1 Benoit Boulet, Ph.D., Eng. Industrial Automation Lab McGill Centre for Intelligent Machines Department of Electrical and Computer Engineering McGill University, Montreal Necessary conditions for Necessary conditions for consistency of noise-free, consistency of noise-free, closed-loop frequency- closed-loop frequency- response data with coprime response data with coprime factor models factor models American Control Conference June 29, 2000
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1 Benoit Boulet, Ph.D., Eng. Industrial Automation Lab McGill Centre for Intelligent Machines Department of Electrical and Computer Engineering McGill.
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Benoit Boulet, Ph.D., Eng.Industrial Automation LabMcGill Centre for Intelligent MachinesDepartment of Electrical and Computer EngineeringMcGill University, Montreal
Necessary conditions for consistency Necessary conditions for consistency of noise-free, closed-loop frequency-of noise-free, closed-loop frequency-response data with coprime factor response data with coprime factor modelsmodels
American Control ConferenceJune 29, 2000
Benoit Boulet, June 29, 2000 2
OutlineOutline
1- Motivation: model validation for robust 1- Motivation: model validation for robust control control
3- Consistency of closed-loop freq. resp. (FR) data with 3- Consistency of closed-loop freq. resp. (FR) data with coprime factor modelscoprime factor models
1- Motivation: model validation 1- Motivation: model validation for robust controlfor robust control
K GController
Plant
+
+
+
+
+
-
reference
Uncertainty
Output dist./Meas. noise
Meas. output
Input Dist.
• Typical feedback control systemTypical feedback control system
/ H
Benoit Boulet, June 29, 2000 4
Design stabilizing LTI K Design stabilizing LTI K such that CL system is such that CL system is stablestable
““robust stability”robust stability”where is the space of stable where is the space of stable transfer functions,transfer functions,
and is a bound on the uncertainty:and is a bound on the uncertainty:
Robust control objectiveRobust control objective
P
K
z
y u
w1, 1r
H
: sup ( )Q Q j
H
r( ) ( )j r j
Motivation: model validation for robust control/ H
Benoit Boulet, June 29, 2000 5
robust controlrobust control
P
K
z
y u
wCondition for robust Condition for robust stability as given by stability as given by small-gain theorem:small-gain theorem:
1closed-loop stable , 1r H
iff , 1Lr P KF
Motivation: model validation for robust control/ H
Benoit Boulet, June 29, 2000 6
robust controlrobust control
P
K
z
y u
w
Condition for robust Condition for robust stabilitystability
provides the provides the motivation to make motivation to make (the uncertainty) (the uncertainty) as small as possible as small as possible through better through better modelingmodeling
, 1Lr P KF
( )r j
Motivation: model validation for robust control/ H
Benoit Boulet, June 29, 2000 7
robust controlrobust control
P
K
z
y u
w
Conclusion:Conclusion:
Robust stability is Robust stability is easier to achieve if the easier to achieve if the size of the uncertainty size of the uncertainty is small.is small.
Same conclusion for Same conclusion for robust performance robust performance ( -( -synthesis)synthesis)
Motivation: model validation for robust control/ H
Benoit Boulet, June 29, 2000 8
……uncertainty modeling is key to uncertainty modeling is key to good controlgood control
• From first principles: Identify From first principles: Identify nominal values of uncertain gains, nominal values of uncertain gains, time delays, time constants, high time delays, time constants, high freq. dynamics, etc. and bounds on freq. dynamics, etc. and bounds on their perturbationstheir perturbationse.g., e.g., ++, |, ||<b|<b
• From experimental I/O dataFrom experimental I/O data
Motivation: model validation for robust control/ H
• Daisy is a large flexible space structure Daisy is a large flexible space structure emulator at Univ. of Toronto Institute for emulator at Univ. of Toronto Institute for Aerospace Studies (46th-order model)Aerospace Studies (46th-order model)
1
23 23
23 23
p p p
p
p
G CM N J
M
N
RH
RH
Coprime factor plant models
Benoit Boulet, June 29, 2000 11
definedefine
: N M RH
1
where bounds the size of the factor uncertainty: ( ) ( ),
: : 1r
r j r j
r
D RH
• Factor perturbationFactor perturbation
• Uncertainty setUncertainty set
• Family of perturbed plants Family of perturbed plants
: :p rG P D
Coprime factor plant models
Benoit Boulet, June 29, 2000 12
block diagram of open-loop block diagram of open-loop perturbed LCFperturbed LCF
1where factor uncertainty is normalized such that,
( ) 1 ( ) ( ),j j r j
r
JNM
rI
C
( )P s
Coprime factor plant models
Benoit Boulet, June 29, 2000 13
Block diagram of closed-loop Block diagram of closed-loop perturbed LCFperturbed LCF
NM
rI
C
1K
2K
( )H s
1 2assumption: , internally stabilize the plant
and provide sufficient damping for FR measurement
K K
Coprime factor plant models
Benoit Boulet, June 29, 2000 14
3- Consistency of closed-loop 3- Consistency of closed-loop frequency-response data with frequency-response data with coprime factor modelscoprime factor models
Given noise-free, (open-loop,closed-Given noise-free, (open-loop,closed-loop) frequency-response data loop) frequency-response data obtained at frequencies , obtained at frequencies , could the data have been produced could the data have been produced by at least one plant model in by at least one plant model in ? ?
1
N p mi i
1, , N
:p rG P D
Benoit Boulet, June 29, 2000 15
open-loopopen-loop model/data model/data consistency problem solved in:consistency problem solved in:
• J. Chen, IEEE T-AC 42(6) June 1997 (general J. Chen, IEEE T-AC 42(6) June 1997 (general solution for uncertainty in LFT form)solution for uncertainty in LFT form)
• B. Boulet and B.A. Francis, IEEE T-AC 43(12) B. Boulet and B.A. Francis, IEEE T-AC 43(12) Dec. 1998 (coprime factor models)Dec. 1998 (coprime factor models)
• R. Smith and J.C. Doyle, IEEE T-AC 37(7) Jul. R. Smith and J.C. Doyle, IEEE T-AC 37(7) Jul. 1992 (uncertainty in LFT form, optimization 1992 (uncertainty in LFT form, optimization approach) approach)
Consistency of closed-loop FR data with coprime factor models
Benoit Boulet, June 29, 2000 16
closed-loop FR data caseclosed-loop FR data case
i
iNiM
ir I
iC
i
I
0
consistency equation at frequency i
where : ( ),
, 0
i i
U i i i
H H j
H
F
iH
1iK
2iK
Consistency of closed-loop FR data with coprime factor models
Consistency of closed-loop FR data with coprime factor models
Benoit Boulet, June 29, 2000 20
This condition is This condition is not sufficient.not sufficient.
For sufficiency, the perturbation For sufficiency, the perturbation would have to be shown to stabilize would have to be shown to stabilize to account for the fact that the closed-to account for the fact that the closed-loop system was stable with the loop system was stable with the original controller(s)original controller(s)
We can’t just We can’t just assumeassume this a priori as it this a priori as it would mean that the original would mean that the original controller(s) is already robust!controller(s) is already robust!
1 2( ), ( )K s K s
( ) rs D( )H s
Consistency of closed-loop FR data with coprime factor models
• Bound on factor uncertaintyBound on factor uncertainty
• one of the plants in family of perturbed one of the plants in family of perturbed plants plants was chosen to be the actual plant generating was chosen to be the actual plant generating the 50 closed-loop FR data pointsthe 50 closed-loop FR data points
• 23 first-order decentralized SISO lead controllers 23 first-order decentralized SISO lead controllers were used as the original controllerwere used as the original controller
• Necessary condition for consistency of noise-Necessary condition for consistency of noise-free FR data with uncertain MIMO coprime factor free FR data with uncertain MIMO coprime factor plant model involves the computation of at plant model involves the computation of at the measurement frequenciesthe measurement frequencies
• Bound on factor uncertainty can be Bound on factor uncertainty can be reshaped to account for all FR measurementsreshaped to account for all FR measurements
• Sufficiency of the condition is difficult to obtain Sufficiency of the condition is difficult to obtain as one would have to prove that the factor as one would have to prove that the factor perturbationperturbation , proven to exist by the boundary , proven to exist by the boundary interpolation theorem, also stabilizes the interpolation theorem, also stabilizes the nominal closed-loop system.nominal closed-loop system.