QUANTIZED OUTPUT FEEDBACK CONTROL of

NONLINEAR SYSTEMS

Daniel Liberzon

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

1 of 11 Mediterranean Control Conference, Athens, Greece, Jun 2007

QUANTIZED OUTPUT FEEDBACK

PLANT

QUANTIZER

CONTROLLER

Motivation:

• limited communication between sensor and actuator• trade-off between communication and computation

Objectives:

• analyze effect of quantization on system stability

• design controllers robust to quantization errors2 of 11

QUANTIZER

is the range, is the quantization error bound

Assume such that:

1.

2.

Encoder Decoderfinite setQUANTIZER

Output space is divided into quantization regions

For , the quantizer saturates3 of 11

LINEAR SYSTEM

Plant:

quantizationerror

Closed-loop system:

Luenberger observer-based controller:

or in short

where is Hurwitz if and are Hurwitz

[Brockett-L]

4 of 11

LINEAR SYSTEM (continued)

level sets of V

Solutions go from the larger level set to the smaller one

Recall:

Hurwitz

For we have

5 of 11

is of class if

• for each fixed

• as for each

INPUT-TO-STATE STABILITY (ISS) [Sontag]

class function

where

ISS:

Equivalent Lyapunov characterization:

when for some

Example:

6 of 11

NONLINEAR SYSTEM

Plant:

Dynamic controller:

Assume: this is ISS w.r.t. quantization error

Closed-loop system:

quantization error

or in short

(so in particular, should have GAS when )7 of 11

NONLINEAR SYSTEM (continued)

level sets of V

Solutions go from the larger level set to the smaller one

ISS

Can recover GAS using dynamic quantization

Lyap. function and class function s.t.

8 of 11

ISS ASSUMPTION: CLOSER LOOK

Closed-loop system

Reason: cascade argument

if for some we have

1.

2.

and

is ISS

9 of 11

Can extend this via a small-gain argument (need )

ISS CONTROLLER DESIGN

Closed-loop system:

1.

• Not always possible to achieve [Freeman ’95, Fah ’99]

• Results exist for classes of systems [Freeman & Kokotovic ’93,

’96, Freeman ’97, Fah ’99, Jiang et al. ’99, Sanfelice & Teel ’05]

• ISS assumption is fundamental in quantized control of nonlinear systems [L ’03]

This is ISS property of control law w.r.t. observation errors:

10 of 11

ISS OBSERVER DESIGN

Closed-loop system:

2.

This is ISS property of observer w.r.t. additive output errors

• No results on design of such ISS observers exist

• This property can be achieved for

with detectable and globally Lipschitz, very restrictive

More research on this problem is needed!11 of 11