Landau RobustAdaptive

I.D.Landau : From Robust Control to Adaptive Control1

I.D. LandauLaboratoire dAutomatique de Grenoble, (INPG/CNRS), France

Mayo 2003

Del Control Robusto al Control Adaptable


I.D. LandauLaboratoire dAutomatique de Grenoble, (INPG/CNRS), France

May 2003

From Robust Control to Adaptive Control


Robust Control

uncertaintiesstructured (parameter variations)

unstructured (often in high frequencies)

Performance may be limited (for large plant uncertainties)

Adaptive Control

-Well suited for handling parameter variations- Should work correctly in the presence of unstructureduncertainties (parasitics)

- Problems for large and abrupt changes in plant parameters


Robust Control plays an important role in Adaptive Control(directly or indirectly)

Adaptive Control can improve the performances of aRobust Controller

Identification in Closed Loop allows to establish linksbetween Robust Control and Adaptive Control


Outline

- Introduction

- Identification in closed loop

- Experimental results (flexible transmission)

- Adaptive control strategies

- Robust control design for adaptive control

- Parameter estimators

- Adaptive control with multiple models

- Experimental results (flexible transmission)

- Adaptive rejection of unknown disturbances

- Experimental results (active suspension)

- Concluding remarks


Plant Identification in Closed Loop

There are systems where open loop operationis not suitable ( instability, drift, .. )

A controller may already exist ( ex . : PID )

Iterative identification and controller redesign

Re-tuning of the controllera) to improve achieved performancesb) controller maintenance

Why ?

Cannot be dissociated from thecontroller and robustness issues

May provide better design models ! !


m

axismotor

d.c.motor

Positiontransducer

axisposition

ref

load

Controlleru(t)

y(t)

ADC

R-S-Tcontroller

DAC+

+

Identification in Closed Loop

The flexible transmission


What is the good model ?

closed loop identified model open loop identified model

fs = 20Hz 0% load


output

control

real system

controller design using theopen loop identified model

output

control

real system

computed polecl. loop syst. pole

Benefits of identification in closed loop (1)

The pattern of identified closed loop poles is different fromthe pattern of computed closed loop poles


real system

output

control

controller computed using theclosed loop identified model

output

control

real system

computed polecl. loop syst. pole

Benefits of identification in closed loop (2)

The computed and the identified closed loop poles are very close


Notations

K G

v(t) p(t)y

v(t)r u

-

)q(A)q(Bq)q(G 1

1d1

=

)q(S)q(R)q(K 1

11

=

Sensitivity functions :

KG11)z(S 1yp +

=

KG1K)z(S 1up +

=

KG1KG)z(S 1yr +

=

KG1G)z(S 1yv +

=

)z(R)z(Bz)z(S)z(A)z(P 11d111 +=

; ; ;

Closed loop poles :

True closed loop system :(K,G), P, SxyNominal simulated(estimated) closed loop : xyS,P),G,K(


-1

1

1

1

crossover

frequency

Re H

Im H

G

|HOL|=1 CR

1

CR2

CR3

Robustness Margins

z = ej

> 29M 0.5 G 2 ;

M 0.5 (-6dB), > Ts

M = 1+HOL(z-1) min = Syp(z-1) max-1 = Syp -1( )

( )

= mini iCR

i

Modulus Margin:

Delay Margin:

Typical values:

The inverse is not necessarily true!

Critical frequency region for control


Syp max= - MSyp dB

0.5fs0

delaymarginnominalperform.

( G = G + Wa )

Sup dB

actuator effort

size of the tolerated additive uncertainty Wa

00.5f s

Sup -1

Templates for the Sensitivity Functions

Output SensitivityFunction

Input SensitivityFunction




noise

q-dBA

w

Plant++

T 1/S

R

+

-r

u y

ru

+

+

q-dBA

w

ASPru

yu +

+

ARP

+

+Syp

-Sup

Identification in Closed Loop

Open loopinterpetation

- take advantage of the improved input spectrum- are insensitive to noise in closed loop operation

Objective : development of algorithms which:


CLOE Algorithms

Objective of the Identification in Closed Loop

(identification for control)

Find the plant model which minimizes the discrepancybetween the real closed loop system and the simulated closed loop system.

noise

Controller Plant

Controller Model

r u y

y

+-

++

ParametricAdaptationAlgorithm

Simulated System

Closed Loop Output Error


Closed Loop Output Error Identification Algorithms(CLOE)

K

P.A.A.

G

r

u

u y

y

CL

+

+

+

K

G

Excitation added to controller output

K G

G

ru

uy

y+

+

+

P.A.A.

CL

K

+

+p

Excitation added to reference signal

Same algorithm but different properties of the estimated model


Iterative Identification in Closed Loop and Controller Re-Design

Repeat 1, 2, 1, 2, 1, 2,CL

CL

Step 1 : Identification in Closed Loop-Keep controller constant-Identify a new model such that

Step 2 : Controller Re Design- Compute a new controller such that

w1/S

RPlant

R

1/S

Model

++

+

+

+

-

-

-CL

r u y

u y

T q-d B/A

q-d B/A


Adaptive Control Basic Schemes

- Indirect adaptive control- Direct adaptive control (the controller is directly estimated)

Performancespecifications

AdjustableController Plant+

-

ControllerDesign

Plant Model

Estimation

Adaptation loop

AdjustableController Plant+

-

Performancespecifications

ControllerEstimation

Adaptation loop

Indirect Adaptive Control Direct Adaptive Control


time

Parameter Estimation+

Controller Computationt t+1

time

Fixed (or time varying)Controller computed at( t)

+Parameter Estimation

t t+N

Controllercomputedat (t +N)

Iterative Identification and Controller Redesign versus (Indirect) Adaptive Control

N = 1 : Adaptive Control

The iterative procedure introduces a time scale separation between identification / control design

N = SmallAdaptive ControlN = LargeIterative Identification in C.L.And Controller Re-design

Plant Identification in C.L. +Controller Re-design

N


The flexible transmission

m

axismotor

d.c.motor

Positiontransducer

axisposition

ref

load

Controlleru(t)

y(t)

ADC

R-S-Tcontroller

DAC

Adaptive Control of a Flexible Transmission


Adaptive Control of a Flexible Transmission

Frequency characteristics for various load

Rem.: the main vibration mode varies by 100%


Robust Control Design for Adaptive Control

parameter variations(low frequency) Adaptation Robust Design

unstructureduncertainities(high frequency)

Basic rule : The input sensitivity function (Sup) should be small inmedium and high frequencies

Pole Placement :- Opening the loop in high frequncies (at 0.5fs)- Placing auxiliary closed loop poles near the high frequency polesof the plant model

Generalized Predictive Control :- Appropriate weighting filter on the control term in the criterion

How to achieve this ?


Robust Control Design for Adaptive Control(Flexible Transmission)

a) Standard pole placement (1 pair dominant poles + h.f. aperiodic poles)b) Opening the loop at 0.5fs (HR = 1 + q-1)c) Auxiliary closed loop poles near high frequency plant poles


Parameter Estimators for Adaptive Control


Classical Indirect Adaptive Control

PLANT

disturb.u y

+

++

q- dBA

CL

q-dBA

-

P.A.A.

y

Reference AdjustableController

q -1Filter Filter

Adaptationmechanism

(design)

- Uses R.L.S. type estimator (equation error)- Sensitive to output disturbances- Requires adaptation freezing in the absence of persistent excitation- The threshhold for adaptation freezing is problem dependent


Closed Loop Output Error Parameter Estimatorfor Adaptive Control

PLANT

disturb.u y+

+ +CL

q-dBA

-1S

R

P.A.A.

y

Reference AdjustableController

AdjustableController

q-1

Adaptationmechanism

(design)

- Insensitive to output disturbances- Remove the need for adaptation freezing in the absence ofpersistent excitation

- CLOE requires stability of the closed loop- Well suited for adaptive control with multiple models

q- dBA

+


Adaptive Control Effect of Disturbances

Classical parameter estimator(filtered RLS) CLOE parameter estimator

Disturbances destabilize the adaptive system when using RLS parameter estimator(in the absence of a variable reference signal)


Adaptive Control with Multiple Models


Supervisory Control

PLANT

MODELS

CONTROLLERS

SUPERVISOR

G1

G2

Gn

Kn

K2

K1

+

+

+

-

-

- 1

2

n...

.. y

Performance criterion:

nijettJ it

j

jtii ...2,1,0,0;)()()(

2

0

)(2=+=

=

)(min tJiiSwitching rule:

Rem. : stability requires the use of hysteresis or time delay in switching



n is small (for the flexible transmission n = 3)Multiple models : improvement of the adaptation transientsCLOE Estimator : reduction of the false swithchings, performance improvement

SUPERVISOR

G1

G2

Gn

+

+

-

-

- 1

2

n..

yPLANT

+

+

-G

0-

+

CL

u

u y

Controller

Controller

r

P.A.A.

GAdaptive model

Fixed models


Adaptive Control versus Robust Control

Load variations : 0% 100% (in several steps)Rem : The robust controller used is the winner of an international

benchmark test for robust control of the flexible transmission (EJC, no.2., 1995)


Adaptation Transients

Adaptive Control with Multiple Models Classical Adaptive Control (simulation)

0 = adaptive ; 1= 0% ; 2 = 50% ; 3 = 100%

Load variations : 100% 0% (in two steps at 19s and 29s)



The plant models are not in the model set

0 = adaptive ; 1= 0% ; 2 = 50% ; 3 = 100%

Load variations : 75% 25%


Adaptive rejection of unknown disturbancesApplication to active suspension


Rejection of unknown disturbancesRejection of unknown disturbances

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

Solution: Direct adaptive feedback control Methodology: Based on the

Internal model principle Sensitivity function Q - parametrization Direct adaptive control algorithm

Objective: Computation of a controller with an adaptive internal model of thedisturbance

Rem: Stationary disturbances models have poles on the unit circle

Hypothesis: Plant model parameters are constant and known


).()(')();()(')(

:Controller

111

111

=

=

qHqSqSqHqRqR

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-

p1-

-1-1-1S

-1t

N p =

++

A / Bq-d S / Ru(t)

Controller Plant )(p1 t

)(t

pp DN / Dirac circle;unit on the poles

edisturbanc ticdeterminis : )()()(

)( 11

1

=

=

(t) D

tqDqN

tp

p

p

p

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

)()()()(

)()()( :Output

11111

11

11

11

+=

==

qRqBzqSqAqP

tpqStpqP

qSqAty

d

yp


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

Bezout: 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).Q(q-1) computed such as [R(q-1),S(q-1)]

contain the internal model of the disturb.

Direct adaptative control (Direct adaptative control (QQ--parameterizationparameterization))

pd MDBQqS = 0

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).

The internal modelcan be tuned with Q


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

).()()(

)()()()()( 11

1

1

10 twqQ

qPqBqtw

qPqSt

d

=

)( of valueestimatedan be Let 11 qQ)(t,qQ -

Direct Adaptive Control (unknown Dp)

)0 termedisturbanc )1((

)1()()(

)()],1()([)1(

thatshowcan We

1

111

=+

+++=+

tv

tvtwqP

qBqqtQqQtd

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

[ ] [ ] )()P(q

))Q(q(qq-)(qS)()(qD)(q

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

e.disturbanc ticdeterminis : )()()(

)(Consider

1-

1-1-d-1-0

1-p

1-

1-

1-1-d-1-0

1-

1

1

1

twBtNB

tqDqN

tp

p

p

p

==

=

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

Leads to a directadaptive control


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 than for the identification of the disturbance model Much simpler than indirect adaptive control


The Active Suspension

Residualforce

(acceleration)measurement

Activesuspension

Primary force(acceleration)(the shaker)


The Active Suspension SystemThe Active Suspension System

controller

residual acceleration (force)

primary acceleration / force (disturbance)

1

23

4

machine

support

elastomere cone

inertia chamber

pistonmainchamber

hole

motor

actuator(pistonposition)

sTs 00125.0=++

A / Bq-d S / R

D / Cq 1-d

u(t)

ce)(disturban

(t)up

Controller

force) (residualy(t)

Plant )(p1 t

Two paths :PrimarySecondary (double differentiator)


Active Suspension

Primary path

Frequency Characteristics of the Identified ModelsSecondary path

0;16;14 === dnn BA


Adaptive disturbance rejection

Closedloop

Open loop

Disturbance : Chirp

Initialization of theadaptive controller

25 Hz47 Hz


Concluding Remarks

- Identification in closed loop establishes a bridge betweenrobustness and adaptation

- Iterative identification in closed loop and controller re-designis a two times scales adaptive control

- Robust linear design in high frequency is needed for adaptivecontrol schemes

- The multiple models approach to adaptive control improvessignificantly the adaptation transients

- Robust control gives hints for adaptive rejection of unknowndisturbances

- High speed simple adaptive direct control scheme for rejectionof unknown disturbances has been proposed and tested.


References

Morse A.S. (1995) Control using logic based switching in Trends inControl (A. Isidori, ed.) Springer Verlag, London, U.K.

Narendra K.S., Balakrishnan (1997) Adaptive control using multiple models IEEE Tr. on Aut. Control, AC-42, pp. 171-187

Karimi A., Landau I.D.(2000) Robust adaptive control of a flexible transmission system using multiple models , IEEE Tr. on Contr.Syst.Technology.March

Landau I.D., Lozano R., MSaad M., (1997) : Adaptive Control, Springer, London,U.K.

I.D.Landau I.D(1999) From robust control to adaptive control Control Eng.Practice,vol 7,no10, pp1113-1124

Landau I.D., (2001) : Identification in closed loop : a powerful design tool(better models, simple controllers) , Control Engineering Practice, vol. 9, no. 1, pp. 51- 65.

A. Constantinescu, I.D. Landau (2002) Adaptive narrow band disturbance rejection in active vibration control, Proceedings of IFAC World Congress 02, Barcelona, Spain

See also: http://www-lag.ensieg.inpg.fr/landau/bookIC


web site :

Commande des systmesconception,identification et mise en oeuvre

http://www-lag.ensieg.inpg.fr/landau/bookIC

Landau RobustAdaptive

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