Create file for simulating tomography experiments
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simscape/sim_nano_station_tomo.slx
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simscape/sim_nano_station_tomo.slx
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tomo-exp/index.org
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tomo-exp/index.org
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#+TITLE: Tomography Experiment
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:DRAWER:
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#+STARTUP: overview
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#+LANGUAGE: en
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#+EMAIL: dehaeze.thomas@gmail.com
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#+AUTHOR: Dehaeze Thomas
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#+HTML_LINK_HOME: ../index.html
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#+HTML_LINK_UP: ../index.html
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/htmlize.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/readtheorg.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="../css/zenburn.css"/>
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#+HTML_HEAD: <script type="text/javascript" src="../js/jquery.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="../js/bootstrap.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="../js/jquery.stickytableheaders.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="../js/readtheorg.js"></script>
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#+HTML_MATHJAX: align: center tagside: right font: TeX
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments org
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :exports both
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :output-dir figs
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#+PROPERTY: header-args:matlab+ :tangle matlab/modal_frf_coh.m
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:shell :eval no-export
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#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/}{config.tex}")
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#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
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#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
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#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
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#+PROPERTY: header-args:latex+ :results raw replace :buffer no
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#+PROPERTY: header-args:latex+ :eval no-export
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#+PROPERTY: header-args:latex+ :exports both
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#+PROPERTY: header-args:latex+ :mkdirp yes
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#+PROPERTY: header-args:latex+ :output-dir figs
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:END:
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* Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :tangle no
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simulinkproject('../');
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#+end_src
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#+begin_src matlab
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open 'simscape/sim_nano_station_tomo.slx'
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#+end_src
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* Initialize Experiment
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#+begin_src matlab
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initializeGround();
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initializeGranite();
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initializeTy();
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initializeRy();
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initializeRz();
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initializeMicroHexapod();
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initializeAxisc();
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initializeMirror();
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initializeNanoHexapod(struct('actuator', 'piezo'));
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initializeSample(struct('mass', 1));
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#+end_src
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#+begin_src matlab
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t = 0:Ts:Tsim;
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#+end_src
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Generate perturbations
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#+begin_src matlab
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win = hanning(ceil(1/Ts));
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[pxx, f] = pwelch(sqrt(1/(2*Ts))*randn(length(t), 1), win, [], [], 1/Ts);
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#+end_src
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#+begin_src matlab
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figure;
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hold on;
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plot(f, sqrt(pxx));
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hold off;
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set(gca, 'xscale', 'log');
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set(gca, 'yscale', 'log');
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xlabel('Frequency [Hz]'); ylabel('ASD of the measured velocity $\left[\frac{m/s}{\sqrt{Hz}}\right]$')
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ylim([0.01, 100]);
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#+end_src
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#+begin_src matlab
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load('./disturbances/mat/dist_psd.mat', 'dist_f');
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Dwx = lsim(dist_f.G_gm, sqrt(1/(2*Ts))*randn(length(t), 1), t);
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Dwy = lsim(dist_f.G_gm, sqrt(1/(2*Ts))*randn(length(t), 1), t);
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Dwz = lsim(dist_f.G_gm, sqrt(1/(2*Ts))*randn(length(t), 1), t);
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Dw = [Dwx, Dwy, Dwz];
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#+end_src
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#+begin_src matlab
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figure;
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hold on;
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plot(t, Dwx);
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plot(t, Dwy);
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plot(t, Dwz);
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hold off;
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xlabel('Time [s]'); ylabel('Displacement [m]');
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#+end_src
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#+begin_src matlab
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Fty_z = lsim(dist_f.G_ty, sqrt(1/(2*Ts))*randn(length(t), 1), t);
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Frz_z = lsim(dist_f.G_rz, sqrt(1/(2*Ts))*randn(length(t), 1), t);
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#+end_src
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#+begin_src matlab
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figure;
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hold on;
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plot(t, Fty_z);
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plot(t, Frz_z);
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hold off;
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xlabel('Time [s]'); ylabel('Force [N]');
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#+end_src
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#+begin_src matlab
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% Spindle
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Rz = 2*pi*t;
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% Axisc
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Rm = zeros(length(t), 2);
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Rm(:, 2) = pi*ones(length(t), 1);
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inputs = struct( ...
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'Ts', Ts, ...
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'Dw', timeseries(Dw, t), ...
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'Dy', timeseries(zeros(length(t), 1), t), ...
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'Ry', timeseries(zeros(length(t), 1), t), ...
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'Rz', timeseries(Rz, t), ...
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'Dh', timeseries(zeros(length(t), 6), t), ...
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'Rm', timeseries(Rm, t), ...
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'Dn', timeseries(zeros(length(t), 6), t), ...
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'Ds', timeseries(zeros(length(t), 6), t), ...
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'Fg', timeseries(zeros(length(t), 3), t), ...
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'Fn', timeseries(zeros(length(t), 6), t), ...
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'Fnl', timeseries(zeros(length(t), 6), t), ...
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'Fty_x', timeseries(zeros(length(t), 1), t), ...
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'Fty_z', timeseries(Fty_z, t), ...
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'Frz_z', timeseries(Frz_z, t), ...
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'Fs', timeseries(zeros(length(t), 6), t) ...
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);
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#+end_src
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* Test
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab
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cd ..
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#+end_src
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We define some parameters:
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- =T0= is the duration in seconds
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- =N= is the number of samples
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#+begin_src matlab
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T0 = 10; % Signal Duration [s]
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N = 10000; % Number of Samples (should be even)
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#+end_src
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Then, we have:
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- $f_s = N/T_0$ is the sampling frequency in Hz
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- $f_c = \frac{1}{2} N/T_0$ is the cut-off frequency in Hz
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- $f_0 = 1/T_0$ is the frequency resolution of the DFT in Hz
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#+begin_src matlab
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Fs = N/T0; % Sampling frequency [Hz]
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Fc = (1/2)*N/T0; % Sampling frequency [Hz]
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df = 1/T0; % Frequency resolution of the DFT [Hz]
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#+end_src
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We then specify the wanted PSD.
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#+begin_src matlab
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phi = ones(N/2, 1);
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% phi = logspace(3, 0, N/2);
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phi(df:df:Fs/2>10) = 0;
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#+end_src
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We create $C_x(k)$ such that:
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\[ C_x(k) = \sqrt{\Phi_{xx}(k\omega_0)\omega_0} \quad k = 1 \dots N/2 \]
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where $\Phi_{xx}$ is the wanted PSD.
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#+begin_src matlab
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C = zeros(N/2, 1);
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for i = 1:N/2
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C(i) = sqrt(phi(i))*2*Fs; % TODO - Why this normalization?
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end
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#+end_src
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We generate some random phase that will be added to =C=.
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#+begin_src matlab
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theta = 2*pi*rand(N/2, 1); % Generate random phase [rad]
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#+end_src
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In order to have
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\[ C_x(N/2+i) = C_x^*(N/2-i) \quad i = 1 \dots N/2 \]
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We do the following
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#+begin_src matlab
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Cx = [0 ; C.*complex(cos(theta),sin(theta))];
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Cx = [Cx; flipud(conj(Cx(2:end)))];;
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#+end_src
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The time domain data is generated by an inverse FFT.
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#+begin_src matlab
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u = ifft(Cx);
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#+end_src
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#+begin_src matlab
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A = fft(u);
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figure; plot(2*A.*conj(A))
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#+end_src
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#+begin_src matlab
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t = linspace(0, T0, N+1); % Time Vector [s]
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#+end_src
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#+begin_src matlab
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figure;
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plot(t, u)
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xlabel('Time [s]');
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ylabel('Amplitude');
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#+end_src
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#+begin_src matlab
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nx = length(u);
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na = 16;
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win = hanning(floor(nx/na));
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[pxx, f] = pwelch(u, win, 0, [], Fs);
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#+end_src
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#+begin_src matlab
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figure;
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hold on;
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plot(f,pxx)
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plot(df:df:Fc,phi)
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hold off;
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xlabel('Frequency [Hz]');
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ylabel('Power Spectral Density');
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set(gca, 'xscale', 'log');
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set(gca, 'yscale', 'log');
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#+end_src
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* Test
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab
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cd ..
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#+end_src
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#+begin_src matlab
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load('./disturbances/mat/dist_psd.mat', 'dist_f');
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#+end_src
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We remove the first value with very high PSD.
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#+begin_src matlab
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dist_f.f = dist_f.f(3:end);
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dist_f.psd_gm = dist_f.psd_gm(3:end);
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#+end_src
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We define some parameters.
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#+begin_src matlab
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Fs = 2*dist_f.f(end); % Sampling Frequency of data is twice the maximum frequency of the PSD vector [Hz]
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N = 2*length(dist_f.f); % Number of Samples match the one of the wanted PSD
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T0 = N/Fs; % Signal Duration [s]
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df = 1/T0; % Frequency resolution of the DFT [Hz]
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% Also equal to (dist_f.f(2)-dist_f.f(1))
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#+end_src
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We then specify the wanted PSD.
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#+begin_src matlab
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phi = dist_f.psd_gm;
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#+end_src
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Create amplitudes corresponding to wanted PSD.
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#+begin_src matlab
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C = zeros(N/2,1);
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for i = 1:N/2
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C(i) = sqrt(phi(i)*df);
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end
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#+end_src
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Add random phase to =C=.
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#+begin_src matlab
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theta = 2*pi*rand(N/2,1); % Generate random phase [rad]
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Cx = [0 ; C.*complex(cos(theta),sin(theta))];
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Cx = [Cx; flipud(conj(Cx(2:end)))];;
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#+end_src
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The time domain data is generated by an inverse FFT.
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We normalize the =ifft= Matlab command.
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#+begin_src matlab
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u = 1/(sqrt(2)*df*1/Fs)*ifft(Cx); % Normalisation of the IFFT
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t = linspace(0, T0, N+1); % Time Vector [s]
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#+end_src
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#+begin_src matlab
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figure;
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plot(t, u)
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xlabel('Time [s]');
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ylabel('Amplitude');
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#+end_src
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#+begin_src matlab
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u_rep = [u;u;u;u;u;u;u;u;u;u;u;u;u;u;u;u;u;u;u;u;u;u;u;u;u];
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#+end_src
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#+begin_src matlab
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nx = length(u_rep);
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na = 16;
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win = hanning(floor(nx/na));
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[pxx, f] = pwelch(u_rep, win, 0, [], Fs);
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#+end_src
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#+begin_src matlab
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figure;
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hold on;
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plot(dist_f.f, dist_f.psd_gm, 'DisplayName', 'Original PSD')
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plot(f, pxx, 'DisplayName', 'Computed')
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% plot(f, pxx./dist_f.psd_gm, 'k-')
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hold off;
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xlabel('Frequency [Hz]');
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ylabel('Power Spectral Density');
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set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
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legend('location', 'northeast');
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#+end_src
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