1. STABILIZING a NONLINEAR SYSTEM with LIMITED INFORMATION FEEDBACK Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign U.S.A. CDC ’03

2. 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

3. ACTIVE PROBING for INFORMATION PLANT QUANTIZER CONTROLLER dynamic (changes at sampling times) (time-varying) EncoderDecoder very small

4. LINEAR SYSTEMS (Baillieul, Brockett-L, Hespanha et. al., Nair-Evans, Petersen-Savkin, Tatikonda, and others)

5. LINEAR SYSTEMS sampling times Zoom out to get initial bound Example: Between sampling times, let

6. 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

7. where is Hurwitz 0 LINEAR SYSTEMS (continued) Pick small enough s.t. sampling frequency vs. open-loop instability amount of static info provided by quantizer grows at most by the factor in one period is divided by 3 at each sampling time The norm

8. NONLINEAR SYSTEMS sampling times Example: Zoom out to get initial bound Between samplings

9. 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

10. 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

11. 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: