Transcript Document 7519241

Decentralized Optimization,
with application to
Multiple Aircraft Coordination
Gökhan Inalhan, Dusan Stipanovic, Claire Tomlin
Hybrid Systems Laboratory
Department of Aeronautics and Astronautics
Stanford University
Decision Making Under Uncertainty MURI Review,
July 2002
Motivating applications
(Source: Boeing X45-A)
(Source: Northrop Grumman-X47A)
(Source: NASA Ames)
Hybrid systems
xç = f ( x; u; d )
continuous systems
(control)
discrete systems
(computer science)
• 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)
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
initial
unsafe
• Methods give definitive answers, unlike simulation
• Often give surprising answers, trajectories which one
might not think to simulate
• Reduces development time, cost of certification
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
initial
unsafe
• Methods give definitive answers, unlike simulation
• Often give surprising answers, trajectories which one
might not think to simulate
• Reduces development time, cost of certification
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
initial
unsafe
• Methods give definitive answers, unlike simulation
• Often give surprising answers, trajectories which one
might not think to simulate
• Reduces development time, cost of certification
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
initial
unsafe
unsafe
unsafe initialization
safe, under appropriate control
Safety Property can be encoded as a condition on the
system’s reachable set of states
Example: Aircraft Collision Avoidance
Two identical aircraft at fixed altitude & speed:
 x
  v  v cos  uy 


d 
 y   f (x, u, d )   v sin   ux 
dt  


d u
 


y
x
u
v

v
d
‘evader’ (control)
‘pursuer’ (disturbance)
Continuous Reachable Set

x
y
Solve:
à
@J(x ;t )
@t
= minf
Display:
@J(x ;t )
0; maxu min d H ( x; @x ; u; d )g
G ( t ) = f x : J ( x; t ) < 0g
Collision Avoidance Filter
Simple demonstration
– Pursuer: turn to head toward evader
– Evader: turn to head right
evader’s actual input
safety filter’s
input modification
unsafe set
collision set
pursuer
evader
evader’s desired input
pursuer’s input
Movies …
Blunder Zones for Closely Spaced Approaches
EEM Maneuver 1: accelerate
EEM Maneuver 2:
turn 45 deg,
accelerate
EEM Maneuver 3:
turn 60 deg
evader
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
Map View showing a blunder
The BZ calculations are
performed in real time (40Hz)
so that the contour is updated
with each video frame.
Verified Mode Switching in Autopilots
Use in Cockpit Interface Verification
• 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
existing interface
controllable TOGA envelope
intersection
flare
TOGA
flaps extended
minimum thrust
flaps retracted
maximum thrust
rollout
flaps extended
reverse thrust
revised interface
controllable flare envelope
flare
TOGA
flaps extended
minimum thrust
flaps retracted
maximum thrust
rollout
slow TOGA
flaps extended
reverse thrust
flaps extended
maximum thrust
A More General Problem Structure
V1
V2
V1
V2
V3
V3
V4
Communication Zone
Safety Assurance Zone
V4
(Decomposed) Centralized Optimization
Neighborhood of ith vehicle
fixed time horizon
t à 2t s
t à ts
t
t + T à 2t s t + T à t s t + T
Bargaining
start
Fixed time horizon – complete global map
Flight Plans published by aircraft 1
Another Example
Flight Plans published by aircraft 1
moving time horizon
t à 2t s
t à ts
t
t + T à 2t s t + T à t s t + T
ts
Bargaining
start
Receding horizon – incomplete global map
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
Decomposition I
Centralized Optimization
Decomposed Centralized
Optimization
Pareto optimality
Nash equilibrium
Nash Equilibrium for Centralized Problem
Define Hamiltonian for each subsystem:
H ci( x; õ; ö) = f i ( x i ) + õ T h ( x ) + ö T g( x )
ã
x =
ã
ã
ã
[x 1; :::; x i ; :::x m ]
is a Nash equilibrium for the centralized optimization
problem if:
c ã
H i ( x ; õ; ö)
where
ô H ci( x ãi ; õ; ö)
x ãi = [x ã1; :::; x i ; :::x ãm ]
Thus, none of the subsystems can improve its solution,
with all other subsystems’ solutions remaining fixed.
Decomposition II
Decomposed Centralized
Optimization
Decentralized Optimization
Nash equilibrium
Local optimal solutions
Nash Equilibrium for Decentralized Problem
Define Hamiltonian for each subsystem:
dc
H i ( x i ; õ i ; ö i jf x j gi )
= f i ( x i ) + õ Ti h i ( x i jf x j gi ) + ö Ti gi ( x i jf x j gi )
ã
ã
ã
=
[
x
x 1; :::; x i ; :::x m ]
ã
is a Nash equilibrium for the decentralized optimization
problem if:
ã
dc ã
H i ( x i ; õ i ; ö i jf x j g)
ô
ã
dc
H i ( x i ; õ i ; ö i jf x j g)
Optimal solutions by each of the subsystems
Proposition: x ã is a Nash equilibrium of the centralized
problem if and only if it is a Nash equilibrium of the
decentralized problem
Example: Nash Equilibrium at (0,0)
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
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
Nash Bargaining with Multiple Threads
Multiple solutions, or “threads”, exist within the system:
Vehicle #1
Iteration #1
Iteration #2
Iteration #3
Iteration #4
Iteration #4
Vehicle #2
Vehicle #3
Vehicle #4
Convergence Results
1. Global convergence to a (not necessarily feasible)
Nash ‘solution’
2. If the gradients of the constraint functions are
linearly independent (Linear Constraint Qualification
Condition, LICQ), then global convergence to a
feasible Nash solution
3. Pareto optimality for convex problems
[Inalhan, Stipanovic, Tomlin.
Decentralized Optimization, with Application to Multiple Aircraft Coordination.
CDC 2002, Submitted to JOTA]
4-Vehicle Example
V1
V2
V3
V4
4-Vehicle Example
PF i (x i jf x j gi ) = PF j ( f x j gi ; x i )
8j 2 N i
Flight Plans published by aircraft 1
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 vehicle’s
solution set
Spectrum of Approaches
Lack of information  Bounded Irrationality
Non-cooperative
No information
Non-cooperative
Full information
Cooperative
incomplete
information
Cooperative
Full information
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
Stanford DragonFly UAV
10 ft wingspan
12 ft wingspan
[Jang, Teo, Inalhan, and Tomlin, DASC 2001], [Jang and Tomlin, AIAA GNC 2002]