Model reduction of dynamical systems Introduction
Lihong Feng
Max Planck Institute for Dynamics of Complex Technical SystemsComputational Methods in Systems and Control Theory
Magdeburg, Germany
MOR Lihong Feng 24/3/2014
Motivation
Example 1.
A
C
D
K C K C C
RL
R
C
LR
C
L
R
C
LR
C
LR
C
L
R
C
LR
C
LR
C
L
Kvia
Cvia
Kcross
Ccross
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 34 / 3 / 2 0 1 4
The interconnect can be modelled by a mathematical model:
With O(106) ordinary d i f f e r e n t i a l equations.
),()(
),()()(
tCxty
tButAxdt
tdxE
�
��
Example 2
� A Gyroscope is a device for measuring ormaintaining orientation and has b e e n u s e d
in various automobiles (aviation, shippingand defense).
� The design of the device is verified by
modelling and simulation.Pic. by Jan Lienemann and
C. Moosmann, IMTEK
Motivation
MOR Lihong Feng 44/3/2014
• 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
tdxK
dt
xdM
=
+++ µµµ
With O(105) ordinary differential equations. is the vector of parameters.
),,( 1 lµµµ K=
Multi-query, real-time simulation?
Motivation
MOR Lihong Feng 54/3/2014
� Petrochemical
� Pharmaceutical
� Fine chemical
Simulated Moving Bed Process
�
Food
SMB chromatography
Example 3 Simulated Moving Bed
Motivation
MOR Lihong Feng 64/3/2014
2
2
1( , ) ( , ) ( , ) ( , ), ,
n n n ni i i i
n n
C t z q t z C t z C t zu D i A B
t t z zε
ε∂ ∂ ∂ ∂−+ = − + =
∂ ∂ ∂ ∂n
n Eq nii i i
q t zKm 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 ni iC t z q t z= = = =
0
0 ,( ( , ) ( )),n
n n ini ni i
nz
C uC t C t
z D=
∂ = −∂
0ni
z L
Cz =
∂ =∂
Coln 1,2,...,N=
( , )niC t z ( , )n
iq t z
Mathematical modeling of SMB process
PDE system (couple NCol columns))
Motivation
porous adsorbent
MOR Lihong Feng 74/3/2014
spatial discretization
A complex system of nonlinear, parametric DAEs:
: operating conditions),()(
,),()(
tCxty
BxAdt
dxM
=
+= µµ
PDE system (couple NCol columns) DAE system
µ
with proper initial conditions.
Mathematical modeling of SMB process
Motivation
MOR Lihong Feng 84/3/2014
Basic Idea of MOR
Original model (discretized)
),()(
,),()(
tCxty
BxAdt
dxM
=
+= µµΣ̂
Σ
Reduced order model
q
m
n
R)(output
R)( inputs
,R states
∈
∈∈
ty
tu
x
q
m
R)(ˆoutput
R)( inputs
,Rˆ states
∈∈
∈
ty
tu
x r
),(ˆ)(ˆ
,ˆ),(ˆ)(ˆ
tzCty
BzAdt
dzM
=
+= µµ
Σ Σ̂),( µtu ),( ty µ ),( µtu ),(ˆ ty µ
Goal: ),(tol||ˆ|| µtuyy ∀<−
MOR Lihong Feng 94/3/2014
Original model Reduce order model (ROM)MOR
Replace the original model with the reduced model.The reduced model is then integrated with othercomponents in the device or circuit; or is integratedinto a process.
The reduced model is repeatedly used in design analysis.
Basic Idea of MOR
MOR Lihong Feng 104/3/2014
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 SMB
Example: Optimization for SMB
Multi-query, real-time simulation?
Basic Idea of MOR
MOR Lihong Feng 114/3/2014
Basic Idea of MOR
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).
MOR Lihong Feng 124/3/2014
Example: Layout of a switch with four microbeams
Basic Idea of MOR
MOR Lihong Feng 134/3/2014
The schematic switchThe microbeam is replaced
by the ROM
Basic Idea of MOR
MOR Lihong Feng 144/3/2014
Enlarged microbeam PartROM of the microbeam
Basic Idea of MOR
MOR Lihong Feng 154/3/2014
Original model (discretized)
),()(
),(),()(
tCxty
tBuxAdt
dxM
=
+= µµΣ̂
Σ
Reduced order model
),(ˆ)(ˆ
),(ˆ),(ˆ)(ˆ
tzCty
tuBzAdt
dzM
=
+= µµ
Find a subspace which includes the trajectory of x, use the projection of x inthe subspace to approximate x.
range(V)
x
Px
Vzx ≈Let:
),()(ˆ
,)(),()(
tCVzty
etBuVzAdt
dzVM
=
++= µµ
Projection based MOR
MOR Lihong Feng 164/3/2014
Basic Idea of MORPetrov-Galerkin projection:
.,...,1 allfor 0)range(in 0 niewWe Ti ==⇔=
}.,,span{)range( ,0 1 rT wwWeW K==
),()(ˆ
,)(),()(
tCVzty
etBuVzAdt
dzVM
=
++= µµ
),()(ˆ
),(),()(
tCVzty
tBuWVzAWdt
dzVMW TTT
=
+= µµ
.ˆ,ˆ),,(ˆ,ˆ CVCBWBVzAWAMVWM TTT ==== µ
Projection based MOR
MOR Lihong Feng 174/3/2014
Basic Idea of MORThe question:
How to construct the ROMs (W, V) for the large-scale complex systems?
Answer:
The lecture will provide many solutions.
Conclusions
�Mathematical basics
�Moment-matching method for linear time invariant systems.
� Balanced truncation method for linear time invariant systems.
� Krylov subspace based method for nonlinear systems and POD
method for nonlinear systems.
� Krylov subspace based method for parametric system.
�Notice: time and location changes
� Lectures on June 12, and June 26 fall out.
� There will be Lectures on June 17 and June 30, 09:00-11:00, in
Room Seminarv0.05 2+3 in MPI.
� http://www.mpi-magdeburg.mpg.de/2492919/mor_ss14
MOR Lihong Feng 184/3/2014
Outline of the Lecture