I.D.Landau :Digital Control/System Identification1

Robust R-S-T Digital Controland

Open Loop System IdentificationA Brief Review

I.D. LandauLaboratoire d’Automatique de Grenoble(INPG/CNRS), France

(landau@lag.ensieg.inpg.fr)ADAPTECH, 4 Rue du Tour de l’Eau, St. Martin d’Hères(38), France

(info@adaptech.com)

IEEE Advanced Process Control

Workshop, Vancouver, April 29-May 1, 2002

I.D.Landau :Digital Control/System Identification2

Applications of R-S-T Controllers

Peugeot (PSA)

Sollac (Florange)Hot Dip Galvanizing

Double Twist Machine(Pourtier)

360° Flexible Arm (LAG)

I.D.Landau :Digital Control/System Identification3

Implementation of R-S-T Digital Controllers

PLC Leroy implementsR-S-T digital controllers andData acquisition modules

ALSPA 320ALSPA 320 implements R-S-T digital controllers andData acquisition modules

I.D.Landau :Digital Control/System Identification4

u yRef.

++

DESIGNMETHOD

MODEL(S)

Performancespecs.

PLANT

IDENTIFICATION

Robustnessspecs.

CONTROLLER

1) Identification of the dynamic model2) Performance and robustness specifications3) Compatible controller design method4) Controller implementation5) Real-time controller validation

(and on site re-tuning)6) Controller maintenance (same as 5)

Controller Design and Validation

I.D.Landau :Digital Control/System Identification5

Outline

Robust digital control-The R-S-T digital controller-Basic design-Robustness issues-An example

Open loop system identification-Data acquisition-Model complexity-Parameter estimation-Validation

I.D.Landau :Digital Control/System Identification6

Robust Digital Control

I.D.Landau :Digital Control/System Identification7

Computer(controller)

D/A+

ZOHPLANT A/D

CLOCK

Discretized Plant

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

The R-S-T Digital Controller

r(t)

m

m

AB

TS1

ABq d−

R

u(t) y(t)

Controller

PlantModel

+

-

)1()(1 −=− tytyq

I.D.Landau :Digital Control/System Identification8

r(t)

m

m

AB

TS1

ABq d−

R

u(t) y(t)

Controller

PlantModel

+

-

The R-S-T Digital Controller

Plant Model:)(

)(*)(

)()(1

11

1

11

−

−−−

−

−−− ==

qAqBq

qAqBqqG

dd

A

A

nn qaqaqA −−− +++= ...1)( 1

11 )(*...)( 111

11 −−−−− =++= qBqqbqbqB B

B

nn

R-S-T Controller: )()()1(*)()()( 111 tyqRdtyqTtuqS −−− −++=

Characteristic polynomial (closed loop poles):

)()()()()( 11111 −−−−−− += qRqBqqSqAqP d

I.D.Landau :Digital Control/System Identification9

rABy mm )/(* =

+

-

R

1 q-d

B

AST

A

B m

m

r(t)

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

q

-(d+1)

P(q -1 )

q

-(d+1)

B*(q )-1

B*(q )

-1

B(1)

q

-(d+1) B m(q )

B*(q )

-1-1

A m(q ) B(1)-1

Pole Placement with R-S-T Controller

';' SHSRHR SR ==Controller : :, SR HH fixed parts

Regulation: R’ and S’ solutions of:

dominantpoles

auxiliarypolesTracking :

FDRd

S PPPRBHqSAH ==+ − ''

)1(/ BPT =

Reference trajectory: y*computer file

I.D.Landau :Digital Control/System Identification10

Connections with other Control Strategies

- Digital PID : 11;2 −−=== qHnn SSR

-Tracking and regulation with independent objectives(MRC):

FD PPBP *= (Hyp.: B* has stable damped zeros)

- Minimum variance tracking and regulation (MVC):

CBP *=noise model

(Hyp.: B* has stable damped zeros)

- Internal Model Control (IMC):

FAPP = (Hyp.: A has stable damped poles)

I.D.Landau :Digital Control/System Identification11

+

-

R

1 q-d

BAS

Tr(t) u(t) y(t)

PlantModel

p(t)

b(t)

+

+

+ +

(disturbance)

(measurement noise)

The Sensitivity Functions

Syp(q-1) = ASAS + q-dBR

= ASP

Output sensitivity function (p -> y)

Sup(q-1) = - ARAS + q-dBR

= - ARP

Input sensitivity function (p -> u)

Syb(q-1) = - q-dBRAS + q-dBR

= - q-dBRP

Noise sensitivity function (b -> y)

Syp - Syb = 1

I.D.Landau :Digital Control/System Identification12

-1

∆Φ1

∆1

∆Μ

1

crossoverfrequency

Re H

Im H

G

|HOL|=1ω CR

1

ω CR2

ω CR3

Robustness Margins

z = ejω

> 29°∆M 0.5 ∆G 2 ; ∆Φ≥ ≥⇒

∆M 0.5 (-6dB), ∆τ > Ts≥

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

-1( ) ∞( )

∆τ = mini ∆Φi

ωCRi

Modulus Margin:

Delay Margin:

Typical values:

The inverse is not necessarily true!

I.D.Landau :Digital Control/System Identification13

Im G

Re Gω = π

G (e )-j ω

uncertaintydisk

δ W

ω = 0

Robust Stability

Family of plant models:

),,(' xyWGFG δ∈

G – nominal model; 1)( 1 ≤∞

−zδ

)( 1−zWxy- size of uncertainty

Robust stability condition:a related sensitivity

functiona type of uncertainty

1<∞xyxyWS

⇓1−

< xyxy WSdefines the size of thetolerated uncertainty

defines an upper templatefor the modulus of the

sensitivity function

There also lower templates (because of the relationship between various sensitivity fct.)

I.D.Landau :Digital Control/System Identification14

Sypmax= - M∆Syp dB

0.5fs0

delay marginnominal

perform.

( G ’ = G + δWa )

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

I.D.Landau :Digital Control/System Identification15

Robust Controller Design

Pole placement with sensitivity functions shaping

FD PPP =

RHRR '=

SHSS '=

Nominal performance: SRD HandHofpartandP

Allow to shape the sensitivity functions

-Iterative Choosing and using band stop filters FjSjFiRi PHPH /,/FP

Several approaches to design :

-Convex optimization(see Langer, Landau, Automatica, June99, Optreg (Adaptech) )

I.D.Landau :Digital Control/System Identification16

MIRROR

DETECTOR

RIGID FRAMES

LIGHTSOURCE

POT.

ENCODER

SERVO.

LOCALPOSITION

COMPUTERTACH.

ALUMINIUM

360° Flexible Arm

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Frequency characteristics Poles-Zeros

360° Flexible Arm

(Identified Model)

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1 2 3 4 5 6 7 8 9-30

-25

-20

-15

-10

-5

0

5

10

15

20

25

Frequence [Hz]

Mod

ule

[dB

]

Syp - Sensibilité perturbation-sortie

A

B

D C

gabarit

1 2 3 4 5 6 7 8 9-30

-20

-10

0

10

20

30

40

50

60

Frequence [Hz]

Mod

ule

[dB

]

Sup - Sensibilité perturbation-entrée

A

D

B

C

gabarit

A- without auxiliary polesB- with auxiliary polesC- with stop band filterD- with stop band filter

11 / FS PH22 / FR PH

Shaping the Sensitivity Functions

Output Sensitivity Function - Syp Input Sensitivity Function - Sup

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Open Loop System Identification

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I/0 Data Acquisition under anExperimental Protocol

Model Complexity Estimation(or Selection)

Choice of the Noise Model

Parameter Estimation

Model Validation

YesNo ControlDesign

System Identification Methodology

I.D.Landau :Digital Control/System Identification21

I/O Data Acqusition

Signal : a P.R.B.S sequenceMagnitude : few % of the input operating pointClock frequency : Length :Largest pulse :

3,2,1;)/1( == pfpf sclock )( frequencysamplingf s =sss

N fTcellsofnumberNpT /1,;)12( 1 ==−−

sNpT

Length : < allowed duration of the experimentLargest pulse : (rise time)Rt≥

P.R.B.S.

NpTs

tr

I.D.Landau :Digital Control/System Identification22

poweramplifier

filter

u(t)

y(t)

inertia

d.c. motor .

tachogenerator

M

TG

An I/O File

I.D.Landau :Digital Control/System Identification23

Complexity Estimation from I/O Data

Objective :To get a good estimation of the model complexitydirectly from noisy data

),,( dnn BA

complexitypenalty term

error term(should be unbiased)

S(n,N)

0 n

CJ(n)

J (n )

minimum

nopt

[ ]),ˆ()ˆ(minminˆˆˆ

NnSnJCJnnnopt +==),max( dnnn BA +=

To get a good order estimation, J should tend to the value fornoisy free data when (use of instrumental variables) ∞→N

I.D.Landau :Digital Control/System Identification24

u(t) y(t)Plant ADC

Discretized plant

+

-

Parameteradaptationalgorithm

y(t)

ε (t)

Estimated modelparameters

)(ˆ tθ

Adjustablediscrete-time

model

DAC +ZOH

Parameter Estimation

I.D.Landau :Digital Control/System Identification25

+

+u(t) y(t)

1A

q-d B

A

e(t)

)()()()()(:1 11 tetuqBqtyqAS d += −−−

(Recursive) Least Squares

+

+u(t) y(t)q-d B

A

w(t)

)()()()()()(:2 111 twqAtuqBqtyqAS d −−−− +=

Ouput Error(O.E.)Instrumental Variable…

+

+u(t) y(t)

1

q-d B

A

e(t)

CA

[ ] )()(/1)()()()(:4 111 teqCtuqBqtyqAS d −−−− +=

Generalized Least Squares

+

+u(t) y(t)A

q-d B

A

e(t)

C

)()()()()()(:3 111 teqCtuqBqtyqAS d −−−− +=

Extended Least SquaresO.E. with Extended Prediction Model(Recursive) Maximum Likelihood

«Plant + Noise » Models

~ 33%

~ 64%

~ 1%

~ 2%

I.D.Landau :Digital Control/System Identification26

Parameter Estimation Methods

)()()(*)()(*)1( 11 tdtuqBtyqAty Tψθ=−+−=+ −−

vectortmeasuremenvectorparameter −− ψθ ;

Plant Model

)()(ˆ)1(ˆ ttty T φθ=+°Estimated model

vectornobservatiovectorparameterestimated −− φθ ;ˆ

)1(ˆ)1()()(ˆ)1()1( 00 +−+=Φ−+=+ tytytttyt TθεPrediction error

Parameter adaptation algorithm

2)(0;1)(0

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

)1()()1()(ˆ)1(ˆ

21

21

11

0

<≤≤<ΦΦ+=+

+Φ++=+−−

tt

ttttFttF

tttFttT

λλλλ

εθθ

[ ])()( tft φ=Φ

I.D.Landau :Digital Control/System Identification27

Parameter Estimation Methods

I- Based on the asymptotic whitening of the prediction error(Recursive Least Squares, Extended Least Squares, RecursiveMax. Likelihood, O.E. with Extended Prediciton Model )

II- Based on the asymptotic decorrelation between the predictionerror and the observation vector(Output Error, Instrumental Variable)

I.D.Landau :Digital Control/System Identification28

1;17.2)( ≥≤ iN

iRN

normalizedcrosscorelation

number of data

1N2.17

N

97%

136.0)(256 ≤→= iRNN

15.0)( ≤iRNpratical value :

+

-

u

y

y

εPlant

Model

q-1

q-1

ARMAX (ARARX) predictor

WhitenessTest

+

-

u

y

y

ε

Plant

Model

q-1

Output Error Predictor

DecorrelationTest

Validation of Identified Models

Statistical Validation

I.D.Landau :Digital Control/System Identification29

Software Tools for Implementing the Methodology

System Identification-Winpim (Adaptech) identification in open loop and closed loop operation -CLID (Adaptech)identification in closed loop (Matlab Toolbox)

Controller Design-Winreg (Adaptech)design and optimisation of R-S-T digital controllers-Optreg (Adaptech)automated design of robust digital controllers (under Matlab)

Real-time implementation-Wintrac (Adaptech): cascade digital control

I.D.Landau :Digital Control/System Identification30

« Personal » References

Landau.I.D.,(1990) System Identification and Control Design, Prentice Hall, N.J.,USA

Landau I.D., (1995) : « Robust digital control of systems with time delay (the Smith predictor revisited) », Int. J. of Control, vol. 62, pp. 325-347.

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

Landau I.D., (1993) Identification et Commande des Systèmes, 2nd edition,Hermes, Paris (June)

Landau I.D., (2002) Commande des Systèmes – Conception, identificationet mise en œuvre, Hermes, Paris (June)