NONLINEAR HYBRID CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois.

NONLINEAR HYBRID CONTROL with

LIMITED INFORMATION

Daniel Liberzon

Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign

Paris, France, April 2008

Plant

Controller

INFORMATION FLOW in CONTROL SYSTEMS

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

• Coarse sensing

[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

Caveat:

This doesn’t 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

QUANTIZATION

Encoder Decoder

QUANTIZER

finite subset

of

is the range, is the quantization error bound

For , the quantizer saturates

Assume such that

is partitioned into quantization regions

QUANTIZATION and ISS

QUANTIZATION and ISS

quantization error

Assume

class

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

LINEAR SYSTEMS

Quantized control law:

9 feedback gain & Lyapunov function

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

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

DYNAMIC QUANTIZATION

– zooming variable

Hybrid quantized control: is discrete state

Can recover global asymptotic stability

ISS from to ISS from to small-gain conditionProof:

QUANTIZATION and DELAY

QUANTIZER DELAY

Architecture-independent approach

Based on the work of Teel

QUANTIZATION and DELAY

Assuming ISS w.r.t. actuator errors:

In time domain:

where

SMALL – GAIN ARGUMENT

hence

ISS property becomes

if

then we recover ISS w.r.t. [Teel ’98]

Small gain:

FINAL RESULT

Need:

small gain true

FINAL RESULT

Need:

small gain true

FINAL RESULT

solutions starting in

enter and remain there

Can use “zooming” to improve convergence

Need:

small gain true

EXTERNAL DISTURBANCES [Nešić–L]

State quantization and completely unknown disturbance

EXTERNAL DISTURBANCES [Nešić–L]

State quantization and completely unknown 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])

EXTERNAL DISTURBANCES [Nešić–L]

State quantization and completely unknown disturbance

After zoom-in:

STABILITY ANALYSIS of

HYBRID SYSTEMS via

SMALL-GAIN THEOREMS

Dragan NešićUniversity of Melbourne, Australia

Daniel LiberzonUniv. of Illinois at Urbana-Champaign, USA

HYBRID SYSTEMS as FEEDBACK CONNECTIONS

continuous

discrete

• Other decompositions possible

• Can also have external signals

See paper for more general setting

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)

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]:

LYAPUNOV – BASED SMALL – GAIN THEOREM

Hybrid system is GAS if:

•

and # of discrete events on is

•

•

quantization error

Zoom in:

where

ISS from to with gain

small-gain condition!

ISS from to with some linear gain

APPLICATION to DYNAMIC QUANTIZATION

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)

http://decision.csl.uiuc.edu/~liberzon