Transcript Robust Industrial Process Control

Necessary conditions for consistency
of noise-free, closed-loop frequencyresponse data with coprime factor
models
American Control Conference
June 29, 2000
Benoit Boulet, Ph.D., Eng.
Industrial Automation Lab
McGill Centre for Intelligent Machines
Department of Electrical and Computer Engineering
McGill University, Montreal
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Outline
1- Motivation: model validation for
robust H /  control
2- Coprime factor plant models
3- Consistency of closed-loop freq. resp.
(FR) data with coprime factor models
4- Example: Daisy LFSS testbed
5- Conclusion
Benoit Boulet, June 29, 2000
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1- Motivation: model validation for
robust H /  control
• Typical feedback control system
Uncertainty

Input Dist.
Output dist./
Meas. noise
Controller
reference
+
-
K
+
+
G
+
Plant
Benoit Boulet, June 29, 2000
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+
Meas. output
Motivation: model validation for robust H /  control
Robust control objective

z
y
P
K
Design stabilizing LTI K such
that CL system is stable
1
  H , r 
w

1
“robust stability”
u
where H  is the space of stable transfer
functions,
Q  : sup Q( j )
and
Benoit Boulet, June 29, 2000

r is a bound on the uncertainty:
( j)  r( j)
4
Motivation: model validation for robust H /  control
robust control
z
y

P
K
Condition for robust
stability as given by
small-gain theorem:
w
u
1
closed-loop stable   H , r 
iff r F L  P, K 
Benoit Boulet, June 29, 2000

5
1

1
Motivation: model validation for robust H /  control
robust control
z
y

P
K
w
u
Benoit Boulet, June 29, 2000
Condition for robust
stability
r F L  P, K 

1
provides the motivation
to make r( j) (the
uncertainty) as small as
possible through better
modeling
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Motivation: model validation for robust H /  control
robust control
z
y
Conclusion:

P
K
w
u
Benoit Boulet, June 29, 2000
Robust stability is easier to
achieve if the size of the
uncertainty is small.
Same conclusion for robust
performance (  -synthesis)
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Motivation: model validation for robust H /  control
…uncertainty modeling is key to
good control
• From first principles: Identify nominal
values of uncertain gains, time delays,
time constants, high freq. dynamics, etc.
and bounds on their perturbations
e.g., +, ||<b
• From experimental I/O data
Benoit Boulet, June 29, 2000
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Coprime factor plant models
2- Coprime factor plant models
• Perturbed left-coprime factorization
1
Gp : CM p N p J
where
M p : M   M , N p : N   N
M ,M  R Hnn , N , N  R Hnm , J R Hmm , C R Hp p
Benoit Boulet, June 29, 2000
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Coprime factor plant models
Aerospace example: Daisy
• Daisy is a large flexible space structure
emulator at Univ. of Toronto Institute for
Aerospace Studies (46th-order model)
1
G p  CM p N p J
M p  R H2323
N p  R H2323
Benoit Boulet, June 29, 2000
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Coprime factor plant models
define
• Factor perturbation
 : N
M  RH
• Uncertainty set

Dr :  RH : r 1


1
where r bounds the size of the factor uncertainty:
• Family of perturbed plants
P : G p :   Dr 
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( j ) r ( j ), 
Coprime factor plant models
block diagram of open-loop
perturbed LCF

P( s )
rI
C

M

J
N
where factor uncertainty is normalized
1
  r  such that,
 ( j )  1   ( j )  r ( j ), 
Benoit Boulet, June 29, 2000
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Coprime factor plant models
Block diagram of closed-loop
perturbed LCF

H (s)
rI
C

M

K2
N


K1
assumption: K1 , K 2 internally stabilize the plant
and provide sufficient damping for FR measurement
Benoit Boulet, June 29, 2000
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3- Consistency of closed-loop frequencyresponse data with coprime factor models
• Model/data consistency problem:
Given noise-free, (open-loop,closed-loop)
N
p m
frequency-response data  i i 1 
obtained at frequencies 1, , N , could
the data have been produced by at least
one plant model in P  G p :   Dr  ?
Benoit Boulet, June 29, 2000
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Consistency of closed-loop FR data with coprime factor models
open-loop model/data consistency
problem solved in:
• J. Chen, IEEE T-AC 42(6) June 1997 (general
solution for uncertainty in LFT form)
• B. Boulet and B.A. Francis, IEEE T-AC 43(12) Dec.
1998 (coprime factor models)
• R. Smith and J.C. Doyle, IEEE T-AC 37(7) Jul. 1992
(uncertainty in LFT form, optimization approach)
Benoit Boulet, June 29, 2000
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Consistency of closed-loop FR data with coprime factor models
closed-loop FR data case
consistency equation at frequency i
i
Hi
ri I
Mi
Ci


K 2i
Ni
K1i

0

i

FU  H i ,  i    i  0
where H i : H ( j i ),
Benoit Boulet, June 29, 2000
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

I
Consistency of closed-loop FR data with coprime factor models
Lemma 1


FU H i ,  i   i  0



rank  I  F L H i ,  i1  i   n
Lemma 2 (Schmidt-Mirsky Theorem)


inf i : rank  I  FL Hi , 
Benoit Boulet, June 29, 2000
    n  
1
i
i
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p

 FL H i , 


1
i
1
Consistency of closed-loop FR data with coprime factor models
Lemma 3 (consistency at i )
 i 
n ( n  p )
, i  1


such that FU H i ,  i   i  0

 p  F L  H i , 
Benoit Boulet, June 29, 2000
1
i

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1
1
Consistency of closed-loop FR data with coprime factor models
Theorem (consistency with CL FR data)
i i 1
N
consistent with LCF model only if
 p  F L H i , 
Proof
1
i

1
 1, i  1,
,N
(using boundary interpolation theorem)
 ( s )  Dr such that ( j i )  ri  i
only if
 p  F L  H i , 
Benoit Boulet, June 29, 2000
1
i

1
  i  1, i  1,
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,N
Consistency of closed-loop FR data with coprime factor models
This condition is not sufficient.
For sufficiency, the perturbation (s) Dr
would have to be shown to stabilize H ( s )
to account for the fact that the closed-loop
system was stable with the original
controller(s) K1 (s), K 2 (s)
We can’t just assume this a priori as it
would mean that the original controller(s) is
already robust!
Benoit Boulet, June 29, 2000
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Example: Daisy LFSS Testbed
Example: Daisy LFSS testbed
• Nominal factorization
G  CM 1 NJ
M , N , J  R H2323 , C 
2323
• Bound on factor uncertainty
r (s) 
0.001s  1.414
2.32 s  1
• Ga , one of the plants in family of perturbed plants
P was chosen to be the actual plant generating
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the 50 closed-loop FR data points  i i1
• 23 first-order decentralized SISO lead controllers
were used as the original controller K1
Benoit Boulet, June 29, 2000
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Example: Daisy LFSS Testbed
Example (continued)
consistency equation at frequency i
i
Hi
ri I
Mi
Ci


Ji
Ni
K1i

0

i

FU  H i ,  i    i  0
where H i : H ( j i ),
Benoit Boulet, June 29, 2000
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

I
Example: Daisy LFSS Testbed
Example (continued)
• Model/data consistency check:


 p FL Hi , i1 
Benoit Boulet, June 29, 2000
1
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5- Conclusion
• Necessary condition for consistency of noise-free FR
data with uncertain MIMO coprime factor plant
model involves the computation of  p at the N
measurement frequencies
• Bound on factor uncertainty r ( s) can be reshaped to
account for all FR measurements
• Sufficiency of the condition is difficult to obtain as
one would have to prove that the factor perturbation
(s) Dr , proven to exist by the boundary
interpolation theorem, also stabilizes the nominal
closed-loop system.
Benoit Boulet, June 29, 2000
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Thank you!
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