1. CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign

2. REASONS for SWITCHING Nature of the control problem Sensor or actuator limitations Large modeling uncertainty Combinations of the above

3. Plant Controller INFORMATION FLOW in CONTROL SYSTEMS

4. Limited communication capacity many control loops share network cable or wireless medium microsystems with many sensors/actuators on one chip Need to minimize information transmission (security) Event-driven actuators Coarse sensing Theoretical interest

5. [ Brockett, Delchamps, Elia, Mitter, Nair, Savkin, Tatikonda, Wong,… ] Deterministic & stochastic models Tools from information theory Mostly for linear plant dynamics BACKGROUND Previous work: Unified framework for quantization time delays disturbances Our goals: Handle nonlinear dynamics

6. Caveat: This doesnt work in general, need robustness from controller OUR APPROACH (Goal: treat nonlinear systems; handle quantization, delays, etc.) Model these effects via deterministic error signals, Design a control law ignoring these errors, Certainty equivalence: apply control, combined with estimation to reduce to zero Technical tools: Input-to-state stability (ISS) Lyapunov functions Small-gain theorems Hybrid systems

7. QUANTIZATION EncoderDecoder QUANTIZER finite subset of is partitioned into quantization regions

8. QUANTIZATION and ISS

9. – assume glob. asymp. stable (GAS) QUANTIZATION and ISS

10. no longer GAS

11. QUANTIZATION and ISS quantization error Assume class

12. Solutions that start in enter and remain there This is input-to-state stability (ISS) w.r.t. measurement errors In time domain: QUANTIZATION and ISS quantization error Assume

13. LINEAR SYSTEMS Quantized control law: 9 feedback gain & Lyapunov function Closed-loop: (automatically ISS w.r.t. )

14. DYNAMIC QUANTIZATION

15. – zooming variable Hybrid quantized control: is discrete state

16. DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state

17. Zoom out to overcome saturation DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state

18. After ultimate bound is achieved, recompute partition for smaller region DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state ISS from to small-gain condition Proof: Can recover global asymptotic stability

19. QUANTIZATION and DELAY Architecture-independent approach Based on the work of Teel Delays possibly large QUANTIZER DELAY

20. QUANTIZATION and DELAY where Can write hence

21. SMALL – GAIN ARGUMENT if then we recover ISS w.r.t. [ Teel 98 ] Small gain: Assuming ISS w.r.t. actuator errors: In time domain:

22. FINAL RESULT Need: small gain true

23. FINAL RESULT Need: small gain true

24. FINAL RESULT solutions starting in enter and remain there Can use zooming to improve convergence Need: small gain true

25. EXTERNAL DISTURBANCES [ Nešić–L ] State quantization and completely unknown disturbance

26. EXTERNAL DISTURBANCES [ Nešić–L ] State quantization and completely unknown disturbance

27. Issue: disturbance forces the state outside quantizer range Must switch repeatedly between zooming-in and zooming-out Result: for linear plant, can achieve ISS w.r.t. disturbance (ISS gains are nonlinear although plant is linear; cf. [ Martins ]) EXTERNAL DISTURBANCES [ Nešić–L ] State quantization and completely unknown disturbance After zoom-in:

28. NETWORKED CONTROL SYSTEMS [ Nešić–L ] NCS: Transmit only some variables according to time scheduling protocol Examples: round-robin, TOD (try-once-discard) QCS: Transmit quantized versions of all variables NQCS: Unified framework combining time scheduling and quantization Basic design/analysis steps: Design controller ignoring network effects Apply small-gain theorem to compute upper bound on maximal allowed transmission interval (MATI) Prove discrete protocol stability via Lyapunov function

29. ACTIVE PROBING for INFORMATION dynamic (changes at sampling times) (time-varying) PLANT QUANTIZER CONTROLLER EncoderDecoder very small

30. NONLINEAR SYSTEMS sampling times Example: Zoom out to get initial bound Between samplings

31. NONLINEAR SYSTEMS is divided by 3 at the sampling time Let Example: Between samplings grows at most by the factor in one period The norm on a suitable compact region (dependent on )

32. Pick small enough s.t. NONLINEAR SYSTEMS (continued) grows at most by the factor in one period is divided by 3 at each sampling time The norm If this is ISS w.r.t. as before, then

33. ROBUSTNESS of the CONTROLLER ISS w.r.t. Same condition as before (restrictive, hard to check) checkable sufficient conditions ([ Hespanha-L-Teel ]) ISS w.r.t. Option 1. Option 2. Look at the evolution of

34. LINEAR SYSTEMS

35. [ Baillieul, Brockett-L, Hespanha, Nair-Evans, Petersen-Savkin,Tatikonda ] Between sampling times, where is Hurwitz 0 divided by 3 at each sampling time grows at most by in one period amount of static info provided by quantizer sampling frequency vs. open-loop instability global quantity:

36. HYBRID SYSTEMS as FEEDBACK CONNECTIONS continuous discrete [ Nešić–L, 05, 06 ] Other decompositions possible Can also have external signals

37. SMALL – GAIN THEOREM Small-gain theorem [ Jiang-Teel-Praly 94 ] gives GAS if: Input-to-state stability (ISS) from to : ISS from to : (small-gain condition)

38. SUFFICIENT CONDITIONS for ISS [ Hespanha-L-Teel 08 ] # of discrete events on is ISS from to if: and ISS from to if ISS-Lyapunov function [ Sontag 89 ] :

39. LYAPUNOV – BASED SMALL – GAIN THEOREM Hybrid system is GAS if: and # of discrete events on is

40. SKETCH of PROOF is nonstrictly decreasing along trajectories Trajectories along which is constant?None! GAS follows by LaSalle principle for hybrid systems [ Lygeros et al. 03, Sanfelice-Goebel-Teel 07 ]

41. quantization error Zoom in: where ISS from to with gain small-gain condition! ISS from to with some linear gain APPLICATION to DYNAMIC QUANTIZATION

42. RESEARCH DIRECTIONS Modeling uncertainty (with L. Vu) Disturbances and coarse quantizers (with Y. Sharon) Avoiding state estimation (with S. LaValle and J. Yu) Quantized output feedback (with H. Shim) Performance-based design (with F. Bullo) Vision-based control (with Y. Ma and Y. Sharon)