part 5: direct and indirect adaptive controlioandore.landau/adaptivecontrol/tra… · digital...

I.D. Landau, A.Karimi: « A Course on Adaptive Control » - 41

Adaptive Control

Part 5: Direct and Indirect Adaptive Control

Adaptive Control – A Basic Scheme

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

AdjustableController

Plant+

SUPERVISION

Performancespecifications

ControllerDesign

Plant Model

Estimation

Adaptation loop

Outline

• Digital control systems•Tracking and regulation with independent objectives(known parameters)

• Adaptive tracking and regulation with independent objectives(direct adaptive control)

• Pole placement (known parameters)• Adaptive pole placement (indirect adaptive control)

Actuator Sensor

ADC DIGITAL

COMPUTER

e(k) y(k) y(t)

u(k) u(t)

Process

Digital Control System

The control law is implemented on a digital computer

ADC: analog to digital converterDAC: digital to analog converterZOH: zero order hold

- Sampling time depends on the system bandwidth

- Efficient use of computer resources

ZOHPLANT ADCCOMPUTER

DISCRETIZED PLANT

e(k) u(k) y(k)y(t)u(t)

Digital Control System

Computer(controller)

ZOHPLANT A/D

Discretized Plant

r(t) u(t) y(t)

The R-S-T Digital Controller

ABq d−

u(t) y(t)

Controller

PlantModel

)1()(1 −=− tytyq

Discrete time model – General form

)()()(11

idtubityatyBA n

ii −−+−−= ∑∑

==d –delay (integer multiple of the sampling period)

)(1)(1 1*

11 −

−−−∑ +==+ qAqqAqaAn

1* ...)( +−−− +++= AA

nn qaqaaqA;

)()( 1*

11 −

−−−∑ == qBqqBqbBn

ii ; 11

211* ...)( +−−− +++= B

nn qbqbbqB

)()()()( 11 tuqBqtyqA d −−− =)()()()( 11 tuqBdtyqA −− =+

)()()( 1 tuqHty −=)(

)()( 1

−−− =

qAqBqqH

)()()( 1

−−− =

zAzBqzH

- pulse transfer operator

(Predictive form)

11 −− → zq - transfer function

(*)(*)

First order systems with delay

ss TLLTd <<+= 0;.τ

Fractional delay

Continuous time model

1)()( −

−−−

−−−−

zazbbz

zazbzbzzH

ddDiscrete time model

ea TT s−

−=1 )1(1

eGb−

−= )1(2 −=−

Remark: For )1(5.01

212 >−⇒>⇒>

bbzerounstablebbTL s

o - zero x - pole

Tracking and regulation with independent objectives

It is a particular case of pole placement(the closed loop poles contain the plant zeros))

Allows to design a RST controller for:• stable or unstable systems• without restrictions upon the degrees of the polynomials A et B• without restriction upon the integer delay d of the plant model• discrete-time plant models with stable zeros!!!

It is a method which simplifies the plant zerosAllows exact achievement of imposed performances

Remarks:•Does not tolerate fractional delay > 0.5 TS (unstable zero)•High sampling frequency generates unstable discrete time zeros !

-1 -0.5 0 0.5 1-1

1Zero Admissible Zone

Real Axis

sf0/fs = 0.4

ζ = 0.1 ζ = 0.2

The model zeros should be stable and enough damped

Admissibility domain for the zeros of the discrete time model

(q )-1

ABq d−

)1(* ++ dty)(tu )(ty

Pq d )1( +−

)1( +− dq

ABq )1( +−

)()()( 111 −−− = qPqPqP FD

)()()(

=++Reference signal (tracking):

)()()1()()()( 1*11 tyqRdtyqTtuqS −−− −++=Controller:

T.F. of the closed loop without T:

)()()(

)()()()()()()( 11*

11*111

−−

−+−

−−+−−−

−+−− ==

qPqBqBq

qRqBqqSqAqBqqH

)()()()()()( 11*11*111 −−−−+−−− =+ qPqBqRqBqqSqA d

The following equation has to be solved :

S should be in the form: )()(...)( 11*110

1 −−−−− ′=+++= qSqBqsqssqS SS

After simplification by B*,(*) becomes:)()()(')( 11111 −−+−−− =+ qPqRqqSqA d

nP = deg P(q-1) = nA+d ; deg S'(q-1) = d ; deg R(q-1) = nA-1Unique solution if:1

101 ...)( −−

−−− ++= A

nn qrqrrqR d

d qsqsqS −−− ++= '...'1)(' 11

Regulation (computation of R(q-1) and S(q-1))

(**) is written as: Mx = p

1 0 a1 1 a2 a1 0: : 1ad ad-1 ... a1 1

ad+1 ad a1ad+2 ad+1 a2

. .0 0 ... 0 anA

1 . 0 .

. 00 0 1

nA + d + 1

d + 1 nA],...,,,,...,,1[ 1101 −′′= nd

T rrrssx ],...,,,...,,,1[ 121 dnnnT

AAApppppp ++=

Use of WinReg or predisol.sci(.m) for solving (**)

x = M-1p

Insertion of pre specified parts in R and S is possible

Regulation ( computation of R(q-1) and S(q-1))

Tracking (computation of T(q-1) )

Closed loop T.F.: r y

)()()()(

−−

+−−−

−+−− ==

qPqAqqTqB

Desired T.F.

It results : T(q-1) = P(q-1)

Controller equation:

)()()()1()()( 1

−− −++=

qStyqRdtyqPtu

[ ])()()1()()1()(1)( 11**1

1tyqRtuqSdtyqP

btu −−− −−−++= (s0 = b1)

)()()(

=++Reference signal (tracking) :

)()()1()()()( 1*11 tyqRdtyqPtuqS −−− −++=

Tracking and regulation with independent objectivesA time domain interpretation

Reformulation of the “design problem”:Find a controller which generate u(t) such that:

[ ] 0)1(*)1()1(0 =++−++=++ dtydtyPdtε

(in case of correct tuning)

y* (t+d+1) u(t) y(t)

- + P)1( +− dq

0)(0 ≡tε

ABq d *)1( +−

Pq d )1( +−

)1(*)(

++= −

)1(*)()( )1(1 ++= +−− dtyqtyqP d

0)1(*)()( )1(1 =++− +−− dtyqtyqP d

Tracking and regulation with independent objectivesSynthesis in the time domain – an example

)(*)( 11 −− = qBqS

)(*1)()(*1)(

)(*)(*)()()()(

−−−

−−−−

qAqqAqPqqP

qAqPqRqPqRqqA

1211 1)();1()()()1( −− +=−++−=+ qpqPtubtubtyaty

For d=0 (S’=1)

[ ][ ]

;)()1()1(*)(

0)1(*)()1()()()1(*)()1()1(*)1()1(

tyrtubtPytu

tPytyptubtubtyatPytyptytytyPt

−=−−−+

=+−+−++−==+−++=+−+=+°ε

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

)()()1()()1(*)(

tytututrbb

ttyrtubtubtyqPTT

=+−+=+−

y* (t+1) u(t) y(t)

- + P1−q

0)(0 ≡tε

ABq *1−

Pq 1−

Example:

Controller satisfies:

Solve for u(t)

Adaptive tracking and regulation with independent objectives

Three techniques:

•Model reference adaptive control (direct)•Plant model estimation + computation of the controller (indirect)•Re-parametrized plant model estimation (direct)

Model Reference Adaptive Control

[ ] 0)1(*)1()(lim)1(lim 10 =+−+=+ −

∞→∞→tytyqPt

)(ˆ)()(ˆ)1()(ˆ)1(*)(

tbtytrtutbtPytu −−−+

[ ] [ ])(),1(),()(;)(ˆ),(ˆ),(ˆ)(ˆ)()(ˆ)1(*)(

tytututtrtbtbt

tttyqPTT

)()1(*)()1()( 11 ttyqPtyqP Tφθ=+=+ −−

[ ] )()(ˆ)1(0 tttTφθθε −=+

[ ] )()1(ˆ)1( tttTφθθε +−=+

Use P.A.A.

However one should show in addition that is bounded (i.e. plantinput and output are bounded)

Objective:

Adjustable controller:

But for the correct values of controller parameters one has:

And therefore one has:

Define the a posteriori adaptation error:

Plant model estimation + computation of the controller (indirect)Step 1 : Plant model estimation

)()1()()()1( 211 ttubtubtyaty TPφθ=−++−=+

[ ] [ ])(),1(),()(;)(ˆ),(ˆ),(ˆ)(ˆ)()(ˆ)1()(ˆ)()(ˆ)()(ˆ)1(ˆ

tytututtatbtbt

tttutbtutbtytatyTT

−=−=

=−++−=+

[ ] )()(ˆ)1(ˆ)1()1( 00 tttytytT

PP φθθε −=+−+=+

[ ] )()1(ˆ)1(ˆ)1()1( tttytytT

PP φθθε +−=+−+=+

Use PAAStep 2 : Computation of the controller

[ ] [ ])(ˆ),(ˆ),(ˆ)(ˆ;)(),1(),()(

)()(ˆ)1(*)(

trtbtbttytutut

tttyqPTT

φθ)(ˆ tPθFrom

)(ˆ)(ˆ 110 taptr −=

Plant model (unknown):

Adjustable predictor:

a priori prediction error:

a posteriori prediction error:

Compute at each instant t :Adjustable controller:

In the general case d > 0 one will have to solve equation (**)

Re-parametrized plant model estimation (direct)

)()()1()()()()(

)()1()()()1(

121111

ttyptubtubtyaptyp

typtubtubtyatyTφθ+−=−++−+−=

±−++−=+

[ ] [ ])(),1(),()(;)(),(ˆ),(ˆ)(ˆ)()(ˆ)(

)1()(ˆ)()(ˆ)()(ˆ)()1(ˆ

tytututtrtbtbt

tutbtutbtytrtypty

−+++−=+

Plant model (unknown):

Re-parametrizedadjustable predictor:

[ ] )()(ˆ)1(ˆ)1()1( 00 tttytytTφθθε −=+−+=+

[ ] )()1(ˆ)1(ˆ)1()1( tttytytTφθθε +−=+−+=+

a priori prediction error:

a posteriori prediction error:

Use PAA

One estimates directly the parameters of the controller

[ ] 0)1(*)1()(lim)1(lim 10 =+−+=+ −

∞→∞→tytyqPt

One has:

Adaptive tracking and regulation with independent objectives

• Easy generalization for the case d > 0• Elegant and simple solution for adaptation (direct)• Unfortunately restricted use in practice because it requires that

the plant zeros remains always stable and well damped

Direct Adaptive Control – Simulations results

Tracking

1)( 1 =−qP )4.01()( 11 −− −= qqP

The choice of the poles for the closed loop (regulation)has a great influence upon adaptation transient behavior!

Parameters change + adaptation

Direct Adaptive Control – Simulations results

Regulation

1)( 1 =−qP )4.01()( 11 −− −= qqP

The choice of the poles for the closed loop (regulation)has a great influence upon adaption transient behavior!

Constant plant parameters

Parameters change + adaptation

Indirect Adaptive Control of non-necessarily zeros- stable plants

Pole placement

It is a method that does not simplify the plant model zeros

The pole placement allows to design a R-S-T controller for• stable or unstable systems• without restriction upon the degrees of A and B polynomials• without restrictions upon the plant model zeros (stable or unstable)• but A and B polynomials should not have common factors(controllable/observable model for design)

Structure

)()()( 1

−−− =

qAqBqqH

nn qaqaqA −−− +++= ...1)( 1

11 )(...)( 1*12

11 −−−−−− =+++= qBqqbqbqbqB B

Plant:

) ------------1

)---------q

R(q- 1

--------------------q

- dB(q

r(t) y(t)+

T( ) q - 1

)()()(

)()()()()()()( 1

−−−

−−−−−

−−−− =

qPqBqTq

qRqBqqSqAqBqTqqH

....1)()()()()( 22

11111 +++=+= −−−−−−−− qpqpqRqBqqSqAqP d

)()()(

)()()()()()()(

−−

−−−−−

−−− =

qPqSqA

qRqBqqSqAqSqAqS

Closed loop T.F. (r y) (reference tracking)

Defines the (desired )closed loop poles

Closed loop T.F. (p y) (disturbance rejection)

Output sensitivity function

Pole placement

Choice of desired closed loop poles (polynomial P)

)()()( 111 −−− = qPqPqP FD

Dominant poles Auxiliary poles

Specificationin continuous time(tM, M)

2nd order (ω0, ζ)discretization

eT )( 1−qPD

5.125.0 0 ≤≤ eTω17.0 ≤≤ ζ

Choice of PD(q-1)(dominant poles)

• Auxiliary poles are introduced for robustness purposes• They usually are selected to be faster than the dominant poles

Auxiliary poles

Regulation( computation of R(q-1) and S(q-1))

)()()()()( 11111 −−−−−− =+ qPqRqBqqSqA d

)(deg 1−= qAnA )(deg 1−= qBnBA and B do not have

common factors

(Bezout)

unique minimal solution for :1)(deg 1 −++≤= − dnnqPn BAP

1)(deg 1 −+== − dnqSn BS 1)(deg 1 −== −AR nqRn

)(*1...1)( 1111

1 −−−−− +=++= qSqqsqsqS S

nn qrqrrqR −−− ++= ...)( 1

Computation of R(q-1) and S(q-1)

Equation (*) is written as: Mx = p

],...,,,...,,1[ 01 RS nnT rrssx = ]0,...,0,,...,,...,,1[ 1 Pni

T pppp =

1 0 ... 0a1 1 .a2 0

anA a20 .

0 ... 0 anA

0 ... ... 0b' 1b' 2 b' 1. b' 2. .

b' nB .0 . . .0 0 0 b' nB

nA + nB + d

nB + d nA

nA + nB + d

x = M-1p

Use of WinReg or bezoutd.sci(.m) for solving (*)b'i = 0 pour i = 0, 1 ...d ; b'i = bi-d pour i > d

Structure of R(q-1) and S(q-1)

R and S may include pre-specified fixed parts (ex: integrator))()(')( 111 −−− = qHqRqR R )()(')( 111 −−− = qHqSqS S

HR, HS, - pre-specified polynomials

''...'')(' 1

nn qrqrrqR −−− ++= '

''...'1)(' 1

nn qsqsqS −−− ++=

•The pre specified filters HR and HS will allow to impose certain properties of the closed loop.•They can influence performance and/or robustness

)()(')()()(')()( 1111111 −−−−−−−− =+ qPqRqHqBqqSqHqA Rd

Fixed parts (HR , HS). Examples

Zero steady state error (Syp should be null at certain frequencies)

Step disturbance : Sinusoidal disturbance : sS TqqH ωαα cos2;1 21 −=++= −−

11 1)( −− −= qqHS

Signal blocking (Sup should be null at certain frequencies)

sR TqqH ωββ cos2;1 21 −=++= −−

2,1;)1( 1 =+=Sinusoidal signal:Blocking at 0.5fS:

− nqH nR

)()()()(

−−−− ′

qSqHqAqS S

)()()()()( 1

−−−− ′

−=qP

qRqHqAqS Rup

------------q

Ideal case

r (t) y* (t)

desiredtrajectory for y (t)

Tracking referencemodel (Hm)

2nd order (ω0, ζ)discretization

sT )( 1−qHm

5.125.0 0 ≤≤ sTω17.0 ≤≤ ζ

)()()( 1

−− =

qAqBqH

The ideal case can not be obtained (delay, plant zeros)Objective : to approach y*(t)

)()( 1

1)1(* tr

−+−

...)( 110

1 ++= −− qbbqB mmm

...1)( 22

1 +++= −−− qaqaqA mmm

Specificationin continuous time(tM, M)

)()()(

=++Build:

Choice of T(q-1) :• Imposing unit static gain between y* and y• Compensation of regulation dynamics P(q-1)

⎩⎨⎧

=0)1(1

0)1()1(/1Bif

BifBGT(q-1) = GP(q-1)

Particular case : P = Am⎪⎩

⎪⎨⎧

≠==−

0)1()1()1(

F.T. r y:)1(

)()()(

qAqBqqH

−+−− ⋅=

Pole placement. Tracking and regulation

y (t+d+1)*u(t) y(t)

-(d+1)

P(q -1 )q

-(d+1)

B*(q )-1

B*(q )

-(d+1) B m(q )

B*(q )

A m(q ) B(1)-1

)1(*)()()()()( 111 ++=+ −−− dtyqTtyqRtuqS

)()()()1()()( 1

−− −++=

qStyqRdtyqTtu

)1()()1()()()()()( *1*111 ++=++=+ −−−− dtyqTdtyqGPtyqRtuqS

)(1)( 1*11 −−− += qSqqS

)()()1()()1()()( 11**1 tyqRtuqSdtGyqPtu −−− −−−++=

)()()(

)(1)( 1*11 −−− += qAqqA mm

)()()()()1( 11** trqBdtyqAdty mm−− ++−=++

...)( 110

1 ++= −− qbbqB mmm ...1)( 22

1 +++= −−− qaqaqA mmm

Pole placement. Control law

Indirect adaptive controlIndirect adaptive control

Step I : Estimation of the plant model ARX identification (Recursive Least Squares)

At each sampling instant:

)ˆ,ˆ( BA

Step II: Computation of the controllerSolving Bezout equation (for S’ and R’)

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

PRHBqSHA Rd

S =+ − 'ˆ'ˆ

⎪⎩

⎪⎨⎧

≠===

0)1(ˆ1ˆ

0)1(ˆ)1(ˆ

)()(')( 111 −−− = qHqRqR R )()(')( 111 −−− = qHqSqS S

Compute:

SupervisionEstimation:

• Check if input is enough “persistently exciting”(if not, do not take in account the estimations)

• Check if and are numerically “sound” (condition number)(no close poles/zeros)

• If necessary, add external excitation (testing signal)

Control:

• Check if desired dominant closed loop poles are compatiblewith estimated plant poles

• Check robustness margins

Additional problem:• How to deal with neglected dynamics ?(filtering of the data, robustification of PAA)

Part 6: Robust Control Design for Adaptive Control

Adaptive Control

Notations

v(t) p(t)y

v(t)r u

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

−−− =

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

−− =

Sensitivity functions :

KG11)z(S 1

yp +=−

KG1K)z(S 1

up +−=−

KG1KG)z(S 1

yr +=−

KG1G)z(S 1

yv +=−

)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(

Templates for the Sensitivity Functions

Output SensitivityFunction

Input SensitivityFunction

Syp max= - MΔSyp dB

0.5fs0

delay marginnominal

perform.

( G ’ = G + δWa )

Sup dB

actuator effort

size of the tolerated additive uncertainty Wa

00.5f s

Critical frequency region for control

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

How to achieve this ?

Pole Placement :- Opening the loop in high frequencies (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

The flexible transmission

axismotor

d.c.motor

Positiontransducer

axisposition

Controlleru(t)

R-S-Tcontroller

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(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

Part 7: Parameter Estimators for Adaptive Control

Objective : to reduce the effect of the disturbancesupon the quality of the estimation

Adaptive Control

Classical Indirect Adaptive Control

disturb.u y

q- dBA

P.A.A.

Reference AdjustableController

Filter 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 threshold for « adaptation freezing » is problem dependent

Closed Loop Output Error Parameter Estimatorfor Adaptive Control

disturb.u y+

+ +εCL

P.A.A.

Reference AdjustableController

AdjustableController

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)

Part 8: Rejection of unknown narrow band disturbances.Application to active vibration control

Adaptive Regulation

The Active Suspension SystemThe Active Suspension System

controller

residual acceleration (force)

primary acceleration / force (disturbance)

machine

support

elastomere cone

inertia chamber

pistonmainchamber

actuator(pistonposition)

sTs m 25.1=++−

A / Bq-d ⋅S / R

D / Cq 1-d ⋅

ce)(disturban

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)

Elastomer

The Active Vibration Control with Inertial ActuatorThe Active Vibration Control with Inertial Actuator

sTs m 25.1=++

−A / Bq-d ⋅S / R

D / Cq 1-d ⋅

ce)(disturban

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

Same control approach but different technology

View of the active vibration control with inertial actuator

Residualforce

(acceleration)measurement

Primary force(acceleration)(the shaker)

Passive damper

Inertial actuator(not visible)

View of a flexible controlled structure using inertial actuators

Inertial actuator

Inertialactuators

Accelerometers

Active Suspension

Primary path

Frequency Characteristics of the Identified ModelsSecondary 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

Initialization of theadaptive controller

Disturbance : Chirp

25 Hz47 Hz

Direct adaptive control

Time Domain ResultsAdaptive Operation

Simultaneous controller initializationand disturbance application

Direct Adaptive Control

19.8 19.9 20 20.1 20.2 20.3 20.4 20.5 20.6-4

Time(s)

19.8 19.9 20 20.1 20.2 20.3 20.4 20.5 20.6

Time(s)

32 Hz 20 Hz

29.8 29.9 30 30.1 30.2 30.3 30.4

Time(s)O

29.8 29.9 30 30.1 30.2 30.3 30.4-2

Time(s)

20 Hz 32 Hz

Output

Initialization of theadaptive controller

Adaptation transient Adaptation transient

Step frequency changes

output

Rejection of unknown finite band 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 (indirect adaptive control)- Use the internal model principle- Use of the Youla parameterization (direct adaptive control)

A robust control design should be considered assuming that the model of the disturbance is known

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

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

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

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

pp DN /

FIRDisturbance

EnvironmentDisturbance

correlatedmeasurement

Disturbance +Plant model

AdaptationAlgorithm

y(t)Obj: minimization of E{y2}

Disadvantages:

Not justified for the rejection of narrow band disturbances

Notations

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

v(t)r(t)

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

−−− =

)()(')()('

)()()( 11

−−

−− ==

qHqSqHqR

qSqRqK

Output Sensitivity function :

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

)()()(')()( 1

−−−− =

zPzHzSzAzS S

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

HR and HS are pre-specified

Dirac circle;unit on the poles havemay

edisturbanc ticdeterminis : )()()(

⋅= −

δ(t) D

Deterministic framework

Stochastic framework

circleunit on the poles havemay )(0, sequence noise hiteGaussian w

edisturbanc stochastic : )()()(

⋅= −

teqDqN

Disturbance model

)/2cos(2;1 21sP ffqqD παα −=++= −−(Sinusoid: )

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

:Controller

−−−

qHqSqSqHqRqR

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

Closed loop system. NotationsClosed loop system. Notations

)()(qD

)(q)(q)S'(q)HA(qy(t) 1-

-1-1-1S

N p δ⋅⋅=

A / Bq-d ⋅S / Ru(t)

Controller Plant )(p t

pp DN / Dirac circle;unit on the poles

edisturbanc ticdeterminis : )()(

)()( 1

⋅= −

δ(t) D

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

)()()()(

)()()( :Output

−−−−−−

−−

⋅=⋅=

qRqBzqSqAqP

tpqStpqP

qSqAty

Two-steps methodology:

1. Estimation of the disturbance model,

2. Computation of the controller, imposing

)( 1−qDp

)(ˆ)( 11 −− = qDqH pS

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

Indirect adaptive regulationIndirect adaptive regulation

Environment

++A / Bq-d ⋅u(t)

Plant )(p t

pp DN /

Disturbance

Disturbancemodel

y(t)Controller

DisturbanceModel

EstimationDesignMethod

Specs. DisturbanceObserver

)(ˆ tp

Indirect adaptive controlIndirect adaptive control

Step II: Computation of the controller

'ˆ'ˆ

PBRqSDA

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)

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)

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

Model = Plantw=Ap1

nn qqqqqqQ −−− ++= ....)( 1

Internal model principle and QInternal model principle and Q--parametrizationparametrization))

The closed loop polesremain unchanged

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).

)()()()( 11 tyqRtuqS −− −=

)()( 11

qBqtyd

+= −

−−

ABq d /−

ABq d /− +-

pp DN /)(1 tp

)(1 tp

)(ty)(tu

Yula-Kucera parametrizationAn interpretation for the case A asympt. stable

)()( tD

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(z-1)=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 (find Q such that S contains the disturbance model):

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 »

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

Direct Adaptive Control (unknown Dp)

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-d-1-0

twBtNB

=⋅⋅=

⋅= −

Define (construct):

S(q-1)

).()()(

)()()()()( 1

10 twqQ

qPqBqtw

⋅−⋅= −−

−−

)()( tD

0SBQqMD dp =+ −

We need to express ε(t)as:

[ ] [ ] )()1(ˆ)(),1(ˆ)()1( 11 tttfqtQqQtT

Φ+−=+−=+ −− θθε

Using:

)(()()()(

)(*)1()()(

10 ++=+ −

−−

qDqMtwqP

qBqQtwqPqS P

(*) becomes:

)1()()(

)()],1(ˆ)([)1( 1

111 ++⋅⋅+−=+ −

−∗−−− tvtw

qPqBqqtQqQt

Vanishing term

Leads to a directadaptive control

)()()()1(

)()()(

+=+=+ −

−−−

−−

qNqAqMtw

qPqDqM

tv pp δ

)()()(

1+⋅=⋅ −

−−

−∗−tw

qPqBqtw

qPqBq dd

Details:

[ ] [ ]),..1(),()(;,..., 10 −=Φ= tftftqq TTθ

:))()()1()(( );()()1()()1( 1111

definetuqBtuqBtuqBqtyqAtw d −∗−−∗−− =+−+=+

)()1(ˆ)1()1( :error adaptation

)()(ˆ)1()1( :error adaptation

tttwtposterioriA

tttwtprioriAT

φθε

+−+=+

−+=+

Parameter adaptation algorithm: ⎪⎩

⎪⎨⎧

+=++++=+

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

ttttFttF Tφφλλεφθθ

Various choices possible for λ1 and λ2 which define the adaptation gain time profile

(For a stability proof see Automatica, 2005, no.4 pp. 563-574)

The Algorithmmeasured

).1()(

)(),(ˆ)1()()()1( 1

10 +−+=+° −

−−−

qBqqtQtwqPqSt

[ ] [ ] )1since 2(for ,)1()()( ;)(q)(q(t)ˆ

)()();1()()()1(

-n nntwtwttt

qBqtwtwqPqStw

PP DQDTT

==−==

⋅=+⋅=+ −

−∗−

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• Operation in “self –tuning” mode (constant unknown disturbance)

or “adaptive” mode (time varying unknown disturbance)

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--time time resultsresults –– Active Suspension (continuation)Active Suspension (continuation)

• 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 – direct adaptive method

Frequency domain results – indirect adaptive method

Self-tuning Mode Adaptive Mode

Direct Adaptive Control

•Direct adaptive control in adaptive mode operation gives betterresults than direct adaptive control in self-tuning mode

•Direct adaptive control leads to a much simpler implementationand better performance than indirect adaptive control

Active vibration control using an inertial actuatorReal-time results

Rejection of two simultaneous sinusoidal disturbances

Active vibration control using an inertial actuator

Primarypath

Secondarypath

0;12;10

Frequency Characteristics of the Identified Models

=== dnn BAComplexity of secondary path:

Frequency domain results – direct adaptive method

Rejection of two simultaneous sinusoidal disturbances

Time Domain Results – Direct adaptive control Adaptive Operation

Simultaneous rejection of two time varying sinusoidal disturbances

Output

Time Domain Results – Direct adaptive control Evolution of the Q parameters

Time Domain Results – Direct adaptive control Evolution of the control input

Comparison direct/indirect adaptive regulationTime domain results – Adaptive regime

Active vibration control using an inertial actuator

Indirect adaptive method Direct adaptive method

Direct adaptive control leads to a much simpler implementation and better performance than Indirect adaptive control

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:• active suspension system • active vibration control with an inertial actuator

part 5: direct and indirect adaptive controlioandore.landau/adaptivecontrol/tra… · digital...

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