control with limited information ; switching adaptive control daniel liberzon coordinated science...
TRANSCRIPT
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CONTROL with LIMITED INFORMATION ;
SWITCHING ADAPTIVE CONTROL
Daniel Liberzon
Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign
IAAC Workshop, Herzliya, Israel, June 1, 2009
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SWITCHING CONTROL
Classical continuous feedback paradigm:
u yP
C
u yPPlant:
But logical decisions are often necessary:
u y
C1
C2
l o g i c
P
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REASONS for SWITCHING
• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above
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REASONS for SWITCHING
• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above
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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 partitioned into quantization regions
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QUANTIZATION and ISS
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– assume glob. asymp. stable (GAS)
QUANTIZATION and ISS
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QUANTIZATION and ISS
no longer GAS
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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:
QUANTIZATION and ISS
quantization error
Assume
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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
ISS from to ISS from to small-gain conditionProof:
Can recover global asymptotic stability
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QUANTIZATION and DELAY
Architecture-independent approach
Based on the work of Teel
Delays possibly large
QUANTIZER DELAY
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QUANTIZATION and DELAY
where
hence
Can write
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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:
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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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HYBRID SYSTEMS as FEEDBACK CONNECTIONS
continuous
discrete
[Nešić–L, ’05, ’06]
• Other decompositions possible
• Can also have external signals
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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 ’08]
# 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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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]
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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
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REASONS for SWITCHING
• Nature of the control problem
• Sensor or actuator limitations
• Large modeling uncertainty
• Combinations of the above
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MODELING UNCERTAINTY
Adaptive control (continuous tuning)
vs. supervisory control (switching)
unmodeleddynamics
0parametricuncertainty
Also, noise and disturbance
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EXAMPLE
Could also take
controller index set
Scalar system:
, otherwise unknown
(purely parametric uncertainty)
Controller family:
stable
not implementable
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SUPERVISORY CONTROL ARCHITECTURE
Plant
Supervisor
Controller
Controller
Controller u1
u2
um
yu...
candidate controllers
...
– switching controller
– switching signal, takes values in
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TYPES of SUPERVISION
• Prescheduled (prerouted)
• Performance-based (direct)
• Estimator-based (indirect)
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TYPES of SUPERVISION
• Prescheduled (prerouted)
• Performance-based (direct)
• Estimator-based (indirect)
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OUTLINE
• Basic components of supervisor
• Design objectives and general analysis
• Achieving the design objectives (highlights)
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OUTLINE
• Basic components of supervisor
• Design objectives and general analysis
• Achieving the design objectives (highlights)
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SUPERVISOR
Multi-Estimator
y1
y2
u
y
...
...yp
estimation errors:
ep
e2
e1
...
Want to be small
Then small indicates likely
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EXAMPLE
Multi-estimator:
exp fast=>
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EXAMPLE
Multi-estimator:disturbance
exp fast=>
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STATE SHARING
Bad! Not implementable if is infinite
The system
produces the same signals
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SUPERVISOR
...Multi-
Estimator
y1
y2
yp ep
e2
e1
u
y
... ...
MonitoringSignals
Generator
1
2
p...
Examples:
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EXAMPLE
Multi-estimator:
– can use state sharing
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SUPERVISOR
SwitchingLogic
...Multi-
Estimator
y1
y2
yp ep
e2
e1
u
y
... ...
MonitoringSignals
Generator
1
2
p...
Basic idea:
Justification?
small => small => plant likely in => gives stable
closed-loop system
(“certainty equivalence”)
Plant , controllers:
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SUPERVISOR
SwitchingLogic
...Multi-
Estimator
y1
y2
yp ep
e2
e1
u
y
... ...
MonitoringSignals
Generator
1
2
p...
Basic idea:
Justification?
small => small => plant likely in => gives stable
closed-loop systemonly know converse!
Need: small => gives stable closed-loop system
This is detectability w.r.t.
Plant , controllers:
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DETECTABILITY
Want this system to be detectable
“output injection” matrix
asympt. stable
view as output
is Hurwitz
Linear case: plant in closedloop with
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SUPERVISOR
SwitchingLogic
...Multi-
Estimator
y1
y2
yp ep
e2
e1
u
y
... ...
MonitoringSignals
Generator
1
2
p...
Switching logic (roughly):
This (hopefully) guarantees that is small
Need: small => stable closed-loop switched system
We know: is small
This is switched detectability
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DETECTABILITY under SWITCHING
need this to be asympt. stable
plant in closedloop with
view as output
Assumed detectable for each frozen value of
Output injection:
• slow switching (on the average)
• switching stops in finite time
Thus needs to be “non-destabilizing”:
Switched system:
Want this system to be detectable:
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SUMMARY of BASIC PROPERTIESMulti-estimator:
1. At least one estimation error ( ) is small
Candidate controllers:
3. is bounded in terms of the smallest
: for 3, want to switch to for 4, want to switch slowly or stop
conflicting
• when • is bounded for bounded &
Switching logic:
2. For each , closed-loop system is detectable w.r.t.
4. Switched closed-loop system is detectable w.r.t. provided this is true for every frozen value of
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SUMMARY of BASIC PROPERTIES
Analysis: 1 + 3 => is small2 + 4 => detectability w.r.t.
=> state is small
Switching logic:
Multi-estimator:
Candidate controllers:
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
2. For each , closed-loop system is detectable w.r.t.
1. At least one estimation error ( ) is small
• when • is bounded for bounded &
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OUTLINE
• Basic components of supervisor
• Design objectives and general analysis
• Achieving the design objectives (highlights)
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Controller
Plant
Multi-estimator
qe
y
uqy
y
fixed
CANDIDATE CONTROLLERS
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CANDIDATE CONTROLLERS
Linear: overall system is detectable w.r.t. if
i. system inside the box is stable
ii. plant is detectable
fixed
Plant
Multi-estimator
qe
y
uqy
y
Controller
P
C
E
Need to show: => P C E , ,
=> EC,i
=> => Pii
=>
![Page 61: CONTROL with LIMITED INFORMATION ; SWITCHING ADAPTIVE CONTROL Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d8d5503460f94a75ce6/html5/thumbnails/61.jpg)
CANDIDATE CONTROLLERS
Linear: overall system is detectable w.r.t. if i. system inside the box is stableii. plant is detectable
fixed
Plant
Multi-estimator
qe
y
uqy
y
Controller
P
C
E
Nonlinear: same result holds if stability and detectability are
interpreted in the ISS / OSS sense: external signal
![Page 62: CONTROL with LIMITED INFORMATION ; SWITCHING ADAPTIVE CONTROL Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d8d5503460f94a75ce6/html5/thumbnails/62.jpg)
CANDIDATE CONTROLLERS
Linear: overall system is detectable w.r.t. if i. system inside the box is stableii. plant is detectable
fixed
Plant
Multi-estimator
qe
y
uqy
y
Controller
P
C
E
Nonlinear: same result holds if stability and detectability are
interpreted in the integral-ISS / OSS sense:
![Page 63: CONTROL with LIMITED INFORMATION ; SWITCHING ADAPTIVE CONTROL Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d8d5503460f94a75ce6/html5/thumbnails/63.jpg)
SWITCHING LOGIC: DWELL-TIME
Obtaining a bound on in terms of is harder
Not suitable for nonlinear systems (finite escape)
Initialize
Find
no ? yes
– monitoring signals – dwell time
Wait time units
Detectability is preserved if is large enough
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SWITCHING LOGIC: HYSTERESIS
Initialize
Find
no yes
– monitoring signals – hysteresis constant
?
or
(scale-independent)
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SWITCHING LOGIC: HYSTERESIS
This applies to exp fast,
finite, bounded switching stops in finite time=>
Initialize
Find
no yes?
Linear, bounded average dwell time=>