1. TOWARDS a UNIFIED FRAMEWORK for NONLINEAR CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign Workshop dedicated to Roger Brockett’s 70 th birthday, Cancun, 12/8/08 1 of 14

2. Plant Controller INFORMATION FLOW in CONTROL SYSTEMS 2 of 14

3. INFORMATION FLOW in CONTROL SYSTEMS Limited communication capacity Need to minimize information transmission Event-driven actuators Coarse sensing Theoretical interest 2 of 14

4. [ Brockett, Delchamps, Elia, Mitter, Nair, Savkin, Tatikonda, Wong,… ] Deterministic & stochastic models Tools from information theory Mostly for linear plant dynamics BACKGROUND Previous work: Unified framework for quantization time delays disturbances Our goals: Handle nonlinear dynamics 3 of 14

5. Caveat: This doesn’t work in general, need robustness from controller OUR APPROACH (Goal: treat nonlinear systems; handle quantization, delays, etc.) Model these effects as deterministic additive error signals, Design a control law ignoring these errors, “Certainty equivalence”: apply control combined with estimation to reduce to zero Technical tools: Input-to-state stability (ISS) Lyapunov functions Small-gain theorems Hybrid systems 4 of 14

6. QUANTIZATION EncoderDecoder QUANTIZER finite subset of is partitioned into quantization regions 5 of 14

7. QUANTIZATION and INPUT-to-STATE STABILITY 6 of 14

8. – assume glob. asymp. stable (GAS) QUANTIZATION and INPUT-to-STATE STABILITY 6 of 14

9. QUANTIZATION and INPUT-to-STATE STABILITY no longer GAS 6 of 14

10. quantization error Assume class QUANTIZATION and INPUT-to-STATE STABILITY 6 of 14

11. Solutions that start in enter and remain there This is input-to-state stability (ISS) w.r.t. measurement errors In time domain: [ Sontag ’89 ] quantization error Assume class QUANTIZATION and INPUT-to-STATE STABILITY class, e.g. 6 of 14

12. LINEAR SYSTEMS Quantized control law: 9 feedback gain & Lyapunov function Closed-loop: (automatically ISS w.r.t. ) 7 of 14

13. DYNAMIC QUANTIZATION 8 of 14

14. DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state 8 of 14

15. DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state 8 of 14

16. Zoom out to overcome saturation DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state 8 of 14

17. After the ultimate bound is achieved, recompute partition for smaller region DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state Can recover global asymptotic stability ISS from to small-gain condition Proof: 8 of 14

18. SMALL-GAIN ANALYSIS of HYBRID SYSTEMS continuous discrete Hybrid system is GAS if we have: (small-gain condition) Can use Lyapunov techniques to check this [ Nešić–L ’06 ] ISS from to : 9 of 14

19. QUANTIZATION and DELAY Architecture-independent approach Based on result of [ Teel ’98 ] Delays possibly large QUANTIZER DELAY 10 of 14

20. SMALL – GAIN ARGUMENT ISS w.r.t. actuator errors gives where ifthen we have ISS w.r.t.Small gain: 11 of 14 hence

21. FINAL RESULT Need: small gain true 12 of 14

22. FINAL RESULT Need: small gain true 12 of 14

23. FINAL RESULT solutions starting in enter and remain there Can use “zooming” to improve convergence Need: small gain true 12 of 14

24. EXTERNAL DISTURBANCES [ Nešić–L ] State quantization and completely unknown disturbance 13 of 14

25. EXTERNAL DISTURBANCES [ Nešić–L ] State quantization and completely unknown disturbance 13 of 14

26. Issue: disturbance forces the state outside quantizer range Must switch repeatedly between zooming-in and zooming-out Result: for linear plant, can achieve ISS w.r.t. disturbance (ISS gain is nonlinear although plant is linear; cf. [ Martins ]) EXTERNAL DISTURBANCES [ Nešić–L ] State quantization and completely unknown disturbance After zoom-in: 13 of 14

27. ONGOING RESEARCH Modeling uncertainty (with L. Vu, ThB08.2 ) Disturbances and coarse quantizers (with Y. Sharon) Coordination with coarse sensing (with S. LaValle and J. Yu, TuC17.2 ) Quantized output feedback and ISS observer design Vision-based control (with Y. Ma and Y. Sharon) http://decision.csl.uiuc.edu/~liberzon 14 of 14