# Active vibration control of aerospace structures using a modified Positive Position Feedback method

**ABSTRACT** A positive position feedback controller is modified and a new active vibration control technique is developed. Unlike the conventional positive position feedback, the new controller separates the damping and stiffness control using two parallel first order and second order compensators. The second order compensator has a damping ratio as low as the damping of flexible structure to provide periodic vibration control. Simultaneously, the high damping is made available through a first order compensator. The new controller is applicable to a strain-based sensing/actuating approach and can be extensively applied to piezoelectrically controlled systems. Control gains are obtained by performing the stability analysis. The controller is verified experimentally using a plate vibration suppression setup. The plate is controlled through two piezoelectric patches and its vibrations are monitored by ten sensors mounted on the surface of the plate. The results confirm that the new controller is able to provide good vibration reduction, with the ability to be used to simultaneously control more than one natural frequency.

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- Journal of Vibration and Acoustics. 01/2006; 128(2).
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**ABSTRACT:**This paper is concerned with vibration control of a flexible spacecraft in the presence of parametric uncertainty/external disturbances as well as control input nonlinearity through distributed piezoelectric sensor/actuator technology. To satisfy pointing requirements and simultaneously suppress vibrations, two separate control loops are adopted. The first uses piezoceramics as sensors and actuators to actively suppress certain flexible modes by designing suboptimal positive position feedback (SOPPF) compensators which add damping to the flexible structures in certain critical modes. The problem of determining the SOPPF gain is formulated as static output feedback problem. The second feedback loop is designed based on an output feedback sliding mode control (OFSMC) design where control input nonlinearity is taken into consideration. The controller has the ability to reject the disturbance, deal with uncertainty and to ensure that the system trajectories globally converge to the sliding mode. Furthermore, an adaptive version of the proposed controller is achieved through releasing the limitation of knowing the bounds of the uncertainties and perturbations in advance. Simulation studies for the proposed control strategy on a flexible spacecraft have been carried out which demonstrate the effectiveness of the proposed approach.Journal of Vibration and Control 01/2007; 13(11):1573-1602. · 4.36 Impact Factor -

Page 1

Abstract—A

modified and a new active vibration control technique is

developed. Unlike the conventional Positive Position Feedback,

the new controller separates the damping and stiffness control

using two parallel first order and second order compensators.

The second order compensator has a damping ratio as low as

the damping of flexible structure to provide periodic vibration

control. Simultaneously, the high damping is made available

through a first order compensator. The new controller is

applicable to a strain-based sensing/actuating approach and

can be extensively applied to piezoelectrically controlled

systems. Control gains are obtained by performing the

stability analysis. The controller is verified experimentally

using a plate vibration suppression setup. The plate is

controlled through two piezoelectric patches and its vibrations

are monitored by ten sensors mounted on the surface of the

plate. The results confirm that the new controller is able to

provide good vibration reduction, with the ability to be used to

simultaneously control more than one natural frequency.

PositivePositionFeedbackcontrolleris

I. INTRODUCTION

CTIVE vibration control of flexible structures by

means of smart materials, especially piezoelectric

patches, is of interest to many researchers. The

application of piezoelectric actuators and shape memory

alloys in vibration control is increasing in many areas, from

microscale actuators in atomic force microscopes to active

vibration control of aircraft bodies [1-2]. Vibrations in

aerospace structures appear due to various issues and there

are different methods to control the vibration. Using

piezoelectric patches to control engine-induced noise and

vibration in the passenger cabin of aircraft is an issue that

has been studied [3]. Active vibration control has also been

used for space structures to control the Solar Array Flight

Experiment (SAFE) structure during deployment [4].

Although the vibration control methods used in those

research projects worked well, an improved method is

required to control the vibrations more efficiently with less

control effort.

There are several classical vibration control methods for

structures such as lead, velocity feedback and acceleration

feedbackthatdependon

displacement, velocity and acceleration, for the feedback

themeasurementofthe

S.N. Mahmoodi is a visiting assistant professor at Center for Vehicle

Systems & Safety at Virginia Tech. and author of correspondence (email:

mahmoodi@vt.edu).

M. Rastgaar Aagaah is a postdoctoral research fellow at Center for Vehicle

Systems & Safety at Virginia Tech. (email: rastgaar@vt.edu).

M. Ahmadian is a professor and director of Center for Vehicle Systems &

Safety at Virginia Tech. (email: ahmadian@vt.edu).

[5]. Positive Position Feedback (PPF) control works based

on velocity feedback control [6]. PPF is a commonly used

control method that is utilized in many applications such as

flexibleantenna[7], flexible manipulators [8], and

spacecrafts [9]. However, in PPF, the control input is

applied through a piezoelectric actuator that generates

strain instead of direct force. PPF uses a second order

compensator in feedback with a large damping value to

suppress the vibrations and uses a gain value that is smaller

than one to keep the system stable [10]. Sometimes a first

order compensator is used in PPF to control the vibration

[11], however, this compensator cannot efficiently control

the induced periodic vibrations in the system.

This paper provides and experimentally verifies a new

technique that is called Modified Positive Position Feedback

(MPPF), which is a developed version of PPF. It will be

shownthat similartoPPF, the

unconditionally stable, i.e., the stability is not dependent on

the damping of the structure. In PPF, the two parameters of

the control gain and damping ratio of the compensator must

be chosen carefully in order to provide desirable vibration

suppression. Sometimes, system identification is required

for choosing the proper control gain and damping ratio for

the compensator [12]. PPF can be combined with other

methods for more effective control such as an adaptive PPF

used for improved control of flexible structures [13]. The

method proposed here shows similar abilities. In addition,

adaptive control was combined with PPF for controlling

vibrations of multi-modes of frequency varying structures

[14]. The PPF controller was also considered as an output

feedback controller to design an improved control system.

The optimal control method was used to develop a control

design algorithm to be used in design of PPF controllers

[15]. In addition, PPF has been used jointly with delayed

feedback for designing more robust controllers [16]. Since

both the control gain and damping ratio in PPF are gathered

in one compensator in the feedback, they influence each

other’s performance. Considering this effect, selecting the

damping ratio and control gain is important for how well

PPF is able to suppress structural vibrations. The proposed

method will use two parallel compensators to ease the

selection of control gains and provide a better control

design.

The MPPF suggested here uses two parallel compensators to

eliminate both effects of transient and periodic dynamics of

the disturbance. One compensator is a second order filter

with very small damping to suppress periodic disturbance.

newmethodis

Active Vibration Control of Aerospace Structures Using a Modified

Positive Position Feedback Method

S. Nima Mahmoodi, Mohammad Rastgaar Aagaah and Mehdi Ahmadian

A

2009 American Control Conference

Hyatt Regency Riverfront, St. Louis, MO, USA

June 10-12, 2009

FrA05.6

978-1-4244-4524-0/09/$25.00 ©2009 AACC4115

Page 2

The other compensator is a first order filter to dissipate

transient disturbance. The MPPF controller provides the

ability to select proper gains that reduce settling time and

controlling effort more efficiently. A plate clamped at four

edges will be disturbed by an electromagnetic shaker and

two piezoelectric actuators will control the plate vibrations.

Two modes will be investigated and controlled using MPPF

method.

II. MODIFIEDPOSITIVEPOSITIONFEEDBACKCONTROL

One

piezoelectric actuators/sensors as a controlling device is the

Positive Position Feedback method. The PPF uses a second

order compensator in the feedback of sensor to suppress the

vibrations, especially at the resonance frequency. There are

three major parameters in this compensator that need to be

precisely determined in order to make it efficiently work.

The first parameter is the compensator frequency that is

usually the same as the resonance frequency if the

disturbance excites the resonance frequency. The other two

parameters are damping and gain of the compensator.

Considering that the compensator frequency is defined then

the other two parameters should be carefully selected. Since

the objective of the method is suppression of vibration, the

damping of the compensator should be large. However, this

causes the deduction of the compensator damping frequency

in comparison with the system that has small damping and

thus the compensator cannot perfectly compensate the

vibration of the flexible structure. By decreasing the

damping, the settling time will increase. In order to

eliminate this disadvantage and improve the efficiency of

the controller, a new modified positive position feedback

method is proposed that uses two compensators with

separate gains. One is a second order compensator with a

low damping and the other is a first order compensator

parallel with the first one to increase the damping of the

closed-loop system.

Here, the general concept of the modified positive position

feedback control is introduced. The controller is a resonant

controller that uses a collocated control system. It will be

shown that it has unconditional stability of the closed-loop

system. Such controllers are of interest because of their

ability to avoid instability due to spill-over effect [17]. The

MPPF controller consists of a second order and a first order

controller that work in parallel. The system includes three

following equations, the first one describes the structure and

the other two describe compensators

(

++=

x Dx Ω xC

Αy

???

Ω y

???

Ω z

Ω Cx

?

ofthesuccessfulcontrolmethodsforusing

)

2

T

+

Βz

(1a)

2

f

=

2

ff

++=

yD y

Ω Cx

(1b)

ff

+

z

(1c)

where x indicates the Nm×1 modal vector, y and z are the

Nf×1 compensator vectors to suppress Nf modes of the

structure, Ω is a Nm× Nm diagonal matrix of modal

frequency, D is a Nm× Nm diagonal matrix of modal

damping, Ωf is a Nf×Nf diagonal matrix of compensator

frequency, Df is a Nf× Nfdiagonal matrix of compensator

damping whose elements are as small as damping elements

of the system, A and B are Nf× Nfdiagonal gain matrices

with positive elements, and C is Nf×Nmparticipation matrix

[14] and is dependent on the modal system.

Theorem 1. The closed-loop system of the structure and

compensators of equations (1) are asymptotically stable if

and only if

()

−+

Ω

CAB C

Proof. The following nonsingular transformations are made

to provide symmetric equations out of equation (1),

1 2

−

=

y

Α

Β Ω ζ (3)

2

0

T

>

.

f

Ω ψ (2)

1 21 2

f

−

=

z

Substituting equations (2) and (3) into (1), the structure and

compensators equations can be obtained as:

(

++=

x Dx Ω xC

Α Ω ψ

ψ

D ψΩ ψ

???

ζΩ ζΩ Β Cx

)

2 1 21 21 2

f

T

f

+

Β Ω ζ

???

(4a)

2

f

1 2

ff

++=

Ω Α Cx

(4b)

1 2

f

1 2

f

+=

?

(4c)

The system of equations (4a-4c) is rewritten in the matrix

form as:

⎡ ⎤⎡

⎢ ⎥⎢

+

⎢ ⎥⎢

⎢ ⎥⎢

⎣ ⎦⎣

⎡

−

⎢

−

⎢

⎢

−

⎣

Defining:

⎡ ⎤⎡

⎢ ⎥⎢

==

⎢ ⎥⎢

⎢ ⎥⎢

⎣ ⎦⎣

⎡

−

⎢

= −

⎢

⎢

−

⎣

The system of equation (5) is asymptotically stable if and

only if both L and P are positive definite [6]. L is positive

definite since D, Dfand Ωfare positive definite. In order to

evaluate P, it should be proven that for any arbitrary and

nonzero vector

12

[

kk

=

k

is valid.

T

>

k Pk

By expansion of equation (7) and adding and subtracting

T

kk

C ΑC

and

11

kk

C ΒC , the

obtained,

() (

12

f

kk

−Ω

Α C

Α C

2 1 21 21 2

f

1 22

f

1 2

f

1 2

00

00

000

00

0

f

TT

f

f

f

⎡ ⎤

⎤⎢ ⎥

⎥

⎢ ⎥

⎥⎢ ⎥

⎥

⎦⎣ ⎦

+

⎤

⎥⎢ ⎥

⎥⎢ ⎥

⎥⎢ ⎥

⎣ ⎦

⎦

−

⎡ ⎤

x

ψ

ζ

=

x

ψ

??

Dx

ψ

ζ

D

I

Ω

C Α Ω

Ω

C Β Ω

Ω Α C

Ω Β C

Ω

???

?

?

(5)

00

;00

00

f

and

⎤

⎥

⎥

⎥

⎦

x

ψ

ζ

D

ρ

LD

I

…….

2 1 21 21 2

f

1 22

f

1 2

f

1 2

0.

0

TT

f

f

f

⎤

⎥

⎥

⎥

⎦

−

Ω

C Α Ω

Ω

C Β Ω

P

Ω Α C

Ω Β C

Ω

(6)

3

]

TT T T

k

the following condition

0

(7)

11

T

followingconditionis

)

() (

(

Α

)

)

1 21 2

12

1 21 2

f

1 21 2

f

1

⎡

⎣

212

2

11

0

T

f

T

TT

kk

kkkk

kk

−Ω

+−Ω−Ω

⎤

⎦

+Ω −+>

Β C

Β C

C

Β C

(8)

4116

Page 3

The first two terms are positive; therefore, the third term

must also be positive, hence

(

−+

Ω

CA

For proving the reverse condition, if the condition of

equation (9) holds, then the inequality in equation (8) and

accordingly equation (7) are correct, meaning P is positive

definite. Since L is positive definite by definition, then the

set of equations (5) are stable and accordingly are the

system of equations (1).

In order to compare the findings with PPF and provide a

better understanding of the gain limits, let’s consider the

case of vibration control of only the first resonant frequency.

To compare the PPF and MPPF stability conditions, the

case of vibration control of mode one is considered.

Therefore, C matrix is only a scalar and the ideal condition

would be

1

[]

= ω

C

; then the equation (9) becomes

()

1111

0

ω −ω α +β>

, which

()

11

1

α +β< . Comparing this with the condition for one

mode PPF control (

PPF gain < ), it is realized that the

)

2

0

T

>

B C

(9)

22

canbesimplifiedas

1

condition of MPPF is similar to PPF, which demands the

control gain to be smaller than 1, but in MPPF the

summation of both gains must be smaller than 1.

III. EXPERIMENTALSETUP ANDMETHOD

To validate the proposed modified positive position

feedback controller, a major set of experiments are

undertaken to generate disturbance on a piezoelectrically

active controlled plate and MPPF is used as the active

control method. The test rig contains a galvanized steel

plate with two piezoelectric patches as control actuators

attached to the plate as shown in Fig. 1. The plate is

clamped at all edges with a frame. There are 10

accelerometers attached to the plate that are shown with

black dots and their corresponding identification number.

These numbers will be used further in explaining the

results. Two sensors are collocated with the piezoelectric

actuators and one is collocated with the point of applying

disturbance.

The plate is clamped at all edges with a frame that is bolted

to the base by 14 bolts as shown in Fig. 2. The disturbance

will be applied using an electromagnetic actuator by Ling

Dynamic Systems®that can produce both periodic and

impulse disturbances. The disturbance is collocated with the

sensor (3) that is on the right side of the piezoelectric

patches. This point is slightly offset from the center line to

provide different influences on each side of the plate (see

Fig. 1); however, it is not that far from center line so the

piezoelectric patches that are on the center line can suppress

the vibration. There are two piezoelectric patches: one in

the center of the plate (PZ 1) and one on the right side of it

(PZ 2). Piezoelectric actuators are produced by Piezo

Systems, Inc.®

(part number T234-A4CL-503X). The

mechanical properties of the plate and piezoelectric patches

are presented in Table 2.

Fig. 1. Schematic of the plate; the unit is millimeter.

Fig 2. Plate and test facilities.

All controlling process and signals are sent and processed

via a dSpace®board that uses the ControlDesk®and Matlab

Simulink®software packages. In order to identify the

resonant frequencies of the plate, a chirp signal is used to

excite the system. This will not only reveal the natural

frequencies but also identify the resonances with the highest

peaks, thus the controller will be used to suppress the

vibrations of these modes. Figure 3 shows the frequency

response of the plate to the chirp signal that sweeps the

frequencies from 0 to 200 Hz.

Table 2. Mechanical properties of plate and piezoelectric material.

PropertiesValue

Plate modulus of elasticity, Ep(GPa)

Plate density, ρp(kg/m3)

200±5

7800

Plate thickness, hp(mm)

0.9±0.05

Piezoelectric modulus of elasticity, Epz(GPa)

Piezoelectric density, ρpz(kg/m3)

65±2

7850

Electromechanical coupling coefficient, d31(pC/N)

-190

Piezoelectric thickness, hpz(mm)

0.8±0.05

All controlling process and signals are sent and processed

via a dSpace®board that uses the ControlDesk®and Matlab

Simulink®software packages. In order to identify the

resonant frequencies of the plate, a chirp signal is used to

excite the system. This will not only reveal the natural

50

50

500

32

64

140

114

102

102

78

Fixed by frame

PZ 1

82

Sensors

Point of

applying

disturbance

PZ 2

(10

)

(9)

(8)

(3)

(2)

(1)

(4)

(5)

(6)

(7)

6

400

4117

Page 4

frequencies but also identify the resonances with the highest

peaks, thus the controller will be used to suppress the

vibrations of these modes. Figure 3 shows the frequency

response of the plate to the chirp signal that sweeps the

frequencies from 0 to 200 Hz.

050 100150200

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequency (Hz)

m/s2

Fig. 3. Frequency response of the plate to a chirp excitation.

The highest peaks occur at 53 Hz, 106 Hz, 126 Hz and 149

Hz. Among these four modes, modes (1,1) and (1,3) that

occur at 53 Hz and 106 Hz, respectively, have the highest

peaks. The mode (1,2) at 80Hz does not provide an

amplitude as high as modes (1,1) and (1,3). In the next

section, the method of MPPF will be used to suppress the

vibrations of one mode i.e., mode (1,1).

The control method is similar to PPF control method. The

reason that piezoelectric patches are mounted at two

locations (on the center and between center and the point of

applying the disturbance) is simply because of interest in

suppressing the vibrations at these two points, in particular

at the center. The center is a point of interest since the

highest amplitude of vibrations (vibrations at the first

resonant frequency) occurs at the center of the clamped

plate as shown in Fig. 1. The piezoelectric patch between

the center and point of applying disturbance provides the

ability to suppress the vibrations that come from the

piezoelectric patch at the center and also the disturbance

that come from the shaker. During the experimental

procedure, the center piezoelectric patch will be adjusted

first to control the vibration at the center, then the controller

on the second piezoelectric patch is activated to suppress the

remaining vibrations on the plate, especially at the point

between the center and disturbance point.

In order to experimentally adjust the control gains, one

should increase the gain of the first order compensator to

provide a proper damping value and then increase the gain

of the second order compensator to completely suppress the

vibration. However, it should be noted that when more than

one frequency is targeted to be suppressed, the compensator

gains of higher frequency need to be adjusted first followed

by adjusting lower frequencies to prevent the spillover

effect.

IV. RESULTS

In this section, the MPPF controller will be experimentally

used to suppress the vibrations of the plate. In the first step,

the first resonance frequency at 53 Hz will be excited,

which has the highest amplitude of vibration, and MPPF

controller will be applied to both piezoelectric patch to

control the vibrations specifically at those two points but

also all over the plate. In order to experimentally apply the

MPPF method to the plate, the MPPF controller is designed,

and the simulated using MATLAB®

software. The controller then was converted to C®and

uploaded onto a dSpace Autobox system. The sampling rate

for the systems is set to be 0.0025 sec. The input voltage for

both piezoelectric actuators and the shaker (to produce

disturbance) and output signals from all the sensors are sent

and received by the dSpace Autobox system.

To determine the control gains for one mode control, the

shaker is set to produce a disturbance at 53 Hz, i.e., the first

resonance frequency of the plate. Referring to equation (2),

the A and B gain matrices for the case of one mode control

are reduced to scalars, α1and β1, respectively. The objective

is to control the vibration at the center and then the rest of

the plate, so the gains for the piezoelectric patch on the

center are adjusted first, then the gains for the other

piezoelectric patch. In the experimental one mode control,

as the first step, the gain β1is adjusted to add damping into

the system, then the gain of second order controller α1is

adjusted. It should be noted that the gains are in the limit

expressed in equation (9). If the conditions are similar to

the case of C1≈ω1, then, as explained at the end of Section 2

of this paper, the gains are limited to(

recommended to start increasing the gain from zero to the

point that the gain can produce the maximum vibration

suppression, since after a certain point, increasing the gain

does not provide better performance and even can increase

the vibration or push the system near the instable region.

Once the optimum gains for first piezoelectric patch are

obtained, a similar procedure can be performed for the

second piezoelectric patch. Figure 4 shows the control

results at all 10 sensors on the plate, as shown and

numbered in Fig.1. The shaker and the piezoelectric patches

are deactivated at the beginning of the experiment. After

1.4 sec, the shaker is turned on and the vibrations are

recorded by the sensors. When the response becomes stable,

the controller of the first piezoelectric actuator (PZ 1) is

activated after approximately 5 sec. It is seen that the

vibrations in the center of the plate (Sensor No.1) have been

very well suppressed by about 92% of the original

amplitude. The vibrations have been also decreased at

sensors 5, 6 and 9, but sensors 2, 4, 7, and 8 show increases

of vibrations. Sensors 3 and 10 have not changed. By

implementing the controller to PZ 2 at 7.6 sec, the

vibrations have started to decrease in more sensors. When

the controller is on for both piezoelectric patches the

vibrations of all points have been decreased, except points 3

and Simulink®

)

11

1

α +β< . It is

53Hz

106Hz

149Hz

126Hz

4118

Page 5

and 7. Sensor 3 is where the excitation has been applied and

sensor 7 still has lower vibration amplitude than some other

sensors. It should be noted that the main objective is

vibration suppression of the center and then the other points

and although the controller on PZ 2 can be actuated to

provide better vibration suppression for point 2 and some

other points, it causes marginal increase in the amplitude of

vibrations of point 1 (center of the plate), which is not

desired. For better observation of vibration suppression on

the center of the plate, the frequency responses of this point

before and after implementing control are shown in Fig. 5.

02468 10

-10

0

10

Sensor No.1

02468 10

-10

0

10

Sensor No.2

0246810

-10

0

10

Sensor No.3

02468 10

-10

0

10

Sensor No.4

0246810

-10

0

10

Acc. (m/s2)

Sensor No.5

02468 10

-10

0

10

Sensor No.6

0246810

-10

0

10

Sensor No.7

02468 10

-10

0

10

Sensor No.8

024

Time (Sec)

68 10

-10

0

10

Sensor No.9

024

Time (Sec)

6810

-10

0

10

Sensor No.10

Fig. 4. Response of all 10 sensors on the plate without control, and after controlling with one and then two piezoelectric actuators.

Figure 5a shows the frequency response of the plate

(response at the center, sensor No.1) when it is excited with

the periodic disturbance at 53 Hz. Fig. 5b shows the

frequency response of the same system when

controllers at PZ 1 and PZ 2 are activated. The response at

53 Hz is 19.96 dB for the system with no controller, while it

is -37.34 dB when both controllers are on. It shows more

both

than a 57 dB decrease in the amplitude of vibrations at the

excitation frequency. At the frequency of 106 Hz, however,

there is a large peak that has increased slightly after

engaging the controller. Since the plate has been excited

just with a 53 Hz periodic signal, and the 106 Hz peak is

twice the first natural frequency, it may be concluded that

there is nonlinear vibrations in the plate. The peak at 159

Hz (3×53 Hz) also confirms this assumption.

4119

Page 6

050100150200

-250

-200

-150

-100

-50

0

50

Frequency (Hz)

Sensor No.1

Acceleration (dB)

Sensor No.1

050100 150 200

-250

-200

-150

-100

-50

0

50

Frequency (Hz)

Acceleration (dB)

Fig. 5. Frequency response of Sensor No.1 to periodic disturbance with 53 Hz

frequency; a) without control, b) controllers for both piezoelectric patches are

on.

I. CONCLUSIONS

A new active vibration control method called Modified

Positive Position Feedback (MPPF) was developed based on

PositivePositionFeedback

experimentally. The controller consists of two parallel

compensators: a first order compensator to provide damping

and a second order compensator to suppress the vibration.

The control effort is applied to the structure using two

piezoelectric patches. The control gains were obtained using

a stability analysis. A plate was used as the test platform for

examining the controller. The dynamic model of the

piezoelectrically actuated plate was constructed and the

transfer function of the closed loop system was obtained.

Using a chirp signal, the resonance frequencies of the plate

were found and MPPF was used to suppress the vibrations

of the fundamental resonant frequencies. The results show

more than 90% suppression of vibration at the center of the

plate for the first mode.

andsuccessfullyverified

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Krishnamurthy,K., and Chao, M., 1992, “Active Vibration Control

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