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An Introduction to the Soft Walls Project

An Introduction to the Soft Walls Project. Adam Cataldo Prof. Edward Lee. University of Pennsylvania Dec 18, 2003 Philadelphia, PA. Outline. The Soft Walls system Objections Control system design Current Research Conclusions. Introduction. On-board database with “no-fly-zones”

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An Introduction to the Soft Walls Project

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  1. An Introduction to the Soft Walls Project Adam Cataldo Prof. Edward Lee University of Pennsylvania Dec 18, 2003 Philadelphia, PA

  2. Outline • The Soft Walls system • Objections • Control system design • Current Research • Conclusions

  3. Introduction • On-board database with “no-fly-zones” • Enforce no-fly zones using on-board avionics • Non-networked, non-hackable

  4. Autonomous control Pilot Aircraft Autonomous controller

  5. Softwalls is not autonomous control Pilot Aircraft + bias pilot control Softwalls

  6. Relation to Unmanned Aircraft • Not an unmanned strategy • pilot authority • Collision avoidance

  7. A deadly weapon? • Project started September 11, 2001

  8. Design Objectives Maximize Pilot Authority!

  9. Unsaturated Control Pilot lets off controls Pilot tries to fly into no-fly zone Pilot turns away from no-fly zone No-fly zone Control applied

  10. Objections • Reducing pilot control is dangerous • reduces ability to respond to emergencies

  11. There is No Emergency That Justifies Landing Here

  12. Objections • Reducing pilot control is dangerous • reduces ability to respond to emergencies • There is no override • switch in the cockpit

  13. Hardwall

  14. 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

  15. Localization Backup • Inertial navigation • Integrator drift limits accuracy range

  16. 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?

  17. Deployment • Fly-by-wire aircraft • a software change • Older aircraft • autopilot level • Phase in • prioritize airports

  18. 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

  19. Not Like Air Traffic Control • Much Simpler • No need for air traffic controller certification

  20. 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

  21. 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

  22. Here’s How It Works

  23. What We Want to Compute Backwards reachable set No-fly zone States that can reach the no-fly zone even with Soft Walls controlller Can prevent aircraft from entering no-fly zone

  24. What We Create • The terminal payoff function l:X -> Reals • The further from the no-fly zone, the higher the terminal payoff payoff + (constant over heading angle) northward position - eastward position no-fly zone

  25. What We Compute terminal payoff optimal payoff No-fly zone Backwards Reachable Set

  26. Our Control Input optimal payoff function dampen optimal control away from boundary optimal control at boundary State Space Backwards Reachable Set

  27. 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) spatial gradient dynamics time derivative terminal payoff

  28. How we computing the optimal payoff (numerically) • (Mitchell) • Computationally intensive: n statesO(2^n) northward position no-fly zone eastward position heading angle time 0 1 M

  29. 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

  30. 2 1 4 3 Current Research • Discrete Abstraction • Discretize time • Discretize space into a finite set • Feasible for Verification northward position eastward position heading angle

  31. Conclusions • Embedded control system challenge • Control theory identified • Future design challenges identified

  32. Acknowledgements • Ian Mitchell • Xiaojun Liu • Shankar Sastry • Steve Neuendorffer • Claire Tomlin

  33. Backup Slides—Problem Model • State space X, a subset of Reals^n • Control space U and disturbance Space D, compact subsets of Reals^u and Reals^d northward position eastward position heading angle U Soft Walls bank right - bank left + D pilot

  34. bank left time (bank right) Actions in Continuous Time • Pilot (Player 1) action d:Reals -> D, a Lebesgue measurable function • The set of all pilot actions Dt

  35. bank left time (bank right) Actions in Continuous Time • Controller (Player 2) action u:Reals -> U, a Lebesgue measurable function • The set of all controller actions Ut

  36. 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)) northward position trajectory eastward position heading angle

  37. pilot bank left time northward position (bank right) trajectory eastward position heading angle 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 controller knows Cataldo

  38. northward position trajectory eastward position heading angle Information Set • At each time t, the pilot knows pilot’s only the aircraft state • We say the pilot uses a feedback strategy pilot knows

  39. What We Want to Compute Backwards reachable set No-fly zone States that can reach the no-fly zone even with Soft Walls controlller Can prevent aircraft from entering no-fly zone

  40. The Terminal Payoff Function • The terminal cost function l:X -> Reals • The further from the no-fly zone, the higher the terminal payoff payoff + (constant over heading angle) northward position - eastward position no-fly zone

  41. 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 trajectories that terminate at y given u and d as inputs distance from no-fly zone payoff at y given u and d time

  42. 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 No-fly zone optimal control strategy optimal disturbancestrategy

  43. 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 the only difference

  44. Computing V* • Assuming J* is continuous, it is the viscosity solution of • J* converges pointwise to V* as T->∞ • (Tomlin, Lygeros, Pappas, Sastry) spatial gradient dynamics finite duration optimal payoff

  45. What Our Computation Gives Us terminal payoff optimal payoff No-fly zone Backwards Reachable Set

  46. Control from Optimal Payoff Function dampen optimal control away from boundary optimal control at boundary State Space Backwards Reachable Set

  47. Numerical Computation of V* • (Mitchell) • Computationally Intensive n states, O(2^n) northward position no-fly zone eastward position heading angle time 0 1 M

  48. northward position eastward position heading angle What About Discrete-Time? • Same state space X • Same control space U and disturbance Space D U Soft Walls bank right - bank left + D pilot

  49. 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 bank left bank left time time (bank right) (bank right)

  50. 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)) northward position trajectory eastward position heading angle

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