CONTROL with LIMITED INFORMATION

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. INFORMATION FLOW in CONTROL SYSTEMS Plant Controller

4. INFORMATION FLOW in CONTROL SYSTEMS • Coarse sensing • 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 • Theoretical interest

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

6. OUR APPROACH • 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 (Goal: treat nonlinear systems; handle quantization, delays, etc.) Caveat: This doesn’t work in general, need robustness from controller

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

8. QUANTIZATION and ISS

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

10. QUANTIZATION and ISS no longer GAS

11. QUANTIZATION and ISS quantization error Assume class

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

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

14. DYNAMIC QUANTIZATION

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

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

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

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

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

20. QUANTIZATION and DELAY where Can write hence

21. SMALL–GAIN ARGUMENT Assuming ISS w.r.t. actuator errors: In time domain: Small gain: if [Teel ’98] then we recover ISS w.r.t.

22. FINAL RESULT small gain true Need:

23. FINAL RESULT Need: small gain true

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

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

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

27. EXTERNAL DISTURBANCES [Nešić–L] State quantization and completelyunknown disturbance After zoom-in: 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])

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 • Prove discrete protocol stability via Lyapunov function • Apply small-gain theorem to compute upper bound on • maximal allowed transmission interval (MATI)

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

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

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

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

33. LINEAR SYSTEMS

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

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

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

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

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

39. 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]

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

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