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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

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Plant Controller INFORMATION FLOW in CONTROL SYSTEMS

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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 Coarse sensing

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

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Caveat: This doesnt work in general, need robustness from controller OUR APPROACH (Goal: treat nonlinear systems; handle quantization, delays, etc.) 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

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

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QUANTIZATION and ISS

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quantization error Assume class

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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 ] QUANTIZATION and ISS quantization error Assume class ; cf. linear: class

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

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DYNAMIC QUANTIZATION

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– zooming variable Hybrid quantized control: is discrete state

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

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

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

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

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

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SMALL – GAIN ARGUMENT hence ISS property becomes if then we recover ISS w.r.t. [ Teel 98 ] Small gain:

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FINAL RESULT Need: small gain true

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FINAL RESULT Need: small gain true

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

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EXTERNAL DISTURBANCES [ Nešić–L ] State quantization and completely unknown disturbance

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EXTERNAL DISTURBANCES [ Nešić–L ] State quantization and completely unknown disturbance

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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]) EXTERNAL DISTURBANCES [ Nešić–L ] State quantization and completely unknown disturbance After zoom-in:

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

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

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

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

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

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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 ]

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

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OUTPUT FEEDBACK: ISS OBSERVER DESIGN Closed-loop system if for some we have and is ISS (ISS property of control law w.r.t. observation errors) (ISS property of observer w.r.t. additive output errors) Open problem (except for very restrictive system classes)

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

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