QUANTIZED CONTROL and GEOMETRIC OPTIMIZATION

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QUANTIZED CONTROL andGEOMETRIC OPTIMIZATION
Francesco Bullo and Daniel Liberzon
Coordinated Science Laboratory Univ. of Illinois
at Urbana-Champaign U.S.A.
CDC 2003
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CONSTRAINED CONTROL
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LIMITED INFORMATION SCENARIO
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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

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QUANTIZER GEOMETRY
Dynamics change at boundaries gt hybrid
closed-loop system
Chattering on the boundaries is possible (sliding
mode)
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QUANTIZATION ERROR and RANGE
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OBSTRUCTION to STABILIZATION
Assume fixed
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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
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LINEAR SYSTEMS (continued)
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NONLINEAR SYSTEMS
For linear systems, we saw that if
gives then
automatically gives
when
This is robustness to measurement errors
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SUMMARY PERTURBATION APPROACH
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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
Compare mailboxes in a city, cellular base
stations in a region
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MULTICENTER PROBLEM

This is the center of enclosing sphere of
smallest radius
iterate
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Play movie step3-animation.fli
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LOCATIONAL OPTIMIZATION REFINED APPROACH
Only applicable to linear systems
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WEIGHTED MULTICENTER PROBLEM
on not containing 0 (annulus)
Lloyd algorithm as before
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Play movie step5-animation.fli
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RESEARCH DIRECTIONS