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QUANTIZATION and DELAY EFFECTS
in NONLINEAR CONTROL SYSTEMS
Daniel Liberzon
Coordinated Science Laboratory andDept. 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]
ISS from to small-gain condition