model reduction of dynamical systems introduction...model reduction of dynamical systems...

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


• 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


� Petrochemical

� Pharmaceutical

� Fine chemical

Simulated Moving Bed Process

�

Food

SMB chromatography

Example 3 Simulated Moving Bed

Motivation


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


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


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 ∀<−


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


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


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).


Example: Layout of a switch with four microbeams

Basic Idea of MOR


The schematic switchThe microbeam is replaced

by the ROM

Basic Idea of MOR


Enlarged microbeam PartROM of the microbeam

Basic Idea of MOR


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


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


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


Outline of the Lecture

model reduction of dynamical systems introduction...model reduction of dynamical systems...

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