Date post: | 22-Jan-2016 |
Category: |
Documents |
Upload: | paul-fowler |
View: | 220 times |
Download: | 0 times |
Hybrid Systems Controller Synthesis Examples
EE291E Tomlin
Backwards Reachable Set
All states for which, for all possible control actions, there is a disturbance action which can drive the
system state into a region G(0) in time t
Backwards Reachable Set
Reachability as game: disturbance attempts to force system into unsafe region, control attempts to stay safe
Reachable Set Propagation
[Mitchell, Bayen, Tomlin 2005]
Theorem [Computing ]:
where is the unique Crandall-Evans-Lions viscosity solution to:
Backwards Reachable Set: Safety
unsafe
Backwards Reachable Set
On boundary, apply control to stay out of red
In red, system may become
unsafe
In blue, system will stay safe
Safety Property can be encoded as a condition on the system’s reachable set of states
Example 1: Aircraft Collision Avoidance
Two identical aircraft at fixed altitude & speed:
‘evader’ (control) ‘pursuer’ (disturbance)
x
y
uv
d
v
Continuous Reachable Set
x
y
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…
Collision Avoidance Control
http://www.cs.ubc.ca/~mitchell/ToolboxLS/
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]
1 2 3 K
modes
1
2
3
n
itera
tion
s
unsafeunsafe
safe
Computing Reach Sets for Hybrid Systems
Reach Sets: uncontrollable predecessor
1 2 3 K
modes
1
2
3
n
itera
tion
s
uncontrolledtransition unsafe
“safe”
Reach Sets: controllable predecessor
1 2 3 K
modes
1
2
3
n
itera
tion
s
safe
controlled transition
“safe”
Reach Sets: Variational Inequality
1 2 3 K
modes
1
2
3
n
itera
tion
s
States which reach G without hitting E first:
where
subject to
Reach Sets: Iterate
1 2 3 K
modes
1
2
3
n
itera
tion
s
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
unsafe set with choiceto maneuver or not?
Example 2: Protocol design
unsafe set with maneuver
unsafe set without maneuver
?
unsafe
safe
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
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
San Francisco Airport750 ft separation
Example 2: Closely Spaced Parallel Approaches
Example 3: Closely Spaced Approaches
evader
EEM Maneuver 1: accelerateEEM Maneuver 2: turn 45 deg, accelerate
EEM Maneuver 3: turn 60 deg
[Rodney Teo]
Sample Trajectories
Segment 1
Segment 2
Segment 3
Dragonfly 3Dragonfly 2
Ground Station
Tested on the Stanford DragonFly UAVs
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
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
Tested at Edwards Air Force Base
T-33 Cockpit
[DARPA/Boeing SEC Final Demonstration:F-15 (blunderer), T-33 (evader)]
Photo courtesy of Sharon Houck;Tests conducted with Chad Jennings
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
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
Map View showing a blunder
The BZ calculations are performed in real time (40Hz) so that the contour is updated with each video frame.
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
Experimental Platform: STARMAC
The Stanford Testbed of Autonomous Rotorcraft for Multi-Agent Control
Example 4: Collision Avoidance
Pilots instructed to attempt to collide vehicles
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 5: Aircraft Autolander
inertial frame
wind frame
body frame
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
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
Aircraft Simulator Tests• Setup
– Commercial flight simulator, B767 pilot– Digital video of primary flight display
• Maneuver– Go-around at low speed, high descent rate
• Goal– Determine whether problematic behavior predicted by our
model is possible in aircraft flight simulator
Aircraft Simulator Results
Produced unexpected behaviorNon-standard procedure; Unable to duplicate
Validated types of problems addressed by this method
Backwards Reachable Set: Safety
unsafe
Backwards Reachable Set
On boundary, apply control to stay out of red
In red, system may become
unsafe
In blue, system will stay safe
Safety Property can be encoded as a condition on the system’s reachable set of states
Backwards Reachable Set: Capture
desired
Backwards Reachable Set
Capture property can also be encoded as a condition on the system’s reachable set of states
Maneuver sequencing, “Reachavoid”
Target Set
Maneuver sequencing is accomplished by stringing together capture sets, starting from the target set and working backwards
Avoid sets can be combined with capture sets to guarantee safety
Unsafe Set
Example 5: Quadrotor Back-Flip
• Divide flip into three modes• Difficult problem:
– Hitting some target sets while avoiding some unsafe sets
• Solution:– Analyze rotational dynamics and vertical dynamics separately
ImpulseDriftRecovery
Back-flip: Method (1)
Recovery Drift Impulse• Identify target region in
rotational state space for each mode
• Use reachable sets to calculate capture basin for each target– Dynamic game
formulation accounts for worst-case disturbances
• Verify that target of each mode is contained by capture basin of next mode
Back-flip: Method (2)• Identify unsafe region in
vertical state space for final mode
• Use reachable sets to propagate unsafe set for each mode– Dynamic game
formulation accounts for worst-case disturbances
• Verify that control keeps state out of unsafe set
Back-Flip: Results
Back-Flip: Results• Assumptions Validated
• Safety Guaranteed
• Reachability Demonstrated
18 20 22 24 26 28 30 32 34-15
-10
-5
0
5
10
time (seconds)
Pitc
h (
de
gre
es)
Pitch vs Time
Ground
Climb
ImpulseDrift
Recovery
Example 6: Automated aerial refueling
Desired Target Set
Capture Set and Unsafe Set Computation Result
Example 7: Teaming up humans and robots
http://www.goforyourlife.vic.gov.au/hav/articles.nsf/pages/Capture_the_Flag
Multiple playersAdversarial gameLimited InformationMultiple objectives
Quadrotor UAVs
GPS-enabled Smartphones
3G Wireless
Computing
Flag Capture Only
Flag Return Only
Full Game
“Capture the Flag”
Action Support For Human Agents
Undergraduate Team
Scott HoagAndrew Sy
The computed solution can be used to guide and assist human agents.
attacker
defender
Supporting Complex Actions
Reachable sets also assist and enable more complex actions and strategic decision making.
attacker
defender
In this case reachability information helps the attacker mislead the defender to win from a losing initial configuration.
Reachability-Guided UAV Search
UAV
UAVVisibility Attacker
Defender
Attacker Goal
Defender Winning Region
Attacker Winning Region
AttackerVisibility
Possible Defender Locations