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Decomposing Uncertain Objects
Decomposing Uncertain Objects
Each uncertain object (umat, uss, ufrd) is a generalized feedback connection (lft) of a
not-uncertain object (e.g., double, ss, frd) with a diagonal augmentation of uncertain
elements (ureal, ultidyn, ucomplex, ucomplexm, udyn). In robust control jargon, if
the uncertain elements are normalized, this decomposition is often called “the M/D form.”
The purpose of the uncertain objects (ureal, ultidyn, umat, uss, etc.) is to
hide this underlying decomposition, and allow the user to focus on modeling and
analyzing uncertain systems, rather than the details of correctly propagating the M/D
representation in manipulations. Nevertheless, advanced users may want access to the
familiar M/D form. The command lftdata accomplishes this decomposition.
Since ureal, ucomplex and ucomplexm do not have their NominalValue necessarily at
zero, and in the case of ureal objects, are not symmetric about the NominalValue, some
details are required in describing the decomposition.
Normalizing Functions for Uncertain Elements
Associated with each uncertain element is a normalizing function. The normalizing
function maps the uncertain element into a normalized uncertain element.
If ρ is an uncertain real parameter, with range [L R] and nominal value N, then the
normalizing function F is
F(r) =
A + Br
C + Dr
with the property that for all ρ satisfying L ≤ ρ ≤ R, it follows that –1 ≤ F(ρ) ≤ 1,
moreover, F(L) = –1, F(N) = 0, and F(R) = 1. If the nominal value is centered in the range,
then it is easy to conclude that
R+L
R-L
2
B=
R-L
C =1
D = 0.
A=
1-73