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How to Stall a Motor: Information-Based Optimization for Safety Refutation of Hybrid Systems. Todd W. Neller Knowledge Systems Laboratory Stanford University. Outline. Defining the problem: Will the critical satellite motor stall? Generalizing the problem: Hybrid Systems

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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

outline
Outline
  • 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 motors
Stepper Motors

a.k.a. “step motors”

t 

the problem
The Problem
  • 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

?

hybrid systems
Hybrid Systems
  • Hybrid = Discrete + Continuous
  • Example: Bouncing Ball
  • Fast Continuous Change  Discrete Change
  • More Interesting Example: Mode Switching Controllers
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 states
  • Initial Safety property - Safety over an initial duration of time
verification refutation
Verification, 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 states
  • Proof by counterexample
stepper motor safety refutation
Stepper 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
general problem statement
General Problem Statement
  • Given:
    • Hybrid system simulator for initial time duration
    • Bounds on initial conditions (parameters and variable assignments)
    • Set of unsafe states
  • Find:
    • Initial conditions such that the system enters an unsafe state during initial time
tools for initial safety refutation of hybrid systems
Tools for Initial Safety Refutation of Hybrid Systems

Nooo!

  • Generate and Test

(There has to be a better way, right?)

distance from unsafe states
Distance from Unsafe States
  • Make use of simple knowledge of problem domain to provide landscape helpful to search
refutation through optimization
Refutation through Optimization
  • Transform refutation problem into an optimization problem with a heuristic (i.e. estimated) measure of relative safety
  • Apply efficient global optimization
problem reformulation
Problem Reformulation
  • 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

simulation

evaluation

initial condition  trajectory  ranking

f

problem simulation isn t cheap
Problem: Simulation isn’t Cheap
  • 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.

f(7.11)=1.85

f(6.35)=0.92

f(6.27)=0.34

f(9.24)=7.90

satisficing
Satisficing
  • 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.
information based approach
Information-Based Approach

Assume: continuous, flat functions more likely

information based optimization
Information-Based Optimization
  • 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
problem only good for one dimension
Problem: 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.
solution shadowing
Solution: 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.
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
solution multilevel optimization
Solution: Multilevel Optimization
  • 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
summary
Summary
  • 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 toolbox
Using 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 Carlo

Optimization

Information-Based

Optimization

Monte Carlo w/

Local Optimization

Information-Based w/

Local Optimization

challenge problem method switching
Challenge 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
conclusion
Conclusion
  • 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!
other approaches
Other Approaches
  • Few minima: Random Local Optimization
  • Many 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 systems1

How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems

Todd W. Neller

Knowledge Systems Laboratory, Stanford University

Gettysburg College, January 21, 2000

how to stall a motor information based optimization for safety refutation of hybrid systems2

How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems

Todd W. Neller

Knowledge Systems Laboratory, Stanford University

Colgate University, January 25, 2000

how to stall a motor information based optimization for safety refutation of hybrid systems3

How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems

Todd W. Neller

Knowledge Systems Laboratory, Stanford University

Lafayette College, January 27, 2000

how to stall a motor information based optimization for safety refutation of hybrid systems4

How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems

Todd W. Neller

Knowledge Systems Laboratory, Stanford University

Bowdoin College, January 31, 2000

how to stall a motor information based optimization for safety refutation of hybrid systems5

How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems

Todd W. Neller

Knowledge Systems Laboratory, Stanford University

Williams College, February 11, 2000