1 introduction to model order reduction thanks to jacob white, peter feldmann ii.1 – reducing...

1

Introduction to Model Order Reduction

Thanks to Jacob White, Peter Feldmann

II.1 – Reducing Linear Time Invariant Systems

Luca Daniel

2

Model Order Reduction

Linear Time Invariant Systems

• II.1.a via Modal Analysis

• II.1.b via Rational Function Fitting (point matching)

• II.1.c. via Quasi Convex Optimization

• II.1.d via Pade’ approximation and AWE

3

Introduction to Model Order Reduction

Thanks to Jacob White, Peter Feldmann

II.1.a – Reduction using Modal Analyis

Luca Daniel

4

State-Space Description

Dynamic Linear case

1 1

( )NxN Nx

scalarinp

T

Nxscalarouu tt put

y tdx t

A x t b u t c x tdt

• Original Dynamical System - Single Input/Output

• Reduced Dynamical System

• q << N, but input/output behavior preserved

1 1

( )rr r

qxq qx scalarinp

Tr r r

scalar qxoutputut

dx tA x yt b u t

dt x

tc t

5

Defining Accuracy Defining Accuracy

• Time-domain response should be “close” Time-domain response should be “close” – For which possible inputs? For which possible inputs?

• Frequency response should match Frequency response should match – At what frequencies? At what frequencies?

6

Matching Frequency Response Matching Frequency Response

• Ensure accuracy for only some inputs? Ensure accuracy for only some inputs? • Example: Example:

– low frequency inputs, low frequency inputs,

– or some band, or some band,

– or some points in the frequency responseor some points in the frequency response

Original matching some

part of the frequency

response

)(log H

log

7

Reminder about Eigenanalysis

1Change of variab (le ) ( ) ( ) (s ): Ew t x t w t E x t

1

1 2 1 2

10 0

0 0

0 0

Eigendecomposition: N N

N

E

A E E E E E E

Substituting: ( )

( ) , (0) 0dEw t

AEw t bu t Ewdt

11 1Multiply by : ( )

( )dw t

E AEw t E bu tdt

E

Consider an ODE( )

( ) , (0) 0: dx t

Ax t bu t xdt

8

Reminder about Eigenanalysis Cont.

1

11 1 1

1

0 0

0 0

0 0N N NN

E bw wd

u tdt

w w E b

b

Decoupled Equations

TT T Ty t c x t c Ew t E c w t

c

Output Equation

9

Reminder about Eigenanalysis Cont.

0

i

tt

i iw t e b u d

Solving Decoupled Equations

1

N

i ii

y t c w t

Output Equation

Assuming Zero Initial Conditions

10

Reduced models via mode truncationDynamic Linear Case

1 1 1 10 0

0 0

0 0q q q q

w w b

u t

w w b

1

q

i ii

y t c w t

Output Equation

11

Reduced models via mode TruncationDynamic Linear Case

Why?

• Certain modes are not affected by the input

• Certain modes do not affect the output

• Keep least negative eigenvalues (slowest modes)– Look at response to a constant input

1, , are all smallk Nb b

1, , are all smallk Nc c

0

1

Small if large

i i

tt t

i i i ii

i

w t e b ud b u b ue

12

Reduced models via mode truncationDynamic Linear Case

Heat Conducting bar Results

N=100

Exact

q=1

q=3

q=10

Keep qth slowest modes

13

Another way to look at Reduction by Modal Analysis

1TH s c sI A b

Transfer Function

Apply Eigendecomposition

1 1TH s c E sI E b

1

10 0

0 0

10 0

T

N

s

c b

s

1

Ni i

i i

c bH s

s

elimitate eachmode for whichthis term is small

14

Model Order Reduction Model Order Reduction via Eigenmode Analysis via Eigenmode Analysis

n

i i

ii

s

bcsH

1

~~)(

)(

)()(

1

1

1

i

n

i

i

n

i

s

ssH

Pole-Residue FormPole-Residue FormPole-Zero Form (SISO)Pole-Zero Form (SISO)

• Ideas for reducing order:Ideas for reducing order:– Drop terms with small residues Drop terms with small residues

– Drop terms with large negative (“fast” modes)Drop terms with large negative (“fast” modes)

– Remove pole/zero near-cancellations Remove pole/zero near-cancellations

– Cluster poles that are “together”Cluster poles that are “together”

iRe

n

i

tii

iebcth1

~~)(

iibc~~

15

Eigenmode Analysis Based Reduction SummaryEigenmode Analysis Based Reduction Summary

• Advantages Advantages – Conceptually familiar Conceptually familiar

– Simple physical interpretation : retains dominant Simple physical interpretation : retains dominant system modes/poles system modes/poles

• Drawbacks Drawbacks – Relatively expensive : Relatively expensive : have to find the eigenvalues firsthave to find the eigenvalues first

– Relatively inefficient. For a given model size, many Relatively inefficient. For a given model size, many other approaches can provide better accuracy for the other approaches can provide better accuracy for the same computational costsame computational cost

• e.g. Hankel Model Order Reduction e.g. Hankel Model Order Reduction • e.g. Truncated Balance Realizatione.g. Truncated Balance Realization

O(n3)

16

Model Order Reduction

Linear Time Invariant Systems

• II.1.a via Modal Analysis

• II.1.b via Rational Function Fitting (point matching)

• II.1.c. via Quasi Convex Optimization

• II.1.d via Pade’ approximation and AWE

17

Introduction to Model Order Reduction

Thanks to Jacob White

II.1.b – Reduction using Fitting

Luca Daniel

18

A canonical form for model order reductionA canonical form for model order reduction

( ) ( )

( ) ( )T

dxE x t b u t

dt

y t c t

Assuming A is non-Assuming A is non-singular we can cast the singular we can cast the dynamical linear system dynamical linear system into one canonical form into one canonical form for model order for model order reductionreduction

Note: not necessarily Note: not necessarily always the best, but the always the best, but the simplest for educational simplest for educational purposespurposes

bAb

EAE1

1

)()(

)()(

txcty

tbutAxdt

dxE

T

19

Original System Transfer Function:Original System Transfer Function:

1

0 1 1

11

NN

NN

b b s b sH s

a s a s

Model Reduction = Find a low order (q << N) Model Reduction = Find a low order (q << N) rational function matchingrational function matching

Model Order Reduction Model Order Reduction via Rational Transfer Function Fittingvia Rational Transfer Function Fitting

rational functionrational function

1

0 1 1

1

ˆ ˆ ˆˆ

ˆ ˆ1

qq

qq

b b s b sH s

a s a s

reduced orderreduced orderrational functionrational function

20

Reduced Model Dynamical SystemReduced Model Dynamical System

1

1

ˆ ˆˆ ˆ

ˆ ˆ ˆ( )

qxq qxscalarinp

T

qxscalaroutput

ut

dxE x t b u t

d

y c x t

t

t

Reduced Model Transfer FunctionReduced Model Transfer Function

2 2q qcoefficientscoefficients

2qcoefficientscoefficients

Rational Transfer Function Fitting: Rational Transfer Function Fitting: Degrees of FreedomDegrees of Freedom

1

0 1 1

1

ˆ ˆ ˆˆ

ˆ ˆ1

qq

qq

b b s b sH s

a s a s

21

Reduced Model Transfer FunctionReduced Model Transfer Function

ˆ ˆˆ ˆ

ˆ ˆ ˆ( ) T

dxE x t b u t

d

y t c x t

t

Apply any invertible change of variables to the stateApply any invertible change of variables to the state

1 ˆˆˆTH s c sE I b

1 1ˆ ˆˆ ˆ

ˆ ˆ ˆ( ) T

dwU EU w t U b

y t c Uw t

u tdt

11 1

11 1

ˆˆˆ

ˆˆˆ

T

T

H s c U sU EU I U b

c UU sE I UU b

Many Dynamical Systems have the same transfer function!!

ˆ ˆ( ) ( )x t U w t

Rational Transfer Function Fitting: Rational Transfer Function Fitting: Degrees of Freedom (cont.)Degrees of Freedom (cont.)

I I

22

H s

Rational Transfer Function Fitting: Rational Transfer Function Fitting: via Point Matchingvia Point Matching

H s

11 0 1 1

ˆ ˆ ˆˆ ˆ1 0q qi q i i i q ia s a s H s b b s b s

For i = 1 to 2q

• cross multiplying generates a linear systemcross multiplying generates a linear system

• Can match 2q pointsCan match 2q points 1

0 1 1

1

ˆ ˆ ˆ

ˆ ˆ1

qi q i

i qi q i

b b s b sH s

a s a s

23

• Columns contain progressively higher powers of the test Columns contain progressively higher powers of the test frequencies: problem is numerically ill-conditionedfrequencies: problem is numerically ill-conditioned

• also... missing data can cause severe accuracy problemsalso... missing data can cause severe accuracy problems

2 111 1 1 1 1 1

122 2

2 11 22 2 2 2 2

q

q

qq qq q q q q

H ss H s s H s s a

H ss a

b H ss H s s H s s

Rational Transfer Function Fitting: Rational Transfer Function Fitting: Point Matching matrix can be ill-conditionedPoint Matching matrix can be ill-conditioned

SMA 2005 MIT 24

Hard to Solve Systems

Fitting Example

Polynomial InterpolationTable of Data

t0 f (t0)t1 f (t1)

tN f (tN)

f

tt0 t1 t2 tN

f (t0)

Problem fit data with an Nth order polynomial2

0 1 2( ) NNf t t t t

SMA 2005 MIT 25

Hard to Solve Systems

Example Problem

Matrix Form2

0 0 0 0 0

21 11 1 1

2

interp

1 ( )

( )1

( )1

N

N

NN NN N N

t t t f t

f tt t t

f tt t t

M

SMA 2005 MIT 26

Hard to Solve Systems

Fitting Example

CoefficientValue

Coefficient number

Fitting f(t) = t

f

t

SMA 2005 MIT 27

Hard to Solve Systems

Perturbation Analysis

Geometric Approach is clearer

1 2 1 1 2 2[ ], Solving is finding M M M M x b x M x M b

2

2|| ||

M

M

1

1|| ||

M

M

2

2|| ||

M

M

1

1|| ||

M

M

When vectors are nearly aligned, difficult to determinehow much of versus how much of 1M

2M

Case 16

1 0

0 10

Case 16

6

1 1 10

1 10 1

Columns orthogonal Columns nearly aligned

1x

2x

1x

2x

SMA 2005 MIT 28

Hard to Solve Systems

Geometric Analysis

Polynomial Interpolation

4 8 16 32

1010

1015

1020

~314~106

~1013

~1020

log(cond(M))

n

The power series polynomialsare nearly linearly dependent

21 1

22 2

2

1

1

1 N N

t t

t t

t t

1

11

t

t 2

t

29

Course Outline

Numerical Simulation Quick intro to PDE Solvers Quick intro to ODE SolversModel Order reduction Linear systems Common engineering practice Optimal techniques in terms of model accuracy Efficient techniques in terms of time and memory Non-Linear SystemsParameterized Model Order Reduction Linear Systems Non-Linear Systems

Yesterday

Today

FridayThursday

Tomorrow

30

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

31

Course Outline

Numerical Simulation Quick intro to PDE Solvers Quick intro to ODE SolversModel Order reduction Linear systems Common engineering practice Optimal techniques in terms of model accuracy Efficient techniques in terms of time and memory Non-Linear SystemsParameterized Model Order Reduction Linear Systems Non-Linear Systems

Monday

Yesterday

FridayTomorrow

Today

32

Model Order Reduction

Linear Time Invariant Systems

• II.1.a via Modal Analysis

• II.1.b via Ratianal Function Fitting (point matching)

• II.1.c. via Quasi Convex Optimization

• II.1.d via Pade’ approximation and AWE

33

Introduction to Model Order Reduction

Thanks to Kin C. Sou, Alexander Megretski

II.1.c – Reduction using Optimization

Luca Daniel

34

Overview

• Optimization based reduction

• Quasi-convex optimization MOR setup

• Solving the MOR setup

• Application examples

• Conclusions

35

H s

Recall Recall Rational Transfer Function Fitting via Point MatchingRational Transfer Function Fitting via Point Matching

H s

11 0 1 1

ˆ ˆ ˆˆ ˆ1 0q qi q i i i q ia s a s H s b b s b s

For i = 1 to 2q

• cross multiplying generates a linear systemcross multiplying generates a linear system

• Can match 2q pointsCan match 2q points 1

0 1 1

1

ˆ ˆ ˆ

ˆ ˆ1

qi q i

i qi q i

b b s b sH s

a s a s

36

Optimization based rational fit Model Order Reduction Setup

p(s),q(s)

( )minimize ( )

( )

p sH s

q s

From field solverOr measurements

Small stable and passivereduced order model

Least Square method• Cast as nonlinear least

squares (solved by Gauss-Newton)

Quasi-convex method• Cast as quasi-convex

program (solved by convex optimization algorithm)

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

• Explicitly take care of stability and passivity while finding poles

37

Change of variables• To make our program tractable, we introduce a change offrequency variables (bilinear transform)

Laplace frequency variablez frequency variable

11

zzs

[s] [z]

38

• 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

Modified optimal H-inf norm MOR setup

331 5q z z 33

2 5q z z

,

( )minimize ( )

( )

deg ,subject to

deg ,

p q

p zH z

q z

q m

p m

31 2( ) ( ) 27

2 2 25

q z q zz z

Stability: q(z) Schur polynomial (roots inside unit circle)

Passivity, and possibly other constraints

39

Overview

• Optimization based reduction

• Quasi-convex optimization MOR setup

• Solving the MOR setup

• Application examples

• Conclusions

40

Relaxation

• Original problem is difficult• Made easier if some constraints are dropped (relaxed)• Solve the relaxed problem • Construct original solution from relaxation• For example, LP relaxation (polynomial time) of IP problems (exponential time).

General idea

-c

feasible set…

optimal solution-c

feasible set

optimal relaxed solution

nearest rounding

41

Relaxation of the H-inf norm MOR setup

1

1, ,

( )minimize ( )

( )

deg , deg ,subject to

deg

p q r

r z

q

p zH z

q z z

q m p m

r m

Benefit: Relaxation equivalent to a quasi-convex program.Drawback: May obtain suboptimal solutions

Anti-stableterm

Stability: q(z) Schur polynomial (roots inside unit circle)

Passivity, and possibly other constraints

42

How bad is this relaxation?

,

1

1,( , , ) arg min

q p r

p zq p r H z

q

r

q zz

z

1m

p zH z m H

q z

Let

such that deg(q) = m, q(z) is Schur polynomial

Then

m+1th Hankel singular value

THEOREM:

43

Change of variables

ˆ , 0,

j j j j

j

j j j

p e r e b e jc eH e

q e q e a e

where a(z) b(z) and c(z) are trigonometric polynomials:

1 11 0( )m m m m

ma z z z a z z a

2cos m whenjz e

0ja e 0, Prop: Stability

44

Passivity

0, 0,jb e

• For SISO systems, passivity means1. H(z) is analytic for |z|>=1 2. H(z)*=H(z*) 3. Re(H(z))>0 for |z|=1 for impedance,

Conclusion: Stability and passivity = positivity of trigonometric polynomials

for all frequencies!

45

Equivalent quasi-convex setup

This is a quasi-convex program, because

1 02cos( ) 2cos(( 1) ) 0jma e m m a a

defines an intersection of halfspaces and sub-level set is

Re j j j j jH e a e b e jc e a e is again intersection of halfspaces parameterized by and

convex set0

1

, ,

( )minimize ( )

( )

deg , deg

deg ,subject to

0, [0, ]

0, [0, ]

j j

jja b c

j

j

b e jc eH e

a e

a m b m

c m

a e

b e

=0

=1

=2

=3

quasi-convex function

convex set

0 0 1 02cos 1 0 1 0mm m a a 1 1 1 02cos 1 0 1 0mm m a a 2 2 1 02cos 1 0 1 0mm m a a 3 3 1 02cos 1 0 1 0mm m a a

46

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

47

Overview

• Optimization based reduction

• Quasi-convex optimization MOR setup

• Algorithm Summary

• Application examples

• Conclusions

48

Summary of QCO algorithm

Step 2: Compute coefficients of q(z) using the relation

1q z q z a z and q(z) being a Schur polynomial

Step 3: Compute coefficients of p(z) by solving

minimize

subject to deg

p zH z

q z

p m

Solved for example by the ellipsoid algorithm

( )( ) min

( )

j j

jj

b e jc eH e

a e

Step 1: Compute optimal solution a(z),b(z),c(z) of the relaxation

subject to stability, passivity…

,stability, passivity…

Solved for example by the ellipsoid algorithm

49

Stability?

Solving quasi-convex programs(a,b,c,) current iteratelocalization set

(e.g. ellipsoid)

?b jc

Ha

Passivity?

…

Generate cut

N

N

N

N

Y

Y

Y

Decrease

All Yes

?c

Qb

Termination?

N

target set

localization set

center

cut

min volume covering ellipsoid

new center

new cut

and so on

Updatelocalization set

Objective oracle, stabilityoracle, passivity oracle…

50

Overview

• Optimization based reduction

• Quasi-convex optimization MOR setup

• Algorithm summary

• Application examples

• Conclusions

51

Example 1: RLC line (MNA)

“PRIMA” (Moment Matching) Model Order Reduction

Quasi Convex OptimizationModel Order Reduction

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

0 2000 4000 6000 8000 10000 12000 14000 160000

0.5

1

1.5

2

2.5

3

3.5

4

frequency (Hz)

mag

nitu

de

Full model

QCOROM

0 2000 4000 6000 8000 10000 12000 14000 160000

0.5

1

1.5

2

2.5

3

3.5

4

frequency (Hz)

mag

nitu

de

Full model

MMROM

4

4

2

52

Example 2: RF inductor with substrate(from field solver)

0 0.5 1 1.5 2 2.5 3

x 109

-2

-1

0

1

2

3

4

5

6

7

8

frequency (Hz)

qual

ity f

acto

r

training data

test pointsROM

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

53

Example 3: RF inductor model (from measurement)

0 1 2 3 4 5 6 7 8 9 10

x 109

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

frequency (Hz)

real

par

t

Fabricated 7 turn spiral inductorBlue: measurementRed: 10th order reduced model (positive real part constraint imposed)

0 0.5 1 1.5 2 2.5 3 3.5

x 109

-5

0

5

10

15

20

25

30

35

40

frequency (Hz)

qual

ity f

acto

r

54

Example 4: Model of graphic card package (from 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

0 1 2 3 4 5 6

0.4

0.5

0.6

0.7

0.8

0.9

1

mag

nitu

de

frequency (GHz)0 1 2 3 4 5 6

0

0.02

0.04

0.06

0.08

0.1

0.12

mag

nitu

de

frequency (GHz)

S11 S13

Solid: ROMDot: measurement

Solid: ROMDot: measurement

55

Example 5: Large IC 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: QCO 40th order reduced model (positive real)

0 10 20 30 40 50 600

500

1000

1500

2000

2500

frequency (GHz)

mag

nitu

de

0 10 20 30 40 50 60-2

-1.5

-1

-0.5

0

0.5

1

1.5

frequency (GHz)

phas

e

3 curves on top of each other

3 curves on top of each other

56

Conclusion

• QCO competes reasonably well in terms of accuracy with moment matching (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 parameterized MOR problems (see IV.2)

57

Model Order Reduction

Linear Time Invariant Systems

• II.1.a via Modal Analysis

• II.1.b via Ratianal Function Fitting (point matching)

• II.1.c. via Quasi Convex Optimization

• II.1.d via Pade’ approximation and AWE

58

Point matching vs. Moment MatchingPoint matching vs. Moment Matching

Point matching:Point matching:can be very inaccurate can be very inaccurate in between pointsin between points

H s

H s

H s

H sMoment (derivatives)Moment (derivatives)matching:matching:accurate around accurate around expansion point, expansion point, but inaccurate on wide but inaccurate on wide frequency bandfrequency band

59

1

0

( ) ( )T T k k

k

H s c I sE b c E b s

1 2 2

01 20

( ) T T T kk

km mmH s c b c E b s c E b s m s

The Taylor coef. = frequency domain moments = The Taylor coef. = frequency domain moments = = derivatives of the transfer function (up to a constant)= derivatives of the transfer function (up to a constant)

Frequency Domain "Moments" (or Taylor Frequency Domain "Moments" (or Taylor coefficients) of the transfer functioncoefficients) of the transfer function

Taylor Series Expansion of the original transfer Taylor Series Expansion of the original transfer function around s=0 function around s=0

2 32 3

2 30 0 0

1 1( ) (0)

2! 3!s s s

dH d H d HH s H s s s

ds ds ds

60

Time domain moments Time domain moments of the impulse responseof the impulse response

Definition:Definition:

0

0

22

0

1

0

0

)(ˆ

)(ˆ

)(ˆ

)(ˆ

dtthtm

dtthtm

dttthm

dtthm

qq

61

Connection to the time-domain moments of the Connection to the time-domain moments of the circuit responsecircuit response

0

2 12 2 2 1 2 1 2

0 0 0 0

( ) ( )

1 ( 1)( ) ( ) ( ) ( ) ( )

2! (2 1)!

st

qq q q

H s e h t dt

h t dt th t dt s t h t dt s t h t dt s O sq

Time-domain momentsTime-domain moments12210 ˆ,,ˆ,ˆ,ˆ qmmmm

2 2 1 20 1 2 2 1( ) ( )q q

qH s m m s m s m s O s

Compare:Compare:

Hence the the Taylor coeff. Hence the the Taylor coeff. are, up to a constant, the are, up to a constant, the time-domain moments of the time-domain moments of the circuit response.circuit response.

( 1)ˆ

!

k

k km mk

62

Rational function fitting via moment matching: Rational function fitting via moment matching: Pade Approximation (AWE)Pade Approximation (AWE)

2 2 1 20 1 2 2 1( ) ( )q q

qH s m m s m s m s O s

1

0 1 1

1

ˆ ˆ ˆˆ

ˆ ˆ1

qq

qq

b b s b sH s

a s a s

0 1 1 1 2

Choose the 2q rational function coefficients ˆ ˆ ˆ ˆ ˆ ˆ, , , , , , ,so that the reduced rational functionmatches the first 2q moments of the original transfer function

q qb b b a a a

10 1 1 2 2 1

0 1 2 2 11

ˆ ˆ ˆ

ˆ ˆ1

qq q

qqq

b b s b sm m s m s m s

a s a s

63

Rational function fitting via moment matching: Rational function fitting via moment matching: Pade Approximation (AWE)Pade Approximation (AWE)

– Step 1:Step 1: calculate the first 2q moments of H(s) calculate the first 2q moments of H(s)

– Step 2:Step 2: calculate the 2q coeff. of the Pade’ approx, calculate the 2q coeff. of the Pade’ approx, matching the first 2q moments of H(s)matching the first 2q moments of H(s)

12210 ,,,, qmmmm

qq aabbb ,,,,,, 1110

0,1, 2 1T kkm c E b k q

64

Step 1: calculation of moments Step 1: calculation of moments simulating equivalent circuits (AWE)simulating equivalent circuits (AWE)

• Historical note Historical note – Electrical engineers calculated freq. domain Taylor coef. by Electrical engineers calculated freq. domain Taylor coef. by

calculating time domain moments, calculating time domain moments,

– synthesizing and simulating circuit networks. synthesizing and simulating circuit networks.

– Specifically the momets can be calculated evaluating the Specifically the momets can be calculated evaluating the asymptotic behaviors of the circuit waveforms, asymptotic behaviors of the circuit waveforms,

– Hence the name AWE (Asymptotic Waveform Evaluation)Hence the name AWE (Asymptotic Waveform Evaluation)

65

Step 1: Calculation of moments (algebraically)Step 1: Calculation of moments (algebraically)

122122

1101

000

ˆ~~

ˆ~~

ˆ

qT

qqq

T

T

wcmwEwA

wcmwEwA

wcmbw

bEAc

bEcmkT

kTk

~~ 1

• For sparse system For sparse system – can use one initial LU decomposition on A can use one initial LU decomposition on A

– then solve 2q linear triangular systems for the 2q moments then solve 2q linear triangular systems for the 2q moments

• For dense systems For dense systems – can use iterative methods and matrix implicit matrix-vector can use iterative methods and matrix implicit matrix-vector

productsproducts

66

Step 2: Calculation of Pade’ coeff. (AWE)Step 2: Calculation of Pade’ coeff. (AWE)

1212

2210

1

1110

1

q

qqq

qq smsmsmm

sasa

sbsbb

012111

0111

00

mamamb

mamb

mb

qqqq

For coeff. a’s For coeff. a’s solve the solve the following following linear system:linear system:

12

2

1

1

2

1

221

2

21

1210

q

q

q

q

q

q

q

qq

q

m

m

m

m

a

a

a

a

mm

m

mm

mmmm

For coeff. b’s For coeff. b’s simply simply calculate:calculate:

67

Heat Conducting BarDemonstration Example State-Space Description

endT

x

1

2

N

h x

h x

h x

b

2 1 0 0

1 2

0 0

2 1

0 0 1 1

A

0

0

1

c

1 1

( )NxN Nx

scalarinp

T

Nxscalarouu tt put

y tdx t

A x t b u t c x tdt

Given the right scaling

Heat In0 0T

A

N=100

Exact

q=1q=3

q=10

Keep qth slowest eigenmodes

Exact

q=1

q=3

Matches q moments

Keeping Eigenmodes versus matching momentsDynamic Linear Case

Heat Flow Results

69

b

Eb

2E b3E b

Vectors will line up with dominant eigenspace!Vectors will line up with dominant eigenspace!

Numerical problem for q >20 (cannot get accuracy)Numerical problem for q >20 (cannot get accuracy)

matrix powers converge to the eigenvector corresponding to the largest eigenvalue.

T kkm c E b 0

kkm m

70

Pade matrix can be very ill-conditionedPade matrix can be very ill-conditioned

• matrix powers converge to the eigenvector matrix powers converge to the eigenvector corresponding to the largest eigenvalue.corresponding to the largest eigenvalue.

T kkm c E b 0

kkm m

12

2

1

1

2

1

221

2

21

1210

q

q

q

q

q

q

q

qq

q

m

m

m

m

a

a

a

a

mm

m

mm

mmmm

Columns become linearly dependent for large qColumns become linearly dependent for large qthe problem is numerically very ill-conditioned!the problem is numerically very ill-conditioned!

71

Pade matrix can be very ill-conditionedPade matrix can be very ill-conditioned

• matrix powers converge to the eigenvector matrix powers converge to the eigenvector corresponding to the largest eigenvalue.corresponding to the largest eigenvalue.

T kkm c E b 0

kkm m

12

2

1

1

2

1

022

012

1

001

1

01

02

10

q

q

q

q

q

q

q

qqq

qq

qq

m

m

m

m

a

a

a

a

mmm

mmm

mmmm

Columns become linearly dependent for large qColumns become linearly dependent for large qthe problem is numerically very ill-conditioned!the problem is numerically very ill-conditioned!

72

Example: simulation of voltage gain of a filter with Example: simulation of voltage gain of a filter with Pade via AWEPade via AWE

73

Model Order Reduction

Summary. Linear Time Invariant Systems

• II.1.a via Modal Analysis

• II.1.b via Rational Function Fitting (point matching)

• II.1.c. via Quasi Convex Optimization

• II.1.d via Pade’ approximation and AWE