OCEAN
France
USA
Control and
Communication:
an overview
Jean-Charles Delvenne
CMI, Caltech
May 5, 2006
Treasure map
Shannon City
Poincaréville
Motivation=Remote Control



Telesurgery
Robot on Mars
Alice project
France
Digital channel
USA
What is a dynamical system ?
(Piece of Nature evolving in time)



Input u = effect of the environment
Output y = effect on the environment
x : f(u, past value of x, noise)

y : g(x, u, noise)
Output y
Input u
State x
What is control ?
(Human vs Nature)
x
0
u
y
Controller

Simple controllers preferred:
Memoryless:
u:=k(y)
If everything is smooth…

Linearization around a fixed point
x : Ax  Bu  Nv

y : Cx  Mw

Easier to analyze than nonlinear.
If information is limited...

Bottleneck for circulation of information
y
u
Encoder
Decoder
Channel
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 inequality
If x,y independent then

Entropy-variance relation:

If u discrete with N values then
Idea for strategy
u
State x
y
Controller
xest


Estimator
Channel
Separation principle
Does not apply in general
Encoder
Nonlinear systems

Practical stability on
S
u1
u4

u3
u2
Lk=Number of possible k-sequences ua…ub
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
-1
-r
-r2 -r3 -r4 0 r4 r3 r2
-





Medium rate, medium time
Optimal
Lyapunov quadratic function
Elia-Mitter (2001)
r
1
Uniform quantizer
-1



-
High rate, low time
Optimal
Nested
0

1
Chaotic strategy
-1




-
0
Low rate, high time
Almost all points stabilized
Nested
Fagnani-Zampieri (2001)

1
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!