Rename Data File, Export matlab scripts and data
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341
matlab/plant_identification.m
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341
matlab/plant_identification.m
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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% Excitation Data
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fs = 1e4;
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Ts = 1/fs;
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% We generate white noise with the "random number" simulink block, and we filter that noise.
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Gi = (1)/(1+s/2/pi/100);
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c2d(Gi, Ts, 'tustin')
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% Input / Output data
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% The identification data is loaded
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ux = load('mat/data_ux.mat', 't', 'ux', 'yx', 'yy');
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uy = load('mat/data_uy.mat', 't', 'uy', 'yx', 'yy');
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% We remove the first seconds where the Cercalo is turned on.
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i0x = 20*fs;
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i0y = 10*fs;
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ux.t = ux.t( i0x:end) - ux.t(i0x);
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ux.ux = ux.ux(i0x:end);
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ux.yx = ux.yx(i0x:end);
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ux.yy = ux.yy(i0x:end);
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uy.t = uy.t( i0y:end) - uy.t(i0x);
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uy.uy = uy.uy(i0y:end);
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uy.yx = uy.yx(i0y:end);
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uy.yy = uy.yy(i0y:end);
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ux.ux = ux.ux-mean(ux.ux);
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ux.yx = ux.yx-mean(ux.yx);
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ux.yy = ux.yy-mean(ux.yy);
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uy.ux = uy.ux-mean(uy.ux);
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uy.yx = uy.yx-mean(uy.yx);
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uy.yy = uy.yy-mean(uy.yy);
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figure;
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ax1 = subplot(1, 2, 1);
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plot(ux.t, ux.ux);
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xlabel('Time [s]');
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ylabel('Amplitude [V]');
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legend({'$u_x$'});
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ax2 = subplot(1, 2, 2);
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hold on;
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plot(ux.t, ux.yx, 'DisplayName', '$y_x$');
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plot(ux.t, ux.yy, 'DisplayName', '$y_y$');
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hold off;
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xlabel('Time [s]');
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ylabel('Amplitude [V]');
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legend()
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linkaxes([ax1,ax2],'x');
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xlim([ux.t(1), ux.t(end)])
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% #+NAME: fig:identification_ux
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% #+CAPTION: Identification signals when exciting the $x$ axis ([[./figs/identification_ux.png][png]], [[./figs/identification_ux.pdf][pdf]])
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% [[file:figs/identification_ux.png]]
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figure;
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ax1 = subplot(1, 2, 1);
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plot(uy.t, uy.uy);
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xlabel('Time [s]');
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ylabel('Amplitude [V]');
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legend({'$u_y$'});
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ax2 = subplot(1, 2, 2);
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hold on;
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plot(uy.t, uy.yy, 'DisplayName', '$y_y$');
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plot(uy.t, uy.yx, 'DisplayName', '$y_x$');
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hold off;
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xlabel('Time [s]');
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ylabel('Amplitude [V]');
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legend()
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linkaxes([ax1,ax2],'x');
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xlim([uy.t(1), uy.t(end)])
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% Estimation of the Frequency Response Function Matrix
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% We compute an estimate of the transfer functions.
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[tf_ux_yx, f] = tfestimate(ux.ux, ux.yx, hanning(ceil(1*fs)), [], [], fs);
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[tf_ux_yy, ~] = tfestimate(ux.ux, ux.yy, hanning(ceil(1*fs)), [], [], fs);
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[tf_uy_yx, ~] = tfestimate(uy.uy, uy.yx, hanning(ceil(1*fs)), [], [], fs);
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[tf_uy_yy, ~] = tfestimate(uy.uy, uy.yy, hanning(ceil(1*fs)), [], [], fs);
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figure;
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ax11 = subplot(2, 2, 1);
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hold on;
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plot(f, abs(tf_ux_yx))
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title('Frequency Response Function $\frac{y_x}{u_x}$')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
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ylabel('Amplitude')
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hold off;
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ax12 = subplot(2, 2, 2);
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hold on;
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plot(f, abs(tf_ux_yy))
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title('Frequency Response Function $\frac{y_x}{u_y}$')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
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hold off;
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ax21 = subplot(2, 2, 3);
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hold on;
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plot(f, abs(tf_uy_yx))
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title('Frequency Response Function $\frac{y_y}{u_x}$')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
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ylabel('Amplitude')
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xlabel('Frequency [Hz]')
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hold off;
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ax22 = subplot(2, 2, 4);
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hold on;
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plot(f, abs(tf_uy_yy))
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title('Frequency Response Function $\frac{y_y}{u_y}$')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
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xlabel('Frequency [Hz]')
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hold off;
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linkaxes([ax11,ax12,ax21,ax22],'x');
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xlim([10, 1000]);
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linkaxes([ax11,ax12,ax21,ax22],'y');
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ylim([1e-2, 1e3])
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% Coherence
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[coh_ux_yx, f] = mscohere(ux.ux, ux.yx, hanning(ceil(1*fs)), [], [], fs);
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[coh_ux_yy, ~] = mscohere(ux.ux, ux.yy, hanning(ceil(1*fs)), [], [], fs);
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[coh_uy_yx, ~] = mscohere(uy.uy, uy.yx, hanning(ceil(1*fs)), [], [], fs);
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[coh_uy_yy, ~] = mscohere(uy.uy, uy.yy, hanning(ceil(1*fs)), [], [], fs);
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figure;
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ax11 = subplot(2, 2, 1);
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hold on;
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plot(f, coh_ux_yx)
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set(gca, 'Xscale', 'log');
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title('Coherence $\frac{y_x}{u_x}$')
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ylabel('Coherence')
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hold off;
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ax12 = subplot(2, 2, 2);
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hold on;
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plot(f, coh_ux_yy)
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set(gca, 'Xscale', 'log');
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title('Coherence $\frac{y_x}{u_y}$')
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hold off;
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ax21 = subplot(2, 2, 3);
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hold on;
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plot(f, coh_uy_yx)
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set(gca, 'Xscale', 'log');
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title('Coherence $\frac{y_y}{u_x}$')
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ylabel('Coherence')
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xlabel('Frequency [Hz]')
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hold off;
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ax22 = subplot(2, 2, 4);
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hold on;
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plot(f, coh_uy_yy)
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set(gca, 'Xscale', 'log');
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title('Coherence $\frac{y_y}{u_y}$')
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xlabel('Frequency [Hz]')
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hold off;
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linkaxes([ax11,ax12,ax21,ax22],'x');
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xlim([10, 1000]);
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linkaxes([ax11,ax12,ax21,ax22],'y');
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ylim([0, 1])
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% Extraction of a transfer function matrix
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% First we define the initial guess for the resonance frequencies and the weights associated.
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freqs_res = [410, 250]; % [Hz]
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freqs_res_weights = [10, 10]; % [Hz]
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% From the number of resonance frequency we want to fit, we define the order =N= of the system we want to obtain.
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N = 2*length(freqs_res);
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% We then make an initial guess on the complex values of the poles.
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xi = 0.001; % Approximate modal damping
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poles = [2*pi*freqs_res*(xi + 1i), 2*pi*freqs_res*(xi - 1i)];
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% We then define the weight that will be used for the fitting.
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% Basically, we want more weight around the resonance and at low frequency (below the first resonance).
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% Also, we want more importance where we have a better coherence.
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weight = ones(1, length(f));
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% weight = G_coh';
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% alpha = 0.1;
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% for freq_i = 1:length(freqs_res)
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% weight(f>(1-alpha)*freqs_res(freq_i) & omega<(1 + alpha)*2*pi*freqs_res(freq_i)) = freqs_res_weights(freq_i);
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% end
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% Ignore data above some frequency.
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weight(f>1000) = 0;
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figure;
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hold on;
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plot(f, weight);
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plot(freqs_res, ones(size(freqs_res)), 'rx');
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hold off;
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xlabel('Frequency [Hz]');
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xlabel('Weight Amplitude');
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set(gca, 'xscale', 'log');
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xlim([f(1), f(end)]);
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% #+NAME: fig:weights
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% #+CAPTION: Weights amplitude ([[./figs/weights.png][png]], [[./figs/weights.pdf][pdf]])
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% [[file:figs/weights.png]]
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% When we set some options for =vfit3=.
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opts = struct();
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opts.stable = 1; % Enforce stable poles
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opts.asymp = 1; % Force D matrix to be null
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opts.relax = 1; % Use vector fitting with relaxed non-triviality constraint
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opts.skip_pole = 0; % Do NOT skip pole identification
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opts.skip_res = 0; % Do NOT skip identification of residues (C,D,E)
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opts.cmplx_ss = 0; % Create real state space model with block diagonal A
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opts.spy1 = 0; % No plotting for first stage of vector fitting
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opts.spy2 = 0; % Create magnitude plot for fitting of f(s)
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% We define the number of iteration.
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Niter = 5;
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% An we run the =vectfit3= algorithm.
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for iter = 1:Niter
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[SER_ux_yx, poles, ~, fit_ux_yx] = vectfit3(tf_ux_yx.', 1i*2*pi*f, poles, weight, opts);
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end
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for iter = 1:Niter
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[SER_uy_yx, poles, ~, fit_uy_yx] = vectfit3(tf_uy_yx.', 1i*2*pi*f, poles, weight, opts);
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end
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for iter = 1:Niter
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[SER_ux_yy, poles, ~, fit_ux_yy] = vectfit3(tf_ux_yy.', 1i*2*pi*f, poles, weight, opts);
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end
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for iter = 1:Niter
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[SER_uy_yy, poles, ~, fit_uy_yy] = vectfit3(tf_uy_yy.', 1i*2*pi*f, poles, weight, opts);
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end
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figure;
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ax11 = subplot(2, 2, 1);
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hold on;
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plot(f, abs(tf_ux_yx))
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plot(f, abs(fit_ux_yx))
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title('Frequency Response Function $\frac{y_x}{u_x}$')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
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ylabel('Amplitude')
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hold off;
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ax12 = subplot(2, 2, 2);
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hold on;
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plot(f, abs(tf_ux_yy))
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plot(f, abs(fit_ux_yy))
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title('Frequency Response Function $\frac{y_x}{u_y}$')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
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hold off;
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ax21 = subplot(2, 2, 3);
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hold on;
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plot(f, abs(tf_uy_yx))
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plot(f, abs(fit_uy_yx))
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title('Frequency Response Function $\frac{y_y}{u_x}$')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
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ylabel('Amplitude')
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xlabel('Frequency [Hz]')
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hold off;
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ax22 = subplot(2, 2, 4);
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hold on;
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plot(f, abs(tf_uy_yy))
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plot(f, abs(fit_uy_yy))
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title('Frequency Response Function $\frac{y_y}{u_y}$')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
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xlabel('Frequency [Hz]')
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hold off;
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linkaxes([ax11,ax12,ax21,ax22],'x');
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xlim([10, 1000]);
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linkaxes([ax11,ax12,ax21,ax22],'y');
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ylim([1e-2, 1e3])
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% #+NAME: fig:identification_matrix_fit
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% #+CAPTION: Transfer Function Extraction of the FRF matrix ([[./figs/identification_matrix_fit.png][png]], [[./figs/identification_matrix_fit.pdf][pdf]])
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% [[file:figs/identification_matrix_fit.png]]
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% And finally, we create the identified state space model:
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G_ux_yx = minreal(ss(full(SER_ux_yx.A),SER_ux_yx.B,SER_ux_yx.C,SER_ux_yx.D));
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G_uy_yx = minreal(ss(full(SER_uy_yx.A),SER_uy_yx.B,SER_uy_yx.C,SER_uy_yx.D));
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G_ux_yy = minreal(ss(full(SER_ux_yy.A),SER_ux_yy.B,SER_ux_yy.C,SER_ux_yy.D));
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G_uy_yy = minreal(ss(full(SER_uy_yy.A),SER_uy_yy.B,SER_uy_yy.C,SER_uy_yy.D));
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G = [G_ux_yx, G_uy_yx;
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G_ux_yy, G_uy_yy];
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save('mat/plant.mat', 'G');
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