John Doyle

1
A new physics?

• John Doyle
• Control and Dynamical Systems, Electrical
  Engineering, Bioengineering
• Caltech

2
For more details
www.cds.caltech.edu/doyle www.aut.ee.ethz.ch/par
rilo
Funding AFOSR MURI Uncertainty Management in
Complex Systems

• ACC workshop
• IFAC workshop
• Both will include physics, biology, and networking

3
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.

4
Biology and advanced technology

• Biology
• Integrates control, communications, computing
• Into distributed control systems
• Built at the molecular level
• Advanced technologies will do the same
• We need new theory and math, plus unprecedented
  connection between systems and devices
• Two challenges for greater integration
• Unified theory of systems (Horizontal)
• Multiscale from devices to systems (Vertical)

5
Surprise!

• Robust control theory provides a surprisingly
  good starting point for a unified systems
  theory (MD)
• Robust control can move from a hidden (and
  peripheral) element to a central and fundamental
  role in science and technology.

6
Bonus!

• Unified systems theory helps resolve
  fundamental unresolved problems at the
  foundations of physics
• Ubiquity of power laws (statistical mechanics)
• Shear flow turbulence (fluid dynamics)
• Macro dissipation and thermodynamics from micro
  reversible dynamics (statistical mechanics)
• Quantum-classical transition
• Quantum measurement
• Thus the new mathematics for a unified theory of
  systems is directly relevant to multiscale
  physics.
• The two challenges are connected.

7
Collaboratorsand contributors(partial list)

• Turbulence Bamieh, Dahleh, Bobba,
• Theory Carlson, Parrilo (ETHZ), .
• Quantum Physics Mabuchi, Doherty,

Caltech faculty
Other Caltech
UCSB faculty
8
Collaboratorsand contributors(partial list)

• Turbulence Bamieh, Dahleh, Bobba, Gharib,
  Marsden,
• Theory Parrilo, Carlson, Paganini, Lall,
  Barahona, DAndrea,
• Physics Mabuchi, Doherty, Marsden,
  Asimakapoulos,
• AfCS Simon, Sternberg, Arkin,
• Biology Csete,Yi, Borisuk, Bolouri, Kitano,
  Kurata, Khammash, El-Samad, Gross, Sauro, Hucka,
  Finney,
• Web/Internet Low, Effros, Zhu,Yu, Chandy,
  Willinger,
• Engineering CAD Ortiz, Murray, Schroder,
  Burdick, Barr,
• Disturbance ecology Moritz, Carlson, Robert,
• Power systems Verghese, Lesieutre,
• Finance Primbs, Yamada, Giannelli, Martinez,
• and casts of thousands

Caltech faculty
Other Caltech
Other
9
Outline

• Illustrate a new, unifying conceptual framework
  plus math tools for complex multiscale physics.
• Combination of robust control and physics
  pioneered by Mohammed Dahleh.
• Concentrate on 2 of many open questions in the
  foundations of theoretical physics
• Coherent structures in shear flow turbulence
  (Bamieh)
• Quantum entanglement (Parrilo)
• Compare the beginning of a new physics circa
  2000 with the origins of robust control circa
  1980.
• Prospects for the future of controls and physics.

10
Topics skipped today

• Other related problems Power laws, origin of
  dissipation and entropy, quantum measurement and
  quantum/classical transition (Carlson, Mabuchi,
  )
• Much work involving control of turbulence
  quantum systems, etc (e.g. Speyer, Kim,
  Cortellezi, Bewley, Burns, King, Krstic, )
• Related multiscale problems in networking
  protocols (Low, Paganini,) and biological
  regulatory networks (Khammash, El Samad, Yi,)

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Caveats

• Not an historical account
• Not a scholarly treatment
• Just the tip of the iceberg
• Lots of details are available in papers and
  online
• See website (URL on the last slide)
• All of this has appeared or will appear outside
  the controls literature
• Emphasize MDs vision

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Turbulence in shear flows
Kumar Bobba, Bassam Bamieh
wings
channels
Thanks to Mory Gharib Jerry Marsden Brian Farrell
pipes
13
Turbulence in shear flows
Kumar Bobba, Bassam Bamieh
wings
channels
Thanks to Mory Gharib Jerry Marsden Brian Farrell
pipes
14
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 perhaps chaotic, certainly
  intrinsically a nonlinear phenomena
• There are certainly many situations where this
  view is useful.
• (But many people believe there is much more to
  the story. See Farrell, et al, etc.)

15
streamwise
Couette flow
16
spanwise
Couette flow
17
high-speed region
From Kline
18
high-speed region
y
flow
position
z
x
19
3d/3c Nonlinear NS
20
3d/3c Linear NS
3d/3c Nonlinear NS
Linearize
3d/3c Linear NS
y
v
flow
flow
position
velocity
z
u
w
x
3 components
3 dimensions
21
3d/3c Linear NS
y
v
flow
flow
position
velocity
z
u
w
x
3 components
3 dimensions
22
streamwise
The mystery.
Thm The first instabilities are spanwise
constant.
All observed flows are largely streamwise
constant.
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Theory
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Theory
This is as different as two flows can be.
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3d/3c Linear NS

• Linearized Navier-Stokes
• Stable for all Reynolds numbers R
• Orthodox wisdom transition must be an inherently
  nonlinear phenomena
• Experimentally no evidence for an attractor or
  subcritical bifurcations
• Theoretically no evidence for
• The mystery deepens.

26
3d/3c Linear NS
Forcing

• Mathematically
• External disturbances
• Initial conditions
• Unmodeled dynamics

• Physically
• Wall roughness
• Acoustics
• Thermo fluctuations
• NonNewtonian
• Upstream disturbances

27
3d/3c Linear NS
Forcing
energy
(Bamieh and Dahleh)
t
28
t
29
The predicted flows are robustly and strongly
streamwise constant.
y
flow
Consistent with experimental evidence.
z
x
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3d/3c Nonlinear NS
3d/3c Linear NS
Linearize
Stable for all R.
y
flow
z
x
2d/3c Linear NS
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3d/3c Nonlinear NS
3d/3c Linear NS
Linearize
Stable for all R.
y
flow
z
x
2d/3c LNS
2d/3c NLNS
Linearize
32

• 2d/3c NLNS solutions to 3d/3c NLNS for streamwise
  constant initial conditions
• 2d/3c NLNS has 3 velocity components depending on
  2 (spanwise) spatial variables

3d/3c NLNS
2d/3c NLNS
y
flow
z
x
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3d/3c NLNS
Thm 2d/3c NLNS
Globally stable for all R.
2d/3c NLNS
Proof can rescale equations to be independent of
R!
34

• High gain, low rank operator
• Implications for
• Model reduction
• Computation
• Control

3d/3c Linear NS
Globally stable for all R.
2d/3c NLNS
Linearize
2d/3c LNS
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The predicted flows are robustly and strongly
streamwise constant.
y
flow
z
x
Consistent with experimental evidence.
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Next Theory and experiment to complete 3d/3c
picture.
?
3d/3c
37
2d/3c
38
?
2d/3c
3d/3c
39
Worst-case amplification is streamwise constant
2d/3c (Bamieh and Dahleh)
?
2d/3c
3d/3c
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Robustness of shear flows
Fragile
Viscosity
Everything else
Robust
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Fragility is a conserved quantity?
Fragile
Random
Viscosity
Everything else
Robust
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Lessons learnedTransition and turbulence

• Be skeptical of orthodox explanations of
  persistent mysteries.
• Listen to the experimentalists.
• Singular values are as important as eigenvalues.
• Interconnection is as important as state.
• Fragility is a conserved quantity?

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Lessons learnedRobust control

• Be skeptical of orthodox explanations of
  persistent mysteries.
• Listen to the experimentalists.
• Singular values are as important as eigenvalues.
• Interconnection is as important as state.
• Fragility is a conserved quantity.

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Lessons learnedRobust control
logS
yet fragile
?
Robust
45
streamlined pipes
flow
HOT turbulence? Robust, yet fragile?
HOT
random pipes

• Through streamlined design
• High throughput
• Robust to bifurcation transition (Reynolds
  number)
• Yet fragile to small perturbations
• Which are irrelevant for more generic flows
• Turbulence is a robustness problem.
• Shear turbulence is a highly linear phenomena.
  (BB)

pressure drop
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Universal
HOT
log(thru-put)
log(demand)
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Highly Optimized Tolerance (HOT)(Jean Carlson,
Physics, UCSB)

• Complex systems in biology, ecology, technology,
  sociology, economics,
• are driven by design or evolution to
  high-performance states which are also tolerant
  to uncertainty in the environment and components.
• This leads to specialized, modular, hierarchical
  structures, often with enormous hidden
  complexity,
• with new sensitivities to unknown or neglected
  perturbations and design flaws.
• Robust, yet fragile!

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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 (the essence of HOT).

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Persistent mystery 2

• The ubiquity of power laws in natural and human
  systems
• Orthodox theories
• Phase transitions and critical phenomena
• Self-organized criticality (SOC)
• Edge of chaos (EOC)
• Single largest topic in physics literature for
  the last decade
• New alternative HOT
• Already a sizeable HOT literature, so this will
  be very brief and schematic

50
Summary

• Power laws are ubiquitous, but not surprising
• 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 system as HOT is just the beginning.

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The real work is

• New Internet protocol design
• Forest fire suppression, ecosystem management
• Analysis of biological regulatory networks
• Convergent networking protocols
• etc

52
Community responseTransition and turbulence 2002

• Enthusiasm from experimentalists and
  mathematicians (and a few visionaries)
• From skepticism to outright hostility from
  mainstream academic theorists
• Combination of solid mathematics and new
  applications will succeed in the end

53
Robust control circa 1980
Community responseTransition and turbulence 2002

• Enthusiasm from experimentalists and
  mathematicians (and a few visionaries)
• From skepticism to outright hostility from
  mainstream academic theorists
• Combination of solid mathematics and new
  applications will succeed in the end

• US academic theoretical community was never
  persuaded, they were simply replaced by younger
  academics with stronger math and more interest in
  applications (e.g. MD)

54
More persistent mysteries

• Lots of mysteries at the foundations of
  statistical and quantum mechanics
• Macro dissipation and entropy versus micro
  reversibility
• Quantum measurement
• Quantum/classical transition and decoherence
• Progress on various aspects, but story incomplete
• Focus hot topic in QM testing entanglement
• Bonus! Not a controversial result!

55
Entangled Quantum States(Doherty, Parrilo,
Spedalieri 2001)

• Entangled states are one of the most important
  distinguishing features of quantum physics.
• Bell inequalities hidden variable theories must
  be non-local.
• Teleportation entanglement classical
  communication.
• Quantum computing some computational problems
  may have lower complexity if entangled states are
  available.

How to determine whether or not a given state is
entangled ?
56

• QM state described by psd Hermitian matrices ?
• States of multipartite systems are described by
  operators on the tensor product of vector spaces
• Product states
• each system is in a definite state
• Separable states
• a convex combination of product states.
• Entangled states those that cannot be written as
  a convex combination of product states.

57
Decision problem find a decomposition of r as a
convex combination of product states or prove
that no such decomposition exists.
(Hahn-Banach Theorem)
Z is an entanglement witness,a generalization
of Bells inequalities
Hard!
58
First Relaxation
Restrict attention to a special type of Z
The bihermitian form Z is a sum of squared
magnitudes.
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First Relaxation

• Equivalent to known condition
• Peres-Horodecki Criterion, 1996
• Known as PPT (Positive Partial Transpose)
• Exact in low dimensions
• Counterexamples in higher dimensions

If minimum is less than zero, r is entangled
60
Further relaxations
Broaden the class of allowed Z to those for which
is a sum of squared magnitudes.
Also a semidefinite program.
Strictly stronger than PPT.
Can directly generate a whole hierarchy of tests.
61
Second Relaxation
minimize
subject to
If the minimum is less than zero then r is
entangled. Detects all the non-PPT entangled
states tried
62
Quantum entanglement and Robust control
63
Quantum entanglement and Robust control
64
Higher order relaxations

• Nested family of SDPs
• Necessary Guaranteed to converge to true answer
• No uniform bound (or PNP)
• Tighter tests for entanglement
• Improved upper bounds in robust control
• Special cases of general approach
• All of this is the work of Pablo Parrilo (PhD,
  Caltech, 2000, now Professor at ETHZ)
• My contribution I kept out of his way.

65
A sample of applications

• Nonlinear dynamical systems
• Lyapunov function computation
• Bendixson-Dulac criterion
• Robust bifurcation analysis
• Nonlinear robustness analysis
• Continuous and combinatorial optimization
• Polynomial global optimization
• Graph problems e.g. Max cut
• Problems with mixed continuous/discrete vars.

In general, any semialgebraic problem.
66
Sums of squares (SOS)
A sufficient condition for nonnegativity

• Convex condition (Shor, 1987)
• Efficiently checked using SDP (Parrilo). Write

where z is a vector of monomials. Expanding and
equating sides, obtain linear constraints among
the Qij. Finding a PSD Q subject to these
conditions is exactly a semidefinite program
(LMI).
67
Nested families of SOS (Parrilo)
Exhausts co-NP!!
68
Stronger µ upper bounds

• Structured singular value µ is NP-hard
• Standard µ upper bound can be interpreted
• As a computational scheme.
• As an intrinsic robustness analysis question
  (time-varying uncertainty).
• As the first step in a hierarchy of convex
  relaxations.
• For the four-block Morton Doyle counterexample
• Standard upper bound 1
• Second relaxation 0.895
• Exact µ value 0.8723

69
Continuous Global Optimization

• Polynomial functions NP-hard problem.
• A simple relaxation (Shor) find the maximum
  ?such that f(x) ? is a sum of squares.
• Lower bound on the global optimum.
• Solvable using SDP, in polynomial time.
• A concise proof of nonnegativity.
• Surprisingly effective (Parrilo Sturmfels 2001).

70

• Much faster than exact algebraic methods (QE,GB,
  etc.).
• Provides a certified lower bound.
• If exact, can recover an optimal feasible point.
• Surprisingly effective
• In more than 10,000 random problems, always the
  correct solution
• Bad examples do exist (otherwise NPco-NP), but
  rare.
• Variations of the Motzkin polynomial.
• Reductions of hard problems.
• None could be found using random search

71
Finding Lyapunov functions

• Ubiquitous, fundamental problem
• Algorithmic LMI solution

Convex, but still NP hard.
Test using SOS and SDP.
After optimization coefficients of V.
A Lyapunov function V, that proves stability.
72
Example
Given
Propose
After optimization coefficients of V.
A Lyapunov function V, that proves stability.
73
Conclusion a certificate of global stability
74
More general framework

• A model co-NP problem
• Check emptiness of semialgebraic sets.
• Obtain LMI sufficient conditions.
• Can be made arbitrarily tight, with more
  computation.
• Polynomial time checkable certificates.

75
Semialgebraic Sets

• Semialgebraic finite number of polynomial
  equalities and inequalities.
• Continuous, discrete, or mixture of variables.
• Is a given semialgebraic set empty?
• Feasibility of polynomial equations NP-hard
• Search for bounded-complexity emptiness proofs,
  using SDP. (Parrilo 2000)

76
Positivstellensatz (Real Nullstellensatz)
if and only if

• Stengle, 1974
• Generalizes Hilberts Nullstellensatz and LP
  duality
• Infeasibility certificates of polynomial
  equations over the real field.
• Parrilo Bounded degree solutions computed via
  SDP!
• ? Nested family of polytime relaxations for
  quadratics, the first level is the S-procedure

77
Combinatorial optimization MAX CUT
Partition the nodes in two subsets
To maximize the number of edges between the two
subsets.
Hard combinatorial problem (NP-complete).
Compute upper bounds using convex relaxations.
78
Standard semidefinite relaxation
Dual problems
This is just a first step. We can do better! The
new tools provide higher order relaxations.

• Tighter bounds are obtained.
• Never worse than the standard relaxation.
• In some cases (n-cycle, Petersen graph),
  provably better.
• Still polynomial time.

79
MAX CUT on the Petersen graph
The standard SDP upper bound 12.5 Second
relaxation bound 12. The improved bound is
exact. A corresponding coloring.
80
Summary

• Single framework with substantial advances in
• Testing entanglement
• MaxCut
• Global continuous optimization
• Finding Lyapunov functions for nonlinear systems
• Improved robustness analysis upper bounds
• Many other applications
• This is just the tip of a big iceberg

81
Nested relaxations and SDP
82

• Huge breakthroughs
• but also a natural culmination of more than 2
  decades of research in robust control.
• Initial applications focus has been CS and
  physics,
• but substantial promise for persistent
  mysteries in controls and dynamical systems
• Completely changes the possibilities for
• robust hybrid/nonlinear control
• interactions with CS and physics

83

• Unique opportunities for controls community
• Resolve old difficulties within controls
• Unify and integrate fragmented disciplines within
• Unify and integrate without comms and CS
• Impact on physics and biology
• Unique capabilities of controls community
• New tools, but built on robust control machinery
• Unique talent and training

84
Problem In general, computation grows
exponentially with m and n.
Key idea systematic search for short proofs.
85
Chemical oscillator (Prajna, Papachristodoulou)
Nondimensional state equations
86
1
0.8
0.6
a
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
b
87
3
2.5
a 0.1, b 0.13
2
1.5
1
0
0.1
0.2
0.3
0.4
0.5
0.6
88
(No Transcript)
89
equilibrium
90
(No Transcript)
91
(No Transcript)
92
a 0.6, b 1.1
1
0.8
y
0.6
0.4
0.2
0
x
1
1.5
2
2.5
93
a 1, b 2
1
0.8
0.6
0.4
0.2
0
2.2
2.6
3
3.4
94
Features of new approach (Parrilo)

• SOS/SDP Based on Sum-of-square (SOS) and
  semidefinite programming (SDP)
• Exist gold standard relaxation algorithms for
  canonical coNP hard problems, such as
• MaxCut
• Quantum entanglement
• Robustness (?) upper bound
• All special cases of first step of SOS/SDP
• Further steps (all in P) converge to answer
• No uniform bound (or PNP)

95

• Standard tools of robust (linear) control
• Unmodeled dynamics, nonlinearities, and IQCs
• Noise and disturbances
• Real parameter variations
• D-K iteration for ?-synthesis
• Are all treated much better
• And generalized to
• Nonlinear
• Hybrid
• DAEs
• Constrained

96
Caveats

• Inherits difficulties from robust control
• High state dimension and large LMIs
• Must find ways to exploit structure, symmetries,
  sparseness
• Note many researchers dont want to get rid of
  the ad hoc, handcrafted core of their approaches
  to control (why take the fun out of it?)

97
Controls will be the physics of the 21st
Century. (Larry Ho)

• Two interpretations (MD)
• Metaphorical
• Literal

98
Metaphorical

• Physics has been the foundation of science and
  technology
• New science and technology
• Ubiquitous, embedded networking
• Integrated controls, comms, computing
• Postgenomics biology
• Global ecosystems management
• Etc. etc
• Controls will be the new foundations of science
  and technology

99
Literal

• Physics has many persistent, unresolved problems
  at its foundations
• New mathematics built on robust control will
  resolve these problems
• Redefine not only the foundations of physics, but
  also fundamentally rethink all of science and
  technology
• Towards a truly rigorous foundation for science

100
Lessons learnedRobust control Physics (MD)

• Dont assume the experts are right (including
  me).
• Listen to the experimentalists.

101
For more details

• Almost nothing so far in controls literatures
• Lots on HOT vs. SOC in physics literature
  (Carlson)
• A few papers on HOT turbulence (Bamieh)
• Parrilo Thesis and papers available online
• ACC workshop
• IFAC workshop
• Both will include physics, biology, and networking

www.cds.caltech.edu/doyle www.aut.ee.ethz.ch/par
rilo