424 lines
14 KiB
Mathematica
424 lines
14 KiB
Mathematica
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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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addpath('./mat/');
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% Load Data
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% The data is loaded in the Matlab workspace.
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id_ol = load('identification_noise_bis.mat', 'd', 'acc_1', 'acc_2', 'geo_1', 'geo_2', 'f_meas', 'u', 't');
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% Then, any offset is removed.
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id_ol.d = detrend(id_ol.d, 0);
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id_ol.acc_1 = detrend(id_ol.acc_1, 0);
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id_ol.acc_2 = detrend(id_ol.acc_2, 0);
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id_ol.geo_1 = detrend(id_ol.geo_1, 0);
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id_ol.geo_2 = detrend(id_ol.geo_2, 0);
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id_ol.f_meas = detrend(id_ol.f_meas, 0);
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id_ol.u = detrend(id_ol.u, 0);
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% Excitation Signal
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% The generated voltage used to excite the system is a white noise and can be seen in Figure [[fig:excitation_signal_first_identification]].
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figure;
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plot(id_ol.t, id_ol.u)
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xlabel('Time [s]'); ylabel('Voltage [V]');
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% Identified Plant
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% The transfer function from the excitation voltage to the mass displacement and to the force sensor stack voltage are identified using the =tfestimate= command.
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Ts = id_ol.t(2) - id_ol.t(1);
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win = hann(ceil(10/Ts));
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[tf_fmeas_est, f] = tfestimate(id_ol.u, id_ol.f_meas, win, [], [], 1/Ts); % [V/V]
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[tf_G_ol_est, ~] = tfestimate(id_ol.u, id_ol.d, win, [], [], 1/Ts); % [m/V]
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% The bode plots of the obtained dynamics are shown in Figures [[fig:force_sensor_bode_plot]] and [[fig:displacement_sensor_bode_plot]].
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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plot(f, abs(tf_fmeas_est), '-')
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hold off;
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
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ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
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ax2 = nexttile;
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hold on;
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plot(f, 180/pi*angle(tf_fmeas_est), '-')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
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ylabel('Phase'); xlabel('Frequency [Hz]');
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hold off;
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2], 'x');
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xlim([1, 1e3]);
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% #+name: fig:force_sensor_bode_plot
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% #+caption: Bode plot of the dynamics from excitation voltage to measured force sensor stack voltage
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% #+RESULTS:
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% [[file:figs/force_sensor_bode_plot.png]]
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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plot(f, abs(tf_G_ol_est), '-')
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hold off;
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
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ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
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ax2 = nexttile;
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hold on;
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plot(f, 180/pi*angle(tf_G_ol_est), '-')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
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ylabel('Phase'); xlabel('Frequency [Hz]');
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hold off;
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2], 'x');
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xlim([1, 1e3]);
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% Simscape Model - Comparison
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% A simscape model representing the test-bench has been developed.
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% The same transfer functions as the one identified using the test-bench can be obtained thanks to the simscape model.
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% They are compared in Figure [[fig:simscape_comp_iff_plant]] and [[fig:simscape_comp_disp_plant]].
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% It is shown that there is a good agreement between the model and the experiment.
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load('piezo_amplified_3d.mat', 'int_xyz', 'int_i', 'n_xyz', 'n_i', 'nodes', 'M', 'K');
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m = 10;
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Kiff = tf(0);
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%% Name of the Simulink File
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mdl = 'sensor_fusion_test_bench_simscape';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N]
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io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m]
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io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage [V]
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io(io_i) = linio([mdl, '/Interferometer'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
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io(io_i) = linio([mdl, '/Voltage_Conditioner'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensor [V]
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options = linearizeOptions('SampleTime', 1e-4);
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G = linearize(mdl, io, options);
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G.InputName = {'Fd', 'w', 'Va'};
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G.OutputName = {'y', 'Vs'};
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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plot(f, abs(tf_fmeas_est), 'DisplayName', 'Identification')
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plot(f, abs(squeeze(freqresp(G('Vs', 'Va'), f, 'Hz'))), 'DisplayName', 'Simscape Model')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
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ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]);
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hold off;
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ylim([1e-1, 1e3]);
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legend('location', 'northwest');
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ax2 = nexttile;
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hold on;
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plot(f, 180/pi*angle(tf_fmeas_est))
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plot(f, 180/pi*angle(squeeze(freqresp(G('Vs', 'Va'), f, 'Hz'))))
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
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ylabel('Phase'); xlabel('Frequency [Hz]');
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hold off;
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2], 'x');
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xlim([1, 5e3]);
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% #+name: fig:simscape_comp_iff_plant
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% #+caption: Comparison of the dynamics from excitation voltage to measured force sensor stack voltage - Identified dynamics and Simscape Model
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% #+RESULTS:
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% [[file:figs/simscape_comp_iff_plant.png]]
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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plot(f, abs(tf_G_ol_est), 'DisplayName', 'Identification')
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plot(f, abs(squeeze(freqresp(G('y', 'Va'), f, 'Hz'))), 'DisplayName', 'Simscape Model')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
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ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
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hold off;
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ylim([1e-8, 1e-3]);
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ax2 = nexttile;
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hold on;
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plot(f, 180/pi*angle(tf_G_ol_est))
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plot(f, 180/pi*angle(squeeze(freqresp(G('y', 'Va'), f, 'Hz'))))
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
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ylabel('Phase'); xlabel('Frequency [Hz]');
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hold off;
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2], 'x');
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xlim([1, 5e3]);
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% Integral Force Feedback
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% The force sensor stack can be used to damp the system.
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% This makes the system easier to excite properly without too much amplification near resonances.
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% This is done thanks to the integral force feedback control architecture.
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% The force sensor stack signal is integrated (or rather low pass filtered) and fed back to the force sensor stacks.
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% The low pass filter used as the controller is defined below:
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Kiff = 102/(s + 2*pi*2);
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% The integral force feedback control strategy is applied to the simscape model as well as to the real test bench.
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%% Name of the Simulink File
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mdl = 'sensor_fusion_test_bench_simscape';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Fd'], 1, 'openinput'); io_i = io_i + 1; % External Vertical Force [N]
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io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m]
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io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage [V]
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io(io_i) = linio([mdl, '/Interferometer'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
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io(io_i) = linio([mdl, '/Voltage_Conditioner'], 1, 'output'); io_i = io_i + 1; % Force Sensor [V]
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options = linearizeOptions('SampleTime', 1e-4);
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G_cl = linearize(mdl, io, options);
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G_cl.InputName = {'Fd', 'w', 'Va'};
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G_cl.OutputName = {'y', 'Vs'};
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% The damped system is then identified again using a noise excitation.
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% The data is loaded into Matlab and any offset is removed.
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id_cl = load('identification_noise_iff_bis.mat', 'd', 'acc_1', 'acc_2', 'geo_1', 'geo_2', 'f_meas', 'u', 't');
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id_cl.d = detrend(id_cl.d, 0);
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id_cl.acc_1 = detrend(id_cl.acc_1, 0);
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id_cl.acc_2 = detrend(id_cl.acc_2, 0);
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id_cl.geo_1 = detrend(id_cl.geo_1, 0);
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id_cl.geo_2 = detrend(id_cl.geo_2, 0);
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id_cl.f_meas = detrend(id_cl.f_meas, 0);
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id_cl.u = detrend(id_cl.u, 0);
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% The transfer functions are estimated using =tfestimate=.
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[tf_G_cl_est, ~] = tfestimate(id_cl.u, id_cl.d, win, [], [], 1/Ts);
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[co_G_cl_est, ~] = mscohere( id_cl.u, id_cl.d, win, [], [], 1/Ts);
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% The dynamics from driving voltage to the measured displacement are compared both in the open-loop and IFF case, and for the test-bench experimental identification and for the Simscape model in Figure [[fig:iff_ol_cl_identified_simscape_comp]].
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% This shows that the Integral Force Feedback architecture effectively damps the first resonance of the system.
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile([2,1]);
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hold on;
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set(gca, 'ColorOrderIndex', 1);
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plot(f, abs(tf_G_ol_est), '-', 'DisplayName', 'OL - Ident.')
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set(gca, 'ColorOrderIndex', 1);
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plot(f, abs(squeeze(freqresp(G('y', 'Va'), f, 'Hz'))), '--', 'DisplayName', 'OL - Simscape')
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set(gca, 'ColorOrderIndex', 2);
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plot(f, abs(tf_G_cl_est), '-', 'DisplayName', 'CL - Ident.')
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set(gca, 'ColorOrderIndex', 2);
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plot(f, abs(squeeze(freqresp(G_cl('y', 'Va'), f, 'Hz'))), '--', 'DisplayName', 'CL - Simscape')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
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ylabel('Amplitude [m/V]'); set(gca, 'XTickLabel',[]);
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legend('location', 'northeast');
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hold off;
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ylim([1e-7, 1e-3]);
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ax2 = nexttile;
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hold on;
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set(gca, 'ColorOrderIndex', 1);
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plot(f, 180/pi*angle(tf_G_ol_est), '-')
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set(gca, 'ColorOrderIndex', 1);
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plot(f, 180/pi*angle(squeeze(freqresp(G('y', 'Va'), f, 'Hz'))), '--')
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set(gca, 'ColorOrderIndex', 2);
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plot(f, 180/pi*angle(tf_G_cl_est), '-')
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set(gca, 'ColorOrderIndex', 2);
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plot(f, 180/pi*angle(squeeze(freqresp(G_cl('y', 'Va'), f, 'Hz'))), '--')
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set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
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ylabel('Phase'); xlabel('Frequency [Hz]');
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hold off;
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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linkaxes([ax1,ax2], 'x');
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xlim([1, 5e3]);
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% Inertial Sensors
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% In order to estimate the dynamics of the inertial sensor (the transfer function from the "absolute" displacement to the measured voltage), the following experiment can be performed:
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% - The mass is excited such that is relative displacement as measured by the interferometer is much larger that the ground "absolute" motion.
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% - The transfer function from the measured displacement by the interferometer to the measured voltage generated by the inertial sensors can be estimated.
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% The first point is quite important in order to have a good coherence between the interferometer measurement and the inertial sensor measurement.
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% Here, a first identification is performed were the excitation signal is a white noise.
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% As usual, the data is loaded and any offset is removed.
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id = load('identification_noise_opt_iff.mat', 'd', 'acc_1', 'acc_2', 'geo_1', 'geo_2', 'f_meas', 'u', 't');
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id.d = detrend(id.d, 0);
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id.acc_1 = detrend(id.acc_1, 0);
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id.acc_2 = detrend(id.acc_2, 0);
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id.geo_1 = detrend(id.geo_1, 0);
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id.geo_2 = detrend(id.geo_2, 0);
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id.f_meas = detrend(id.f_meas, 0);
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% Then the transfer functions from the measured displacement by the interferometer to the generated voltage of the inertial sensors are computed..
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Ts = id.t(2) - id.t(1);
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win = hann(ceil(10/Ts));
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[tf_acc1_est, f] = tfestimate(id.d, id.acc_1, win, [], [], 1/Ts);
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[co_acc1_est, ~] = mscohere( id.d, id.acc_1, win, [], [], 1/Ts);
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[tf_acc2_est, ~] = tfestimate(id.d, id.acc_2, win, [], [], 1/Ts);
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[co_acc2_est, ~] = mscohere( id.d, id.acc_2, win, [], [], 1/Ts);
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|
[tf_geo1_est, ~] = tfestimate(id.d, id.geo_1, win, [], [], 1/Ts);
|
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[co_geo1_est, ~] = mscohere( id.d, id.geo_1, win, [], [], 1/Ts);
|
||
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|
[tf_geo2_est, ~] = tfestimate(id.d, id.geo_2, win, [], [], 1/Ts);
|
||
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|
[co_geo2_est, ~] = mscohere( id.d, id.geo_2, win, [], [], 1/Ts);
|
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% The same transfer functions are estimated using the Simscape model.
|
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|
||
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|
||
|
|
m = 10;
|
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|
|
Kiff = tf(0);
|
||
|
|
|
||
|
|
%% Name of the Simulink File
|
||
|
|
mdl = 'sensor_fusion_test_bench_simscape';
|
||
|
|
|
||
|
|
%% Input/Output definition
|
||
|
|
clear io; io_i = 1;
|
||
|
|
io(io_i) = linio([mdl, '/Va'], 1, 'openinput'); io_i = io_i + 1; % Actuator Voltage [V]
|
||
|
|
io(io_i) = linio([mdl, '/Interferometer'], 1, 'openoutput'); io_i = io_i + 1; % Vertical Displacement [m]
|
||
|
|
io(io_i) = linio([mdl, '/Vertical_Accelerometer_1'], 1, 'openoutput'); io_i = io_i + 1; % Accelerometer [V]
|
||
|
|
io(io_i) = linio([mdl, '/Voltage_Ampl_geo_1'], 1, 'openoutput'); io_i = io_i + 1; % Geophone [V]
|
||
|
|
|
||
|
|
options = linearizeOptions('SampleTime', 1e-4);
|
||
|
|
G = linearize(mdl, io, options);
|
||
|
|
|
||
|
|
G.InputName = {'Va'};
|
||
|
|
G.OutputName = {'y', 'a', 'v'};
|
||
|
|
|
||
|
|
G_acc = G('a', 'Va')*inv(G('y', 'Va')); % [V/m]
|
||
|
|
G_geo = G('v', 'Va')*inv(G('y', 'Va')); % [V/m]
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
% The obtained dynamics of the accelerometer are compared in Figure [[fig:comp_dynamics_accelerometer]] while the one of the geophones are compared in Figure [[fig:comp_dynamics_geophone]].
|
||
|
|
|
||
|
|
|
||
|
|
freqs = logspace(-1, 4, 1000)';
|
||
|
|
|
||
|
|
figure;
|
||
|
|
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||
|
|
|
||
|
|
ax1 = nexttile([2,1]);
|
||
|
|
hold on;
|
||
|
|
plot(f, abs(tf_acc1_est./(1i*2*pi*f).^2), '.')
|
||
|
|
plot(f, abs(tf_acc2_est./(1i*2*pi*f).^2), '.')
|
||
|
|
plot(freqs, abs(squeeze(freqresp(G_acc, freqs, 'Hz'))./(1i*2*pi*freqs).^2), 'k-')
|
||
|
|
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
|
||
|
|
ylabel('Amplitude $\left[\frac{V}{m/s^2}\right]$'); set(gca, 'XTickLabel',[]);
|
||
|
|
hold off;
|
||
|
|
|
||
|
|
ax2 = nexttile;
|
||
|
|
hold on;
|
||
|
|
plot(f, 180/pi*angle(tf_acc1_est./(1i*2*pi*f).^2), '.')
|
||
|
|
plot(f, 180/pi*angle(tf_acc2_est./(1i*2*pi*f).^2), '.')
|
||
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_acc, freqs, 'Hz'))./(1i*2*pi*freqs).^2), 'k-')
|
||
|
|
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
|
||
|
|
ylabel('Phase'); xlabel('Frequency [Hz]');
|
||
|
|
hold off;
|
||
|
|
ylim([-180, 180]);
|
||
|
|
yticks([-180, -90, 0, 90, 180]);
|
||
|
|
|
||
|
|
linkaxes([ax1,ax2], 'x');
|
||
|
|
xlim([2, 2e3]);
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
% #+name: fig:comp_dynamics_accelerometer
|
||
|
|
% #+caption: Comparison of the measured accelerometer dynamics
|
||
|
|
% #+RESULTS:
|
||
|
|
% [[file:figs/comp_dynamics_accelerometer.png]]
|
||
|
|
|
||
|
|
|
||
|
|
freqs = logspace(-1, 4, 1000)';
|
||
|
|
|
||
|
|
figure;
|
||
|
|
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||
|
|
|
||
|
|
ax1 = nexttile([2,1]);
|
||
|
|
hold on;
|
||
|
|
plot(f, abs(tf_geo1_est./(1i*2*pi*f)), '.')
|
||
|
|
plot(f, abs(tf_geo2_est./(1i*2*pi*f)), '.')
|
||
|
|
plot(freqs, abs(squeeze(freqresp(G_geo, freqs, 'Hz'))./(1i*2*pi*freqs)), 'k-')
|
||
|
|
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
|
||
|
|
ylabel('Amplitude $\left[\frac{V}{m/s}\right]$'); set(gca, 'XTickLabel',[]);
|
||
|
|
hold off;
|
||
|
|
|
||
|
|
ax2 = nexttile;
|
||
|
|
hold on;
|
||
|
|
plot(f, 180/pi*angle(tf_geo1_est./(1i*2*pi*f)), '.')
|
||
|
|
plot(f, 180/pi*angle(tf_geo2_est./(1i*2*pi*f)), '.')
|
||
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_geo, freqs, 'Hz'))./(1i*2*pi*freqs)), 'k-')
|
||
|
|
set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'lin');
|
||
|
|
ylabel('Phase'); xlabel('Frequency [Hz]');
|
||
|
|
hold off;
|
||
|
|
ylim([-180, 180]);
|
||
|
|
yticks([-180, -90, 0, 90, 180]);
|
||
|
|
|
||
|
|
linkaxes([ax1,ax2], 'x');
|
||
|
|
xlim([0.5, 2e3]);
|