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 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 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 Problem ?

Hybrid Systems Hybrid = Discrete + Continuous Example: Bouncing Ball Fast Continuous Change  Discrete Change More Interesting Example: Mode Switching Controllers

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 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 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 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

Generate and Test Tools for Initial Safety Refutation of Hybrid Systems (There has to be a better way, right?)

Distance from Unsafe States Make use of simple knowledge of problem domain to provide landscape helpful to search

Refutation through Optimization Transform refutation problem into an optimization problem with a heuristic (i.e. estimated) measure of relative safety Apply efficient global optimization

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 Reformulation initial condition  trajectory  ranking f simulationevaluation

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 Cheap f(6.27)=0.34 f(6.35)=0.92 f(7.11)=1.85 f(9.24)=7.90

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 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 Optimization

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 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 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

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 Optimization

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 You have a set of optimization methods. You have a set of observations during optimization (e.g. function evals, local minima). Monte Carlo Optimization Monte Carlo w/ Local Optimization Information-Based Optimization Information-Based w/ Local Optimization

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

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! Conclusion

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.

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