STABILIZATION by QUANTIZED FEEDBACK :

HYBRID CONTROL APPROACH

Daniel Liberzon

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

IFAC ‘02, Barcelona

QUANTIZED FEEDBACK

PLANT

QUANTIZER

CONTROLLER

u x

q(x) x

q(x)

sensitivity

M values

OBSTRUCTION to STABILIZATION

Assume: fixed,M

Asymptotic stabilization is impossible

MOTIVATING EXAMPLES

1. Temperature sensor

normal

too low

too high

2. Camera with zoom

Tracking a golf ball

3. Coding and decoding

VARYING the SENSITIVITY

zoom out zoom in

Why switch ?• More realistic• Easier to design and analyze• Robust to time delays

LINEAR SYSTEMS

K BKxAxx is GAS: 0V

BuAxx

Assume:

Along solutions of )( exBKAxx

quantization error

0|||| Vxe

Then can achieve GAS

we have for some 0

s.t.

SWITCHING POLICY

We have 2|| /ne

2|| /nx

level sets of V

0|||| Vxe

.

NONLINEAR SYSTEMS

k ))(,( xkxfx is GAS: 0V

),( uxfx

Assume: s.t.

Need: along solutions of ))(,( exkxfx

where is pos. def., increasing, and unbounded

0)||(|| Vxe

(this is input-to-state stability wrt measurement error)

Then can achieve GAS

quantization error

EXTENSIONS and APPLICATIONS

• Arbitrary quantization regions

• Active probing for information

• Output and input quantization

• Relaxing the assumptions

• Performance-based design

• Application to visual servoing