Model Order Reduction for Nonlinear Systems...Max Planck Institute Magdeburg Model Order Reduction...
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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
Transcript of 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.
Max Planck Institute Magdeburg 3
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~
Max Planck Institute Magdeburg 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
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
Max Planck Institute Magdeburg 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;
voltage a
t node 1
+ q=10 __ full simulation
Max Planck Institute Magdeburg 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;
voltage a
t node 1
+ q=10
-- q=20
__full simulation
Max Planck Institute Magdeburg 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;
voltage a
t node 1
+ q=10
-- q=20
...linear, no reduction
__ full simulation
Max Planck Institute Magdeburg 8
Quadratic MOR
)(Xf )(Xg
)(Xf
)(Xg
nqRZVZX q ,,
)()(ˆ
)(~
/
tLVZty
tuBVWVZVZVAVZVdtCXdZV TTTTTT
Approximate by a quadratic polynomial
)()(
)()(/
tLXty
tBuXfdtCdX
Taylor series expansion:
)()(~)(
~/
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
Max Planck Institute Magdeburg 9
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
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
Max Planck Institute Magdeburg 10
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
)(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.
Max Planck Institute Magdeburg 11
Example
1 2 3 N-3 N-2 N-1 N
C C C C C C Ci(t)
Y. Chen, MIT thesis 1999.
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
Max Planck Institute Magdeburg 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;
voltage a
t node 1
-- quadratic MOR, q=10
…linear, no reduction
__ full simulation
Max Planck Institute Magdeburg 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;
voltage a
t node 1
-- quadratic MOR, q=10
…quadratic MOR, q=20
__ full simulation
Max Planck Institute Magdeburg 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;
voltage a
t node 1
-- quadratic MOR, q=10
…quadratic, no reduction
__ full simulation
Max Planck Institute Magdeburg 15
Bilinearization MOR
)(Xf )(Xg
)(Xf
)(XgnqRZVZX q ,,
)()(ˆ
)()(/
tLVZty
tuBVtVZuNVVZAVdtdZ TTT
Approximate by quadratic polynomial , but written into Kronecker
product
)()(
)()(/
tLXty
tBuXfdtdX
Taylor series expansion:
XXAXAXf
XXXHXXXXDXf
XXXHXXXXDXfXf
f
T
f
f
T
f
210
00000
00000
)(
))(()(2
1)()(
))(()(2
1)()()(
)()(
)()(/
tXLty
tuBtuXNXAdtdX
2, nNRX N
Max Planck Institute Magdeburg 16
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
Max Planck Institute Magdeburg 17
)()(
)()(/
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
Max Planck Institute Magdeburg 18
)()(
)()(/
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
Max Planck Institute Magdeburg 19
)()(
)()(/
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
Max Planck Institute Magdeburg 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;
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
Max Planck Institute Magdeburg 21
Variational analysis MOR
XXXAXXAXAXf
XXXHXXXXDXfXf f
T
f
3210
0000
)(
))(()(2
1)()()(
Taylor series expansion:
)()(
)()(/
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:
Max Planck Institute Magdeburg 22
Variational analysis MOR
)()()()( 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
Max Planck Institute Magdeburg 23
Variational analysis MOR
)()()()( 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
Max Planck Institute Magdeburg 24
Variational analysis MOR
Original system:
)()(
)()(/
tLXty
tBuXfdtdX
)()(
)(~~/ 321
tLXty
tuBXXXAXXAXAdtdX
},,{)( 321 VVVspantX
Compute V: },,{)( 321 VVVspanVrange
Reduced model: )()(ˆ
)(~~/ 321
tLVZty
tuBVVZVZVZAVVZVZAVVZAVdtdZ TTTT
VZtX )(
Max Planck Institute Magdeburg 25
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
Max Planck Institute Magdeburg 26
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
Max Planck Institute Magdeburg 27
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
)(),(
Max Planck Institute Magdeburg 28
Trajectory piece-wise linear MOR
)()(
)()(/
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
Max Planck Institute Magdeburg 29
Trajectory piece-wise linear MOR
)()(
)()(/
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
Max Planck Institute Magdeburg 30
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
Max Planck Institute Magdeburg 31
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
Max Planck Institute Magdeburg 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
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
Max Planck Institute Magdeburg 33
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
Max Planck Institute Magdeburg 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
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
Max Planck Institute Magdeburg 35
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
Max Planck Institute Magdeburg 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.
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.
