QUANTIZATION and DELAY EFFECTS

in NONLINEAR CONTROL SYSTEMS

Daniel Liberzon

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

Notre Dame, Oct 18, 2005

PROBLEM FORMULATION

QUANTIZER DELAY

Architecture-independent approach

QUANTIZATION ONLY

quantization error

Assume

QUANTIZATION ONLY

Solutions that start in

enter and remain there

quantization error

Assume

This is input-to-state stability (ISS) w.r.t. actuator errors

In time domain: [Sontag ’89]

QUANTIZATION and DELAY

Assuming ISS w.r.t. actuator errors as before:

In time domain:

where

SMALL – GAIN ARGUMENT

hence

ISS property becomes

if

then we recover ISS w.r.t. [Teel, IEEE TAC, Jul ’98]

Small gain:

FINAL RESULT

small gain true

Need:

FINAL RESULT

small gain true

Need:

FINAL RESULT

small gain true

Need:

FINAL RESULT

Need:

small gain true

FINAL RESULT

solutions starting in

enter and remain there

Need:

small gain true

VARIATIONS

Relaxing ISS assumption

Previous approach works if is sufficiently small

Different system architecture

globally asymptotically stabilizing we have

External disturbances

DELAY

QUANTIZER

Need ISS with respect

to measurement errors

(or work with small )

DYNAMIC QUANTIZATION

zoom in

Can recover global asymptotic stability

– zooming variable

Hybrid quantized control: is discrete state

zoom out

ISS from to

[Nesic-Liberzon ’05]

ISS from to small-gain condition