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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