nonlinear hybrid control with limited information daniel liberzon coordinated science laboratory and...
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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
Plant
Controller
INFORMATION FLOW in CONTROL SYSTEMS
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
[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
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
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
QUANTIZATION and ISS
QUANTIZATION and ISS
quantization error
Assume
class
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
LINEAR SYSTEMS
Quantized control law:
9 feedback gain & Lyapunov function
Closed-loop:
(automatically ISS w.r.t. )
DYNAMIC QUANTIZATION
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
Zoom out to overcome saturation
DYNAMIC QUANTIZATION
– zooming variable
Hybrid quantized control: is discrete state
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:
QUANTIZATION and DELAY
QUANTIZER DELAY
Architecture-independent approach
Based on the work of Teel
QUANTIZATION and DELAY
Assuming ISS w.r.t. actuator errors:
In time domain:
where
SMALL – GAIN ARGUMENT
hence
ISS property becomes
if
then we recover ISS w.r.t. [Teel ’98]
Small gain:
FINAL RESULT
Need:
small gain true
FINAL RESULT
Need:
small gain true
FINAL RESULT
solutions starting in
enter and remain there
Can use “zooming” to improve convergence
Need:
small gain true
EXTERNAL DISTURBANCES [Nešić–L]
State quantization and completely unknown disturbance
EXTERNAL DISTURBANCES [Nešić–L]
State quantization and completely unknown disturbance
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:
STABILITY ANALYSIS of
HYBRID SYSTEMS via
SMALL-GAIN THEOREMS
Dragan NešićUniversity of Melbourne, Australia
Daniel LiberzonUniv. of Illinois at Urbana-Champaign, USA
HYBRID SYSTEMS as FEEDBACK CONNECTIONS
continuous
discrete
• Other decompositions possible
• Can also have external signals
See paper for more general setting
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)
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]:
LYAPUNOV – BASED SMALL – GAIN THEOREM
Hybrid system is GAS if:
•
and # of discrete events on is
•
•
quantization error
Zoom in:
where
ISS from to with gain
small-gain condition!
ISS from to with some linear gain
APPLICATION to DYNAMIC QUANTIZATION
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