STABILIZING a NONLINEAR SYSTEM with

LIMITED INFORMATION FEEDBACK

Daniel Liberzon

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

CDC ’03

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

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 Hurwitz

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

sampling times

Example:

Zoom out to get initial bound

Between samplings

NONLINEAR SYSTEMS

• is divided by 3 at the sampling time

Let

Example:

Between samplings

where is Lipschitz constant of

• grows at most by the factor in one period

The norm

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

Need ISS w.r.t. measurement errors

SUMMARY

Derived a sufficient condition for stabilization:

• Similar to known results for linear systems

• Involves alphabet size, sampling period, and Lipschitz constant

• Relies on input-to-state stabilizability w.r.t. measurement errors

• Relaxing the ISS assumption (De Persis)

• Outputs: how many variables to transmit?

• Necessary conditions for stabilization

• Performance

Research directions: