Automatic qualitative analysis of dynamic systems using piecewise linear approximations

摘要

This paper explores automating the qualitative analysis of physical systems. It describes a program, called PLR (for Piecewise Linear Reasoner), that formalizes an analysis strategy employed by experts. PLR takes parameterized ordinary differential equations as input and produces a qualitative description of the solutions for all initial values. It approximates intractable nonlinear systems with piecewise linear ones, analyzes the approximations, and draws conclusions about the original systems. It chooses approximations that are accurate enough to reproduce the essential properties of their nonlinear prototypes, yet simple enough to be analyzed completely and efficiently. It derives additional properties, such as boundedness or periodicity, by theoretical methods. PLR's analysis depends on abstract properties of systems rather than on specific numeric values. This makes its conclusions more robust and enables it to handle parameterized equations transparently. I demonstrate PLR on several common nonlinear systems and on published examples from mechanical engineering.