A QuasiConvex Optimization Approach to Parameterized Model Order Reduction

1
A Quasi-Convex Optimization Approach to
Parameterized Model Order Reduction
(Supported by GSRC, SRC and NSF)

• Kin Cheong Sou
• Alexandre Megretski
• Luca Daniel
• Massachusetts Institute of Technology

www.rle.mit.edu
2
Why PMOR?
Parameterized model order reduction can
facilitate design
A ? Rm?m
m is small (lt100)
D wire separation W wire width
3
Previous work (MOR,PMOR)
Parameterized MOR

• Moment matching (PRIMA, PVL) Odabasioglu ICCAD
  97,Freund DATE 98
• Parameterized moment matching Liu DAC99, Pullela
  TCAD97, Daniel TCAD04, Li DATE05
• TBR and approximationsMoore IEEE TAC 81
• Optimal Hankel norm reduction Glover IJC 84
• Proper orthogonal decomposition Sirovich QAM 87
• Optimization based rational fit Gustavsen99,
  Coelho DAC 99

Our method is quasi-convex and parameterized
4
Previous work (optimization based)
Can be interpreted as 2 step approach

• Find poles of reduced model

• Iterative linear least squares problems Coelho
  DAC 99
• Vector fitting (partial fraction) Gustavsen
  IEEE Trans Power Delivery 99

Post-processing modification for stability

• Flipping real parts of poles
• PVL? algorithm Bai Bell Lab Tech Report 97

2. Find gain and zeros of (passive) reduced model

• Improving residues (linear least squares)
  Gustavsen 99
• Finding C and D matrices (semi-definite
  programming) Coelho ICCAD 01

5
Step 1 (previous vs. proposed)

• Available methods
• Cast as nonlinear least squares (solved by
  Gauss-Newton or iterative LLS)

• Our method
• Cast as quasi-convex program (solved by convex
  optimization algorithm)

• Explicitly take care of stability and passivity
  while finding poles

• Do not consider stability or passivity while
  finding poles (need post-processing)

6
Overview

• Motivations
• Previous Work
• Background
• Quasi-convex optimization MOR setup
• Solving the MOR setup
• Extension to parameterized MOR
• Application examples
• Conclusion

7
Continuous time/Discrete time transform

• To make our program tractable, we introduction
  change of frequency variables (bilinear
  transform)

z frequency variable
Laplace frequency variable
z
s
8
Modified optimal H-inf norm MOR setup
Stability q(z) Schur polynomial (roots of inside
unit circle) Passivity, and possibly other
constraints

• Desirable MOR setup to solve
• Feasible set is not convex if m ? 3
• For example,
  
• but
• Problem has not been proved to be NP complete
  either

9
Quasi-convex optimization MOR setup

• Motivations
• Previous Work
• Background
• Quasi-convex optimization MOR setup
• Solving the MOR setup
• Extension to parameterized MOR
• Application examples
• Conclusion

10
Relaxation of the H-inf norm MOR setup
Anti-stable term
Stability q(z) Schur polynomial (roots of inside
unit circle) Passivity, and possibly other
constraints
Benefit Relaxation equivalent to a quasi-convex
program. Drawback May obtain suboptimal solutions
11
How bad is our relaxation?
Let
such that deg(q) m, q(z) is Schur polynomial
Then
m1th Hankel singular value
12
Change of variables
real part
imaginary part
denominator
where a(z) b(z) and c(z) are trigonometric
polynomials
when
Prop Stability ?
13
Stability

• In the two relaxations, a and q are related by

a(z) q(z)q(1/z)

• If q(z) is a Schur polynomial (q(z) ? 0, z1)

then

• If trigonometric polynomial a(z)

then there exists a unique Schur polynomial q(z)

(by extracting the stable poles of a(z) )
Conclusion
Stability ?
14
Passivity

• For SISO systems, passivity means
• H(z) is analytic for zgt1
• H(z)H(z)
• Re(H(z))gt0 for z1 for impedance,

(i.e. H is stable)
(i.e. impulse response is real)
since real part b/a
Positive real condition
for all frequencies!
Conclusion Stability and passivity
positivity of trigonometric polynomials
15
Equivalent quasi-convex setup
For example
convex set
This is a quasi-convex program, because
defines an intersection of halfspaces and ?
sub-level set is
is again intersection of halfspaces parameterized
by ? and ?
16
Additional constraints

• Can model additional constraints such as

• Bounded real passivity (for scatter parameters)
• Explicit minimization of quality factor error
  (for inductors)
• Weighting of frequency responses
• Point-wise transfer function (and/or
  derivatives) matching

• See paper for more detail

17
Summary of our algorithm
Step 1 Compute optimal solution a(z),b(z),c(z)
of the relaxation
subject to stability, passivity
Solved by the ellipsoid algorithm, for example
Step 2 Compute coefficients of q(z) using the
relation
and q(z) being a Schur polynomial
Step 3 Compute coefficients of p(z) by solving
,stability, passivity
18
Construction of oracles

• Motivations
• Previous Work
• Background
• Quasi-convex optimization MOR setup
• Solving the MOR setup
• Extension to parameterized MOR
• Application examples
• Conclusion

19
Solving quasi-convex programs
Part of our work ofthe MOR algorithm
(a,b,c,?) current iterate
localization set (e.g. ellipsoid)
Objective oracle, stabilityoracle, passivity
oracle
Standard technique!
N
Termination?
Y
N
Stability?
Update localization set
N
Y
and so on
Passivity?
Generate cut
N
N
Y
Decrease ?

All Yes
20
Objective oracle

• TF matching imposed only at discrete frequency
  samples
• Objective oracle involves only computation of
  vector norm

If (a,b,c) satisfies
then objective constraint is satisfied
Then
Otherwise let
defines a cut w.r.t. (a,b,c).
21
Stability oracle
Since
Need 2 conditions (considering a(z) being
continuous)
1. Since
If a0 ? 0, then
for some ?0
a0 gt 0 defines a cut. Now assume a0 gt 0
by condition 1
implies
2.
or
defines a cut
Otherwise, ?0 can be found s.t.
22
Passivity oracle

• Real part of the relaxation

• Given stability

positive realness means

• Can use the same oracle as a(z) gt 0

23
Extension to PMOR

• Motivations
• Previous Work
• Background
• Quasi-convex optimization MOR setup
• Solving the MOR setup
• Extension to parameterized MOR
• Application examples
• Conclusion

24
Parameterized MOR setup

• Construct guaranteed stable parameterized
  reduced model

• Let design parameter p explicitly enters (a,b,c)

• Objective oracle same as the non-parameterized
  case
• More difficult to construct stability oracle

25
Parameterized stability oracle

• Need to check a(z,p) gt 0 for all z, for all p
• Cannot use the trick in the non-parameterized
  case
• Make a(z,p) a multivariate trigonometric
  polynomial

trig poly
ordinary poly
trig poly
trig polys
with

• Check positivity of multivariate trigonometric
  polynomial by using sum-of-squares argument
• Oracle should also return linear cut if
  positivity is not met

26
Stability oracle as an SDP
Given a(z,p), solving the semi-definite program

• If feasible and y lt 0, then

• Otherwise, construct a cut using the Farkas
  dual solution ?

27
Complexity

• 2 computationally expensive parts Computation
  of frequency response and solving the relaxation

• Frequency response samples requires O(nlog(n)
  ns) operations

full model order
of frequency samples(e.g. 20 200)

• Solving relaxation requires
• For non-parameterized case O(m4) operations
• For parameterized case O((m ?mpk)4) operations

reduced model order (e.g. 10, 20)
order in a(z,p) of pk (e.g. 2,3)
28
Application examples

• Motivations
• Previous Work
• Background
• Quasi-convex optimization MOR setup
• Solving the MOR setup
• Extension to parameterized MOR
• Application examples
• Conclusion

29
Example 1 RLC line (MNA)

• RLC line full model 20th order Vasilyev 2004
• Open circuit terminal
• 10th order reduced model by PRIMA and QCO

4
2
4
PRIMA reduction
QCO reduction
30
Example 2 RF inductor model (from field solver)

• 7 turn spiral inductor (wire width 18 µm, wire
  separation 18 µm)
• Full model (order 1576) generated by an EMQS-MPIE
  field solver
• PRIMA 20th order reduced model (expansion point s
  j2p6e9 (6GHz), matching 10 moments)
• Proposed method 20th order reduced model (20
  frequency response samples, 20 derivative samples)

3 curves on top of each other
3 curves on top of each other
31
Example 3 RF inductor with substrate(from field
solver)

• RF inductor with substrate effect captured by
  layered Greens function Hu Dac 05
• System matrices are frequency dependent
• Full model has infinite order
• Reduced model has order 6

32
Example 4 RF inductor identification (from
measurement)
Fabricated 7 turn spiral inductor Blue
measurement Red 10th order reduced model
(positive real part constraint imposed)
33
Example 5 Graphic card package model (S
parameter measurement)

• Industry example of a multi-port device (390
  frequency samples)
• 12th order SISO reduced models are constructed
• Bounded realness constraint is imposed
• Frequency weight is employed

S13
S11
Solid ROM Dot measurement
Solid ROM Dot measurement
34
Example 6 Power distribution grid(from field
solver)

• Power distribution grid (dimension size 7mm,
  wire width 2 µm)
• Blue full model (order 2046)
• Green PRIMA 40th order reduced model
• Red proposed method 40th order reduced model
  (positive real)

3 curves on top of each other
3 curves on top of each other
35
Example 7 Parameterized RF inductor
Quality factor for W16.5 µm, D 1,5,18,20 µm
Circle training points
----- full model ___ our QCO PMOR
Triangle test points
36
Conclusion

• QCO competes reasonably well with existing alg
  (e.g. PRIMA) for reducing large systems
• But in addition QCO can reduce models with
  frequency dependent matrices
• QCO is very flexible in imposing constraints such
  as stability and passivity
• QCO can be extended to parameterizedMOR problems
• Numerical examples have validated the practical
  value of QCO