TOWARDS a UNIFIED FRAMEWORK for
NONLINEAR CONTROL with
LIMITED INFORMATION
Daniel Liberzon
Coordinated Science Laboratory and
Dept. of Electrical & Computer Eng.,
Univ. of Illinois at Urbana-Champaign
Workshop dedicated to Roger Brockett’s 70th birthday, Cancun, 12/8/08
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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
• Need to minimize information transmission
• 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
Our goals:
• Handle nonlinear dynamics
• Unified framework for
• quantization
• time delays
• disturbances
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OUR APPROACH
(Goal: treat nonlinear systems; handle quantization, delays, etc.)
• Model these effects as deterministic additive error signals,
• Design a control law ignoring these errors,
• “Certainty equivalence”: apply control
combined with estimation to reduce to zero
Caveat:
This doesn’t work in general, need robustness from controller
Technical tools:
• Input-to-state stability (ISS) • Small-gain theorems
• Lyapunov functions
• Hybrid systems
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QUANTIZATION
Encoder
finite subset
of
Decoder
QUANTIZER
is partitioned into quantization regions
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QUANTIZATION and INPUT-to-STATE STABILITY
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QUANTIZATION and INPUT-to-STATE STABILITY
– assume glob. asymp. stable (GAS)
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QUANTIZATION and INPUT-to-STATE STABILITY
no longer GAS
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QUANTIZATION and INPUT-to-STATE STABILITY
quantization error
Assume
class
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QUANTIZATION and INPUT-to-STATE STABILITY
quantization error
Assume
Solutions that start in
enter
and remain there
This is input-to-state stability (ISS) w.r.t. measurement errors
[Sontag ’89]
In time domain:
class
, e.g.
class
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LINEAR SYSTEMS
9 feedback gain
& Lyapunov function
Quantized control law:
Closed-loop:
(automatically ISS w.r.t. )
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DYNAMIC QUANTIZATION
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DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control:
is discrete state
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DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control:
is discrete state
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DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control:
is discrete state
Zoom out to overcome saturation
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DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control:
is discrete state
After the ultimate bound is achieved,
recompute partition for smaller region
Can recover global asymptotic stability
Proof: ISS from
to
ISS from
to
small-gain condition
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SMALL-GAIN ANALYSIS of HYBRID SYSTEMS
continuous
discrete
Hybrid system is GAS if we have:
• ISS from
to
:
• ISS from
to
:
•
(small-gain condition)
Can use Lyapunov techniques to check this [Nešić–L ’06]
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QUANTIZATION and DELAY
QUANTIZER
DELAY
Architecture-independent approach
Delays possibly large
Based on result of [Teel ’98]
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SMALL – GAIN ARGUMENT
where
hence
ISS w.r.t. actuator errors gives
Small gain: if
then we have ISS w.r.t.
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FINAL RESULT
Need:
small gain true
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FINAL RESULT
Need:
small gain true
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FINAL RESULT
Need:
small gain true
enter
solutions starting in
and remain there
Can use “zooming” to improve convergence
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EXTERNAL DISTURBANCES
[Nešić–L]
State quantization and completely unknown disturbance
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EXTERNAL DISTURBANCES
[Nešić–L]
State quantization and completely unknown disturbance
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EXTERNAL DISTURBANCES
[Nešić–L]
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 gain is nonlinear although plant is linear; cf. [Martins])
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ONGOING RESEARCH
• Quantized output feedback and ISS observer design
• Disturbances and coarse quantizers (with Y. Sharon)
• Modeling uncertainty (with L. Vu, ThB08.2 )
• Coordination with coarse sensing (with S. LaValle and J. Yu,
TuC17.2 )
• Vision-based control (with Y. Ma and Y. Sharon)
http://decision.csl.uiuc.edu/~liberzon
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