1 introduction to model order reduction luca daniel massachusetts institute of technology...
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Introduction to Model Order ReductionIntroduction to Model Order Reduction
Luca Daniel
Massachusetts Institute of Technology
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)