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I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances 1 Adaptive rejection of unknown disturbances Application to active vibration control I.D. Landau, A. Constantinescu Laboratoire d’Automatique de Grenoble(INPG/CNRS), France Sevilla October 2004
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Page 1: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances1

Adaptive rejection of unknown disturbances Application to active vibration control

I.D. Landau, A. ConstantinescuLaboratoire d’Automatique de Grenoble(INPG/CNRS), France

SevillaOctober 2004

Page 2: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances2

OutlineOutline

• Introduction• Rejection of unknown stationary disturbances. Basic facts• Rejection of unknown narrow band disturbances in active

vibration control. Real-time results• Adaptive control solutions (indirect/direct)• Further experimental results

(comparison direct/indirect adaptive control)• Conclusions

Page 3: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances3

Unknown disturbance rejection Unknown disturbance rejection –– classical solutionclassical solution

• requires the use of an additional transducer• difficult choice of the location of the transducer• adaptation of many parameters

++A / Bq -d ⋅u(t)

Plant )(p1 t

)(tδ

pp DN /

FIRDisturbance

Environment

Disturbance correlated

measurementDisturbance +Plant

model

AdaptationAlgorithm

y(t)

Obj: minimization of E{y2}

Disadvantages:

Page 4: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances4

Rejection of unknown disturbances

• Assumption: Plant model almost constant and known (obtained by systemidentification)

• Problem: Attenuation of unknown and/or variable stationary disturbanceswithout using an additional measurement

• Solution: Adaptive feedback control- Estimate the model of the disturbance- Use the internal model principle- Use of the Youla parameterization (direct adaptive control)

Attention :The area is “dominated “ by adaptive signal processing solutions(Widrow’s adaptive noise cancellation) which require an additional transducer

Most surprising : there is an elegant “direct adaptive control” solution

Remainder : Models of stationary disturbances have poles on the unit circle

A class of applications : suppression of unknown vibrations(active vibration control)

Page 5: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances5

Internal Model Principle

For asymptotic rejection of a disturbance the controllershould incorporate the model for the disturbance

Remember:

Step disturbance model : )1

1(

11−− q

ors

The controller should incorporate an « integrator »(which is the model of the disturbance)

DistrubanceModel

Distrubanceδ

(Dirac)

Page 6: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances6

Indirect adaptive controlIndirect adaptive control

Two-steps methodology:

1. Estimation of the disturbance model,

2. Compute the controller using the « internal model principle »(the controller contains the model of the disturbance)

)( 1−qDp

++A / Bq-d ⋅

u(t)Plant )(p1 t

)(tδ

pp DN /

Disturbance

Environment

Disturbancemodel

y(t)Controller

DisturbanceModel

Estimation

DesignMethod

Specs.

It can be time consuming (if the plant model B/A is of large dimension)

Page 7: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances7

Direct adaptive rejection of unknown disturbances

Plant

ModelModel

^Adaptationalgorithm

•One directly adapts a filter which is part of the controller.•It tunes the « internal model » inside the controller•Does not changes the poles of the closed loop (Y-K param.)

Model = Plantw=Ap

Page 8: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances8

Plant

Model

0/1 S

0R

ABq d−

ABq d−

A

ParameterAdaptationAlgorithm

y(t)u(t)

w(t)

p(t)+

++

-

- -

Direct adaptive rejection of unknown disturbances

Equivalent representation of the scheme (case A = as.stable)

Page 9: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances9

Rejection of unknown narrow band disturbancesin active vibration control

Page 10: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances10

The Active Suspension System

controller

residual acceleration (force)

primary acceleration / force (disturbance)

1

23

4

machine

support

elastomere cone

inertia chamber

piston

mainchamber

hole

motor

actuator(pistonposition)

sTs m 25.1=++−

A / Bq-d ⋅S / R

D / Cq 1-d ⋅

u(t)

ce)(disturban

(t)up

Controller

force) (residualy(t)

Plant )(p1 t

Two paths :•Primary•Secondary (double differentiator)

Objective:•Reject the effect of unknownand variable narrow banddisturbances•Do not use an aditionalmeasurement

Page 11: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances11

The Active Suspension

Residualforce

(acceleration)measurement

Activesuspension

Primary force(acceleration)(the shaker)

Page 12: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances12

Active Suspension

Primary path

Frequency Characteristics of the Identified Models

Secondary path

0;16;14 === dnn BA

Further details can be obtained from : http://iawww.epfl.ch/News/EJC_Benchmark/

Page 13: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances13

Direct Adaptive Control : disturbance rejection

Closedloop

Open loop

Disturbance : Chirp

Initialization of theadaptive controller

25 Hz47 Hz

Page 14: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances14

Direct adaptive control

Time Domain ResultsAdaptive Operation

Page 15: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances15

19.8 19.9 20 20.1 20.2 20.3 20.4 20.5 20.6-4

-2

0

2

4

Time(s)

Out

put

(res

idua

l for

ce)

19.8 19.9 20 20.1 20.2 20.3 20.4 20.5 20.6

-2

0

2

Time(s)

Inpu

t (c

ontro

l act

ion)

32 Hz 20 Hz

29.8 29.9 30 30.1 30.2 30.3 30.4

-2

0

2

Time(s)O

utpu

t (re

sidu

al fo

rce)

29.8 29.9 30 30.1 30.2 30.3 30.4-2

-1

0

1

2

Time(s)

Inpu

t (co

ntro

l act

ion)

20 Hz

32 Hz

0 10 20 30 40 50 60-4

-2

0

2

4

Time(s)

Out

put (

resi

dual

forc

e)

0 10 20 30 40 50 60-2

-1

0

1

2

Time(s)

Inpu

t (co

ntro

l act

ion)

32 Hz 20 Hz 32 Hz 42 Hz 32 Hz

Direct Adaptive Control

Output

Input

Initialization of theadaptive controller

Adaptation transient Adaptation transient

Page 16: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances16

Notations

K G

v(t) p(t)y(t)

v(t)r(t) u(t)

-

)q(A)q(Bq)q(G

1

1d1

−−− =

)()(')()('

)()(

)( 11

11

1

11

−−

−−

−− ==

qHqSqHqR

qSqR

qKR

S

Output Sensitivity function :

)z(R)z(Bz)z(S)z(A)z(P 11d111 −−−−−− +=Closed loop poles :

)()()(')(

)( 1

1111

−−−− =

zPzHzSzA

zS Syp

The gain of Syp is zero at the frequencies where Syp(e-jω)=0(perfect attenuation of a disturbance at this frequency)

+

ypS

HR and HS are pre-specified

Page 17: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances17

Dirac circle;unit on the poles

edisturbanc ticdeterminis : )()(

)()( 1

1

=→

⋅= −

d(t) D

tqD

qNtp

p

p

p δ

Deterministic framework

Stochastic framework

)(0, sequence noise hiteGaussian w circle;unit on the poles

edisturbanc stochastic : )()(

)()( 1

1

σ=→

⋅= −

e(t) D

teqD

qNtp

p

p

p

Disturbance model

Page 18: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances18

).()(')(

);()(')(

:Controller

111

111

−−−

−−−

⋅=

⋅=

qHqSqS

qHqRqR

S

R

Internal model principle: HS(z-1)=Dp(z-1)

Closed loop system. NotationsClosed loop system. Notations

)()(qD

1

)P(q

)(q)(q)S'(q)HA(qy(t)

1-p

1-

-1-1-1S

-1

tN p δ⋅⋅=

++−

A / Bq -d ⋅S / Ru(t)

Controller Plant )(p t

)(tδ

pp DN / Dirac circle;unit on the poles

edisturbanc ticdeterminis : )()(

)()( 1

1

=→

⋅= −

d(t) D

tqD

qNtp

p

p

p δ

)()()()()( :poles CL

)()()()(

)()()( :Output

11111

11

11

−−−−−−

−−

−−

+=

⋅=⋅=

qRqBzqSqAqP

tpqStpqP

qSqAty

d

yp

Page 19: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances19

Indirect adaptive controlIndirect adaptive control

Two-steps methodology:

1. Estimation of the disturbance model,

2. Computation of the controller, considering

)( 1−qDp

)(ˆ)( 11 −− = qDqH pS

++A / Bq-d ⋅

u(t)Plant )(p1 t

)(tδ

pp DN /

Disturbance

Environment

Disturbancemodel

y(t)Controller

DisturbanceModel

Estimation

DesignMethod

Specs.

It can be time consuming (if the plant model B/A is of large dimension)

Page 20: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances20

Indirect adaptive controlIndirect adaptive control

Step II: Computation of the controller

'

ˆ

SDS

PBRqSDA

DH

p

dp

pS

=

=+

=−

Solving Bezout equation (for S’ and R)

Remark:It is time consuming for large dimension of the plant model

Step I : Estimation of the disturbance model ARMA identification (Recursive Extended Least Squares)

Page 21: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances21

Plant

ModelModel

Central contr: [R0(q-1),S0(q-1)].

CL poles: P(q-1)=A(q-1)S0(q-1)+q-dB(q-1)R0(q-1).

Control: S0(q-1) u(t) = -R0 (q-1) y(t)

Control: S0(q-1) u(t) = -R0 (q-1) y(t) - Q (q-1) w(t),

where w(t) = A (q-1) y(t) - q-dB (q-1) u(t).

CL poles: P(q-1)=A(q-1)S0(q-1)+q-dB(q-1)R0(q-1).

Q-parameterization :R(z1)=R0(q-1)+A(q-1)Q(q-1);S(q-1)=S0(z-1)-q-dB(q-1)Q(q-1).

InternalInternal modelmodel principleprinciple andand QQ--parameterizationparameterization))

Model= Plantw=Ap

Page 22: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances22

Internal model principle and Q- parameterization

Central contr: [R0(q-1),S0(q-1)].

CL Poles: P(q-1)=A(q-1)S0(q-1)+q-dB(q-1)R0(q-1).

Control: S0(q-1) u(t) = -R0 (q-1) y(t)

Q-parameterization :R(z1)=R0(q-1)+A(q-1)Q(q-1);S(q-1)=S0(z-1)-q-dB(q-1)Q(q-1).

Closed Loop Poles remain unchanged

pd MDBQqSS =−= −

0 0SBQqMD dp =+ −Solve:

? ?

Internal model assignment on Q

Q(q-1) computed such as [S(q-1)] contains the internal model of the disturbance

Q can be used to “directly”tune the internal model without changing the closed loop poles(see next)

Will lead also to an « indirect adaptive control solution »

BUT:

Page 23: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances23

Goal: minimize y(t) (according to a certain criterion).

).()()(

)()(

)()(

)( 11

1

1

10 twqQ

qPqBq

twqPqS

td

⋅−⋅= −−

−−

ε

)( of valueestimatedan be ˆLet 11 −qQ)(t,qQ -

Direct Adaptive Control (unknown Dp)

)0 termedisturbanc )1((

)1()()(

)()],1(ˆ)([)1(

thatshowcan We

1

111

→=+

++⋅⋅+−=+−

−∗−−−

tv

tvtwqP

qBqqtQqQt

Hypothesis: Identified (known) plant model (A,B,d).

[ ] [ ])(

)P(q))Q(q(qq-)(qS

)()(qD)(q

)P(q))Q(q(qq-)(qS)A(q

y(t)

e.disturbanc ticdeterminis : )()()(

)(Consider

1-

1-1-d-1-0

1-p

1-

1-

1-1-d-1-0

1-

1

1

1

twB

tNB

tqDqN

tp

p

p

p

=⋅⋅=

⋅=−

δ

δ

(Based on an ideea of Y. Z. Tsypkin)

Leads to a directadaptive control

Define:w

ε+

-

PS /0

S

Page 24: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances24

[ ] [ ]

).()()1()(

);()()1()(A1) w(t

);()(

)()(

);1()(

)()1(

and

1)(for ,)1()()( ;)(q̂)(q̂(t)ˆ

where

),()1(ˆ)1()1( :error adaptation

);()(ˆ)1()1( :error adaptation

11

11

1

1

2

1

10

1

2210

1

10

tuqBtuqB

tuqBqtyq

twqP

qBqtw

twqP

qStw

ntwtwttt

tttwtposterioriA

tttwtprioriA

d

d

QTT

T

T

⋅=+⋅

⋅−+⋅=+

⋅=

+⋅=+

=−==

+−+=+

−+=+

−∗−

−∗−−

−∗−

φθ

φθε

φθε

Parameter adaptation algorithm:

+=++++=+

−− ).()()()()()1(1);(t(t)1)F(t(t)ˆ1)(tˆ

21

11

0

ttttFttF Tφφλλεφθθ

The Algorithm

1−=PDQ nn

Page 25: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances25

Plant

ModelModel

^Adaptationalgorithm

Direct adaptive rejection of unknown disturbances

• The order of the Q polynomial depends upon the order of the disturbance modeldenominator (DP) and not upon the complexity of the plant model

• Less parameters to estimate then for the identification of the disturbance model

Page 26: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances26

Further experimental results on the active suspensionComparison between direct/indirect adaptive control

Page 27: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances27

Narrow band disturbances = variable frequency sinusoid ⇒ nQ = 1Frequency range: 25 ÷ 47 Hz

Nominal controller [R0(q-1),S0(q-1)]: nR0=14, nS0=16

Evaluation of the two algorithms in real-time

RealReal--timetime resultsresults

• The algorithm stops when it converges and the controller is applied.• It restarts when the variance of the residual force is bigger than a given threshold.• As long as the variance is not bigger than the threshold, the controller is constant.

Implementation protocol 1: Self-tuning

Implementation protocol 2: Adaptive• The adaptation algorithm is continuously operating• The controller is updated at each sample

Page 28: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances28

Frequency domain results – indirect adaptive method

Page 29: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances29

Frequency domain results – direct adaptive method

Page 30: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances30

Indirect adaptive method Direct adaptive method

Time Domain ResultsSelf-tuning Operation

Page 31: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances31

Indirect adaptive method Direct adaptive method

Time Domain ResultsAdaptive Operation

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

-6

-4

-2

0

2

4

6

Indirect method in adaptative operation

Samples

Res

idua

l for

ce [

V]

32 Hz 25 Hz 32 Hz 47 Hz 32 Hz

•Direct adaptive control leads to a much simpler implementationand better performance than indirect adaptive control•Direct adaptive control in adaptive mode operation gives betterresults than direct adaptive control in self-tuning mode

Page 32: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances32

ConclusionsConclusions

-Using internal model principle, adaptive feedback control solutions can be provided for the rejection of unknown disturbances

-Both direct and indirect solutions can be provided

-Two modes of operation can be used : self-tuning and adaptive

-Direct adaptive control is the simplest to implement

-Direct adaptive control offers better performance

-The methodology has been extensively tested on an activesuspension system

Open problem:Theoretical study of the plant – model mismatch

Page 33: Adaptive rejection of unknown disturbances Application to active vibration controleduardo/TOK/PerturbSev.pdf · 2004-10-21 · Unknown disturbance rejection – classical solution

I.D. Landau, A. Constantinescu: Adaptive rejection of unknown disturbances33

).()()(

)()(

)()(

)( 11

1

1

10 twqQ

qPqBq

twqPqS

td

⋅−⋅= −−

−−

−ε

Direct Adaptive Control (unknown Dp)

0SBQqMD dp =+ −

We need to express ε(t)as:

[ ] )(),1(ˆ)()1( 11 tqtQqQt Ψ+−=+ −−ε

(*)

Using: , (*) becomes

)1()(

)()()(

)()()],1(ˆ)([)1( 1

11

1

111 ++⋅⋅+−=+ −

−−

−∗−−− tp

qP

qDqMtw

qPqBqqtQqQt p

Vanishing term

)1()(

)()()1(

)(

)()(1

11

1

11

+=+ −

−−

−−

tqP

qNqMtp

qP

qDqM pp δ

)1()(

)()(

)()(

1

1

1

1+⋅=⋅ −

−−

−∗−tw

qPqBq

twqP

qBq dd

Details:

[ ]QiqtQt

iQˆ,minarg)*,(ˆ

0

1 ∑==

− ε

search recursively for :Instead of solving 0SBQqMD dp =+ −


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