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Control and Communication: an overview Jean-Charles Delvenne CMI, Caltech May 5, 2006

Treasure map Poincaréville Shannon City

Motivation=Remote Control Telesurgery Robot on Mars Alice project USAFranceDigital channel

What is a dynamical system ? (Piece of Nature evolving in time) State x Input u Output y 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 x 0 y u

If everything is smooth… Linearization around a fixed point Easier to analyze than nonlinear.

If information is limited... Bottleneck for circulation of information Channel u y Encoder Decoder

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 Separation principle Does not apply in general State x Channel uy Encoder Estimator x est Controller

Nonlinear systems Practical stability on L k =Number of possible k-sequences u a …u b u1u1 u2u2 u3u3 u4u4 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 2 R symbols over noiseless channel Trade-offs Bound: Achieved by zooming strategy (Tatikonda)

Memoryless strategies Fixed partition of [-1,1] 2 R =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) 01 r -rr2r2 r3r3 r4r4 -r 4 -r 3 -r 2 

Uniform quantizer High rate, low time Optimal Nested 0 1 

Chaotic strategy   1 0 Low rate, high time Almost all points stabilized Nested Fagnani-Zampieri (2001)

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!