52 lines
1.4 KiB
Matlab
52 lines
1.4 KiB
Matlab
clear;
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s = tf('s');
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k = ureal('k', 2.5, 'Range', [2, 3]);
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tau = ureal('tau', 2.5, 'Range', [2, 3]);
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theta = ureal('theta', 2.5, 'Range', [2, 3]);
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% delay is approximated using pade:
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% e^(-theta*s) = e^(-theta*s)/2 1 - theta*s/2
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% ------------- = --------------
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% e^(theta*s)/2 1 + theta*s/2
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G = k/(tau*s+1) * (1-theta*s)/2 / ((1+theta*s)/2);
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G_nom = k.NominalValue/(tau.NominalValue*s+1);
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% G = (1+I_m)G_nom => I_m = (G-G_nom)/G_nom
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I_m = (G-G_nom)/G_nom;
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% get wcgain
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w = logspace(-3, 3, 40);
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[wcg, wcu, info] = wcgain(I_m, w);
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% fit worst-case magnitude to obtain a closed form of the multiplicative
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% uncertainty
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data = frd(info.Bounds(:, 2), w);
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% use a second-order model
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I_m_analytic = tf(fitmagfrd(data, 2));
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% For MIMO systems, the division in the expression for I_m doesn't make
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% sense
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% This is another way to compute it:
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% generate some instances of the uss
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systems = usample(G, 20);
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data = frd(systems, w);
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% cover the frequency response data using an order 2 multiplicative
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% uncertainty
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% ucover returns [G, Info] where G is the uncertain system given by
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% G(s) = [I+W1(s) Delta(s)] G_nom(s)
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% W1(s) can be recovered from Info.W1
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[G2, Info] = ucover(data, G_nom, 2);
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I_m_analytic_2 = tf(Info.W1);
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sigma(I_m, w); hold;
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semilogx(w, 20*log10(info.Bounds(:, 2)), 'green');
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sigma(I_m_analytic, 'red');
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sigma(I_m_analytic_2, 'yellow');
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I_m_analytic
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