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NONLINEAR HYBRID CONTROL with LIMITED INFORMATION

NONLINEAR HYBRID CONTROL with LIMITED INFORMATION. Daniel Liberzon. Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign. Paris, France, April 2008. INFORMATION FLOW in CONTROL SYSTEMS. Plant. Controller.

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NONLINEAR HYBRID CONTROL with LIMITED INFORMATION

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  1. NONLINEAR HYBRID CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign Paris, France, April 2008

  2. INFORMATION FLOW in CONTROL SYSTEMS Plant Controller

  3. 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

  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

  5. 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

  6. QUANTIZATION finite subset of is partitioned into quantization regions QUANTIZER Assume such that Encoder Decoder is the range, is the quantization error bound For , the quantizer saturates

  7. QUANTIZATION and ISS

  8. QUANTIZATION and ISS quantization error Assume class

  9. 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 class In time domain: [Sontag ’89] class ; cf. linear:

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

  11. DYNAMIC QUANTIZATION

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

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

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

  15. 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

  16. QUANTIZATION and DELAY QUANTIZER DELAY Architecture-independent approach Based on the work of Teel

  17. QUANTIZATION and DELAY where Assuming ISS w.r.t. actuator errors: In time domain:

  18. SMALL–GAIN ARGUMENT if [Teel ’98] then we recover ISS w.r.t. hence ISS property becomes Small gain:

  19. FINAL RESULT small gain true Need:

  20. FINAL RESULT Need: small gain true

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

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

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

  24. 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])

  25. STABILITY ANALYSIS of HYBRID SYSTEMS via SMALL-GAIN THEOREMS Daniel Liberzon Univ. of Illinois at Urbana-Champaign, USA Dragan Nešić University of Melbourne, Australia

  26. HYBRID SYSTEMS as FEEDBACK CONNECTIONS See paper for more general setting • Other decompositions possible • Can also have external signals continuous discrete

  27. 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:

  28. 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]

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

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

  31. RESEARCH DIRECTIONS http://decision.csl.uiuc.edu/~liberzon • Quantized output feedback • Performance-based design • 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)

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