Adaptive rejection of unknown disturbances Application to active vibration...

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


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


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:


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)


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)


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)


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


Plant

Model

0/1 S

0R

ABq d−

Q̂

ABq d−

A

ParameterAdaptationAlgorithm

y(t)u(t)

w(t)

p(t)+

++

-

- -


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


Rejection of unknown narrow band disturbancesin active vibration control


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


The Active Suspension

Residualforce

(acceleration)measurement

Activesuspension

Primary force(acceleration)(the shaker)


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/


Direct Adaptive Control : disturbance rejection

Closedloop

Open loop

Disturbance : Chirp

Initialization of theadaptive controller

25 Hz47 Hz


Direct adaptive control

Time Domain ResultsAdaptive Operation


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


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


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


).()(')(

);()(')(

: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



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)



Step II: Computation of the controller

'

'ˆ

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)


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


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:


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

dε

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+

-

Q̂

PS /0

S


[ ] [ ]

).()()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


Plant

ModelModel

^Adaptationalgorithm


• 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


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


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


Frequency domain results – indirect adaptive method


Frequency domain results – direct adaptive method


Indirect adaptive method Direct adaptive method

Time Domain ResultsSelf-tuning Operation


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


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


).()()(

)()(

)()(

)( 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

dε

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

2ˆ

1 ∑==

− ε

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

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Adaptive rejection of unknown disturbances Application to active vibration...

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