QUANTIZED CONTROL and GEOMETRIC OPTIMIZATION Francesco Bullo and Daniel Liberzon Coordinated Science...
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![Page 1: QUANTIZED CONTROL and GEOMETRIC OPTIMIZATION Francesco Bullo and Daniel Liberzon Coordinated Science Laboratory Univ. of Illinois at Urbana-Champaign U.S.A.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649db95503460f94aa9333/html5/thumbnails/1.jpg)
QUANTIZED CONTROL and
GEOMETRIC OPTIMIZATION
Francesco Bullo and Daniel Liberzon
Coordinated Science LaboratoryUniv. of Illinois at Urbana-ChampaignU.S.A.
CDC 2003
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0
Control objectives: stabilize to 0 or to a desired set
containing 0, exit D through a specified facet, etc.
CONSTRAINED CONTROL
Constraint: – given
control commands
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LIMITED INFORMATION SCENARIO
– partition of D
– points in D,
Quantizer/encoder:
Control:
for
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MOTIVATION
• Limited communication capacity
• many systems/tasks share network cable or wireless medium
• microsystems with many sensors/actuators on one chip
• Need to minimize information transmission (security)
• Event-driven actuators
• PWM amplifier
• manual car transmission
• stepping motor
Encoder Decoder
QUANTIZER
finite subset
of
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QUANTIZER GEOMETRY
is partitioned into quantization regions
uniform logarithmic arbitrary
Dynamics change at boundaries => hybrid closed-loop system
Chattering on the boundaries is possible (sliding mode)
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QUANTIZATION ERROR and RANGE
1.
2.
Assume such that:
is the range, is the quantization error bound
For , the quantizer saturates
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OBSTRUCTION to STABILIZATION
Assume: fixed,M
Asymptotic stabilization is usually lost
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BASIC QUESTIONS
• What can we say about a given quantized system?
• How can we design the “best” quantizer for stability?
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BASIC QUESTIONS
• What can we say about a given quantized system?
• How can we design the “best” quantizer for stability?
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STATE QUANTIZATION: LINEAR SYSTEMS
Quantized control law:
where is quantization error
Closed-loop system:
is asymptotically stable
9 Lyapunov function
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LINEAR SYSTEMS (continued)
Recall:
Previous slide:
Lemma: solutions
that start in
enter in
finite time
Combine:
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NONLINEAR SYSTEMS
For nonlinear systems, GAS such robustness
For linear systems, we saw that if
gives then
automatically gives
when
This is robustness to measurement errors
This is input-to-state stability (ISS) for measurement errors
To have the same result, need to assume
when
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SUMMARY: PERTURBATION APPROACH
1. Design ignoring constraint
2. View as approximation
3. Prove that this still solves the problem
Issue:
error
Need to be ISS w.r.t. measurement errors
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BASIC QUESTIONS
• What can we say about a given quantized system?
• How can we design the “best” quantizer for stability?
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LOCATIONAL OPTIMIZATION: NAIVE APPROACH
This leads to the problem:
for Also true for nonlinear systemsISS w.r.t. measurement errors
Smaller => smaller
Compare: mailboxes in a city, cellular base stations in a region
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MULTICENTER PROBLEM
Critical points of satisfy
1. is the Voronoi partition :
2.
This is the
center of enclosing sphere of smallest radius
Lloyd algorithm:
iterate
Each is the Chebyshev center
(solution of the 1-center problem).
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Play movie: step3-animation.fli
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LOCATIONAL OPTIMIZATION: REFINED APPROACH
only need thisratio to be smallRevised problem:
. .. ..
.
.
...
.
. ..Logarithmic quantization:
Lower precision far away, higher precision close to 0
Only applicable to linear systems
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WEIGHTED MULTICENTER PROBLEM
This is the center of sphere enclosing
with smallest
Critical points of satisfy
1. is the Voronoi partition as before
2.
Lloyd algorithm – as before
Each is the weighted center
(solution of the weighted 1-center problem)
on not containing 0 (annulus)
Gives 25% decrease in for 2-D example
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Play movie: step5-animation.fli
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RESEARCH DIRECTIONS
• Robust control design
• Locational optimization
• Performance
• Applications