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QUANTIZED CONTROL and
GEOMETRIC OPTIMIZATION
Francesco Bullo and Daniel Liberzon
Coordinated Science LaboratoryUniv. of Illinois at Urbana-ChampaignU.S.A.
CDC 2003
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
LIMITED INFORMATION SCENARIO
– partition of D
– points in D,
Quantizer/encoder:
Control:
for
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
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)
QUANTIZATION ERROR and RANGE
1.
2.
Assume such that:
is the range, is the quantization error bound
For , the quantizer saturates
OBSTRUCTION to STABILIZATION
Assume: fixed,M
Asymptotic stabilization is usually lost
BASIC QUESTIONS
• What can we say about a given quantized system?
• How can we design the “best” quantizer for stability?
BASIC QUESTIONS
• What can we say about a given quantized system?
• How can we design the “best” quantizer for stability?
STATE QUANTIZATION: LINEAR SYSTEMS
Quantized control law:
where is quantization error
Closed-loop system:
is asymptotically stable
9 Lyapunov function
LINEAR SYSTEMS (continued)
Recall:
Previous slide:
Lemma: solutions
that start in
enter in
finite time
Combine:
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
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
BASIC QUESTIONS
• What can we say about a given quantized system?
• How can we design the “best” quantizer for stability?
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
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).
Play movie: step3-animation.fli
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
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
Play movie: step5-animation.fli
RESEARCH DIRECTIONS
• Robust control design
• Locational optimization
• Performance
• Applications