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Hybrid Systems Controller Synthesis Examples
EE291E Tomlin/Sastry
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Example 1: Aircraft Collision Avoidance
Two identical aircraft at fixed altitude & speed:
‘evader’ (control) ‘pursuer’ (disturbance)
x
y
uv
d
v
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Continuous Reachable Set
x
y
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Collision Avoidance Filter
Simple demonstration– Pursuer: turn to head toward evader– Evader: turn to head right
pursuer
safety filter’s input modification
pursuer’s inputevader’s desired input
evader
evader’s actual input
unsafe setcollision set
Movies…
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Collision Avoidance Control
http://www.cs.ubc.ca/~mitchell/ToolboxLS/
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Overapproximating Reachable Sets
[Khrustalev, Varaiya, Kurzhanski]
Overapproximative reachable set:
Exact:
Approximate:
~1 sec on 700MHz Pentium III (vs 4 minutes for exact)
• Polytopic overapproximations for nonlinear games• Subsystem level set functions• “Norm-like” functions with identical strategies to exact
[Hwang, Stipanović, Tomlin]
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Can separation assurance be automated?
Requires provably safe protocols for aircraft interaction
Must take into account:• Uncertainties in sensed information, in actions of the other vehicle• Potential loss of communication• Intent, or non-intent
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unsafe set with choiceto maneuver or not?
Example 2: Protocol design
unsafe set with maneuver
unsafe set without maneuver
?
unsafe
safe
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Protocol Safety Analysis• Ability to choose maneuver start time further reduces unsafe set
safe without switchunsafe to switch
safe with switch
unsafe with or without switch
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Implementation: a finite automaton• It can be easier to analyze discrete systems than continuous:
use reachable set information to abstract away continuous details
q1
safe at presentwill become unsafe
unsafe to 1
q5
safe at presentalways safesafe to 1
q3
safe at presentwill become unsafe
safe to 1
q4
safe at presentalways safeunsafe to 1
q2
unsafe at presentwill become unsafe
unsafe to 1
qs
SAFE
qu
UNSAFE
forced transitioncontrolled transition (1)
q1
q5
q3
qu
q4 q2
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Example 3: Closely Spaced Approaches
Photo courtesy of Sharon Houck
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Example 3: Closely Spaced Approaches
evader
EEM Maneuver 1: accelerateEEM Maneuver 2: turn 45 deg, accelerate
EEM Maneuver 3: turn 60 deg
[Rodney Teo]
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Sample Trajectories
Segment 1
Segment 2
Segment 3
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Dragonfly 3Dragonfly 2
Ground Station
Tested on the Stanford DragonFly UAVs
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EEM alert
Sep
arat
ion
dist
anc
e (m
)N
orth
(m
)
East (m)
time (s)
Above threshold
Accelerate and turn EEM
Put video here
Tested at Moffett Federal Airfield
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EEM alert
Sep
arat
ion
dist
anc
e (m
)N
orth
(m
)
East (m)
time (s)
Above threshold
Put video here
Coast and turn EEM
Tested at Moffett Federal Airfield
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Tested at Edwards Air Force Base
T-33 Cockpit
[DARPA/Boeing SEC Final Demonstration:F-15 (blunderer), T-33 (evader)]
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Photo courtesy of Sharon Houck;Tests conducted with Chad Jennings
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Implementation: Display design courtesy of
Chad Jennings, Andy Barrows, David Powell
R. Teo’s Blunder Zone is shown by the yellow contour
Red Zone in the green tunnel is the intersection of the BZ with approach path.
The Red Zone corresponds to an assumed 2 second pilot delay. The Yellow Zone corresponds to an 8 second pilot delay
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R. Teo’s Blunder Zone is shown by the yellow contour
Red Zone in the green tunnel is the intersection of the BZ with approach path.
The Red Zone corresponds to an assumed 2 second pilot delay. The Yellow Zone corresponds to an 8 second pilot delay
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Map View showing a blunder
The BZ calculations are performed in real time (40Hz) so that the contour is updated with each video frame.
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Map View with Color Strips
The pilots only need to know which portion of their tunnel is off limits. The color strips are more efficient method of communicating the relevant extent of the Blunder zone
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Aircraft must stay within safe flight envelope during landing:– Bounds on velocity ( ), flight path angle (), height ( )– Control over engine thrust ( ), angle of attack (), flap settings– Model flap settings as discrete modes – Terms in continuous dynamics depend on flap setting
Example 4: Aircraft Autolander
inertial frame
wind frame
body frame
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Autolander: Synthesizing Control
For states at the boundary of the safe set, results of reach-avoid computation determine– What continuous inputs (if any) maintain safety– What discrete jumps (if any) are safe to perform– Level set values and gradients provide all relevant data
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Application to Autoland Interface• Controllable flight envelopes for landing and Take Off / Go
Around (TOGA) maneuvers may not be the same• Pilot’s cockpit display may not contain sufficient information to
distinguish whether TOGA can be initiated
flareflaps extendedminimum thrust
rolloutflaps extendedreverse thrust
slow TOGAflaps extended
maximum thrust
TOGAflaps retracted
maximum thrust
flareflaps extendedminimum thrust
rolloutflaps extendedreverse thrust
TOGAflaps retracted
maximum thrust
revised interface
existing interface
controllable flare envelope
controllable TOGA envelopeintersection