John Doyle Control and Dynamical Systems Caltech.

John DoyleControl and Dynamical

Systems Caltech

Research interests

• Complex networks applications– Ubiquitous, pervasive, embedded control,

computing, and communication networks– Biological regulatory networks

• New mathematics and algorithms– robustness analysis – systematic design– multiscale physics

Collaboratorsand contributors

(partial list)

Biology: Csete,Yi, Borisuk, Bolouri, Kitano, Kurata, Khammash, El-Samad, …

Alliance for Cellular Signaling: Gilman, Simon, Sternberg, Arkin,…HOT: Carlson, Zhou,…Theory: Lall, Parrilo, Paganini, Barahona, D’Andrea, …Web/Internet: Low, Effros, Zhu,Yu, Chandy, Willinger, …Turbulence: Bamieh, Dahleh, Gharib, Marsden, Bobba,…Physics: Mabuchi, Doherty, Marsden, Asimakapoulos,…Engineering CAD: Ortiz, Murray, Schroder, Burdick, Barr, …Disturbance ecology: Moritz, Carlson, Robert, …Power systems: Verghese, Lesieutre,…Finance: Primbs, Yamada, Giannelli,……and casts of thousands…

Background reading online• On website accessible from SFI talk abstract• Papers with minimal math

– HOT and power laws– Chemotaxis, Heat shock in E. Coli– Web & Internet traffic, protocols, future issues

• Thesis: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization

• Recommended books– A course in Robust Control Theory, Dullerud and

Paganini, Springer– Essentials of Robust Control, Zhou, Prentice-Hall– Cells, Embryos, and Evolution, Gerhart and Kirschner

+ Regulatory InteractionsMass Transfer in Metabolism*

Biochemical Network: E. Coli Metabolism

* from: EcoCYC by Peter Karp

From Adam Arkin

SuppliesMaterials &

Energy

SuppliesMaterials &

Energy

SuppliesRobustness

SuppliesRobustness

Complexity RobustnessComplexity Robustness

ComplexityRobustness

Transcription/translation

MicrotubulesNeurogenesisAngiogenesis

Immune/pathogenChemotaxis

….

Regulatory feedback control

An apparent paradox

Component behavior seems to be gratuitously uncertain, yet the systems have robust performance.

Mutation

Selection

Darwinian evolution uses selection on random mutations

to create complexity.




….


• Such feedback strategies appear throughout biology (and advanced technology).

• Gerhart and Kirschner (correctly) emphasis that this “exploratory” behavior is ubiquitous in biology…

• …but claim it is rare in our machines.

• This is true of primitive, but not advanced, technologies.

• Robust control theory provides a clear explanation.


Overview

• Without extensive engineering theory and math, even reverse engineering complex engineering systems would be hopeless. (Let alone actual design.)

• Why should biology be much easier? • With respect to robustness and complexity, there is too

much theory, not too little.

Overview

• Two great abstractions of the 20th Century:– Separate systems engineering into control, communications,

and computing• Theory

• Applications

– Separate systems from physical substrate

• Facilitated massive, wildly successful, and explosive growth in both mathematical theory and technology…

• …but creating a new Tower of Babel where even the experts do not read papers or understand systems outside their subspecialty.

“Any sufficiently advanced technology is indistinguishable from magic.”

Arthur C. Clarke

“Those who say do not know, those who know do not say.”

Zen saying

“Any sufficiently advanced technology is indistinguishable from magic.”

Arthur C. Clarke

Today’s goal• Introduce basic ideas about robustness and complexity• Minimal math• Hopefully familiar (but unconventional) example

systems• Caveat: the “real thing” is much more complicated• Perhaps any such “story” is necessarily misleading• Hopefully less misleading than existing popular

accounts of complexity and robustness

Complexity and robustness

• Complexity phenotype : robust, yet fragile• Complexity genotype: internally complicated• New theoretical framework: HOT (Highly optimized

tolerance, with Jean Carlson, Physics, UCSB)• Applies to biological and technological systems

– Pre-technology: simple tools– Primitive technologies use simple strategies to build fragile

machines from precision parts.– Advanced technologies use complicated architectures to create

robust systems from sloppy components…– … but are also vulnerable to cascading failures…

Robust, yet fragile phenotype

• Robust to large variations in environment and component parts (reliable, insensitive, resilient, evolvable, simple, scaleable, verifiable, ...)

• Fragile, often catastrophically so, to cascading failures events (sensitive, brittle,...)

• Cascading failures can be initiated by small perturbations (Cryptic mutations,viruses and other infectious agents, exotic species, …)

• There is a tradeoff between – ideal or nominal performance (no uncertainty) – robust performance (with uncertainty)

• Greater “pheno-complexity”= more extreme robust, yet fragile

Robust, yet fragile phenotype

• Cascading failures can be initiated by small perturbations (Cryptic mutations,viruses and other infectious agents, exotic species, …)

• In many complex systems, the size of cascading failure events are often unrelated to the size of the initiating perturbations

• Fragility is interesting when it does not arise because of large perturbations, but catastrophic responses to small variations

Complicated genotype

• Robustness is achieved by building barriers to cascading failures

• This often requires complicated internal structure, hierarchies, self-dissimilarity, layers of feedback, signaling, regulation, computation, protocols, ...

• Greater “geno-complexity” = more parts, more structure• Molecular biology is about biological simplicity, what

are the parts and how do they interact.• If the complexity phenotypes and genotypes are linked,

then robustness is the key to biological complexity.• “Nominal function” may tell little.




….


An apparent paradox


Mutation

Selection

Darwinian evolution uses selection on random mutations

to create complexity.

Tempenviron

Tempcell

Folded Proteins

Unfolded Proteins Aggregates

Loss of ProteinFunction

Networkfailure

Death

Cell

Tempenviron

Tempcell

Folded Proteins


Loss of ProteinFunction

Networkfailure

Death

Cell

How does the cell build “barriers” (in state space) to stop

this cascading failure event?

Tempenviron

Folded Proteins

Tempcell

Insulate &Regulate

Temp

Tempenviron

Folded Proteins

Tempcell

Thermo-tax

Tempenviron

Tempcell

Folded Proteins


More robust ( Temp stable)

proteins

Tempenviron

Tempcell

Folded Proteins


• Key proteins can have multiple (allelic or paralogous) variants• Allelic variants allow populations to adapt• Regulated multiple gene loci allow individuals to adapt

-1/T

21o

Log of E. ColiGrowthRate

37o

46o

Heat Shock Response

RTAEev

42o

-1/T

21o

Log of E. ColiGrowthRate

37o

42o

46o

Robustness/performance tradeoff?

Tempenviron

Tempcell

Folded Proteins

Unfolded Proteins

Refold denatured proteins

Heat shock response involves complex feedback

and feedforward control.

Alternative strategies

• Robust proteins– Temperature stability

– Allelic variants

– Paralogous isozymes

• Regulate temperature• Thermotax• Heat shock response

– Up regulate chaperones and proteases

– Refold or degraded denatured proteins

Why does biology (and advanced technology)

overwhelmingly opt for the complex control

systems instead of just robust components?

E. Coli Heat Shock (with Kurata, El-Samad, Khammash, Yi)

unfoldPDnaK :

dependent T

DnaK:32

320 32

free

0FtsH

FtsHDnaK ::32

protease:32

0DnaK freeDnaK

1k 2k

distk

3k

03.0

1

s

D n a k t r a n s l a t i o n & t r a n s c r i p t i o n

d y n a m i c s

1r

2r

rateon translati

dependent T 32

raten degradatio 32

protease

rpoH gene

Transcription

32 mRNA

hsp1 hsp2

Transcription & Translation

FtsHLonDnaKGroLGroS

Chaperones

Proteases

-

- Translation

32

Heat

Heat stabilizes32

Heat

Outer Feedback Loop

Local Loop

Feedforward

Heater

Thermostat

Added mass

Moves the center of mass forward.

Tail

Moves the center of pressure aft.

Thus stabilizing forward flight.

At the expense of extra weight and drag.

For minimum weight & drag, (and other performance issues)

eliminate fuselage and tail.

Why do we love building robust systems from highly uncertain

and unstable components?

P- +

(disturbance)d

r( )y P r d

Assumptions on components:• Everything just numbers • Uncertainty in P• Higher gain = more uncertain

( )y P P r d

1 21 2

1 2

P PP P

P P

G-

K

+

dr

P- +

(disturbance)d

r

11y GSr Sd S r Sd

K

1

1S

GK

Negative feedback

( )y G r GK y d

( )y P r d

11y GSr Sd S r Sd

K

1

1S

GK

G-

K

+

dr y

11 1

11

G GKK

S y rK

Results for y (1/K )r:• high gain• low uncertainty• d attenuated

S = sensitivity function

Design recipe:• 1 >> K >> 1/G • G >> 1/K >> 1• G maximally uncertain!• K small, low uncertainty

Results for y (1/K )r:• high gain• low uncertainty• d attenuated

Extensions to:• Dynamics• Multivariable• Nonlinear• Structured uncertainty

All cost more computationally.

G-

K

+

dr y

Design recipe:• 1 >> K >> 1/G • G >> 1/K >> 1• G maximally uncertain!• K small, low uncertainty

G-

K

r y

Transcription/translationMicrotubule formation

NeurogenesisAngiogenesis

Antibody productionChemotaxis

….


Uncertain high gain

Summary

• Primitive technologies build fragile systems from precision components.

• Advanced technologies build robust systems from sloppy components.

• There are many other examples of regulator strategies deliberately employing uncertain and stochastic components…

• …to create robust systems.• High gain negative feedback is the most powerful

mechanism, and also the most dangerous.• In addition to the added complexity, what can go

wrong?

G-

K

y

1

1y d

F

1d if F 1

F

( )y F y d

+

(disturbance)d

F

y+

d

F GK

1

1y d

F

F

y+

d

If y, d and F are just numbers:

S = sensitivity function

S measures disturbance rejection.

It’s convenient to study ln(S).

1

1

yS

d F

P

N

o

eg

si

ativ

tive

e ( 0) ln

( 0) ln( ) 0 Disturbance ampli

( ) 0 Disturbance attenuated

fiedF

F S

S

F

F

P

N

o

eg

si

ativ

tive

e ( 0) ln



fiedF

F S

S

F

F

ln(S)

F

F < 0ln(S) < 0

attenuation

F > 0ln(S) > 0

amplification

ln( |S| )

1

1

yS

d F

P

N

o

eg

si

ativ

tive

e ( 0) ln



fiedF

F S

S

F

F

P

N

o

eg

si

ativ

tive

e ( 0) ln



fiedF

F S

S

F

F

ln(S)

F ln(S)

extreme robustnessextreme robustness

F 1 ln(S)

extreme sensitivityextreme sensitivity

F

1

1

yS

d F

If these model physical processes, then d and y are signals and F is an operator. We can still define

S( = |Y( /D( |where E and D are the Fourier transforms of y and d. ( If F is linear, then S is independent of D.)

Under assumptions that are consistent with F and d modeling physical systems (in particular, causality), it is possible to prove that:

0)(log dS

(Bode, ~1940)

Fy+d

1

1S

F

log|S |he amplification (F>0) must atleast balance the attenuation (F<0).

( 0) ln( ) 0 attenuate

( 0) ln( ) 0 amplify

F

F S

S

( 0) ln( ) 0 attenuate

( 0) ln( ) 0 amplify

F

F S

S

log|S |

ln|S|

F

Negative feedback

Positive feedback

log|S |

ln|S|

F

Negative feedbackRobust

Positive feedback

…yetfragile

Robustness of HOT systems

Robust

Fragile

Robust(to known anddesigned-foruncertainties)

Fragile(to unknown

or rareperturbations)

Uncertainties

Feedback and robustness

• Negative feedback is both the most powerful and most dangerous mechanism for robustness.

• It is everywhere in engineering, but appears hidden as long as it works.

• Biology seems to use it even more aggressively, but also uses other familiar engineering strategies:– Positive feedback to create switches (digital systems)– Protocol stacks– Feedforward control– Randomized strategies– Coding

ComplexityRobustness

Current research

• So far, this is all undergraduate level material• Current research involves lots of math not

traditionally thought of as “applied”• New theoretical connections between robustness,

evolvability, and verifiability• Beginnings of a more integrated theory of control,

communications and computing• Both biology and the future of ubiquitous,

embedded networking will drive the development of new mathematics.


Robust

Fragile

Robust(to known anddesigned-foruncertainties)

Fragile(to unknown

or rareperturbations)

Uncertainties


Robust

Fragile

Chess Meteors

Humans

Archaea


Robust

Fragile

Chess Meteors

Humans

Archaea

Humans + machines?

Machines

Robust

Fragile

Uncertainty

Diseases of complexity

CancerEpidemics

Viral infectionsAuto-immune disease

Robust

Fragile

Sources of uncertainty

• In a system– Environmental perturbations– Component variations

• In a model– Parameter variations– Unmodeled dynamics– Assumptions– Noise

( )F

Robust

Fragile

Sources of uncertainty

( ) ?F

( ) ?F

Typically NP hard.

• If true, there is always a short proof.• Which may be hard to find.

, ( ) ?F

Typically coNP hard.

• More important problem.• Short proofs may not exist.

Fundamental asymmetries* • Between P and NP• Between NP and coNP

Fundamental asymmetries* • Between P and NP• Between NP and coNP

* Unless they’re the same…

• Standard techniques include relaxations, Grobner bases, resultants, numerical homotopy, etc…

• Powerful new method based on real algebraic geometry and semidefinite programming (Parrilo, Shor, …)

• Nested series of polynomial time relaxations search for polynomial sized certificates

• Exhausts coNP (but no uniform bound)• Relaxations have both computational and physical

interpretations• Beats gold standard algorithms (eg MAX CUT)

handcrafted for special cases• Completely changes the P/NP/coNP picture

How do we prove that , ( ) ?F

Bacterial chemotaxis

Random walk

Ligand Motion Motor

Bacterial chemotaxis (Yi, Huang, Simon, Doyle)

pCheY

Ligand

SignalTransduction

gradient

Biased random walk

Motion Motor

pCheYSignal

Transduction

MotorLigand Motion

High gain (cooperativity)

“ultrasensitivity”

References:Cluzel, Surette, Leibler

http://www.sciencemag.org/content/vol287/issue5458/images/large/se070831002a.jpeg

pCheYSignal

Transduction

+CH3R

ATP ADPP

~

flagellarmotor

Z

Y

PY

~

PiB

B~P

Pi

CW-CH3

ATP

WA

MCPs

WA

+ATT

-ATT

MCPsSLOW

FAST

ligand binding motor

Motor

References:Cluzel, Surette, Leibler + Alon, Barkai, Bray, Simon, Spiro, Stock, Berg, …

http://www.sciencemag.org/content/vol287/issue5458/images/large/se070831002a.jpeg

+CH3R

ATP ADPP

~

flagellarmotor

Z

Y

PY

~

PiB

B~P

Pi

CW-CH3

ATP

WA

MCPs

WA

+ATT

-ATT

MCPsSLOW

FAST

ligand binding

motor

ATP ADPP

~

flagellarmotor

Z

Y

PY

~

Pi

CW

ATP

WA

MCPs

WA

+ATT

-ATT

MCPs

FAST

ligand binding

motor

Fast (ligand and phosphorylation)

0 1 2 3 4 5 6

0

1

0 1 2 3 4 5 6

Time (seconds)

No methylation

Barkai, et al

Short time Yp response

Che Yp

Ligand

Extend run(more ligand)

+CH3R

ATP ADPP

~B

B~P

Pi

-CH3

ATP

WA

MCPs

WA

MCPsSLOW

Slow (de-) methylation dynamics

+CH3R

ATP ADPP

~

flagellarmotor

Z

Y

PY

~

PiB

B~P

Pi

CW-CH3

ATP

WA

MCPs

WA

+ATT

-ATT

MCPsSLOW

FAST

ligand binding

motor

0 1000 2000 3000 4000 5000 6000 7000

01

3

5

0 1000 2000 3000 4000 5000 6000 7000Time (seconds)

No methylation

B-L

Long time Yp response

No methylation

Extend run(more ligand)

Tumble(less ligand)

Ligand

Biologists call this “perfect adaptation”

• Methylation produces “perfect adaptation” by integral feedback.• Integral feedback is ubiquitous in both engineering systems and

biological systems.• Integral feedback is necessary for robust perfect adaptation.

Tumbling bias

pCheY

SignalTransduction

Motor

Perfect adaptation is necessary …

pCheYligand

pCheY

Tumbling bias

ligand

Perfect adaptation is necessary …

…to keep CheYp in the responsive range of the motor.

Fine tuned or robust ?

• Maybe just not the right question.

• Fine tuned for robustness…

• …with resource costs and new fragilities as the price.

+ Regulatory InteractionsMass Transfer in Metabolism*

Biochemical Network: E. Coli Metabolism

* from: EcoCYC by Peter Karp

From Adam Arkin

SuppliesMaterials &

Energy

SuppliesMaterials &

Energy

SuppliesRobustness

SuppliesRobustness

Complexity RobustnessComplexity Robustness

What about ?

• Information & entropy

• Fractals & self-similarity

• Chaos

• Criticality and power laws

• Undecidability

• Fuzzy logic, neural nets, genetic algorithms

• Emergence

• Self-organization

• Complex adaptive systems

• New science of complexity

• Not really about complexity

• These concepts themselves are “robust, yet fragile”

• Powerful in their niche

• Brittle (break easily) when moved or extended

• Some are relevant to biology and engineering systems

• Comfortably reductionist

• Remarkably useful in getting published

Criticality and power laws

• Tuning 1-2 parameters critical point• In certain model systems (percolation, Ising, …) power

laws and universality iff at criticality.• Physics: power laws are suggestive of criticality• Engineers/mathematicians have opposite interpretation:

– Power laws arise from tuning and optimization.

– Criticality is a very rare and extreme special case.

– What if many parameters are optimized?

– Are evolution and engineering design different? How?

• Which perspective has greater explanatory power for power laws in natural and man-made systems?

-6 -5 -4 -3 -2 -1 0 1 2-1

0

1

2

3

4

5

6

Size of events

Frequency

Decimated dataLog (base 10)

Forest fires1000 km2

(Malamud)

WWW filesMbytes

(Crovella)

Data compression

(Huffman)

Los Alamos fire

Cumulative

Size of events x vs. frequency

log(size)

)1()( xxpdx

dPlog(probability)

log(Prob > size)

xPlog(rank)

-6 -5 -4 -3 -2 -1 0 1 2-1

0

1

2

3

4

5

6

Size of events

FrequencyFires

Web filesCodewords

Cumulative

Log (base 10)

-1/2

-1

The HOT view of power laws

• Engineers design (and evolution selects) for systems with certain typical properties:

• Optimized for average (mean) behavior

• Optimizing the mean often (but not always) yields high variance and heavy tails

• Power laws arise from heavy tails when there is enough aggregate data

• One symptom of “robust, yet fragile”

Source coding for data compression

Based on frequencies of source word occurrences,

Select code words.

To minimize message length.

Shannon coding

• Ignore value of information, consider only “surprise”• Compress average codeword length (over stochastic

ensembles of source words rather than actual files)• Constraint on codewords of unique decodability• Equivalent to building barriers in a zero dimensional tree• Optimal distribution (exponential) and optimal cost are:

DataCompression

length log( )

exp( )i i

i i

l p

p cl

Avg. length =

log( )

i i

i i

p l

p p

0 1 2-1

0

1

2

3

4

5

6

DC

Data

Avg. length =

log( )

i i

i i

p l

p p

How well does the model predict the data?

length log(

exp( )

)i i

i i

l p

p cl

0 1 2-1

0

1

2

3

4

5

6

DC

Data + Modellength log(

exp( )

)i i

i i

l p

p cl

Avg. length =

log( )

i i

i i

p l

p p

How well does the model predict the data?

Not surprising, because the file was compressed using

Shannon theory.

Small discrepancy due to integer lengths.

Web layout as generalized “source coding”

• Keep parts of Shannon abstraction:– Minimize downloaded file size– Averaged over an ensemble of user access

• But add in feedback and topology, which completely breaks standard Shannon theory

• Logical and aesthetic structure determines topology

• Navigation involves dynamic user feedback • Breaks standard theory, but extensions are

possible• Equivalent to building 0-dimensional

barriers in a 1- dimensional tree of content

document

split into N files to minimize download time

A toy website model(= 1-d grid HOT design)

# links = # files

split into N files to minimize download time

Forest fires dynamics

IntensityFrequency

Extent

WeatherSpark sources

Flora and fauna

TopographySoil type

Climate/season

A HOT forest fire abstraction…

Burnt regions are 2-d

Fire suppression mechanisms must stop a 1-d front.

Optimal strategies must tradeoff resources with risk.

Generalized “coding” problems

Fires

Web

Data compression

• Optimizing d-1 dimensional cuts in d dimensional spaces…

• To minimize average size of files or fires, subject to resource constraint.

• Models of greatly varying detail all give a consistent story.

• Power laws have 1/d.• Completely unlike criticality.

d = 0 data compressiond = 1 web layoutd = 2 forest fires

1

(1 )0d

i ip l c d

1

d

1

( ) dP l l

exp( )

0i ip cl

d

Theory

-6 -5 -4 -3 -2 -1 0 1 2-1

0

1

2

3

4

5

6

FF

WWWDC

Data

-6 -5 -4 -3 -2 -1 0 1 2-1

0

1

2

3

4

5

6

FF

WWWDC

Data + Model/Theory

Forest fires?

Burnt regions are 2-d

Fire suppression mechanisms must stop a 1-d front.

Forest fires?

Geography could make d <2.

California geography:further irresponsible speculation

• Rugged terrain, mountains, deserts• Fractal dimension d 1?• Dry Santa Ana winds drive large ( 1-d) fires

-6 -5 -4 -3 -2 -1 0 1 2-1

0

1

2

3

4

5

6

FF(national)

d = 2

Data + HOT Model/Theory

d = 1

California brushfires

-6 -5 -4 -3 -2 -1 0 1 2-1

0

1

2

3

4

5

6

Data + HOT+SOC

d = 1

SOC FFd = 2

.15

Critical/SOC exponents are way off

SOC < .15

Data: > .5

Forest Fires: An Example of Self-Organized Critical BehaviorBruce D. Malamud, Gleb Morein, Donald L. Turcotte

18 Sep 1998

4 data sets

10-2

10-1

100

101

102

103

104

100

101

102

103

SOC FF

HOT FFd = 2

Additional 3 data sets

Fires 1991-1995

Fires 1930-1990

HOT

SOC

d=1

dd=1d

• HOT decreases with dimension.• SOC increases with dimension.

SOC and HOT have very different power laws.

1

d 1

10

d

• HOT yields compact events of nontrivial size.• SOC has infinitesimal, fractal events.

HOT

SOC

sizeinfinitesimal large

HOT

SOC

SOC HOT Data

Max event size Infinitesimal Large Large

Large event shape Fractal Compact Compact

Slope Small Large Large

Dimension d d-1 1/d 1/d

SOC and HOT are extremely different.

SOC HOT & Data

Max event size Infinitesimal LargeLarge event shape Fractal Compact

Slope Small LargeDimension d d-1 1/d

SOC and HOT are extremely different.

HOT

SOC

yetfragile

Robust

Gaussian,Exponential

Log(event sizes)

Log(freq.) cumulative

Gaussian

log(size)

log(prob>size)

Power laws are inevitable.

Improved design,more resources

Power laws summary

• Power laws are ubiquitous• HOT may be a unifying perspective for many• Criticality, SOC is an interesting and extreme

special case…• … but very rare in the lab, and even much rarer still

outside it.• Viewing a complex system as HOT is just the

beginning of study.• The real work is in new Internet protocol design,

forest fire suppression strategies, etc…

Universal network behavior?

demand

throughputCongestion

induced “phase

transition.”

Similar for:• Power grid?• Freeway traffic?• Gene regulation?• Ecosystems?• Finance?

Web/Internet?demand

thro

ughp

utCongestion induced “phase transition.”

Power laws

log(file size)

log(

P>

)

2

3 H

random networks

log(thru-put)

log(demand)

Networks Making a “random network:”• Remove protocols

– No IP routing

– No TCP congestion control

• Broadcast everything

Many orders of magnitude slower

BroadcastNetwork

Networks

random networks

real networks

HOTlog(thru-put)

log(demand)

BroadcastNetwork

HOT

Turbulence

flow

pressure drop

random pipes

streamlined pipes

HOT turbulence?Robust, yet

fragile?

• Through streamlined design• High throughput• Robust to bifurcation transition (Reynolds number)• Yet fragile to small perturbations• Which are irrelevant for more “generic” flows

HOT

flow

pressure drop

random pipes

streamlined pipes

Shear flow turbulence summary

• Shear flows are ubiquitous and important

• HOT may be a unifying perspective

• Chaos is interesting, but may not be very important for many important flows

• Viewing a turbulent or transitioning flow as HOT is just the beginning of study

random

designed

HOTYield,flow, …

Densities, pressure,…

The yield/density curve predicted using random ensembles is way off.

Similar for:• Power grid• Freeway traffic• Gene regulation• Ecosystems• Finance?

pipes

channelswings

Turbulence in shear flows

Turbulence is thegraveyard of theories.

Hans Liepmann Caltech

Kumar Bobba, Bassam Bamieh

Chaos and turbulence

• The orthodox view:

• Adjusting 1 parameter (Reynolds number) leads to a bifurcation cascade to chaos

• Turbulence transition is a bifurcation

• Turbulent flows are chaotic, intrinsically nonlinear

• There are certainly many situations where this view is useful.

velocitylow high

equilibriumequilibrium periodicperiodic chaoticchaotic

pressure drop

averageflow

speed

“random” pipe

pressure (drop)

flow(averagespeed)

laminar

turbulent

bifurcation

Random pipes are like bluff bodies.

pressure

flowTypical flow

pipes

channels

wingsStreamline

log(pressure)

log(flow)laminar

turbulent

“theory”

experiment

Random pipe

streamlined pipe

log(Re)

log(flow)

Random pipe

streamlined pipe

210 310 410 510

log(Re)

Random pipe

streamlined pipe

210 310 410 510

It can be promoted (or delayed!)with tiny perturbations.

This transition is extremely delicate(and controversial).

Transition to turbulence is promoted (occurs at lower speeds) by

Surface roughnessInlet distortionsVibrationsThermodynamic fluctuations?Non-Newtonian effects?

None of which makes much difference for “random” pipes.

Random pipe

210 310 410 510

Shark skin delays transition to turbulence

log(pressure)

log(flow)

water

80 ppm Guar

It can be reduced with small amounts of polymers.


fragile?


HOT

flow

pressure drop

random pipes

streamlined pipes

streamwise

Couette flow

upflow

high-speedregion

downflow

low speedstreaks

From Kline

Streamwiseconstantperturbation

Spanwiseperiodic

w

vu

flow

velocity

z

yx

flow

position

z

y

x

flowposition

flow

w

v

u

velocity

0u

2 2 2

2 2 2

/1

/

/

x y z

x y z

x y z

u u u u x

v v v v y pt R x y z

w w w w z

1uu u p u

t R

( , , )u u v w0u v w

x y z

z

yx

flow

w

vu

flow

velocityposition

0x

2d NS

0u

2 2 2

2 2 2

/1

/

/

x y z

x y z

x y z

u u u u x

v v v v y pt R x y z

w w w w z

1uu u p u

t R

( , , )u u v w

,

( , , )

v wz y

y x t

2

1

1

u u uu

t z y y z R

t z y y z R

0u v w

x y z

2

1

1

u u uu

t z y y z R

t z y y z R

,

( , , )

v wz y

y x t

2d-3c model

z

yx

flow

position

0x

2 dimensionsw

vu

flow

velocity

3 components

2

1

1

u u uu

t z y y z R

t z y y z R

,

( , , )

v wz y

y x t

2d-3c model

0x

These equations are globally stable!Laminar flow is global attractor.

t

energy

2RR

Total energy3R

(Bamieh and Dahleh)

0 200 400 600 800 100010

-10

10-5

100

105

t

ener

gyenergyN=10R=1000t=1000alpha=2

Total energy

vortices

What you’ll see next.

( , , )z y t

( , , )u z y t

( , , )z y t

( , , )u z y t

Log-log plot of time response.

Random initial conditions on

( , , 0)z y t concentrated at lower boundary.

( , , )z y t

( , , )u z y t

( , , )z y t

( , , )u z y t

Exponential decay.

Long range correlation.

Streamwise streaks.


fragile?


HOT

flow

pressure drop

random pipes

streamlined pipes

Complexity, chaos and criticality

• The orthodox view:– Power laws suggest criticality

– Turbulence is chaos

• HOT view:– Robust design often leads to power laws

– Just one symptom of “robust, yet fragile”

– Shear flow turbulence is noise amplification

• Other orthodoxies:– Dissipation, time irreversibility, ergodicity and mixing

– Quantum to classical transitions

– Quantum measurement and decoherence

Epilogue

• HOT may make little difference for explaining much of traditional physics lab experiments,

• So if you’re happy with orthodox treatments of power laws, turbulence, dissipation, quantum measurement, etc then you can ignore HOT.

• Otherwise, the differences between the orthodox and HOT views are large and profound, particularly for…

• Forward or reverse (eg biology) engineering complex, highly designed or evolved systems,

• But perhaps also, surprisingly, for some foundational problems in physics

John Doyle Control and Dynamical Systems Caltech.

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Transcript of John Doyle Control and Dynamical Systems Caltech.