IEEE TRANS ON AUTOMATIC CONTROL, FEBRUARY, 2011 Sandberg, Delvenne, and Doyle http://arxiv.org/abs/1009.2830 today
Jan 04, 2016
IEEE TRANS ON AUTOMATIC CONTROL, FEBRUARY, 2011Sandberg, Delvenne, and Doyle
http://arxiv.org/abs/1009.2830
today
w white, unit intensity (J-N noise)k Boltzmann’s constant
Phenomenology
R
Resistor RTemperature T
Capacitor CVoltage v
2
dvCv C i
dt
v Ri kTRw
Dissipation Fluctuation
w white, unit intensityk Boltzmann’s constant
2
dvCv C i
dt
v Ri kTRw
Dissipation Fluctuation
1 2kTCv v w
R R
Origins?Consequences?
Assume
2 1
1
kT
R
R
y
ideal sensor
v i w
Dissipation
Sensor noise
y v w
Cv i
Fluctuation
Measurement
R
y
ideal sensor
v i w
Dissipation
Sensor noise
y v w
Cv i
Fluctuation
1 1v v w
C C
Back actionMeasurement
R
y
ideal sensor
v i w
Cv i
1 1v v w
C C
Back action
y v w
Sensor noise
y v w
Measurement
R
y
more ideal sensor
v i w
Cv i
1 1v v w
C C
Back action
y v w
Sensor noise
-R Active
Assume active device has infinite power supply
Measurement
1v w
C
Back action
y v w Sensor noise
R
y
-R y
2ˆmin E v v v̂Optimal estimator
2E v
1v w
C
y v w
R
y
-R y
2ˆmin E v v
Optimal estimator
SoftwareHardware
DigitalAnalog
Active
Lumped
Computers
1v w
C
Upside down from other pictures
1v w
C
y v w
y
2ˆmin E v v v̂
Optimal estimator
2E v
2
2
1 1ˆ
1 tE v v
te
22
( )t
E vC
2ˆAssume (0) (0)E v v
0Assume (0) unknownv v
1v w
C
y v w
y
2ˆmin E v v v̂
2E v
2
2
2 1
1 YtE e
te
22
tE v
C
ˆe v v 2ˆAssume (0) (0)E v v
0Assume (0) unknownv v
Can compute everything analytically because of special structure.
1 2kTv w
C R
2y v kTRw
y
2ˆmin E v v v̂
2E v
2
2
2 1
1 YtE e
te
22
tE v
C
back-action
error
v
ˆe v v
ˆe v v
2
2
2 1
1 tE e
te
22
tE v
C
back-action
error
v
ˆe v v
1
22 2
2
1 2 1
1 t
tE v E e
C Ce
Cold sensors are uniformly easier
2 RE e
t
22
tE v
RC
back-action
error
v
ˆe v v
2 2 1E v E e
C
2 RE e
t
22
1tE v
R C
2
2 2 1E v E e
C
time t 2 2E e
2 RE e
t
22
tE v
RC
back-action
error
v
ˆe v v
2 2 1E v E e
C
t small
vary R
Particle mass
Position
Velocity
m
x
v
Dectector at 0
Friction
x t
R
x v
0x
m vm v
Heisenberg?
1 2
x v
kTv w
m R
Back action
2y v kTRw Sensor noise
y
2ˆmin E v v
Optimal estimator
2E v
Particle mass
Position
Velocity
m
x
v
Dectector at 0
Friction
x t
R
2E x
2
2
ˆ
ˆ
E x x
E v v
1
x v
v wm
y v w
2
2
ˆ
ˆ
E x x
E v v
2
2
2
2 1ˆ
1
t
t
eE x x t
e
2
2
ˆAssume (0) (0) 0
ˆ (0) (0)
E x x
E v v
0
Assume x(0) 0 known
(0) unknownv v
2
2
2 1ˆ
1 tE v v
te
1
x v
v wm
y v w
2
2
ˆ
ˆ
E x x
E v v
2ˆE x x t
2 2
2
ˆ ˆ
2 11
1
t
t
E v v E x x
e
e
2 1ˆE v v
t
error
error
2ˆE x x t
2 1ˆE v v
t
22
tE v
m back
action
Cold sensors (and large masses) are uniformly easier
error
error
2ˆE x x t
2 1ˆE v v
t
Let 2 1kT R
22
tE v
m back
action
2 2ˆ ˆ 1E v v E x x
2 22
1ˆE v v E v
m
R
y
more ideal sensor
1 1 2kTv v w
CR C R
Back action
2y v kTRw
Sensor noise
-R Active
Active device has infinite power supply
2ˆE x x Rt
2ˆR
E v vt
22
1tE v
R m
2 2ˆ ˆE v v E x x R
2 22
1ˆE v v E v
m
Next steps• Estimation to control• Efficiency of devices, enzymes• Classical to quantum
w white, unit intensityk Boltzmann’s constant
2
dvCv C i
dt
v Ri kTRw
Dissipation Fluctuation
1 2kTCv v w
R R
Origins?Consequences?
Resistor RTemperature T
Capacitor CVoltage v
w white, unit intensityk Boltzmann’s constant
T=0
R
1 2kTCv v w
R R
Temporarily
1
Cs
Y
+
1Y
R
R
step response
1v v
CRy v
y v
Caution: this is a visualization of the equations, the “signals” are not physical(“virtual”)
1
Cs
Y
+
1Y
R
R
step response
1v v
CRy v
y v
Caution: this is a visualization of the equations, the “signals” are not physical
Step response is easier to visualize than impulse…
0 100
0.5
1
1.5
Time (sec)
Am
plitu
de
1 te
1
1
C
Y
dissipative,lossy
But the microscope world is lossless (energy is conserved). Where does dissipation come from?
1
s
1
+step response
0Y
1v v
CRy v
+step response
2 2
4
k
s
T s
LosslessApproximate
1
+step response
dissipative,lossy
LosslessApproximate
1dissipative,
lossy
step response
2 2
4
k
s
T s
1: 2 : 2k nT
step response
Step response
Emphasize the differences
Cosine series
LosslessApproximate
1dissipative,
lossy
2 2
4
k
s
T s 1: 2 : 2k n
Step response
0 0.5 1 1.5
-1
0
1
n=10, =1
Time (sec)
Time (sec)
n=5, =1 dissipative,lossy
0 0.2 0.4 0.6 0.8 1 1.2
0
0.2
0.4
0.6
0.8
1
1.2
n=5, =1
n=10, =1
n=10, =2
For n/, step
2 2
4
k
s
T s 1: 2 : 2k n
=1
0 1 2
-1
0
1
n=10
n/
LosslessApproximate
step response
2 2
4
k
s
T s
1: 2 : 2k nT
age of universe 4e26 nanosecs
=1
n=10
0 0.2 0.4 0.6 0.8 1-1.5
-1
-0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1-1.5
-1
-0.5
0
0.5
1
1.5
n=100
Theorem: Linear dissipative (passive)iff linear lossless approximation
For n/,
Theorem: Linear dissipative (passive)iff linear lossless approximation
Corollary: Linear active needs nonlinear lossless approximation
Proof: Essentially Fourier series plus elementary control theory.
Question: what nonlinearities can be fabricated?
For n/,
0 2 4 6 8 100
0.5
1
1.5
Time (sec)
1
Cs+step
responsev(t)
2 2
4
k
s
T s
LosslessApproximate
=10n=10
n=4
1
1
C
R
0 0.5 1 1.5 2 2.5-1.5
-1
-0.5
0
0.5
1
1.5
Time (sec)
Am
plitu
de
n=10
step response
2 2
4
k
s
T s
LosslessApproximate
2 2
4
k
s
T s
random initial
conditions
0 0.2 0.4 0.6 0.8 1-0.5
0
0.5
n=10
random initial
conditions
2 2
4
k
s
T s
0 0.5 1 1.5 2 2.5-1.5
-1
-0.5
0
0.5
1
1.5
Time (sec)
Am
plitu
de
n=10
step response
2 2
4
k
s
T s
0 0.2 0.4 0.6 0.8 1-0.5
0
0.5
0 0.5 1 1.5 2 2.5-1.5
-1
-0.5
0
0.5
1
1.5
Time (sec)
Am
plitu
de
n=10
0 0.2 0.4 0.6 0.8 1-0.5
0
0.5
Impulse responseg t
Autocorrelationa t
fluctuation
dissipation
a(t) = kT g(t) (all n)
k = Boltzmann constant, T=temperature
Theorem:
T=1
n=100
n=10
0 0.2 0.4 0.6 0.8 1
0
0 0.2 0.4 0.6 0.8 1
0
“white” for n large
n=1000 0.2 0.4 0.6 0.8 1
0
T=10 0.2 0.4 0.6 0.8 1
-1.5
-1
-0.5
0
0.5
1
1.5
Dissipation
Fluctuation
Theorem: Fluctuation Dissipation
Theorem: Fluctuation Dissipation
Theorem: Linear passive iff linear lossless approximation
Corollary: Linear active needs nonlinear lossless approximation
“New”
“Old”
Resistor RTemperature T
Capacitor CVoltage v
1 1 2kTv v w
CR C R
w white, unit intensityk Boltzmann’s constant
1
Cs
Y
+
1Y
R
T>0
1v w
C
Back action
y v w Sensor noise
R
y
-R y
2ˆmin E v v v̂Optimal estimator
2E v
2 RE e
t
22
tE v
RC
back-action
error
v
ˆe v v
t small
vary R
1
22 2
2
1 2 1
1 t
tE v E e
C Ce