model order reduction for nonlinear systems...max planck institute magdeburg model order reduction...

Max Planck Institute Magdeburg

Model Order Reduction for Nonlinear Systems

Lihong Feng

MAX-PLANCK-INSTITUT

DYNAMIK KOMPLEXER

TECHNISCHER SYSTEME

MAGDEBURG

Max Planck Institute for Dynamics of Complex Technical Systems

Computational Methods in Systems and Control Theory

Max Planck Institute Magdeburg 2

Overview

• Linearization MOR.

• Qudratic MOR.

• Bilinearization MOR.

• Variational analysis MOR.

• Trajectory piece-wise linear MOR.

• Proper orthogonal decomposition (POD).

• References.


Linearization MOR

Original large ODE

)()(

)()(/

tLXty

tBuXfdtCdX

)(Xf

AXXf )(~

Linearization: approximate

by a linear function )(Xf

)()(~

1

)(])(,[/ 00

tLXty

tuXDXfBAXdtCdX f

)()(

))(()(2

1)()()(

00

00000

XXDXf

XXXHXXXXDXfXf

f

f

T

f

Taylor series expansion:

)()(~

)()()(/ 00

tLXty

tBuXXDXfdtCdX f

.,1,0,])[(,~

},,,{

1

0

1

21

irCACsMBAr

rMrMrMrizationorthogonalV

i

i

j

,...1,0,)( 1 irCAM i

i

B~


Example

1 2 3 N-3 N-2 N-1 N

C C C C C C Ci(t)

Y. Chen, MIT thesis 1999.

)()(

),(

0

0

1

)(

)()(

)()(

)()(

1

11

3221

211

tLXty

tu

xxg

xxgxxg

xxgxxg

xxgxg

dt

dX

nn

kkkk

1)( 40 xexg x

xxgg

xg

xggxg

41)0(')0(

!2

)0('')0(')0()( 2

)()(

),(

0

0

1

4141

418241

418241

4282

1

11

321

21

tLXty

tu

xx

xxx

xxx

xx

dt

dX

nn

kkk


Example

0 2 4 6 8 100

0.005

0.01

0.015

0.02

0.025

time, u(t)=0, t<3, u(t)=1, t>=3;

voltage a

t node 1

+ q=10 __ full simulation


Example

0 2 4 6 8 100

0.005

0.01

0.015

0.02

0.025

time, u(t)=0, t<3, u(t)=1, t>=3;

voltage a

t node 1

+ q=10

-- q=20

__full simulation


Example

0 2 4 6 8 100

0.005

0.01

0.015

0.02

0.025

time, u(t)=0, t<3, u(t)=1, t>=3;

voltage a

t node 1

+ q=10

-- q=20

...linear, no reduction

__ full simulation


Quadratic MOR

)(Xf )(Xg

)(Xf

)(Xg

nqRZVZX q ,,

)()(ˆ

)(~

/

tLVZty

tuBVWVZVZVAVZVdtCXdZV TTTTTT

Approximate by a quadratic polynomial

)()(

)()(/

tLXty

tBuXfdtCdX


)()(~)(

~/

tLXty

tuBWXXAXdtCdX T

))(()(2

1)()(

))(()(2

1)()()(

00000

00000

XXXHXXXXDXf

XXXHXXXXDXfXf

f

T

f

f

T

f

.,1,0,])[(,~

)(

},,,{

1

0

1

0

21

irCACsMBACsr

rMrMrMrizationorthogonalV

i

i

j


Example

1 2 3 N-3 N-2 N-1 N

C C C C C C Ci(t)


)()(

),(

0

0

1

)(

)()(

)()(

)()(

1

11

3221

211

tLXty

tu

xxg

xxgxxg

xxgxxg

xxgxg

dt

dX

nn

kkkk

1)( 40 xexg x

22'

2'

80041!2

)0('')0()0(

!2

)0('')0()0()(

xxxg

xgg

xg

xggxg

)()(

),(

0

0

1

)(800

)(800)(800

)(800)(800

)(800800

4141

418241

418241

4282

2

1

2

1

2

1

2

32

2

21

2

21

2

1

1

11

321

21

tLXty

tu

xx

xxxx

xxxx

xxx

xx

xxx

xxx

xx

dt

dX

nn

kkkk

nn

kkk


Example

1 2 3 N-3 N-2 N-1 N

C C C C C C Ci(t)


)()(

),(

0

0

1

)(800

)(800)(800

)(800)(800

)(800800

4141

418241

418241

4282

2

1

2

1

2

1

2

32

2

21

2

21

2

1

1

11

321

21

tLXty

tu

xx

xxxx

xxxx

xxx

xx

xxx

xxx

xx

dt

dX

nn

kkkk

nn

kkk

)()(~)(

~/

tLXty

tuBWXXAXdtCdX T

W

is a tensor, it has n matrices, the ith matrix

corresponds to the ith element of the

nonlinear vector.


Example

1 2 3 N-3 N-2 N-1 N

C C C C C C Ci(t)


0000

0000

00800800

008001600

1

nnRW

W

1

1

000000

0000

08008000

08000800

000800800

0000

i

i

i

RW nni

1,,1 iii

80080000

800800

0000

00

0000

nnn RW

XWX

XWX

XWX

WXX

nT

iT

T

T

1

VZWVZ

VZWVZ

VZWVZ

WVZVZ

nTT

iTT

TT

TT

1


Example

0 2 4 6 8 100

0.005

0.01

0.015

0.02

0.025

time, u(t)=0, t<3, u(t)=1, t>=3;

voltage a

t node 1

-- quadratic MOR, q=10

…linear, no reduction

__ full simulation


Example

0 2 4 6 8 100

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

time, u(t)=0, t<3, u(t)=1, t>=3;

voltage a

t node 1


…quadratic MOR, q=20

__ full simulation


Example

0 2 4 6 8 100

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

time, u(t)=0, t<3, u(t)=1, t>=3;

voltage a

t node 1


…quadratic, no reduction

__ full simulation


Bilinearization MOR

)(Xf )(Xg

)(Xf

)(XgnqRZVZX q ,,

)()(ˆ

)()(/

tLVZty

tuBVtVZuNVVZAVdtdZ TTT

Approximate by quadratic polynomial , but written into Kronecker

product

)()(

)()(/

tLXty

tBuXfdtdX


XXAXAXf

XXXHXXXXDXf

XXXHXXXXDXfXf

f

T

f

f

T

f

210

00000

00000

)(

))(()(2

1)()(

))(()(2

1)()()(

)()(

)()(/

tXLty

tuBtuXNXAdtdX

2, nNRX N


XX

XX

0

BB 0LL

11

21

0 AIIA

AAA

0

00

BIIBN

)()(

)()(/

tXLty

tuBtuXNXAdtdX

)()(

)()(/

tLXty

tBuXfdtdX

Carleman bilinearization

Carleman bilinearization:

[1] W.J. Rugh, Nolinear System Theory, The John Hopkins University Press, Boltimore, 1981.

[2] S. Sastry, Nonlinear Systems: Analysis, Stability and Control, Springer, New York, 1999.

Bilinearization MOR

BaBa

BaBa

BA

mnm

n

1

111

Kronecker product


)()(

)()(/

tXLty

tuBtuXNXAdtdX

How to compute V?

1

)()(n

n tyty

nnn

t

n

reg

nn dtdtttuttttuttthty 1210

21

)( )()(),,,()(

BeNeNeLttthtAtAtAT

n

reg

nnn 11),,,( 21

)(

Volterra series expression of bilinear system According to the theory in [Rugh 1981], the output response of the bilinear system can be expressed into Volterra series,

BAAsINAAsINAAsINAAsIL

BAIsNAIsNAIsNAIsLsssh

nn

Tn

nn

T

n

reg

n

111

1

111

2

111

1

111

1

1

1

2

1

1

1

21

)(

)()()()()1(

)()()()(),,,(

Laplace transform (drop for simplicity):

Bilinearization MOR


)()(

)()(/

tXLty

tuBtuXNXAdtdX

How to compute V?

BAAsINAAsINAAsINAAsIL

BAIsNAIsNAIsNAIsLsssh

nn

Tn

nnn

reg

n

111

1

111

2

111

1

111

1

1

1

2

1

1

1

21

)(

)()()()()1(

)()()()(),,,(

Laplace transform:

i

n

i

nn sAsAIAsI 111)(

1 1

1

1

1

21

)(

1

111)1(),,,(n

nnn

l l

lllll

n

n

n

reg

n BANNALAsssssh

BANNALAllmllln

nnn 11)1(),,( 1

Multimoments:

Bilinearization MOR


)()(

)()(/

tXLty

tuBtuXNXAdtdX

How to compute V?

1 1

1

1

1

21

)(

1

111)1(),,,(n

nnn

l l

lllll

n

n

n

reg

n BANNALAsssssh

BANNALAllmllln

nnn 11)1(),,( 1

Multimoments:

},,{},{}{ 1

1

111

1 BABAspanbAAKVrangeq

q

},,{},{}{ 11

1

,1

1

1

11

j

q

jjjqj NVANVANVANVAAKVrange j

j

},,{}{ 1 JVVcolspanVrange

)()(ˆ

)()(/

tLVZty

tuBVtVZuNVVZAVdtdZ TTT

Reduced model:

Bilinearization MOR


Example

0 2 4 6 8 100

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

time, u(t)=0, t<3, u(t)=1, t>=3;

voltage a

t node 1

original system

quadratic MOR, q=20

quadratic, no reduction

bilinearization, q=20

V for bilinearization MOR: },{}{ 21 VVcolspanVrange

})(,,){(}{ 191

1 BABAspanVrange

)}1(:,){(}{ 1

1

2 NVAspanVrange

V for quadratic MOR:

},,{}{ 201 BABAspanVrange


Variational analysis MOR

XXXAXXAXAXf

XXXHXXXXDXfXf f

T

f

3210

0000

)(

))(()(2

1)()()(


)()(

)()(/

tLXty

tBuXfdtdX

)()(

)(~~/ 21

tLXty

tuBXXAXAdtdX

)()(

)(~~/ 321

tLXty

tuBXXXAXXAXAdtdX

)()()(),( 3

3

2

2

1 tXtXatXtX

Variational analysis:

)()(

)(~~/ 21

tLXty

tuBXXAXAdtdX

)()(

)(~~/ 321

tLXty

tuBXXXAXXAXAdtdX

or

or

Original system:



)()()()( 3

3

2

2

1 tXtXatXtX

Variational analysis [11]:

)()(

)(~~/ 321

tLXty

tuBXXXAXXAXAdtdX

)()(

)(~~)]()()[(

)]()[(

)(/)(

3

3

2

2

13

3

2

2

13

3

2

2

13

3

3

2

2

13

3

2

2

12

3

3

2

2

113

3

2

2

1

tLXty

tuBXXXXXXXXXA

XXXXXXA

XXXAdtXXXd

)(~~)(/)(: 111 tuBtXAdttdX

)()(/)(: 112212

2 XXAtXAdttdX

)()()(/)(: 111312212313

3 XXXAXXXXAtXAdttdX

0)( if ,0)( :Assume tutX



)()()()( 3

3

2

2

1 tXtXatXtX

Variational analysis:

)()(

)(~~/ 321

tLXty

tuBXXXAXXAXAdtdX

)(~~)(/)(: 111 tuBtXAdttdX

)()(/)(: 112212

2 XXAtXAdttdX

)()()(/)(: 111312212313

3 XXXAXXXXAtXAdttdX

}~

,,~

{ 1

1

1

11111 BABAspanVZVXq

},,{ 212

1

122222 AAAAspanVZVX

q

]},[,],,[{ 32132

1

133332 AAAAAAspanVZVX

q

33

3

22

2

11

3

3

2

2

1

3

3

2

2

1 )()()()(

ZVZVZV

XXX

tXtXatXtX

},,{)( 321 VVVspantX



Original system:

)()(

)()(/

tLXty

tBuXfdtdX

)()(

)(~~/ 321

tLXty

tuBXXXAXXAXAdtdX

},,{)( 321 VVVspantX

Compute V: },,{)( 321 VVVspanVrange

Reduced model: )()(ˆ

)(~~/ 321

tLVZty

tuBVVZVZVZAVVZVZAVVZAVdtdZ TTTT

VZtX )(


Example

V for bilinearization MOR:

},{}{ 21 VVcolspanVrange

},,{}{ 191

1 BABAspanVrange

)}1(:,{}{ 1

2 NVAspanVrange

V for Variational analysis MOR:

},{}{

)}6:1(:,{}{

}{

},,{}{

21

0

2

1

12

2

0

2

41

11

VVspanVrange

VAspanVrange

AorthV

BABAspanVrange

0 2 4 6 8 100

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

time, u(t)=0, t<3; u(t)=1, t>=3

voltage a

t node 1

original exact output response

Bilinearization MOR, q=20, relative error=0.0078

Variational analysis MOR, q=9, relative error=0.0073


Example

V for [Phillips 2000] method:

V for our method [Feng 2014]:

},{}{

)]}6:1(:,[{}{

}{

},,{}{

21

0

2

1

12

2

0

2

41

11

VVspanVrange

VAspanVrange

AorthV

BABAspanVrange

},{}{

)}6:1(:,{}{

)(

},,{}{

21

0

22

112

10

2

41

11

VVspanVrange

VspanVrange

VVAAV

BABAspanVrange

0 2 4 6 8 100

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

time, u(t)=0, t<0.3; u(t)=1, t>=0.3

voltage a

t node 1

original

our method, q=9, relative error=0.0073

Phillips 2003 method, q=9, relative error=0.0076


Trajectory piece-wise linear MOR

)(Xf

)(1 Xg

)(2 Xg

)(4 Xg

)(3 Xg

0X1X

2X

)()(

)()(/

tLXty

tBuXfdtdX

Original system:

)()(

,)(~/1

0

tLXty

BuXgwdtdXs

i

ii

1,,1,0),()()( siXXAXfXg iiii

iXk

j

jkjkix

XfaaA

)(),(



)()(

)()(/

tLXty

tBuXfdtdX

Original system:

)()(

,)(/1

0

tLXty

BuXgdtdXs

i

i

))((~~

1,,1,0),()(~)()(~)(

iiiiii

iiiiii

XAXfwXAw

siXXAXwXfXwXg

)()(

),()~~(/ 0

1

0

tLXty

tBuwBXAwdtdX i

s

i

ii

ii AXXfBBBB )(],,[~

00

How to compute V?

},,{}{

1,,1,0}~

,,~

{}{

11

1

s

i

q

iiii

VVspanVrange

siBABAspanVrange i



)()(

)()(/

tLXty

tBuXfdtdX

Original system:

)()(ˆ

),(~~(/ 0

1

0

tLVZty

tBuVwBVVZAVwdtdZ T

i

Ts

i

i

T

i

Trajectory piece-wise linear system:

Reduced model:

)()(

),()~~(/ 0

1

0

tLXty

tBuwBXAwdtdX i

s

i

ii


Proper orthogonal decomposition (POD)

POD and SVD

SVD:

nmTT RD

YVUVUY

:

00

0or

s.t. ,),,( and ),,(exist there,matrix any For 11

nn

n

mm

m

nm RvvVRuuURY

ly.respective and

of columns first theincluding matrices thebe and Let ).,,( Here, 1

VU

dVUdiagD dd

d

It is obvious,

d

i

d

i

iRji

d

i

iRiji

m

k

kj

d

ki

d

i

iij

Td

d

i

iij

TddTdd

i

iij

Td

ij

Td

i

d

ij

Tdd

n

uyuuuyuYUuYU

uVDUUuVDVDuy

VDUyyY

mm

1 1111

111

1

.,,))((

))()(())(())((

)(),,(

Y can be represented in terms of d linearly independent columns of . dU


Proper orthogonal decomposition (POD)

Definition

.,1 ,~,~ s.t. minarg}{1

~,,~11

ljiuuu ijRji

n

j

jRuu

l

ii mml

.rank of basis POD called are }{ vectors the},,1{For 1 ludl l

ii

: of columns theingapproximatin

ions,approximat rank all among optimal, is }{ basis POD The 1

Y

lu l

ii

2

1||~~,|| Here, mm RiRij

l

ijj uuyy


Model Order Reduction using POD

)())(()(

ROM get the to Use.3

)~,,~( ,~~

of from rank of vectorsPOD Get the 2.

),(

snapshots get the tosystemnonlinear original theSolve 1.

1

1

tBuVtVzfVdt

tdzEVV

V

uuVVUX

XSVDq

xxX

TTT

q

T

tt m

Algorithm MOR using POD

How to deal with ?

An effective way is to approximate the nonlinear function by projecting it onto

a subspace that approximates the space generated by the nonlinear function,

and with dimension

))(( tVzf

.nl

),,( ),()( 1 luuUtUctf


DEIM

).()()()(

that,so

)()()()()(

thenr,nonsingula is Suppose

,],,[

matrix aconsider ,particularIn

).()(

system inedoverdeterm thefrom rows heddistinguis select we,)( determine To

1

1

1

tfPUPUtUctf

tfPUPtctUcPtfP

UP

ReeP

tUctf

mtc

TT

TTTT

T

mn

l

?,,1, indices especify th tohow and compute toHow liU i

Compute :

U

).,,( 3.

)( : toApply 2.

)).(,),((matrix a into ))(( of snapshots eCollect th 1.

1

1

F

l

F

TFF

tt

uuU

VUFFSVD

xfxfFtxfm


DEIM

Using DEIM to decide the indices:

Algorithm Discrete Empirical Interpolation Method (DEIM)

for end .8

], [ ], [ .7

||max]|,[| .6

.5

),,( where,for )( Solve 4.

do to2for .3

][ ],[ ],[ .2

|| max]|,.[|1

],[ :Output

Ffor basis POD :Input

T

11

11

11

1

1

1

i

F

i

l

F

i

i

F

i

TT

F

F

lT

l

l

i

F

i

iePPuUU

r

Uur

uPUP

li

ePuU

u

R

u


DEIM

Come back to :

))(( tVzfV T

)).(()())(( 1 tVzfPUPUtVzf TT

. oft independen still is ROM solving during ))(( ofn computatio then ),( of entries

few aon dependsonly entry each but above, as evaluatedisely componentwnot is ))(( If

. oft independen is ROM solving during ))(( ofn Computatio

))(()())((

that so

))(())((

then))),((,)),((())(( If

1

11

ntVzfVtx

ftxf

ntVzfV

tVzPfUPUVtVzfV

tVzPftVzfP

txftxftxf

T

i

T

TTTT

TT

nn

can be precomputed before solving the ROM


References

Quadratic MOR:

[1] Yong Chen, “Model order reduction for nonlinear systems”, MIT master thesis, 1999.

Bilinearization MOR:

[2] J. R. Phillips, “Projection Frameworks for Model Reduction of Weakly Nonlinear

Systems”, Proc. IEEE/ACM DAC, 184-189, 2000.

[3] Z. Bai, and D. Skoogh, “A projection method for model reduction of bilinear dynamical systems”, Linear Algebra Appl., 415: 406-425, 2006.

[4] L. Feng and P. Benner, A Note on Projection Techniques for Model Order Reduction of

Bilinear Systems, AIP Conf. Proc. 936: 208-211, 2007.

Max Planck Institute Magdeburg Model Order Reduction of Parametric Systems 37 30.06.2014

Variational analysis MOR:

[5] J. R. Phillips, “Automated Extraction of Nonlinear Circuit Macromodels ”, Proc. of IEEE

Custom Itegrated Circuits Conference, 451-454, 2000.

[6] L. Feng, X. Zeng, Ch. Chiang, D. Zhou, Q, Fang “Direct nonlinear order reduction with

variational analysis”, Proc. of Design, Automation and Test in Europe Conference and

Exhibition, 2:1316-1321, 2004.

Trajectory piece-wise linear (polynomial) MOR:

[7] M. Rewienski and J. White, “ A Trajectory Piecewise-Linear Approach to Model Order

Reduction and Fast Simulation of Nonlinear Circuits and Micromachined Devices”, IEEE

Trans. Comput.-Aided Des. Integr. Circuits Syst., 22(2):155-170, 2003.

[8] M. Rewienski and J. White, “ Model order reduction for nonlinear dynamical systems based on trajectory piecewise-linear approximations ”, Linear Algebra Appl., 415(2-3):426-454,

2006.

References

Max Planck Institute Magdeburg Model Order Reduction of Parametric Systems 38 30.06.2014

References

POD:

[9] S. Volkwein, “Model reduction using proper orthogonal decomposition ”, Lecture notes, June 7,

2010.

[10] S. Schaturantabut and D. C. Sorensen, “Nonlinear model reduction via discrete empirical

interpolation”, SIAM J. Sci. Comput. 32(5): 2737-2764, 2010.

A very good book for nonlinear system:

[11] W. J. Rugh, “Nonlinear system theory, the Volterra/Wiener Approach”, The Johns Hopkins

University Press, 1981.

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