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Unified Framework for Nonlinear Control with Limited Information

Explore an innovative approach toward controlling nonlinear systems with limited information flow. Learn about quantization, time delays, disturbances, and more. Find out how to apply certainty equivalence and small-gain theorems for effective control.

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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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