LFRT version 2.0: µ-analysis



This page describes four functions that generate from an LFR-object the input arguments of four standard Matlab functions for µ-analysis (mu, mustab, mubnd and mussv).


Function lfr2mu

Generation of a model with an uncertain pole close to the imaginary axis at about 10 Rd/s.

lfrs a b c
A = [-1+5*a -10*(1+b);10*(1+c^2) -1-a*b*c];
B = [1+b;1*c];
C = [1 1];
D = 0;

sys = abcd2lfr([A B;C D],2);
sys = minlfr(sys);

µ-analysis using the function mu.

frequ = logspace(0,2,20);

[M,blk] = lfr2mu(sys,frequ);
bnds = mu(M,blk,'wlc');

semilogx(frequ,bnds(1:20,1),'c+'); hold on
semilogx(frequ,bnds(1:20,2),'g.');

See figure below.

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Function lfr2mubnd

Same as above but using an LMI-based µ upper bound.

[M,delta] = lfr2mubnd(sys,frequ);
for ii = 1:length(frequ);
   mu2(ii) = mubnd(M{ii},delta,1e-6,'off');
end;

semilogx(frequ,mu2,'rd');

See figure below.

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Function lfr2mussv

Same as above using the function mussv of version 3 of the Robust Control Toolbox. Here we use the low precision default options.

[M,blk] = lfr2mussv(sys,frequ);
[bnds2,muinfo] = mussv(M,blk);

plot(bnds2);

muanalysis

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Function lfr2mustab

Computation of the stability margin (inverse of the maximum value of the µ-curve) using the function mustab (the function mustab doesn't work for this example on some platforms).

[P,delta] = lfr2mustab(sys);
[margin1,frequ0] = mustab(P,delta);

...............
 Mu upper bound (peak value): 2.556e+00
 Robust stability margin    : 3.912e-01

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