NONLINEAR HYBRID CONTROL with
LIMITED INFORMATION
Daniel Liberzon
Coordinated Science Laboratory and
Dept. of Electrical & Computer Eng.,
Univ. of Illinois at Urbana-Champaign
Paris, France, April 2008
INFORMATION FLOW in CONTROL SYSTEMS
Plant
Controller
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
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
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
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
QUANTIZATION
Encoder
Decoder
QUANTIZER
is partitioned into quantization regions
Assume
such that
is the range,
For
is the quantization error bound
, the quantizer saturates
finite subset
of
QUANTIZATION and ISS
QUANTIZATION and ISS
quantization error
Assume
class
QUANTIZATION and ISS
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
; cf. linear:
class
LINEAR SYSTEMS
9 feedback gain
& Lyapunov function
Quantized control law:
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
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
Can recover global asymptotic stability
Proof: ISS from
to
ISS from
to
small-gain condition
QUANTIZATION and DELAY
QUANTIZER
Architecture-independent approach
Based on the work of Teel
DELAY
QUANTIZATION and DELAY
where
Assuming ISS w.r.t. actuator errors:
In time domain:
SMALL – GAIN ARGUMENT
hence
ISS property becomes
Small gain: if
then we recover ISS w.r.t.
[Teel ’98]
FINAL RESULT
Need:
small gain true
FINAL RESULT
Need:
small gain true
FINAL RESULT
Need:
small gain true
enter
solutions starting in
and remain there
Can use “zooming” to improve convergence
EXTERNAL DISTURBANCES
[Nešić–L]
State quantization and completely unknown disturbance
EXTERNAL DISTURBANCES
[Nešić–L]
State quantization and completely unknown disturbance
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 gains are nonlinear although plant is linear [cf. Martins])
STABILITY ANALYSIS of
HYBRID SYSTEMS via
SMALL-GAIN THEOREMS
Daniel Liberzon
Univ. of Illinois at Urbana-Champaign, USA
Dragan Nešić
University of Melbourne, Australia
HYBRID SYSTEMS as FEEDBACK CONNECTIONS
continuous
discrete
See paper for more general setting
• Other decompositions possible
• Can also have external signals
SMALL – GAIN THEOREM
Small-gain theorem [Jiang-Teel-Praly ’94] gives GAS if:
• Input-to-state stability (ISS) from
• ISS from
•
to
to
:
:
(small-gain condition)
SUFFICIENT CONDITIONS for ISS
• ISS from
to
• ISS from
to
if
ISS-Lyapunov function [Sontag ’89]:
if:
and
# of discrete events on
[Hespanha-L-Teel]
is
LYAPUNOV – BASED SMALL – GAIN THEOREM
Hybrid system is GAS if:
•
•
and # of discrete events on
•
is
APPLICATION to DYNAMIC QUANTIZATION
ISS from
to
with some linear gain
quantization
error
Zoom in:
where
ISS from
to
with gain
small-gain condition!
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)
http://decision.csl.uiuc.edu/~liberzon