QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science...
-
Upload
dwight-hutchinson -
Category
Documents
-
view
215 -
download
0
Transcript of QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science...
![Page 1: QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.](https://reader035.fdocuments.in/reader035/viewer/2022062423/56649eb35503460f94bbaefe/html5/thumbnails/1.jpg)
QUANTIZED OUTPUT FEEDBACK CONTROL of
NONLINEAR SYSTEMS
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
1 of 11 Mediterranean Control Conference, Athens, Greece, Jun 2007
![Page 2: QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.](https://reader035.fdocuments.in/reader035/viewer/2022062423/56649eb35503460f94bbaefe/html5/thumbnails/2.jpg)
QUANTIZED OUTPUT FEEDBACK
PLANT
QUANTIZER
CONTROLLER
Motivation:
• limited communication between sensor and actuator• trade-off between communication and computation
Objectives:
• analyze effect of quantization on system stability
• design controllers robust to quantization errors2 of 11
![Page 3: QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.](https://reader035.fdocuments.in/reader035/viewer/2022062423/56649eb35503460f94bbaefe/html5/thumbnails/3.jpg)
QUANTIZER
is the range, is the quantization error bound
Assume such that:
1.
2.
Encoder Decoderfinite setQUANTIZER
Output space is divided into quantization regions
For , the quantizer saturates3 of 11
![Page 4: QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.](https://reader035.fdocuments.in/reader035/viewer/2022062423/56649eb35503460f94bbaefe/html5/thumbnails/4.jpg)
LINEAR SYSTEM
Plant:
quantizationerror
Closed-loop system:
Luenberger observer-based controller:
or in short
where is Hurwitz if and are Hurwitz
[Brockett-L]
4 of 11
![Page 5: QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.](https://reader035.fdocuments.in/reader035/viewer/2022062423/56649eb35503460f94bbaefe/html5/thumbnails/5.jpg)
LINEAR SYSTEM (continued)
level sets of V
Solutions go from the larger level set to the smaller one
Recall:
Hurwitz
For we have
5 of 11
![Page 6: QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.](https://reader035.fdocuments.in/reader035/viewer/2022062423/56649eb35503460f94bbaefe/html5/thumbnails/6.jpg)
is of class if
• for each fixed
• as for each
INPUT-TO-STATE STABILITY (ISS) [Sontag]
class function
where
ISS:
Equivalent Lyapunov characterization:
when for some
Example:
6 of 11
![Page 7: QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.](https://reader035.fdocuments.in/reader035/viewer/2022062423/56649eb35503460f94bbaefe/html5/thumbnails/7.jpg)
NONLINEAR SYSTEM
Plant:
Dynamic controller:
Assume: this is ISS w.r.t. quantization error
Closed-loop system:
quantization error
or in short
(so in particular, should have GAS when )7 of 11
![Page 8: QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.](https://reader035.fdocuments.in/reader035/viewer/2022062423/56649eb35503460f94bbaefe/html5/thumbnails/8.jpg)
NONLINEAR SYSTEM (continued)
level sets of V
Solutions go from the larger level set to the smaller one
ISS
Can recover GAS using dynamic quantization
Lyap. function and class function s.t.
8 of 11
![Page 9: QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.](https://reader035.fdocuments.in/reader035/viewer/2022062423/56649eb35503460f94bbaefe/html5/thumbnails/9.jpg)
ISS ASSUMPTION: CLOSER LOOK
Closed-loop system
Reason: cascade argument
if for some we have
1.
2.
and
is ISS
9 of 11
Can extend this via a small-gain argument (need )
![Page 10: QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.](https://reader035.fdocuments.in/reader035/viewer/2022062423/56649eb35503460f94bbaefe/html5/thumbnails/10.jpg)
ISS CONTROLLER DESIGN
Closed-loop system:
1.
• Not always possible to achieve [Freeman ’95, Fah ’99]
• Results exist for classes of systems [Freeman & Kokotovic ’93,
’96, Freeman ’97, Fah ’99, Jiang et al. ’99, Sanfelice & Teel ’05]
• ISS assumption is fundamental in quantized control of nonlinear systems [L ’03]
This is ISS property of control law w.r.t. observation errors:
10 of 11
![Page 11: QUANTIZED OUTPUT FEEDBACK CONTROL of NONLINEAR SYSTEMS Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of.](https://reader035.fdocuments.in/reader035/viewer/2022062423/56649eb35503460f94bbaefe/html5/thumbnails/11.jpg)
ISS OBSERVER DESIGN
Closed-loop system:
2.
This is ISS property of observer w.r.t. additive output errors
• No results on design of such ISS observers exist
• This property can be achieved for
with detectable and globally Lipschitz, very restrictive
More research on this problem is needed!11 of 11