Parameterized Model Order Reduction via Quasi …hsan/modred_files/QCO.pdf · Reduction via...
Transcript of Parameterized Model Order Reduction via Quasi …hsan/modred_files/QCO.pdf · Reduction via...
Parameterized Model Order Reduction via Quasi-Convex Optimization
Kin Cheong Souwith Luca Daniel and Alexandre Megretski
Systems on Chip or PackageInterconnect & Substrate
Courtesy of Harris semiconductor
RF Inductors
MEMresonators
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DSP
Digital
LNA
LO
Analog RF
ADC
ADC
Mixed SignalI
Q
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
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From 3D Geometry to Circuit Model
dtdEH
dtdHE
4 2 2
4 2 20
( )w
elec au u uEI S F p p dy
x x t
3 ( )( (1 6 ) ) 1 2 p uK u p pt
dtdEH
dtdHE
inBvvGdtdvvC )()(
Z(f) Z(f) Z(f) Z(f)
•Model generated by available field solver•Field solver models usually high order
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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 104
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
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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
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RF Inductor Parameter Dependency
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
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d
w
DSPLNA
LO
ADC
ADC
I
Q
,d w
reducedmodel
Parameterized Reduced Model Example
•Parameter dependent complex system
•Parameterized reduced order model
•Coefficients depend explicitly on d•Low order, inexpensive to simulate
2, ( , )rsG s G ss
dd dd
99
99
0.5 2,
0.499
s sG s
d dd
ds s d
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Continuous/Discrete-time Setups
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Continuous-time Discrete-time
left-half plane & imaginary axis unit disk & unit circle
& G s G j & jG z G e
11
zsz
szs
•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
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Parameterized Model Reduction Methods
Moment Matching Method
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1G z C zI A B D 1r V U VG z C z A B DUI
Projection withUV = I
Full model Reduced model
( ) ( )k kn n
k r kd dG z G zdz 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
Mag
nitu
de (a
bs)
Bode Diagram
Frequency (rad/sec)
8th order full4th order MM
moments matched
user specified
Rational Transfer Function Fitting
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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
Explanations in Two Steps
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•Will present a rational transfer function fitting method
•First describe basic non-parameterized reduction
•Then extend basic method to parameterized setup
Non-Parameterized Model Order Reduction
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 zq z q z q
p zG z
q z
subject to stableq z
,minimize
p q
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•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)
Difficulty with H Norm Reduction
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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 2 272 2 25q q z z but
branchingsolutions
convex combo. of stable poly.not necessarily stable
Idea From Optimal Hankel Reduction
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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
Anti-Stable Relaxation in Rational Fit
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•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)
Combine Stable and Anti-stable Terms
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•Combine stable and anti-stable terms in reduced model
1
1
f zp z b z jc zq 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
Stability and Positivity
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•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
Quasi-Convex Relaxation
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•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
From Relaxation Back to H Reduction
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•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
Quality of Suboptimal Reduced Model
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•Minimizing upper bound of Hankel norm error
1
1H
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
Back to Big Picture – Model Reduction
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minrG
rG G
stablerGs.t. order rG r
, ,mina b c
b jcGa
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?
Quasi-Convex Optimization
Quasi-Convex Optimization
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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 method2) cutting plane method
Cutting Plane Methodoptimal pointcovering set
iterate 1
iterate 2
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•Optimization problem data described by oracle
•What is the oracle in our model reduction problem?
Oracle
kept
removedkeptremoved
Model Reduction Oracles
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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!
Positivity Check
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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
Back to Big Picture – Model Reduction
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minrG
rG G
stablerGs.t. order rG r
, ,mina b c
b jcGa
0, 1a z z s.t.
optimal a(z), b(z), c(z) suboptimal p(z), q(z)
discussed
discussed
discussed
Solved withcutting planemethod
Parameterized Model Order Reduction
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
dp
G zq z q z qd d
max , ,d rd dG z G z
subject to ,q z d,
minimizep q
stable for all d
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•Reduced model found as the solution design parameter
Parameterized Reduced Model Example
•Parameter dependent complex system
•Parameterized reduced order model
•Coefficients depend explicitly on d•Low order, inexpensive to simulate
2, ( , )rzG z G zz
dd dd
99
99
0.5 2,
0.499
z zG z
d dd
dz z d
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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
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Parameterized Decision Variables
•Decision variables = parameterized trig. poly.
•When evaluated on unit circle, i.e.
Parameterized Quasi-Convex Relaxation
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, ,,
,b z jd d
dc z
G za 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!
Parameterized Positivity Check
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0 1 2, cos cos 2ja e d a d a d a d denominator
1 2
poly of
cosia 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
…
Positivity Check of Multivariate Polynomials
Checking Polynomial Positivity –Special Cases
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•Univariate case simple, check the roots of derivative4 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 22 3 6 4 3 1 0 0 ?
2 0 3
Tx xx x x x x x x x x
x x
polynomial nonnegative matrix positive semidefinite
Checking Polynomial Positivity –General Case
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•Positivity check of general multivariate polynomial is hard2 2 3
1 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 xx x q q q x x
= Q (Gram matrix)Monomials of relevant degrees
•What if we still write out “quadratic form”?
Checking Polynomial Positivity –General Case
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•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 xx x x x
Generally, linear constraintson Q, i.e. L(Q) = 0
Semidefinite Program/LMI Optimization
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0minimize
subject to 00
Q
T
L Q
L QQ 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
Positivity Check is Sufficient Only
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2 2 21 2 3 1 2 1 3
1 1
2 2
3 3
2 3 6 4
2 3 23 1 02 0 3
T
x x x x x x x
x xx xx 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 xx 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?
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 03 5 0 3 3 1 1
1 0 5 1 1 3
12 2
3
1T 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 32 2
x x x x x x x x x x x x x
sum of squares positive semidefinite Q nonnegativity
Wrap Up
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
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•4 turn RF inductor with substrate•Circle: training data•Triangle: test data
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
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
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 2009Lefteriu, 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
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