Robustness and Complexity

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Robustness and Complexity
John Doyle Control and Dynamical Systems Caltech
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Mathematical Infrastructure for Robust Virtual
Engineering
Mathematical Foundations for Robust Virtual
Engineering
Mathematical Foundations for Uncertainty
Management of Complex Systems
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Motivation from engineering

• We are building increasingly complex systems,
  with
• increasing emphasis on modeling, simulation, and
  virtual prototyping.
• We are on the verge of a huge revolution

Titan IV
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Motivation from engineering

• ...but revolutions are often bloody, messy,
  miserable affairs with uncertain outcomes.
• Current complex systems technology is on the
  ragged edge.
• Inadequate math/science foundation for
  simulation-based design of complex systems.

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Motivation from engineering

• But the good news is that were finally starting
  to work on it, and making progress.
• It is becoming clear that the key to
  understanding complexity is robustness.

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Motivation from biology

• If molecular biology was a revolution at the
  component level (fueled by physics and
  chemistry)...
• we may be on the verge of an even bigger
  revolution at the systems level
• with all that implies
• Can we help?

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Motivation from biology

• Physics and chemistry wont help much in dealing
  with complexity...
• ...and may even impede progress if they dont
  realize this.
• Theoretical biology and complex systems
  theory have until recently been oxymorons.
• Math and systems engineering can help, but it
  will be hard to get their attention.

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Motivation from biology

• The math/science foundation needed for complex
  systems engineering appears to be exactly what is
  needed for biology as well.
• Thus the futures of biology and engineering are
  converging in many ways, but...
• in particular, they are motivating similar math.
• Biology offers a rich experimental testbed.

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Cartoon evolution of the program
Uncertainty management
CAD
CFD
Physics
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Cartoon evolution of the program
Uncertainty management
CAD
CFD
Physics
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Cartoon evolution of the program
Uncertainty management
Uncertainty management
CAD
CFD
Physics
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Cartoon evolution of the program
Uncertainty management
Uncertainty management Structured multiscale
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Outline of this morning

• Are there core, fundamental concepts about
  complex systems in engineering and biology?
• Claim Robustness is the key issue
• Cartoon review of motivating anecdotes and basic
  concepts
• An introduction to new theoretical results

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Outline of the rest

• Because our research has resulted in fairly
  radical new directions...
• ...we first have to develop a shared language and
  set of core concepts...
• And then we can discuss the organization,
  connections, and highlights of the rest of the
  MURI program.
• The latter will begin this afternoon.

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Uncertainty and Robustness
Complexity
Dynamics Interconnection/ Feedback Hierarchical/ M
ultiscale Heterogeneous Nonlinearity
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Uncertainty and Robustness
Complexity
Dynamics Interconnection/ Feedback Hierarchical/ M
ultiscale Heterogeneous Nonlinearity
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Robustness
Complexity
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Complexity

• Organisms 1000 to 100,000 genes
• Boeing 777 gt100,000 subsystems and ...
• many subsystems are highly complex.
• Engine gt 10,000 subsystems
• Laptop gt1,000,000,000 transistors
• Refinery gt 10,000 integral feedback loops
• High rise heating-ventilation-AC gt 1000
• Internet, power grid, etc. ?

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Complexity

• The gas in this room has 1e30 molecules
• ... but few distinctly different components...
• O2, N2, CO22, O3,

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Convergent networking
Data Voice Video
Integrated services
ubiquitious networking/ computing
Energy Transportation Utilities Food Waste Finance

Single Inter-internet
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Robustness
the environment is uncertain
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Robustness and uncertainty
Sensitive
Error, sensitivity
Robust
Types of uncertainty
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Robustness and uncertainty
Sensitive
Error, sensitivity
Robust
Meteor impact
speech/ vision
Types of uncertainty
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Robustness and uncertainty
yet fragile
Sensitive
Error, sensitivity
Robust
Meteor impact
speech/ vision
Types of uncertainty
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Complex systems
yet fragile
Sensitive
Error, sensitivity
Robust
Robust
Types of uncertainty
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Automobile air bags
Error, sensitivity
Types of uncertainty
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Heating system
Sensitive
Error, sensitivity
Robust
Thermostat
Environment
Heater
Types of uncertainty
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Heating system
disturbances
Room temperature
heat
desired temperature
thermostat
heater
temperature error
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Uncertainty and robustness in heating system
Environment
Heater
Thermostat
Uncertainty
Sensitivity
The critical sensitivities occur at the lowest
signal levels.
Robustness
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Uncertainty and robustness in chemotaxis
Rate constants
Environment
Concentrations
Uncertainty
Sensitivity
Robustness
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Uncertainty/robustness in complex systems
Uncertainty
Sensitivity
Robustness
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Heating system
What if we turn off the system?
Sensitive
Error, sensitivity
Robust
Thermostat
Environment
Heater
Types of uncertainty
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Conservation of robustness
Error, sensitivity
is balanced by
Types of uncertainty
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Robust, yet fragile

• Robust to uncertainties
• that are common,
• the system was designed for, or
• has evolved to handle,
• yet fragile otherwise
• This is the most important feature of complex
  systems (we call it HOT).

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Biological systems

• Extreme robustness, from DNA repair mechanisms to
  survival at the cell, organism, and population
  levels.
• Grow, persist, and reproduce despite large
  uncertainties in environments and components.
• Yet perturbation of a specie or gene can be
  fatal.
• Only 1 in 1000 species have avoided extinction.

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Boeing 777

• Robust to large scale atmospheric disturbances,
  variations in cargo loads and fuels, turbulent
  boundary layers, inhomogeneities and aging of
  materials, etc
• ...but could be catastrophically disabled by
  microscopic alterations in a handful of
  components (eg. 4 carefully chosen transistors).
• This is, fortunately, very unlikely.

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Laptop

• Robust to variations in temperature, power
  supply, user behavior, such as presentations,
  email, etc
• Catastrophically disabled by microscopic
  alterations in hardware or small software bugs.
• Hardware failures are fortunately rare.
• Software bugs are unfortunately not.

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Typical corporate costs per year per PC

• Hardware 1K
• Software 2K
• Network mgmt. 1K
• Tech support 4K
• Futzing (own PC) 6K
• Futzing (others PC) 10K
• Total 24K

Hardware 1K Robustness
Futzing (own PC) 6K Lack of
Futzing (others PC) 10K
robustness
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Typical corporate costs per year per PC

• Hardware 1K
• Software 2K
• Network mgmt. 1K
• Tech support 4K
• Futzing (own PC) 6K
• Futzing (others PC) 10K
• Total 24K

Hardware 1K Robustness
Futzing (own PC) 6K Lack of
Futzing (others PC) 10K
robustness
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2020 Cost/year/person for systems.

• Hardware 100
• Software 200
• Network mgmt. 300
• Tech support 400
• Everything else 10,000
• Total 11,000

Can we guess what everything else might be?
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Obstacles

• Few examples, poorly understood outside narrow
  technical discipline.
• Mathematical theory is emerging, but inaccessible
  to nonexperts, and is fragmented (controls,
  dynamical systems, communications, computational
  complexity, statistical physics,...).

Can we guess what everything else might be?
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Robust, yetfragile

• Organisms thousands of genes
• Boeing 777 gt100,000 subsystems
• Engine gt 10,000 subsystems
• Laptop gt1,000,000,000 transistors
• Refinery gt 10,000 integral feedback loops
• High rise heating-ventilation-AC gt 1000
• Internet, power grid, etc. ?

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complexity?
drive
Does robustness
the environment is uncertain
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In an idealized lab environment with no
uncertainty?
Thought experiment.
How complex would systems have to be?
the environment is uncertain
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Complexity
Organisms ? 100,000 ? 400? Boeing 777?
100,000 ? 1,000 ? Engine? 10,000 ?
100? Laptop? 1e9 ? 1e5? Refinery?
10,000 ? 0 High rise HVAC? 1000 ?
0 Internet? ? ? 0
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Claim
Complexity is overwhelmingly due to robustness.
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Complexity

• The gas in this room has 1e30 molecules
• The thermodynamic properties (temperature,
  pressure) are uniformly robust
• completely unlike robust, yet fragile.

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Complexity management
Complexity management
Implicit to explicit
Materials
Energy
Entropy
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Implicit to explicit
For example,entropy was always there.
Materials
Energy
Entropy
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Implicit to explicit
For example,entropy was always there.
But until steam engines, it was not treated
explicitly.
Materials
Energy
Entropy
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Implicit to explicit
For example,entropy was always there.
Efficiency once drove complexity, and still
contributes.
But until steam engines, it was not treated
explicitly.
Materials
Energy
Entropy
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Efficiency
System
environment
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Is efficiency just a symptom of robustness?
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Coal
waste
electricity
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Coal
waste
electricity
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Coal
waste
electricity
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Electricity generation and consumption
http//phe.rockefeller.edu/Daedalus/Elektron/
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log(complexity)
But efficiency aggravates robustness tradeoffs.
time
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Complexity management
Complexity management
Implicit to explicit
Materials
Energy
Entropy
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Complexity management
Information/ Computation
Robustness/ Uncertainty
Implicit to explicit
Materials
Energy
Entropy
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Robustness is already our dominant technical
challenge.

• Would you rather have a faster computer/network
  or more robust software?
• You probably spend more on car insurance than on
  gasoline.
• The real costs of energy are side-effects in the
  form of pollution, climate change, etc.

Materials
Energy
Entropy
Information/ Computation
Robustness/ Uncertainty
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Complexity and science
Claim
Complexity and uncertainty management has always
been the aim of science.
Until recently, the strategy has always been
denial and avoidance.
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Scientific foundation?
Control, optimization, identification, feedback
Computational complexity
Dynamical systems
Information, communications
Statistical physics
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Complex adaptive systems?
new science of complexity
Artificial, emergent, adaptive, etc etc
Soft computing
Edge-of-chaos (EOC)
Self-organized criticality (SOC)
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Some typical titles

• Networks - Fractals reemerge in the new math of
  the Internet, Science
• Scaling and criticality in a stochastic
  multi-agent model of a financial market, Nature
• Scaling behavior in the dynamics of an economic
  index, Nature
• Self-similarity of extinction statistics in the
  fossil record, Nature

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SCIsearch
Physical Review Letters has published 22,000
papers since 1990
papers ________ in abstract 1651 criti
ca 151 self-organized critica
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• Life at the edge of chaos, PRL
• Self-organized criticality induced by diversity,
  PRL
• Adaptive competition, market efficiency, and
  phase transitions, PRL
• Species-area relation and self-similarity in a
  biogeographical model of speciation and
  extinction , PRL
• Environmental changes, coextinction, and
  patterns in the fossil record, PRL
• Pulse-coupled relaxation-oscillators - from
  biological synchronization to self-organized
  criticality, PRL
• Punctuated equilibrium and criticality in
  asimple-model of evolution, PRL

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Complexity
Life at the Edge of Chaos
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The Search for the Laws of Self-Organization and
Complexity
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the science of self-organized criticality
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Complex adaptive systems?
The criticisms
The attraction

• Too ambitious
• Pretentious
• Outrageous claims
• Useless
• Wrong

Ambitious Provocative Pioneering Post-reductionist
Integrative
Our goal
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Physics, math, and hacker cultures
Physics
Math
Hacker
Complexity
Need
Reality
Rigor
Low
Medium
High
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Complex adaptive systems
Physics
Hacker
Math
Complexity
Reality
Rigor
Low
Medium
High
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Our starting point
Physics
Math
Hacker
Complexity
Need
Reality
Rigor
Low
Medium
High
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Is there a role for physics?
Physics
Math
Hacker
Complexity
Need
Reality
Rigor
Low
Medium
High
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Complex adaptive systems?
new science of complexity
Artificial, emergent, adaptive, etc etc
Soft computing
Well return to EOC/SOC shortly
Edge-of-chaos (EOC)
Self-organized criticality (SOC)
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Theme
Uncertainty and robustness
are the unifying concepts.
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Computational complexity
Control, optimization, identification
Uncertainty and robustness
Information, communications
Dynamical systems
Statistical physics
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Computational complexity
Control, optimization, identification
undecidability
Uncertainty and robustness
Limitations
Information, communications
Dynamical systems
Statistical physics
entropy
chaos
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Computational complexity
Control, optimization, identification
undecidability
protocols
feedback
Uncertainty and robustness
Strategies
Information, communications
Dynamical systems
averaging
Statistical physics
entropy
chaos
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Morning focus
Control, optimization, identification, feedback
Uncertainty and robustness
Statistical physics
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Is robustness a conserved quantity?
Information/ Computation
Robustness/ Uncertainty
Materials
Energy
Entropy
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Uncertainty and Robustness
Complexity
Dynamics Interconnection/ Feedback Hierarchical/ M
ultiscale Heterogeneous Nonlinearity
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Prediction the most basic scientific question.
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x(k) uncertain sequence
-
e(k)
u(k-1)
u(k)
delay
predictor
u(k) prediction of x(k1) e(k) error
e(k) x(k) - u(k-1)
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Prediction is a special case of feedforward. For
known stable plant, these are the same
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For simplicity, assume x, u, and e are finite
sequences.
x(k)
u(k)
k
e(k)
k
Then the discrete Fourier transform X, U, and E
are polynomials in the transform variable z.
If we set z ei? , ? ? 0,?? then X(w) measures
the frequency content of x at frequency w.
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x
x(k)
-
e
u(k-1)
u(k)
u(k)
delay
C
e(k) x(k) - u(k-1)
e(k)
How do we measure performance of our predictor C
in terms of x, e, X, and E?
Typically want ratios of norms
or
to be small.
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Good performance (prediction) means
or
Equivalently,
or
For example,
Plancheral Theorem
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Interesting alternative
Or to make it closer to existing norms
Not a norm, but a very useful measure of signal
size, as well see. (The b in ?b is in honor
of Bode.)
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A useful measure of performance is in terms of
the sensitivity function S(z) defined by Bode as
If we set z ei? , ? ? 0,?? then S(w)
measures how well C does at each frequency. (If C
is linear then S is independent of x, but in
general S depends on x.) It is convenient to
study log S(w) and then
u ? 0 ( u(k)0 ? k) ? S ? 1, and logS ? 0.
log S(w) lt 0 ? C attenuates x at frequency
w.
log S(w) gt 0 ? C amplifies x at frequency w.
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Note as long as we assume that for any possible
sequence x(k) it is equally likely that -x(k)
will occur, then guessing ahead can never help.
Assume u is a causal function of x.
x(k)
u(k)
k
0
This will be used later.
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-
e(k)
x(k)
u(k-1)
u(k)
delay
C
e(k) x(k) - u(k-1)
For any C, an unconstrained worst-case x(k)
is x(k) -u(k-1), which gives e(k) x(k) -
u(k-1) - 2u(k-1) 2x(k)
Thus, if nothing is known about x(k), the
safest choice is u ? 0. Any other choice of u
does worse in any norm.
If x is white noise, then u ? 0 is also the best
choice for optimizing average behavior in almost
any norm.
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Summary so far

• Some assumptions must be valid about x in order
  that it be at all predictable.
• Intuitively, there appear to be fundamental
  limitations on how well x can be predicted.
• Can we give a precise mathematical description
  of these limitations that depends only on
  causality and require no further assumptions?

-
e(k)
x(k)
u(k-1)
u(k)
delay
C
e(k) x(k) - u(k-1)
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• Recall that S(z) E(z)/X(z) and S(?) 1.
• Denote by ek and xk the complex zeros for z
  gt 1 of E(z) and X(z), respectively. Then

Proof Follows directly from Jensens formula, a
standard result in complex analysis (advanced
undergraduate level).
If x is chosen so that X(z) has no zeros in z gt
1 (this is an open set), then
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logS gt 0 amplified logS lt 0 attenuated
?
?he amplification must at least balance the
attenuation.
logS
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yet fragile
?
Robust
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• Originally due to Bode (1945).
• Well known in control theory as a property of
  linear systems.
• But its a property of causality, not linearity.
• Many generalizations in the control literature,
  particularly in the last decade or so.
• Because it only depends on causality, it is in
  some sense the most fundamental known
  conservation principle.
• This conservation of robustness and related
  concepts are as important to complex systems as
  more familiar notions of matter, energy, entropy,
  and information.

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Recall
is equivalent to
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Uncertainty and Robustness
Complexity
Dynamics Interconnection/ Feedback Hierarchical/ M
ultiscale Heterogeneous Nonlinearity
105
What about feedback?
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Two major strategies for building robust systems
Redundancy (and averaging)
Redundancy (and averaging)
Redundancy (and averaging)
Redundancy (and averaging)
Feedback
Feedback
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Simple case of feedback.
e error d disturbance c control
e d c d F (e)
(1-F )e d
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F gt 0 ln(S) gt 0
ln(S)
amplification
F
F lt 0 ln(S) lt 0
attenuation
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F ? 1 ln(S) ? ?
ln(S)
extreme sensitivity
F
extreme robustness
F ? ?? ln(S) ? ??
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Uncertainty and Robustness
Complexity
Dynamics Interconnection/ Feedback Hierarchical/ M
ultiscale Heterogeneous Nonlinearity
Dynamics
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If these model physical processes, then d and e
are signals and F is an operator. We can still
define S(?? E(?? /D(?? where E and D are
the Fourier transforms of e 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
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logS gt 0 amplified logS lt 0 attenuated
?
?he amplification must at least balance the
attenuation.
logS
Positive and negative feedback are balanced.
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Negative feedback
?
lnS
logS
Positive feedback
F
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yet fragile
Negative feedback
?
lnS
logS
Positive feedback
Robust
F
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Feedback is very powerful, but there are
limitations.
It gives us remarkable robustness, as well as
recursion and looping.
Formula 1 The ultimate high technology sport
But can lead to instability, chaos, and
undecidability.
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• In development
• drive-by-wire
• steering/traction control
• collision avoidance

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• Electronic fuel injection
• Computers
• Sensors
• Telemetry/Communications
• Power steering

Formula 1 allows
sensors
actuators
driver
computers
telemetry
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uncertain sequence
error
-
delay
predictor
This is a natural departure point for
introduction of chaos and undecidability.