CONTROL of NONLINEAR SYSTEMS under

COMMUNICATION CONSTRAINTS

Daniel Liberzon

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

Caltech, Apr 1, 2005

LIMITED INFORMATION SCENARIO

– partition of D

– points in D,

Quantizer:

Control:

for

OBSTRUCTION to STABILIZATION

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?

• What can we do with very coarse quantization?

• What are the difficulties for nonlinear systems?

STATE QUANTIZATION: LINEAR SYSTEMS

Quantized control law:

Closed-loop:

9 feedback gain & Lyapunov function

quantization error

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

when

To have the same result, need to assume pos.def. incr. :

LOCATIONAL OPTIMIZATION

This leads to the problem:

for

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

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

Small => small

[Bullo-L]

MULTICENTER PROBLEM

Critical points of satisfy

1. is the Voronoi partition :

2.

This is the

center of enclosing sphere of smallest radius

Lloyd algorithm:

Each is the Chebyshev center

(solution of the 1-center problem).

iterate

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

DYNAMIC QUANTIZATION

zoom in

After ultimate bound is achieved,recompute partition for smaller region

Can recover global asymptotic stability

– zooming variable

Hybrid quantized control: is discrete state

Zoom out to overcome saturation

zoom out

ACTIVE PROBING for INFORMATION

PLANT

QUANTIZER

CONTROLLER

dynamic

dynamic

(changes at sampling times)

(time-varying)

Encoder Decoder

very small

LINEAR SYSTEMS

(Baillieul, Brockett-L, Hespanha et. al., Nair-Evans,

Petersen-Savkin, Tatikonda, and others)

LINEAR SYSTEMS

sampling times

Zoom out to get initial bound

Example:

Between sampling times, let

LINEAR SYSTEMS

Consider

• is divided by 3 at the sampling time

Example:

Between sampling times, let

• grows at most by the factor in one period

The norm

where is stable

0

LINEAR SYSTEMS (continued)

Pick small enough s.t.

sampling frequency vs. open-loop instability

amount of static infoprovided by quantizer

• grows at most by the factor in one period

• is divided by 3 at each sampling time

The norm

NONLINEAR SYSTEMS

• is divided by 3 at the sampling time

Let

Example:

Between samplings

• grows at most by the factor in one period

The norm

on a suitable compact region

Pick small enough s.t.

NONLINEAR SYSTEMS (continued)

• grows at most by the factor in one period

• is divided by 3 at each sampling time

The norm

What properties of guarantee GAS ?

ROBUSTNESS of the CONTROLLER

ISS w.r.t.

ISS w.r.t. measurement errors – quite restrictive...

ISS w.r.t.

Option 1.

Option 2. [Hespanha-L] Look at the evolution of

Easier to verify (e.g., GES & glob. Lip.)

SOME RESEARCH DIRECTIONS

• ISS control design

• ISS of impulsive systems (work with Hespanha, Teel)

• Performance and robustness (work with Nesic)

• Applications

• Other?