Model Reduction for Dynamical Systems Lecture 8 · 2019. 6. 23. · [1] W.J. Rugh, Nolinear System...

v

Model Reduction for Dynamical Systems Lecture 8 Lihong Feng

Otto-von-Guericke Universität Magdeburg Faculty of Mathematics Summer term 2019

Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory

Magdeburg, Germany [email protected]

https://www.mpi-magdeburg.mpg.de/3668354/mor_ss19

mailto:[email protected]

2

Overview

Linearization-based MOR

Quadratic MOR

Bilinearization-based MOR

Variational analysis-based MOR

Trajectory piece-wise linear MOR

Proper orthogonal decomposition (POD)

References

3

Linearization-based MOR

Original large ODE

)()()()(/

tCXtytBuXfdtEdX

=+=

)(Xf

AXXf =)(~

Linearization: approximate by a linear function )(Xf

)()(~1

)(])(,[/ 00

tCXty

tuXDXfBAXdtEdX f

=

−+=

)()(

))(()(21)()()(

00

00000

XXDXf

XXXHXXXXDXfXf

f

fT

f

−+≈

+−−+−+=

Taylor series expansion:

)()(~)()()(/ 00

tCXtytBuXXDXfdtEdX f

=

+−+=

.,1,0,])[(,~},,,{izationorthogonal

10

1

21

=−==

=−− irEAEsMBAr

rMrMrMrVi

i

j

,...1,0,)( 1 == − irEAM ii

B~

4

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

01

)(

)()(

)()()()(

1

11

3221

211

tLXty

tu

xxg

xxgxxg

xxgxxgxxgxg

dtdX

nn

kkkk

=

+

−

−−−

−−−−−−

=

−

+−

1)( 40 −+= xexg x

xxgg

xgxggxg

41)0(')0(!2

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

=+≈

+++=

)()(

),(

0

01

4141

418241

4182414282

1

11

321

21

tLXty

tu

xx

xxx

xxxxx

dtdX

nn

kkk

=

+

−

−−

+−+−

=

−

+−

5

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;

volta

ge a

t nod

e 1

+ q=10 __ full simulation

6

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;

volta

ge a

t nod

e 1

+ q=10 -- q=20 __full simulation

7

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;

volta

ge a

t nod

e 1

+ q=10 -- q=20 ...linear, no reduction __ full simulation

8

Quadratic MOR )(Xf )(Xg

)(Xf

)(Xg

nqRZVZX q <<∈≈ ,,

)()(ˆ)(~/

tCVZtytuBVWVZVZVAVZVdtEVdZV TTTTTT

=++=

Approximate by a quadratic polynomial

)()()()(/

tCXtytBuXfdtEdX

=+=


)()(~)(~/

tCXtytuBWXXAXdtEdX T

=++=

))(()(21)()(

))(()(21)()()(

00000

00000

XXXHXXXXDXf

XXXHXXXXDXfXf

fT

f

fT

f

−−+−+≈

+−−+−+=

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

},,,{izationorthogonal1

01

0

21

=−=−=

=−− irEAEsMBAEsr

rMrMrMrVi

i

j

9

Example

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

C C C C C C Ci(t)


)()(

),(

0

01

)(

)()(

)()()()(

1

11

3221

211

tLXty

tu

xxg

xxgxxg

xxgxxgxxgxg

dtdX

nn

kkkk

=

+

−

−−−

−−−−−−

=

−

+−

1)( 40 −+= xexg x

22'

2'

80041!2

)0('')0()0(

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

xxxgxgg

xgxggxg

+=++≈

+++=

)()(

),(

0

01

)(800

)(800)(800

)(800)(800)(800800

4141

418241

4182414182

21

21

21

232

221

221

21

1

11

321

21

tLXty

tu

xx

xxxx

xxxxxxx

xx

xxx

xxxxx

dtdX

nn

kkkk

nn

kkk

=

+

−

−−−

−−−−−−

+

−

−−

+−+−

=

−

+−

−

+−

10

Example

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

C C C C C C Ci(t)


)()(

),(

0

01

)(800

)(800)(800

)(800)(800)(800800

4141

418241

4182414282

21

21

21

232

221

221

21

1

11

321

21

tLXty

tu

xx

xxxx

xxxxxxx

xx

xxx

xxxxx

dtdX

nn

kkkk

nn

kkk

=

+

−

−−−

−−−−−−

+

−

−−

+−+−

=

−

+−

−

+−

)()(~)(~/

tLXtytuBWXXAXdtdX T

=++=

W is a tensor, it has n matrices, the ith matrix corresponds to the ith element of the nonlinear vector.

11

Example

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

C C C C C C Ci(t)


−

−

=∈ ×

0000

000000800800008001600

1

nnRW

W

1

1

000000

00000800800008000800

000800800

0000

+

−

−−

−

=∈ × ii

i

RW nni

1,,1 +− iii

−−

=∈ ×

800800008008000000

000000

nnn RW

=

XWX

XWX

XWX

WXX

nT

iT

T

T

1

=

VZWVZ

VZWVZ

VZWVZ

WVZVZ

nTT

iTT

TT

TT

1

12

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;

volta

ge a

t nod

e 1

-- quadratic MOR, q=10 …linear, no reduction __ full simulation

13

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;

volta

ge a

t nod

e 1

-- quadratic MOR, q=10 …quadratic MOR, q=20 __ full simulation

14

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;

volta

ge a

t nod

e 1

-- quadratic MOR, q=10 …quadratic, no reduction __ full simulation

15

Bilinearization-bsed MOR

)(Xf )(Xg

)(Xf

)(XgnqRZVZX q <<∈≈⊗ ,,

)()(ˆ)()(/

tLVZtytuBVtVZuNVVZAVdtdZ TTT

=++= ⊗⊗⊗

Approximate by quadratic polynomial , but written into Kronecker product

)()()()(/

tLXtytBuXfdtdX

=+=


XXAXAXf

XXXHXXXXDXf

XXXHXXXXDXfXf

fT

f

fT

f

⊗++=

−−+−+≈

+−−+−+=

210

00000

00000

)(

))(()(21)()(

))(()(21)()()(

)()()()(/

tXLtytuBtuXNXAdtdX

⊗⊗

⊗⊗⊗⊗⊗⊗

=

++=

2, nNRX N ≈∈⊗

16

⊗

=⊗

XXX

X

=⊗

0B

B [ ]0LL =⊗

⊗+⊗

=⊕

11

21

0 AIIAAA

A

⊗+⊗

=⊕

000

BIIBN

)()()()(/

tXLtytuBtuXNXAdtdX

⊗⊗

⊗⊗⊗⊗⊗⊗

=

++=)()(

)()(/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.

=⊗

BaBa

BaBaBA

mnm

n

1

111

Kronecker product

Bilinearization-bsed MOR Exercise

17

)()(

)()(/tXLty

tuBtuXNXAdtdX⊗⊗

⊗⊗⊗⊗⊗⊗

=

++=How to compute V?

∑∞

=

=1

)()(n

n tyty

nnn

t

nreg

n

t

n dtdtttuttttuttthty 1210 21)(

0)()(),,,()( −−−−−= ∫ ∫

⊗⊗⊗⊗ ⊗−

⊗⊗

= BeNeNeLttth tAtAtATn

regn

nn 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,

BAAsINAAsINAAsINAAsILBAIsNAIsNAIsNAIsLsssh

nnTn

nnT

nreg

n111

1111

2111

1111

11

12

11

121

)(

)()()()()1(

)()()()(),,,(−−−−−−−−−

−−−−

−−−−

−

−−−−−=

−−−−=

Laplace transform (drop for simplicity): ⊗


18

)()(

)()(/tXLty

tuBtuXNXAdtdX⊗⊗

⊗⊗⊗⊗⊗⊗

=


BAAsINAAsINAAsINAAsILBAIsNAIsNAIsNAIsLsssh

nnTn

nnnreg

n111

1111

2111

1111

11

12

11

121

)(

)()()()()1(

)()()()(),,,(−−−−−−−−−

−−−−

−−−−

−

−−−−−=

−−−−=

Laplace transform:

++++=− −−−− in

inn sAsAIAsI 111)(

∑ ∑∞

=

∞

=

−−−−− −−=1 1

11

121

)(

1

111)1(),,,(n

nnn

l l

llllln

nn

regn BANNALAsssssh

BANNALAllm lllnn

nn 11)1(),,( 1−−− −−=

Multimoments:


19

)()(

)()(/tXLty

tuBtuXNXAdtdX⊗⊗

⊗⊗⊗⊗⊗⊗

=


∑ ∑∞

=

∞

=

−−−−− −−=1 1

11

121

)(

1

111)1(),,,(n

nnn

l l

llllln

nn

regn BANNALAsssssh

BANNALAllm lllnn

nn 11)1(),,( 1−−− −−=

Multimoments:

},,{span},{}{range 1

1

1111 BABABAAKV q

q−−−− ==

},,{span},{}{range 112

,11

111

−−

−−

−−

−−− == j

qjjjqj NVANVANVANVAAKV j

j

},,{colspan}{range 1 JVVV =

)()(ˆ)()(/

tLVZtytuBVtVZuNVVZAVdtdZ TTT

=++= ⊗⊗⊗

Reduced model:


20

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;

volta

ge a

t nod

e 1

original systemquadratic MOR, q=20quadratic, no reductionbilinearization, q=20

V for bilinearization MOR: },{colspan}{range 21 VVV =

})(,,){(span}{range 1911 BABAV −⊗−⊗=

)}1(:,){(span}{range 11

2 NVAV −⊗=

V for quadratic MOR:

},,{span}{range 201 BABAV −−=

21


+⊗⊗+⊗++≈

+−−+−+=

XXXAXXAXAXf

XXXHXXXXDXfXf fT

f

3210

0000

)(

))(()(21)()()(


)()()()(/

tLXtytBuXfdtdX

=+=

)()()(~~/ 21

tLXtytuBXXAXAdtdX

=+⊗+=

)()()(~~/ 321

tLXtytuBXXXAXXAXAdtdX

=+⊗⊗+⊗+=

++++== )()()(),0(),( 33

22

1 tXtXtXtXtX ααααα

Variational analysis:

)()()(~~/ 21

tLXtytuBXXAXAdtdX

=+⊗+= α

)()()(~~/ 321


=+⊗⊗+⊗+= α

or

or

Original system:

.0),0( that so ,0)( if ,0)( :Assume ==== tXtutX α

22

+++= )()()()( 33

22

1 tXtXatXtX αα

Variational analysis [11]:

)()()(~~/ 321


=+⊗⊗+⊗+= α

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

)]()[(

)(/)(

33

22

133

22

133

22

13

33

22

133

22

12

33

22

1133

22

1

tLXtytuBXXXXXXXXXA

XXXXXXAXXXAdtXXXd

=++++⊗+++⊗++++

+++⊗++++

+++=+++

αααααααααα

αααααα

αααααα

)(~~)(/)(: 111 tuBtXAdttdX +=α

)()(/)(: 1122122 XXAtXAdttdX ⊗+=α

)()()(/)(: 1113122123133 XXXAXXXXAtXAdttdX ⊗⊗+⊗+⊗+=α


23

+++= )()()()( 33

22

1 tXtXatXtX αα

Variational analysis:

)()()(~~/ 321


=+⊗⊗+⊗+= α

)(~~)(/)(: 111 tuBtXAdttdX +=α

)()(/)(: 1122122 XXAtXAdttdX ⊗+=α

)()()(/)(: 1113122123133 XXXAXXXXAtXAdttdX ⊗⊗+⊗+⊗+=α

}~,,~{span 11

111111 BABAVZVX q−−=≈

},,{span 2121

122222 AAAAVZVX q−−=≈

]},[,],,[{span 321321

133332 AAAAAAVZVX q−−=≈

333

222

11

33

22

1

33

22

1 )()()()(

ZVZVZVXXX

tXtXatXtX

ααα

ααα

αα

++≈

++≈

+++=

},,{span)( 321 VVVtX ≈∈


24

Original system:

)()()()(/

tLXtytBuXfdtdX

=+=

)()()(~~/ 321


=+⊗⊗+⊗+=

≈

},,{span)( 321 VVVtX ≈∈

Compute V: },,{orth)(range 321 VVVV =

Reduced model: )()(ˆ

)(~~/ 321

tLVZtytuBVVZVZVZAVVZVZAVVZAVdtdZ TTTT

=+⊗⊗+⊗+=

VZtX ≈)(


25

Example

V for bilinearization MOR:

},{colspan}{range 21 VVV =

)}1(:,{span}{range 12 NVAV −=

V for Variational analysis MOR:

},{span}{range)}6:1(:,{span}{range

}{orth

},,{span}{range

21

02

112

20

2

4111

VVVVAV

AVBABAV

==

=

=

−

−−

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

volta

ge a

t nod

e 1

original exact output responseBilinearization MOR, q=20, relative error=0.0078Variational analysis MOR, q=9, relative error=0.0073

})(,,){(span}{range 1911 BABAV −⊗−⊗=

26

Example

V for [Phillips 2000] method:

V for our method [Feng 2014]:

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

}{orth

},,{span}{range

21

02

112

20

2

4111

VVVVAV

AVBABAV

==

=

=

−

−−

},{span}{range)}6:1(:,{span}{range

)(

},,{span}{range

21

022

11210

2

4111

VVVVV

VVAAVBABAV

==

⊗=

=−

−−

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

volta

ge a

t nod

e 1

originalour method, q=9, relative error=0.0073Phillips 2003 method, q=9, relative error=0.0076

27


)(Xf)(1 Xg

)(2 Xg

)(4 Xg

)(3 Xg

2X 3X

4X

)()()()(/

tLXtytBuXfdtdX

=+=

Original system:

)()(

,)(/1

0

tLXty

BuXgwdtdXs

iii

=

+=∑−

=

1,,1,0),()()( −=−+= siXXAXfXg iiii

iXk

jjkjki x

XfaaA

∂

∂==

)(),(

1X

28


)()()()(/

tLXtytBuXfdtdX

=+=

Original system:

)()(

,)(/1

0

tLXty

BuXgwdtdXs

iii

=

+=∑−

=

))((

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

iiii

iiii

XAXfXA

siXXAXfXg

−+=

−=−+=

)()(

),()(/1

0

tLXty

tBuwBXAwdtdX ii

s

iii

=

++=∑−

=

iiiiii XAXfBBBB −== )(],,[~

How to compute V?

},,{span}{range1,,1,0}~,,~{span}{range

11

1

−

−−

=−==

s

iq

iiii

VVVsiBABAV i

29


)()()()(/

tLXtytBuXfdtdX

=+=

Original system:

)()(ˆ

),((/1

0

tLVZty

tBuVwBVVZAVwdtdZ Tii

Ts

ii

Ti

=

++=∑−

=

Trajectory piece-wise linear system:

Reduced model:

)()(

),()(/1

0

tLXty

tBuwBXAwdtdX ii

s

iii

=

++=∑−

=

30


POD and SVD SVD:

nmTT RD

YVUVUY ×∈Σ=

=Σ= :

000

or

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

nmm

mnm RvvVRuuURY ××× ∈=∈=∈

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

VUdVUD dd

dσσ =

It is obvious,

∑ ∑∑∑∑

∑∑

= ====

===

==

==

==Σ=⇒

==

d

i

d

iiRji

d

iiRiji

m

kkj

dki

d

iiij

Td

d

iiij

TddTdd

iiij

Tdij

Tdi

d

ij

Tddn

uyuuuyuYUuYU

uVDUUuVDVDuy

VDUyyY

mm

1 1111

111

1

.,,))((

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

)(),,(

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

31


Definition

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

~,,~11

ljiuuu ijRji

n

jj

Ruu

lii mm

l

≤≤== ∑=∈

= δε

.rank of basis POD called are }{ vectors the},,,1{For 1 ludl lii =∈

: of columns theingapproximatin ions,approximat rank all among optimal, is }{ basis POD The 1

Ylu l

ii =

2

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

l

ijj uuyy=Σ−=ε

32

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

VuuVVUX

XSVDq

xxX

TTT

qT

tt N

+=

=Σ=

=

Algorithm MOR using POD

How to deal with ? An effective way is to approximate the nonlinear function by projecting it onto a subspace with dimension , that approximates the subspace spanned by the snapshots of the nonlinear function.

))(( tVzf

nl <<

),,( ),())(( 1f

lfff uuUtcUttxf =≈

33

DEIM

).()( s.t. ,],,[ matrix a find : toequivalent is This

1tcUPtfPReeP fTTmn

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

Fl

Ff

TFF

Nt

uuUVUFFSVD

xfxfFtxf

=

Σ=

=

:points at )( esinterpolat )( that require we,)( determine To nltftcUtc f <<

thenr,nonsingula is Suppose UPT

).()()()( that,so

)()()()()(

1

1

tfPUPUtcUtf

tfPUPtctcUPtfP

TfTff

TfTfTT

−

−

=≈

=⇒=

34

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

T11

11

11

1

1

1

℘℘

←℘←←

=℘−=

==

=

℘=℘==

=℘

∈℘℘=℘

℘

−

℘

=

i

Fi

ff

i

fFi

iFi

TfT

Ff

F

lTl

li

Fi

iePPuUU

rUur

uPUPli

ePuUu

Ru

ρα

ααααα

ρ

35

DEIM

Come back to :

))(( tVzfV T

)).(()())(( 1 tVzfPUPUtVzf TT −≈

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

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

))(()())(( Finally,

1

1

ntVzfVtVzx

tVzftVzftVzfP

tVzfPUPUVtVzfV

T

TT

TTTT

l

=

=

≈

℘℘

−

can be precomputed before solving the ROM

36

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.

Model Order Reduction of Parametric Systems 37 21.06.2019

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

Model Order Reduction of Parametric Systems 38 21.06.2019

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.

References

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