parameterized model order reduction via quasi-convex optimization kin cheong sou with luca daniel...
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Parameterized Model Order Reduction via Quasi-Convex Optimization
Kin Cheong Sou with Luca Daniel and Alexandre Megretski
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Systems on Chip or PackageInterconnect & Substrate
Courtesy of Harris semiconductor
RF Inductors
MEMresonators
210/22/2010
DSP
Digital
LNA
LO
Analog RF
ADC
ADC
Mixed SignalI
Q
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From 3D Geometry to Circuit Model
Fig. thanks to Coventor
•Need accurate mathematical models of components•Describe components using Maxwell equations, Navier-
Stokes equations, etc
310/22/2010
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From 3D Geometry to Circuit Model
dt
dEH
dt
dHE
4 2 2
4 2 20
( )w
elec a
u u uEI S F p p dy
x x t
3 ( )((1 6 ) ) 12
puK u p p
t
dt
dEH
dt
dHE
inBvvGdt
dvvC )()(
Z(f)Z(f) Z(f)Z(f) Z(f)Z(f) Z(f)Z(f)
•Model generated by available field solver•Field solver models usually high order
410/22/2010
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RF Inductor Model Reduction
•Spiral radio frequency (RF) inductor•Impedance•State space model has 1576 states•Reduced model has 8 states
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
x 1010
0
0.5
1
1.5
2
2.5
3x 10
4
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
x 1010
-1.5
-1
-0.5
0
0.5
1
1.5x 10
-7
R L
f f
x full 1576 states- reduced 8 states
x full 1576 states- reduced 8 states
2 Z f R f j f L f
510/22/2010
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•Parameter dependent RF inductor•Two design parameters:
- Wire width w- Wire separation d
d
w
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 1010
0
2000
4000
6000
8000
10000
12000
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 1010
-5
-4
-3
-2
-1
0
1
2
3
4
5x 10
-8
f f
R L
d = 1umd = 3umd = 5um
d = 1umd = 3umd = 5um
610/22/2010
RF Inductor Parameter Dependency
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Parameterized Reduced Modeling
Parameterizedreduced model
Gr(d,w)
•One reduced model with explicit dependency on parameters
•Fast generation of reduced model for all parameter values
710/22/2010
d
w
DSPLNA
LO
ADC
ADC
I
Q
,d w
reducedmodel
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Parameterized Reduced Model Example
•Parameter dependent complex system
•Parameterized reduced order model
•Coefficients depend explicitly on d•Low order, inexpensive to simulate
2, ( , )r
sG s G s
s
dd d
d
99
99
0.5 2,
0.499
s sG s
d dd
ds s d
810/22/2010
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Continuous/Discrete-time Setups
10/22/2010 9
Continuous-time Discrete-time
left-half plane & imaginary axis unit disk & unit circle
& G s G j & jG z G e
1
1
zs
z
s
zs
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•Parameterized moment matching methods- References:
• [Grimme et al. AML 99]• [Daniel et al. TCAD 04]• [Pileggi et al. ICCAD 05]• [Bai et al. ICCAD 07]
- Reduced model order increases with number of parameters rapidly- Require knowledge of state space model
•Rational transfer function fitting methods- Does not require state space model- Reduced model order does not increase with number of parameters- More expensive than moment matching in general
1010/22/2010
Parameterized Model Reduction Methods
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Moment Matching Method
10/22/2010 11
1G z C zI A B D
1
r V U VG z C z A B DUI
Projection withUV = I
Full model Reduced model
( ) ( )k kn n
k r k
d dG z G z
dz dz
with the moment matching properties
10-1
100
101
102
103
104
105
10-10
10-8
10-6
10-4
10-2
100
102
Ma
gn
itud
e (
ab
s)
Bode Diagram
Frequency (rad/sec)
8th order full 4th order MM
moments matched
user specified
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Rational Transfer Function Fitting
10/22/2010 12
r
p zG z
q z
input output
•Idea from system ID – reduced model matching I/O data
? ?p z q z
•Data in time domain or frequency domain•Data from state space model or experiment measurement
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Explanations in Two Steps
10/22/2010 13
•Will present a rational transfer function fitting method
•First describe basic non-parameterized reduction
•Then extend basic method to parameterized setup
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Non-Parameterized Model Order Reduction
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Non-parameterized Problem Statement
•Given transfer function G(z)
•Find parameterized reduced model of order r
0
0
rr
r rr
p z p z pG z
q z q z q
p zG z
q z
subject to stableq z
,minimize
p q
1510/22/2010
•Reduced model found as the solution
dec. vars.
roots inside unit circle
H norm error
•Can obtain state space realization from p(z) and q(z)
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Difficulty with H Norm Reduction
10/22/2010 16
p zG z
q z
j j j jG e q e p e q e
•Difficulty #2: abs. value on the “wrong” side
iff
•Difficulty #1: stability constraint not convex if r >2
331 5q z z 33
2 5q z z
31 2 27
2 2 25
q qz z but
branchingsolutions
convex combo. of stable poly.not necessarily stable
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Idea From Optimal Hankel Reduction
10/22/2010 17
minrG
rG G
stablerGs.t.
order rG r
,min
rG FrG FG
stable, anti-stabler FGs.t.
order rG r
minQ
G Q
( )Q H rs.t.
,
,1
Obtain s.t.r Han
n
r Han ii r
G
G G G
anti-stablerelaxation
redefinedec. var.Solve AAK problem
efficiently (Glover)
suboptimalsolution
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Anti-Stable Relaxation in Rational Fit
10/22/2010 18
•Similar to Hankel reduction, add anti-stable term
1
1
f zp zG z
q z q z
subject to stable, degq z f r
,minimize
p q
added DOF
•In Hankel MR, entire anti-stable term is decision variable•Here, only numerator f is decision variable
flip polesof q(z)
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Combine Stable and Anti-stable Terms
10/22/2010 19
•Combine stable and anti-stable terms in reduced model
1
1
f zp z b z jc z
q z a zq z
10 1
10 1
111
r rr
r rr
r rrj
a z a a z z a z z
b z b b z z b z z
c z c z z c z z
•New decision variables are trigonometric polynomials
0 1
0 1
1
cos
cos
sin
j
j
j
a e a a
b e b b
c e c
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Stability and Positivity
10/22/2010 20
•Can show
stableq z 0, ja e
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
•Overcome Difficulty #1, trigonometric positivity convex constraint 1 2e.g. 1 cos cos 2 0a a
a1
a 2
•Overcome Difficulty #2, the trouble making abs. value is gone!
j
j
j
j
j jG e a e b e j
a e
c e
a e
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Quasi-Convex Relaxation
10/22/2010 21
•Quasi-convex relaxation
b z jc zG z
a z
subject to 0, for 1a z z
, ,minimize
a b c
•Original optimal H norm model reduction problem
p zG z
q z
subject to stableq z
,minimize
p q
Quasi-convex problem,easy to solve
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From Relaxation Back to H Reduction
10/22/2010 22
•Obtain (a,b,c) by solving quasi-convex relaxation
•Spectral factorize a to obtain stable denominator q*
* * 1z K za q q z for some K
•With q* found, search for numerator p* by solving
*
* arg minp
p zG z
zp
q
convex problem
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Quality of Suboptimal Reduced Model
10/22/2010 23
•Minimizing upper bound of Hankel norm error
1
1
H
f zb z jc z p z p zG z G z G z
a z q z q zq z
•H norm error upper bound
*
* , ,1 min
a b c
p z b z jc zG z r G z
q z a z
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Back to Big Picture – Model Reduction
10/22/2010 24
minrG
rG G
stablerGs.t.
order rG r
, ,mina b c
b jcG
a
0, 1a z z s.t.
optimal a(z), b(z), c(z) suboptimal p(z), q(z)
discussed
discussed
discussed
How to solve it?
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Quasi-Convex Optimization
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Quasi-Convex Optimization
10/22/2010 26
J(x)
x
All sub-level setsare convex sets
•Quasi-convex function is “almost convex”
Local (also global) minima Local (but not global) minima
Function not necessarily convex
•[Outer loop] Bisection search for objective value•[Inner loop] Convex feasibility problem (e.g. LP, SDP)
•Convex problem algorithms: 1) interior-point method 2) cutting plane method
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Cutting Plane Method
optimal point
covering set
iterate 1
iterate 2
2710/22/2010
•Optimization problem data described by oracle
•What is the oracle in our model reduction problem?
Oraclecall oracle
retu
rn c
ut
kept
removed
call oracle
return cut
keptremoved
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Model Reduction Oracles
10/22/2010 28
Oracle #1 (objective value):
b z jc zG z
a z
j j j j jG e a e b e c e a e
Oracle #2 (positive denominator):
Discretize frequency finite number of linear inequalities, “easy”
for any fixed
0, ja e
•Given candidate a(z), b(z), c(z), check two conditions
Cannot discretize frequency!
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Positivity Check
10/22/2010 29
0 0.5 1 1.5 2 2.5 3 3.5-2
-1
0
1
2
3
4
5
6
t
stationary pointsr = 8 case
ja e
•Check only finite number of stationary points
•Much harder to check in the parameterized case
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Back to Big Picture – Model Reduction
10/22/2010 30
minrG
rG G
stablerGs.t.
order rG r
, ,mina b c
b jcG
a
0, 1a z z s.t.
optimal a(z), b(z), c(z) suboptimal p(z), q(z)
discussed
discussed
discussed
Solved withcutting planemethod
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Parameterized Model Order Reduction
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Problem Statement
•Given parameter dependent transfer function G(z,d)
•Find parameterized reduced model of order r
0
0
,,
,
rr
r rr
p z p zd d dd
d
pG z
q z q z qd d
max , ,d
rd dG z G z
subject to ,q z d
,minimize
p q
stable for all d
3210/22/2010
•Reduced model found as the solution design parameter
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Parameterized Reduced Model Example
•Parameter dependent complex system
•Parameterized reduced order model
•Coefficients depend explicitly on d•Low order, inexpensive to simulate
2, ( , )r
zG z G z
z
dd d
d
99
99
0.5 2,
0.499
z zG z
d dd
dz z d
3310/22/2010
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10 1
10 1
111
,
,
,
r rr
r rr
r rrj
d d d d
d d
a z a a z z a z z
b z b b z zd db z z
c z c z z c z zd d d
1
0 1
, 2 sin 2 sin
, 2 cos 2 cos
jr
jr
d d d
d d d
c e c c r
a de a a a r
jz e
3410/22/2010
Parameterized Decision Variables
•Decision variables = parameterized trig. poly.
•When evaluated on unit circle, i.e.
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Parameterized Quasi-Convex Relaxation
10/22/2010 35
, ,,
,
b z jd dd
c zG z
a dz
subject to , 0, for 1 and a z d z d
, ,minimize
a b c
•Parameterized quasi-convex relaxation
•Solution technique similar to non-parameterized case•Some extension requires more care, e.g.
check , 0, for 1 and a z zd d
Parameterized positivity check is hard!
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Parameterized Positivity Check
10/22/2010 36
0 1 2, cos cos 2ja e d a d a d a d denominator
1 2
poly of
cos
ia d d
d d d
a simple parameterdependency
denominator = multivariate trigonometric polynomial
cos 3 cos 5cos 2 cos e.g.
•Positivity check of multivariable trig. poly. is hard•Another variant is multivariable ordinary polynomial our focus
…
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Positivity Check of Multivariate Polynomials
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Checking Polynomial Positivity – Special Cases
10/22/2010 38
•Univariate case simple, check the roots of derivative
4 3 23 2 4 0 ?x x x Is it true for all x,
•Multivariate quadratic form is easy but important
1 1
1 2 3 1 2 1 3 2 2
3
2 2 2
3
2 3 2
2 3 6 4 3 1 0 0 ?
2 0 3
Tx x
x x x x x x x x x
x x
polynomial nonnegative matrix positive semidefinite
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Checking Polynomial Positivity – General Case
10/22/2010 39
•Positivity check of general multivariate polynomial is hard
2 2 31 24
2 24
1 12 5 2 0 ?x x x x x x Question: [from Parrilo & Lall]
2 21 11 12 13 1
4 4 2 2 3 2 21 2 1 2 1 2 2 12 22 23 2
1 2 13 23 33 1 2
2 5 2
Tx q q q x
x x x x x x x q q q x
x x q q q x x
= Q (Gram matrix)Monomials of relevant degrees
•What if we still write out “quadratic form”?
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Checking Polynomial Positivity – General Case
10/22/2010 40
•To find Q, equate coefficients of all monomials
132 2q31 2 :x x2 21 2 :x x 12 331 2q q
230 2q31 2 :x x
112 q41 :x
225 q42 :x
•Gram matrix Q is typically not unique. If we can find Q ≥ 0
2 21 1
4 4 2 2 3 2 21 2 1 2 1 2 2 2
1 2 1 2
2 5 2 0
Tx x
x x x x x x x Q x
x x x x
Generally, linear constraintson Q, i.e. L(Q) = 0
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Semidefinite Program/LMI Optimization
10/22/2010 41
0minimize
subject to 0
0
Q
T
L Q
L Q
Q Q
linear objective
linear constraints
pos. def. matrix variable
•Standard form:
•Efficiently solvable in theory and practice•Polynomial-time algorithm available•Efficient free solvers: SeDuMi, SDPT3, etc.
•Lots of applications•KYP lemma, Lyapunov function search, filter design,
circuit sizing, MAX-CUT, robust optimization …
Read Boyd and Vandenberghe’s SIAM review
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Positivity Check is Sufficient Only
10/22/2010 42
2 2 21 2 3 1 2 1 3
1 1
2 2
3 3
2 3 6 4
2 3 2
3 1 0
2 0 3
T
x x x x x x x
x x
x x
x x
4 4 2 2 31 2 1 2 1 2
2 21 12 22 2
1 2 1 2
2 5 2T
x x x x x x
x x
x Q x
x x x x
spans R3 does not span R3
Quadratic case General case
•Requiring Q ≥ 0 sufficient but not necessary!2 4 4 2 2 21 2 1 2 1 21 3x x x x x x Positive? Can you find Q ≥ 0?
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Sum of Squares (SOS)
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•Finding Q ≥ 0 equivalent to sum of squares decomposition•In our example, we can find
2 3 1 2 2 0 0
3 5 0 3 3 1 1
1 0 5 1 1 3
1
2 23
1
T T
Q
2 24 4 2 2 3 2 2 21 2 1 2 1 2 1 2 1 2 2 1 2
1 12 5 2 2 3 3
2 2x x x x x x x x x x x x x
sum of squares positive semidefinite Q nonnegativity
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Wrap Up
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4 Turn RF Inductor PMOR
d
w
1 1.5 2 2.5 3 3.5 4 4.5 51
1.5
2
2.5
3
3.5
4
4.5
5
W ( m)
d (
m)
0 1 2 3 4 5 6 7 8 9 10
x 109
0
5
10
15
f (Hz)
Q
x full model- QCO PROM
4510/22/2010
•4 turn RF inductor with substrate•Circle: training data•Triangle: test data
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Summary (1)
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•Motivation for model reduction in design automation•PDE high order ODE reduced ODE•Parameterized reduced modeling facilitates design
•Model reduction based on rational transfer function fitting
•H problem difficult, resort to anti-stable relaxation
•Relaxation easy to solve, closely related to H problem
•Quasi-convex optimization•Efficient algorithms exist (e.g. cutting plane method)•Cutting plane method in model reduction setting
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Summary (2)
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•Parameterized model reduction•Reduced rational transfer function, coefficients are
function of design parameters•Easily extended from non-parameterized case, except
positivity check is difficult
•Positivity check of multivariate polynomials•Univariate case easy, quadratic case easy•General case requires semidefinite programs, only
sufficient•Related to sum of squares optimization
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Some References (1)
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•Parameterized reduced modeling•Moment matching: Eric Grimme’s PhD thesis•Parameterized moment matching:
L. Daniel, O. Siong, C. L., K. Lee, and J. White, “A multiparameter moment matching model reduction approach for generating geometrically parameterized interconnect performance models,” IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems, vol. 23, no. 5, pp. 678–693.
•Parameterized rational fitting:Kin Cheong Sou; Megretski, A.; Daniel, L.; , "A Quasi-Convex Optimization Approach to Parameterized Model Order Reduction," Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on , vol.27, no.3, pp.456-469, March 2008
•MIMO rational fitting/interpolation:A. Sootla, G. Kotsalis, A. Rantzer, “Multivariable Optimization-Based Model Reduction”, IEEE Transactions on Automatic Control, 54:10, pp. 2477-2480, October 2009
Lefteriu, S. and Antoulas, A. C. 2010. A new approach to modeling multiport systems from frequency-domain data. Trans. Comp.-Aided Des. Integ. Cir. Sys. 29, 1 (Jan. 2010), 14-27
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Some References (2)
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•Convex/quasi-convex optimization•Convex optimization:
S. Boyd and L. Vandenberghe, “Convex Optimization”, Cambridge University Press, 2004.
•Ellipsoid Cutting plane method:Bland, Robert G., Goldfarb, Donald, Todd, Michael J. Feature Article--The Ellipsoid Method: A SurveyOPERATIONS RESEARCH 1981 29: 1039-1091
•Multivariate polynomials and sum of squares•Ordinary polynomial case: Pablo Parrilo’s PhD thesis•Trigonometric polynomial case:
B. Dumitrescu, “Positive Trigonometric Polynomials and Signal Processing Applications”, Springer, 2007