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
PROBLEM FORMULATION QUANTIZER DELAY Architecture-independent approach
QUANTIZATION ONLY quantization error Assume
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 ]
QUANTIZATION and DELAY Assuming ISS w.r.t. actuator errors as before: In time domain: where
SMALL – GAIN ARGUMENT hence ISS property becomes if then we recover ISS w.r.t. [ Teel, IEEE TAC, Jul ’98 ] Small gain:
FINAL RESULT small gain true Need:
FINAL RESULT small gain true Need:
FINAL RESULT small gain true Need:
FINAL RESULT Need: small gain true
FINAL RESULT solutions starting in enter and remain there Need: small gain true
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 )
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 ] iuc.edu/ liberzon ISS from to small-gain condition