IEEE TRANS ON AUTOMATIC CONTROL, FEBRUARY, 2011 Sandberg, Delvenne, and Doyle today.

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