I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction1

I.D. Landau*, A. Karimi°, A. Constantinescu*

* Laboratoire d ’Automatique de Grenoble, (INPG/CNRS), Fr.° Ecole Polytechnique Fédérale de Lausanne, Switzerland

April 2002

DIRECT CONTROLLER REDUCTION BY IDENTIFICATION IN CLOSED LOOP

( The Daphné Algorithms)

DIRECT CONTROLLER REDUCTION BY IDENTIFICATION IN CLOSED LOOP

( The Daphné Algorithms)

Application to Active Suspension Control

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction2

CONTROLLER REDUCTION. Why ?

- Complex Models- Robust Control Design

High Order Controllers

Example : The Flexible Transmission(Robust control benchmark, EJC no. 2/1995 and no.2-4/1999)

Model complexity : nA= 4; nB= 2; d =2)q(A

)q(Bq)q(G1

1d1

−

−−− =

Fixed controller part : Integrator

Pole placement design : nR= 4; nS= 4)q(S)q(R)q(K

1

11

−

−− =

Max : nR= 9; nS= 9 (Nordin) Min : nR= 7; nS= 7 (Langer)

Complexity of controllers achieving 100 % of specifications:

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction3

Approaches to Controller Reduction

Indirect Approach

PlantModel

ControllerDesign

Specs.

ControllerReduction

ReducedOrder

Controller

-Does not guarantee resulting controllers of desired order- Propagation of model errorsDirect Approach

- Approximation carried in the final step- Further controller reduction for “indirect approach”

PlantModel

ModelReduction

ControllerDesign

Specs.

Additional Specs.

ReducedOrder

Controller

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction4

Controller reduction should be done with the aim topreserve as much as possible the closed loop properties.

Reminder :Controller reduction without taking into account theclosed loop properties can be a disaster !

Some basic references :- Anderson &Liu : IEEE-TAC, August 1989- Anderson : IEEE Control Magazine, August 1993

Basic rule :

Controller Reduction

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction5

Identification in Closed Loop and Controller Reduction

- Identification in closed loop is an efficient tool for control oriented model order reduction

- Closed loop identification techniques can be used (with smallchanges) for direct estimation of reduced order controllers

Identification of reduced order modelsin closed loop

Identification of reduced order controllersin closed loop

Duality

- Possibility of using “real data” for controller reduction

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction6

Outline

- Introduction- Notations- Specific Objectives- Basic Schemes - The Daphné Algorithms- Properties of the algorithms- Properties of the estimated reduced order controllers- Validation of reduced order controllers- Experimental results ( Active Suspension Control)- Practical Hints- Conclusions

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction7

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 closed loop :Simulated C.L. using reduced order controller :

xyS,P),G,K(xyS,P),G,K(

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction8

K

G

G

rc u

c u

y

y

+

+

+−

−

−

K

minimize

Controller Reduction - Objectives

K G

G

rc u

c u y

y+

+

+−

−

−K

minimize

Input matching

Ouput matching

ypypKupupK

* S)KK(SminargSSminargK −=−=

yvypKypypKyryrK

* S)KK(SminargSSminargSSminargK −=−=−=

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction9

K

G

P.A.A.

G

rc u

c u

y

y

system loop-closed true

controllerorder reduced

CLε

+

+

+

−

−

−

v+

+

p+

+

K

Plant

Input Matching (CLIM)Use of simulated data Use of real data

Remarks on the use of real data:- new identified in closed loop can be used (can be better than the design model)

- will try to minimize the discrepancy between the two loops(will take into account )

G

K)GG( −

K

G

P.A.A.

G

rc u

c u

y

y

loop-closed simulated nominal

CLε

+

+

+

−

−

−

K

controllerorder reduced

Identification of Reduced Order Controllers

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction10

Identification of Reduced Order Controllers

Ouput Matching (CLOM)

K

G

P.A.A.G

r

c yu =

c u

controllerorder reduced

CLε

+

+

+

−

−

−

K

G

An alternative realization :

- CLIM algorithmbut on filtered data

- Same asymptotic steadystate properties as CLOM

KGrx u

x u

+

+

+−

−

−K

P.A.A.

CLε

G

controllerorder reduced

loop-closed simulated nominal

-The transfer fct. between r and u is Syp-The position of K andcan be interchanged

G

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction11

K

P.A.A.

G

r

u

u y

y

CLε

+

+

+

−

−

−

K

G K

G

P.A.A.

G

rc u

c u

y

y CLε

+

+

+

−

−

−

KGK

KG

KG

→

→

→

Duality

CLOE Daphné (CLIM)

Plant model identificationin closed loop

Reduced order controlleridentification in closed loop

Remark : one should take care of the structure of R and B

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction12

CLIM Algorithm

[ ][ ]

2)t(0;1)t(0;)t()t()t()t(F)t()1t(F

)1t()t()1t(F)t(ˆ)1t(ˆ)1t(u)1t(u)1t(

)dt(uB)t(yA)1t(r)1t(y)1t(r)1t(c

)1nt(c),..1t(c),1nt(u),..t(u)t(

)t(r),...t(r),t(s),...t(s)t(ˆ)t()t(ˆ)1t(c)q,t(R)t(u)q,t(S)1t(u

))q(Sq1)q(S(;)1t(c)q(R)t(u)q(S)1t(u

21T

21

11

0CL

00CL

**

RS

T

n0n1T

T11*0

1*1111*

Rs

<λ≤≤λ<ΦΦλ+λ=+

+εΦ++θ=+θ

+−+=+ε

−−++=+−+=+

+−++−−−=φ

=θ

φθ=++−=+

+=++−=+

−−

−−

−−−−−

:)t(ofChoice Φ

)t()t(:CLIM φ=Φ )t()q(P)q(A

)t(:CLIMF1

1

φ=Φ−−

−

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction13

CLOM Algorithm

Idem CLIM but r is added to the plant input

Forcing fixed parts in the reduced order controller:

knownisK,'KKK FF=

cKbyreplacedisc FSame algorithm but

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction14

Stability Analysis

SSRRnn;nn ==

0)1t(lim)1t(lim 0CL

tCL

t=+ε=+ε

∞→∞→

2)t(max;2

)z(H)z('H(*) 2t

11 <λ≤λλ

−= −−

is a strictly positive real transfer function where:

−=

CLIMFfor1CLIMforP/A

H

if:

)1t()1t(c)q(R)t(u)q(S)1t(u 11* +η+++−=+ −−

boundednorm)t(),t(r =ηAll signals are norm bounded under the passivity condition (*)

Hypotheses:

A)

SSRRnn;nn <<B)

A stabilizing controller with orders exists SR

nandn

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction15

Asymptotic Properties of the Estimated Controller

*θ - vector of the estimated controller parameters

Closed Loop Input MatchingSimulated Data

ωωφ−∫=ωωφ∫ −=θπ

π−θ

π

π−θd)(SKKSminargd)(SSminargˆ

r

2

yp

22

ypr

2

upup*

2upup SS − is minimized if r(t) is white noise-

- The frequency distribution of is weighted by theoutput sensitivity functions for the nominal and forthe reduced order controller

- The frequency distribution of can be tuned bythe choice of r(t)

2KK −

2KK −

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction16

Asymptotic Properties of the Estimated Controller

Closed Loop Input MatchingUse of Real Data

ω∫

ωφ+ωφ−=θ

π

π−θd)(S)(SSminargˆ

'v

2

ypr

2

upup*

)t(Kp)t(v)t('v −= : equivalent input noise

- The noise does not affect estimation of controller parameters

- When using real data, the closed loop system with reducedorder controller approximates the real closed loop system(instead of the nominal simulated system)

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction17

Asymptotic Properties of the Estimated Controller

Closed Loop Output MatchingSimulated Data

ωωφ−∫=ωωφ∫ −=θπ

π−θ

π

π−θd)(SKKSminargd)(SSminargˆ

r

2

yv

22

ypr

2

ypyp*

2ypyp SS − is minimized if r(t) is white noise-

- The frequency distribution of is weighted by

- The frequency distribution of can be tuned bythe choice of r(t)

2KK −

2KK −

yvyp SandS

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction18

Asymptotic Properties of the Estimated Controller

Closed Loop Output MatchingUse of Real Data

} } ωωφωφωωφ

ωφθ

π

πθ

π

πθ

dSSSdS

SKKSSGGS

vyprypypvyp

ryvypypup

)()(ˆminarg)(

)()ˆ(ˆ)ˆ(ˆminargˆ

'

22

'

2

2*

+

−=+

−−−=

∫

∫

−

−

- The noise does not affect estimation of controller parameters- Minimization of in the frequency regions where the

- Minimization of the gain of at the frequencies whereimportant additive modeling errors exist and the gain of theestimated model is low

2KK −

upS

highareSandS yvyp

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction19

Validation of Estimated Reduced Order Controllers

Simulated Data

-The reduced order controller should stabilize the nominal model

- The (reduced) sensitivity functions should be close to the nominalones in the critical regions for performance and robustness

- The generalized stability margin for the reduced order systemshould be close to the nominal one

Validation tools :− ν-gap between nominal and “reduced order” sensitivity fct.

( ) 1)S'S1)(SS()S'S1(S,S 21

21

<+−+=δ∞

−−

ν

(+ winding number condition. S’ denotes complex conjugate of S)

- Visual comparisonOne assumes: ! (as everybody in reduction business)GG =

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction20

Vinnicombe Stability Test

( ) ),(, 121 GKbGG nom≤νδ

=−

∞

otherwise0

stableis)G,K(if)G,K(T)G,K(b

1

−

=ypup

yvyr

SS

SS)G,K(T

Initial robust design:

We would like to have: ( ) ),ˆ(, 121 GKbGG ≤νδ

0;),ˆ(),( 11 ><− εεGKbGKbValidation test:

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction21

Validation of Estimated Reduced Order Controllers

Use of Real Data- Statistical tests (like in closed loop identification)- variance of residual closed loop error- cross-correlations

- Vinnicombe gap between :)u/( CLε

K

G

G

rc u

c u

y

y

system loop-closed true

controllerorder reduced

CLε

+

+

+

−

−

−

v+

+

p+

+

K

Plant

Identified T.F. ofthe true nominal

closed loop

Computed T.F. of thesimulated closed loop withreduced order controller

and

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction22

The Active Suspension

Residualforce

(acceleration)measurement

Activesuspension

Primary force(acceleration)(the shaker)

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction23

Experimental Results - Control of an Active Suspension

controller

residual acceleration

primary acceleration (disturbance)

1

23

4

machine

support

elastomere cone

inertia chamber

piston

mainchamber

hole

motor

actuator(pistonposition)

- controller: PC- sampling freq.: 800 Hz

Interesting frequencyrange for

vibration attenuation:0 - 200 Hz

++−

A / Bq-d ⋅S / R

D / Cq 1-d ⋅

u(t)Controller Plant

Disturbance(primary acc.)

y(t)

(residual acc.)

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction24

Active Suspension

- Open loop identified model(design model)

- Closed loop identified modelused for controller reduction(better C.L. validation)

Control objectives :- Minimize residual acc. aroundfirst vibration mode

- Distribute amplification of disturb.over high frequency region

Primary path Secondary pathFrequency Characteristics of the Identified Model

2d;11n;12n BA ===

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction25

The Nominal Controller

Important attenuation of Syp at the frequency of the firstvibration mode (32 Hz)

Design method: Pole placement with sensitivity shaping usingconvex optimization

Dominant poles : first vibration mode with ξ=0.8 (instead of 0.078)Opening of the loop at 0.5fs :

Nominal controller complexity :

Pole placement complexity :

)'RHR(;q1H R1

R =+= −

28n;27n SR ==

13n;12n SR ==

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction26

Direct Controller Reduction

CLIM algorithm/ simulated data

Controller2827

==

S

R

n

nn

K

2019

1

==

S

R

nn

K

1312

2

==

S

R

nn

K

109

3

==

S

R

nn

K

),( in KKνδ 0 0.1810 0.5049 0.5180),( i

upnup SSνδ 0 0.1487 0.4388 0.4503

),( iyp

nyp SSνδ 0 0.0928 0.1206 0.1233

)(kb 0.0800 0.0786 0.0685 0.0810))(),(( in KCLKCLνδ 0.1296 0.2461 0.5435 0.5522

C.L. error variance 0.0023 0.0083 0.0399 0.0398real time

experiments{

r(t) = PRBS, L= 4096 , clock = 0.5fS , N=10P.A.A.: variable forgetting factor

)'KHK(;q1H R1

R =+= −

Performances of the reduced order controllers are very closeto those of the nominal controller (see next slide)

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction27

Direct Controller Reduction

CLIM algorithm/ simulated dataSpectral density of the residual acceleration (performance)

Open loop

Closed loop (nominal controller):

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction28

Direct Controller Reduction

CLIM algorithm/ simulated data

ypS upS

K

Spectraldensity ofresidualacceleration(performance)

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction29

Direct Controller Reduction

CLIM algorithm/ use of real data

Controller

2827

==

S

R

n

nn

K

2019

1

==

S

Rnn

K

1312

2

==

S

Rnn

K

109

3

==

S

R

nn

K

),( in KKνδ 0 0.1500 0.4870 0.5216 ),( i

upnup SSνδ 0 0.1285 0.4197 0.4560

),( iyp

nyp SSνδ 0 0.1719 0.1639 0.1150

)(kb 0.0800 0.0722 0.0605 0.0823 ))(),(( in KCLKCLνδ 0.1296 0.1959 0.5230 0.5602

C.L. error variance 0.0023 0.0072 0.0359 0.0422

real timeexperiments{

Results are very close to thoseobtained with simulated data

Explanation : Quality of the model used for controller reduction

Spectraldensity ofresidualacceleration(performance)

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction30

Direct Controller Reduction

CLOM algorithm/ simulated data

Controller2827

==

S

R

n

nn

K

1312

2

==

S

R

nn

K

109

3

==

S

R

nn

K

),( in KKνδ 0 0.7287 0.7743),( i

upnup SSνδ 0 0.7144 0.7709

),( iyp

nyp SSνδ 0 0.0975 0.1007

)(kb 0.0800 0.0786 0.0796

(coherent with the theory)

CLIM and CLOMprovide reduced order controllers

with good performances

Spectraldensity ofresidualacceleration(performance)

( )( ) CLIMwithS,SSmaller

CLOMwithS,SSmalleriup

nup

iyp

nyp

ν

ν

δ−

δ−

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction31

Practical Hints

A) No access to real-time dataClassical situation for controller reduction techniquesGiven : nominal plant model, nominal controller

B) Access to the real system- Improve the quality of the model by identification in closed loop

- Use also real data for direct controller reduction- Do real time validation of the reduced order controllers

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction32

COHERENCE

What closed loop plant model identification scheme should beused when a criterion for controller reduction is given ?

Answer: Same criterion for identification in closed loop andcontroller reduction

Tracking and output disturbance rejection

CLOE with excitation added to controller input

Model identification orModel reduction

CLIMwith excitation added to plant input Controller reduction

2ypyp SS − is minimizedIn both schemes:

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction33

Concluding Remarks

- The Daphné algorithms (CLIM,CLOM) allow to directly estimate reduced order controllers

- The algorithms achieve a two norm minimization betweennominal and reduced order sensitivity functions

- They have the unique feature of using also real data (this allows to take in account to a certain extent the modeling error)

- Direct estimation of reduced order controllers can be interpretedas the dual of reduced order plant model identification in closed loop

- Successful use in practice - A MATLAB Toolbox is available (REDUC)- There is an interaction between closed loop identification and direct controller reduction (coherence)

Future work :- multivariable case

I.D. Landau, A. Karimi, A. Constantinescu: Direct controller reduction34

Anderson B.D.O., Liu Y., (1989) : « Controller reduction : concepts and approaches »,IEEE Trans. on Automatic Control, vol. 34, no. 8, pp. 802-812.

Anderson B.D.O., (1993) : « Controller reduction : moving from theory to practice »,IEEE Control Magazine, vol. 13, pp. 16-25.

Landau I.D., Karimi A., Constantinescu A., (2001) : « Direct controller reduction byidentification in closed loop », Automatica, vol. 37, no. 11, pp. 1689-1702.

Landau I.D., Karimi A., (2001a) : « A unified approach to closed-loop plant identificationand direct controller reduction », Proceedings European Control Conference 2001 (ECC01),Porto, Portugal, Septembre.

Landau I.D., (2002) Identification et Commande des Systèmes, 3rd edition,Hermes, Paris (June)

Vinnicombe G., (1993) : « Frequency domain uncertainty and the graph topology », IEEE Trans. on Automatic Control, vol. 38, no. 9, pp. 1371-1383.

References