model reduction for dynamical systems -lecture 3- · -lecture 3- peter benner and lihong feng ....

v

Model Reduction for Dynamical Systems

-Lecture 3- Peter Benner and Lihong Feng

Otto-von-Guericke Universität Magdeburg Faculty of Mathematics Summer term 2017

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

Magdeburg, Germany [email protected] [email protected]

http://www.mpi-magdeburg.mpg.de/csc/teaching/17ss/mor

mailto:[email protected]






Outline

• Mathematical basics II Systems and control theory

• Controllability measures • Observability measures

• Infinite Gramians

Motivation

Balanced truncation: first balancing, then truncate.

Given a LTI system:

)()(

)()(/)(

txLty

tButAxdttdxT=

+=

For convenience of discussion, we denote the system as a block form:

TL

BA

=

TTT

LLBAABAA

LBA

21

22221

11211

~~~~~~~~

~~~

TL

BA

1

111~

~~

Balancing T

truncate reduced model The unimportant

part is truncated

What’s the unimportant part?

The states which are difficult to control and difficult to observe correspond the unimportant part.

In system theory, the unknown vector x is called the state (vector) of the system. Actually, the entries in x depict the system variables, such as branch currents, node voltages in the interconnect model, and therefore describe the state of the system.

Motivation

Outline

• Controllability measures • Observability measures


Controllability measure

Reachability

Definition: Given a system , a state x is reachable from the zero state if there exist an input function of finite energy such that x can be obtain from the zero state and within a finite period of time .

TL

BA

)(tu∞<t

C

A

B

x (0)=0 1 y x )(tu

∞<t

reachX


Denote the subspace spanned by the reachable states, then

XX reach =

XX reach ⊆

The system is reachable : every state in the state space is reachable.

is the whole state space, e.g.

:)( nCRtxX →= +

X


Example 1

x +

_ +

_

Ω1

Ω2

Ω4

Ω8

V

x denotes the voltage drop along the capacitor, and is the state of the system. In this circuit, x=0 at any time.

Conclusion: In this circuit, 0 state is a reachable state, but any nonzero state is a unreachable state! Therefore the whole system is unreachable.

Picture referred to [Chi-Tsong Chen, Linear system Theory and Design, 3rd edition, New York Oxford, Oxford University Press, 1999]

Wheatstone bridge

A Wheatstone bridge is a measuring instrument invented by Samuel Hunter Christie in 1833 and improved and popularized by Sir Charles Wheatstone in 1843. (http://en.wikipedia.org/wiki/Wheatstone_bridge)

Example 1 is actually the Wheatstone bridge.

+

_

1R

2R

3R

xR

VGV

is adjustable, it is adjusted till becomes zero. It means there is no voltage drop through . Therefore, we have

3R

GVGV

31

2

RR

RR x=

xR can be easily adjusted by the above equation.



Example 2

+

_

F1

Ω1 Ω1

V

F1)(1 tx +

_ _

+ )(2 tx

)()(

)()(/)(

txLty

tButAxdttdxT=

+=

=

)()(

)(2

1

txtx

tx

voltage drops through the two capacitors.

Those states with are reachable, but those states with are not reachable. Because whatever the input is, the voltage drops through the two capacitors are always identical. Therefore the whole system is unreachable.

)(tx )()( 21 txtx =)()( 21 txtx ≠


Reachability matrix of the system:

],,[),( 12 BABAABBBAR n−=

By the Cayley-Halmilton theorem, the rank of the reachability matrix and the span of its columns are determined (at most) by the first n terms (not the first n columns), i.e. Thus for computational purpose the following (finite) reachability matrix is of importance:

.1,,2,1, −= ntBAt

],,[),( 12 BABAABBBAR nn

−=

Sometimes is directly defined as the reachability matrix. ),( BARn

• Why it is called reachability matrix? • Any connection between and reachability? ),( BARn


Notice the analytical solution of system state equation is BuAxdtdx +=/

∫ ≥+= −tt

tAAt ttdBuexetxux0

,,)(),,( 0)(

00 τττ

The reachability of a state x of the system is tested by the zero initial state, , we look at the above analytical solution with , 00 =x

∫ −=t tA dBuetux0

)(),0,( )( τττ

Notice:

+++++= kk

nAt A

ktAtAtIe

!!2!12

2

00 =x

,)()()()(

)(!2

)()()()(

)()!2

)()(()(),0,(

22

10

0 0

22

0

22

0

)(

0

+++++=

−+−+=

+−

+−+==

∫ ∫∫

∫∫ −

tBAtBAtABtB

dutBAdutABduB

duBAtABtBdBuetux

kk

t tt

tt tA

αααα

ττττττττ

τττττττ

which means a reachable state x is the linear combination of the terms:

,,,,, 2 BABAABB k

Therefore is defined as the reachability Matrix.

),,(),( 12 BABAABBAR n−=



Actually there is a Theorem (Theorem 4.5 in Chapter 4 in [Antoulas05]):

Theorem 1 If is the subspace spanned by the reachable states, then reachXcolumns. by the spanned subspace :),( im BARX reach =

The theorem tells us the subspace spanned by all reachable states is exactly the subspace spanned by the columns of the reachability matrix . ),( BAR

The finite reachability gramian at time is defined as : ∞<t

∞<<= ∫ tfordeBBetPt ATA T

0,)(0

τττ


Connection between reachability matrix and reachability gramians

Proposition 1 The finite reachability gramians have the following properties: (a) and (b) their columns span the reachability subspace, i.e.,

,0)()( ≥= tPtP T

).,( im)( im BARtP =

Proof An easier way is to prove nn CBARBARCtPtP =⊕=⊕ ⊕⊕ ),( im),( imand)( im)( im

where),,( im)( im BARtP ⊕⊕ =

⊕∈∀ Px im we have

,0||||)(0

2 == ∫t ATT dxeBxtPx

T

ττ

0,0 ≥=⇔ tallforxeB tAT T

We first prove ),( im)( im BARxtPx ⊕⊕ ∈⇒∈∀

( Proposition 4.8 in [Antoulas 05] )


+++++= kTk

TTn

tA AktAtAtIe

T)(

!)(

!2!12

2

.0 allfor ,0)(0 Therefore, 1 >=⇔= − ixABxeB iTTtAT T

BAx i 1−⊥

),( im BARx ⊥

),( im BARx ⊕∈

We have proved: ),( im)( im BARxtPx ⊕⊕ ∈⇒∈∀

Controllability measure ⊕⊕ ∈⇒∈∀ PxBARx im),( imNext we prove:

),( im BARx ⊕∈ ),( im BARx ⊥ 0,1 >⊥ − iallforBAx i

.0,0)( 1 >=− iallforxAB iTT

0,0 ≥= tallforxeB tAT T

0=xeBBe tATAt T

,0)(0

== ∫t ATA xdeBBextP

Tτττ)(Pnullx∈

)( im Px ⊥

⊕∈ Px im

P symmetric

why?


)( null Px∈∀ 0=Px

==

Tn

T

T

Tn

T

T

p

pp

Pand

xp

xpxp

2

1

2

1

0

)( null Ppi ⊥ )( null)( im PPT ⊥

P symmetric

,,span of columnsspan )( im 1 nTT ppPP ==

)( im)( im TPP =

)( null)( im PP ⊥

The states using large minimal energy are difficult to reach and will be truncated during MOR based on balanced truncation.

The relation provides a way to derive the minimal energy which is needed to reach a state x.

),( im)( im BARtP =


Therefore, the minimal energy for reaching a reachable state x is a key concept for model order reduction based on balanced truncation.

Next, we will derive the minimal energy for reaching a state x.


From the analytical solution, if a state x is reached at time , then with finite energy, such that

τττ dBuexT TA∫ −=0

)( )(

T

How much must the input u(t) be?

We have proved if x is reachable, then , i.e. ))(( im tPx∈

τ

τξξξ

τ

ττ

duBe

deBBedteBBexTPxT TA

TATT TAT tATAt TT

∫

∫∫−

−−

=

−==⇒=

0

)(

)(

0

)(

0)()(

ξτ τ )()( −−= TAT T

eBuand

,,Tξ∃

)(tu∃

This means x can be reached at time with input T u


C

A

B

x (0)=0 1 y x )(tu

∞<t

The input u(t) is the excitation of the system, its energy is the energy required to reach the state x .

Energy of a function is defined as: dttutuuT

)()(||||0

2 ∫ ∗=


Actually the energy of is the minimal energy to reach the state x at the given time period . (Proposition 4.10 in [Antoulas 05])

u

ξξξξ )()()(|||| )()(

00

2 tPdteBBedttutuu TttATttAt Tt T T

=== −−∫∫

We see from above analysis, if x is reachable at time , x can be represented as:

ττ duBext tA∫ −=0

)(

t

22 ||)(||||)(|| tutu > Any other input can also reach x. However if , it cannot reach x at time , but needs longer time.

t

t

22 ||)(||||)(|| tutu <

)( )( ξτ−−= tAT T

eBu

Energy of : u

x

relation to x?


A system is reachable means every state x in the whole state space is reachable.

),( im),( im BARBARX nreach ==

nBARn =)),((rank

From theorem 1:

From Proposition 1: ),( im)( im BARtP =

Therefore the system is reachable

Therefore the system is reachable 0 ,))((rank >∀= tntP

Therefore, is nonsingular for any t, if the system is reachable. )(tP


Energy of (notice ) : ξτ )( −= tAT TeBu

xtPxxtPtPxtPtPu TTT )())()(())(()(|||| 1112 −−− === ξξ

ξ)(tPx =

Controllability measure!

xtPxu T )(|||| 12 −=

Only for reachable systems.

Remark 1: Reachability is a generic property for LTI systems with the form: This means, intuitively, that almost every LTI system with the form above is reachable. If there are any unreachable systems, they are very rare. The unreachable LTI systems like examples 1,2 are rare.


BuAxdtdx +=/

Remark 2: The reachability of the system can be more easily checked by the criteria:

The system is reachable nBARrank n =)),((


A concept which is closely related to reachability is that of controllability. Here, instead of driving the zero state to a desired state, a given non-zero state is steered to the zero state. More precisely we have:

Definition of controllability: Given a LTI system as above, a non-zero state x is controllable if there exist an input u(t) with finite energy such that the state of the system goes to zero from x within a finite time: . ∞<t


Theorem 2 For time continuous systems . (Theorem 4.16 in [Antoulas 05])

contrreach XX =

Similarly, is the subspace spanned by the controllable states.

It has been proved that for time continuous LTI systems (as discussed in this lecture), the concepts of reachability and controllability are equivalent.

contrX

The system is controllable

From the property of reachable system, we have

nBARn =)),((rank


Spring Constant: 1

Damping Coefficient: 1

Damping Coefficient: 2

2u(t)

1x

dashpot dashpot

2xExample: Platform system

The system is described by the following linear time invariant (LTI) system:

)(15.0

)(10

05.0/)( tutxdttdx

+

−

−=

A B

Spring Constant: 1

002

22

11

=−−=−−

xxuxxu

makxvF =−−ηassume mass of the platform is zero, then from Newton’s law:


Is the platform system controllable?

nBARrank n =)),((The system is controllable

],,[),( ABBBARn =

=

15.0

B

−

−=

−

−=

125.0

15.0

1005.0

AB

ABB, are linearly independent!

nBARrank n == 2)),((

Therefore, the platform system is controllable.


Associated with controllability, there is the concept of observability.

Controllability: input u(t) state x(t).

Possibility of steering the state using the input.

Observability: output y(t) state x(t).

Possibility of estimating the state from the output.

Outline

• Observability measures


Observability measure

Observability is a measure for how well internal states of a system can be estimated by knowledge of its external outputs.

Definition of Observability: Given any input u(t) , a state x of the system is observable, if starting with the state x (x(0)=x), and after a finite period of time , x can be uniquely determined by the output . ∞<t

C

A

B u(t)

x(0)=x

1

)(ty

∞<t

)(ty

Observability measure

Observability matrix? Observability Gramian? Output energy?

=

2),(ALAL

L

ALO T

T

T

Observability measure Derivation of Observability matrix

τττ dBuexetxt tAtA ∫ −+=0

)(0 )()(~

From the analytical solution of , we see that after time : BuAxdtdx +=/ ∞<t

The system starting with x(0)=x, therefore

τττ dBuexetxt tAtA ∫ −+=0

)( )()(~

And the output corresponding to is: )(~ tx

∫∫

∫

−

−

−

+==

+=

+==

t AtAT

t AtATtAT

t tATtATT

dBuexxandxeL

dBueeLxeL

dBueLxeLtxLty

0

0

0

)(

)(

)(

)()(~)(

ττ

ττ

ττ

τ

τ

τ

If x is observable, then for any u(t), x can be uniquely determined by the corresponding y :

∫−+== t AtAT dBuexxandxeLty 0 )()( τττ

Since x can be uniquely determined by , it is sufficient to prove that can be uniquely determined by .

Let us see under what condition can be uniquely determined by ? x


x x)(ty

)(ty

xeLty tAT=)(

Differentiate the above equation on both sides and get the derivatives at t=0:

xLy T=)0(

xALy T=)0('

xALy T 2'' )0( =

xALy kTk =)0()(

=

)0(

)0()0(

)(

'

kkT

T

T

y

yy

x

AL

ALL

(#) has a unique solution if is square and has full rank n.

(#)

x

kT

T

T

AL

ALL


=

kT

T

T

k

AL

ALL

Q

Denote:

yQx k1−=

=

)0(

)0()0(

)(

'

ky

yy

y

can be uniquely determined, with k being at most n-1. x

if m>1, then k<n-1, if m=1, k=n-1. nmT RL ×∈


Observability matrix:

=

2),(ALAL

L

ALO T

T

T

Therefore:

=

−1

),(

nT

T

T

n

AL

ALL

ALO

From above analysis, actually the finite Observability matrix is enough to determine observability:

Therefore we define

The system is observable nALOrank n =)),((


The output energy associated with the initial state x is:

dtxeLLexdttytyty AtTtAt Tt T T

∫∫ ==00

2 )()(||)(||

xtQx

xdteLLexT

t AtTtAT T

)(0

=

= ∫

1. Energy of observation produced by an observable state x. 2. Observability measure!

Observability measure Output energy

Finite Observability Gramian at time is defined as: ∞<t

∞<<= ∫ tdeLLetQt ATAT

0,)(0

τττ

Observability measure Observability Gramian

Recall the minimal energy to reach a state x at time is txtPxu T )(|||| 12 −=

Notice both energies are related to time.

xtQxty T )(||)(|| 2=xtPxu T )(|||| 12 −=

Finite (reachability) controllability Gramian and observability Gramian will be used to derive the infinite Gramians which 1. Make the two measures computable. 2. will be directly used for truncation in MOR.

∞<<= ∫ tdeBBetPt ATA T

0,)(0

τττ ∞<<= ∫ tdeLLetQt ATA

T

0,)(0

τττ

Outline


Under which condition, and are bounded when time goes to infinity: ?

)(tQ )(tP∞→t

Roughly speaking, and can be bounded when , if

∞<<= ∫ tdeLLetQt ATA

T

0,)(0

τττ

∞<<= ∫ tdeBBetPt ATA T

0,)(0

τττ

Ate∞→t

Infinite Gramians make the two measures computable

∞→tis bounded when .

)(tQ )(tP

is bounded if the real parts of all the eigenvalues of A are negative. Ate

Why? Let be the eigen-decomposition of A, SSA Λ=−1

SeeSSeSSeSee tttttStSAt imreimre ΛΛ−Λ+Λ−Λ−Λ ====− 1111

=Λ

ren

re

re

re

λ

λλ

2

1

=Λ

imn

im

im

im

j

jj

λ

λλ

2

1

nij imi

reii ,2,1, =+= λλλ are eigenvalues of A.


SeeSSeSee imre tttStSAt ΛΛ−Λ−Λ ===− 111

=Λ

ren

re

re

re

t

t

t

t

e

ee

e

λ

λ

λ

2

1

=Λ

imn

im

im

im

tj

tj

tj

t

e

ee

e

λ

λ

λ

2

1

∞→t

0

00

0<reiλ

∞→t

)sin()cos( imi

imi

tj jteimi λλλ +=

bounded


0111→=== ΛΛ−Λ−Λ−

SeeSSeSee imre ttSSAt Therefore,

if the real parts of all the eigenvalues of A are negative.

dteBBedeBBetPP tATAtt ATAtt

TT

∫∫∞

∞→∞→===

00lim)(lim τττ

Therefore the follow limits exists if all the eigenvalues of A are negative, i.e. if the system is stable:

dteLLedeLLetQQ AtTtAt ATAtt

TT

∫∫∞

∞→∞→===

00lim)(lim τττ

where P and Q are the infinite Gramians (only for stable systems).


The infinite Gramians:

dteBBedeBBetPP tATAtt ATA

tt

TT

∫∫∞

∞→∞→===

00lim)(lim τττ

dteLLedeLLetQQ AtTtAt ATAtt

TT

∫∫∞

∞→∞→===

00lim)(lim τττ

From the property of integral, we have

ttPP ∀≥ ),( ttQQ ∀≥ ),(


In the meaning of inner product: ),)((),()( xxtPxPxtPP ≥⇔≥

For stable systems, lower bound of the minimal energy necessary for reaching a reachable state x is:

xPxxtPxu TT 112 )(|||| −− ≥=

The minimal energy necessary for reaching a reachable state x at time t is:

xtPxu T )(|||| 12 −=

because ttPP ∀≥ ),(

For stable systems, the upper bound of the energy produced by the observable state x is:

xQxxtQxty TT ≤= )(||)(|| 2 because ttQQ ∀≥ ),(

Only suitable for stable systems!


Computable measures!

For stable systems, the minimal energy necessary for reaching any state is:

xPxu T 12||||min −=

For stable systems, the maximal energy produced by any state x is:

xQxty T=2||)(||max


Because the MOR method we will introduce uses P and Q to derive the reduced-order model, and therefore is only suitable for stable systems.

xPxu T 12||||min −= xQxty T=2||)(||max

The eigenspaces of P and Q make the two measurements practically computable!


The states which are difficult to reach are included in the subspace spanned by those eigenvectors of P that corresponds to small eigenvalues.

The states which are difficult to observe are included in the subspace spanned by those eigenvectors of Q that corresponds to small eigenvalues.

why and how?

Eigenspaces of P and Q make the two measures parctically computable

nξξξ ,,, 21 Denote as the n eigenvectors of P, the corresponding eigenvalues are . (P is symmetric positive definite, it has positive eigenvalues.)

nC

The state x can therefore be represented by : nξξξ ,,, 21

nnx ξαξαξα +++= 2211


If a matrix is nonsingular, then its inverse has the same eigenvectors, but the eigenvalues are the reciprocals:

ξλξξλξλξξ 111 / −−− =⇒=⇒= PPPPP

are linearly independent, therefore they constitute a basis of the whole space .

nξξξ ,,, 21

nλλλ ≥≥≥ 21

Eigenspaces of P and Q make the two measures practically computable



nn

nxP ξλ

αξλ

αξλ

α 1112

221

11

1 +++=−

nTn

nn

TTT xPx ξξλ

αξξλ

αξξλ

α 111 222

2

2211

1

21

1 +++=−

nnuλ

αλ

αλ

α 111||||min 2

2

22

1

21 +++=

indicates the minimal energy needed to reach the state x, therefore the larger is, the more difficult the state x to reach.

2||||min u2||||min u

Therefore is orthogonal. P is symmetric, ],,[~1 nQ ξξ =


nn λλλ

λλλ 111

2121 ≤≤≤⇒≥≥≥

2||||min u

This means if x is difficult to reach ( is large), x should have large components in the subspace spanned by the eigenvectors corresponding to the small eigenvalues of P. Or x should almost locates in the subspace spanned by the eigenvectors corresponding to the small eigenvalues.


is larger if and

2|||| u

nkk λλλλλ ≥≥≥>>≥≥ + 121

nkk ααααα ,,,,, 121 +<<

nnuλ

αλ

αλ

α 111||||min 2

2

22

1

21 +++=


than if

nkk λλλλλ ≥≥≥>>≥≥ + 121 and

nkk ααααα ,,,,, 121 +>>

Similarly, if x is difficult to observe ( is small) x should have large components in the subspace spanned by the eigenvectors corresponding to the small eigenvalues of Q. Or x should almost locates in the subspace spanned by the eigenvectors corresponding to the small eigenvalues.

xQxty T=2||)(||

kξ2ξ

1ξ

1+kξ2+kξ nξ

x

niP iii ,2,1, == ξλξ

niQ iii ,2,1,~~~== ξλξ

nkk λλλλλ ≥≥≥>>≥≥ + 121

nkk λλλλλ ~~~~~121 ≥≥≥>>≥≥ +


[1] A.C. Antoulas, “Approximation of large-scale Systems”, SIAM Book Series: Advances in Design and Control, 2005. [2] Chi-Tsong Chen, Linear system Theory and Design, 3rd edition, New York Oxford, Oxford University Press, 1999.

References

model reduction for dynamical systems -lecture 3- · -lecture 3- peter benner and lihong feng ....

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