Model Reduction for Dynamical Systems -Lecture 1-...v Model Reduction for Dynamical Systems -Lecture...

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

mailto:[email protected]

v

• 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


v

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


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

https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Gyroscope

https://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Gyroscope


• 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


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


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


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.


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 <<


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


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


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


Example: Layout of a switch with four microbeams

Basic Idea of MOR


The schematic switch The microbeam is replaced by the ROM

Basic Idea of MOR


Enlarged microbeam Part ROM of the microbeam

Basic Idea of MOR


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


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


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


Outline of the Lecture





Model Reduction for Dynamical Systems -Lecture 1-...v Model Reduction for Dynamical Systems -Lecture...

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