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How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems
How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems
Todd W. Neller
Knowledge Systems Laboratory
Stanford University
OutlineOutline
Defining the problem: Will the critical satellite motor stall?
Generalizing the problem: Hybrid Systems Reformulating the problem: Optimizing for failure Describing the tool we need: Information-Based
Optimization Exciting Conclusion: Why should a power
screwdriver be inspiring?
Stepper MotorsStepper Motors
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a.k.a. “step motors”
t
Dan Goldin, head of NASA: “Smaller, Faster, Better, Cheaper” microsatellites, autonomy, C.O.T.S.
SSDL’s OPAL: Orbiting Picosatellite Automated Launcher
Problem: Will the motor stall while accelerating the picosatellite?
How to find good research problems: specific general
The ProblemThe Problem
?
Hybrid SystemsHybrid Systems
Hybrid = Discrete + Continuous Example: Bouncing Ball Fast Continuous Change Discrete Change More Interesting Example: Mode Switching
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Safety Safety
Safety property - Something that is always true about a system
Another view: A set of states the system never leaves
Safe/unsafe states, desired/undesired statesInitial Safety property - Safety over an
initial duration of time
Verification, RefutationVerification, Refutation
Verification of safety: Proving that the system can never leave safe states
Verification through simulation?Refutation of safety: Proving that the
system can leave safe statesProof by counterexample
Stepper Motor Safety RefutationStepper Motor Safety Refutation
Given: Stepper motor simulator and acceleration table Bounds on stepper motor system parameters
and initial state Set of stall states
Find: Parameters and initial conditions such that the
motor enters a stall state during acceleration
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General Problem StatementGeneral Problem Statement
Given: Hybrid system simulator for
initial time duration Bounds on initial conditions
(parameters and variable assignments)
Set of unsafe states
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Find: Initial conditions such that the system enters an unsafe
state during initial time
Generate and Test
Tools for Initial Safety Refutation of Hybrid Systems
Tools for Initial Safety Refutation of Hybrid Systems
(There has to be a better way, right?)
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Distance from Unsafe StatesDistance from Unsafe States
Make use of simple knowledge of problem domain to provide landscape helpful to search
Refutation through OptimizationRefutation through Optimization
Transform refutation problem into an optimization problem with a heuristic (i.e. estimated) measure of relative safety
Apply efficient global optimization
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Given: Hybrid system simulator for initial time t Possible initial conditions I Heuristic evaluation function f which takes an initial
condition as input and returns a relative safety ranking of the resulting trajectory
Find: Initial condition x in I, such that f(x) = 0
Problem ReformulationProblem Reformulation
initial condition trajectory ranking
f
simulation evaluation
f(x) is usually assumed cheap to compute. Most methods store and use very little data.
Solution: Use simulation intelligently. General principle: Information gained at great cost
should be treated with great value.
Problem: Simulation isn’t CheapProblem: Simulation isn’t Cheap
f(6.
27)=
0.34
f(6.35)=0.92f(7.11)=1.85
f(9.24)=7.90
SatisficingSatisficing
General optimization seeks an unknown optimum.
We don’t know our optimum, but we have a goal value we’re seeking to satisfy.
Satisficing (= “satisfying”, economist Herbert Simon)
This knowledge can be leveraged to make our optimization more efficient.
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Information-Based ApproachInformation-Based Approach
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Assume: continuous, flat functions more likely
Information-Based Optimization (Neimark and Strongin, 1966; Strongin and Sergeyev, 1992; Mockus, 1994)
Previous function evaluations shape probability distribution over possible functions.
But we needn’t deal with probabilities. Ranking candidates is enough.
Prefer smooth functions Prefer candidate which minimizes slope at goal value
Information-Based OptimizationInformation-Based Optimization
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Problem: Only Good for One DimensionProblem: Only Good for One Dimension
In 1-D, candidates are ranked with respect to immediate neighbors.
What are “immediate neighbors” in multi-dimensional space?
Intuition: Closer points have greater relevance.
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Solution: ShadowingSolution: Shadowing
Point b shadows point a from point d if: b is closer to d than a, and the slope between a and b is
greater than the slope between a and d.
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Multidimensional Information-Based Optimization
Multidimensional Information-Based Optimization
Choose initial point x and evaluate f(x)
Iterate: Pick next point x according to ranking function g(x) and evaluate f(x)
Excellent for efficiently finding zeros when not rare.
Problem: Slow convergence for rare zeros, points clustered near minima
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Perform a local optimization for each top level function evaluation
Summarize information tractability
Multilevel Optimization: Generalize to n levels, with each level expediting search for level above
Solution: Multilevel OptimizationSolution: Multilevel Optimization
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SummarySummary
Initial safety refutation of hybrid system can be reformulated as satisficing optimization given a heuristic measure of relative safety.
Information-based optimization is suited to such optimization, and can be extended to multidimensions with shadowing
and sampling.
Convergence to rare unsafe trajectories: Multilevel optimization
Using an Optimization ToolboxUsing an Optimization Toolbox
You have a set of optimization methods. You have a set of observations during optimization (e.g.
function evals, local minima).
Monte CarloOptimization
Monte Carlo w/Local Optimization
Information-BasedOptimization
Information-Based w/Local Optimization
Challenge Problem: Method SwitchingChallenge Problem: Method Switching
Given: a set of iterative optimization procedures a distribution of optimization problems a set of optimization features
Learn: a policy for dynamically switching between
procedures which minimizes time to solution for such a distribution
The computer is a power tool for the mind. Power screwdrivers with Phillips bits don’t
work well with slotted screws. Understand the assumptions of the tools you
apply. You can design new bits suited to new tasks. One new bit can change the world of
computing!
ConclusionConclusion
Other ApproachesOther Approaches
Few minima: Random Local OptimizationMany minima: Simulated Annealing with
Local Optimization (Desai and Patil, 1996)For higher dimensions, you’re forever
searching corners! Direction Set Methods: Successive 1D
minimizations in different directions.
How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems
How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems
Todd W. NellerKnowledge Systems Laboratory, Stanford University
Gettysburg College, January 21, 2000
How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems
How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems
Todd W. NellerKnowledge Systems Laboratory, Stanford University
Colgate University, January 25, 2000
How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems
How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems
Todd W. NellerKnowledge Systems Laboratory, Stanford University
Lafayette College, January 27, 2000
How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems
How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems
Todd W. NellerKnowledge Systems Laboratory, Stanford University
Bowdoin College, January 31, 2000
How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems
How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems
Todd W. NellerKnowledge Systems Laboratory, Stanford University
Williams College, February 11, 2000