1. QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign Notre Dame, Oct 18, 2005

2. PROBLEM FORMULATION QUANTIZER DELAY Architecture-independent approach

3. QUANTIZATION ONLY quantization error Assume

4. QUANTIZATION ONLY Solutions that start in enter and remain there quantization error Assume This is input-to-state stability (ISS) w.r.t. actuator errors In time domain: [ Sontag ’89 ]

5. QUANTIZATION and DELAY Assuming ISS w.r.t. actuator errors as before: In time domain: where

6. SMALL – GAIN ARGUMENT hence ISS property becomes if then we recover ISS w.r.t. [ Teel, IEEE TAC, Jul ’98 ] Small gain:

7. FINAL RESULT small gain true Need:

8. FINAL RESULT small gain true Need:

9. FINAL RESULT small gain true Need:

10. FINAL RESULT Need: small gain true

11. FINAL RESULT solutions starting in enter and remain there Need: small gain true

12. VARIATIONS  Relaxing ISS assumption Previous approach works if is sufficiently small  Different system architecture globally asymptotically stabilizing we have  External disturbances DELAY QUANTIZER Need ISS with respect to measurement errors (or work with small )

13. DYNAMIC QUANTIZATION zoom in Can recover global asymptotic stability – zooming variable Hybrid quantized control: is discrete state zoom out ISS from to [ Nesic-Liberzon ’05 ] http://decision.csl.u iuc.edu/ liberzon ISS from to small-gain condition