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Introduction to Model Order ReductionIntroduction to Model Order Reduction

Luca Daniel

Massachusetts Institute of Technology

luca@mit.edu

http://onigo.mit.edu/~dluca/2006PisaMOR

www.rle.mit.edu/cpg

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Model Order Reduction of Linear SystemsModel Order Reduction of Linear Systems

via Modal Analysis

via Rational function fitting (point matching)

via Quasi Convex Optimization

via Pade approximation (AWE)

Projection Framework

SVD, PCA, LVD, POD

Krylov Subspace Moment Matching Projection MethodsArnoldiPVLPRIMA

Truncated Balance Realization (TBR)Positive Real TBR

Distributed Systems (with Frequency Dependent Matrices)

Distributed Linear Systems Distributed Linear Systems

Examples:ODE’s with delays (e.g. full-wave integral equation solvers) frequency-dependent basis functions frequency dependent discretizationssolvers using layered-media Green functions (e.g. for

handling substrate or dielectrics)

NOTE: Distributed systems may have infinite order (e.g. delay)!!

xcy

buxsAsxT

)(

Polynomial interpolation [Phillips96]Polynomial interpolation [Phillips96]

Polynomial approximation e.g. Taylor expansion, or a polynomial interpolation for A(s)

Convert to non-distributed model reduction problem

Performance: Fast and accurate in the frequency band of interest

Problem: Can not be used in a time domain circuit simulator because does not guarantee stability and passivity

][~ 2 xsxssxxx M

buxAsAssAAsx MM 2

210

buxAxs ~~~

xcybuxsAsx T )(

Passivity condition on transfer functionPassivity condition on transfer function

For systems with immittance matrix representation, passivity is equivalent to positive-realness of the transfer function

0Refor 0,)()(

0Refor )()(

0Refor analytic is )(

(s)ss

(s)ss

(s)s

HHH

HH

H

)()()( susHsy (no unstable poles)

(no negative resistors)

(impulse response is real)

It means its real part is a positive for any frequency.Note: it is a global property!!!! FOR ANY FREQUENCY

allfor 0)()( HjHjH

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Positive real transfer functionPositive real transfer functionin the complex plane for different frequenciesin the complex plane for different frequencies

)}(Re{ jH

sfrequencie allfor ,0)}(Re{ jHPassive regionActive

region

)}(Im{ jH

original system )( jH

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Why does polynomial interpolation failWhy does polynomial interpolation failwhen applied to the Laplace parameter ‘s’?when applied to the Laplace parameter ‘s’?

original system )( jH

)}(Im{ jH

Passive regionActiveregion

Although accurate in the frequency band of interest

Polynomial interpolation is unlikely to preserve GLOBAL properties such as positive realness because it is GLOBALLY not well-behaved

)}(Re{ jH

sfrequencie allfor ,0)}(Re{ jH

Observation: practical systems Observation: practical systems have some loss at any frequencyhave some loss at any frequency

Most systems are non-ideal i.e. contain some small loss at any frequency i.e. can be described by strictly positive real functions

Re

Im

sfrequencie allfor ,0)}(Re{ jE

original system )( jE

Passive regionActiveregion

Using global uniformly convergent interpolantsUsing global uniformly convergent interpolants

If A(s) is strictly positive real, a GLOBALLY and UNIFORMLY convergent interpolant will eventually get close enough (for a large enough order M of the interpolant) and be positive-real as well.

Re

Im

original system )( jA

reduced system )(ˆ jA

Passive regionActiveregion

Proof: just choose

accuracy of interpolation smaller than minimum distance from imaginary axis

M

kkk

M sAsA0

)( )()(

A good example of uniformly convergent A good example of uniformly convergent interpolants: the Laguerre basis functionsinterpolants: the Laguerre basis functions

Consider the family of basis functions:

They form a complete, rational, orthonormal basis over the imaginary axis which gives a uniformly convergent interpolant

no poles in RHP (stable)

(real time-domain representation)

,,1,0;;)( kjss

ss

k

k

)()( ss kk

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Calculation of interpolation coefficientsCalculation of interpolation coefficients

Note: it is a bilinear transform that maps the Laguerre basis to Fourier series on the unit circle.

kk

k zs

ss

)(

M

kkk

M sAsA0

)( )()(

Re{s}

Im{s}

Re{z}

Im{z}

Hence in practice one can use FFT to calculate the interpolation coefficients: very efficient!

Note: FFT coefficients typically drop quickly and the series can be truncated to the first few M coefficients because field solver matrices A(s) are often smooth.

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An implementation example:An implementation example:Two wires on a MCM package [D. DAC02]Two wires on a MCM package [D. DAC02]

Discretize Maxwell equations in integral form using PEEC

NOTE: system matrices are frequency dependent because the substrate is handled by layered Green functions

packagepackage

s x=- A(s) x+b u

Step 2: Interpolation (example) Step 2: Interpolation (example)

FFT coefficients of A (s)

0 10 20 30 40 50 60

10-4

10-5

10-6

10-7

10-8

Step 2: Interpolation (example) Step 2: Interpolation (example)

A (s) reconstructed fromreconstructed from first 5 out of 64first 5 out of 64 FFT coefficients FFT coefficients and compared to original and compared to original A (s)

0 10 20 30 40 50 60

5

4

3

2

1

0

nH Real part Imaginary part

Reduction procedure [D. and Phillips DAC01]Reduction procedure [D. and Phillips DAC01]

Matrix sizes

System order

Start from original system described by causal, strictly positive-real matrices

~ 3,000 infinite

1) Evaluate and squash them at uniformly spaced points on the unit circle using e.g. POD with congruence transformation which preserves positive realness

UTA(zk)U, k=1,2,...,64

br=UTb

~ 6 ~ 6 x 64

buxsAsx )(

Reduction procedure [D. and Phillips DAC01]Reduction procedure [D. and Phillips DAC01]

Matrix sizes

System order

3) Calculate first few (e.g 5) FFT coef of the reduced system matrix 6 6 x 5

4) Introduce extended state and realize a single matrix discrete time system 6 x 5 6 x 5

5) Transform to continuous time 6 x 5 6 x 5

ubxzAzx rk

kk

~~4

0

][~ 2 xzxzzxxx MubxAxz

~~~~

ubxAxs ˆˆˆˆ

Step 3: Realization (Multichip Module MCM example)Step 3: Realization (Multichip Module MCM example)

Real part of frequency response

Inductive part of Inductive part of frequency responsefrequency response

5

4

3

2

nH

frequencyfrequency frequencyfrequency

5

4

3

2

x104Ohm

106 107 108 109 1010 106 107 108 109 1010

original system 3000 distrib

reduced system 30

original system

reduced system

Open issues for distributed systemsOpen issues for distributed systems

Guaranteeing positive realness relies on accuracy of the uniform interpolant. Hence if the matrices are NOT smooth, we might need a large order of the interpolant.

working on internal matrices might give smoother matrices

Laguerre basis functions are efficient since they use FFT. However equally spaced points on the unit circle correspond to non-equally spaced points on the imaginary axis accumulating around a reference center frequency.

Model Order Reduction of Linear SystemsModel Order Reduction of Linear Systems

via Modal Analysis via Rational function fitting (point matching) via Quasi Convex Optimization

via Pade approximation (AWE) Projection Framework SVD, PCA, LVD, POD Krylov Subspace Moment Matching Projection Methods

Arnoldi PVL PRIMA

Truncated Balance Realization (TBR) Positive Real TBR

Laguerre interpolation for Distributed Systems

Model Order Reduction of Linear SystemsModel Order Reduction of Linear Systems

via Modal Analysis via Rational function fitting (point matching) via Quasi Convex Optimization (use this for LARGE and passive

systems, or for model construction from measurements, or for distributed systems)

via Pade approximation (AWE) Projection Framework SVD, PCA, LVD, POD Krylov Subspace Moment Matching Projection Methods

Arnoldi PVL (use this for HUGE systems if passivity is not an issue) PRIMA (use this for HUGE and passive systems)

Truncated Balance Realization (TBR) (use this for SMALL systems) Positive Real TBR (use this for SMALL and passive systems)

Laguerre interpolation for Distributed Systems (use this for LARGE systems with frequency dependent matrices, e.g. delays)