landau robustadaptive

46
I.D.Landau : From Robust Control to Adaptive Control 1 I.D. Landau Laboratoire d’Automatique de Grenoble, (INPG/CNRS), France Mayo 2003 Del Control Robusto al Control Adaptable

Upload: ali-zayn-al-abidin

Post on 15-Nov-2015

26 views

Category:

Documents


1 download

DESCRIPTION

Robust Adaptive Control

TRANSCRIPT

  • 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.Landau : From Robust Control to Adaptive Control2

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

    May 2003

    From Robust Control to Adaptive Control

  • I.D.Landau : From Robust Control to Adaptive Control3

    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

  • I.D.Landau : From Robust Control to Adaptive Control4

    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

  • I.D.Landau : From Robust Control to Adaptive Control5

    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

  • I.D.Landau : From Robust Control to Adaptive Control6

    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 ! !

  • I.D.Landau : From Robust Control to Adaptive Control7

    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

  • I.D.Landau : From Robust Control to Adaptive Control8

    What is the good model ?

    closed loop identified model open loop identified model

    fs = 20Hz 0% load

  • I.D.Landau : From Robust Control to Adaptive Control9

    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

  • I.D.Landau : From Robust Control to Adaptive Control10

    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

  • I.D.Landau : From Robust Control to Adaptive Control11

    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(

  • I.D.Landau : From Robust Control to Adaptive Control12

    -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

  • I.D.Landau : From Robust Control to Adaptive Control13

    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

    Critical frequency region for control

    Critical frequency region for control

  • I.D.Landau : From Robust Control to Adaptive Control14

    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:

  • I.D.Landau : From Robust Control to Adaptive Control15

    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

  • I.D.Landau : From Robust Control to Adaptive Control16

    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

  • I.D.Landau : From Robust Control to Adaptive Control17

    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

  • I.D.Landau : From Robust Control to Adaptive Control18

    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

  • I.D.Landau : From Robust Control to Adaptive Control19

    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

  • I.D.Landau : From Robust Control to Adaptive Control20

    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

  • I.D.Landau : From Robust Control to Adaptive Control21

    Adaptive Control of a Flexible Transmission

    Frequency characteristics for various load

    Rem.: the main vibration mode varies by 100%

  • I.D.Landau : From Robust Control to Adaptive Control22

    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 ?

  • I.D.Landau : From Robust Control to Adaptive Control23

    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

  • I.D.Landau : From Robust Control to Adaptive Control24

    Parameter Estimators for Adaptive Control

  • I.D.Landau : From Robust Control to Adaptive Control25

    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

  • I.D.Landau : From Robust Control to Adaptive Control26

    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

    +

  • I.D.Landau : From Robust Control to Adaptive Control27

    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)

  • I.D.Landau : From Robust Control to Adaptive Control28

    Adaptive Control with Multiple Models

  • I.D.Landau : From Robust Control to Adaptive Control29

    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

  • I.D.Landau : From Robust Control to Adaptive Control30

    Adaptive Control with Multiple Models

    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

  • I.D.Landau : From Robust Control to Adaptive Control31

    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)

  • I.D.Landau : From Robust Control to Adaptive Control32

    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)

  • I.D.Landau : From Robust Control to Adaptive Control33

    Adaptive Control with Multiple Models

    The plant models are not in the model set

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

    Load variations : 75% 25%

  • I.D.Landau : From Robust Control to Adaptive Control34

    Adaptive rejection of unknown disturbancesApplication to active suspension

  • I.D.Landau : From Robust Control to Adaptive Control35

    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

  • I.D.Landau : From Robust Control to Adaptive Control36

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

    :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

  • I.D.Landau : From Robust Control to Adaptive Control37

    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

  • I.D.Landau : From Robust Control to Adaptive Control38

    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

  • I.D.Landau : From Robust Control to Adaptive Control39

    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

  • I.D.Landau : From Robust Control to Adaptive Control40

    The Active Suspension

    Residualforce

    (acceleration)measurement

    Activesuspension

    Primary force(acceleration)(the shaker)

  • I.D.Landau : From Robust Control to Adaptive Control41

    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)

  • I.D.Landau : From Robust Control to Adaptive Control42

    Active Suspension

    Primary path

    Frequency Characteristics of the Identified ModelsSecondary path

    0;16;14 === dnn BA

  • I.D.Landau : From Robust Control to Adaptive Control43

    Adaptive disturbance rejection

    Closedloop

    Open loop

    Disturbance : Chirp

    Initialization of theadaptive controller

    25 Hz47 Hz

  • I.D.Landau : From Robust Control to Adaptive Control44

    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.

  • I.D.Landau : From Robust Control to Adaptive Control45

    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

  • I.D.Landau : From Robust Control to Adaptive Control46

    web site :

    Commande des systmesconception,identification et mise en oeuvre

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