Validating a HamiltonJacobi Approximation to Hybrid System Reachable Sets

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(No Transcript)
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Air Traffic Control Separation Assurance
Data from NASA Ames
Safety 5 mile lateral, 1000 ft vertical
separation
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Flight Management Systems Autoland
what the pilot sees
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Systems Biology Differentiation in Xenopus
Safety biologically feasible equilibria are
reachable
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Hybrid System Model
Sontag, Brockett, Branicky, Lygeros
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Outline

• Controller synthesis for hybrid systems using
  game theory
• Algorithm for computing reachable sets and
  synthesis of controllers
• for continuous systems
• for hybrid systems
• Applications
• aircraft collision avoidance
• flight management system design
• Fast overapproximations
• Controller synthesis using predicate abstraction
• Analysis of biological regulatory networks

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Controller Synthesis
unsafe
Controller synthesis to guarantee that a system
satisfies a safety property
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Controller Synthesis
unsafe
Controller synthesis to guarantee that a system
satisfies a safety property
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Controller Synthesis
Alur, Aubin, Cardaliaguet, Dang, Dill,
Egerstedt, Falcone, Greenstreet, Henzinger,
Jadbabaie, Krogh, Kurzhanski, Maler, Morari,
Pappas, Prajna, Saint-Pierre, Varaiya, Wong-Toi
safe, under appropriate control
unsafe
Safety Property can be encoded as a condition on
the systems reachable set of states
In blue, system will stay safe
In red, system may become unsafe
On boundary, apply control to stay out of red
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A discrete game
Player 1 (control)
Player 2 (disturbance)
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A discrete game
Player 1 (control)
Player 2 (disturbance)
unsafe
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A discrete game
Player 1 (control)
Player 2 (disturbance)
unsafe
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A discrete game
Player 1 (control)
Player 2 (disturbance)
unsafe
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Continuous reachable set calculation
is the set of states for which, for all
control actions, there exists a disturbance
action which can drive the system to in
at most
Mitchell, Bayen, Tomlin 02, Tomlin, Lygeros,
Sastry 00
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Numerical computation of reach sets
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Numerical computation of reach sets

• Level set methods
• Convergent numerical algorithms to compute
  viscosity solution
• Non-oscillatory, high accuracy spatial derivative
  approximation
• Stable, consistent numerical Hamiltonian
• Variation diminishing, high order, explicit time
  integration
• Example (2 player zero sum game)

y
x
v
y
y
d
u
v
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Mitchell, Tomlin 01, based on Osher and Sethian
88
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Collision Avoidance Control
http//www.cs.ubc.ca/mitchell/ToolboxLS/
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Computing Reach Sets for Hybrid Systems
modes
2
3
K
1
1
2
iterations
3
n
safe
unsafe
unsafe
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Reach Sets uncontrollable predecessor
modes
2
3
K
1
1
safe
2
iterations
3
n
uncontrolled transition
unsafe
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Reach Sets controllable predecessor
modes
2
3
K
1
1
safe
2
iterations
3
n
controlled transition
safe
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Reach Sets Variational Inequality
States which reach G without hitting E first
modes
2
3
K
1
1
2
iterations
3
where
n
subject to
Mitchell, Tomlin 01, Tomlin, Lygeros, Sastry
00
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Reach Sets Iterate
modes
2
3
K
1
1
2
iterations
3
n
Mitchell, Tomlin 01, Tomlin, Lygeros, Sastry
00
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Outline

• Controller synthesis for hybrid systems using
  differential games
• Algorithm for computing reachable sets and
  synthesis of controllers
• for continuous systems
• for hybrid systems
• Applications
• flight management system design
• aircraft collision avoidance
• Fast overapproximations
• Controller synthesis using predicate abstraction
• Analysis of biological regulatory networks

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Example Aircraft Autolander

• Aircraft must stay within safe flight envelope
  during landing
• Bounds on velocity ( ), flight path angle (
  ), height ( )
• Control over engine thrust ( ), angle of
  attack ( ), flap settings
• Model flap settings as discrete modes
• Terms in continuous dynamics depend on flap
  setting

body frame
wind frame
inertial frame
Bayen, Mitchell,Tomlin 02
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Autolander Synthesizing Control

• For states at the boundary of the safe set,
  results of reach-avoid computation determine
• What continuous inputs (if any) maintain safety
• What discrete jumps (if any) are safe to perform
• Level set values and gradients provide all
  relevant data

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Application to Autoland Interface

• 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
Oishi, Mitchell, Bayen, Tomlin 02
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5D Safe Landing
X distance, Altitude, and Pitch (?)
0,1) nm
1,10) nm
J. Sprinkle, M. Eklund (UC Berkeley)
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5D Safe Landing
X distance, Y distance, and heading (?)
0,1) nm
1,10) nm
J. Sprinkle, M. Eklund (UC Berkeley)
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Example Closely Spaced Parallel Approaches
San Mateo Bridge
San Francisco Airport
750 ft separation
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Example Closely Spaced Parallel Approaches
evader

• Three emergency escape maneuvers (EEMs)
• Evader accelerates straight ahead
• Evader accelerates, turns to the right 45 deg
• Evader turns to the right 60 deg

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Tested on the Stanford DragonFly UAVs
Dragonfly 3
Dragonfly 2
Ground Station
Jang, Teo, Tomlin
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Moffett Federal Airfield, September 2003
Accelerate and turn EEM
DF 2, the evader, is the larger blob
Evader, DF 2 (red and yellow aircraft)
Put video here
North (m)
East (m)
Separation distance (m)
EEM alert
Above threshold
time (s)
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Moffett Federal Airfield, September 2003
DF 2, the evader, is the larger blob
Coast and turn EEM
Evader, DF 2 (red and yellow aircraft)
Put video here
North (m)
East (m)
Separation distance (m)
EEM alert
Above threshold
time (s)
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Edwards Air Force Base June 2004
T-33 Cockpit
DARPA/Boeing SEC Final Demonstration F-15
(blunderer), T-33 (evader)
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Overapproximating Reachable Sets
Exact
Approximate
Overapproximative reachable set
Khrustalev, Varaiya, Kurzhanski

• Polytopic overapproximations for nonlinear games
• Subsystem level set functions
• Norm-like functions with identical strategies
  to exact

Hwang, Stipanovic, Tomlin
1 sec on 700MHz Pentium III (vs 4 minutes for
exact)
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Outline

• Controller synthesis for hybrid systems using
  differential games
• Algorithm for computing reachable sets and
  synthesis of controllers
• for continuous systems
• for hybrid systems
• Applications
• aircraft collision avoidance
• flight management system design
• Fast overapproximations
• Controller synthesis using predicate abstraction
• Analysis of biological regulatory networks

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Delta-Notch Signaling Pathway

• Observables
• Transmembrane proteins direct contact required
• Notch
• Receptor ? production
• promoted by localized high
• Delta concentrations
• Delta
• Ligand ? activates Notch, production promoted by
  low intracellular Notch
  concentrations
• Produces lateral inhibition
• Delta hair

Deconstructing Hairy, Ghosh, Tomlin 01
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Previous work

• Accepted influence model
• Reaction-diffusion Collier 96, Marnellos 00
• Parameters are unknown.
  

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and may be hard to identify
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Hybrid Model of a Cell

• Idea approximate the nonlinear switch by a
  piecewise linear switch
• For example, using , each biological cell
  has four discrete states

• Notch off/delta off
• Notch on/delta off
• Notch off/delta on
• Notch on/delta on

Ghosh, Tomlin 01
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Reachable set calculation for piecewise affine
systems
Morari, Bemporad, Johansson, Petit, Wellstead

• Piecewise affine dynamics
• Boundaries between discrete states described by
  polynomials
• Predicate abstraction used to determine
  transition relations

Graf Saidi, Tiwari, Kuipers
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Visualization of Reach Sets for 2-cell

• Implementation
• Symbolic manipulations done
  in MATLAB
• Decision procedure on polynomials done in QEPCAD
• Currently 8 continuous, 256 discrete variables

Ghosh, Tomlin 04
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Simulation using viable parameters and IC
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Drosophila wing epithelium

• Drosophila wing hairs point distally, virtually
  error free
• Planar Cell Polarity (PCP)

proximal
distal
Courtesy Dali Ma, Stanford University
Rousset, et al., Genes Dev 15 658-71, 2001
Human pathology cochlear hair cells spina
bifida oncogenic Wnt pathway
Amonlirdviman, Khare, Tree, Chen, Axelrod,
Tomlin 04
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Signaling Molecules

• Includes Frizzled (Fz), Dishevelled (Dsh),
  Prickle (Pk), Flamingo (Fmi) and Van Gogh (Vang)
• Dsh and Fz localize on the distal portion of each
  cell
• Pk and Vang localize on the proximal portion of
  each cell
• Hair grows at Dsh localization

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Mutant Wings

• Loss of Fz disrupts polarity in distal non-mutant
  cells
• Loss of Vang disrupts polarity in proximal
  non-mutant cells

Mutant Fz clones
Mutant Vang clones
Vinson and Adler, Nature 329, 549-51, 1987
Taylor, et al., Genetics 150, 199-210, 1998
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Influence model

• Fz promotes recruitment of Dsh to a membrane
• Dsh stabilizes Fz localization
• Fz acts indirectly to promote the localization of
  Vang and Pk on the membrane of a neighboring cell
• Pk (and Vang) suppresses the recruitment of Dsh
  to a membrane
• Network amplifies unknown directional cue

Tree, et al., Cell 109, 371-81, 2002
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Loss-of-Fz Numerical Results
Dsh Distribution w/ Resulting Hair Pattern
Domineering non-autonomy distal of cloned mutant
cells
Mutant Fz clones
Vinson and Adler, Nature 329, 549-51, 1987
Fz mutant clone
Amonlirdviman, Axelrod, Tomlin 04
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Loss-of-Pk (Vang) Numerical Results
Fz Distribution w/ Resulting Hair Pattern
Domineering non-autonomy proximal of cloned
mutant cells
Mutant Vang clones
Taylor, et al., Genetics 150, 199-210, 1998
Vang mutant clone
Amonlirdviman, Axelrod, Tomlin 04
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Biological Insights

• Demonstrates sufficiency
• Explains even non-intuitive results

Suppose you overexpress Pk in part of the wing
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Summary and new directions

• The development of a reach set toolkit for hybrid
  systems
• Software C http//www.cs.ubc.ca/mitchell/Tool
  boxLS/
• Matlab/QEPCAD
• The toolkit is useful for
• Engineering when (not) to switch modes, which
  mode(s) to switch to, and provides a set-valued
  feedback control law to remain in safe set
• Biology can be used in reverse engineering of
  biological circuits, used to extract ranges of
  parameters, determine feasibility of a particular
  influence connection often as much system
  identification as modeling
• New algorithms, tools for (fast) analysis in high
  dimensions
• Geometric approximation
• Approximation algorithms based on optimization
  relaxations
• Decentralization
• Estimation and learning
• Optimization control of PDE Systems Bayen,
  Raffard (FrP11-1, FrP11-4)

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Collaborators

• Ian Mitchell, Alex Bayen, Inseok Hwang, Meeko
  Oishi, Rodney Teo, Jung Soon Jang, Ronojoy
  Ghosh, Gökhan Inalhan, Hamsa Balakrishnan, Keith
  Amonlirdviman, Robin Raffard, Gabe Hoffmann,
  Kaushik Roy, Peter Brende, Dusan
  Stipanovic, Sriram Shankaran, Jianghai Hu

Stanford Hybrid Systems Lab
Stanford Medical School
Jeff Axelrod, Harley McAdams
The Berkeley gang
John Lygeros, George Pappas, Datta Godbole,
Shankar Sastry
NASA
George Meyer, Michael Heymann, Asaf Degani
Boeing
David Corman, Jim Paunicka, Brian Mendel, Don
Winter
Honeywell
Datta Godbole, Tariq Samad
DARPA
John Bay, Eric Eisenstadt, Sri Kumar
NSF
Helen Gill, Kishan Baheti
ONR
Behzad Kamgar-Parsi
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Other research Approximation algorithms for
hybrid systems

• Approximation algorithms for hybrid trajectory
  optimization
• Applied to routing/scheduling aircraft in
  vicinities of airports
• Results Bayen, Zhang, Ye, Tomlin 03,04
• 5-approximation for minimum sum of arrival times
• 3-approximation for makespan

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Aircraft Simulator Tests

• Setup
• Commercial flight simulator, B767 pilot
• Digital video of primary flight display
• Maneuver
• Go-around at low speed, high descent rate
• Goal
• Determine whether problematic behavior predicted
  by our model is possible in aircraft flight
  simulator

(movie)
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Aircraft Simulator Results
Produced unexpected behavior Non-standard
procedure Unable to duplicate Validated types of
problems addressed by this method
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Biological Insights

• Suggests mechanism for explaining phenotypes of
  different mutant Fz alleles experimentally
  verifiable

Fz autonomous allele FzDsh interaction reduced
to 0
Fz non-autonomous allele All Fz function removed
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Biological Insights

• Suggests mechanism for explaining phenotypes of
  different mutant Fz alleles experimentally
  verifiable