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![Page 1: NONLINEAR HYBRID CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois.](https://reader034.fdocuments.in/reader034/viewer/2022051614/55195da1550346a5698b4748/html5/thumbnails/1.jpg)
NONLINEAR HYBRID CONTROL with
LIMITED INFORMATION
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
Paris, France, April 2008
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Plant
Controller
INFORMATION FLOW in CONTROL SYSTEMS
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INFORMATION FLOW in CONTROL SYSTEMS
• 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
• Coarse sensing
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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 via deterministic 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 the range, is the quantization error bound
For , the quantizer saturates
Assume such that
is partitioned into quantization regions
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QUANTIZATION and ISS
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QUANTIZATION and ISS
quantization error
Assume
class
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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 and ISS
quantization error
Assume
class ; cf. linear:
class
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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 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:
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QUANTIZATION and DELAY
QUANTIZER DELAY
Architecture-independent approach
Based on the work of Teel
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QUANTIZATION and DELAY
Assuming ISS w.r.t. actuator errors:
In time domain:
where
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SMALL – GAIN ARGUMENT
hence
ISS property becomes
if
then we recover ISS w.r.t. [Teel ’98]
Small gain:
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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 gains are nonlinear although plant is linear [cf. Martins])
EXTERNAL DISTURBANCES [Nešić–L]
State quantization and completely unknown disturbance
After zoom-in:
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STABILITY ANALYSIS of
HYBRID SYSTEMS via
SMALL-GAIN THEOREMS
Dragan NešićUniversity of Melbourne, Australia
Daniel LiberzonUniv. of Illinois at Urbana-Champaign, USA
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HYBRID SYSTEMS as FEEDBACK CONNECTIONS
continuous
discrete
• Other decompositions possible
• Can also have external signals
See paper for more general setting
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SMALL – GAIN THEOREM
Small-gain theorem [Jiang-Teel-Praly ’94] gives GAS if:
• Input-to-state stability (ISS) from to :
• ISS from to :
• (small-gain condition)
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SUFFICIENT CONDITIONS for ISS
[Hespanha-L-Teel]
# of discrete events on is
• ISS from to if:
and
• ISS from to if ISS-Lyapunov function [Sontag ’89]:
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LYAPUNOV – BASED SMALL – GAIN THEOREM
Hybrid system is GAS if:
•
and # of discrete events on is
•
•
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quantization error
Zoom in:
where
ISS from to with gain
small-gain condition!
ISS from to with some linear gain
APPLICATION to DYNAMIC QUANTIZATION
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RESEARCH DIRECTIONS
• Modeling uncertainty (with L. Vu)
• Disturbances and coarse quantizers (with Y. Sharon)
• Avoiding state estimation (with S. LaValle and J. Yu)
• Quantized output feedback
• Performance-based design
• Vision-based control (with Y. Ma and Y. Sharon)
http://decision.csl.uiuc.edu/~liberzon