QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science...
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Transcript of QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science...
![Page 1: QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.](https://reader033.fdocuments.us/reader033/viewer/2022042822/56649ebe5503460f94bc843e/html5/thumbnails/1.jpg)
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
![Page 2: QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.](https://reader033.fdocuments.us/reader033/viewer/2022042822/56649ebe5503460f94bc843e/html5/thumbnails/2.jpg)
PROBLEM FORMULATION
QUANTIZER DELAY
Architecture-independent approach
![Page 3: QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.](https://reader033.fdocuments.us/reader033/viewer/2022042822/56649ebe5503460f94bc843e/html5/thumbnails/3.jpg)
QUANTIZATION ONLY
quantization error
Assume
![Page 4: QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.](https://reader033.fdocuments.us/reader033/viewer/2022042822/56649ebe5503460f94bc843e/html5/thumbnails/4.jpg)
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]
![Page 5: QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.](https://reader033.fdocuments.us/reader033/viewer/2022042822/56649ebe5503460f94bc843e/html5/thumbnails/5.jpg)
QUANTIZATION and DELAY
Assuming ISS w.r.t. actuator errors as before:
In time domain:
where
![Page 6: QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.](https://reader033.fdocuments.us/reader033/viewer/2022042822/56649ebe5503460f94bc843e/html5/thumbnails/6.jpg)
SMALL – GAIN ARGUMENT
hence
ISS property becomes
if
then we recover ISS w.r.t. [Teel, IEEE TAC, Jul ’98]
Small gain:
![Page 7: QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.](https://reader033.fdocuments.us/reader033/viewer/2022042822/56649ebe5503460f94bc843e/html5/thumbnails/7.jpg)
FINAL RESULT
small gain true
Need:
![Page 8: QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.](https://reader033.fdocuments.us/reader033/viewer/2022042822/56649ebe5503460f94bc843e/html5/thumbnails/8.jpg)
FINAL RESULT
small gain true
Need:
![Page 9: QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.](https://reader033.fdocuments.us/reader033/viewer/2022042822/56649ebe5503460f94bc843e/html5/thumbnails/9.jpg)
FINAL RESULT
small gain true
Need:
![Page 10: QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.](https://reader033.fdocuments.us/reader033/viewer/2022042822/56649ebe5503460f94bc843e/html5/thumbnails/10.jpg)
FINAL RESULT
Need:
small gain true
![Page 11: QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.](https://reader033.fdocuments.us/reader033/viewer/2022042822/56649ebe5503460f94bc843e/html5/thumbnails/11.jpg)
FINAL RESULT
solutions starting in
enter and remain there
Need:
small gain true
![Page 12: QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.](https://reader033.fdocuments.us/reader033/viewer/2022042822/56649ebe5503460f94bc843e/html5/thumbnails/12.jpg)
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 )
![Page 13: QUANTIZATION and DELAY EFFECTS in NONLINEAR CONTROL SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ.](https://reader033.fdocuments.us/reader033/viewer/2022042822/56649ebe5503460f94bc843e/html5/thumbnails/13.jpg)
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