2. How to do control theory and discrete maths at the same time?

3. Control and information: linear, memoryless and noiseless Jean-Charles Delvenne CMI, Caltech CMI seminar May 12, 2006 May 12, 2006

4. What is a dynamical system ? (Piece of Nature) Maps Discrete time Interaction with environment State x Input u Output y

5. What is control ? Simple controllers preferred: Memoryless: u:=k(y) Controller x 0 y u

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

7. 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)

8. 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

9. Main framework: no noise Deterministic systems Volumes multiplied by at least We know: x in set S 0 We want: x in subset S F within average timeT We can: measure x, transmit a symbol, choose u Noiseless feedback, memory allowed Contraction C=  (S 0 )/  (S F ) If rate R, then 2 R symbols allowed T=average time to reach S F provably

10. 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

11. Lower bounds: Fundamental limitations Proof Time: we reach subset S F 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

12. 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 2 R symbols are allowed. Formulation: Fagnani and Zampieri But: we do not require intervals

13. What is possible? We want to contract [-1,1] into [- ,  ] in time T with 2 R values of u We want , T, R small (good, fast, economical) We want a memoryless feedback What is the trade-off ?

14. Main result We can contract [-1,1] into [- ,  ] in time T with 2 R symbols iff ‘Only if’: See above ‘If’: Find a optimal strategies… …that span the surface. Strategy with 2 degrees of freedom

15. An idea:The Separation Principle and Certainty Equivalent strategies Start from perfect-information strategy Insert quantizer + channel State x Quantizer Controller x x est (2 R values) x est u est = - x est

16. Certainty-equivalent strategies Choose the quantizer Example: logarithmic strategy (Elia-Mitter) =2, 2 R =8,  =1/8, T≈3 T (R-log ) > log  -1 In general: T ~ 2 R ~ log  -1 Not optimal 01 1/21/4 1/8 -1/2-1/4-1/8 x est

17. The uniform strategy: certainty-equivalent and optimal Dynamics: (| | > 1) We want to go from [-1,1] to [-ε,ε] Control: T = 1; 2 R = /ε T (R-log  = log ε -1 01 x est

18. Uniform strategy x ax u (feedback law) ax+u 1 a

19. An optimal strategy How to contract [0, 1] into [0, 10 -5 ] ? (10-ary expansion) Measure digits c 1 and c 6 We apply

20. An optimal strategy Effect: =10; 2 R = 10 2 ; R = log 2 100 bits;  = 10 -5 ; T=5 Optimal: T (R - log  = log ε -1

21. 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... (13 th digit) then u=-2 -12 If both then u = -1-2 -12 If none then u = 0 After 12 steps, x = 0,0 12...

22. 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

23. Quantization subsets are disconnected x 1 u 4 levels: 0, -2 -12, -1, -1-2 -12 Constant on intervals of length 2 -13 u

24. If we want to go faster… Feedback law: If x = 0,1... then u=-1 If x = 0,.............1... (13 th digit) then u=-2 -12 If x = 0,........1... (9 th digit) then u=-2 -8 If x = 0,....1... (5 th digit) then u=-2 -4 After 4 steps, x = 0,0 12... 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

25. Non integral eigenvalue  -expansion: Ambiguous: take greedy expansion Shift property preserved

26. Stable eigenvalue Bound remains valid Bound remains tight Example: =0.1,  =10 -10, R= log 10, T=5 If x=0.c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 c 9 c 10 … then u=-c 5 10 -6

27. Marginal eigenvalue The order of inputs does not matter Hence the bound is not tight Unstable systems are better!

28. Optimal strategy: scalar case Suppose Given any two of T,N, , we can choose the third s.t.

29. The vector case We want: Contract unit ball into  -ball, 2 R 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

30. 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   = k 2 log |  R= (k+1) log E.g., for every k, T=k, log   = k log |  R= 2 log

31. 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)

32. 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 x 0 2 +…+x T 2

33. 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?

34. Thank you for your attention !