Download Lecture 4 - Daniel Liberzon

CONTROL with LIMITED INFORMATION ;
SWITCHING ADAPTIVE CONTROL
Daniel Liberzon
Coordinated Science Laboratory and
Dept. of Electrical & Computer Eng.,
Univ. of Illinois at Urbana-Champaign
IAAC Workshop, Herzliya, Israel, June 1, 2009
SWITCHING CONTROL
Plant:
Classical continuous
feedback paradigm:
u
P
u
y
y
P
C
u
But logical decisions
are often necessary:
y
P
C1
C2
logic
REASONS for SWITCHING
• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above
REASONS for SWITCHING
• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above
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
finite subset
of
QUANTIZATION and ISS
QUANTIZATION and ISS
– assume glob. asymp. stable (GAS)
QUANTIZATION and ISS
no longer GAS
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
In time domain:
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
Delays possibly large
Based on the work of Teel
DELAY
QUANTIZATION and DELAY
where
Can write
hence
SMALL – GAIN ARGUMENT
Assuming ISS w.r.t. actuator errors:
In time domain:
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])
HYBRID SYSTEMS as FEEDBACK CONNECTIONS
continuous
discrete
• Other decompositions possible
• Can also have external signals
[Nešić–L, ’05, ’06]
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 ’08]
is
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]
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
REASONS for SWITCHING
• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above
MODELING UNCERTAINTY
unmodeled
dynamics
parametric
uncertainty
 0
Also, noise
and disturbance
Adaptive control (continuous tuning)
vs. supervisory control (switching)
EXAMPLE
Scalar system:
, otherwise unknown
(purely parametric uncertainty)
stable 
not implementable
Controller family:
Could also take
controller index set
SUPERVISORY CONTROL ARCHITECTURE
Supervisor
candidate controllers
Controller
Controller
...
Controller
u1
u2

u
Plant
um
...

– switching signal, takes values in
– switching controller
y
TYPES of SUPERVISION
• Prescheduled (prerouted)
• Performance-based (direct)
• Estimator-based (indirect)
TYPES of SUPERVISION
• Prescheduled (prerouted)
• Performance-based (direct)
• Estimator-based (indirect)
OUTLINE
• Basic components of supervisor
• Design objectives and general analysis
• Achieving the design objectives (highlights)
OUTLINE
• Basic components of supervisor
• Design objectives and general analysis
• Achieving the design objectives (highlights)
SUPERVISOR
y
u
y1 
e1

Multi- y2  e2
... 
Estimator y
 ep
p
...  ...
Want
to be small
Then
small indicates
estimation errors:
likely
EXAMPLE
Multi-estimator:
=>
exp fast
EXAMPLE
disturbance
Multi-estimator:
=>
exp fast
STATE SHARING
Bad! Not implementable if
The system
produces the same signals
is infinite
SUPERVISOR
y
u
y1 
e1
1
  e Monitoring
y
2
Multi2
2
... 
Signals
Estimator y
 ep Generator  p
p
...  ...
...
Examples:
EXAMPLE
Multi-estimator:
– can use state sharing
SUPERVISOR
y
u
y1 
e1
1
  e Monitoring
y
 2 Switching
Multi2
2
... 
Signals
Estimator y
Logic
 ep Generator  p
p
...  ...
...

Basic idea:
Justification? Plant
small =>
, controllers:
small => plant likely in
(“certainty equivalence”)
gives stable
closed-loop system
=>
SUPERVISOR
y
u
y1 
e1
1
  e Monitoring
y
 2 Switching
Multi2
2
... 
Signals
Estimator y
Logic
 ep Generator  p
p
...  ...
...

Basic idea:
, controllers:
Justification? Plant
small =>
small => plant likely in
only know converse!
Need:
small =>
gives stable
closed-loop system
=>
gives stable closed-loop system
This is detectability w.r.t.
DETECTABILITY
Linear case:
plant in closed
loop with
view as output
Want this system to be detectable
“output injection”
matrix
is Hurwitz
asympt. stable
SUPERVISOR
y
u
y1 
e1
1

Multi- y2  e2 Monitoring  2 Switching
.
Signals
Estimator ..y 
Logic
 ep Generator  p
p
...  ...
...
We know:

is small
Switching logic (roughly):
This (hopefully) guarantees that
Need:
is small
small => stable closed-loop switched system
This is switched detectability
DETECTABILITY under SWITCHING
Switched system:
plant in closed
loop with
view as output
Want this system to be detectable:
Assumed detectable for each frozen value of
Output injection:
need this to be asympt. stable
Thus
needs to be “non-destabilizing” :
• switching stops in finite time
• slow switching (on the average)
SUMMARY of BASIC PROPERTIES
Multi-estimator:
1. At least one estimation error (
•
when
•
is bounded for bounded
) is small
&
Candidate controllers:
2. For each
, closed-loop system is detectable w.r.t.
Switching logic:
3.
is bounded in terms of the smallest
4. Switched closed-loop system is detectable w.r.t.
provided this is true for every frozen value of
conflicting : for 3, want to switch to
for 4, want to switch slowly or stop
SUMMARY of BASIC PROPERTIES
Multi-estimator:
1. At least one estimation error (
•
when
•
is bounded for bounded
) is small
&
Candidate controllers:
2. For each
, closed-loop system is detectable w.r.t.
Switching logic:
3.
is bounded in terms of the smallest
4. Switched closed-loop system is detectable w.r.t.
provided this is true for every frozen value of
Analysis: 1 + 3 =>
is small
2 + 4 => detectability w.r.t.
=> state is small 
OUTLINE
• Basic components of supervisor
• Design objectives and general analysis
• Achieving the design objectives (highlights)
CANDIDATE CONTROLLERS
fixed
Controller
y
Plant
u
Multiestimator
y
yq

 eq
CANDIDATE CONTROLLERS
fixed
y
Plant
P
Controller
C
u
Multiestimator
E
yq
y 


Linear: overall system is detectable w.r.t.
i. system inside the box is stable
ii. plant is detectable
Need to show:
=>
i
C, E
=>
=>
=>
P,
C,
 eq
if
E
=>
ii
P
CANDIDATE CONTROLLERS
fixed
y
Plant
P
Controller
u
C
Multiestimator
E
yq
y 

 eq

Linear: overall system is detectable w.r.t.
i. system inside the box is stable
ii. plant is detectable
if
Nonlinear: same result holds if stability and detectability are
interpreted in the ISS / OSS sense:
external signal
CANDIDATE CONTROLLERS
fixed
y
Plant
P
Controller
C
u
Multiestimator
E
yq
y 

 eq

Linear: overall system is detectable w.r.t.
i. system inside the box is stable
ii. plant is detectable
if
Nonlinear: same result holds if stability and detectability are
interpreted in the integral-ISS / OSS sense:
SWITCHING LOGIC: DWELL-TIME
– dwell time
– monitoring signals
Initialize
Wait
time units
Find
?
no
Detectability is preserved if
Obtaining a bound on
yes
is large enough 
in terms of
is harder
Not suitable for nonlinear systems (finite escape)
SWITCHING LOGIC: HYSTERESIS
– hysteresis constant
– monitoring signals
Initialize
Find
no
or
(scale-independent)
?
yes
SWITCHING LOGIC: HYSTERESIS
Initialize
Find
no
finite,
This applies to
Linear,
?
yes
bounded => switching stops in finite time
exp fast,
bounded => average dwell time