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Feedback Loop for the mechanical Stabilisation

Jacques Lottin*(jacques.lottin@univ-savoie.fr)

Laurent Brunetti*(laurent.brunetti@univ-savoie.fr)

Mihai Corduneanu**Vlad Cozma **

*LISTIC-ESIA, Université de Savoie, Annecy, France**Universitatea Politehnica, Bucuresti, Romania

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Overview :

- Presentation of active vibration reduction- Principle of rejection- Control scheme- Programs- Experiments- Results- Conclusion

Collaboration with and

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Mechanical part

control

Measurement :(continuous + resonances)

Disturbance : -ground: cultural noise…-equipment: motors, flows…

Excitation : (strength applied with actuators…)

Presentation of Active Vibration Reduction

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Principle of rejection

Assumptions:

- There are a few resonances which are independentsmall amplitudes, linearity, …

- amplitude and phase of each resonance are constant or slowly varying with respect to the signal period

- frequency of each resonance is knowncomputed by means of Fourier transform

- there is no accurate model available

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Principle of rejection

Current objectives: Independently reduce the main resonances

For example, an identification of one mock-up…

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Principle of rejection

- rule 2: make the components of excitation converge to values such that the global effect of disturbance and excitation is null at sensor location

lumped / distributed

- rule 1: decompose each resonance as a weighted combination of sine and cosine measurement, disturbance, excitation

with:

( ) sin( ) sin( ) cos( )s cf t t f t f t cos( )

sin( )s

c

f

f

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Principle of rejection

An exact direct compensation would require the knowledge of eight elementary transfer functions:

at least at one frequency

Mechanical part

disturbance

excitation

measurementpc

ps

fc

fs yc

ys

Disturbance is not well definedMeasurement/Excitation transfers are badly known

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Model (State space representation for one frequency):

Control scheme

Disturbance

With the different components are decomposed in sine and cosine:

Input(Strength)

State

(1)

( ) ( ) ( ) ( )

( ) ( )n n n

n

x t A x t B u t G p t

y t C x t

( ) ( ), ( ) , ( )

( )

)

( ) ))

(

(( s

c

s s

c c

f t p tu t p t

f

yx t

t p t

t

y t

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An : represents the dynamic of the system.

Control scheme

Dynamic response

open loop : first order-> Setting time = 3 * time-constant

: time-constant

10

,1

0nA

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Control scheme

Bn ,Gn : Transfer’s matrix of the strength and the disturbance.

(1) Steady state then :

1st case: disturbance = 0

y = kyf ( fs sin (t + yf ) + fc cos (t + yf ) )

(2)

(2) & (4):

(3)

also:

(4)

(5)

(6)

( ) ( )

( )

cos sin

sin cos

cos sin

s

( ) ( )

( ) ( )

(

n os

)

i cyp yp

ypyp

yf yf

yfy

y

f yn n

n n

f

p

B A

G A

k

k

1 1s s s sn n n n

c c c c

x y f pA B A G

x y f p

cos sin

s

( ) ( )

( ) o ( )in c syf yf

yfyf y

s s

c cf

y f

y fk

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Estimation of the disturbance using a state observer :

Control scheme

Requirement: the new model should include the disturbance in the components of the state:

The new state vector:

The model:(7)

(8)

e e e e e

e e

x A x B u

y C x

s

ce

s

c

y

yx

p

p

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State observer:

, , 00 0 0

n n ne e e n

A G BA B C C

Control scheme

Where:

Final relation:

(9)

(10)

(Luenberger)

ˆ ˆ ˆ( )

ˆ ˆe e e e

e e

x A x B u L y y

y C x

ˆ ˆ

ˆ ˆ

ee e e e e

e

e e

ux A LC x B L

y

y C x

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Control law:

Control schemeState feedback control :

(1) =>

To reject the disturbance:

(11)

(12)

(13)1p n nK B G

( ) 0n p nB K G

( ) ( ) ( )

ˆ( ) ( )

x p

x p

u t K x t K p t

K x t K p t

( ) ( )n n x n p nx A B K x B K G p

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Control scheme

Mechanicalpart

spectralanalysis

signalprocessing

observer

statefeedback

signalrebuilding

actu

ator

sens

or

yc(wi)ys(wi)

fc(wi)

fs(wi)

y(wi)f

)(ˆ is wp)(ˆ ic wp

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ProgramMain program:(Matlab / Simulink / XPC Target toolboxes)

Analog Input board

Analog Output board

Control of disturbances

Summation of each command

Algorithm forone frequency

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ProgramProgram for one frequency :(Matlab / Simulink / XPC Target toolboxes)

State feedback

State observerSignal

processingBand-pass filter

for the disturbanceSensor

command rebuilding

Actuator

Algorithm for one frequency :

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« a steel beam »

2 loudspeakers2 opposite PZT

Accelerometer(only for

monitoring)

ExperimentsMock-up:

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ExperimentsMock-up:

Sensor PZT(bottom)

Actuator PZT (on top)

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Experiments

Layout :

Development PC (host)

Matlab + Simulink+ XPC Target

Toolboxes

Dedicated PC (target)

XPC Target

Ethernet network

Input / Output analog boardTexas InstrumentsPlant

Sample time : 0.0007 s

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ResultsRejection of 6 resonances : (without and with rejection)

Resonances of : -beam-support

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Results

Some zooms…

Without rejectionWith rejection

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ResultsRejection of 6 resonances : (disturbances and control)

Resonances of :-beam-support

Disturbances

Control

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ResultsPhase robustness of yf (matrix Bn) : 1st resonance : margin of /2

Robustness OK :Identified phase

Stability limit :Identified phase +2/20Identified phase -8/20

Without rejection

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ResultsGain robustness of kyf (matrix Bn) : 1st resonance : factor 10.

Identified gainIdentified gain * 0.1Identified gain * 10

Beginningof control

Worse in setting time

Time

Right convergence in

steady time(stability limit…)

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Conclusion

Status:- Validation is OK- Robustness is ~OK

Current and future works:-Spectral analysis of the disturbance in real time-Translation of the last studies on a new model with a parallelepiped beam (2.5 meters long).(testing the algorithm and choosing the appropriate actuators….)

Jacques LOTTIN : jacques.lottin@univ-savoie.frLaurent BRUNETTI : laurent.brunetti@univ-savoie.fr