An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania...
-
date post
19-Dec-2015 -
Category
Documents
-
view
216 -
download
0
Transcript of An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania...
![Page 1: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/1.jpg)
An Introduction to the Soft Walls Project
Adam CataldoProf. Edward Lee
University of PennsylvaniaDec 18, 2003Philadelphia, PA
![Page 2: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/2.jpg)
Outline
• The Soft Walls system• Objections• Control system design• Current Research• Conclusions
![Page 3: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/3.jpg)
Introduction
• On-board database with “no-fly-zones”• Enforce no-fly zones using on-board
avionics• Non-networked, non-hackable
![Page 4: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/4.jpg)
Autonomous control
Pilot
Aircraft
Autonomouscontroller
![Page 5: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/5.jpg)
Softwalls is not autonomous control
Pilot
Aircraft
Softwalls
+
bias pilotcontrol
![Page 6: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/6.jpg)
Relation to Unmanned Aircraft
• Not an unmanned strategy– pilot authority
• Collision avoidance
![Page 7: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/7.jpg)
A deadly weapon?
• Project started September 11, 2001
![Page 8: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/8.jpg)
Design Objectives
Maximize Pilot Authority!
![Page 9: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/9.jpg)
Unsaturated Control
No-flyzone
Control applied
Pilot lets off controls
Pilot turns away from no-fly zone
Pilot tries to fly into no-fly
zone
![Page 10: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/10.jpg)
Objections
• Reducing pilot control is dangerous– reduces ability to respond to emergencies
![Page 11: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/11.jpg)
There is No Emergency That Justifies Landing Here
![Page 12: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/12.jpg)
Objections
• Reducing pilot control is dangerous– reduces ability to respond to emergencies
• There is no override– switch in the cockpit
![Page 13: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/13.jpg)
Hardwall
![Page 14: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/14.jpg)
Objections
• Reducing pilot control is dangerous– reduces ability to respond to emergencies
• There is no override– switch in the cockpit
• Localization technology could fail– GPS can be jammed
![Page 15: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/15.jpg)
Localization Backup
• Inertial navigation• Integrator drift
limits accuracy range
![Page 16: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/16.jpg)
Objections
• Reducing pilot control is dangerous– reduces ability to respond to emergencies
• There is no override– switch in the cockpit
• Localization technology could fail– GPS can be jammed
• Deployment could be costly– Software certification? Retrofit older
aircraft?
![Page 17: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/17.jpg)
Deployment
• Fly-by-wire aircraft– a software change
• Older aircraft– autopilot level
• Phase in– prioritize airports
![Page 18: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/18.jpg)
Objections
• Reducing pilot control is dangerous– reduces ability to respond to emergencies
• There is no override– switch in the cockpit
• Localization technology could fail– GPS can be jammed
• Deployment could be costly– how to retrofit older aircraft?
• Complexity– software certification
![Page 19: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/19.jpg)
Not Like Air Traffic Control
• Much Simpler• No need for
air traffic controller certification
![Page 20: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/20.jpg)
Objections
• Reducing pilot control is dangerous– reduces ability to respond to emergencies
• There is no override– switch in the cockpit
• Localization technology could fail– GPS can be jammed
• Deployment could be costly– how to retrofit older aircraft?
• Deployment could take too long– software certification
• Fully automatic flight control is possible– throw a switch on the ground, take over plane
![Page 21: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/21.jpg)
Potential Problems with Ground Control
• Human-in-the-loop delay on the ground– authorization for takeover– delay recognizing the threat
• Security problem on the ground– hijacking from the ground?– takeover of entire fleet at once?– coup d’etat?
• Requires radio communication– hackable– jammable
![Page 22: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/22.jpg)
Here’s How It Works
![Page 23: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/23.jpg)
What We Want to Compute
No-fly zone
Backwards reachable set
States that canreach the no-flyzone even with Soft Walls controlllerCan prevent aircraft from
entering no-fly zone
![Page 24: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/24.jpg)
What We Create
• The terminal payoff function l:X -> Reals
• The further from the no-fly zone, the higher the terminal payoff
payoff
northwardposition
eastwardpositionno-fly zone
(constant over heading angle)
-
+
![Page 25: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/25.jpg)
What We Compute
No-fly zone
terminal payoff
Backwards Reachable Set
optimal payoff
![Page 26: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/26.jpg)
Our Control Input
Backwards Reachable Set
optimal control at boundary
dampen optimal control away from boundary
State Space
optimal payoff function
![Page 27: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/27.jpg)
How we computing the optimal payoff (analytically)
• We solve this equation for J*: Reals^n x (-∞, 0] -> Reals
• J* is the viscosity solution of this equation• J* converges pointwise to the optimal
payoff as T->∞• (Tomlin, Lygeros, Pappas, Sastry)
)()0,(
0),,(),(maxmin,0min),( **
ylyJ
evyfTyJTyJT y
UvDe
dynamicsspatial gradient
time derivativeterminal payoff
![Page 28: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/28.jpg)
• (Mitchell)
• Computationally intensive: n statesO(2^n)
How we computing the optimal payoff (numerically)
northwardposition
eastwardposition
heading angle
no-fly zone
time0 1 M
![Page 29: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/29.jpg)
Current Research
• Model predictive controller– Linearize the system– Discretize time– Compute an optimal control input for the
next N steps– Use the optimal input at the current step– Recompute at the next step
• Feasible in real time
![Page 30: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/30.jpg)
Current Research
• Discrete Abstraction– Discretize time– Discretize space into a finite set
• Feasible for Verification
northwardposition
eastwardposition
heading angle
1
2
3
4
![Page 31: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/31.jpg)
Conclusions
• Embedded control system challenge• Control theory identified• Future design challenges identified
![Page 32: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/32.jpg)
Acknowledgements
• Ian Mitchell• Xiaojun Liu• Shankar Sastry• Steve Neuendorffer• Claire Tomlin
![Page 33: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/33.jpg)
• State space X, a subset of Reals^n
• Control space U and disturbance Space D, compact subsets of Reals^u and Reals^d
Backup Slides—Problem Model
northwardposition
eastwardpositionheading angle
U
D
bank right-
bank left+
pilot
Soft Walls
![Page 34: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/34.jpg)
Actions in Continuous Time
• Pilot (Player 1) action d:Reals -> D, a Lebesgue measurable function
• The set of all pilot actions Dt
time
bank left
(bank right)
![Page 35: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/35.jpg)
Actions in Continuous Time
• Controller (Player 2) action u:Reals -> U, a Lebesgue measurable function
• The set of all controller actions Ut
time
bank left
(bank right)
![Page 36: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/36.jpg)
Dynamics
• The state trajectory x:Reals -> X• The dynamics equation f:X x U x D ->
Reals^n• (d/dt)x(t) = f(x(t), u(t), d(t))
eastwardposition
northwardposition
heading angle
trajectory
![Page 37: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/37.jpg)
Cataldo
Information Set
• At each time t, the controller knows pilot’s current action and the aircraft state
• We say s:Dt -> Ut is the controller’s nonanticipative strategy
• S is the set of nonanticipative strategies
eastwardposition
northwardposition
heading angle
trajectorytime
pilot bank left
(bank right)
controller knows
![Page 38: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/38.jpg)
Information Set
• At each time t, the pilot knows pilot’s only the aircraft state
• We say the pilot uses a feedback strategy
eastwardposition
northwardposition
heading angle
trajectory
pilot knows
![Page 39: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/39.jpg)
What We Want to Compute
No-fly zone
Backwards reachable set
States that canreach the no-flyzone even with Soft Walls controlllerCan prevent aircraft from
entering no-fly zone
![Page 40: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/40.jpg)
The Terminal Payoff Function
• The terminal cost function l:X -> Reals
• The further from the no-fly zone, the higher the terminal payoff
payoff
northwardposition
eastwardpositionno-fly zone
(constant over heading angle)
-
+
![Page 41: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/41.jpg)
The Payoff Function
• Given the pilot action d and the control action u, the payoff is V: X x Ut X Dt -> Reals
• Aircraft state at time t is x(t)• y is any state yxtxlduyV
t
)0())((inf),,(
]0,(
distance from no-fly zone
time
trajectories that terminate at ygiven u and d as inputs
payoff at ygiven u and d
![Page 42: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/42.jpg)
The Optimal Payoff
• At state y, the optimal payoff V*:X -> Reals comes from the control strategy s*:X ->S which maximizes the payoff for a disturbance which minimizes the payoff
)),(,(infsuparg)(
)),(,(infsup)(
*
*
ddsyVys
ddsyVyV
DtdSs
DtdSs
No-flyzone
optimalcontrol
strategy
optimaldisturbance
strategy
![Page 43: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/43.jpg)
The Finite Duration Payoff
• We look back no more than T seconds to compute the finite duration payoff J: X x Ut x Dt x (-∞, 0] -> Reals and the optimal finite duration payoff J*:X x (-∞, 0] -> Reals
yxtxlduyVt
)0())((inf),,(]0,(
),),(,(infsup),(
)0())((inf),,,(
*
]0,[
TddsyJTyJ
yxtxlTduyJ
DtdSs
Tt
the only difference
![Page 44: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/44.jpg)
Computing V*
• Assuming J* is continuous, it is the viscosity solution of
• J* converges pointwise to V* as T->∞• (Tomlin, Lygeros, Pappas, Sastry)
)()0,(
0),,(),(maxmin,0min),( **
ylyJ
evyfTyJTyJT y
UvDe
dynamicsspatial gradient
finite duration optimal payoff
![Page 45: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/45.jpg)
What Our Computation Gives Us
No-fly zone
terminal payoff
Backwards Reachable Set
optimal payoff
![Page 46: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/46.jpg)
Control from Optimal Payoff Function
Backwards Reachable Set
optimal control at boundary
dampen optimal control away from boundary
State Space
![Page 47: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/47.jpg)
• (Mitchell)
• Computationally Intensive n states, O(2^n)
Numerical Computation of V*
northwardposition
eastwardposition
heading angle
no-fly zone
time0 1 M
![Page 48: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/48.jpg)
• Same state space X
• Same control space U and disturbance Space D
What About Discrete-Time?
northwardposition
eastwardpositionheading angle
U
D
bank right-
bank left+
pilot
Soft Walls
![Page 49: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/49.jpg)
Actions in Discrete Time
• Pilot (Player 1) action d:Integers -> D • Control (Player 2) action u:Integers -> U
• The set of all pilot actions Dk• The set of all control actions Uk
time
bank left
(bank right)
time
bank left
(bank right)
![Page 50: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/50.jpg)
Dynamics
• The state trajectory x:Integers -> X• The dynamics equation f:X x U x D ->
Reals^n• x(k+1) = f(x(k), u(k), d(k))
eastwardposition
northwardposition
heading angle
trajectory
![Page 51: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/51.jpg)
Same Information Set
eastwardposition
northwardposition
heading angle
trajectorytime
pilot bank left
(bank right)
controller knows
pilot knows
• We say s:Dk -> Uk is the controller’s nonanticipative strategy
• S is the set of nonanticipative strategies
![Page 52: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/52.jpg)
Again We Compute
No-fly zoneBackwards reachable set
Can prevent aircraft from entering no-fly zone
![Page 53: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/53.jpg)
Same Terminal Payoff Function• The terminal cost function l:X -> Reals
• The further from the no-fly zone, the higher the terminal payoff
payoff
northwardposition
eastwardpositionno-fly zone
(constant over heading angle)
-
+
![Page 54: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/54.jpg)
The Payoff Function
• Given the pilot action d and the control action u, the payoff is V: X x Ut X Dt -> Reals
• Aircraft state at time k is x(k)• y is any state yxkxlduyV
k
)0())((inf),,(
}0,1{...,
distance from no-fly zone
time
trajectories that terminate at ygiven u and d as inputs
payoff at ygiven u and d
![Page 55: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/55.jpg)
The Optimal Payoff
• At state y, the optimal payoff V*:X -> Reals comes from the control strategy s*:X ->S which maximizes the payoff for a disturbance which minimizes the payoff
)),(,(infsuparg)(
)),(,(infsup)(
*
*
ddsyVys
ddsyVyV
DkdSs
DkdSs
No-flyzone
optimalcontrol
strategy
optimaldisturbance
strategy
![Page 56: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/56.jpg)
The Finite Duration Payoff
• We look back no more than K seconds to compute the finite duration payoff J: X x Ut x Dt x {…, -1, 0} -> Reals and the optimal finite duration payoff J*:X x {…, -1, 0} -> Reals
yxkxlduyVk
)0())((inf),,(}0,1{...,
),),(,(infsup),(
)0())((inf),,,(
*
}0,1,...,{
KddsyJKyJ
yxkxlKduyJ
DkdSs
Kk
the only difference
![Page 57: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/57.jpg)
Computing V*
• J* is the solution of
• J* converges pointwise to V* as K->∞
)()0,(
)}),,,((maxmin),,(min{)1,(
*
***
ylyJ
kevyfJkyJkyJUvDe
dynamicsfinite duration optimal payoff
![Page 58: An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d375503460f94a10727/html5/thumbnails/58.jpg)
How Do We Solve This Numerically?
)()0,(
)}),,,((maxmin),,(min{)1,(
*
***
ylyJ
kevyfJkyJkyJUvDe
No-fly zone
terminal payoff
Backwards Reachable Set
optimal payoff