-intergral-derivative (PID) and

notch ﬁlter are adequate (Fleming, 2010). As the ref-

erence signal frequency increases, model based con-

troller are alternativelydesigned, such as H

∞

feedback

control (Wu and Zou, 2009), inversion-based feed-

forward control (Liu and Lee, 2013b; Tan and Ang,

2009; Liu and Lee, 2013a). Accurate modeling over

a broad frequency range is necessary to enhance the

performance of model-based control.

At broadband frequencies, the hysteretic dynam-

ics of PMS has multi-ﬁeld effects. It is required to

model the coupled hysteresis, creep, electric and vi-

bration dynamics. Hysteresis is a strongly nonlinear

element with global memory (Brokate and Sprekels,

1996). Preisach model is typically to describe the

static hysteresis (Mayergozy, 2003). Creep is slow

dynamics and can be represented by spring-damping

model (Devasia and Moheimani, 2007).

In this paper, the multi-ﬁeld dynamics with phys-

ical meanings is developed for PMS. The non-

hysteretic creep model is used. Electrical and vi-

bration dynamics of PMS are fast dynamics and can

be represented using transfer functions. In PMS, the

time constant of electric dynamics is in order of 0.002

seconds, and the ﬁrst resonance frequency of the vi-

bration dynamics is generally in the order of 1kHz.

To represent the PMS at broadband frequencies, this

paper employs a cascade connection of static and

dynamic components. The static hysteresis is rep-

resented using classical Preisach model. The non-

hysteretic creep, electric and vibration dynamics are

represented using transfer functions.

To compensate PMS dynamics at broadband fre-

quencies, various modern controllers were investi-

gated. Clayton reviewed feedforward approaches

441

Tang L., Liu L. and Guan X..

Modeling and H∞ Composite Control of the Coupled Hysteretic Dynamics in Piezoelectric Micro-displacement Systems.

DOI: 10.5220/0005015204410449

In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 441-449

ISBN: 978-989-758-039-0

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

which were mainly based on linear dynamical mod-

els(Clayton and Devasia, 2009). Wu presented a 2-

DOF feedforward-feedback controller (Wu and Zou,

2009). Leaning also proposed a notch ﬁlter and a

inversion-based feedforward controller to enhance the

high-gain feedback(Leaning and Devasia, 2007). In-

telligent feedback controllers were also investigated.

Liaw used neural network to enhance the motion

tracking of piezo-based ﬂexible mechanisms(Liaw

and Shirinzadeh, 2009). Shieh and Hsu investigates

the adaptive control (Shieh and Hsu, 2008). Ad-

ditionally, dynamic hysteresis models were investi-

gated to achieve high bandwidth tracking(Jiang and

Chen, 2010; Janaiden and Rakheja, 2008). Based on

rate-dependent Prandtl-Ishlinskii (P-I) hysteresis, Tan

proposed the hysteresis-based inversion to extend the

tracking bandwidth(Tan and Ang, 2009), but it is dif-

ﬁcult to design modern control techniques using rate-

dependent hysteresis. Alternatively, most modern

controllers are designed using non-hysteretic models.

In this paper, the H

∞

composite control is designed

using the proposed hysteretic dynamics of PMS. The

proposed composite controller comprises of an sep-

arate H

∞

feedback controller and an inversion-based

feedforwardcontroller. More accurate tracking is thus

presented at high frequencies.

This paper is organized as follows. First, Section

2 presents the modeling of the coupled hysteretic dy-

namics with physical meanings. Next, Section 3 pro-

vides the analysis of the composite control strategy of

the hysteretic dynamics. Then, the H

∞

composite con-

trol strategy is developed in Section 4. To validate the

proposed modeling and control approaches, the ex-

perimental studies are provided in Section 5. Finally,

Section 6 makes a conclusion of this paper.

2 COUPLED HYSTERETIC

DYNAMICS IN PMS

In this section, the multi-ﬁeld modeling of the hys-

teretic dynamics in PMS is presented. The hysteretic

dynamics model is derived from the material, elec-

trical and mechanical ﬁelds. The complete model of

the hysteretic dynamics consists of the static Preisach

hysteresis effect, creep effect, electrical and vibration

dynamics.

Fig. 1 shows the complete hysteretic model struc-

ture of PMS. The hysteretic model is derived as fol-

lows. First, the electrical model of the voltage ampli-

ﬁer is presented. Next, the hysteresis effect due to the

lead zirconate titanate (PZT) stack is proposed using

the classical Preisach model. Additionally, the creep

effect is presented using a transfer function. Then, the

electrical model of PZT stack is proposed. Moreover,

the mechanical vibration dynamics is derived using

stiff and damping parameters. Finally, the character-

istics of the hysteretic dynamics are proposed.

RLC effect

(Amplifier)

Hysteresis

(PZT)

RC effect

˄PZT˅

Mechanical

vibration

Creep effect

(PZT)

Figure 1: Model structure of PMS(u

denotes the input volt-

age of the voltage ampliﬁer, u denotes the output voltage

of the voltage ampliﬁer, u

denotes the voltage of the PZT

stack, F denotes the actuating force due to the inverse piezo-

electric effect, x denotes the displacement of PMS, and y

denotes the drifted displacement of PMS).

2.1 Electrical Model of the Voltage

Ampliﬁer

The power and bandwidth of voltage ampliﬁers are

limited. As the input frequency increases, the current

reduces and the phase delay increases. To describe

this dynamic response, the deduced electrical dynam-

ics of the voltage ampliﬁer is presented.

Fig. 2 shows the sketch of deduced RLC electrical

dynamics of the voltage ampliﬁer where the amplify-

ing factor is not presented. R

, L

and C

represent

the resistance, inductance and capacitance of the volt-

age ampliﬁer, respectively. u

and u represents the

input and output voltage, respectively. Then,

U(s)

(s)

1/(L

)

+ s/(R

) + 1/(L

)

, (1)

Let ω

= 1/

√

LC and ξ

= 1/(2R

)

√

, equa-

tion (1) can be rewritten as

U(s)

(s)

+ 2ξ

s+ ω

. (2)

2.2 Preisach Hysteresis Model

The hysteresis effect of PZT material (stack) are

described using Preisach model. Fig. 3 shows the

hysteresis effect and RC electrical dynamics in the

PZT stack. Γ represents the hysteresis effect. R and

C represent the resistance and capacitance of the PZT

stack, respectively. T

represents the electromechan-

ical transformer ratio of the PZT material. i is the

conductor current, u is the input voltage of the PZT

stack, u

is the effective voltage for the PZT stack.

The hysteresis between the input voltage and

the effective PZT voltage can be represented as

the following Preisach model (Mayergozy, 2003)

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Figure 2: Electrical dynamics of the voltage ampliﬁer.

Figure 3: Hysteresis effect and electrical dynamics in the

PZT stack.

= Γ(u) =

µ(α,β)γ

αβ

[u(t)]dαdβ, (3)

where µ(α,β) and γ

αβ

are the density function and

hysteron output of point α,β on the Preisach plane,

respectively. The Preisach model is rate-independent,

i.e. it is a static model.

Fig. 4 shows the Preisach plane. The shadowing

area S

is activated with the γ

αβ

of one. The blank

area S

−

is unactivated with the γ

αβ

of zero.

2.3 RC Electrical Model of the PZT

Stack

In the electrical ﬁeld of the PZT stack, the voltage

drop u

is represented by

= iR + u

, (4)

where R is the resistance and i is the current. u

the voltage of the equivalent capacitor C of the PZT

stack. u

can be represented by

= QC, (5)

where Q is the charge.

−

( )

Figure 4: Preisach plane: S

represents the actuated area

with the γ

αβ

of one, S

−

represents the unactuated area with

the γ

αβ

of zero, and µ(α,β) represents the density function

at the point (α,β).

Additionally, the conduction current i is repre-

sented by

i =

, (6)

By combining equations (4), (5) and (6), the elec-

trical dynamics is written as

Q(s)

(s)

(1+ τs)

, (7)

where s is the Laplace operator τ = RC.

In summary, the electrical dynamics in PMS con-

sists of the electrical dynamics of the voltage ampli-

ﬁer and the electrical dynamics of the PZT material.

By combining equations (2) and (7), the electrical

model of PMS is given by

(s) =

1+ τs

+ 2ξ

s+ ω

. (8)

The force F due to the inverse piezoelectricity ef-

fect of the PZT stack is written as

F = T

Q, (9)

where T

the electromechanical transformer ratio

due to the inverse piezoelectric effect.

2.4 Mechanical Vibration Dynamics

A typical mechanical strut with motion ampliﬁcation

is considered in this paper, as shown in Fig. 5. The

proposed PMS can be used to compensate jitters and

micro vibrations of spacecrafts. The stiff and damp-

ing of the PZT stack as well as the passive isolator and

ﬂexible joints are contained in the mechanical vibra-

tion dynamics. Rubber cushion between the working

platform and the base can be used to introduce damp-

ing for passive isolation.

The mechanical motions of PMS are driven by the

ModelingandH∞CompositeControloftheCoupledHystereticDynamicsinPiezoelectricMicro-displacementSystems

443

force F due to the inverse piezoelectricity effect. Fig.

5 shows the mechanical vibration dynamics. K

and

represent the stiff and damping of the PZT stack,

respectively. K

and C

represent the stiff and damp-

ing of the passive isolation and the ﬂexure guide. x

represents the displacement of the arm tip which is

generally attached to a working platform. θ repre-

sents the tilt angle due to the piezo displacement. L

represents the length of the motion amplifying arm.

N represents the amplifying value of the motion am-

plifying arm.

L N

Figure 5: Mechanical vibration dynamics of the PMS with

motion ampliﬁcation.

For PMS, only micro displacement is provided.

For instance, the maximum displacement of a typ-

ical PMS is 50µm. Compared with its arm length

L = 10cm, tanθ = x



L < 0.0005.Thus, θ

= tanθ.

x = Lθ. According to Newton’s law, the dynamics

of M can be written as

θ+ K

xL+C

˙xL+ K

˙x

= F

, (10)

where J = ML

and x

= x/N.

Equation (10) can be written as

M ¨x+





˙x+





x =

. (11)

Then,

(s) = k

+ 2ξ

s+ ω

, (12)

where G

(s) = X (s)/Q(s), ω



.

2ξ



.

M, K

= T



+ K



Finally, there exists the creep effect (also named

drift) in PZT material. The creep effect can be repre-

sented by (Devasia and Moheimani, 2007)

(s) = k

∏

i=1

s+ z

s+ p

, (13)

where k

is the creep gain when s goes to inﬁnity, i.e.

represents the creep gain at inﬁnite frequencies. m

is the creep order. p

and z

are the poles and zeros

of the creep dynamics, respectively.

2.5 Multi-ﬁeld Hysteretic Dynamics

The multi-ﬁeld hysteretic dynamics of PMS can be di-

vided into the static hysteresis and the non-hysteretic

dynamics. In this paper, the static hysteresis is

represented by classical Preisach model. The non-

hysteretic dynamics comprises of the creep, electrical

and vibration dynamics.

By combining equations (8) and (12), the electric

and mechanical dynamics can be presented as

(s) =

1+ τs

+ 2ξ

s+ ω

+ 2ξ

s+ ω

(14)

where k

= k

Fig. 6 shows the model sketch of the multi-ﬁeld

hysteretic dynamics in PMS. The cascade connection

is used to represent the relationship among the com-

ponents of the hysteretic dynamics. The hysteresis ef-

fect and creep effect are built in the material ﬁeld. The

electrical dynamics is built in the electrical ﬁeld. The

vibration dynamics is built in the mechanical ﬁeld.

The non-hysteretic dynamics G can be summarized

as G = G

Electrical field

Material field

Mechanical field

Figure 6: Multi-ﬁeld dynamics of PMS.

2.6 Characteristics of the Coupled

Hysteretic Dynamics

In this section, the responses of typical PMS are pro-

posed. Fig. 7 shows the response of the static Preisach

hysteresis under sinusoidal inputs. Compared with

phase delay in linear dynamics, the Preisach hystere-

sis achieve its peak value simultaneously with the in-

put signal, i.e., there is not delay at the peak point.

Moreover, the Preisach output is not differential at the

peak point.

Fig. 8 shows the response of the creep, electri-

cal and vibration dynamics under square inputs. It

can be seen that the electrical and vibration dynamics

behaves fast, but the creep dynamics behaves slow.

Moreover, the creep, electrical and vibration dynam-

ics are coupled.

Fig. 9 shows the response of the hysteresis, creep,

electrical and vibration dynamics under slow sinu-

soidal inputs with varying amplitudes. As the input

continues, the drift due to the creep and low frequency

electrical and vibration dynamics are obvious.

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98 98.2 98.4 98.6 98.8 99 99.2 99.4 99.6 99.8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time (s)

Normalized voltage/displacement

Input voltage

Piezo displacement

No delay at

peak point

Figure 7: Response of the Preisach hysteresis under sinu-

soidal inputs.

0 1 2 3 4 5 6

100

120

140

160

180

Time (s)

Displacement (nm)

Electrical and vibration

Creep

Figure 8: Response of the creep, electrical and vibration

dynamics.

0 1 2 3 4 5 6 7 8 9 10

Time (s)

Piezo displacement with creep suppression (µm)

Creep displacement

Creep suppressed displacement

Figure 9: Response of creep effect under the sinusoidal in-

put with varying amplitudes.

3 COMPOSITE CONTROL

ANALYSIS

The proposed composite controller is analyzed in this

section. First, using the reference signal r, the model-

based inversion feedforward controller K

of PMS

can be written as

−1

(r), (15)

where the hysteresis estimation

Γ is strong nonlinear-

ities with global memories(Mayergozy,2003).

Γ(r) is

computed using Preisach model in equation (3).

−1

and

−1

can be represented as



−1

= G

−1

(1+ δ

)

−1

= Γ

−1

(1+ δ

)

where δ

denotes the inversion error of the non-

hysteretic dynamics and δ

denotes the inversion er-

ror of the rate-independent hysteresis. δ

and δ

are

bounded uncertainties and determined by the identiﬁ-

cation accuracy of PMS. Then,

−1

(r) and

−1

can

be rewritten as

−1

(r) = (1+ δ

+ δ

−1

(r). (16)

Let δ = δ

+ δ

, the model-based inversion

feedforward controller of PMS is rewritten as

= (1 + δ)G

−1

(r). (17)

With only the feedforward controller K

in (17),

the relative error in e/r is given by

= δ +

. (18)

Equation (18) indicates that the tracking perfor-

mance of feedforward relies on the identiﬁcation ac-

curacy and the output disturbances are not suppressed.

Thus, feedback control is necessary to guarantee the

stability and robustness under modeling error δ and

disturbance d.

Fig. 10 shows the proposed composite control

strategy where

−1

and

−1

are represented by G

−1

and Γ

−1

according to (17), respectively. With the pro-

posed composite control, the relationship between the

reference r and PMS displacement output y is written

= 1 +

GΓ(u)K

+ 1

δ+

GΓ(u)K

+ 1

GΓ(u)K

+ 1

, (19)

where K

denotes the feedback controller, n and d

are the measurement noise and output disturbance, re-

spectively. Γ(u) is computed using Preisach model in

equation (3).

ModelingandH∞CompositeControloftheCoupledHystereticDynamicsinPiezoelectricMicro-displacementSystems

445

−

Figure 10: Analysis of proposed composite control strategy.

Then, the relationship between the reference r and

tracking error e is represented as

GΓ(r)K

+ 1

δ+

GΓ(u)K

+ 1

GΓ(u)K

+ 1

. (20)

The feedback controller K

is designed to sup-

press the output disturbance d and the feedforward

error δ, but the measurement noise n is ampliﬁed in

the feedback bandwidth. If the feedback bandwidth is

too large, some signals in noise n may coincide with

mode frequencies of PMS, which will result in chat-

tering and unstable responses. Thus, multi-objective

robust H

∞

control is necessary to design K

. The dif-

ferent objectives of the feedback controller are speci-

ﬁed at different frequencies.

4 H

∞

COMPOSITE CONTROL

The proposed H

∞

composite control consists of a H

∞

feedback controller and an inversion-based feedfor-

ward controller.

4.1 H

∞

Controller Design

The loop shaping is employed to design the feedback

controller as shown in Fig. 11. The performance and

stability requirements are satisﬁed by specifying L

and L

. ω

is the cross frequency of GK

and is

related and close to feedback bandwidth, ω

is re-

lated to disturbance rejection performance, and ω

related to the robust stability under modeling errors,

disturbances and measurement noise at high frequen-

cies.

Weighting functions are suitable for specifying

different requirements at different frequencies as

shown in Fig. 11. It is convenient to achieve multi

objectives using weighting functions(Skogestad and

Postlethwaite, 2005). The robust H

∞

controller is de-

signed based on the non-hysteresis dynamics, while

Robust stability range

Performance

range

(

)

Magnitude (dB)

Figure 11: Illustration of loop shaping.

the rate-independent hysteresis Γ can be regarded as

an input uncertainty consisting of the nominal gain k

and the weighting function w

Fig. 12 shows the sketch of multi-objective robust

∞

control. w

is the performance weighting function

to specify performance requirements and achieve ﬁne

tracking. Signiﬁcant vibrations are easily induced by

high gain at high frequencies. Then, an integral ac-

tion is added to w

to reduce the feedback bandwidth

and enhance the disturbance suppressing at low fre-

quencies. w

and w

denote the noise and reference

weighting functions, respectively, w

is the control

weighting function to limit the control gain and sup-

press noise at high frequencies, w

denotes the uncer-

tainty due to the hysteresis nonlinearity, ∆

is an unit

complex uncertainty with norm k∆

k < 1.

∆

−

Input uncertainty

Figure 12: Illustration of multi-objective H

∞

control.

Weighting functions w

and w

are used to satisfy

the trajectory of GK

that is bounded by L

and L

The relationships are as follow



ω≤ω

= L

ω≥ω

= 1/L

. (21)

4.2 Inversion-based Feedforward

Compensation

The feedforward controller is used to overcome the

bandwidth limitation of the feedback controller. In

this section, an inversion-based feedforward con-

troller is used to enhance the H

∞

feedback perfor-

mance.

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The inversion-based feedforward controller en-

compasses the inverse non-hysteretic dynamics and

the inverse hysteresis. First, the reference signals pass

through the inverse non-hysteresis dynamics

−1

then the inverse hysteresis

−1

. The details of the

model-based inversion

−1

can be found in Refs.(Liu

and Lee, 2013b). The Preisach-based inversion

−1

is shown in (Liu and Lee, 2012). The inversion of the

non-hysteretic dynamics can be represented as

−1

(s) =

τs+ 1

∏

i=1

s+ ˆp

s+ ˆz

∏

+ 2

)

(22)

where

τ,

and

are the identiﬁed parameters

of the electric and vibration dynamics, respectively, ˆz

and ˆp

are the estimated zeros and poles of the creep

dynamics, respectively.

5 EXPERIMENTAL STUDIES

5.1 Experimental Setup

The experimental setup consists of a piezoelectric ac-

tuator with motion ampliﬁcation, an voltage ampliﬁer,

a linear variable differential transformer(LVDT) and

a DSPACE 1104 board. Fig. 13 shows the piezoelec-

tric actuator. The actuator has a travel span of 80µm.

The ampliﬁer is E-662 with the output voltage range

of [−20,120]V. The LVDT sensor has white noise,

and the RMS value of sensor noise is 0.01µm. MAT-

LAB/Simulink and a dSPACE DS1104 board are used

to implement the model-based controller.

Figure 13: Piezoelectric actuator (PA) in the experiment.

The electric and vibration dynamics are identiﬁed

(s) =

0.000474s+1

8.111×10

+ 3786s+8.111×10

2.478×10

+ 809.1s+2.478×10

(23)

The creep dynamics is identiﬁed as

(s) =

(s+ 0.0146)(s+ 0.172)(s+ 0.241)

(s+ 0.0142)(s+ 0.169)(s+ 0.2402)

(s+ 1.07)(s+ 18.29)

(s+ 1.053)(s+ 17.57)

. (24)

Fig. 14 shows the identiﬁed density function µ(α,β)

in equation (3).

Figure 14: Identiﬁed density function µ(α,β).

5.2 Controller Parameters

The performance weighting function w

and the con-

trol weighting function w

are set to

350π

s+ 0.0001

. (25)

= 0.1

s+ 1000π

s+ 10000π

. (26)

The reference signal and measurement noise are

represented using the weighting functions w

, w

and

, respectively

= 0.1,w

= 0.0001.

To reduce the conservation, the discrete D-K it-

eration with structured singular value (SSV) is used

to solve the controller(Skogestad and Postlethwaite,

2005). After 6 iterations, the SSV is less than 0.98,

and the order of the H

∞

controller is 9. To easily im-

plement the controller in DSP, the H

∞

controller with

order of 4 is given by

= 1903075

(s+ 31360)(s+ 566.9)(s+ 0.313)

(s+ 19830)(s+ 2951)(s+ 923.7)

(27)

5.3 Experimental Result

In this paper, square and sinusoidal references are

used to demonstrate the effectiveness of the proposed

ModelingandH∞CompositeControloftheCoupledHystereticDynamicsinPiezoelectricMicro-displacementSystems

447

composite control. Further, the root-mean-square

(RMS) error e

rms

is used to measure the tracking er-

rors.

Fig. 15 shows the tracking performance of the

square reference at 20Hz. The RMS tracking error

is 0.19µm (To grantee the differential of the reference

signal, a pre-ﬁler is used for inversion-based feedfor-

ward). Fig. 16 shows the control voltage. Further,

Fig. 17 shows the tracking performance of the sinu-

soidal trajectory at 600Hz. The RMS tracking error

is 0.78µm. Fig. 18 shows the control voltage of the

sinusoidal tracking at 600Hz.

The experimental results demonstrate that the pro-

posed composite control provides precision tracking

performance at broadband frequencies.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0.5

1.5

2.5

Piezo displacement (µm)

Measured displacement

Reference displment

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

−2

−1

Time (s)

Tracking error (µm)

Figure 15: Tracking performance of square inputs at 20Hz.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Time (s)

Control voltage (V)

Figure 16: Control voltage at 20Hz.

6 CONCLUSIONS

It is increasingly demanded to present broadband ac-

curate tracking of PMS. The modeling and H

∞

com-

posite control of the coupled hysteretic dynamics is

thus provided in this paper. The Preisach hystere-

sis, creep, electrical and vibration dynamics are devel-

oped to describe the complex behaviors of PMS. The

proposed hysteretic dynamics has physical meanings

which is useful for deep developments of PMS. The

1.8 1.802 1.804 1.806 1.808 1.81 1.812 1.814 1.816 1.818 1.82

Piezo displacement (µm)

1.8 1.802 1.804 1.806 1.808 1.81 1.812 1.814 1.816 1.818 1.82

−2

−1

Time (s)

Tracking error (µm)

Measured displacement

Reference dispacelment

Figure 17: Tracking performance at 600Hz

1.8 1.802 1.804 1.806 1.808 1.81 1.812 1.814 1.816 1.818 1.82

−20

Time (s)

Control voltage (V)

Figure 18: Control voltage at 600Hz.

proposed H

∞

composite control provides high-speed

and precision tracking. The experimental studies

demonstrate the effectiveness of the proposed mod-

eling and control approaches.

The proposed modeling and control approaches

of PMS are beneﬁcial to the suppression of jitters

and micro-vibrations in precision spacecrafts, such as

inter-satellite laser communication, staring cameras,

space-based interferometers and space telescopes.

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