1 parasitic extraction step 2: model order reduction luca daniel massachusetts institute of...

1

Parasitic ExtractionParasitic ExtractionStep 2: Model Order ReductionStep 2: Model Order Reduction

Luca Daniel

Massachusetts Institute of Technology

[email protected]

Slides available at: http://onigo.mit.edu/~dluca/2009MOMiNE

www.rle.mit.edu/cpg

mailto:[email protected]

http://onigo.mit.edu/~dluca/2009MOMiNE

2

Conventional Design FlowConventional Design Flow

Funct. Spec

Logic Synth.

Gate-level Net.

RTL

Layout

Floorplanning

Place & Route

Front-end

Back-end

Behav. Simul.

Gate-Lev. Sim.

Stat. Wire Model

Parasitic Extrac.

3

Layout parasiticsLayout parasitics

• Wires are not ideal. Wires are not ideal. Parasitics:Parasitics:– ResistanceResistance

– CapacitanceCapacitance

– InductanceInductance

• Why do we care? Why do we care? – Impact on delayImpact on delay

– noisenoise

– energy consumptionenergy consumption

– power distributionpower distribution

Picture from “Digital Integrated Circuits”, Rabaey, Chandrakasan, Nikolic

4

Conventional Design FlowConventional Design Flow

Funct. Spec

Logic Synth.

Gate-level Net.

RTL

Layout

Floorplanning

Place & Route

Front-end

Back-end

Behav. Simul.

Gate-Lev. Sim.

Stat. Wire Model

Parasitic Extrac.

5

)()( tuBtxAdt

dxE

dt

dEH

dt

dHE

dt

dEH

dt

dHE

dt

dEH

dt

dHE

dt

dEH

dt

dHE

Parasitic Extraction: Step 1 Electromagnetic Field Solvers (review from last time)

Step 1. Electromagnetic Field SolversStep 1. Electromagnetic Field Solvers

nanophotonics

carbon nanotubes

RF inductorsinterconnect

6

Overview Step1 : Field SolversOverview Step1 : Field Solvers

• Formulation of Parasitic Extraction Problems:Formulation of Parasitic Extraction Problems:– Capacitance Extraction (electrostatic)Capacitance Extraction (electrostatic)

– Inductance Extraction (Magneto-Quasi-Static MQS)Inductance Extraction (Magneto-Quasi-Static MQS)

– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)

– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)

• Electromagnetic solvers: Electromagnetic solvers: – Classification (time vs. frequency, differential vs. integral)Classification (time vs. frequency, differential vs. integral)

– Integral equation solvers in detailsIntegral equation solvers in details• basis functionsbasis functions• equation testing (collocation vs. Galerkin)equation testing (collocation vs. Galerkin)• linear system solution (direct vs. iterative methods)linear system solution (direct vs. iterative methods)• fast matrix-vector products (eg. fastmultipole, pFFT)fast matrix-vector products (eg. fastmultipole, pFFT)• example: FastMaxwell a EMQS/Fullwave solver with pFFTexample: FastMaxwell a EMQS/Fullwave solver with pFFT

– Current Field Solvers Research DirectionsCurrent Field Solvers Research Directions

yesterday

7

)()( tuBtxAdt

dxE

dt

dEH

dt

dHE

dt

dEH

dt

dHE

dt

dEH

dt

dHE

dt

dEH

dt

dHE

The two steps Parasitic Extraction process

Step 1. Electromagnetic Field SolversStep 1. Electromagnetic Field Solvers

Step 2. Parameterized Model Order ReductionStep 2. Parameterized Model Order Reduction

)(ˆ)(ˆ),(ˆˆ),(ˆ tuBtxWhA

dt

xdWhE

nanophotonics

carbon nanotubes

RF inductorsW

interconnect

(10(1066 x 10 x 1066)) (10(1066 x 10 x 1066))

),( Wh ),( Wh

• small (10-15 equations)small (10-15 equations)• accuracy of PDE field solvers accuracy of PDE field solvers • geometrically parameterizedgeometrically parameterized• generated automatically!generated automatically!

8

ReferencesReferencesfrom Field Solvers to Reduced Order Modelsfrom Field Solvers to Reduced Order Models

[PEEC Ruehli 74] A. E. Ruehli, "Equivalent circuit models for three dimensional multiconductor systems", Trans Microwave Theory and Techniques, 1974, v 22, p 216-221.

[Kamon Trans Packaging 98] M. Kamon, N. Marques, L. M. Silveira, J. K. White, Automatic Generation of Accurate Circuit Models", Trans on Packaging, Aug 1998.

[MIT course 6.336J] http://stellar.mit.edu/S/course/6/fa04/6.336J/

[MIT course 16.920J] http://ocw.mit.edu/OcwWeb/Aeronautics-and-Astronautics/16-920JNumerical-Methods-for-Partial-Differential-EquationsSpring2003/CourseHome/

[Grimme PhD97] Eric Grimme, “Krylov Projection Methods for Model Reduction", PhD Thesis, University of Illinois at Urbana-Champaign", 1997.

[PRIMA TCAD98] A. Odabasioglu, M. Celik, L. Pileggi , "PRIMA: passive reduced-order interconnect macromodeling algorithm", TCAD, v 17, n 8, Aug 1998.

[Phillips96] J. R. Phillips, E. Chiprout, D. D. Ling, "Efficient full-wave electromagnetic analysis via model-order reduction of fast integral transforms", DAC, June 1996.

[Willems72] J. C. Willems, “Dissipative Dynamical Systems”, Arch. Rational Mechanics and Analysis, 1972, v 45, p 321-393.

9

ReferencesReferencesParameterized Model Order ReductionParameterized Model Order Reduction

[Pullela97] S. Pullela N. Menezes L.T. Pileggi, "Moment-Sensitivity-Based Wire Sizing for Skew Reduction in On-Chip Clock Nets", Trans on CAD, v 16, n 2, p 210-215, Feb, 1997.

[Weile99] D. S. Weile E. Michielssen Eric Grimme K. Gallivan, "A Method for Generating Rational Interpolant Reduced Order Models of Two-Parameter Linear Systems", Applied Mathematics Letters, v 12, p 93-102, 1999.

[Prud’homme 02] C. Prud'homme, D. Rovas, K. Veroy, Y. Maday, A.T. Patera, G. Turinici, "Reliable real-time solution of parameterized partial differential equations: Reduced-basis output bounds methods", Journal of Fluids Engineering, 2002.

[Liu DAC99] Y Liu, L T. Pileggi, A J. Strojwas", Model Order-Reduction of RCL Interconnect Including Variational Analysis", DAC, p 201-206, 1999.

[Heydari ICCAD01] P. Heydari, M. Pedram, “Model Reduction of Variable-Geometry Interconnects Using Variational Spectrally-Weighted Balanced Truncation", ICCAD, 586—591, 2001.

[Rutenbar DAC02] H Liu, Singhee, A, Rutenbar, R.A., Carley, L.R, “Remembrance of circuits past: macromodeling by data mining in large analog design spaces”, DAC 2002, p 437-42.

[Li DATE05] P Li, F Liu, S Nassif, L Pileggi, Modeling Interconnect Variability using Efficient Parametric Model Order Reduction, DATE 2005.

10

ReferencesReferencesParameterized Model Order Reduction (cont.)Parameterized Model Order Reduction (cont.)

[D. TCAD04] L. Daniel, C. S. Ong, S. C. Low, K. H. Lee, 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, v 23, n 5, p 678-93, May 2004.

[D. BMAS03] L. Daniel, J. White, "Automatic generation of geometrically parameterized reduced order models for integrated spiral RF-inductors", Proceedings of the 2003 IEEE International Workshop on Behavioral Modeling and Simulation, p 18-23, San Jose, CA, 2003.

[D. APS04] L. Daniel, J. White, “Numerical Techniques for Extracting Geometrically Parameterized Reduced Order Interconnect Models from Full-wave Electromagnetic Analysis”, IEEE AP-S Symposium, June 2004.

[D. DAC02] L. Daniel, J. Phillips, "Model Order Reduction for Strictly Passive and Causal Distributed Systems", IEEE/ACM 39th Design Automation Conference, New Orleans, Jun 2002.

[D. PhD04] Luca Daniel, "Simulation and Modeling Techniques for Signal Integrity and Electromagnetic Interference on High Frequency Electronic Systems," Ph.D. Thesis, University of California at Berkeley, May 2003.

11

OverviewOverview

• Problem SetupProblem Setup• Connection between circuits and State Space models Connection between circuits and State Space models • Reduction via eigenmode truncation methodReduction via eigenmode truncation method• Reduction via transfer function fittingReduction via transfer function fitting

– point matchingpoint matching– least squareleast square– quasi-convex optimization methodquasi-convex optimization method

• Reduction via Projection FrameworkReduction via Projection Framework– Truncated Balance RealizationsTruncated Balance Realizations– Krylov Subspace Moment MatchingKrylov Subspace Moment Matching

• need for orthogonalization (Arnoldi process)need for orthogonalization (Arnoldi process)• computational complexitycomputational complexity• passivity preservationpassivity preservation

• Reduction of Non-Linear SystemsReduction of Non-Linear Systems

12

State-Space Models State-Space Models

• Linear system of ordinary differential equations Linear system of ordinary differential equations (ABCD form) (ABCD form)

State Input

Output

)()()(

)()(

tDutCxty

tButAxdt

dxE

13

State-Space Model Example:State-Space Model Example:Interconnect Segment Interconnect Segment

• Step 1: Identify internal state variablesStep 1: Identify internal state variables– Example : MNA uses node voltages & inductor current Example : MNA uses node voltages & inductor current

1v 3v

LI

2v

14


• Step 2: Identify inputs & outputs Step 2: Identify inputs & outputs – Example : For Z-parameter representation, choose port Example : For Z-parameter representation, choose port

currents inputs and port voltage outputs currents inputs and port voltage outputs

in1I

in2I

out2vout

1v

1v 3v

LI

2v

15


• Step 3: Write state-space & I/O equations Step 3: Write state-space & I/O equations – Example : KCL + inductor equation Example : KCL + inductor equation

01211 inI

Rvv

dtdv

C

1out1 vv

3out2 vv

012 LI

Rvv 02

3 inL II

dtdv

C

32 vvdt

dIL L

in1I

in2I

out2vout

1v LI

16

A linear circuit can be expressed A linear circuit can be expressed as a state space modelas a state space model

• Step 4: Identify state variables & matrices Step 4: Identify state variables & matrices

L

C

C

E0

00

10

00

01

B

LI

v

v

v

x3

2

1

0110

1000

10

0011

11

RR

RR

A

0100

0001C

00

00D

in

in

I

Iu

2

1

2

1

v

vy

inIR

vv

dt

dvC 1

211

LIR

vv

120

inL II

dt

dvC 2

3

32 vvdt

dIL L

)()(

)()(

txCty

tButAxdt

dxE

T

LARGE!LARGE!LARGE!LARGE!

17

OverviewOverview






18

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

E

S S

SS S

S S S1S

1S 2S NS 1S 2S NS

S

19

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

Output Equation

S

S

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

c

S S

20

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

21

Reduced models via mode truncationDynamic Linear Case

1

q

i ii

y t c w t

Output Equation

)(

~

~

~1111

tu

b

b

b

w

w

w

w

w

w

N

q

N

q

N

q

N

q

22

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

23


Single Wire Results

N=100

Exact

q=1

q=3

q=10

Keep qth slowest modes

24

An Aside on Transfer Functions – Laplace Transform

x tBilateral Laplace Transform: stX s e dt

Consider an ODE: ( )

( )dx t

Ax t bu tdt

Key Transform Property: dx t

dtstsX s e dt

Rewrite the ODE in transformed variables

TsX s AX s bU s Y s c X s

1TY s c sI A bU s

H s

Transfer Function

25

An Aside on Transfer Functions – Meaning of H(s)

then ( ) j ty t H j e

For Stable Systems, H(jw) is the frequency response

If ( ) j tu t e

Sinusoid with shifted phase and amplitude

Sinusoid

H j

26

An Aside on Transfer Functions – EigenAnalysis

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

1 SSA S S

27

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

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 Drop terms with small

– Drop terms with small residuesDrop 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~~iiibc /~~

28

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)

29

OverviewOverview






30

Dynamical SystemDynamical System

Transfer FunctionTransfer Function

coefficientscoefficients


Counting Degrees of Freedom...Counting Degrees of Freedom...

qq

qq

q

q

q

q

sasa

sbsbb

ss

ssk

s

r

s

rsH

1

1110

1

11

1

1

1

)()(

)()(

)()()(

2q

q2+2q

)(ˆˆ)(ˆ

)(ˆ)(ˆˆˆ

txcty

tubtxAdt

xd

T

Degrees of Freedom?Degrees of Freedom?

Degrees of Freedom?Degrees of Freedom?

31

Reduced Model Transfer FunctionReduced Model Transfer Function

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

Many Dynamical Systems have the same transfer function!!

Fully Invertible Change of CoordinatesFully Invertible Change of Coordinates

)(ˆˆ)(

)(ˆ)(ˆˆˆ

txcty

tubtxAdt

xd

T

bAsIcsH T ˆˆˆ)(1

)(~)(ˆ txUtx

)(~ˆ)(

)(ˆ)(~ˆ~

txUcty

tubtxUAdt

xdU

T

)(~ˆ)(

)(ˆ)(~ˆ~

111

txUcty

tubUtxUAUdt

xdUU

T

I

)(ˆˆˆ)(~ 1

sHbAsIcsH T

32

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

33

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

34

• 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 35

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 36


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 37


Fitting Example

CoefficientValue

Coefficient number

Fitting f(t) = t

38

H s

Rational Transfer Function Fitting via Point MatchingRational Transfer Function Fitting via Point MatchingLeast Square MethodLeast Square Method

H s

• cross multiplying generates a cross multiplying generates a linear TALL SKINNY systemlinear TALL SKINNY system

• solve with Least Square methods, solve with Least Square methods, QR or Gauss-NewtonQR or Gauss-Newton

• get better H(s) accuracy, but get better H(s) accuracy, but produced model can be unstable: produced model can be unstable: needs post-processingneeds post-processing

Use many morepoints than order2q<<m

)(

)(

)(

)( 1

1

1

1

1111

mqqmmm

q

sH

sH

b

a

ssHs

ssHs

m

39

OverviewOverview






40

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 e.g. 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

41

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]

42

• 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

43

A standard optimization technique:Relaxation

• Original problem is difficult• Made easier if some constraints are dropped (relaxed)• Solve the relaxed problem • Construct original solution from relaxation

General idea

-c

feasible set…

optimal solution-c

feasible set

optimal relaxed solution

nearest rounding

44 44

Relaxation of the H-inf norm MOR setup [Sou, Megretski, Daniel DAC05 TCAD08]

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 programDrawback: May obtain suboptimal solutions

Anti-stableterm

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

Passivity, and possibly other constraints

45 45

How bad is the 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:

47 47

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

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

Stability:

Passivity:

48 48

Solving the quasi-convex program

, ,

( )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

convex setStability:

Passivity:

quasi-convex set

Standard problem.Use for example by the ellipsoid algorithm

50

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…

51

Example 1: 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

52

Example 2: 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

53

Example 3: 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

54

Example 4: Large IC power distribution grid(from field solver)

• Power distribution grid (dimension size = 7mm, wire width = 2 µm)• Blue: full model (order 2046)• 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

2 curves on top of each other

3 curves on top of each other

55

Parameterized stability [Sou TCAD08]

• Checking stability can be done checking positivity of a multivariable trigonometric polynomial• Check positivity by sum of squares argument (semidefinite program)

,min

subject to W ,

0

y W Wy

y a z p

W

*, W W 0a z p y

• Otherwise, claim a(z,p) 0 at some point

• If feasible and y* < 0, then

56

Example: Parameterized RF inductor

0 2 4 6 8 10 12

x 109

-5

0

5

10

15

20

25

30

35

40

frequency (Hz)

qual

ity fa

ctor

Wire width = 16.5 µm,separation = 1,5,18,20µm

----- full model___ QCO PMOR

Worst case errorin amplitude = 4%

57

OverviewOverview






58

Dynamical SystemDynamical System

Transfer FunctionTransfer Function



Remember Remember Counting Degrees of Freedom...Counting Degrees of Freedom...

nn

nn

n

n

n

n

sasa

sbsbb

ss

ssk

s

r

s

rsH

1

1110

1

11

1

1

1

)()(

)()(

)()()(

2n

n2+2n

)()(

)()(

txcty

tubtxAdt

dx

T

Degrees of FreedomDegrees of Freedom

Degrees of FreedomDegrees of Freedom

59

Reduced Model Transfer FunctionReduced Model Transfer Function

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

Many Dynamical Systems have the same transfer function!!

Remember the idea of theRemember the idea of theFully Invertible Change of Coordinates?Fully Invertible Change of Coordinates?

)()(

)()(

txcty

tubtxAdt

dx

T

bAsIcsH T 1)(

)(~)( txUtx

)(~)(

)()(~~

txUcty

tubtxAUdt

xdU

T

)(~)(

)()(~~

111

txUcty

tubUtxAUUdt

xdUU

T

I

)()(~ 1 sHbAsIcsH T

60

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

E

S S

SS S

S S S1S

1S 2S NS 1S 2S NS

S

61


)(

~

~

~1111

tu

b

b

b

w

w

w

w

w

w

N

q

N

q

N

q

N

q

62

1

1

1

ˆ

ˆq

q

N

q

x

x

u u

x

x

U

Projection Framework:Projection Framework:Non invertible Change of CoordinatesNon invertible Change of Coordinates

Note: q << NNote: q << N

reduced statereduced state

original stateoriginal state

63

• Original System Original System

Projection FrameworkProjection Framework

• Note: now few variables (q<<N) in the state, but still Note: now few variables (q<<N) in the state, but still thousands of equations (N)thousands of equations (N)

)()(

)()(

txcty

tubtxAdt

dx

T

• SubstituteSubstitute

)(ˆ)(ˆ

)()(ˆ)(ˆ

txUcty

tubtxUAdt

txdU

qT

qq

)(ˆ txUx q

64

)(ˆ)(

)()(ˆ)(ˆ

txUcty

tubVtxUAVdt

txdUV

qT

Tqq

Tqq

Tq

Projection Framework (cont.)Projection Framework (cont.)

• If If VqT and and Uq

T are are

chosen biorthogonal chosen biorthogonal

Tq qV U I

)(ˆ)(

)()(ˆ)(ˆ

txUcty

tubtxUAdt

txdU

qT

qq

Reduction of number of equations: test by multiplying by Vq

T

)(ˆˆ)(

)(ˆ)(ˆˆ)(ˆ

txcty

tubtxAdt

txd

T

65

Projection Framework (graphically)Projection Framework (graphically)

nxnnxn

x TqV bu

nxqnxq

qU xdt

dxA

Eqxqqxq

qxnqxn

TqV

dt

xdˆ

nxqnxq

qU

)(ˆ)(ˆˆˆtubtxA

dt

xd

66

• Use Eigenvectors of the system matrix (modal analysis)Use Eigenvectors of the system matrix (modal analysis)

• Use Frequency Domain DataUse Frequency Domain Data– ComputeCompute

– Use the SVD to pick Use the SVD to pick q < kq < k important vectors important vectors

• Use Time Series DataUse Time Series Data– ComputeCompute

– Use the SVD to pick Use the SVD to pick q < kq < k important vectors important vectors1 2( ), ( ), , ( )kx t x t x t

1 2( ), ( ), , ( )kx s x s x s

Approaches for picking V and UApproaches for picking V and U

II.2.b SVD Singular Value Decomposition other names: POD Proper Orthogonal Decompos.or KLD Karhunen-Lo`eve Decompositionor PCA Principal Component Analysisor Poor Man TBR

Point Matching

67







• Use Singular Vectors of System Grammians Product Use Singular Vectors of System Grammians Product (Truncated Balance Realizations)(Truncated Balance Realizations)

• Use Krylov Subspace Vectors (Moment Matching)Use Krylov Subspace Vectors (Moment Matching)

1 2( ), ( ), , ( )kx t x t x t

1 2( ), ( ), , ( )kx s x s x s

68

Energy of the output Energy of the output y(t)y(t) starting from state starting from state x x with no input:with no input:

Observability GrammianObservability Grammian

2)(

xty

0

xdtCexCe AtTAt xdtCeCex AtTtAT T

0

0W

0

)()( dttyty T

Observability Grammian:

Note: If Note: If x=xx=xii the i-th eigenvector of the i-th eigenvector of WWo o ::

ioT

ixxWxty

i

2)( io,

Hence: eigenvectors of Hence: eigenvectors of WWoo corresponding to corresponding to smallsmall

eigenvalues do NOT produce much energy at the outputeigenvalues do NOT produce much energy at the output(i.e. they are not very observable):(i.e. they are not very observable): Idea: let’s get rid of them!Idea: let’s get rid of them!

CCAWWA To

T 0Note: it is also the solution of

69

Minimum amount input energy required to drive theMinimum amount input energy required to drive thesystem to a specific state system to a specific state x x ::

Controllability GrammianControllability Grammian

xdteBBex tATAtT T

0

1cWInverse of Controllability Grammian:

Note: If Note: If x=xx=xii the i-th eigenvector of the i-th eigenvector of WWcc::

ic

Tix

xWxtui

12)(min

0

)()(min dttutu T

Hence: eigenvectors of Hence: eigenvectors of WWcc corresponding to corresponding to small small

eigenvalues do require a lot of input energy in ordereigenvalues do require a lot of input energy in orderto be reached (i.e. they are not very controllable): to be reached (i.e. they are not very controllable): Idea: let’s get rid of them!Idea: let’s get rid of them!

TTCC BBAWAW

It is also the solution of

ic,

1

70

Naïve Controllability/Observability MORNaïve Controllability/Observability MOR

• Suppose I could compute a basis for the strongly Suppose I could compute a basis for the strongly observable and/or strongly controllable spaces. observable and/or strongly controllable spaces. Projection-based MOR can give a reduced model that Projection-based MOR can give a reduced model that deletes weakly observable and/or weakly controllable deletes weakly observable and/or weakly controllable modes. modes.

• Problem: Problem: – What if the same mode is strongly controllable, but weakly What if the same mode is strongly controllable, but weakly

observable?observable? – Are the eigenvalues of the respective Gramians even Are the eigenvalues of the respective Gramians even

unique? unique?

71

Changing coordinate systemChanging coordinate system

• Consider an invertible change of coordinates: Consider an invertible change of coordinates: • We know that the input/output relationship will be We know that the input/output relationship will be

unchanged.unchanged.• But what about the the Grammians, and their eigenvalues?But what about the the Grammians, and their eigenvalues?

• Grammians and their eigenvalues change! Hence the Grammians and their eigenvalues change! Hence the relative degrees of observability and controllability are relative degrees of observability and controllability are properties of the coordinate systemproperties of the coordinate system

• A bad choice of coordinates will lead to bad reduced A bad choice of coordinates will lead to bad reduced models if we look at controllability and observability models if we look at controllability and observability separately. separately.

• What coordinate system should we use then?What coordinate system should we use then?

)(~)( txUtx

UWUW oT0

~ Tcc UWUW 1~

72

Fortunately the eigenvalues of the Fortunately the eigenvalues of the product product (Hankel singular (Hankel singular values) do not change when changing coordinates:values) do not change when changing coordinates:

BalancingBalancing

120

SSWWc

)(~)( txUtx

TcUWU 1 UWWU c 0

1 1121 )()( SUSU

Diagonal matrix with eigenvalues of the product

The eigenvectors change

And since And since WWc c and and WWoo are symmetric a are symmetric a

change of coordinate matrix change of coordinate matrix UU can be can be found that diagonalizes both:found that diagonalizes both:

2 In Balanced coordinates the Grammians In Balanced coordinates the Grammians are equal and diagonalare equal and diagonal

But not the eigenvalues

UWU T0

73

Selection of vectors for the columns of Selection of vectors for the columns of the reduced order projection matrix.the reduced order projection matrix.

20

1 UWUUWU TTc

In balanced coordinates it is easy to select the best In balanced coordinates it is easy to select the best vectors for the reduced model: we want the subspace of vectors for the reduced model: we want the subspace of vectors that are at the same time most controllable and vectors that are at the same time most controllable and observable: observable:

In other words the ones corresponding to the largestIn other words the ones corresponding to the largesteigenvalues of the controllability and observability eigenvalues of the controllability and observability Grammians product.Grammians product.

simply pick the eigenvectors simply pick the eigenvectors corresponding to the largest corresponding to the largest entries on the diagonal entries on the diagonal (Hankel singular values).(Hankel singular values).

74

Truncated Balance Realization SummaryTruncated Balance Realization Summary

• The good news:The good news:– we even have we even have bounds for the errorbounds for the error– Can do even a bit better with the optimal Hankel Reduction

• The bad news:The bad news:– it is it is expensiveexpensive::

• need to compute the Gramians (solve Lyapunov equation)need to compute the Gramians (solve Lyapunov equation)• need to compute eigenvalues of the product: need to compute eigenvalues of the product: O(NO(N33))

• The bottom line:The bottom line:– If the size of your system allows you O(NIf the size of your system allows you O(N33) computation, ) computation,

Truncated Balance Realization is a much better choice than the Truncated Balance Realization is a much better choice than the any other reduction methodany other reduction method..

– But if you cannot afford O(NBut if you cannot afford O(N33) computation (e.g. dense matrix ) computation (e.g. dense matrix with N > 5000) then PRIMA or PVL or Quasi-Convex-Optimization with N > 5000) then PRIMA or PVL or Quasi-Convex-Optimization are better choicesare better choices

NNqqq jHjH ,1,1 ...2)()(

CCAWWA To

T 0

75







• Use Singular Vectors of System Grammians Product Use Singular Vectors of System Grammians Product (Truncated Balance Realizations)(Truncated Balance Realizations)

• Use Krylov Subspace Vectors (Moment Matching)Use Krylov Subspace Vectors (Moment Matching)

1 2( ), ( ), , ( )kx t x t x t

1 2( ), ( ), , ( )kx s x s x s

76

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

Assuming Assuming AA is non-singular is non-singular we can cast the dynamical we can cast the dynamical linear system into a linear system into a canonical form for moment canonical form for moment matching model order matching model order reductionreduction

Note: this step is not Note: this step is not necessary, it just makes the necessary, it just makes the notation simple for notation simple for educational purposeseducational purposes

xcy

ubAxsxT

xcy

ubxsExT

bAb

AE1

1

77

H s

H s

Taylor series expansion:

2 xx b Ab A b

UqUq

The Moment Matching idea [Grimme PhD97]The Moment Matching idea [Grimme PhD97]

2span , , ,x b Eb E b

change basis: use only first q vectors of the Taylor series expansion matching first q derivatives around expansion point

0

k k

k

x s E b u

sEx x bu 1x I sE bu

2b Eb E b

78

Krylov Subspaces - DefinitionKrylov Subspaces - Definition

2 1, , , , kk E b span b Eb E b E b

The order The order kk Krylov subspace generated Krylov subspace generated from matrix E and vector b is defined asfrom matrix E and vector b is defined as

79

Moment matching around non-zero frequenciesMoment matching around non-zero frequencies

In stead of expanding around only s=0 we can expand around another points 1 20 Js s s s s s s

xcy

buAxsxT

xcy

buAxxssT

h

)~(

xcy

ubxxEsT

hh

~

bIsAb

IsAE

hh

hh

1

1

For each expansion point the problem can then be put again in the canonical form

hsss ~

80

IfIf

andand

ThenThen

Projection Framework: Projection Framework: Moment Matching Theorem (E. Grimme 97)Moment Matching Theorem (E. Grimme 97)

hh

J

hkq bEU b

h,)(Range

1

hTh

J

hkq cEV c

h,)(Range

1

Total of 2q moment of the transfer function will match

,...,1

1,...,0for

ˆ

Jh

kkl

s

H

s

H bh

bh

s

l

l

s

l

l

hh

81

Combine point and moment matching: Combine point and moment matching: multipoint moment matchingmultipoint moment matching

H s

H s

• Multipole expansion points give larger bandMultipole expansion points give larger band• Moment (derivates) matching gives more Moment (derivates) matching gives more accurate behavior in between expansion pointsaccurate behavior in between expansion points

82

Moment Matching Moment Matching Parameterized Model Order ReductionParameterized Model Order Reduction

ubxEsWdEWE 1,1,10,2,02

0,0,0 ......

21 2 1 1 2 2 1, , , , , , ,px span b E b E b E b E b E E E E b

ˆq xx U

Uq

It is a p-variables Taylor series expansion

buEsEsEsxm

mpp 2211

Once again change basis:Once again change basis: use first few vectors of the use first few vectors of the

Taylor expansion,Taylor expansion, matching first few matching first few

derivatives with respect to derivatives with respect to each parametereach parameter

complexity O(q N log N)complexity O(q N log N)

83

ubxEsWdEWE ˆˆˆ...ˆ...ˆ1,1,10,2,0

20,0,0

ubxEsWdEWE

1,1,10,2,02

0,0,0 ......

TqU

nxnnxn

qxqqxq

nxqnxq

qxnqxn

Congruence transformations on each of the matrices

kjiE ,,

kjiE ,,ˆ

qU

Moment Matching Moment Matching Parameterized Model Order ReductionParameterized Model Order Reduction

84 84

Example: Parameterized RF inductor[Daniel, White, BMAS03]

Wire width = 5umseparation = 1, 2, 3, 4, 5um

frequency [ x10GHz ]

Worst case errorin amplitude = 4%__ full system (field solver)--- moment matching

Q

85

OverviewOverview






86

b

Eb

2E b3E b

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

Need for Orthonormalization of UNeed for Orthonormalization of U

Vectors {b,Eb,...,Ek-1b} cannot be computed directly

87

1 1

1

1,

1i i

i

i i

u uu

Normalize new vectorNormalize new vector

1 /u b b

For i = 1 to k

1i iu Eu

Generates new Krylov Generates new Krylov subspace vectorsubspace vector

1 1 1T

i i i j j

ji

u u u u u

Orthogonalize new vectorOrthogonalize new vector

For j = 1 to i

Orthonormalization of U: The Arnoldi AlgorithmOrthonormalization of U: The Arnoldi Algorithm

Normalize first vectorO(n)

O(?????????)O(?????????)

O(kO(k22n)n)

O(n)O(n)

Computational ComplexityComputational Complexity

88

Most of the computation cost is spent in calculating:

Set up and solve a linear system using GCR

If we have a good preconditioners and a fast matrix vector product each new vector is calculated in O(n)

The total complexity for calculating the projection

matrix Uq is O(qn)

Generating vectors for the Krylov subspaceGenerating vectors for the Krylov subspace

ih uIsA1)(

iih uuIsA

1)(

ihi uEu

1

89

What about computing the reduced matrix ?What about computing the reduced matrix ?

iq

i

qi uuuE

,

,1

1 ...

qTq EUUE ˆ

qq uuuuE

...... 11 qq UEU

qTq EUU

E

Orthonormalization of the i-th column of Uq

Orthonormalization of all columns of Uq

So we don’t need to computethe reduced matrix. We have it already:

90

Model Order ReductionComputational Complexity (time and memory)

MOR technique Computational Complexity

TBR (‘81) Hankel (‘84)(have error bounds)

Bottleneck: SVDO(n3)

e.g. 10months, 80GB, for N=100,000

Moment Matching + pFFT matrix-vector product

O(q n log n)e.g. 8hours, 0.3GB for N=100,000 q=10

Moment matching (’97)(no error bounds)

Bottleneck: matrix-vector productO(q n2)

e.g. 7days, 80GB, for N=100,000 q=10

QuasiConvex Optimization + pFFT field solver

O(m n log n)Same as above, but have error bound!

)()(),(),( tuBtxwdAdt

dxwdE

Two Complementary Approaches • Moment Matching Approaches

– Accurate over a narrow band. • Matching function values and

derivatives.

– Cheap: O(qn).– Use it as a FIRST STAGE

REDUCTION

• Truncated Balanced Realization and Hankel Reduction

– Optimal (best accuracy for given size q, and apriori error bound.

– Expensive: O(n3)– USE IT AS A SECOND STAGE

REDUCTION

92

OverviewOverview






93

Need to preserve passivityNeed to preserve passivityfor models of passive interconnectfor models of passive interconnect

PCB, package, IC PCB, package, IC interconnects interconnects

Analog or digital IP Analog or digital IP blocksblocks

Z(f)Z(f)

Note: passive! Note: passive! Designers will connect models use Designers will connect models use them in ODE time domain simulators.them in ODE time domain simulators.If the models are not passive they If the models are not passive they can generate energy and thecan generate energy and thesimulation may explode!! simulation may explode!!

Picture byPicture byM. ChouM. Chou

QD

C

-

+

+-

Picture byPicture byJ. PhillipsJ. Phillips

94

Interconnecting Passive Systems Interconnecting Passive Systems

QDC

-++-

QDC

-++-

QDC

-++-

QDC

-++-

The interconnection of stable models is not necessarily stable

BUT the interconnection of passive models is a passive model:

95

Sufficient conditions for passivitySufficient conditions for passivity

Sufficient conditions for passivity:

Cxy

BuAxsEx

xAxx

xExx

BC

T

T

T

allfor ,0)3

allfor ,0)2

)1

Note that these are NOT necessary conditions (common misconception)

i.e. E is positive and A is negative semidefinite

Example IC wire discretized into RC line

x

1 1

( )NxN Nx

scalarinp

T

Nxscalarouu tt put

y tdx t

A x t b u t c x tdt

2 1 0 0

1 2

0 0

2 1

0 0 1 1

A

A

1

0

0

b

1

0

0

c

Matrix is negative semidefinite.

97

Congruence Transformation Preserves Definiteness of Congruence Transformation Preserves Definiteness of E and A (hence passivity and stability) [PRIMA97]E and A (hence passivity and stability) [PRIMA97]

If we use Tq

Tq UV

• Then we loose half of the degrees of freedom i.e. we match only q moments instead of 2q• But if the original matrix E is negative semidefinite so is the reduced hence the system is passive and stable

nxnnxn

nxqnxq

qU xs x TqU bu

nxqnxq

qUx xEqxnqxn

TqV

nxnnxn

s Aqxnqxn

TqV

98

Step 2. Model Order Reduction

Fitting Based,System ID

Least SquareCoelho99

PRIMA97(only if A>0)Moment,

MatchingAWE Pillage90,PVL Feldman94

OptimalTBR 81

Hankel 84

Proof Daniel TCAD04

BasicTechnique

Parameterized PassivityPreservation

Guaranteed Passive Quasi-Convex Optimiz. Daniel TCAD08

Daniel ICCAD08any A

best paper

Daniel DAC02 best paper,

Daniel TCAD04

POD, KVL,PCA, SVD

Wilcox Peraire91,PMTBR04 PMTBR04

Weile99,Gunupudi00,

Heydari01

Coelho01if poles given

Daniel ICCAD08any A

best paper

99

dt

dEH

dt

dHE

dt

dEH

dt

dHE

dt

dEH

dt

dHE

dt

dEH

dt

dHE

The two steps modeling process

Step 1. Field SolversStep 1. Field Solvers

Step 2. Model Order ReductionStep 2. Model Order Reduction

)()( tuBtxAdt

dxE

)(ˆ)(ˆˆˆˆ tuBtxAdt

xdE

(10(1066 x 10 x 1066))

nanophotonics

carbon nanotubes

RF inductorswinterconnect

Very difficult to control stability/passivity at this step

Potentiallyunstable and unstructured

Need to enforce stability/passivity at this step

(10(1066 x 10 x 1066))

100

Optimal Stabilizing Projection: Stability Constraints

AVzUzEVU TT E A

Stable Reduced Model

AxxE

V

U

AE,

0U

sUU 0min

Stability Constraints

Field-Solver Model

Any Projection-BasedReduction Technique

Q

f

Reduced ModelsFull model

xxV matches moments

Space of stabilizing

U

Space of U

0U

101


AVzUzEVU TT E A


AxxE

V

U

AE,

0U

sUU 0min


Field-Solver Model



U0U

0ˆˆ EVUPUAVAVUPUEV TTTTTT Difficult to solve: Quadratic in U

Stability Constraints0ˆ EVUPUEV TTT

102


AVzUzEVU TT E A


AxxE

V

U

AE,

0U

sUU 0min


Field-Solver Model



U0U


0UAVAVU TTT 0EVU T

Constraints now linear in U

Force reduced model to be definite

103

109

1010

-50

0

50

100

150

200

250

300

350

Large system should be stable and passive

Extracted with EMQS field solver

0 1 2 3 4 5 6 7

x 10-10

-0.02

0

0.02

0.04

0.06

0.08

0.1

Example: RF Inductor

-2.5 -2 -1.5 -1 -0.5 0 0.5

x 10-9

-1.5

-1

-0.5

0

0.5

1

1.5x 10

-12

-10 -5 0 5

x 10-15

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

-12

xx

N=961

q=10

Re

Im

Re

Im

)(tyQ

f

Field Solver ModelPRIMA

Proposed Method

1.0 3.0 5.0 time (ns)

910 1010

104

109

1010

101

102

103

Example: Power Grid

N = 1566

q = 10

Large system should be stable and passive

Extracted with EMQS field solver

Re

Imxx

)(ty

f)(Re jH

)(Im jH

Field Solver Model

PRIMA

Proposed Method

time (ns)1 3 5 7

910 1010

105

OverviewOverview






106

Parameterized Model Order ReductionParameterized Model Order Reductionof Non-Linear Systemsof Non-Linear Systems

Non-Linear analog components

e.g. MEMs, VCO, LNADistributed Amplifiers

)()),(( tuBptxFdt

dx

)(ˆ)),(ˆ(ˆˆtuBptxF

dt

xd

Parameterized NonLinear ReductionParameterized NonLinear Reduction

PDE Field Solvers or Circuit SimulatorsPDE Field Solvers or Circuit Simulators

107

Previous work on Non-ParameterizedPrevious work on Non-ParameterizedMOR for Non-Linear systemsMOR for Non-Linear systems

Representation of Representation of F(x)F(x) using using a polynomiala polynomial (e.g. Taylor’s (e.g. Taylor’s expansions, Volterra Series) [Phillips00]expansions, Volterra Series) [Phillips00]

Representation of Representation of F(x)F(x) with with several polynomialsseveral polynomials (PWP (PWP PieceWise Polynomial) [Dong03]PieceWise Polynomial) [Dong03]

Representation of Representation of F(x)F(x) using using several linearizationsseveral linearizations (Trajectory Piece-Wise Linear TPWL) [Rewienski01](Trajectory Piece-Wise Linear TPWL) [Rewienski01]

...)())(()(

))(()(

000

000

xxxtxxW

xxxJxFxF

...

))(()()(

))(()()(

222222

111111

xxxJxFxw

xxxJxFxwxF

108

x3

x11

x7

x10

x6

x8 x9

x5x4

x0

x1

x2

x1 x20J 3J2J1J

4J 5J 6J7J

8J 9J 10J 11J

#linearization#linearizationss =#samples=#samplesnn

n = 10n = 1044

#samples = 100#samples = 100

Background – TPWL [Rewienski01]Construction of Linearized Systems

109

Background – TPWL [Rewienski01] Background – TPWL [Rewienski01] Construction of Linearized SystemsConstruction of Linearized Systems

x1

x2

t

y(t)

Use training trajectories to pick linearization points

Error

Linearization at current state xi

State SpaceTime Domain Simulation

110

Background – TPWL [Rewienski01] Background – TPWL [Rewienski01] Reduction of the Linearized SystemsReduction of the Linearized Systems

dt

dx tbu

Model from linearization 1

A1A1 K1K1x1w 2w kw

Model from linearization 2

A2A2 K2K2x

Model from linearization k

AkAk KkKkx

dt

xdˆ1A 2A kA

1K 2KkK

tub x x x1w2w kw

=UTU

iA iA

Moment matching projection into a single subspace

111

x0

Background – TPWL [Rewienski01]Background – TPWL [Rewienski01]Possible Behaviors and LimitationsPossible Behaviors and Limitations

x1

x2

x1

x2x3

A

C – poorlyapproximated

B

wellapproximated

112

Non-Linear Parameterized Reduction[Bond, Daniel TCAD07]

x1

x2

time

y(t)

Use training trajectories to pick linearization points

s1

s2

1s

2sTrain again at different points in parameter space

Parameter Space

Repeat at different points in parameter space to populate state space

State Space

113

Non-Linear Parameterized Reduction[Bond, Daniel TCAD07]

1s

2s

4s

5s 3s

Populate relevant regions of state-space with linear models from training at different inputs or different points in parameter-space

Parameter Space

s2

s1

x1

x2

State Space

114

Parameterized Moment Matching Projection into a single subspace

For qth order model with p parameters and k linear models, total number of vectors for U is O(kpm)

Keeping all vectors would results in a huge “reduced” model

Hence we perform an SVD on U and keep only the q most important vectors

SVD

NN

O(kpm) q

U U

115

Model from a Discretized Non-Linear Transmission Line

1exp

Tdd v

VIVi

Threshold voltageOff state current

POSSIBLE PARAMETERS:

Resistance per unit length r

Capacitance per unit length C

116

Model from a Discretized Non-Linear Transmission Line

u(t) = 1 + sin(wt) + sin(5wt)

Our model

117

Micromachined Switch Example

t

pzppzK

t

zhwdypp

z

wv

x

zhwS

x

zwhEI

w

a

1261

2

3

2

2

0

02

20

2

2

04

43

0

Governing Equations

Picture by Michał Rewienski

118

State-Space System

3331

1131

323

2

0

041

43

021

2

0

0

30

31

31

222

31

21

6112

1

3

2

33

3

2

3

xxxxxx

xx

dt

dx

vhx

xwhEI

x

xhwSdypx

hw

x

x

x

dt

dx

x

x

dt

dx

w

a

Beam width

Material Properties

Beam height

Possible Parameters

Micromachined Switch ExampleDiscretization by Finite Difference PDE solver

119

Micromachined Switch Example

Our model

120

SummarySummary






1 parasitic extraction step 2: model order reduction luca daniel massachusetts institute of...

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