model reduction for dynamical systems 6

7/25/2019 Model Reduction for Dynamical Systems 6

1/29

Otto-von-Guericke Universitat MagdeburgFaculty of Mathematics

Summer term 2012

Model Reductionfor Dynamical Systems

Lecture 6

Peter Benner

Max Planck Institute for Dynamics of Complex Technical SystemsComputational Methods in Systems and Control Theory

Magdeburg, Germany

[email protected]

www.mpi-magdeburg.mpg.de/research/groups/csc/lehre/2012 SS MOR/


2/29

Introduction Mathematical Basics MOR by Projection

Outline

1 IntroductionModel Reduction for Dynamical SystemsApplication AreasMotivating Examples

2 Mathematical BasicsNumerical Linear AlgebraSystems and Control TheoryQualitative and Quantitative Study of the Approximation Error

3 Model Reduction by ProjectionProjection and InterpolationModal Truncation

Max-Planck-Institute Magdeburg Peter Benner, MOR for Dynamical Systems 2/7


3/29


Model Reduction by ProjectionGoals

Automatic generation of compact models.

Satisfy desired error tolerance for all admissible input signals, i.e.,want

y y


4/29





y y


5/29





y y


6/29





y y


7/29





y y


8/29





y y


9/29


Model Reduction by ProjectionProjection Basics

Definition 3.1 (Projector)

A projector is a matrix P Rnn with P2 =P. LetV= range (P), then P isprojector onto V. On the other hand, if{v1, . . . , vr} is a basis ofV andV = [ v1 , . . . , vr], then P= V(V

TV)1VT is a projector onto V.




10/29

y j





Lemma 3.2: Projector PropertiesIfP= PT, then P is anorthogonal projector(aka: Galerkin projection),otherwise anoblique projector(aka: Petrov-Galerkin projection).

P is the identity operator on V, i.e., Pv= v v V.

I P is the complementary projector onto ker P.

IfV is an A-invariant subspace corresponding to a subset ofAs spectrum,then we call P aspectral projector.

LetW Rn be another r-dimensional subspace and W = [w1 , . . . , wr]be a basis matrix for W, then P=V(WTV)1WT is anobliqueprojector onto V along W.




11/29

y j













12/29













13/29













14/29













15/29

Model Reduction by Projectionsubsecname

Methods:

1 Modal Truncation

2 Rational Interpolation (Pade-Approximation and (rational) KrylovSubspace Methods)

3 Balanced Truncation

4 many more. . .Joint feature of these methods:computation of reduced-order model (ROM) by projection!




16/29


Joint feature of these methods:

computation of reduced-order model (ROM) by projection!

Assume trajectory x(t; u) is contained in low-dimensional subspace V. Thus, useGalerkinorPetrov-Galerkin-type projectionof state-space onto Valong comple-mentary subspace W: xVWTx=: x, where

range (V) =V, range (W) =W, WT

V =Ir.

Then, withx= WTx, we obtain x Vxso that

x x= x Vx,

and the reduced-order model is

A:= WTAV, B :=WTB, C := CV, (D :=D).




17/29




Assume trajectory x(t; u) is contained in low-dimensional subspace V. Thus, useGalerkinorPetrov-Galerkin-type projectionof state-space onto Valong comple-mentary subspace W: x VWTx =: x, and the reduced-order model isx=WTx

A:= WTAV, B :=WTB, C := CV, (D :=D).

Important observations:

The state equation residual satisfies x Ax Bu W, since

WT x Ax Bu = WT VWTx AVWTx Bu




18/29





A:= WTAV, B :=WTB, C := CV, (D :=D).



WT x Ax Bu = WT VWTx AVWTx Bu= WTx

x

WTAV =A

WTx

=x

WTB =B

u




19/29





A:= WTAV, B :=WTB, C := CV, (D :=D).



WT x Ax Bu = WT VWTx AVWTx Bu= WTx

x

WTAV =A

WTx

=x

WTB =B

u

= xAx Bu= 0.




20/29


Projection Rational Interpolation

Given the ROM

A= WTAV, B=WTB, C= CV, (D=D),

the error transfer function can be written as

G(s) G(s) =C(sIn A)

1B+D

C(sIrA)

1B+D




21/29



Given the ROM



G(s) G(s) =C(sIn A)

1B+D

C(sIrA)

1B+D

= C

(sIn A)1 V(sIrA)

1W

TB




22/29



Given the ROM



G(s) G(s) =C(sIn A)

1B+D

C(sIrA)

1B+D

= C

(sIn A)1 V(sIrA)

1W

TB

= CIn V(sIrA)

1W

T(sIn A) =:P(s)

(sIn A)

1B.




23/29



Given the ROM



G(s) G(s) =C(sIn A)

1B+D

C(sIrA)

1B+D

= CIn V(sIrA)

1W

T(sIn A) =:P(s)

(sIn A)

1B.

Ifs C \ ( (A) (A)), then P(s) is a projector onto V:

range (P(s)) range (V), all matrices have full rank =, andP(s)2 = V(sIrA)

1W

T(sIn A)V(sIrA)1W

T(sIn A)




24/29



Given the ROM



G(s) G(s) =C(sIn A)

1B+D

C(sIrA)

1B+D

= CIn V(sIrA)

1W

T(sIn A) =:P(s)

(sIn A)

1B.

Ifs C \ ( (A) (A)), then P(s) is a projector onto V:

range (P(s)) range (V), all matrices have full rank =, andP(s)2 = V(sIrA)

1W

T(sIn A)V(sIrA)1W

T(sIn A)

= V(sIrA)1 (sIrA)(sIrA)

1

=Ir

WT(sIn A) = P(s).



M d l R d i b P j i


25/29



Given the ROM



G(s) G(s) =C(sIn A)

1B+D

C(sIrA)

1B+D

= CIn V(sIrA)

1W

T(sIn A) =:P(s)

(sIn A)

1B.

Ifs C \ ( (A) (A)), then P(s) is a projector onto V =

if(sIn A)1B V, then (In P(s))(sIn A)1B= 0,

hence

G(s) G(s) = 0 G(s) = G(s), i.e., G interpolates G in s!



M d l R d i b P j i


26/29



Given the ROM



G(s) G(s) =C(sIn A)

1B+D

C(sIrA)

1B+D

= CIn V(sIrA)

1W

T(sIn A) =:P(s)

(sIn A)

1B.

Analogously, = C(sIn A)1

In (sIn A)V(sIrA)1

WT

=:Q(s)

B.

Ifs C \ ( (A) (A)), then Q(s) is a projector onto W =

if(sIn A)

CT W, thenC(sIn A)

1(In Q(s)) = 0,

hence

G(s) G(s) = 0 G(s) = G(s), i.e., G interpolates G in s!



M d l R d ti b P j ti


27/29


Theorem 3.3 [Grimme 97, Villemagne/Skelton 87]Given the ROM

A= WTAV, B=WTB, C=CV, (D=D),

and s

C

\ ( (A) (A)), if either

(sIn A)1B range (V), or

(sIn A)CT range (W),

then the interpolation condition

G(s) = G(s).

in s holds.

Note: extension to Hermite interpolation conditions later!



Modal Truncation


28/29

Modal TruncationTransfer Functions in C



Modal Truncation


29/29

Modal TruncationExample


model reduction for dynamical systems 6

Documents