Fundamental Physics from Control Theory? Reduced Order Models in Physics:

Fundamental Physics from Control Theory?

Reduced Order Models in Physics:

The “Really Big” Picture

D eterm in in gC on s titu en t

m atte r

D yn am ics ,E q u ilib riu m ,an d sys tem

m an ip u la tion

P rop ertieso f su ch sys tem s

w ith m an y d eg reeso f freed om

P h en om en o log y M atch in g

Directions in Theoretical Physics (not exhaustive and highly subjective)

small normal large

Physical Focus of Talk

• Systems With Many Degrees of Freedom

• “Natural” bulk characteristics• Theoretical techniques used to find

bulk characteristics (physicists methods)

Control Theory

S ystem Id en tif ica tionan d R ealization

S tab ility P erf orm an ce R ob u stn ess

C on tro l D esign C on tro l O b jectives

P h ysica l L aw s(o r resp on se o f

sys tem )

from an applied perspective

Control Theory Focus of Talk

• Distributed Systems

(high order systems, usually governed by PDE’s)

• Model Reduction

(related to finding approximate reduced order realizations)

An issue of practicaldesign

Phenomena

Physics

• The Quantum-Classical transition

• Molecular Dynamics (simulations) leading to STZ theory (understanding shearing in amorphous materials)

• Statistical Mechanics -- Thermodynamics

Focus of Talk

Stat Phys and Thermo• Many Degrees of Freedom =

Micro Statistical Description of System

• “Natural” Bulk Characteristics =

Pressure, Volume, Temperature, Energy, …

(i.e. thermodynamic quantities)

• Theoretical Techniques: Mean Field, Projection-operator methods, Renormalization Group (RG), …..

An Example

micromacro

Tennis ballMonomers & Molecules

Polymers (fibers)

Reduction: The System-Environment Split

Ingredients:

• Many state variables X=(x1,x2, …., xN)

• Energy Conservation (i.e. for linear systems -- only has strictly imaginary eigenvalues)

• Insulating Walls (walls cannot act as an energy sink)

Mathematical Caricature:

• Dynamics

• System-Environment split occurs when some state variables effectively decouple.

(can result when there is an invariant subspace)

i.e. X = (Xs,Xe)

• System = Bulk Properties that are observed

• Environment = effectively is noise, the source of fluctuations in the system

)X(X f

•New Effective Dynamics:

•Approximate the environment contribution by a stochastic driving term, F(XS)

•New Stochastic Dynamics:

•Result: A Langevin type equation

(motivated by work by M. Kac and R. Zwanzig)

)X,X()X(~

X eSSS gf

eff )X(F)X(~

X SSS

0)X(F)X,X( SeS gfe

RG: In a picture

“fine grained”

“coarse grained”

Coarse Grained Variables = “averaged” variables

RG: HeuristicsSystem-Environment split in RG context

• System = Averaged variables

• Environment = The “details” that are “ignored”

Example: i

iio aL φ

i

iif aL φ~~

Fine grained functional

Coarse grained functional

where )(~~ aaa

Physics Reduction Comments:Pros:• Quite generally applicable for closed systems• A great calculational apparatus – may be applied to linear

and nonlinear systems• Quite algorithmic – easy to put on the computer• Many implementations: Path integral RG, Density Matrix

RG, Wilsonian RG, etc.Caveats:• Open systems?• Not often implemented for non-homogeneous systems• Uncontrolled approximation• Not very rigorous (at least in majority of literature)

• Noise gets translated into• In physics: The coupling constants (in the

Lagrangian) get RENORMALIZED• Example: Charge Screening in electronic

systemsASIDE: The above transformation may not

be invertible (i.e. the RG transformations form a semi-group)

)(~~ aaa

A Control Theory Tutorial:Linear Systems

u

X

0C

BA

y

X X = The “internal” state of the system

y = The output

u = The inputnRt )(XmRt )(upRt )(y

A = Determines the “internal” dynamics of the system

B = Determines which states get “externally” excited

C = Determines what quantities are “measured”

Solution to such a system is:

))(B)0(X()(X AA dueet t If X(0)=0, then

duett

0

)-A(t )(B)(X

0

t

dutG

)(),(

)(~

y uG ),(),(:~

22 LLG

The Names of G:

•Impulse Response

•Greens Function

.

.

.

*

*

*....

.*...

..*..

..0**

...0*

.

.

.

*

*

For Linear Time Invariant Causal Systems:

•Schematic form of the above equation

•Zero above diagonal

•Equal along the diagonals

G~

y = u

2

1 0~T

TG

]0,(]0,(: 221 LLT

),0[]0,(: 22 LL

Ti = Toeplitz operator Γ = Hankel operatorGT

~1

0

0

0

1 2 2

0 20

12

0 02

: ( ) ( ) ( )

( )

minimizes ( )

Tn At At

T

TA t At A t n n

TA t

opt

opt

u e Bu t dt e Bu t dt

x B e x e BB e dt

u B e Cx u u t dt

u x x

C C

C CC

CC

CC

-L R

R

t-T

0nx R

Control from x(-T) = 0 to x(0) = x0, with minimal input.

0

0

0

( ) ( )T

At At

T

x e Bu t dt e Bu t dt

u - L

NOTE: later C=Ψc

Quantifying Controllability• ΨcΨc

* has the same range as Ψc

• If the matrix ΨcΨc* is invertible, then the

system is controllable

• Small eigenvalues of ΨcΨc* correspond to

directions (states) that aren’t very controllable

• Singular values of Ψc are related to the eigenvalues of ΨcΨc

* as so:)()( *

CCiCi

0 0

2 2

0 020

1 2

2

: ( )

( ) ( )

( )

( ) minimizes ( ) given y

n At

T TA t A t At

T

opt

x Ce x

y e C y t dt e C Ce dt

y y t dt x x

x y y x

O O

O O O

O O

O O O O

+

+

R L

L

t T

0Aty Ce x0

nx R

Observe output.

NOTE: later O=Ψo

Quantifying Observability• Ψo

*Ψo has the same null space as Ψo

• If the matrix Ψo*Ψo is invertible, then the

system is observable • Small eigenvalues of ΨoΨo correspond to

directions (states) that aren’t very observable

• Singular values of Ψo are related to the eigenvalues of Ψo

*Ψo as so:)()( *

ooioi

t-T t T

0Aty Ce x0

0

( )T

Atx e Bu t dt 0nx R

Simple input-output system

• Past inputs (t < 0) create state x(0) = x0 at time t = 0.• The input is shut off for t > 0. • The output is observed for t > 0.• Separating forcing from observing makes the math simple

and accessible• Key conclusions are relevant to more complicated

situations

t-T

( ,0)

u

T

2

L

x Ax bu

y cx

y u

tT

(0, )y T 2 L

Hankel operators and singular values

O Cu

t-T

( ,0)

u

T

2

L

x Ax bu

y cx

y u

0 C

0( )

T A

x u

e bu d

2

0Nx R

tT

(0, )y T 2 LO 0

0At

y x

ce x

Impulse response

O Cu

t-T

( ,0)

u

T

2

L

y u

tT

(0, )y T 2 L

1 2 3 4

2 3 4 5

3 4 5

4 5

(0) ( 1)

(1) ( 2)

(2) ( 3)

(3) ( 4)

y h h h h u

y h h h h u

y h h h u

y h h u

Singular values:

O C( ), ( ), ( )i i i

measure gain and approximate rank

1 2 3 4

2 3 4 5

3 4 5

4 5

(0) ( 1)

(1) ( 2)

(2) ( 3)

(3) ( 4)

y h h h h u

y h h h h u

y h h h u

y h h u

Intuition: H is a high-gain, low-rank operator (matrix).

t-T

( ,0)

u

T

2

L

tT

(0, )y T 2 L

(future ) = (past )y H u

O Cu

t-T

( ,0)

u

T

2

L

y u

tT

(0, )y T 2 L

O C( ) ( ) ( )i i i

1 - =k kH H

Optimal kth order model

1 2 3 4

2 3 4 5

3 4 5

4 5

(0) ( 1)

(1) ( 2)

(2) ( 3)

(3) ( 4)

y h h h h u

y h h h h u

y h h h u

y h h u

(future ) = (past )y H u

Model Reduction:Goal: Approximate the impulse response by a lower

rank operator by using information from the Hankel operator (this scheme generalizes)

Fact: When a system is controllable and observable, then one can find coordinates such that:

Ψo*Ψo= ΨcΨc

*

WHAT ADVANTAGE DO THESE COORDINATES GIVE US?

)()()( iCioi and

ANSWER:• Controllability and observability are on the same footing

• The Hankel Singular Values may be directly interpreted in terms of oberservability and controllability

RESULT:

• System = state variables that are very controllable and observable (i.e. correspond to large HSV)

• Environment = state variables that correspond to small HSV

Example 1: The Heat Equation

ln(σn) vs n

More observable

Less observable

σn vs n

σn vs n

1 1 0 0

1 2 1 0

0 1

2 1

0 0 1 2

K

21 N

u=force

y=velocity

Mass=1Spring K=1

( , ) ( ) / 2H p q p p q Kq

0

0 0

p K p Fu

q I q

y Mp

Homogeneous N masses, N+1 springs(Work by Caltech group)

0

0 0

p K p Fu

q I q

y Mp

0 200 400 600 800 1000

0

0.5

1

Impulseresponse

N=100

21 N u=force

y=velocity

0 200 400 600 800 1000

N=100

N=200

N=4002N

0 10 20 30 40 50

0

0.5

1

3/ 2 sin( )t t

0 200 400 600 800 1000-0.5

0

0.5

0 5 10 15 20 25 30 35 40 45 50-0.5

0

0.5

1

O C, reversible 40 states

Full order

0 200 400 600 800 1000-0.5

0

0.5

0 5 10 15 20 25 30 35 40 45 50-0.5

0

0.5

1

dissipative 6 statesFull order

( )i

0 20 40 6010

-10

10-5

100

O

C

( )

( )i

i

Can get low order models with guaranteed error bounds.

dissipative

O C,

reversible

6 states

40 states

1 - =k kH H

Semi-Summary• Small HSV’s correspond to environmental degrees of freedom

Small HSV’s are related to entropy and carry information about uncertainty and noise

• Small HSV’s related to the observed dissipation

Hints of fluctuation-dissipation theorem – without stochastic processes!

Oddities:

σn vs n

N = 20springs in chain

Approximate over reasonably short time scale

C=B=I

B=rank 1C=I

B=rank 1

C=rank 1

σn vs n

N = 100 springs in chain

Time scale on the order of the system length (mid scale)

C=B=I

B = rank 1C=I

σn vs nN = 20

springs in chain

Quite a long time scale (going like N2)

C=B=IC= rank 1B= rank 1

The End