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Validating a HamiltonJacobi Approximation to Hybrid System Reachable Sets

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Continuous systems controlled by a discrete logic: embedded systems ... in Cockpit Interface ... Pilot's cockpit display may not contain sufficient ... – PowerPoint PPT presentation

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Title: Validating a HamiltonJacobi Approximation to Hybrid System Reachable Sets


1
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Motivating applications
(Source Boeing X45-A)
(Source Northrop Grumman-X47A)
(Source NASA Ames)
3
Hybrid systems
  • Continuous systems controlled by a discrete
    logic embedded systems (autopilot logic)
  • Coordinating processes multi-vehicle systems
    interfacing continuous control with coordination
    protocols
  • Continuous systems with a phased operation
    (biological cell growth and division)

discrete systems (computer science)
continuous systems (control)
4
Verification and Controller Synthesis
  • Verification a mathematical proof that the
    system satisfies a property
  • Controller synthesis the design of control laws
    to guarantee that the system satisfies the
    property
  • Methods give definitive answers, unlike
    simulation
  • Often give surprising answers, trajectories which
    one might not think to simulate
  • Reduces development time, cost of certification

initial
unsafe
5
Verification and Controller Synthesis
  • Verification a mathematical proof that the
    system satisfies a property
  • Controller synthesis the design of control laws
    to guarantee that the system satisfies the
    property
  • Methods give definitive answers, unlike
    simulation
  • Often give surprising answers, trajectories which
    one might not think to simulate
  • Reduces development time, cost of certification

initial
unsafe
6
Verification and Controller Synthesis
  • Verification a mathematical proof that the
    system satisfies a property
  • Controller synthesis the design of control laws
    to guarantee that the system satisfies the
    property
  • Methods give definitive answers, unlike
    simulation
  • Often give surprising answers, trajectories which
    one might not think to simulate
  • Reduces development time, cost of certification

initial
unsafe
7
Verification and Controller Synthesis
  • Verification a mathematical proof that the
    system satisfies a property
  • Controller synthesis the design of control laws
    to guarantee that the system satisfies the
    property
  • Safety Property can be encoded as a condition on
    the systems reachable set of states

initial
unsafe
unsafe
unsafe initialization
safe, under appropriate control
8
Example Aircraft Collision Avoidance
  • Two identical aircraft at fixed altitude speed

y
v
y
u
x
v
d
9
Continuous Reachable Set
Solve
Display
10
Collision Avoidance Filter
  • Simple demonstration
  • Pursuer turn to head toward evader
  • Evader turn to head right

Movies
11
Blunder Zones for Closely Spaced Approaches
EEM Maneuver 1 accelerate
EEM Maneuver 2 turn 45 deg, accelerate
EEM Maneuver 3 turn 60 deg
evader
12
Implementation Display design courtesy of
Chad Jennings, Andy Barrows, David Powell
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
13
Map View showing a blunder The BZ calculations
are performed in real time (40Hz) so that the
contour is updated with each video frame.
14
Verified Mode Switching in Autopilots
15
Use in Cockpit Interface Verification
  • Controllable flight envelopes for landing and
    Take Off / Go Around (TOGA) maneuvers may not be
    the same
  • Pilots cockpit display may not contain
    sufficient information to distinguish whether
    TOGA can be initiated

existing interface
controllable TOGA envelope
intersection
revised interface
controllable flare envelope
16
A More General Problem Structure
Communication Zone
Safety Assurance Zone
17
(Decomposed) Centralized Optimization
Neighborhood of ith vehicle
18
fixed time horizon
Bargaining start
Fixed time horizon complete global map
19
Flight Plans published by aircraft 1
20
Another Example
21
Flight Plans published by aircraft 1
22
moving time horizon
Bargaining start
Receding horizon incomplete global map
23
Local Optimization with Constraints
  • Constraints embed
  • local dynamics coordinated turn and straight
    flight hdi
  • input constraints limited turn rate and
    velocity gei
  • global coordination constraints minimum safety
    assurance gsij for all j within neighborhood of
    i

24
Decomposition I
Centralized Optimization
Decomposed Centralized Optimization
Pareto optimality Nash equilibrium
25
Nash Equilibrium for Centralized Problem
  • Define Hamiltonian for each subsystem
  • is a Nash equilibrium for the centralized
    optimization problem if
  • where
  • Thus, none of the subsystems can improve its
    solution, with all other subsystems solutions
    remaining fixed.

26
Decomposition II
Decomposed Centralized Optimization
Decentralized Optimization
Nash equilibrium Local optimal solutions
27
Nash Equilibrium for Decentralized Problem
  • Define Hamiltonian for each subsystem
  • is a Nash equilibrium for the decentralized
    optimization problem if
  • Optimal solutions by each of the
    subsystems

Proposition is a Nash equilibrium of the
centralized problem if and only if it is a Nash
equilibrium of the decentralized problem
28
Example Nash Equilibrium at (0,0)
29
Using Penalty function methods
  • Global contraction function from the local
    optimization structures
  • For a particular solution, local optimization of
    the ith vehicle only affects the portion of F
    tied to its own local optimization

30
Cooperation Assumptions
  • Eliminates cases in which a subsystem is
    artificially acting against a constraint dictated
    by another group
  • Eliminates cases in which two subsystems act
    against each other with non-identical constraints

31
Nash Bargaining with Multiple Threads
  • Multiple solutions, or threads, exist within
    the system

Vehicle 1
Vehicle 2
Vehicle 3
Vehicle 4
Iteration 1
Iteration 2
Iteration 3
Iteration 4
Iteration 4
32
Convergence Results
  • Global convergence to a (not necessarily
    feasible) Nash solution
  • If the gradients of the constraint functions are
    linearly independent (Linear Constraint
    Qualification Condition, LICQ), then global
    convergence to a feasible Nash solution
  • Pareto optimality for convex problems

Inalhan, Stipanovic, Tomlin. Decentralized
Optimization, with Application to Multiple
Aircraft Coordination. CDC 2002, Submitted to
JOTA
33
4-Vehicle Example
34
4-Vehicle Example
35
Flight Plans published by aircraft 1
36
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37
Applied to other problems of interest
  • Decentralized Initialization Procedure Heuristics
  • Multiple-Depots (Vehicles), Time-windows for
    access, Priority on objectives and the vehicles
  • Iterative selection process carried at each
    vehicle
  • Best solution in the fleet is then selected from
    each vehicles solution set

38
Spectrum of Approaches
Lack of information ?? Bounded Irrationality
Cooperative incomplete information
Non-cooperative Full information
Cooperative Full information
Non-cooperative No information
39
Research Goals
  • Design of provably correct and safe decentralized
    control protocols
  • Adapt to coordination
  • Allow for dynamic reconfiguration
  • Treatment of information
  • Multi-scale provisioning of data based on inputs
    from various sensing modalities
  • Urgency of the need for sensed data
  • Available bandwidth
  • Verification algorithms
  • Used during design phase, to reduce the time
    spent during the validation phase

40
Stanford DragonFly UAV
10 ft wingspan
12 ft wingspan
Jang, Teo, Inalhan, and Tomlin, DASC 2001,
Jang and Tomlin, AIAA GNC 2002
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