Validating a HamiltonJacobi Approximation to Hybrid System Reachable Sets

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Personnel

• Investigators
• Claire Tomlin, Stephen Boyd, David Dill
• Research Associates
• Haitham Hindi, Henny Sipma, Dusan Stipanovic
• Graduate Students
• Alexandre Bayen, Jung Soon Jang, Ian Mitchell,
  Meeko Oishi, Rodney Teo , Lin Xiao
• Research Collaborators
• Boeing OCP (David Corman, Jim Paunicka, Brian
  Mendel, Don Winter), NASA, Boeing CAG
• UC Berkeley (Hybrid Systems, Reachable Set
  Computation)

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Technology Transfer

• Boeing (OCP)
• NASA Ames, Honeywell Labs, Boeing CAG (777
  Autoland)
• NASA Ames (Automated Air Traffic Management)

Announcement Hybrid Systems 2002 (HSCC
2002) Stanford University, March 25-27,
2002 Organizers Claire Tomlin and Mark
Greenstreet www.stanford.edu/group/hscc2002
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Schedule

• Tomlin group 1.30-3.00
• Boyd group 3.00-4.00
• Sipma/Dill group 4.00-4.30
• Discussion 4.30-5.00

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Project Objectives

• Design of computational methods to analyze and
  control hybrid systems (Tomlin, Dill, Boyd)
• Design of communications and control strategies
  for networked control systems (Boyd)
• Embedded software realizations (Tomlin, Dill)
• Applications to
• Verification of autopilot logic
• Multiple aircraft collision avoidance maneuver
  design
• Multiple sensor localization

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Project Objectives
A
Verification and control design using hybrid
systems
B
Embedded software design
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Discrete Codes for Verification and Synthesis
1. A Discrete Game
unsafe set
(i,j)
Player 2
Player 1
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Discrete Codes for Verification and Synthesis
unsafe set after 1 play
(i,j)
Player 2
Player 1
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Discrete Codes for Verification and Synthesis
unsafe set after 2 plays
(i,j)
Player 2
Player 1
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Discrete Codes for Verification and Synthesis
unsafe set after 2 plays
Fixed point
(i,j)
Player 2
Player 1
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Continuous Dynamic Games

• Approximation methods
• Predicate abstraction
• Extract a discrete transition graph from
  signs of and

Example
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Continuous Codes for Verification and Synthesis
2. A Continuous Game
unsafe set
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Continuous Codes for Verification and Synthesis
unsafe set
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Continuous Codes for Verification and Synthesis
unsafe set
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Continuous Codes for Verification and Synthesis
unsafe set
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Continuous Codes for Verification and Synthesis
unsafe set
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Continuous Codes for Verification and Synthesis
unsafe set
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Continuous Codes for Verification and Synthesis
unsafe set
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Continuous Codes for Verification and Synthesis
unsafe set

• Compute the set of variables (aircraft positions,
  velocities ) for which for all control
  actions, the system can be driven into the unsafe
  set by uncertainty or disturbance.
• Efficient data structures to represent sets
  EXTREMELY important.

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Continuous Reachable Set Calculation
Computation provides, at each
Distance to boundary Vector pointing to boundary
Theorem Computing
where is the unique
Crandall-Evans-Lions viscosity
solution to
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A. Hybrid Codes for Verification and Synthesis
glide slope
3. A Hybrid Game
altitude
flare
TOGA
capture
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Hybrid Codes for Verification and Synthesis
glide slope
altitude
flare
TOGA
capture
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Hybrid Codes for Verification and Synthesis
glide slope
altitude
flare
TOGA
capture
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Hybrid Codes for Verification and Synthesis
glide slope
altitude
flare
TOGA
capture
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Hybrid Codes for Verification and Synthesis
glide slope
altitude
flare
TOGA
capture
Implementation in Boeings Open Control Platform
(used in control logic design Test A/C F-15,
T-33)
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B. Embedded Software Design

• Process 1 GPS
• gps_position
• 1 Hz
• data storage

embedded software
real time operating system
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B. Embedded Software Design

• Process 1 GPS
• gps_position
• 1 Hz
• data storage

• Process 2 control
• gps_position
• 20 Hz
• data retrieval

embedded software
real time operating system
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Ian Mitchell
y
y
x
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Alexandre Bayen
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Meeko Oishi
what the pilot sees
with Boeing CAG, NASA Ames
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Jung Soon Jang
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Dusan Stipanovic
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Rodney Teo
control
estimator
estimator
control
estimator
control
Each aircraft broadcasts its estimated
state Analyze performance with respect to link
reliability