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TOWARDS a UNIFIED FRAMEWORK for NONLINEAR CONTROL with LIMITED INFORMATION

TOWARDS a UNIFIED FRAMEWORK for NONLINEAR CONTROL with LIMITED INFORMATION. Daniel Liberzon. Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign. Workshop dedicated to Roger Brockett’s 70 th birthday, Cancun, 12/8/08.

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TOWARDS a UNIFIED FRAMEWORK for NONLINEAR CONTROL with LIMITED INFORMATION

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  1. TOWARDS a UNIFIED FRAMEWORK forNONLINEAR CONTROL withLIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign Workshop dedicated to Roger Brockett’s 70th birthday, Cancun, 12/8/08 1 of 14

  2. INFORMATION FLOW in CONTROL SYSTEMS Plant Controller 2 of 14

  3. INFORMATION FLOW in CONTROL SYSTEMS • Coarse sensing • Limited communication capacity • Need to minimize information transmission • Event-driven actuators • Theoretical interest 2 of 14

  4. 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 3 of 14

  5. OUR APPROACH • Model these effects as deterministic additive 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 4 of 14

  6. QUANTIZATION finite subset of is partitioned into quantization regions QUANTIZER Encoder Decoder 5 of 14

  7. QUANTIZATION and INPUT-to-STATE STABILITY 6 of 14

  8. QUANTIZATION and INPUT-to-STATE STABILITY – assume glob. asymp. stable (GAS) 6 of 14

  9. QUANTIZATION and INPUT-to-STATE STABILITY no longer GAS 6 of 14

  10. QUANTIZATION and INPUT-to-STATE STABILITY quantization error Assume class 6 of 14

  11. QUANTIZATION and INPUT-to-STATE STABILITY quantization error Assume Solutions that start in enter and remain there This is input-to-state stability (ISS) w.r.t. measurement errors class In time domain: [Sontag ’89] class , e.g. 6 of 14

  12. LINEAR SYSTEMS 9 feedback gain & Lyapunov function Quantized control law: (automatically ISS w.r.t. ) Closed-loop: 7 of 14

  13. DYNAMIC QUANTIZATION 8 of 14

  14. DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state 8 of 14

  15. DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state 8 of 14

  16. DYNAMIC QUANTIZATION Hybrid quantized control: is discrete state – zooming variable Zoom out to overcome saturation 8 of 14

  17. DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state Proof: ISS from to small-gain condition ISS from to After the ultimate bound is achieved, recompute partition for smaller region Can recover global asymptotic stability 8 of 14

  18. SMALL-GAIN ANALYSIS of HYBRID SYSTEMS continuous discrete • ISS from to : • ISS from to : (small-gain condition) Hybrid system is GAS if we have: Can use Lyapunov techniques to check this [Nešić–L ’06] 9 of 14

  19. QUANTIZATION and DELAY QUANTIZER DELAY Architecture-independent approach Delays possibly large Based on result of [Teel ’98] 10 of 14

  20. SMALL–GAIN ARGUMENT where hence ISS w.r.t. actuator errors gives then we have ISS w.r.t. Small gain: if 11 of 14

  21. FINAL RESULT small gain true Need: 12 of 14

  22. FINAL RESULT Need: small gain true 12 of 14

  23. FINAL RESULT solutions starting in enter and remain there Need: small gain true Can use “zooming” to improve convergence 12 of 14

  24. EXTERNAL DISTURBANCES [Nešić–L] State quantization and completelyunknown disturbance 13 of 14

  25. EXTERNAL DISTURBANCES [Nešić–L] State quantization and completelyunknown disturbance 13 of 14

  26. 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 gain is nonlinear although plant is linear; cf.[Martins]) 13 of 14

  27. http://decision.csl.uiuc.edu/~liberzon ONGOING RESEARCH • Quantized output feedback and ISS observer design • Disturbances and coarse quantizers (with Y. Sharon) • Modeling uncertainty (with L. Vu, ThB08.2) • Coordination with coarse sensing (with S. LaValle and J. Yu, • TuC17.2) • Vision-based control (with Y. Ma and Y. Sharon) 14 of 14

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