1. Polyhedral Containment Check for a Linear Hybrid Automata Reachability Procedure Industry mentor: Steve Vestal Honeywell, Inc. Team 5: Sonja Petrović (presenter), Jessica Conway, Ali Khoujmane, Gary Kilper, Harun Kurkcu, Rochelle Pereira

2. Presentation Outline Background info: Finite state & Hybrid automatons Motivation for project 3 approaches –LP approach LSPD Redundancy reduction –QP approach –Vertex enumeration & vertex containment

3. A Finite State Automaton Discrete states and connecting edges Example: a vending machine A hybrid automaton is augmented by continuous variables S1S1 S2S2 e1e1 e2e2

4. A Linear Hybrid Automaton if x≥7 if x≤5 x x = 7 x = 5 Control Box Faucet State: Faucet OFF, x = 6

5. MOTIVATION A hybrid state is a region consisting of a discrete state and a polyhedron –Polyhedron  set of possible values for continuous variables –Reachable polyhedron  can be reached from a previously reached polyhedron Reachability problem: decide if a given hybrid state is reachable by any path from the initial state

6. Reachability Problem Difficulty: new hybrid state may have already been visited before Solution: check against the list of previously visited states –Is the current polyhedron contained in any previously visited one? –Set Outer  {x | Ax  b}, Inner  {x | Cx  d}. Does x  Inner  x  Outer?

7. Containment Check Initial approach: check inner polyhedron against outer facet by solving LP problem Improved idea: solve one QP problem by checking inner against the whole outer New approach: vertex enumeration and point containment

8. Solving LP: Simplex Find a feasible basic solution (a vertex of the polyhedron). If there is an “adjacent” feasible basic solution that is more optimal, then move to that vertex along a connecting edge. If not, then stop. y ≤ 1 x + y ≤ 2 x ≥ 0 y ≥ 0 Maximize 2x + y

9. Problem with Simplex Degeneracy due to redundant constraints! Degeneracy: have more edges at a vertex than there are variables This may result in stalling or cycling around the same vertex when attempting vertex- vertex travel

10. Why Reduce Redundancies? A positive slope means: More constraints correspond to more stalling.

11. SOLUTIONS Least squares primal dual algorithm to avoid stalling (LSPD) Redundancy reduction QP formulation New idea: Vertex enumeration & containment

12. Solution – LSPD LSPD solves the standard LP problem: Find a vector x that minimizes cx such that Ax = b and x ≥ 0. Finishes in finitely many steps Uses Non-Negative Least Squares (NNLS) algorithm to solve a series of least squares sub- problems Guarantees travel in the optimal direction towards the best feasible solution, not along the edges

13. Solution – Redundancy Reduction Can help diminish the amount of degeneracy in a problem This can be done using the brute force approach – removing one constraint at a time. This is inefficient. Future Research: explore other ways to reduce redundancy in a system of linear inequalities

14. Solution – QP Formulation Goal: show that there doesn’t exist a point that is both inside the Inner and outside the Outer Objective function is maximized when the largest constraint violation is recognized: max [y T (Ax – b)], where x in I, y  0, and ||y|| =1. Inner  Outer when max [y T (Ax – b)]  0. So there is no containment if the objective function is positive. Future Research: Implement the quadratic programming approach using CPLEX

15. Solution – NEW IDEA Reformulate the containment problem into two steps: 1. Vertex enumeration –Transform constraint polyhedron P = {Ax  b} into a set of vertices P = {v 1, …, v n } using LRS or DD 2. Vertex containment check –Want Inner fully contained in Outer

16. Solution – Vertex Enumeration There are known algorithms and codes available !! –Lrs –DD –Even implemented in Mathematica!

17. Solution – Vertex Containment Check if each vertex of inner polyhedron is within outer polyhedron: –For Outer defined by inequalities, simply test to see if the vertex satisfies them all –Or, also enumerate the outer vertices and solve a feasibility problem using LP (but: how many vertices do we have?)

18. Future Research Proposal: Containment of Projections Phase I –Project inner vertices onto facets of outer polyhedron –Point-in-polygon test (3x3 determinants & Boolean op.) Phase II –Extend the idea of projectively invariant containment check from three to n dimensions Although vertex enumeration may be expensive, together with a “projected containment test” it may yield a more efficient algorithm.

19. Many thanks to: Steve Vestal and AFOSR IMA, NSF, U. of Minnesota, U. of Kentucky, Texas Tech, U. of Chicago, Northwestern Whoever else paid for us