CONTROL of NONLINEAR SYSTEMS with LIMITED INFORMATION

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CONTROL of NONLINEAR SYSTEMS with LIMITED
INFORMATION
Daniel Liberzon
Coordinated Science Laboratory and Dept. of
Electrical Computer Eng., Univ. of Illinois at
Urbana-Champaign
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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?

• What can we do with very coarse quantization?

• What are the difficulties for nonlinear systems?

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BASIC QUESTIONS

• What can we say about a given quantized system?

• How can we design the best quantizer for
  stability?

• What can we do with very coarse quantization?

• What are the difficulties for nonlinear systems?

• What are the difficulties for nonlinear systems?

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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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INPUT QUANTIZATION
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BASIC QUESTIONS

• What can we say about a given quantized system?

• How can we design the best quantizer for
  stability?

• What can we do with very coarse quantization?

• What are the difficulties for nonlinear systems?

• What are the difficulties for nonlinear systems?

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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
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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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DYNAMIC QUANTIZATION
After ultimate bound is achieved, recompute
partition for smaller region
Zoom out to overcome saturation
Can recover global asymptotic stability
(also applies to input and output quantization)
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BASIC QUESTIONS

• What can we say about a given quantized system?

• How can we design the best quantizer for
  stability?

• What can we do with very coarse quantization?

• What are the difficulties for nonlinear systems?

• What are the difficulties for nonlinear systems?

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ACTIVE PROBING for INFORMATION
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LINEAR SYSTEMS
(Baillieul, Brockett-L, Hespanha et. al.,
Nair-Evans, Petersen-Savkin, Tatikonda, and
others)
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LINEAR SYSTEMS
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LINEAR SYSTEMS
Example

• is divided by 3 at the sampling time

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LINEAR SYSTEMS (continued)
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NONLINEAR SYSTEMS
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NONLINEAR SYSTEMS

• is divided by 3 at the sampling time

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NONLINEAR SYSTEMS (continued)
The norm

• grows at most by the factor in
  one period

• is divided by 3 at each sampling time

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ROBUSTNESS of the CONTROLLER
ISS w.r.t. measurement errors quite
restrictive...
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RESEARCH DIRECTIONS
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REFERENCES
Brockett L, 2000 (IEEE TAC) Bullo L, 2003, L
Hespanha, 2004 (http//decision.csl.uiuc.edu/li
berzon)