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 1 of 15 Workshop on Control and Optimization, UIUC, Dec 5, 2007

2. Plant Controller INFORMATION FLOW in CONTROL SYSTEMS 2 of 15

3. INFORMATION FLOW in CONTROL SYSTEMS Limited communication capacity many control loops share network cable or wireless medium microsystems with many sensors/actuators on one chip Need to minimize information transmission (security) Event-driven actuators 2 of 15

4. [ Brockett, Delchamps, Elia, Mitter, Nair, Savkin, Tatikonda, Wong,… ] Deterministic & stochastic models Tools from information theory Mostly for linear plant dynamics BACKGROUND Earlier work: Unified framework for quantization time delays disturbances Our goals: Handle nonlinear dynamics 3 of 15

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 via deterministic 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 15

6. QUANTIZATION EncoderDecoder QUANTIZER finite subset of is the range, is the quantization error bound For, the quantizer saturates Assume such that is partitioned into quantization regions 5 of 15

7. QUANTIZATION and ISS 6 of 15

8. QUANTIZATION and ISS quantization error Assume 6 of 15

9. 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 and ISS 6 of 15 quantization error Assume cf. linear:

10. LINEAR SYSTEMS Quantized control law: 9 feedback gain & Lyapunov function Closed-loop: (automatically ISS w.r.t. ) 7 of 15

11. DYNAMIC QUANTIZATION 8 of 15

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

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

14. Zoom out to overcome saturation DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state 8 of 15

15. After 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 Switching strategy based on dwell time or quantized state 8 of 15

16. 9 of 15 EXTERNAL DISTURBANCES [ Nešić-L ] State quantization and completely unknown disturbance

17. EXTERNAL DISTURBANCES [ Nešić-L ] State quantization and completely unknown disturbance 9 of 15

18. Issue: disturbance forces the state outside quantizer range Must switch repeatedly between zooming-in and zooming-out EXTERNAL DISTURBANCES [ Nešić-L ] State quantization and completely unknown disturbance Developed two different strategies: continuous-time vs. sampled-data implementation Lyapunov-based vs. trajectory-based analysis Result: for linear plant, can achieve ISS w.r.t. disturbance (ISS gains are nonlinear although plant is linear [cf. Martins]) 9 of 15

19. 10 of 15 QUANTIZATION and DELAY QUANTIZER DELAY Architecture-independent approach Based on the work of Teel

20. QUANTIZATION and DELAY Assuming ISS w.r.t. actuator errors: In time domain: where 11 of 15

21. SMALL – GAIN ARGUMENT hence ISS property becomes if then we recover ISS w.r.t. [ Teel ’98 ] Small gain: 12 of 15

22. FINAL RESULT Need: small gain true 13 of 15

23. FINAL RESULT Need: small gain true 13 of 15

24. FINAL RESULT solutions starting in enter and remain there 13 of 15 Can use “zooming” to recover global asymptotic stability Need: small gain true

25. LOCATIONAL OPTIMIZATION This leads to the problem: Compare: mailboxes in a city, cellular base stations in a region Also true for nonlinear systems ISS w.r.t. measurement errors Small => small [ Bullo-L ] for 14 of 15

26. RESEARCH DIRECTIONS Modeling uncertainty (robust, adaptive control) Disturbances and coarse quantizers [ Sharon-L ] Moving away from estimation-based approach Quantized output feedback Performance-based design Vision-based control and other applications how many variables to transmit? nonlinear ISS observer design 15 of 15 http://decision.csl.uiuc.edu/~liberzon