CONTROL with LIMITED INFORMATION

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CONTROL with LIMITED INFORMATION
Daniel Liberzon
Coordinated Science Laboratory and Dept. of
Electrical Computer Eng., Univ. of Illinois at
Urbana-Champaign
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REASONS for SWITCHING

• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above

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INFORMATION FLOW in CONTROL SYSTEMS
Plant
Controller
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INFORMATION FLOW in CONTROL SYSTEMS

• Coarse sensing

• Limited communication capacity
• many control loops share network cable or
  wireless medium
• microsystems with many sensors/actuators on one
  chip

• Need to minimize information transmission
  (security)

• Event-driven actuators

• Theoretical interest

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BACKGROUND
Previous work
Brockett, Delchamps, Elia, Mitter, Nair, Savkin,
Tatikonda, Wong,

• Deterministic stochastic models
• Tools from information theory

• Mostly for linear plant dynamics

• Unified framework for
• quantization
• time delays
• disturbances

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OUR APPROACH
(Goal treat nonlinear systems handle
quantization, delays, etc.)
Caveat This doesnt work in general, need
robustness from controller
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QUANTIZATION
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QUANTIZATION and ISS
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QUANTIZATION and ISS
assume glob. asymp. stable (GAS)
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QUANTIZATION and ISS
no longer GAS
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QUANTIZATION and ISS
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QUANTIZATION and ISS
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LINEAR SYSTEMS
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DYNAMIC QUANTIZATION
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DYNAMIC QUANTIZATION
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DYNAMIC QUANTIZATION
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DYNAMIC QUANTIZATION
Zoom out to overcome saturation
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DYNAMIC QUANTIZATION
After ultimate bound is achieved, recompute
partition for smaller region
Can recover global asymptotic stability
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QUANTIZATION and DELAY
Architecture-independent approach
Delays possibly large
Based on the work of Teel
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QUANTIZATION and DELAY
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SMALL GAIN ARGUMENT
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FINAL RESULT
Need
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FINAL RESULT
Need
small gain true
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FINAL RESULT
Need
small gain true
Can use zooming to improve convergence
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State quantization and completely unknown
disturbance
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State quantization and completely unknown
disturbance
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State quantization and completely unknown
disturbance
After zoom-in
Issue disturbance forces the state outside
quantizer range
Must switch repeatedly between zooming-in and
zooming-out
Result for linear plant, can achieve ISS w.r.t.
disturbance
(ISS gains are nonlinear although plant is
linear cf. Martins)
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NETWORKED CONTROL SYSTEMS
NeicL
NCS Transmit only some variables according to
time scheduling protocol
Examples round-robin, TOD (try-once-discard)
QCS Transmit quantized versions of all variables
NQCS Unified framework combining time scheduling
and quantization
Basic design/analysis steps

• Design controller ignoring network effects

• Prove discrete protocol stability via Lyapunov
  function

• Apply small-gain theorem to compute upper bound
  on
• maximal allowed transmission interval (MATI)

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ACTIVE PROBING for INFORMATION
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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
Same condition as before (restrictive, hard to
check)
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LINEAR SYSTEMS
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LINEAR SYSTEMS

• divided by 3 at each sampling time

Baillieul, Brockett-L, Hespanha, Nair-Evans,
Petersen-Savkin,Tatikonda
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HYBRID SYSTEMS as FEEDBACK CONNECTIONS

• Other decompositions possible
• Can also have external signals

NeicL, 05, 06
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SMALL GAIN THEOREM
Small-gain theorem Jiang-Teel-Praly 94 gives
GAS if
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SUFFICIENT CONDITIONS for ISS
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LYAPUNOV BASED SMALL GAIN THEOREM
Hybrid system is GAS if
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SKETCH of PROOF
None!
GAS follows by LaSalle principle for hybrid
systems Lygeros et al. 03, Sanfelice-Goebel-Tee
l 07
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APPLICATION to DYNAMIC QUANTIZATION
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RESEARCH DIRECTIONS

• Quantized output feedback

• Performance-based design

• Disturbances and coarse quantizers (with Y.
  Sharon)

• Modeling uncertainty (with L. Vu)

• Avoiding state estimation (with S. LaValle and
  J. Yu)

• Vision-based control (with Y. Ma and Y. Sharon)