NONLINEAR HYBRID CONTROL with LIMITED INFORMATION Daniel Liberzon

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

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

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

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 Caveat: This doesn’t work in general, need robustness from controller Technical tools: • Input-to-state stability (ISS) • Small-gain theorems • Lyapunov functions • Hybrid systems

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

QUANTIZATION and ISS

QUANTIZATION and ISS quantization error Assume class

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

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

DYNAMIC QUANTIZATION

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

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

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

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

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

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

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

FINAL RESULT Need: small gain true

FINAL RESULT Need: small gain true

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

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

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

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

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

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

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

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

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

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

RESEARCH DIRECTIONS • Quantized output feedback • Performance-based design • Disturbances and coarse quantizers (with Y. Sharon) • Modeling uncertainty (with L. Vu) • Avoiding state estimation (with S. La. Valle and J. Yu) • Vision-based control (with Y. Ma and Y. Sharon) http: //decision. csl. uiuc. edu/~liberzon