v
Model Reduction for Dynamical Systems
-Lecture 1- 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
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• Ask questions. • Take notes when necessary, and only when necessary.
• Lecture: Lihong Feng • Exercise: Sridhar Chellappa • Lecture and exercises/homework information: https://www.mpi-magdeburg.mpg.de/3668354/mor_ss19
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What problem does this Lecture try to solve? • Not
• Not
• Not
• Not
• But:
.,
222
111
cxacxa==
.)(
,)(
22
2
11
1
cdt
tdxa
cdt
tdxa
=
=
4
111
1
111111
11
10
.
,
≥
++=+
++=+
n
bxaxadt
dxedtdxe
bxaxadt
dxedtdxe
nnnnnn
nnn
nann
n
.bax =
.baxdtdxe +=
MOR Lihong Feng 4 4/3/2019
Motivation
Example 1.
A
C
D
K C K C C
R L R
C
L R
C
L
R
C
L R
C
L R
C
L
R
C
L R
C
L R
C
L
KviaCvia
KcrossCcross
Rvia
C
K
Picture from [N.P can der Meijs’01]
Copper interconnect pattern(IBM)
Signal integrity: noise, delay.
Large-scale dynamical systems are omnipresent in science and technology.
Picture from Encarta http://encarta.msn.com/media_461519585/pentium_microprocessor.html
MOR Lihong Feng 5 4/3/2019
The interconnect can be modelled by a mathematical model: With O(106) ordinary differential equations.
),()(
),()()(
tCxty
tButAxdt
tdxE
=
+=
Example 2 A Gyroscope is a device for measuring or maintaining orientation and has been used in various automobiles (aviation, shipping and defense). The design of the device is verified by modelling and simulation. https://morwiki.mpi- magdeburg.mpg.de/morwiki/index.php/Gyroscope
Pic. by Jan Lienemann and C. Moosmann, IMTEK
Motivation
MOR Lihong Feng 6 4/3/2019
• The butterfly Gyroscope can be modeled by partial differential equations (PDEs).
• Using finite element method, the PDE is discretized into (in space) ODEs:
),()(
),()()()()()( 2
2
tCxty
tButxDdt
tdxKdt
xdM
=
+++ µµµ
With O(105) ordinary differential equations. is the vector of parameters. ),,( 1 lµµµ =
Motivation
MOR Lihong Feng 7 4/3/2019
Petrochemical
Pharmaceutical
Fine chemical
Feed (A+B) Raffinate (A)
Extract (B) Desorbent
Zone III
Zone IVZone
IIZone I
Direction of liquid flow and port
switching
Simulated Moving Bed Process
Food
SMB chromatography
Example 3: Simulated Moving Bed
Motivation
MOR Lihong Feng 8 4/3/2019
2
2
1( , ) ( , ) ( , ) ( , ) , ,n n n ni i i i
n nC t z q t z C t z C t zu D i A B
t t z zε
ε∂ ∂ ∂ ∂−
+ = − + =∂ ∂ ∂ ∂
nn Eq ni
i i iq t z Km q t z q t z i A B
t,( , ) ( ( , ) ( , )), ,∂
= − =∂
1 2
1 1 2 21 1
n nn Eq n n i i i ii i A B n n n n
A A B B A A B B
H C H Cq f C C
K C K C K C K C, , ,
, , , ,
( , )= = ++ + + +
Initial and boundary conditions: 0 0 0 0( , ) , ( , )n n
i iC t z q t z= = = =
0
0 ,( ( , ) ( )),n
n n ini ni i
nz
C u C t C tz D
=
∂= −
∂0
ni
z L
Cz
=
∂=
∂
Coln 1,2,...,N=
( , )niC t z ( , )n
iq t z
Mathematical modeling of SMB process
PDE system (couple NCol columns))
column n
porous adsorbent
Motivation
MOR Lihong Feng 9 4/3/2019
spatial discretization
A complex system of nonlinear, parametric DAEs:
: operating conditions ),()(
,),()(
tCxty
BxfdtdxM
=
+= µµ
PDE system (couple NCol columns) DAE system
µ
with proper initial conditions.
Mathematical modeling of SMB process
Motivation
0)0( xx =
Multiplying on both sides of yields Ate−
)()()( tBuetAxedt
tdxe AtAtAt −−− =−
)()(/)( tButAxdttdx +=
which implies,
)())(( tBuetxedtd AtAt −− =
Its integration from 0 to t yields,
τττ ττ
τ dBuexet AtA ∫ −
=− =
00 )()(
)()(/)( tButAxdttdx +=LTI System:
Analytical solution of the LTI System
Analytical solution of the LTI System Thus we have τττ dBuexetxe
t AAt ∫ −− =−0
00 )()(
Because the inverse of is and , (1) implies Ate− Ate IeA =⋅0
(1)
τττ dBuexetxt tAAt ∫ −+=0
)(0 )()(
• It is difficult to compute x(t) by following the analytical formulation in (2) if A is very large. We need to solve the LTI system numerically with some numerical methods, like backward Euler, ...etc.
• If the system is very large, then MOR is necessary!
(2)
• It is impossible to plot the waveform of x(t) by hand, we need computers to compute x(t) numerically and plot x(t) at many samples of time.
MOR Lihong Feng 12 4/3/2019
Basic Idea of MOR
Original model (discretized)
),()(
,),()(
tCxty
BxfdtdxM
=
+= µµΣ̂Σ
Reduced order model
q
m
n
R)(output R)( inputs,R states
∈
∈
∈
tytu
x
q
m
R)(ˆoutput R)( inputs
,R)( states
∈
∈
∈
tytu
tz r
),(ˆ)(ˆ
,ˆ),(ˆ)(ˆ
tzCty
BzfdtdzM
=
+= µµ
Σ Σ̂),( µtu ),( ty µ ),( µtu ),(ˆ ty µ
),(tol||ˆ|| µtuyy ∀<−
nr <<
MOR Lihong Feng 13 4/3/2019
Original model Reduce order model (ROM) MOR
Replace the original model with the reduced model. The reduced model is then integrated with other components in the device or circuit; or is integrated into a process.
The reduced model is repeatedly used in design analysis.
Basic Idea of MOR
MOR Lihong Feng 14 4/3/2019
5max Fp R
Q∈s.t.
Cyclic steady state (CSS) constraints SMB model (PDEs)
,A A,min B B,minPur Pur Pur Pur≥ ≥ Product purity constraints:
Operational constraints on p
ROM-based optimization
ROMs
Feed (A+B) Raffinate (A)
Extract (B) Desorbent
Zone III
Zone IVZone
II
Zone I
SMB
Example: Optimization for SMB
Basic Idea of MOR
MOR Lihong Feng 15 4/3/2019
Axial concentration profiles of the full-order DAE model with order of 672.
Axial concentration profiles reproduced by POD-based ROM (with reduced order of only 2).
Basic Idea of MOR
MOR Lihong Feng 16 4/3/2019
Example: Layout of a switch with four microbeams
Basic Idea of MOR
MOR Lihong Feng 17 4/3/2019
The schematic switch The microbeam is replaced by the ROM
Basic Idea of MOR
MOR Lihong Feng 18 4/3/2019
Enlarged microbeam Part ROM of the microbeam
Basic Idea of MOR
MOR Lihong Feng 19 4/3/2019
Original model (discretized)
),()(
),(),()(
tCxty
tBuxAdtdxM
=
+= µµΣ̂
Σ
Reduced order model
),(ˆ)(ˆ
),(ˆ),(ˆ)(ˆ
tzCty
tuBzAdtdzM
=
+= µµ
Find a subspace which includes the trajectory of x, use the projection of x in the subspace to approximate x.
range(V)
x
Px
Vzx ≈Let:
),()(ˆ
,)(),()(
tCVzty
etBuVzAdtdzVM
=
++= µµ
Projection based MOR
MOR Lihong Feng 20 4/3/2019
Basic Idea of MOR Petrov-Galerkin projection:
0 i.e. ,,...,1 allfor 0)range(in 0 ===⇔= eWniewWe TTi
],,[: 1 rwwW =
),()(ˆ
),(),()(
tCVzty
tBuWVzAWdtdzVMW TTT
=
+= µµ
.ˆ,ˆ),,(ˆ,ˆ CVCBWBVzfWfMVWM TTT ==== µ
Projection based MOR
MOR Lihong Feng 21 4/3/2019
Basic Idea of MOR The question: How to construct the ROMs (W, V) for large-scale complex systems?
Answer: The lecture will provide many solutions.
Conclusion
Lecture 1: Introduction Lecture 2-5: Mathematical basics Lecture 6: Balanced truncation method for linear time invariant systems. Lecture 7: Moment-matching and rational interpolation methods for linear time invariant systems. Lecture 8: Krylov subspace based method for nonlinear systems and POD method for nonlinear systems. Lecture 9: Krylov subspace based method for linear parametric systems. Lecture 10: POD and reduced basis method for nonlinear parametric systems. Notice: Lecture slides, excercises, time and location changes can be found at: https://www.mpi-magdeburg.mpg.de/3668354/mor_ss19
MOR Lihong Feng 22 4/3/2019
Outline of the Lecture