how to stall a motor: information-based optimization for safety refutation of hybrid systems todd w....

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.



Todd W. NellerKnowledge Systems Laboratory, Stanford University

Gettysburg College, January 21, 2000




Colgate University, January 25, 2000




Lafayette College, January 27, 2000




Bowdoin College, January 31, 2000




Williams College, February 11, 2000

how to stall a motor: information-based optimization for safety refutation of hybrid systems todd w....

Documents

initial time slide

refutation of safety

refutation problem

initial duration of

optimization problem

acceleration slide

safety safety property

counterexample slide