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TOWARDS a UNIFIED FRAMEWORK for NONLINEAR CONTROL with LIMITED INFORMATION

This workshop presentation discusses the challenges and approaches for controlling nonlinear systems with limited information, including quantization, time delays, and external disturbances.

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TOWARDS a UNIFIED FRAMEWORK for NONLINEAR CONTROL with LIMITED INFORMATION

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  1. 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 on Control and Optimization, UIUC, Dec 5, 2007 1 of 15

  2. INFORMATION FLOW in CONTROL SYSTEMS Plant Controller 2 of 15

  3. INFORMATION FLOW in CONTROL SYSTEMS • 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 2 of 15

  4. BACKGROUND Our goals: • Handle nonlinear dynamics Earlier 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 15

  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 4 of 15

  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 5 of 15

  7. QUANTIZATION and ISS 6 of 15

  8. QUANTIZATION and ISS quantization error Assume 6 of 15

  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 In time domain: [Sontag ’89] cf. linear: 6 of 15

  10. LINEAR SYSTEMS 9 feedback gain & Lyapunov function Quantized control law: (automatically ISS w.r.t. ) Closed-loop: 7 of 15

  11. DYNAMIC QUANTIZATION 8 of 15

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

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

  14. DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state Zoom out to overcome saturation 8 of 15

  15. DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state Switching strategy based on dwell time or quantized state ISS from to small-gain condition ISS from to After ultimate bound is achieved, recompute partition for smaller region Can recover global asymptotic stability 8 of 15

  16. EXTERNAL DISTURBANCES [Nešić-L] State quantization and completelyunknown disturbance 9 of 15

  17. EXTERNAL DISTURBANCES [Nešić-L] State quantization and completelyunknown disturbance 9 of 15

  18. EXTERNAL DISTURBANCES [Nešić-L] Developed two different strategies: • continuous-time vs. sampled-data implementation • Lyapunov-based vs. trajectory-based analysis State quantization and completelyunknown disturbance 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]) 9 of 15

  19. QUANTIZATION and DELAY QUANTIZER DELAY Architecture-independent approach Based on the work of Teel 10 of 15

  20. QUANTIZATION and DELAY where Assuming ISS w.r.t. actuator errors: In time domain: 11 of 15

  21. SMALL–GAIN ARGUMENT if [Teel ’98] then we recover ISS w.r.t. hence ISS property becomes Small gain: 12 of 15

  22. FINAL RESULT small gain true Need: 13 of 15

  23. FINAL RESULT Need: small gain true 13 of 15

  24. FINAL RESULT solutions starting in enter and remain there Need: small gain true Can use “zooming” to recover global asymptotic stability 13 of 15

  25. RESEARCH DIRECTIONS • nonlinear ISS observer design • how many variables to transmit? http://decision.csl.uiuc.edu/~liberzon • Quantized output feedback • Disturbances and coarse quantizers [Sharon-L] • Modeling uncertainty (robust, adaptive control) • Performance-based design • Moving away from estimation-based approach • Vision-based control and other applications 15 of 15

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