introduction to model order reduction ii.2 the projection framework methods luca daniel...

Introduction to Model Order ReductionIntroduction to Model Order Reduction

II.2 The Projection Framework Methods II.2 The Projection Framework Methods

Luca DanielLuca Daniel

Massachusetts Institute of TechnologyMassachusetts Institute of Technology

with contributions from:with contributions from:

Alessandra Nardi, Joel Phillips, Jacob WhiteAlessandra Nardi, Joel Phillips, Jacob White

2

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

3

• 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

4

)(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

5

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

6

spaceqV

Equation TestingEquation Testing(Projection)(Projection)

ˆqx xU

x

spaceqU

Non-invertible changeNon-invertible changeof coordinates (Projection)of coordinates (Projection)

x

Projection FrameworkProjection Framework

)()( tubtxAdt

dx )(ˆ)(ˆˆˆ

tubtxAdt

xd

xAAxVxAUV Tqq

Tq ˆˆˆ

AAx

xAˆˆ

qT

q AUVA ˆ

7

• 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 POD Principal Component Analysisor SVD Singular Value Decompositionor KLD Karhunen-Lo`eve Decompositionor PCA Principal Component Analysis

Point Matching

8

• Use Eigenvectors of the system matrixUse Eigenvectors of the system matrix

• POD or SVD or KLD or PCA.

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

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


9

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

10

H s

H s

Taylor series expansion:Taylor series expansion:

2 xx b Ab A b

UU

Intuitive view of Krylov subspace choice for Intuitive view of Krylov subspace choice for change of base projection matrixchange of base projection matrix

2span , , ,x b Eb E b

• change base and use only the first few change base and use only the first few vectors of the Taylor series expansion: vectors of the Taylor series expansion: equivalent to match first derivatives equivalent to match first derivatives around expansion pointaround expansion point

0

k k

k

x s E b u

sEx x bu 1x I sE bu

2b Eb E b

11

Aside on Krylov Subspaces - DefinitionAside on Krylov Subspaces - Definition

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

The order k Krylov subspace generated The order k Krylov subspace generated from matrix A and vector b is defined asfrom matrix A and vector b is defined as

12

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 ~

13

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

14

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

15

Compare Pade’ ApproximationsCompare Pade’ Approximationsand Krylov Subspace Projection Frameworkand Krylov Subspace Projection Framework

Krylov Subspace Krylov Subspace Projection Framework:Projection Framework:• multipoint moment multipoint moment matchingmatching• AND numerically very AND numerically very stable!!!stable!!!

H s

H s

H s

H s

Pade approximations:Pade approximations:• moment matching at moment matching at single DC pointsingle DC point• numerically very numerically very ill-conditioned!!!ill-conditioned!!!

16



• Use Krylov Subspace Vectors (Moment Matching)Use Krylov Subspace Vectors (Moment Matching)– general Krylov Subspace methodsgeneral Krylov Subspace methods

– case 1: Arnoldicase 1: Arnoldi

– case 2: PVLcase 2: PVL

– case 3: multipoint moment matchingcase 3: multipoint moment matching

– moment matching preserving passivity: PRIMAmoment matching preserving passivity: PRIMA



17

IfIf U and V are such that: U and V are such that:

ThenThen the first q moments (derivatives) of the the first q moments (derivatives) of the reduced system matchreduced system match

1,..., qU V u u

11,..., ( , ) , , , q

q qspan u u E b span b Eb E b

TU U I

Special simple case #1: Special simple case #1: expansion at s=0,V=U, orthonormal Uexpansion at s=0,V=U, orthonormal UTTU=I U=I

0 s=0

ˆ for k 0, ,

k k

k k

s

H Hq

s s

ˆˆˆ T k T kc E b c E b

18

1

0

T T k k

k

H s c I sE b c E b s

1

0

ˆ ˆˆ ˆ ˆˆ ˆT T k k

k

H s c I sE b c E b s

ˆˆˆ

kT k T T Tq q q q

T T T T Tq q q q q q q q

k

c E b c U U EU U b

c U U EU U EU U EU U b

Algebraic proof of case #1: Algebraic proof of case #1: expansion at s=0, V=U, orthonormal Uexpansion at s=0, V=U, orthonormal UTTU=IU=I

T T k

k

c EE E b c E b

apply k times lemma in next slideapply k times lemma in next slide

19

Lemma: . Lemma: . Tq qU U b b

T Tq q q q q qU U b U U U g U g b

1,..., s.t.

q qb span u u g b U g

Note in general: Note in general:

Tqq nUU I

BUT...BUT...

Substitute:Substitute:

IIqq U is orthonormalU is orthonormal

20

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

21

Need for Orthonormalization of U Need for Orthonormalization of U (cont.)(cont.)

• In "change of base matrix" U In "change of base matrix" U transforming to the new transforming to the new reduced state space, we can use reduced state space, we can use ANY columns that span the ANY columns that span the reduced state spacereduced state space

• In particular we can In particular we can ORTHONORMALIZE the Krylov ORTHONORMALIZE the Krylov subspace vectorssubspace vectors

1

1

1

q

r

q

r

N

q

x

x

u u

x

x

U

i ju u i j

22

1 1

1

1,

1i i

i

i i

u uu

Normalize new vectorNormalize new vector

1 /u b b

For i = 1 to q

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)

sparse: O(n) dense:O(nsparse: O(n) dense:O(n22))

O(qO(q22n)n)

O(n)O(n)

Computational ComplexityComputational Complexity

23

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

iuA1

ii uuA

1

ihi uEu

1

24

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:

25










26

ThenThen the first the first 2q2q moments of reduced system match moments of reduced system match

IfIf U and V are such that: U and V are such that:

11,..., ( , ) , , , ( )

��T T T q

q qspan v v E c span c E c E c

TV U I

Special case #2: Special case #2: expansion at s=0, biorthogonal Vexpansion at s=0, biorthogonal VTTU=IU=I

11,..., ( , ) , , , q


0 s=0

ˆ for 0, , 2

k k

k k

s

H Hk q

s s

ˆˆˆ T k T kc E b c E b

27

1 2 2

0

( ) ( )

Tk kT T k

k

H s c I sE b E c E b s

Proof of special case #2: expansion at s=0, Proof of special case #2: expansion at s=0, biorthogonal Vbiorthogonal VTTU=UU=UTTV=IV=Iqq (cont.) (cont.)

12 2

0

ˆ ˆˆ ˆ ˆ ˆˆ ˆ( ) ( )

Tk kT T k

k

H s c I sE b E c E b s

2 2 2 2

2 2

ˆˆ ˆ ˆ ˆˆ( ) ( ) ( ) ( )

ˆ ˆ ˆ ˆ

( ) ( )

T Tk k k kT T T T T T

TT T T T T T T T

Tk kT T

E c E b U E V U c V EU V b

U E V U E VU c V EU V EUV b

E c E b

apply k times the lemma in next slideapply k times the lemma in next slide

28

Lemma: . Lemma: . Tq qU V b b

T Tq q q q q qU V b U V U g U g b

1,..., s.t.

q qb span u u g b U g


Tq qV U c c

T Tq q q q q qV U c V U V f V f c

1,..., s.t.

q qc span v v f c V f


IIqq biorthonormalitybiorthonormality

IIqq biorthonormalitybiorthonormality

29

PVL: Pade Via LanczosPVL: Pade Via Lanczos[P. Feldmann, R. W. Freund TCAD95][P. Feldmann, R. W. Freund TCAD95]

• PVL is an implementation of the biorthogonal case 2:PVL is an implementation of the biorthogonal case 2:

11,..., ( , ) , , , ( )

��T T T q

q qspan v v E c span c E c E c

TV U I

11,..., ( , ) , , , q


Use Lanczos process to Use Lanczos process to biorthonormalize the columns of U biorthonormalize the columns of U and V: and V: gives very good numerical gives very good numerical stabilitystability

30

Example: Simulation of voltage gain of a filter Example: Simulation of voltage gain of a filter with PVL (Pade Via Lanczos)with PVL (Pade Via Lanczos)

31

Compare to Pade via AWE Compare to Pade via AWE (Asymptotic Waveform Evaluation)(Asymptotic Waveform Evaluation)

32










33

Case #3: Intuitive view of subspace choice for general Case #3: Intuitive view of subspace choice for general expansion pointsexpansion points

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 ~

34

xcy

ubIsAxxIsAsT

hh

11~

Case #3: Intuitive view of Krylov subspace choice Case #3: Intuitive view of Krylov subspace choice for general expansion points (cont.)for general expansion points (cont.)

matches first matches first kkjj of transfer of transfer

function function around each around each expansion expansion point spoint sjj

Hence choosing Krylov subspaceHence choosing Krylov subspace

ss11=0=0

ss11ss22

ss33

bIsAIsAU hh

J

hkq b

h

11

1

,)(Range

35

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

36










37

Sufficient conditions for passivitySufficient conditions for passivity

Sufficient conditions for passivity:

sx Ax Bu

y Cx

1)

2) 0, for all

T

T

C B

x Ax x

Note that these are NOT necessary conditions (common misconception)

i.e. A is negative semidefinite

Example Finite Difference System from on Poisson Equation (heat problem)

endT

x

1 1

( )NxN Nx

scalarinp

T

Nxscalarouu tt put

y tdx t

A x t b u t c x tdt

Heat In0 0T

2 1 0 0

1 2

0 0

2 1

0 0 1 1

A

A

1

0

0

b

1

0

0

c

We already know the Finite Difference matrices is positive semidefinite. Hence A or E=A-1 are negative semidefinite.

39



Cxy

BuxsEx

xExx

BCT

T

allfor ,0)2

)1


i.e. E is negative semidefinite

40

Congruence Transformations Congruence Transformations Preserve Negative (or positive) SemidefinitnessPreserve Negative (or positive) Semidefinitness

• Def. congruence transformationDef. congruence transformation EUUE Tˆ

same matrix

• Note: case #1 in the projection framework V=U produces Note: case #1 in the projection framework V=U produces congruence transformationscongruence transformations

• Lemma: a congruence transformation preserves the Lemma: a congruence transformation preserves the negative (or positive) semidefiniteness of the matrixnegative (or positive) semidefiniteness of the matrix

• Proof. Just renameProof. Just rename

x 0x then 0 if EUxUxExx TTT

yUx

41

Congruence Transformation Preserves Negative Congruence Transformation Preserves Negative Definiteness of E (hence passivity and stability)Definiteness of E (hence passivity and stability)

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

42



Cxy

BuAxsEx

xAxx

xExx

BC

T

T

T

allfor ,0)3

allfor ,0)2

)1


i.e. A is negativesemidefinite

i.e. E is positivesemidefinite

43

Example. Example. hState-Space Model from MNA of R, L, C circuitshState-Space Model from MNA of R, L, C circuits

)(

)(

)(

)(

0100

0001

)(

)(

00

10

00

01

)(

)(

)(

)(

11

1

111

11

0

3

2

1

2

1

2

1

3

2

1

3

2

1

ti

tv

tv

tv

tv

tv

I

I

ti

tv

tv

tv

RR

RR

i

v

v

v

dt

d

L

C

C

L

out

out

in

in

LL

in1I in

2I

out2vout

1v LI

E is PositiveSemidefinite

A is NegativeSemidefinite TCB

When using MNA

For immittance systemsin MNA form

xxAAx T allfor ,0)( Lemma: A is negative semidefinite if and only ifLemma: A is negative semidefinite if and only if

44

PRIMA (for preserving passivity) PRIMA (for preserving passivity) (Odabasioglu, Celik, Pileggi TCAD98)(Odabasioglu, Celik, Pileggi TCAD98)

A different implementation of case #1:A different implementation of case #1:V=U, UV=U, UTTU=I, ArnoldiU=I, Arnoldi Krylov Projection Framework: Krylov Projection Framework:

b

ˆTb

Use Arnoldi: Numerically very stableUse Arnoldi: Numerically very stable

xby

buAxsExT

xUby

buUxAUUxEUsU

qT

Tqq

Tqq

Tq

E A

IUU

uuVU

bEAEAbspanbEAuuspan

T

q

qqq

},...,{

})(,...,,{),(},...,{

1

11111

45

PRIMA preserves passivityPRIMA preserves passivity

• The main difference between and case #1 and PRIMA:The main difference between and case #1 and PRIMA:• case #1 applies the projection framework tocase #1 applies the projection framework to

• PRIMA applies the projection framework to PRIMA applies the projection framework to

• PRIMA preserves passivity becausePRIMA preserves passivity because– uses Arnoldi so that U=V and the projection becomes a uses Arnoldi so that U=V and the projection becomes a

congruence transformation congruence transformation

– E and -A produced by electromagnetic analysis are typically E and -A produced by electromagnetic analysis are typically positive semidefinite positive semidefinite

– input matrix must be equal to output matrixinput matrix must be equal to output matrix

xByBuAxsEx T

xByBuAxExsA T 11

46

Algebraic proof of moment matching for PRIMA Algebraic proof of moment matching for PRIMA expansion at s=0, V=U, orthonormal Uexpansion at s=0, V=U, orthonormal UTTU=IU=I

k

k

kTTT

k

k

kTTT

sbAEAbbAEAsIbbEsAbsH

sbAEAbbAEsAIbbsEAbsH

0

111111

0

111111

ˆˆˆˆˆˆˆ)ˆˆ(ˆˆ)ˆˆ(ˆ)(ˆ

)()()(

bAEAbbEAAEAb

bUAUUEUUAUUEUUAUUUb

bUAUUEUUAUUUbbAEAb

kTT

TTTTTTT

TTk

TTTkT

11111

111

1111 ˆˆˆˆˆ

Used Lemma: If U is orthonormal (UTU=I) and b is a vector such that

bAUbUAUUUcolspanbA TTT 111 then )(

47

Proof of lemmaProof of lemma

bAUbUAUUUcolspanbA TTT 111 )(

Proof:

bAUUgUg

AUgUAUU

bAAUAUU

bUAUU

TT

TT

TT

TT

1

1

11

1

UgbAgUcolspanbA 11 s.t. )(

48

Compare methodsCompare methods

number of number of moments moments matched by matched by model of order qmodel of order q

preserving passivitypreserving passivity

case #1 (Arnoldi, case #1 (Arnoldi, V=U, UV=U, UTTU=I on U=I on

sAsA-1-1Ex=x+Bu)Ex=x+Bu)qq nono

PRIMAPRIMA (Arnoldi, (Arnoldi,

V=U, UV=U, UTTU=I on U=I on

sEx=Ax+Bu)sEx=Ax+Bu)qq

yesyes

necessary when model necessary when model is used in a time domain is used in a time domain

simulatorsimulator

case #2 (case #2 (PVLPVL, , Lanczos,VLanczos,V≠≠U, VU, VTTU=I U=I on sAon sA-1-1Ex=x+Bu)Ex=x+Bu)

2q2q

more efficientmore efficient

no no

(good only if model is (good only if model is used in frequency used in frequency

domain)domain)

49

ConclusionsConclusions

• Reduction via eigenmodesReduction via eigenmodes– expensive and inefficientexpensive and inefficient

• Reduction via rational function fitting (point matching)Reduction via rational function fitting (point matching)– inaccurate in between points, numerically ill-conditionedinaccurate in between points, numerically ill-conditioned

• Reduction via Quasi-Convex OptimizationReduction via Quasi-Convex Optimization– quite efficient and accuratequite efficient and accurate

• Reduction via moment matching: Pade approximationsReduction via moment matching: Pade approximations– better behavior but covers small frequency bandbetter behavior but covers small frequency band– numerically very ill-conditionednumerically very ill-conditioned

• Reduction via moment matching: Krylov Subspace Projection Reduction via moment matching: Krylov Subspace Projection FrameworkFramework– allows multipoint expansion moment matching (wider frequency allows multipoint expansion moment matching (wider frequency

band)band)– numerically very robust and computationally very efficientnumerically very robust and computationally very efficient– use PVL is more efficient for model in frequency domainuse PVL is more efficient for model in frequency domain– use PRIMA to preserve passivity if model is for time domain use PRIMA to preserve passivity if model is for time domain

simulatorsimulator

Case study: Passive Reduced Models Case study: Passive Reduced Models from an Electromagnetic Field Solverfrom an Electromagnetic Field Solver

dielectric layerdielectric layerlong coplanar T-line,long coplanar T-line,shorted on other sideshorted on other side

frequency [Hz]frequency [Hz]

__ with dielectrics__ with dielectrics

- - w/o dielectrics- - w/o dielectrics

Importance of including dielectrics:Importance of including dielectrics:a simple transmission line examplea simple transmission line example

1010-4-4

1010-3-3

1010-2-2

1010-1-1

101000

1100 22 33 44 55 66x 10x 1088

admittance [S]admittance [S]

Can guarantee passivityCan guarantee passivityCan guarantee passivityCan guarantee passivity

Techniques for including dielectricsTechniques for including dielectrics

• Finite Element MethodFinite Element Method

• Green’s Functions for dielectric bodiesGreen’s Functions for dielectric bodies

• Surface Formulations using Equivalent TheoremSurface Formulations using Equivalent Theorem– (substitute dielectrics with equivalent surface currents (substitute dielectrics with equivalent surface currents

and use free space Green’s functions)and use free space Green’s functions)

• Volume Formulations using Polarization CurrentsVolume Formulations using Polarization Currents

EjEj

EjH

r

r

)1(00

0

pJ

'),()'(

4'),()'(

4

)(drr'rrJdrr'rrJ

rJdc V pV c

c KjKj

cJ cJcJ

Volume Integral Formulation Volume Integral Formulation including Dielectrics including Dielectrics

)()(ˆ0)( srrJnrJ j current and charge conservationcurrent and charge conservation

A B

dielectricsdielectrics

')()'(4

')()'(4)(

)(

0

drr'r,rJdrr'r,rJrJ

dc V pV cp KjKj

j

pJ pJ pJ pJ

)(')()(4

1')()(

4

1SSsssSsss rdr'r,r'rdr'r,r'r

dc S dS c KK

c

d

c c

dd

conductorsconductors

Frequency independent kernel Frequency independent kernel approximationapproximation

• Note: in this work we used a classical frequency Note: in this work we used a classical frequency independent approximation for the integration independent approximation for the integration kernel:kernel:

r'rr'r

1),(K

d

c

d

c

d

c

d

c

dddc

cdcc

dddc

cdccc

VV

qqII

PPPP

PolsLLLL

sR

0

00001

000

Reducing to algebraic formReducing to algebraic form

• Surface and Volume discretization both for conductors and Surface and Volume discretization both for conductors and dielectrics + dielectrics + Galerkin gives branch equations:Galerkin gives branch equations:

conductorsconductors

dielectricsdielectrics

dielectric layerdielectric layer

conductorconductor

A mesh formulation for both A mesh formulation for both conductors and dielectricsconductors and dielectrics

0 0 010

0 0 0

0

cc cdc c c

dc dd d d

c ccc cd

d ddc dd

L LR I Vs

L L Pols I V

qP P

qP P

0 0 010

0 0 0

10

cc cdc

dc dd Tm ms

cc cd

dc dd

L LRs

L L PolsM M I V

P P

P Ps

Mesh analysis guarantees passivityMesh analysis guarantees passivity

)()(

)()(

tCxty

tButRxdt

dxL

where:where:

Can prove that:Can prove that:1)

2) ( ) 0, for all

3) ( ) 0, for all

T

T T

T T

C B

x L L x x

x R R x x

0 0

0 0

0 0

Tcc cd fc

fc fd Tdc dd fd

cc cd

dc dd

L L MM M

L L M

P PL

P P

Pol

Mesh analysis guarantees passivity Mesh analysis guarantees passivity (cont.)(cont.)

)()(

)()(

tCxty

tButRxdt

dxL

where:where:

Can prove that:Can prove that:

00

00

Tfd

T

Tpd

Tpc

T

dddc

cdcc

fddddc

cdccpdpc

Tfccfc

MPol

M

MPPPP

PolMPPPP

MMMRM

R

1)

2) ( ) 0, for all

3) ( ) 0, for all

T

T T

T T

C B

x L L x x

x R R x x

( ) 0, for all T Tx L L x x Proof :

T

T

dddc

cdcc

Tfd

Tfc

dddc

cdccfdfc

Pol

PPPP

M

MLLLL

MM

L

00

00

00

diagonal with diagonal with positive coef.positive coef.

positive definite when positive definite when using Galerkinusing Galerkin

congruence transformationcongruence transformationpreserves positive definitenesspreserves positive definiteness

is block diagonal and the blocks are all positive, is block diagonal and the blocks are all positive, hence is positive semidefinite and so ishence is positive semidefinite and so isL

TL LL

( ) 0, for all T Tx R R x x Proof :

00

00

Tfd

T

Tpd

Tpc

T

dddc

cdcc

fddddc

cdccpdpc

Tfccfc

MPol

M

MPPPP

PolMPPPP

MMMRM

R

2 0 0

0 0 0

0 0 0

Tfc c fc

T

M R M

R R

( ) 0, for all T Tx R R x x Proof :

congruence transformationcongruence transformationpreserves positive definitenesspreserves positive definiteness

diagonal with positive coef.diagonal with positive coef.

is block diagonal and the blocks are all is block diagonal and the blocks are all positive semidefinite, hence is also positive positive semidefinite, hence is also positive semidefinitesemidefinite

TR RTR R

Example 1: frequency responseExample 1: frequency responseof the coplanar transmission lineof the coplanar transmission line


__ with dielectrics, reduced model__ with dielectrics, reduced model

o with dielectrics, full systemo with dielectrics, full system

1010-4-4

1010-3-3

1010-2-2

1010-1-1

101000

1100 22 33 44 55 66x 10x 1088

admittance [S]admittance [S](order 16)(order 16)

(order 700)(order 700)

Example2: frequency responseExample2: frequency responseof the line with opposite stripsof the line with opposite strips


__ with dielectrics, reduced model__ with dielectrics, reduced model

o with dielectrics, full systemo with dielectrics, full system1010-4-4

1010-3-3

1010-2-2

1010-1-1

101000

1100 22 33 44 55 66x 10x 1088




Example2: Current distributionsExample2: Current distributions

Note: NOT TO SCALE!Note: NOT TO SCALE!reduced filament widths reduced filament widths for visualization purposesfor visualization purposes

Example 3: current distributionsExample 3: current distributionsfor two bus wires on an MCMfor two bus wires on an MCM

Frequency response for Frequency response for the reduced model of the MCM busthe reduced model of the MCM bus

__ with dielectrics, reduced model (order __ with dielectrics, reduced model (order 12)12)

o with dielectrics, full system (order 600)o with dielectrics, full system (order 600)

- - without dielectrics- - without dielectrics

frequency [Hz]frequency [Hz]1010-4-4

1010-3-3

1010-2-2

1010-1-1

101000

1100 22 33 44 55 66x 10x 1088


Conclusions Electromagnetic Example Conclusions Electromagnetic Example

• Volume formulation with full mesh analysis (both Volume formulation with full mesh analysis (both conductors conductors and dielectricsand dielectrics) produces ) produces

– well conditioned well conditioned

– and positive semidefinite matricesand positive semidefinite matrices

• Hence Hence guaranteed passive modelsguaranteed passive models are generated are generated when using congruence transformationwhen using congruence transformation

68






69

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 GramianObservability Gramian

2)(

xty

0

xdtCexCe AtTAt xdtCeCex AtTtAT T

0

0W

0

)()( dttyty T

Observability Gramian:

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 Lyapunov equation

70

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 GramianControllability Gramian

xdteBBex tATAtT T

0

1cWInverse of Controllability Gramian:

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

71

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?

72

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 Gramians, and their eigenvalues?But what about the the Gramians, and their eigenvalues?

• Gramians and their eigenvalues change! Hence the Gramians 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~

73

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, are symmetric,

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

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

But not the eigenvalues

UWU T0

74

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).

75

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

76




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

– Guaranteed Passive TBRGuaranteed Passive TBR


77

TBR: Passivity Preserving?

• TBR does not generally preserve passivity

‑ Not guaranteed PR-preserving

‑ Not guarateed BR-preserving

• A special case: “symmetrizable” models

‑ Suppose the system is transformable to symmetric and internally PR

‑ TBR will generate PR models! (via congruence!)

‑ Stronger property than for PRIMA: TBR is coordinate-invariant

s.p.d. is and AsECB T

78

Positive-Real Lemma

• Lur’e equations :

• The system is positive-real if and only if is positive semidefinite

• A dual set of equations can be written for with

TT

TT

TT

DDWW

QWXBC

QQXAXA

X

TTTT DDBCCBAA ,,,Y

79

PR Preserving TBR

• Lur’e equations for “Grammians” : Lyapunov + Constraints

• Insight : from the PR lemma Can be used in a TBR procedure ‑ “Balance” the Lur’e equations then truncate

• By similar partitioning argument, truncated (reduced) system will be PR/BR (passive) iff the original is!

TT

TT

TT

DDWW

QWXBC

QQXAXA

YX ,

YX ,

80

Physical Interpretation

• Consider Y-parameter model ‑ Inputs: voltages. Outputs: currents.

‑ Dissipated energy

• Lur’e Equation for PR-“Controllability” Grammian ‑ Singular values represent: gains from dissipated energy to state

‑ Minimum energy dissipation to reach a given state:

• Lur’e Equation for PR-“Observability” Grammian‑ Singular values represent: gains from state to output

‑ Energy dissipated, given initial state:

s.p.d! iff 0 )()( 01

0 XxXxdttuty TT

uyVI TT

00)()( Yxxdttuty TT

81

Computational Procedure

• Put system into standard form

‑ If is singular, requires an eigendecomposition

• Solve the PR/BR Lur’e equations

‑ Solve a generalized eigenproblem of 2X size

‑ Special treatment for singular

• Balance & Truncate as in standard TBR

BAsICsKDBAsECDsH ~)~

(~

)()( 11

E

D

82

Alternate Hybrid Procedure

• Perform standard TBR

• Use Positive-Real Lemma to check passivity of models generated

• If model is not acceptable, proceed to PR-TBR

• Why?

‑ Usually costs less

‑ May get better models

83

Example : RLC Model

TBR Model Not Positive Real

84

Example : Integrated Spiral Inductor

Order 60 PRIMA

Order 5 PR-TBR

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

Combined Krylov-TBR algorithm

Initial Model: (A B C), n

Intermediate Model: (Ai Bi Ci), ni

Reduced Model: (Ar Br Cr), q

Krylov reduction (Wi , Vi):

Ai = WiTAVi

Bi = WiTB

Ci = CVi

TBR reduction (Wt , Vt):

Ar = WtTAVt

Br = WtTB

Cr = CVt

87

Conclusions

• Moment Matching Projection Methods

‑ e.g. PVL, PRIMA, Arnoldi

‑ are suitable for application to VERY large systems O(qn)

‑ but do not generate optimal models

• PR/BR-TBR

‑ Independent of system structure

‑ Guarantee passive models

‑ but computationally O(n3) usable only on model size < 3000

• Combination of projection methods and new TBR technique provides near-optimal compression and guaranteed passive models -- in reasonable time

• Quasi-Convex Optimization Reduction is also a good alternative specially when building models from measurements

88

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

introduction to model order reduction ii.2 the projection framework methods luca daniel...

Documents