A Theory of Discontinuities in Physical System Models
Pieter J. Mosterman and Gautam Biswas
Center for Intelligent Systems
Abstract
Physical systems are by nature continuous, but often display
nonlinear behaviors that makes them hard to analyze.
Typically, these nonlinearities occur at a time scale that
is much smaller than the time scale at which gross system behavior needs
to be described. In other
situations, nonlinear effects are small and of a parasitic nature.
To achieve efficiency and clarity in building complex system models,
and to reduce computational complexity in the analysis of system
behavior, modelers often abstract away any parasitic component
parameter effects, and analyze the system at more abstract time
scales. However, these abstractions often introduce
abrupt, instantaneous changes in system behavior. To accommodate mixed
continuous and discrete behavior, this paper develops a
hybrid modeling formalism that dynamically constructs bond graph
model fragments that govern system behavior during continuous
operation. When threshold values are crossed, a meta-level control
model invokes discontinuous state and model configuration changes.
Discontinuities violate physical principles of conservation of energy
and continuity of power, but the principle of invariance of state
governs model behavior
when the control module is active. Conservation of energy and continuity
of power again govern behavior generation as soon as a new model configuration
is established.
This allows for maximally constrained continuous model fragments.
The two primary contributions of this paper are an algorithm for inferring
the correct new mode and state variable values in the hybrid modeling
framework, and a
verification scheme that ensures hybrid models conform to physical system
principles based on the principles of divergence of time
and temporal evolution in behavior transtions. These principles are
employed in energy phase space analysis to verify physical consistency
of models.
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