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Simplify representation of uncertain object
B = simplify(A) B = simplify(A,'full') B = simplify(A,'basic') B = simplify(A,'class')
B = simplify(A) performs model-reduction-like techniques to detect and eliminate redundant copies of uncertain elements. Depending on the result, the class of 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.
B = simplify(A,'full') overrides all uncertain element's AutoSimplify property, and uses 'full' reduction techniques.
B = simplify(A,'basic') overrides all uncertain element's AutoSimplify property, and uses 'basic' reduction techniques.
B = simplify(A,'class') does not perform reduction. However, any uncertain elements in 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.