Model uncertainty leads to uncertainty in predicting system performance,
which is the justification for a variety of robust control analysis and
design methods. Analyzing the frequency response envelope (value set)
of a model with parametric uncertainty is a precursor to using these
methods, and is itself a challenging problem. Despite considerable
progress in extending Kharitonov's theorem to analyzing models with a
variety of uncertainty structures, most uncertain physical systems will
not lead to a structure for which results are directly applicable. We
have developed two distinct approaches to synthesizing value sets of
uncertain models. The first uses projection, optimization, and an
adaptive angular sweep algorithm to obtain a convex hull approximation
of a value set. The second uses bond graphs to synthesize
tree-structured transfer functions with disjoint parameters, which
greatly simplify value set synthesis. In addition to facilitating the
standard problem of value set synthesis, the tree-structured transfer
functions have other applications, for example, model order reduction.
The background, development, analysis, and application of uncertain
physical system models will be discussed.