How to do control theory and How to do control theory and discrete maths at the same discrete maths at the same

time?time?

Control and information: Control and information: linear, memoryless and linear, memoryless and

noiselessnoiseless

Jean-Charles DelvenneJean-Charles DelvenneCMI, CaltechCMI, CaltechCMI seminar CMI seminar May 12, 2006May 12, 2006

What is a dynamical system ?(Piece of Nature)

Maps

Discrete time Interaction with environment

State xInput u Output y

noise)u,g(x,:y

noise)x,f(u,:x

What is control ?

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

Controller

x0

yu

If information is limited...

Bottleneck for circulation of information

Channel

u y

EncoderDecoder

What do we mean by…

Channel Noiseless

digital

Noisy digital

Analog Gaussian

Packet drop

System Discrete-time or

continuous-time

Deterministic or stochastic

Linear or non-linear

Feedback Unlimited

memory

Limited memory

No memory (and time- invariant)

What do we mean by…

Performances time

rate

distance to 0

final value of Nth moment

sum of squares

Stability x 0

x neighb. of 0

Nth moment of x bounded

Main framework: no noise

Deterministic systems Volumes multiplied by at least We know: x in set S0

We want: x in subset SF within average timeT We can: measure x, transmit a symbol, choose u Noiseless feedback, memory allowed Contraction C= (S0)/(SF) If rate R, then 2R symbols allowed T=average time to reach SF provably

Summary

Lower bound for deterministic systems with noiseless feedback

Tight for unstable scalar linear systems, memoryless feedback

Tight for stable scalar linear systems, memoryless feedback

Tight for hyperbolic real-eigenvalue linear systems

Lower bounds: Fundamental limitations

Proof Time: we reach subset SF and we know it Observe sequences of inputs Shannon’s noiseless theorem

See also: Fagnani-Zampieri, Touchette-Lloyd Information acquired ≥ Information forced Maxwell’s demon

Scalar linear system

Scalar system We know: x in [-1,1] We want: x in [-,] within average time T We can: measure x, transmit a symbol, choose u Input u depends on transmitted symbol only Symbols induce partition of [-1,1] If rate R, then 2R symbols are allowed.

Formulation: Fagnani and Zampieri But: we do not require intervals

What is possible?

We want to contract [-1,1] into [-, ] in time T with 2R values of u

We want , T, R small (good, fast, economical)

We want a memoryless feedback

What is the trade-off ?

Main result

We can contract [-1,1] into [-, ] in time T with 2R symbols iff

‘Only if’: See above ‘If’:

Find a optimal strategies… …that span the surface. Strategy with 2 degrees of freedom

An idea:The Separation Principle and Certainty Equivalent strategies

Start from perfect-information strategy Insert quantizer + channel

State x

Quantizer

Controller

x

xest (2R values)xest

uest = - xest

Certainty-equivalent strategies

Choose the quantizer Example: logarithmic strategy (Elia-Mitter)

=2, 2R=8, =1/8, T≈3 T (R-log ) > log -1

In general: T ~ 2R ~ log -1

Not optimal

0 11/21/41/8-1 -1/2 -1/4 -1/8

xest

The uniform strategy: certainty-equivalent and optimal

Dynamics: (|| > 1) We want to go from [-1,1] to [-ε,ε] Control: T = 1; 2R= /ε T (R-log = log ε-1

0 1-1

xest

Uniform strategy

x

ax

u (feedback law)

ax+u

1

a

An optimal strategy

How to contract [0, 1] into [0, 10-5] ?

(10-ary expansion)

Measure digits c1 and c6

We apply

u10x:x

76543210.x ccccccc

61 0000.0c 10x :x c

An optimal strategy Effect:

=10; 2R = 102; R = log2 100 bits; = 10-5; T=5

Optimal: T (R - log = log ε-1

00000340.x

00000230.x

50000120.x

45000910.x

34500890.x

23450780.x

12345670.x

6

5

4

3

2

1

0

Another optimal strategy

x:=2x+u From [0,1] to [0,2-12] Binary expansion of x Feedback law

If x = 0,1... then u = -1 (to keep x in [0,1]) If x = 0,.............1... (13th digit) then u=-2-12

If both then u = -1-2-12

If none then u = 0 After 12 steps, x = 0,012...

Performances

Performances: Time T = 12 4 values of u (R=2 bits to transmit) Contraction ε=2-12

The bound: T (R-log 2) ≥ log ε-1

is met with equality: optimal solution

Quantization subsets are disconnected

x

1

u-1

4 levels: 0, -2-12, -1, -1-2-12

Constant on intervals of length 2-13

u

If we want to go faster…

Feedback law: If x = 0,1... then u=-1 If x = 0,.............1... (13th digit) then u=-2-12

If x = 0,........1... (9th digit) then u=-2-8

If x = 0,....1... (5th digit) then u=-2-4

After 4 steps, x = 0,012... T = 4; R=4 ; ε = 2-12

Optimal again: T (R- log 2)= log ε-1 The same for ε=power of 2 ; R divides -log ε

Non integral eigenvalue

-expansion:

Ambiguous: take greedy expansion Shift property preserved

Stable eigenvalue

Bound remains valid Bound remains tight Example: =0.1, =10-10 , R= log 10 , T=5 If x=0.c1c2c3c4c5c6c7c8c9c10… then u=-c5 10-6

Marginal eigenvalue

The order of inputs does not matter Hence the bound is not tight

Unstable systems are better!

Optimal strategy: scalar case

Suppose

Given any two of T,N,, we can choose the third s.t.

The vector case We want:

Contract unit ball into -ball, 2R values of u, time T

Suppose eigenvalues real, ≠±1 Possible iff

(up to constant that depends on A and the norm) If =1: only unstable part of A

Intervals are a constraint If subsets of [-1, 1] are intervals, then (Fagnani-Zampieri)

If no interval constraint then we can beat it E.g., for every k, T=k, log = k2 log |R= (k+1) log

E.g., for every k, T=k, log = k log | R= 2 log

What if noise?

A small noise on the state can be amplified exponentially and disappear suddenly

Not desirable

Easy to fix? (if noise bounded and small)

How to treat noise?

Logarithmic strategy behaves well with noise Because has a quad. Lyapunov function:

Norm decreases at every step Logarithmic strategy optimal when noise? How to prove it? Either add a bounded noise at every step Or change T, N, E.g., replace T with energy x0

2+…+xT2

Conclusions

How to control a linear system with limited information on the state?

Separation principle, interval quantizers not optimal Discrete maths solution Scalar case: Characterization of possible

performances Vector case partially solved But: noise Interval strategies optimal with noise? with energy?

Thank you for your Thank you for your attention !attention !