Transcript Reduced Order Models in Physics:

Reduced Order Models in
Physics:
Fundamental Physics from Control
Theory?
The “Really Big” Picture
Directions in Theoretical Physics (not exhaustive and
highly subjective)
Phenomenology Matching
Determining
Constituent
matter
Dynamics,
Equilibrium,
and system
manipulation
Properties
of such systems
with many degrees
of freedom
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
from an applied perspective
Physical Laws
(or response of
system)
System Identification
and Realization
Stability
Control Design
Performance
Control Objectives
Robustness
Control Theory
Focus of Talk
• Distributed Systems
(high order systems, usually governed by
PDE’s)
An issue of practical
• Model Reduction
design
(related to finding approximate reduced
order realizations)
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
macro
Tennis ball
Polymers (fibers)
micro
Monomers &
Molecules
Reduction: The SystemEnvironment 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:
  f (X)
• Dynamics
X
• 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
•New Effective Dynamics:
~

XS  f (XS )  g (XS , Xe )
•Approximate the environment contribution by a
stochastic driving term, F(XS)
•New Stochastic Dynamics:
~
  f (X )  F(X )  f
X
S
S
S
e
f e  g (XS , Xe )  F(XS )  0
•Result: A Langevin type equation
(motivated by work by M. Kac and R. Zwanzig)
RG: In a picture
“fine grained”
“coarse grained”
Coarse Grained Variables = “averaged” variables
RG: Heuristics
System-Environment split in RG context
• System = Averaged variables
• Environment = The “details” that are
“ignored”
Example:
i
Lo   ai φ
Fine grained functional
i
~i
L f   a~i φ
Coarse grained functional
i
where
~
~
a  a (a)
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)
~  a~(a)
• Noise gets translated into a
• In physics: The coupling constants (in the
Lagrangian) get RENORMALIZED
• Example: Charge Screening in electronic
systems
ASIDE: The above transformation may not
be invertible (i.e. the RG transformations
form a semi-group)
A Control Theory Tutorial:
Linear Systems
   A B  X 
X = The “internal” state of
X
the system
 



 y  C 0  u 
y = The output
n
u = The input
X(t )  R
m
A = Determines the “internal”
u(t )  R
dynamics of the system
p
y(t )  R
B = Determines which states
get “externally” excited
C = Determines what quantities
are “measured”
Solution to such a system is:
t
X(t )  e (X(0)   e
At
 A
Bu ( )d )
0
If X(0)=0, then
t
X(t )   e
0
A(t - )
Bu( )d

  G(t , )u( )d

~
y  G(u)
The Names of G:
•Impulse Response
•Greens Function
~
G : L2 (, )  L2 (, )
For Linear Time Invariant Causal Systems:
* * 0 . . .  *
* * * 0 . .  *
  
 
.   . . * . . . 
  
 
.
.
.
.
*
.
  
. 
 .   . . . . *  . 
y =
~
G
~ T1 0 
G

  T2 
~
  T1  G
•Schematic form of the above
equation
•Zero above diagonal
•Equal along the diagonals
u
T1 : L2 (,0]  L2 (,0]
 : L2 (,0]  L2 [0, )
Ti = Toeplitz operator
Γ = Hankel operator
Control from x(-T) = 0 to x(0) = x0, with minimal input.
u L-
0
T
T
0
x0 Rn
 At
At
e
Bu
(
t
)
dt

e

 Bu(t )dt
x0 
t
-T
C: L-  R n
C(u ) 
0
T
T
0
 At
At
e
Bu
(
t
)
dt

e

 Bu (t )dt
T
C ( x)  B e  At x CC   e At BB e At dt  R nn
0


uopt  CC

1
B e
 At
T
Cx0 minimizes
u
2
2
  u ( t ) dt
2
0
uopt
2
2

 x0 CC


1
x0
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:
 i (C )  i ( C C )
*
Observe output.
x0 Rn
y  Ce At x0
t
O : R n  L+
T
T
O( x)  Ce At x
T
O ( y )   e C y (t )dt O O   e At C Ce At dt
At
0
0
T
y
2
2


  y (t ) dt x0 O O x0
2
0


xopt  O O

1
O ( y ) minimizes
y  O( x)
NOTE: later O=Ψo
2
2
given y  L +
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:
 i (o )  i ( o o )
*
Simple input-output system
T
x0   e At Bu (t )dt
0
t
-T
•
•
•
•
x0 R
n
y  Ce At x0
t
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
u
L 2 (T , 0)
Hankel operators and
singular values
t
-T
y  u
x  Ax  bu
y  cx
y  L 2 (0, T )
t
T
x0   Cu
u
L 2 (T , 0)
T
  e bu ( )d
A
0
x0 R
t
-T
y  u
  O  Cu
x  Ax  bu
y  cx
y  L 2 (0, T )
y   O x0
 ce x0
At
t
T
2N
u
L 2 (T , 0)
Impulse response
t
-T
y  u
  O  Cu
 y (0)   h1
 y (1)   h

  2
 y (2)    h3

 
 y (3)   h4

 
h3
h4
h3
h4
h4
h5
h5
h5
  u (1) 
 u (2) 


 u (3) 


 u (4) 
 

Singular values:
y  L 2 (0, T )
t
h2
T
 i (),  i (O ),  i (C )
measure gain and
approximate rank
u
L 2 (T , 0)
t
-T
 y (0)   h1
 y (1)   h

  2
 y (2)    h3

 
 y (3)   h4

 
h2
h3
h4
h3
h4
h4
h5
h5
h5
  u (1) 
 u (2) 


 u (3) 


 u (4) 
 

(future y) = H (past u)
y  L 2 (0, T )
t
T
Intuition: H is a
high-gain, low-rank
operator (matrix).
u
L 2 (T , 0)
 i ()   i (O )   i (C )
t
-T
y  u
  O  Cu
 y (0)   h1
 y (1)   h

  2
 y (2)    h3

 
 y (3)   h4

 
h3
h4
h3
h4
h4
h5
h5
h5
  u (1) 
 u (2) 


 u (3) 


 u (4) 
 

(future y) = H (past u)
y  L 2 (0, T )
t
h2
T
Optimal kth order model
H - Hk
= k 1
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*
and
 i (o )   i (C )   i ()
WHAT ADVANTAGE DO THESE
COORDINATES GIVE US?
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
Less
observable
More
observable
ln(σn) vs n
σn vs n
σn vs n
Homogeneous N masses, N+1 springs
Mass=1
Spring K=1
(Work by Caltech group)
y=velocity
1
2
N

u=force
H ( p, q)  ( pp  qKq) / 2
 p  0  K   p   F 
 q   I 0   q    0  u
  
   
y  Mp
0
 1 1 0
  1 2 1

0



K   0 1


2 1

 0 0
1 2 
Impulse
response
y=velocity
1
2

N
u=force
1
0.5
0
0
200
400
 p  0  K   p   F 
 q   I 0   q    0  u
  
   
y  Mp
600
800
1000
N=100
N=100
N=200
 2N
N=400
0
200
400
600
800
1000
1
0.5
t
3/ 2
sin(t )
0
0
10
20
30
40
50
O , C  reversible
40 states
0.5
Full order
0
-0.5
0
200
400
600
800
1000
1
0.5
0
-0.5
0
5
10
15
20
25
30
35
40
45
50
  dissipative
6 states
Full order
0.5
0
-0.5
0
200
400
600
800
1000
1
0.5
0
-0.5
0
5
10
15
20
25
30
35
40
45
50
0
10
 i ( O )
  i ( C )
Can get low order
models with
guaranteed error
-5
bounds.
10

O , C 
reversible
 i ()
40 states
dissipative
-10
10
0
6 states
20
40
H - Hk
60
= k 1
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:
N = 20
springs in chain
B=rank 1
C=I
B=rank 1
σn vs n
C=B=I
C=rank 1
Approximate over reasonably short time scale
σn vs n
N = 100 springs
in chain
B = rank 1
C=I
C=B=I
Time scale on the order of the system length (mid scale)
σn vs n
N = 20
springs in chain
C=B=I
Quite a long time scale (going like N2)
C= rank 1
B= rank 1
The End