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TOWARDS a UNIFIED FRAMEWORK for
NONLINEAR CONTROL with
LIMITED INFORMATION
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
Workshop dedicated to Roger Brockett’s 70th birthday, Cancun, 12/8/081 of 14
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Plant
Controller
INFORMATION FLOW in CONTROL SYSTEMS
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INFORMATION FLOW in CONTROL SYSTEMS
• Limited communication capacity
• Need to minimize information transmission
• Event-driven actuators
• Coarse sensing
• Theoretical interest2 of 14
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[Brockett, Delchamps, Elia, Mitter, Nair, Savkin, Tatikonda, Wong,…]
• Deterministic & stochastic models
• Tools from information theory
• Mostly for linear plant dynamics
BACKGROUND
Previous work:
• Unified framework for
• quantization
• time delays
• disturbances
Our goals:
• Handle nonlinear dynamics
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Caveat:
This doesn’t work in general, need robustness from controller
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
Technical tools:
• Input-to-state stability (ISS)
• Lyapunov functions
• Small-gain theorems
• Hybrid systems
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QUANTIZATION
Encoder Decoder
QUANTIZER
finite subset
of
is partitioned into quantization regions
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QUANTIZATION and INPUT-to-STATE STABILITY
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– assume glob. asymp. stable (GAS)
QUANTIZATION and INPUT-to-STATE STABILITY
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QUANTIZATION and INPUT-to-STATE STABILITY
no longer GAS
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quantization error
Assume
class
QUANTIZATION and INPUT-to-STATE STABILITY
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Solutions that start in
enter and remain there
This is input-to-state stability (ISS) w.r.t. measurement errors
In time domain: [Sontag ’89]
quantization error
Assume
class
QUANTIZATION and INPUT-to-STATE STABILITY
class , e.g. 6 of 14
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LINEAR SYSTEMS
Quantized control law:
9 feedback gain & Lyapunov function
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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Zoom out to overcome saturation
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
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After the ultimate bound is achieved,recompute partition for smaller region
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
Can recover global asymptotic stability
ISS from to ISS from to small-gain conditionProof:8 of 14
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SMALL-GAIN ANALYSIS of HYBRID SYSTEMS
continuous
discrete
Hybrid system is GAS if we have:
• (small-gain condition)
Can use Lyapunov techniques to check this [Nešić–L ’06]
• ISS from to :
• ISS from to :
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QUANTIZATION and DELAY
Architecture-independent approach
Based on result of [Teel ’98]
Delays possibly large
QUANTIZER DELAY
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SMALL – GAIN ARGUMENT
ISS w.r.t. actuator errors gives
where
if then we have ISS w.r.t. Small gain:11 of 14
hence
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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
solutions starting in
enter and remain there
Can use “zooming” to improve convergence
Need:
small gain true
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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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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])
EXTERNAL DISTURBANCES [Nešić–L]
State quantization and completely unknown disturbance
After zoom-in:
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ONGOING RESEARCH
• Modeling uncertainty (with L. Vu, ThB08.2 )
• Disturbances and coarse quantizers (with Y. Sharon)
• Coordination with coarse sensing (with S. LaValle and J. Yu, TuC17.2 )
• Quantized output feedback and ISS observer design
• Vision-based control (with Y. Ma and Y. Sharon)
http://decision.csl.uiuc.edu/~liberzon
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