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A new physics?

- John Doyle
- Control and Dynamical Systems, Electrical

Engineering, Bioengineering - Caltech

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

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.

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)

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.

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.

Collaboratorsand contributors(partial list)

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

Caltech faculty

Other Caltech

UCSB faculty

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

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.

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,)

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

Turbulence in shear flows

Kumar Bobba, Bassam Bamieh

wings

channels

Thanks to Mory Gharib Jerry Marsden Brian Farrell

pipes

Turbulence in shear flows

Kumar Bobba, Bassam Bamieh

wings

channels

Thanks to Mory Gharib Jerry Marsden Brian Farrell

pipes

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.)

streamwise

Couette flow

spanwise

Couette flow

high-speed region

From Kline

high-speed region

y

flow

position

z

x

3d/3c Nonlinear NS

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

3d/3c Linear NS

y

v

flow

flow

position

velocity

z

u

w

x

3 components

3 dimensions

streamwise

The mystery.

Thm The first instabilities are spanwise

constant.

All observed flows are largely streamwise

constant.

Theory

Theory

This is as different as two flows can be.

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.

3d/3c Linear NS

Forcing

- Mathematically
- External disturbances
- Initial conditions
- Unmodeled dynamics

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

3d/3c Linear NS

Forcing

energy

(Bamieh and Dahleh)

t

t

The predicted flows are robustly and strongly

streamwise constant.

y

flow

Consistent with experimental evidence.

z

x

3d/3c Nonlinear NS

3d/3c Linear NS

Linearize

Stable for all R.

y

flow

z

x

2d/3c Linear NS

3d/3c Nonlinear NS

3d/3c Linear NS

Linearize

Stable for all R.

y

flow

z

x

2d/3c LNS

2d/3c NLNS

Linearize

- 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

3d/3c NLNS

Thm 2d/3c NLNS

Globally stable for all R.

2d/3c NLNS

Proof can rescale equations to be independent of

R!

- 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

The predicted flows are robustly and strongly

streamwise constant.

y

flow

z

x

Consistent with experimental evidence.

Next Theory and experiment to complete 3d/3c

picture.

?

3d/3c

2d/3c

?

2d/3c

3d/3c

Worst-case amplification is streamwise constant

2d/3c (Bamieh and Dahleh)

?

2d/3c

3d/3c

Robustness of shear flows

Fragile

Viscosity

Everything else

Robust

Fragility is a conserved quantity?

Fragile

Random

Viscosity

Everything else

Robust

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?

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.

Lessons learnedRobust control

logS

yet fragile

?

Robust

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

Universal

HOT

log(thru-put)

log(demand)

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!

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).

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

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.

The real work is

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

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

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)

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!

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 ?

- 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.

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!

First Relaxation

Restrict attention to a special type of Z

The bihermitian form Z is a sum of squared

magnitudes.

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

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.

Second Relaxation

minimize

subject to

If the minimum is less than zero then r is

entangled. Detects all the non-PPT entangled

states tried

Quantum entanglement and Robust control

Quantum entanglement and Robust control

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.

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.

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).

Nested families of SOS (Parrilo)

Exhausts co-NP!!

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

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).

- 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

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.

Example

Given

Propose

After optimization coefficients of V.

A Lyapunov function V, that proves stability.

Conclusion a certificate of global stability

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.

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)

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

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.

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.

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.

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

Nested relaxations and SDP

- 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

- 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

Problem In general, computation grows

exponentially with m and n.

Key idea systematic search for short proofs.

Chemical oscillator (Prajna, Papachristodoulou)

Nondimensional state equations

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

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

(No Transcript)

equilibrium

(No Transcript)

(No Transcript)

a 0.6, b 1.1

1

0.8

y

0.6

0.4

0.2

0

x

1

1.5

2

2.5

a 1, b 2

1

0.8

0.6

0.4

0.2

0

2.2

2.6

3

3.4

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)

- 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

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?)

Controls will be the physics of the 21st

Century. (Larry Ho)

- Two interpretations (MD)
- Metaphorical
- Literal

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

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

Lessons learnedRobust control Physics (MD)

- Dont assume the experts are right (including

me). - Listen to the experimentalists.

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