CONTROL with LIMITED INFORMATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign
Mar 30, 2015
CONTROL with LIMITED INFORMATION
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
REASONS for SWITCHING
• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above
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
• Theoretical interest
[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 partitioned into quantization regions
QUANTIZATION and ISS
– assume glob. asymp. stable (GAS)
QUANTIZATION and ISS
QUANTIZATION and ISS
no longer GAS
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:
QUANTIZATION and ISS
quantization error
Assume
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
ISS from to ISS from to small-gain conditionProof:
Can recover global asymptotic stability
QUANTIZATION and DELAY
Architecture-independent approach
Based on the work of Teel
Delays possibly large
QUANTIZER DELAY
QUANTIZATION and DELAY
where
Can write
hence
SMALL – GAIN ARGUMENT
if
then we recover ISS w.r.t. [Teel ’98]
Small gain:
Assuming ISS w.r.t. actuator errors:
In time domain:
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:
NETWORKED CONTROL SYSTEMS [Nešić–L]
NCS: Transmit only some variables according to time scheduling protocol
Examples: round-robin, TOD (try-once-discard)
QCS: Transmit quantized versions of all variables
NQCS: Unified framework combining time scheduling and quantization
Basic design/analysis steps:
• Design controller ignoring network effects
• Apply small-gain theorem to compute upper bound on maximal allowed transmission interval (MATI)
• Prove discrete protocol stability via Lyapunov function
ACTIVE PROBING for INFORMATION
dynamic
dynamic
(changes at sampling times)
(time-varying)
PLANT
QUANTIZER
CONTROLLER
Encoder Decoder
very small
NONLINEAR SYSTEMS
sampling times
Example:
Zoom out to get initial bound
Between samplings
NONLINEAR SYSTEMS
• is divided by 3 at the sampling time
Let
Example:
Between samplings
• grows at most by the factor in one period
The norm
on a suitable compact region(dependent on )
Pick small enough s.t.
NONLINEAR SYSTEMS (continued)
• grows at most by the factor in one period
• is divided by 3 at each sampling time
The norm
If this is ISS w.r.t. as before, then
LINEAR SYSTEMS
LINEAR SYSTEMS
[Baillieul, Brockett-L, Hespanha, Nair-Evans, Petersen-Savkin,Tatikonda]
Between sampling times,
where is Hurwitz0
• divided by 3 at each sampling time
• grows at most by in one period
amount of static infoprovided by quantizer
sampling frequency vs. open-loop instability
global quantity:
HYBRID SYSTEMS as FEEDBACK CONNECTIONS
continuous
discrete
[Nešić–L, ’05, ’06]
• 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 to :
• ISS from to :
• (small-gain condition)
SUFFICIENT CONDITIONS for ISS
[Hespanha-L-Teel ’08]
# 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
•
•
SKETCH of PROOF
is nonstrictly decreasing along trajectories
Trajectories along which is constant? None!
GAS follows by LaSalle principle for hybrid systems [Lygeros et al. ’03, Sanfelice-Goebel-Teel ’07]
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