CONTROL of NONLINEAR SYSTEMS

with LIMITED INFORMATION

Daniel Liberzon

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

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

is the range, is the quantization error bound

For , the quantizer saturates

Assume such that:

1.

2.

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?

• What can we do with very coarse quantization?

• What are the difficulties for nonlinear systems?

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?• What are the difficulties for nonlinear systems?

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

when

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

SUMMARY: PERTURBATION APPROACH

1. Design ignoring constraint

2. View as approximation

3. Prove that this still solves the problem

(in a weaker sense)

Issue:

error

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

INPUT QUANTIZATION

where

Control law:

Closed-loop system:

Analysis – same as before

Control law:

where

Need ISS with respect to actuator errors

Closed-loop system:

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?• What are the difficulties for nonlinear systems?

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:

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

Zoom out to overcome saturation

Can recover global asymptotic stability

(also applies to input and output quantization)

– zooming variable

Hybrid quantized control: is discrete state

zoom out

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?• What are the difficulties for nonlinear systems?

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

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

• 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. Look at the evolution of

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

RESEARCH DIRECTIONS

• ISS control design

• Locational optimization

• Performance and robustness

• Applications

REFERENCES

Brockett & L, 2000 (IEEE TAC)Bullo & L, 2003, L & Hespanha, 2004(http://decision.csl.uiuc.edu/~liberzon)