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

1. How to Stall a Motor:Information-Based Optimization for Safety Refutation of Hybrid Systems Todd W. Neller Knowledge Systems Laboratory Stanford University

2. 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?

3. Stepper Motors   a.k.a. “step motors” t 

4. 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 ?

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

6. 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

7. 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

8. 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

9. 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

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

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

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

13. 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

14. 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

15. 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.

16. Information-Based Approach Assume: continuous, flat functions more likely

17. 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

18. 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.

19. 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.

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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!

26. 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.

27. 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

28. 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

29. 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

30. 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

31. 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