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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
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