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OCEAN

Control and Communication:

an overview

Jean-Charles DelvenneCMI, Caltech May 5, 2006

Treasure map

Poincaréville

Shannon City

Motivation=Remote Control

Telesurgery Robot on Mars Alice project

USAFrance Digital channel

What is a dynamical system ?(Piece of Nature evolving in time)

State xInput u Output y

noise) u,g(x,:y

noise) x,of past value f(u,:x

Input u = effect of the environment Output y = effect on the environment

What is control ?(Human vs Nature)

Simple controllers preferred:Memoryless: u:=k(y)

Controller

x0

yu

If everything is smooth…

Linearization around a fixed point

Easier to analyze than nonlinear.

MwCx:y

:x NvBuAx

If information is limited...

Bottleneck for circulation of information

Channel

u y

EncoderDecoder

A first observation

Delchamps (1991) Linear system, memoryless strategy

x:= x+u, ||>1 Solution: u=- x

… with noiseless digital channel Finitely many values for u Convergence to zero impossible We settle for ‘practical stability’:

neighborhood of zero

What kind of channel?

Noiseless digital

Noisy digital

Analog Gaussian

Packet drop (digital or analog)

What kind of system?

Discrete-time or continuous-time

Deterministic or stochastic

Linear versus non-linear

What kind of objective?

State converges to 0

State goes to/remains in a neighborhood of 0

Nth moment of the state bounded

Time needed to reach a neighborhood of 0

Summary

Many models, many results

Lower bounds : what we can’t hope

Strategies : what we can hope

Stability and performance

Stability

The fundamental lower bound

We want to (practically) stabilize

We need a channel rate

The lower bound is tight

If digital noiseless channel, then

is sufficient

Proof: cut and paste, volume preserved

Nair and Evans, Tatikonda, Liberzon, 2002

With noise

If additive noise:

Practical stability impossible in general Second moment stability

iff

Tools for lower bound

Entropy

Entropy power inequalityIf x,y independent then

Entropy-variance relation:

If u discrete with N values then

Idea for strategy

Separation principle Does not apply in general

State x

Channel

u y

EncoderEstimatorxest

Controller

Nonlinear systems

Practical stability on

Lk=Number of possible k-sequences ua…ub

u1

u2u3

u4

S

Topological feedback entropy

Set S is not stabilizable if

Set S is stabilizable with noiseless channel if

Topological feedback entropy

If S small neighborhood of differentiable fixed point, then

Similarity with topological entropy for dynamical systems

Nair, Evans, Mareels and Moran (2004)

What do we mean by rate?

Ok if noiseless If noisy: Shannon capacity Justified by Shannon channel coding theorem Relies on block coding Unsuitable for control: cannot afford delay More refined: Anytime Capacity (Sahai) Moment-stabilizable iff

AnytimeCapacity >

Performance

Time, rate, contraction

System From [-1,1] to [-] Average time T 2R symbols over noiseless channel Trade-offs Bound:

Achieved by zooming strategy (Tatikonda)

Memoryless strategies

Fixed partition of [-1,1]

2R=number of intervals

Intervals ~ Separation principle

Lower bounds

Fagnani, Zampieri (2001)

The logarithmic strategy

Medium rate, medium time Optimal Lyapunov quadratic function Elia-Mitter (2001)

0 1-1 r-r r2r3r4-r4-r3-r2

Uniform quantizer

High rate, low time Optimal Nested

0-1 1

Chaotic strategy

1-1 0

Low rate, high time Almost all points stabilized Nested Fagnani-Zampieri (2001)

Conclusions

Conclusions

Broad rather than deep Control, dynamical systems, information theory Stability vs performance Steady state vs transient What if quantization subsets are not intervals?

No separation principle Simpler theory Next week!