Decentralized Model Order Reduction of Linear Networks with Massive Ports

Boyuan Yan, Lingfei Zhou, Sheldon X.-D. Tan, Jie Chen

University of California, Riverside

Bruce McGaughy

Cadence Design Systems Inc.

Outline

Model order reduction Measure of interaction Decentralized model order reduction Examples Conclusion

RLC circuit model

L

s

s

LL

i

V

V

V

u

i

i

V

V

V

RR

RR

i

V

V

V

L

C

3

2

1

3

2

1

3

2

1

0001

0

0

0

1

0110

1000

1011

0011

000

000

0000

00001v 2v 3v

1bi

3bi

2bi

R LCsi sv

State-space model (MNA)

Transfer function

C G B

LT

C BLT Gs+Y (s) = ( )-1 U (s)

H (s) Frequency

|H(s)|

Model order reduction (MOR)

C BLT Gs +H (s) = ( )-1

BrLrT Gr

s+Hr (s) = ( )-1Cr

G V

VTGr

How to pick a projection matrix V

C V

VTCr

Br

B

VT

LT

VLr

T

such that H (s) ≈ Hr(s)To find

V

Moment-matching

Taylor expansion:

M0LT s+H (s) = ( )M1 + M2 s2 …

M0 M1 M2

Frequency

Magnitude Hr(s)H(s)

Projection matrix:

V=

Krylov subspace method

M0LT s+H (s) = ( )M1 + M2 s2 …

Km( A, R ) = span( A, A2R,…Am-1R)

= K m(G-1C,G-1B)M0 M1 M2span ( )… Mm-1

Lanzos, Arnoldi: [Feldmann, TCAD 95, Silveira, ICCAD 96, Odabasioglu, TCAD 98]

M0=G-1B M1=(G-1C)G-1B Mk=(G-1C)kG-1B… Moments formula:

Krylov subspace:

To find an orthogonal basis V for Krylov subspace:

A fundamental problem

M0 M1 M2V = M0 M1 M2V =B B

EKS [Wang, DAC 00] Dependent on input

SVDMOR [Feldmann, DATE 04, ICCAD 04] Still not compact enough

Problem: Centrality Each input-output pair is implicitly assumed to be equally interacted!

Explore input information as well as system information:

Explore system information only:

Existing solutions

MOR degrades as the number of inputs increases!

Motivation

u1

u2

y1

y2g22

g21

g12

y1 = g11u1 + g21u2

g11

G

Solution:1. Introduce some tools to measure interaction2. A decentralized framework is needed.

1 0.001

Outline

Model order reduction Measure of interaction Decentralized model order reduction Examples Conclusion

Relative gains

u1

u2

y1

y2g22

g21

g12

g11

Cr

Open loop gain:

Relative gain: If λ11=0 , y1 is NOT influenced by u1 at all

Δu1 Δy11

Δy21Δu2

g11|u2= |u2

Δy11

Δu1

Closed loop gain:

g11|y2= |y2

Δy11+ Δy21

Δu1

g11|u2

g11|y2

λ11 =

+ Δy21

If λ11=1 , y1 is influenced by u1 Only.

Relative gain array (RGA)

11 12 1

21 22 2

1 1

p

p

p p pp

Choose pairings corresponding to RGA elements close to 1

Columns and rows always sum to 1

u1

u2

u3

y1

y2

y3

Relative gain array:

Pairing in decentralized control:

0.176471 1.17647 0.

0.00297089 0.00713012 1.0101

1.17944 0.16934 0.010101

u1 u2 u3

y1

y2

y3

[Bristol, IEEE Trans. Automatic Control, 1966]

Scaled RGA

| | (| | 1)

1(| | 1)

| |

ij ij ij

ij ijij

Scaled to [0,1]:

The closer λij is to 1, the more important uj is in terms of yi

RGA

010

2030

40

0

10

20

30

400

0.2

0.4

0.6

0.8

1

OutputInput

Re

lative

ga

in

RGA in terms of one output (one row)

0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Input

Re

lative

ga

in

( ) ( ) ( ) Ts H s H s

(0) (0) (0) TH H

Computation of RGA

A function of frequency

Typically evaluated at zero frequency Higher frequency components tend to be more localized!

Pseudoinverse is used for non-square system

010

2030

40

0

10

20

30

400

0.2

0.4

0.6

0.8

1

s = 0

OutputInput

Re

lative

ga

in

010

2030

40

0

10

20

30

400

0.2

0.4

0.6

0.8

1

s = 1 GHz

Output Input

Re

lative

ga

in

RGA at DC is conservative and valid at higher frequencies

Outline

Model order reduction Measure of interaction Decentralized model order reduction Examples Conclusion

C BGs +H (s) = ( )-1

Decentralize

BrLrT Grs +Hir(s) = ( )-1Cr

MOR

Decentralized MOR framework

Partition the output matrix and decentralize the system into a set of subsystems corresponding to each output of interests.

LiT

LpT

L1T

:

:

C BGs +Hi(s) = ( )-1LiT

Spatial dominant Krylov subspace method

LT

:

:

:

:

C BGs +Hi(s) = ( )-1LiT

BrLirT Grs +Hir (s) = ( )-1Cr

0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

K m(G-1C,G-1B)

Spatial dominant Krylov subspace

M0 M1 M2V =B M0Vi =Bi M1 M2

Bi

Spatial dominant: K m(G-1C,G-1Bi)

The moments of dominant inputs are exactly preserved. The energy transfers from other inputs are also coarsely preserved.

Electrical distance and locality

RC network can be viewed as a cascaded low-pass RC filter. Far away nodes have little electrical impact on each other beca

use of the attenuation. The voltage response at a node is only dominated by a small n

umber of inputs nearby.

RC network

Observed node

Dominant input

Minor input

Principle components in terms of both frequency and space

Existing MOR preserves the principle components in terms of frequency (time) only and ignores the other degree of freedom.

In decentralized MOR, the principle components are in terms of both frequency (time) and space.

As a result , the reduced model can be made much more compact!Observed node

Dominant input

Minor inputFrequency

Magnitude

Frequency (time)

Space

Projection subspace

Outline

Model order reduction Measure of interaction Decentralized model order reduction Examples Conclusion

Frequency domain evaluation

2 21 1( ) ( )

p p

j r jj jh s h s

11 12 1

21 22 2

1 1

( ) ( ) ( )

( ) ( ) ( )( )

( ) ( ) ( )

p

p

p p pp

h s h s h s

h s h s h sH s

h s h s h s

2 21 22 2( ) ( ) ( ) ( )r r r r pH s h s h s h s

Original transfer function matrix:

Reduced transfer function matrix corresponding to 2nd output:

Instead of comparing each element in the transfer function matrix, we compare the sum of the corresponding row.

We hope

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Port

RG

AA simple RC mesh

105

106

107

108

109

1010

1011

-50

-40

-30

-20

-10

0

10

20

30

40

50

Frequency (Hz)

Ma

gn

itud

e (

dB

)

Original

PRIMA

DeMOR

0 1 2 3 4 5 6

x 10-11

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (s)

Vo

ltag

e (

V)

Original

PRIMADeMOR

Node #: 1600Port #: 33Reduced order : 7Dominant input #: 1Expansion point: 0 Hz

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-10

0

0.5

1

1.5

2

2.5

TIme (s)

Vo

ltag

e (

V)

Original

PRIMA 8

PRIMA 100

PRIMA 200

DeMOR 8

A larger RC mesh

Nodes: 10000Ports: 100Reduced order : 8Dominant inputs: 2Expansion point: 0 Hz

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Port

RG

A

An RLC mesh

Node #: 17500Port #: 250Reduced order : 400Dominant input #: 10Expansion point: 10 GHz

0 50 100 150 200 2500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Port

RGA

Since RLC circuit is less localized and much more complicated, it is really hard to match the wide-band frequency response!

100

102

104

106

108

1010

1012

-120

-100

-80

-60

-40

-20

0

20

40

60

Frequency (Hz)

Ma

gn

itud

e (

dB

)

Original

PRIMA

DeMOR

Outline

Model order reduction Measure of interaction Decentralized model order reduction Examples Conclusion

Conclusion

Propose a decentralized model order reduction framework.

Introduce relative gain array to evaluate the relative importance of ports.

Propose the concept of spatial dominant subspace to create one more degree of freedom in addition to frequency.

The proposed method can take advantage of parallel computation in modeling and simulation.

The proposed method is more efficient if only a small number of nodes are to be observed.

Thank You!