Simplify representation of uncertain object

B = simplify(A) B = simplify(A,'full') B = simplify(A,'basic') B = simplify(A,'class')

performs model-reduction-like techniques to detect and eliminate redundant
copies of uncertain elements. Depending on the result, the class of `B = simplify(A)`

`B`

may
be lower than `A`

. The `AutoSimplify`

property
of each uncertain element in `A`

governs what reduction
methods are used. After reduction, any uncertain element which does
not actually affect the result is deleted from the representation.

overrides all uncertain element's `B = simplify(A,'full')`

`AutoSimplify`

property,
and uses `'full'`

reduction techniques.

overrides all uncertain element's `B = simplify(A,'basic')`

`AutoSimplify`

property,
and uses` 'basic'`

reduction techniques.

does not perform reduction. However, any uncertain elements in `B = simplify(A,'class')`

`A`

with
zero occurences are eliminated, and the class of `B`

may
be lower than the class of `A`

.

Create a simple `umat`

with a single uncertain
real parameter. Select specific elements, note that result remains
in class `umat`

. Simplify those same elements, and
note that class changes.

p1 = ureal('p1',3,'Range',[2 5]); L = [2 p1]; L(1) UMAT: 1 Rows, 1 Columns L(2) UMAT: 1 Rows, 1 Columns p1: real, nominal = 3, range = [2 5], 1 occurrence simplify(L(1)) ans = 2 simplify(L(2)) Uncertain Real Parameter: Name p1, NominalValue 3, Range [2 5]

Create four uncertain real parameters, with a default value
of `AutoSimplify('basic')`

, and define a high order
polynomial [1].

m = ureal('m',125000,'Range',[100000 150000]); xcg = ureal('xcg',.23,'Range',[.15 .31]); zcg = ureal('zcg',.105,'Range',[0 .21]); va = ureal('va',80,'Range',[70 90]); cw = simplify(m/(va*va)*va,'full') UMAT: 1 Rows, 1 Columns m: real, nominal = 1.25e+005, range = [100000 150000], 1 occurrence va: real, nominal = 80, range = [70 90], 1 occurrence cw = m/va; fac2 = .16726*xcg*cw*cw*zcg - .17230*xcg*xcg*cw ... -3.9*xcg*cw*zcg - .28*xcg*xcg*cw*cw*zcg ... -.07*xcg*xcg*zcg + .29*xcg*xcg*cw*zcg ... + 4.9*xcg*cw - 2.7*xcg*cw*cw ... +.58*cw*cw - 0.25*xcg*xcg - 1.34*cw ... +100.1*xcg -14.1*zcg - 1.91*cw*cw*zcg ... +1.12*xcg*zcg + 24.6*cw*zcg ... +.45*xcg*xcg*cw*cw - 46.85 UMAT: 1 Rows, 1 Columns m: real, nominal = 1.25e+005, range = [100000 150000], 18 occurrences va: real, nominal = 80, range = [70 90], 8 occurrences xcg: real, nominal = 0.23, range = [0.15 0.31], 18 occurrences zcg: real, nominal = 0.105, range = [0 0.21], 1 occurrence

The result of the high-order polynomial is an inefficient representation
involving 18 copies of `m`

, 8 copies of `va`

,
18 copies of `xcg`

and 1 copy of `zcg`

.
Simplify the expression, using the `'full'`

simplification
algorithm

fac2s = simplify(fac2,'full') UMAT: 1 Rows, 1 Columns m: real, nominal = 1.25e+005, range = [100000 150000], 4 occurrences va: real, nominal = 80, range = [70 90], 4 occurrences xcg: real, nominal = 0.23, range = [0.15 0.31], 2 occurrences zcg: real, nominal = 0.105, range = [0 0.21], 1 occurrence

which results in a much more economical representation.

Alternatively, change the `AutoSimplify`

property
of each parameter to `'full'`

before forming the
polynomial.

m.AutoSimplify = 'full'; xcg.AutoSimplify = 'full'; zcg.AutoSimplify = 'full'; va.AutoSimplify = 'full';

You can form the polynomial, which immediately gives a low order representation.

cw = m/va; fac2f = .16726*xcg*cw*cw*zcg - .17230*xcg*xcg*cw ... -3.9*xcg*cw*zcg - .28*xcg*xcg*cw*cw*zcg ... -.07*xcg*xcg*zcg + .29*xcg*xcg*cw*zcg ... + 4.9*xcg*cw - 2.7*xcg*cw*cw ... +.58*cw*cw - 0.25*xcg*xcg - 1.34*cw ... +100.1*xcg -14.1*zcg - 1.91*cw*cw*zcg ... +1.12*xcg*zcg + 24.6*cw*zcg ... +.45*xcg*xcg*cw*cw - 46.85 UMAT: 1 Rows, 1 Columns m: real, nominal = 1.25e+005, range = [100000 150000], 4 occurrences va: real, nominal = 80, range = [70 90], 4 occurrences xcg: real, nominal = 0.23, range = [0.15 0.31], 2 occurrences zcg: real, nominal = 0.105, range = [0 0.21], 1 occurrence

Create two real parameters, `da`

and `dx`

,
and a 2-by-3 matrix, `ABmat`

, involving polynomial
expressions in the two real parameters .

da = ureal('da',0,'Range',[-1 1]); dx = ureal('dx',0,'Range',[-1 1]); a11 = -.32 + da*(.8089 + da*(-.987 + 3.39*da)) + .15*dx; a12 = .934 + da*(.0474 - .302*da); a21 = -1.15 + da*(4.39 + da*(21.97 - 561*da*da)) ... + dx*(9.65 - da*(55.7 + da*177)); a22 = -.66 + da*(1.2 - da*2.27) + dx*(2.66 - 5.1*da); b1 = -0.00071 + da*(0.00175 - da*.00308) + .0011*dx; b2 = -0.031 + da*(.078 + da*(-.464 + 1.37*da)) + .0072*dx; ABmat = [a11 a12 b1;a21 a22 b2] UMAT: 2 Rows, 3 Columns da: real, nominal = 0, range = [-1 1], 19 occurrences dx: real, nominal = 0, range = [-1 1], 2 occurrences

Use `'full'`

simplification to reduce the complexity
of the description.

ABmatsimp = simplify(ABmat,'full') UMAT: 2 Rows, 3 Columns da: real, nominal = 0, range = [-1 1], 7 occurrences dx: real, nominal = 0, range = [-1 1], 2 occurrences

Alternatively, you can set the parameter's `AutoSimplify`

property
to `'full'`

.

da.AutoSimplify = 'full'; dx.AutoSimplify = 'full';

Now you can rebuild the matrix

a11 = -.32 + da*(.8089 + da*(-.987 + 3.39*da)) + .15*dx; a12 = .934 + da*(.0474 - .302*da); a21 = -1.15 + da*(4.39 + da*(21.97 - 561*da*da)) ... + dx*(9.65 - da*(55.7 + da*177)); a22 = -.66 + da*(1.2 - da*2.27) + dx*(2.66 - 5.1*da); b1 = -0.00071 + da*(0.00175 - da*.00308) + .0011*dx; b2 = -0.031 + da*(.078 + da*(-.464 + 1.37*da)) + .0072*dx; ABmatFull = [a11 a12 b1;a21 a22 b2] UMAT: 2 Rows, 3 Columns da: real, nominal = 0, range = [-1 1], 7 occurrences dx: real, nominal = 0, range = [-1 1], 2 occurrences

Multidimensional model reduction and realization theory are
only partially complete theories. The heuristics used by `simplify`

are
that - heuristics. The order in which expressions involving uncertain
elements are built up, eg., distributing across addition and multiplication,
can affect the details of the representation (i.e., the number of
occurences of a `ureal`

in an uncertain matrix).
It is possible that `simplify`

's naive methods cannot
completely resolve these differences, so one may be forced to work
with "nonminimal" representations of uncertain systems.

[1] Varga, A. and G. Looye, "Symbolic
and numerical software tools for LFT-based low order uncertainty
modeling," *IEEE International Symposium on Computer
Aided Control System Design, *1999, pp. 5-11.

[2] Belcastro, C.M., K.B. Lim and E.A. Morelli,
"Computer aided uncertainty modeling for nonlinear parameter-dependent
systems Part II: F-16 example," *IEEE International
Symposium on Computer Aided Control System Design,* 1999,
pp. 17-23.

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