Non-Intersection of Trajectories In Qualitative Phase Space: A Global Constraint for Qualitative Simulation

Wood W. Lee, Benjamin J. Kuipers

The QSIM algorithm is useful for predicting the possible qualitative behaviors of a system, given a qualitative differential equation (QDE) describing its structure and an initial state. Although QSIM is guaranteed to predict all real possibilities, it may also predict spurious behaviors which, if uncontrolled, can lead to an intractably branching tree of behaviors. Prediction of spurious behaviors is due to an interaction between the qualitative level of description and the local state-tostate perspective on the behavior taken by the algorithm. In this paper, we describe the non-intersection constraint, which embodies the requirement that a trajectory in phase space cannot intersect itself. We develop a criterion for applying it to all second order systems. It eliminates a major source of spurious predictions. Using it with the curvature constraint tightens simulation to the point where system-specific constraints can be applied more effectively. We demonstrate this on damped oscillatory systems with potentially nonlinear monotonic restoring force and damping terms. Its introduction represents significant progress towards tightening QSIM simulation.

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