QUANTIZED CONTROL and

GEOMETRIC OPTIMIZATION

Francesco Bullo and Daniel Liberzon

Coordinated Science LaboratoryUniv. of Illinois at Urbana-ChampaignU.S.A.

CDC 2003

0

Control objectives: stabilize to 0 or to a desired set

containing 0, exit D through a specified facet, etc.

CONSTRAINED CONTROL

Constraint: – given

control commands

LIMITED INFORMATION SCENARIO

– partition of D

– points in D,

Quantizer/encoder:

Control:

for

MOTIVATION

• Limited communication capacity

• many systems/tasks share network cable or wireless medium

• microsystems with many sensors/actuators on one chip

• Need to minimize information transmission (security)

• Event-driven actuators

• PWM amplifier

• manual car transmission

• stepping motor

Encoder Decoder

QUANTIZER

finite subset

of

QUANTIZER GEOMETRY

is partitioned into quantization regions

uniform logarithmic arbitrary

Dynamics change at boundaries => hybrid closed-loop system

Chattering on the boundaries is possible (sliding mode)

QUANTIZATION ERROR and RANGE

1.

2.

Assume such that:

is the range, is the quantization error bound

For , the quantizer saturates

OBSTRUCTION to STABILIZATION

Assume: fixed,M

Asymptotic stabilization is usually lost

BASIC QUESTIONS

• What can we say about a given quantized system?

• How can we design the “best” quantizer for stability?

BASIC QUESTIONS

• What can we say about a given quantized system?

• How can we design the “best” quantizer for stability?

STATE QUANTIZATION: LINEAR SYSTEMS

Quantized control law:

where is quantization error

Closed-loop system:

is asymptotically stable

9 Lyapunov function

LINEAR SYSTEMS (continued)

Recall:

Previous slide:

Lemma: solutions

that start in

enter in

finite time

Combine:

NONLINEAR SYSTEMS

For nonlinear systems, GAS such robustness

For linear systems, we saw that if

gives then

automatically gives

when

This is robustness to measurement errors

This is input-to-state stability (ISS) for measurement errors

To have the same result, need to assume

when

SUMMARY: PERTURBATION APPROACH

1. Design ignoring constraint

2. View as approximation

3. Prove that this still solves the problem

Issue:

error

Need to be ISS w.r.t. measurement errors

BASIC QUESTIONS

• What can we say about a given quantized system?

• How can we design the “best” quantizer for stability?

LOCATIONAL OPTIMIZATION: NAIVE APPROACH

This leads to the problem:

for Also true for nonlinear systemsISS w.r.t. measurement errors

Smaller => smaller

Compare: mailboxes in a city, cellular base stations in a region

MULTICENTER PROBLEM

Critical points of satisfy

1. is the Voronoi partition :

2.

This is the

center of enclosing sphere of smallest radius

Lloyd algorithm:

iterate

Each is the Chebyshev center

(solution of the 1-center problem).

Play movie: step3-animation.fli

LOCATIONAL OPTIMIZATION: REFINED APPROACH

only need thisratio to be smallRevised problem:

. .. ..

.

.

...

.

. ..Logarithmic quantization:

Lower precision far away, higher precision close to 0

Only applicable to linear systems

WEIGHTED MULTICENTER PROBLEM

This is the center of sphere enclosing

with smallest

Critical points of satisfy

1. is the Voronoi partition as before

2.

Lloyd algorithm – as before

Each is the weighted center

(solution of the weighted 1-center problem)

on not containing 0 (annulus)

Gives 25% decrease in for 2-D example

Play movie: step5-animation.fli

RESEARCH DIRECTIONS

• Robust control design

• Locational optimization

• Performance

• Applications