armin@ai.toronto.edu (Armin Haken) (02/12/88)
There will be an AI seminar on Tuesday 23 February at 2PM in room
SF 1105, given by Dr. Elisha Sacks of MIT. The abstract follows.
Mr. Sacks is a candidate for a faculty position. Hosting is Hector
Levesque.
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Automatic Qualitative Analysis of Ordinary Differential Equations
Using Piecewise Linear Approximations
by Elisha Sacks
This talk explores automating the qualitative analysis of physical
systems. Scientists and engineers model many physical systems with
ordinary differential equations. They deduce the behavior of the
systems by analyzing the equations. Most realistic models are
nonlinear, hence difficult or impossible to solve explicitly. Analysts
must resort to approximations or to sophisticated mathematical
techniques. I describe 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.
PLR uses the standard phase space representation. It builds a
composite phase diagram for a piecewise linear system by pasting
together the local phase diagrams of its linear regions. It employs a
combination of geometric and algebraic reasoning to determine whether
the trajectories in each linear region cross into adjoining regions and
summarizes the results in a transition graph. Transition graphs
explicitly express many qualitative properties of systems. PLR 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.
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|| Armin Haken armin@ai.toronto.edu ||
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