2064 lines
77 KiB
Org Mode
2064 lines
77 KiB
Org Mode
#+TITLE: Active Damping with an uni-axial model
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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 no
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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/org/}{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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* Introduction :ignore:
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First, in section [[sec:undamped_system]], we will looked at the undamped system.
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Then, we will compare three active damping techniques:
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- In section [[sec:iff]]: the integral force feedback is used
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- In section [[sec:rmc]]: the relative motion control is used
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- In section [[sec:dvf]]: the direct velocity feedback is used
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For each of the active damping technique, we will:
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- Compare the sensitivity from disturbances
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- Look at the damped plant
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The disturbances are:
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- Ground motion
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- Direct forces
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- Motion errors of all the stages
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* Undamped System
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:PROPERTIES:
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:header-args:matlab+: :tangle ../matlab/undamped_system_uniaxial.m
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:header-args:matlab+: :comments none :mkdirp yes
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:END:
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<<sec:undamped_system>>
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** ZIP file containing the data and matlab files :ignore:
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#+begin_src bash :exports none :results none
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if [ matlab/undamped_system.m -nt data/undamped_system.zip ]; then
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cp matlab/undamped_system.m undamped_system.m;
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zip data/undamped_system \
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undamped_system.m
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rm undamped_system.m;
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fi
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#+end_src
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#+begin_note
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All the files (data and Matlab scripts) are accessible [[file:data/undamped_system.zip][here]].
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#+end_note
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** Introduction :ignore:
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We first look at the undamped system.
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The performance of this undamped system will be compared with the damped system using various techniques.
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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('active_damping_uniaxial/matlab/sim_nano_station_id.slx')
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#+end_src
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** Init
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We initialize all the stages with the default parameters.
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The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
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#+begin_src matlab
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initializeReferences();
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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('actuator', 'piezo');
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initializeSample('mass', 50);
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#+end_src
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All the controllers are set to 0.
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#+begin_src matlab
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K = tf(zeros(6));
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save('./mat/controllers_uniaxial.mat', 'K', '-append');
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K_iff = tf(zeros(6));
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save('./mat/controllers_uniaxial.mat', 'K_iff', '-append');
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K_rmc = tf(zeros(6));
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save('./mat/controllers_uniaxial.mat', 'K_rmc', '-append');
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K_dvf = tf(zeros(6));
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save('./mat/controllers_uniaxial.mat', 'K_dvf', '-append');
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#+end_src
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** Identification
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We identify the various transfer functions of the system
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#+begin_src matlab
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G = identifyPlant();
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#+end_src
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And we save it for further analysis.
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#+begin_src matlab
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save('./mat/active_damping_uniaxial_plants.mat', 'G', '-append');
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#+end_src
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** Sensitivity to disturbances
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The sensitivity to disturbances are shown on figure [[fig:sensitivity_dist_undamped]].
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#+begin_src matlab :exports none
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freqs = logspace(0, 3, 1000);
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figure;
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subplot(2, 1, 1);
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title('$D_g$ to $D$');
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hold on;
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plot(freqs, abs(squeeze(freqresp(G.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / D_{g,x}\right|$');
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plot(freqs, abs(squeeze(freqresp(G.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / D_{g,y}\right|$');
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plot(freqs, abs(squeeze(freqresp(G.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / D_{g,z}\right|$');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
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legend('location', 'southeast');
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subplot(2, 1, 2);
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title('$F_s$ to $D$');
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hold on;
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plot(freqs, abs(squeeze(freqresp(G.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{s,x}\right|$');
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plot(freqs, abs(squeeze(freqresp(G.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{s,y}\right|$');
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plot(freqs, abs(squeeze(freqresp(G.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{s,z}\right|$');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
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legend('location', 'northeast');
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#+end_src
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#+HEADER: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/sensitivity_dist_undamped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+NAME: fig:sensitivity_dist_undamped
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#+CAPTION: Undamped sensitivity to disturbances ([[./figs/sensitivity_dist_undamped.png][png]], [[./figs/sensitivity_dist_undamped.pdf][pdf]])
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[[file:figs/sensitivity_dist_undamped.png]]
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#+begin_src matlab :exports none
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freqs = logspace(0, 3, 1000);
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figure;
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hold on;
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plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{rz, z}\right|$');
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plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{ty, z}\right|$');
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plot(freqs, abs(squeeze(freqresp(G.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{ty, x}\right|$');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
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legend('location', 'northeast');
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#+end_src
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#+HEADER: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/sensitivity_dist_stages.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+NAME: fig:sensitivity_dist_stages
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#+CAPTION: Sensitivity to force disturbances in various stages ([[./figs/sensitivity_dist_stages.png][png]], [[./figs/sensitivity_dist_stages.pdf][pdf]])
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[[file:figs/sensitivity_dist_stages.png]]
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** Undamped Plant
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The "plant" (transfer function from forces applied by the nano-hexapod to the measured displacement of the sample with respect to the granite) bode plot is shown on figure [[fig:sensitivity_dist_undamped]].
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#+begin_src matlab :exports none
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freqs = logspace(0, 3, 1000);
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figure;
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ax1 = subplot(2, 1, 1);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))));
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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/N]'); set(gca, 'XTickLabel',[]);
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ax2 = subplot(2, 1, 2);
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'DisplayName', '$D_x / F_x$');
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plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))), 'DisplayName', '$D_y / F_y$');
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plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'DisplayName', '$D_z / F_z$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180, -90, 0, 90, 180]);
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legend('location', 'southwest');
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linkaxes([ax1,ax2],'x');
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#+end_src
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#+HEADER: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/plant_undamped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+NAME: fig:plant_undamped
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#+CAPTION: Transfer Function from cartesian forces to displacement for the undamped plant ([[./figs/plant_undamped.png][png]], [[./figs/plant_undamped.pdf][pdf]])
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[[file:figs/plant_undamped.png]]
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* Integral Force Feedback
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:PROPERTIES:
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:header-args:matlab+: :tangle ../matlab/iff_uniaxial.m
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:header-args:matlab+: :comments none :mkdirp yes
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:END:
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<<sec:iff>>
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** ZIP file containing the data and matlab files :ignore:
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#+begin_src bash :exports none :results none
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if [ matlab/iff.m -nt data/iff.zip ]; then
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cp matlab/iff.m iff.m;
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zip data/iff \
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mat/plant.mat \
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iff.m
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rm iff.m;
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fi
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#+end_src
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#+begin_note
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All the files (data and Matlab scripts) are accessible [[file:data/iff.zip][here]].
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#+end_note
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** Introduction :ignore:
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Integral Force Feedback is applied.
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In section [[sec:iff_1dof]], IFF is applied on a uni-axial system to understand its behavior.
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Then, it is applied on the simscape model.
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** One degree-of-freedom example
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:PROPERTIES:
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:header-args:matlab+: :tangle no
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:END:
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<<sec:iff_1dof>>
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*** Equations
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#+begin_src latex :file iff_1dof.pdf :post pdf2svg(file=*this*, ext="png") :exports results
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\begin{tikzpicture}
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% Ground
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\draw (-1, 0) -- (1, 0);
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% Ground Displacement
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\draw[dashed] (-1, 0) -- ++(-0.5, 0) coordinate(w);
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\draw[->] (w) -- ++(0, 0.5) node[left]{$w$};
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% Mass
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\draw[fill=white] (-1, 1.4) rectangle ++(2, 0.8) node[pos=0.5]{$m$};
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\node[forcesensor={0.4}{0.4}] (fsensn) at (0, 1){};
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\draw[] (-0.8, 1) -- (0.8, 1);
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\node[left] at (fsensn.west) {$F_m$};
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% Displacement of the mass
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\draw[dashed] (-1, 2.2) -- ++(-0.5, 0) coordinate(x);
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\draw[->] (x) -- ++(0, 0.5) node[left]{$x$};
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% Spring, Damper, and Actuator
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\draw[spring] (-0.8, 0) -- (-0.8, 1) node[midway, left=0.1]{$k$};
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\draw[damper] (0, 0) -- (0, 1) node[midway, left=0.2]{$c$};
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\draw[actuator={0.4}{0.2}] (0.8, 0) -- (0.8, 1) coordinate[midway, right=0.1](F);
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% Displacements
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\node[block={0.8cm}{0.6cm}, right=0.6 of F] (Kiff) {$K$};
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\draw[->] (Kiff.west) -- (F) node[above right]{$F$};
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\draw[<-] (Kiff.east) -- ++(0.5, 0) |- (fsensn.east);
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\end{tikzpicture}
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#+end_src
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#+name: fig:iff_1dof
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#+caption: Integral Force Feedback applied to a 1dof system
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#+RESULTS:
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[[file:figs/iff_1dof.png]]
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The dynamic of the system is described by the following equation:
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\begin{equation}
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ms^2x = F_d - kx - csx + kw + csw + F
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\end{equation}
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The measured force $F_m$ is:
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\begin{align}
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F_m &= F - kx - csx + kw + csw \\
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&= ms^2 x - F_d
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\end{align}
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The Integral Force Feedback controller is $K = -\frac{g}{s}$, and thus the applied force by this controller is:
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\begin{equation}
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F_{\text{IFF}} = -\frac{g}{s} F_m = -\frac{g}{s} (ms^2 x - F_d)
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\end{equation}
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Once the IFF is applied, the new dynamics of the system is:
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\begin{equation}
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ms^2x = F_d + F - kx - csx + kw + csw - \frac{g}{s} (ms^2x - F_d)
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\end{equation}
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And finally:
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\begin{equation}
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x = F_d \frac{1 + \frac{g}{s}}{ms^2 + (mg + c)s + k} + F \frac{1}{ms^2 + (mg + c)s + k} + w \frac{k + cs}{ms^2 + (mg + c)s + k}
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\end{equation}
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We can see that this:
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- adds damping to the system by a value $mg$
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- lower the compliance as low frequency by a factor: $1 + g/s$
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If we want critical damping:
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\begin{equation}
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\xi = \frac{1}{2} \frac{c + gm}{\sqrt{km}} = \frac{1}{2}
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\end{equation}
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This is attainable if we have:
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\begin{equation}
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g = \frac{\sqrt{km} - c}{m}
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\end{equation}
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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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*** Matlab Example
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Let define the system parameters.
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#+begin_src matlab
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m = 50; % [kg]
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k = 1e6; % [N/m]
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c = 1e3; % [N/(m/s)]
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#+end_src
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The state space model of the system is defined below.
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#+begin_src matlab
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A = [-c/m -k/m;
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1 0];
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B = [1/m 1/m -1;
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0 0 0];
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C = [ 0 1;
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-c -k];
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D = [0 0 0;
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1 0 0];
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sys = ss(A, B, C, D);
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sys.InputName = {'F', 'Fd', 'wddot'};
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sys.OutputName = {'d', 'Fm'};
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sys.StateName = {'ddot', 'd'};
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#+end_src
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The controller $K_\text{IFF}$ is:
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#+begin_src matlab
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Kiff = -((sqrt(k*m)-c)/m)/s;
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Kiff.InputName = {'Fm'};
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Kiff.OutputName = {'F'};
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#+end_src
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And the closed loop system is computed below.
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#+begin_src matlab
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sys_iff = feedback(sys, Kiff, 'name', +1);
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#+end_src
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#+begin_src matlab :exports none
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freqs = logspace(0, 3, 1000);
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figure;
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subplot(2, 2, 1);
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title('Fd to d')
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hold on;
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plot(freqs, abs(squeeze(freqresp(sys('d', 'Fd'), freqs, 'Hz'))));
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plot(freqs, abs(squeeze(freqresp(sys_iff('d', 'Fd'), freqs, 'Hz'))));
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
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xlim([freqs(1), freqs(end)]);
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subplot(2, 2, 3);
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title('Fd to x')
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hold on;
|
|
plot(freqs, abs(squeeze(freqresp(sys('d', 'Fd'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(sys_iff('d', 'Fd'), freqs, 'Hz'))));
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
xlim([freqs(1), freqs(end)]);
|
|
|
|
subplot(2, 2, 2);
|
|
title('w to d')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(sys('d', 'wddot')*s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(sys_iff('d', 'wddot')*s^2, freqs, 'Hz'))));
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
xlim([freqs(1), freqs(end)]);
|
|
|
|
subplot(2, 2, 4);
|
|
title('w to x')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(1+sys('d', 'wddot')*s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(1+sys_iff('d', 'wddot')*s^2, freqs, 'Hz'))));
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
xlim([freqs(1), freqs(end)]);
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/iff_1dof_sensitivitiy.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:iff_1dof_sensitivitiy
|
|
#+CAPTION: Sensitivity to disturbance when IFF is applied on the 1dof system ([[./figs/iff_1dof_sensitivitiy.png][png]], [[./figs/iff_1dof_sensitivitiy.pdf][pdf]])
|
|
[[file:figs/iff_1dof_sensitivitiy.png]]
|
|
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no
|
|
simulinkproject('../');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('active_damping_uniaxial/matlab/sim_nano_station_id.slx')
|
|
#+end_src
|
|
|
|
** Control Design
|
|
Let's load the undamped plant:
|
|
#+begin_src matlab
|
|
load('./mat/active_damping_uniaxial_plants.mat', 'G');
|
|
#+end_src
|
|
|
|
Let's look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor (figure [[fig:iff_plant]]).
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i=1:6
|
|
plot(freqs, abs(squeeze(freqresp(G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i=1:6
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/iff_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:iff_plant
|
|
#+CAPTION: Transfer function from forces applied in the legs to force sensor ([[./figs/iff_plant.png][png]], [[./figs/iff_plant.pdf][pdf]])
|
|
[[file:figs/iff_plant.png]]
|
|
|
|
The controller for each pair of actuator/sensor is:
|
|
#+begin_src matlab
|
|
K_iff = -1000/s;
|
|
#+end_src
|
|
|
|
The corresponding loop gains are shown in figure [[fig:iff_open_loop]].
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i=1:6
|
|
plot(freqs, abs(squeeze(freqresp(K_iff*G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i=1:6
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(K_iff*G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/iff_open_loop.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:iff_open_loop
|
|
#+CAPTION: Loop Gain for the Integral Force Feedback ([[./figs/iff_open_loop.png][png]], [[./figs/iff_open_loop.pdf][pdf]])
|
|
[[file:figs/iff_open_loop.png]]
|
|
|
|
** Identification of the damped plant
|
|
Let's initialize the system prior to identification.
|
|
#+begin_src matlab
|
|
initializeReferences();
|
|
initializeGround();
|
|
initializeGranite();
|
|
initializeTy();
|
|
initializeRy();
|
|
initializeRz();
|
|
initializeMicroHexapod();
|
|
initializeAxisc();
|
|
initializeMirror();
|
|
initializeNanoHexapod('actuator', 'piezo');
|
|
initializeSample('mass', 50);
|
|
#+end_src
|
|
|
|
All the controllers are set to 0.
|
|
#+begin_src matlab
|
|
K = tf(zeros(6));
|
|
save('./mat/controllers_uniaxial.mat', 'K', '-append');
|
|
K_iff = -K_iff*eye(6);
|
|
save('./mat/controllers_uniaxial.mat', 'K_iff', '-append');
|
|
K_rmc = tf(zeros(6));
|
|
save('./mat/controllers_uniaxial.mat', 'K_rmc', '-append');
|
|
K_dvf = tf(zeros(6));
|
|
save('./mat/controllers_uniaxial.mat', 'K_dvf', '-append');
|
|
#+end_src
|
|
|
|
We identify the system dynamics now that the IFF controller is ON.
|
|
#+begin_src matlab
|
|
G_iff = identifyPlant();
|
|
#+end_src
|
|
|
|
And we save the damped plant for further analysis
|
|
#+begin_src matlab
|
|
save('./mat/active_damping_uniaxial_plants.mat', 'G_iff', '-append');
|
|
#+end_src
|
|
|
|
** Sensitivity to disturbances
|
|
As shown on figure [[fig:sensitivity_dist_iff]]:
|
|
- The top platform of the nano-hexapod how behaves as a "free-mass".
|
|
- The transfer function from direct forces $F_s$ to the relative displacement $D$ is equivalent to the one of an isolated mass.
|
|
- The transfer function from ground motion $D_g$ to the relative displacement $D$ tends to the transfer function from $D_g$ to the displacement of the granite (the sample is being isolated thanks to IFF).
|
|
However, as the goal is to make the relative displacement $D$ as small as possible (e.g. to make the sample motion follows the granite motion), this is not a good thing.
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 1, 1);
|
|
title('$D_g$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / D_{g,x}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / D_{g,y}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / D_{g,z}\right|$');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
|
|
subplot(2, 1, 2);
|
|
title('$F_s$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{s,x}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{s,y}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{s,z}\right|$');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/sensitivity_dist_iff.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:sensitivity_dist_iff
|
|
#+CAPTION: Sensitivity to disturbance once the IFF controller is applied to the system ([[./figs/sensitivity_dist_iff.png][png]], [[./figs/sensitivity_dist_iff.pdf][pdf]])
|
|
[[file:figs/sensitivity_dist_iff.png]]
|
|
|
|
#+begin_warning
|
|
The order of the models are very high and thus the plots may be wrong.
|
|
For instance, the plots are not the same when using =minreal=.
|
|
#+end_warning
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{rz, z}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{ty, z}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{ty, x}\right|$');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(minreal(prescale(G_iff.G_dist('Dz', 'Frzz'), {2*pi, 2*pi*1e3})), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(minreal(G_iff.G_dist('Dz', 'Ftyz')), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(minreal(G_iff.G_dist('Dx', 'Ftyx')), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/sensitivity_dist_stages_iff.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:sensitivity_dist_stages_iff
|
|
#+CAPTION: Sensitivity to force disturbances in various stages when IFF is applied ([[./figs/sensitivity_dist_stages_iff.png][png]], [[./figs/sensitivity_dist_stages_iff.pdf][pdf]])
|
|
[[file:figs/sensitivity_dist_stages_iff.png]]
|
|
|
|
** Damped Plant
|
|
Now, look at the new damped plant to control.
|
|
|
|
It damps the plant (resonance of the nano hexapod as well as other resonances) as shown in figure [[fig:plant_iff_damped]].
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 2, 1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
|
|
ax2 = subplot(2, 2, 2);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [rad/(Nm)]'); xlabel('Frequency [Hz]');
|
|
|
|
ax3 = subplot(2, 2, 3);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{n,x}\right|$');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{n,y}\right|$');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{n,z}\right|$');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
legend('location', 'northwest');
|
|
|
|
ax4 = subplot(2, 2, 4);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), 'DisplayName', '$\left|R_x / M_{n,x}\right|$');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))), 'DisplayName', '$\left|R_y / M_{n,y}\right|$');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), 'DisplayName', '$\left|R_z / M_{n,z}\right|$');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
legend('location', 'northwest');
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/plant_iff_damped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:plant_iff_damped
|
|
#+CAPTION: Damped Plant after IFF is applied ([[./figs/plant_iff_damped.png][png]], [[./figs/plant_iff_damped.pdf][pdf]])
|
|
[[file:figs/plant_iff_damped.png]]
|
|
|
|
However, it increases coupling at low frequency (figure [[fig:plant_iff_coupling]]).
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
for ix = 1:6
|
|
for iy = 1:6
|
|
subplot(6, 6, (ix-1)*6 + iy);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart(ix, iy), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart(ix, iy), freqs, 'Hz'))), 'k--');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylim([1e-12, 1e-5]);
|
|
end
|
|
end
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/plant_iff_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:plant_iff_coupling
|
|
#+CAPTION: Coupling induced by IFF ([[./figs/plant_iff_coupling.png][png]], [[./figs/plant_iff_coupling.pdf][pdf]])
|
|
[[file:figs/plant_iff_coupling.png]]
|
|
|
|
** Conclusion
|
|
#+begin_important
|
|
Integral Force Feedback:
|
|
- Robust (guaranteed stability)
|
|
- Acceptable Damping
|
|
- Increase the sensitivity to disturbances at low frequencies
|
|
#+end_important
|
|
|
|
* Relative Motion Control
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle ../matlab/rmc_uniaxial.m
|
|
:header-args:matlab+: :comments none :mkdirp yes
|
|
:END:
|
|
<<sec:rmc>>
|
|
|
|
** ZIP file containing the data and matlab files :ignore:
|
|
#+begin_src bash :exports none :results none
|
|
if [ matlab/rmc.m -nt data/rmc.zip ]; then
|
|
cp matlab/rmc.m rmc.m;
|
|
zip data/rmc \
|
|
mat/plant.mat \
|
|
rmc.m
|
|
rm rmc.m;
|
|
fi
|
|
#+end_src
|
|
|
|
#+begin_note
|
|
All the files (data and Matlab scripts) are accessible [[file:data/rmc.zip][here]].
|
|
#+end_note
|
|
|
|
** Introduction :ignore:
|
|
In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input.
|
|
|
|
** One degree-of-freedom example
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle no
|
|
:END:
|
|
<<sec:rmc_1dof>>
|
|
*** Equations
|
|
#+begin_src latex :file rmc_1dof.pdf :post pdf2svg(file=*this*, ext="png") :exports results
|
|
\begin{tikzpicture}
|
|
% Ground
|
|
\draw (-1, 0) -- (1, 0);
|
|
|
|
% Ground Displacement
|
|
\draw[dashed] (-1, 0) -- ++(-0.5, 0) coordinate(w);
|
|
\draw[->] (w) -- ++(0, 0.5) node[left]{$w$};
|
|
|
|
% Mass
|
|
\draw[fill=white] (-1, 1) rectangle ++(2, 0.8) node[pos=0.5]{$m$};
|
|
|
|
% Displacement of the mass
|
|
\draw[dashed] (-1, 1.8) -- ++(-0.5, 0) coordinate(x);
|
|
\draw[->] (x) -- ++(0, 0.5) node[left]{$x$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.8, 0) -- (-0.8, 1) node[midway, left=0.1]{$k$};
|
|
\draw[damper] (0, 0) -- (0, 1) node[midway, left=0.2]{$c$};
|
|
\draw[actuator={0.4}{0.2}] (0.8, 0) -- (0.8, 1) coordinate[midway, right=0.1](F);
|
|
|
|
% Measured deformation
|
|
\draw[dashed] (1, 0) -- ++(2, 0) coordinate(d_bot);
|
|
\draw[dashed] (1, 1) -- ++(2, 0) coordinate(d_top);
|
|
\draw[<->] (d_bot) --coordinate[midway](d) (d_top);
|
|
|
|
% Displacements
|
|
\node[block={0.8cm}{0.6cm}, right=0.6 of F] (Krmc) {$K$};
|
|
\draw[->] (Krmc.west) -- (F) node[above right]{$F$};
|
|
\draw[->] (d)node[above left]{$d$} -- (Krmc.east);
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:rmc_1dof
|
|
#+caption: Relative Motion Control applied to a 1dof system
|
|
#+RESULTS:
|
|
[[file:figs/rmc_1dof.png]]
|
|
|
|
The dynamic of the system is:
|
|
\begin{equation}
|
|
ms^2x = F_d - kx - csx + kw + csw + F
|
|
\end{equation}
|
|
In terms of the stage deformation $d = x - w$:
|
|
\begin{equation}
|
|
(ms^2 + cs + k) d = -ms^2 w + F_d + F
|
|
\end{equation}
|
|
The relative motion control law is:
|
|
\begin{equation}
|
|
K = -g s
|
|
\end{equation}
|
|
Thus, the applied force is:
|
|
\begin{equation}
|
|
F = -g s d
|
|
\end{equation}
|
|
And the new dynamics will be:
|
|
\begin{equation}
|
|
d = w \frac{-ms^2}{ms^2 + (c + g)s + k} + F_d \frac{1}{ms^2 + (c + g)s + k} + F \frac{1}{ms^2 + (c + g)s + k}
|
|
\end{equation}
|
|
|
|
And thus damping is added.
|
|
|
|
If critical damping is wanted:
|
|
\begin{equation}
|
|
\xi = \frac{1}{2}\frac{c + g}{\sqrt{km}} = \frac{1}{2}
|
|
\end{equation}
|
|
This corresponds to a gain:
|
|
\begin{equation}
|
|
g = \sqrt{km} - c
|
|
\end{equation}
|
|
|
|
*** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
*** Matlab Example
|
|
Let define the system parameters.
|
|
#+begin_src matlab
|
|
m = 50; % [kg]
|
|
k = 1e6; % [N/m]
|
|
c = 1e3; % [N/(m/s)]
|
|
#+end_src
|
|
|
|
The state space model of the system is defined below.
|
|
#+begin_src matlab
|
|
A = [-c/m -k/m;
|
|
1 0];
|
|
|
|
B = [1/m 1/m -1;
|
|
0 0 0];
|
|
|
|
C = [ 0 1;
|
|
-c -k];
|
|
|
|
D = [0 0 0;
|
|
1 0 0];
|
|
|
|
sys = ss(A, B, C, D);
|
|
sys.InputName = {'F', 'Fd', 'wddot'};
|
|
sys.OutputName = {'d', 'Fm'};
|
|
sys.StateName = {'ddot', 'd'};
|
|
#+end_src
|
|
|
|
The controller $K_\text{RMC}$ is:
|
|
#+begin_src matlab
|
|
Krmc = -(sqrt(k*m)-c)*s;
|
|
Krmc.InputName = {'d'};
|
|
Krmc.OutputName = {'F'};
|
|
#+end_src
|
|
|
|
And the closed loop system is computed below.
|
|
#+begin_src matlab
|
|
sys_rmc = feedback(sys, Krmc, 'name', +1);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(-1, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 2, 1);
|
|
title('Fd to d')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(sys('d', 'Fd'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(sys_rmc('d', 'Fd'), freqs, 'Hz'))));
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
xlim([freqs(1), freqs(end)]);
|
|
|
|
subplot(2, 2, 3);
|
|
title('Fd to x')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(sys('d', 'Fd'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(sys_rmc('d', 'Fd'), freqs, 'Hz'))));
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
xlim([freqs(1), freqs(end)]);
|
|
|
|
subplot(2, 2, 2);
|
|
title('w to d')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(sys('d', 'wddot')*s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(sys_rmc('d', 'wddot')*s^2, freqs, 'Hz'))));
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
xlim([freqs(1), freqs(end)]);
|
|
|
|
subplot(2, 2, 4);
|
|
title('w to x')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(1+sys('d', 'wddot')*s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(1+sys_rmc('d', 'wddot')*s^2, freqs, 'Hz'))));
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
xlim([freqs(1), freqs(end)]);
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/rmc_1dof_sensitivitiy.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:rmc_1dof_sensitivitiy
|
|
#+CAPTION: Sensitivity to disturbance when RMC is applied on the 1dof system ([[./figs/rmc_1dof_sensitivitiy.png][png]], [[./figs/rmc_1dof_sensitivitiy.pdf][pdf]])
|
|
[[file:figs/rmc_1dof_sensitivitiy.png]]
|
|
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no
|
|
simulinkproject('../');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('active_damping_uniaxial/matlab/sim_nano_station_id.slx')
|
|
#+end_src
|
|
|
|
** Control Design
|
|
Let's load the undamped plant:
|
|
#+begin_src matlab
|
|
load('./mat/active_damping_uniaxial_plants.mat', 'G');
|
|
#+end_src
|
|
|
|
Let's look at the transfer function from actuator forces in the nano-hexapod to the measured displacement of the actuator for all 6 pairs of actuator/sensor (figure [[fig:rmc_plant]]).
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i=1:6
|
|
plot(freqs, abs(squeeze(freqresp(G.G_dleg(['Dm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i=1:6
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_dleg(['Dm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/rmc_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:rmc_plant
|
|
#+CAPTION: Transfer function from forces applied in the legs to leg displacement sensor ([[./figs/rmc_plant.png][png]], [[./figs/rmc_plant.pdf][pdf]])
|
|
[[file:figs/rmc_plant.png]]
|
|
|
|
The Relative Motion Controller is defined below.
|
|
A Low pass Filter is added to make the controller transfer function proper.
|
|
#+begin_src matlab
|
|
K_rmc = s*50000/(1 + s/2/pi/10000);
|
|
#+end_src
|
|
|
|
The obtained loop gains are shown in figure [[fig:rmc_open_loop]].
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i=1:6
|
|
plot(freqs, abs(squeeze(freqresp(K_rmc*G.G_dleg(['Dm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i=1:6
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(K_rmc*G.G_dleg(['Dm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/rmc_open_loop.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:rmc_open_loop
|
|
#+CAPTION: Loop Gain for the Integral Force Feedback ([[./figs/rmc_open_loop.png][png]], [[./figs/rmc_open_loop.pdf][pdf]])
|
|
[[file:figs/rmc_open_loop.png]]
|
|
|
|
** Identification of the damped plant
|
|
Let's initialize the system prior to identification.
|
|
#+begin_src matlab
|
|
initializeReferences();
|
|
initializeGround();
|
|
initializeGranite();
|
|
initializeTy();
|
|
initializeRy();
|
|
initializeRz();
|
|
initializeMicroHexapod();
|
|
initializeAxisc();
|
|
initializeMirror();
|
|
initializeNanoHexapod('actuator', 'piezo');
|
|
initializeSample('mass', 50);
|
|
#+end_src
|
|
|
|
And initialize the controllers.
|
|
#+begin_src matlab
|
|
K = tf(zeros(6));
|
|
save('./mat/controllers_uniaxial.mat', 'K', '-append');
|
|
K_iff = tf(zeros(6));
|
|
save('./mat/controllers_uniaxial.mat', 'K_iff', '-append');
|
|
K_rmc = -K_rmc*eye(6);
|
|
save('./mat/controllers_uniaxial.mat', 'K_rmc', '-append');
|
|
K_dvf = tf(zeros(6));
|
|
save('./mat/controllers_uniaxial.mat', 'K_dvf', '-append');
|
|
#+end_src
|
|
|
|
We identify the system dynamics now that the RMC controller is ON.
|
|
#+begin_src matlab
|
|
G_rmc = identifyPlant();
|
|
#+end_src
|
|
|
|
And we save the damped plant for further analysis.
|
|
#+begin_src matlab
|
|
save('./mat/active_damping_uniaxial_plants.mat', 'G_rmc', '-append');
|
|
#+end_src
|
|
|
|
** Sensitivity to disturbances
|
|
As shown in figure [[fig:sensitivity_dist_rmc]], RMC control succeed in lowering the sensitivity to disturbances near resonance of the system.
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 1, 1);
|
|
title('$D_g$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / D_{g,x}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / D_{g,y}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / D_{g,z}\right|$');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'southeast');
|
|
|
|
subplot(2, 1, 2);
|
|
title('$F_s$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{s,x}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{s,y}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{s,z}\right|$');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/sensitivity_dist_rmc.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:sensitivity_dist_rmc
|
|
#+CAPTION: Sensitivity to disturbance once the RMC controller is applied to the system ([[./figs/sensitivity_dist_rmc.png][png]], [[./figs/sensitivity_dist_rmc.pdf][pdf]])
|
|
[[file:figs/sensitivity_dist_rmc.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{rz, z}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{ty, z}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{ty, x}\right|$');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/sensitivity_dist_stages_rmc.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:sensitivity_dist_stages_rmc
|
|
#+CAPTION: Sensitivity to force disturbances in various stages when RMC is applied ([[./figs/sensitivity_dist_stages_rmc.png][png]], [[./figs/sensitivity_dist_stages_rmc.pdf][pdf]])
|
|
[[file:figs/sensitivity_dist_stages_rmc.png]]
|
|
|
|
** Damped Plant
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 2, 1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
|
|
ax2 = subplot(2, 2, 2);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [rad/(Nm)]'); xlabel('Frequency [Hz]');
|
|
|
|
ax3 = subplot(2, 2, 3);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{n,x}\right|$');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{n,y}\right|$');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{n,z}\right|$');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
legend('location', 'northwest');
|
|
|
|
ax4 = subplot(2, 2, 4);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), 'DisplayName', '$\left|R_x / M_{n,x}\right|$');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))), 'DisplayName', '$\left|R_y / M_{n,y}\right|$');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), 'DisplayName', '$\left|R_z / M_{n,z}\right|$');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
legend('location', 'northwest');
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/plant_rmc_damped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:plant_rmc_damped
|
|
#+CAPTION: Damped Plant after RMC is applied ([[./figs/plant_rmc_damped.png][png]], [[./figs/plant_rmc_damped.pdf][pdf]])
|
|
[[file:figs/plant_rmc_damped.png]]
|
|
|
|
** Conclusion
|
|
#+begin_important
|
|
Relative Motion Control:
|
|
-
|
|
#+end_important
|
|
|
|
* Direct Velocity Feedback
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle ../matlab/dvf_uniaxial.m
|
|
:header-args:matlab+: :comments none :mkdirp yes
|
|
:END:
|
|
<<sec:dvf>>
|
|
|
|
** ZIP file containing the data and matlab files :ignore:
|
|
#+begin_src bash :exports none :results none
|
|
if [ matlab/dvf.m -nt data/dvf.zip ]; then
|
|
cp matlab/dvf.m dvf.m;
|
|
zip data/dvf \
|
|
mat/plant.mat \
|
|
dvf.m
|
|
rm dvf.m;
|
|
fi
|
|
#+end_src
|
|
|
|
#+begin_note
|
|
All the files (data and Matlab scripts) are accessible [[file:data/dvf.zip][here]].
|
|
#+end_note
|
|
|
|
** Introduction :ignore:
|
|
In the Relative Motion Control (RMC), a feedback is applied between the measured velocity of the platform to the actuator force input.
|
|
|
|
** One degree-of-freedom example
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle no
|
|
:END:
|
|
<<sec:dvf_1dof>>
|
|
*** Equations
|
|
#+begin_src latex :file dvf_1dof.pdf :post pdf2svg(file=*this*, ext="png") :exports results
|
|
\begin{tikzpicture}
|
|
% Ground
|
|
\draw (-1, 0) -- (1, 0);
|
|
|
|
% Ground Displacement
|
|
\draw[dashed] (-1, 0) -- ++(-0.5, 0) coordinate(w);
|
|
\draw[->] (w) -- ++(0, 0.5) node[left]{$w$};
|
|
|
|
% Mass
|
|
\draw[fill=white] (-1, 1) rectangle ++(2, 0.8) node[pos=0.5]{$m$};
|
|
|
|
% Velocity Sensor
|
|
\node[inertialsensor={0.3}] (velg) at (1, 1.8){};
|
|
\node[above] at (velg.north) {$\dot{x}$};
|
|
|
|
% Displacement of the mass
|
|
\draw[dashed] (-1, 1.8) -- ++(-0.5, 0) coordinate(x);
|
|
\draw[->] (x) -- ++(0, 0.5) node[left]{$x$};
|
|
|
|
% Spring, Damper, and Actuator
|
|
\draw[spring] (-0.8, 0) -- (-0.8, 1) node[midway, left=0.1]{$k$};
|
|
\draw[damper] (0, 0) -- (0, 1) node[midway, left=0.2]{$c$};
|
|
\draw[actuator={0.4}{0.2}] (0.8, 0) -- (0.8, 1) coordinate[midway, right=0.1](F);
|
|
|
|
% Control
|
|
\node[block={0.8cm}{0.6cm}, right=0.6 of F] (Kdvf) {$K$};
|
|
\draw[->] (Kdvf.west) -- (F) node[above right]{$F$};
|
|
\draw[<-] (Kdvf.east) -- ++(0.5, 0) |- (velg.east);
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:dvf_1dof
|
|
#+caption: Direct Velocity Feedback applied to a 1dof system
|
|
#+RESULTS:
|
|
[[file:figs/dvf_1dof.png]]
|
|
|
|
The dynamic of the system is:
|
|
\begin{equation}
|
|
ms^2x = F_d - kx - csx + kw + csw + F
|
|
\end{equation}
|
|
In terms of the stage deformation $d = x - w$:
|
|
\begin{equation}
|
|
(ms^2 + cs + k) d = -ms^2 w + F_d + F
|
|
\end{equation}
|
|
The direct velocity feedback law shown in figure [[fig:dvf_1dof]] is:
|
|
\begin{equation}
|
|
K = -g
|
|
\end{equation}
|
|
Thus, the applied force is:
|
|
\begin{equation}
|
|
F = -g \dot{x}
|
|
\end{equation}
|
|
And the new dynamics will be:
|
|
\begin{equation}
|
|
d = w \frac{-ms^2 - gs}{ms^2 + (c + g)s + k} + F_d \frac{1}{ms^2 + (c + g)s + k} + F \frac{1}{ms^2 + (c + g)s + k}
|
|
\end{equation}
|
|
|
|
And thus damping is added.
|
|
|
|
If critical damping is wanted:
|
|
\begin{equation}
|
|
\xi = \frac{1}{2}\frac{c + g}{\sqrt{km}} = \frac{1}{2}
|
|
\end{equation}
|
|
This corresponds to a gain:
|
|
\begin{equation}
|
|
g = \sqrt{km} - c
|
|
\end{equation}
|
|
|
|
*** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
*** Matlab Example
|
|
Let define the system parameters.
|
|
#+begin_src matlab
|
|
m = 50; % [kg]
|
|
k = 1e6; % [N/m]
|
|
c = 1e3; % [N/(m/s)]
|
|
#+end_src
|
|
|
|
The state space model of the system is defined below.
|
|
#+begin_src matlab
|
|
A = [-c/m -k/m;
|
|
1 0];
|
|
|
|
B = [1/m 1/m -1;
|
|
0 0 0];
|
|
|
|
C = [1 0;
|
|
0 1;
|
|
0 0];
|
|
|
|
D = [0 0 0;
|
|
0 0 0;
|
|
0 0 1];
|
|
|
|
sys = ss(A, B, C, D);
|
|
sys.InputName = {'F', 'Fd', 'wddot'};
|
|
sys.OutputName = {'ddot', 'd', 'wddot'};
|
|
sys.StateName = {'ddot', 'd'};
|
|
#+end_src
|
|
|
|
Because we need $\dot{x}$ for feedback, we compute it from the outputs
|
|
#+begin_src matlab
|
|
G_xdot = [1, 0, 1/s;
|
|
0, 1, 0];
|
|
G_xdot.InputName = {'ddot', 'd', 'wddot'};
|
|
G_xdot.OutputName = {'xdot', 'd'};
|
|
#+end_src
|
|
|
|
Finally, the system is described by =sys= as defined below.
|
|
#+begin_src matlab
|
|
sys = series(sys, G_xdot, [1 2 3], [1 2 3]);
|
|
#+end_src
|
|
|
|
The controller $K_\text{DVF}$ is:
|
|
#+begin_src matlab
|
|
Kdvf = tf(-(sqrt(k*m)-c));
|
|
Kdvf.InputName = {'xdot'};
|
|
Kdvf.OutputName = {'F'};
|
|
#+end_src
|
|
|
|
And the closed loop system is computed below.
|
|
#+begin_src matlab
|
|
sys_dvf = feedback(sys, Kdvf, 'name', +1);
|
|
#+end_src
|
|
|
|
The obtained sensitivity to disturbances is shown in figure [[fig:dvf_1dof_sensitivitiy]].
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(-1, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 2, 1);
|
|
title('Fd to d')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(sys('d', 'Fd'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(sys_dvf('d', 'Fd'), freqs, 'Hz'))));
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
xlim([freqs(1), freqs(end)]);
|
|
|
|
subplot(2, 2, 3);
|
|
title('Fd to x')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(sys('xdot', 'Fd')/s, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(sys_dvf('xdot', 'Fd')/s, freqs, 'Hz'))));
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
xlim([freqs(1), freqs(end)]);
|
|
|
|
subplot(2, 2, 2);
|
|
title('w to d')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(sys('d', 'wddot')*s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(sys_dvf('d', 'wddot')*s^2, freqs, 'Hz'))));
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
xlim([freqs(1), freqs(end)]);
|
|
|
|
subplot(2, 2, 4);
|
|
title('w to x')
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(1+sys('d', 'wddot')*s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(1+sys_dvf('d', 'wddot')*s^2, freqs, 'Hz'))));
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
xlim([freqs(1), freqs(end)]);
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/dvf_1dof_sensitivitiy.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:dvf_1dof_sensitivitiy
|
|
#+CAPTION: Sensitivity to disturbance when DVF is applied on the 1dof system ([[./figs/dvf_1dof_sensitivitiy.png][png]], [[./figs/dvf_1dof_sensitivitiy.pdf][pdf]])
|
|
[[file:figs/dvf_1dof_sensitivitiy.png]]
|
|
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no
|
|
simulinkproject('../');
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
open('active_damping_uniaxial/matlab/sim_nano_station_id.slx')
|
|
#+end_src
|
|
|
|
** Control Design
|
|
Let's load the undamped plant:
|
|
#+begin_src matlab
|
|
load('./mat/active_damping_uniaxial_plants.mat', 'G');
|
|
#+end_src
|
|
|
|
Let's look at the transfer function from actuator forces in the nano-hexapod to the measured velocity of the nano-hexapod platform in the direction of the corresponding actuator for all 6 pairs of actuator/sensor (figure [[fig:dvf_plant]]).
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i=1:6
|
|
plot(freqs, abs(squeeze(freqresp(G.G_geoph(['Vm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i=1:6
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_geoph(['Vm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/dvf_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:dvf_plant
|
|
#+CAPTION: Transfer function from forces applied in the legs to leg velocity sensor ([[./figs/dvf_plant.png][png]], [[./figs/dvf_plant.pdf][pdf]])
|
|
[[file:figs/dvf_plant.png]]
|
|
|
|
The controller is defined below and the obtained loop gain is shown in figure [[fig:dvf_open_loop_gain]].
|
|
|
|
#+begin_src matlab
|
|
K_dvf = tf(3e4);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
for i=1:6
|
|
plot(freqs, abs(squeeze(freqresp(K_dvf*G.G_geoph(['Vm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
for i=1:6
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(K_dvf*G.G_geoph(['Vm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/dvf_open_loop_gain.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:dvf_open_loop_gain
|
|
#+CAPTION: Loop Gain for DVF ([[./figs/dvf_open_loop_gain.png][png]], [[./figs/dvf_open_loop_gain.pdf][pdf]])
|
|
[[file:figs/dvf_open_loop_gain.png]]
|
|
|
|
** Identification of the damped plant
|
|
Let's initialize the system prior to identification.
|
|
#+begin_src matlab
|
|
initializeReferences();
|
|
initializeGround();
|
|
initializeGranite();
|
|
initializeTy();
|
|
initializeRy();
|
|
initializeRz();
|
|
initializeMicroHexapod();
|
|
initializeAxisc();
|
|
initializeMirror();
|
|
initializeNanoHexapod('actuator', 'piezo');
|
|
initializeSample('mass', 50);
|
|
#+end_src
|
|
|
|
And initialize the controllers.
|
|
#+begin_src matlab
|
|
K = tf(zeros(6));
|
|
save('./mat/controllers_uniaxial.mat', 'K', '-append');
|
|
K_iff = tf(zeros(6));
|
|
save('./mat/controllers_uniaxial.mat', 'K_iff', '-append');
|
|
K_rmc = tf(zeros(6));
|
|
save('./mat/controllers_uniaxial.mat', 'K_rmc', '-append');
|
|
K_dvf = -K_dvf*eye(6);
|
|
save('./mat/controllers_uniaxial.mat', 'K_dvf', '-append');
|
|
#+end_src
|
|
|
|
We identify the system dynamics now that the RMC controller is ON.
|
|
#+begin_src matlab
|
|
G_dvf = identifyPlant();
|
|
#+end_src
|
|
|
|
And we save the damped plant for further analysis.
|
|
#+begin_src matlab
|
|
save('./mat/active_damping_uniaxial_plants.mat', 'G_dvf', '-append');
|
|
#+end_src
|
|
|
|
** Sensitivity to disturbances
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
subplot(2, 1, 1);
|
|
title('$D_g$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / D_{g,x}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / D_{g,y}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / D_{g,z}\right|$');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
|
|
subplot(2, 1, 2);
|
|
title('$F_s$ to $D$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{s,x}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{s,y}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{s,z}\right|$');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/sensitivity_dist_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:sensitivity_dist_dvf
|
|
#+CAPTION: Sensitivity to disturbance once the DVF controller is applied to the system ([[./figs/sensitivity_dist_dvf.png][png]], [[./figs/sensitivity_dist_dvf.pdf][pdf]])
|
|
[[file:figs/sensitivity_dist_dvf.png]]
|
|
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{rz, z}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{ty, z}\right|$');
|
|
plot(freqs, abs(squeeze(freqresp(G.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{ty, x}\right|$');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/sensitivity_dist_stages_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:sensitivity_dist_stages_dvf
|
|
#+CAPTION: Sensitivity to force disturbances in various stages when DVF is applied ([[./figs/sensitivity_dist_stages_dvf.png][png]], [[./figs/sensitivity_dist_stages_dvf.pdf][pdf]])
|
|
[[file:figs/sensitivity_dist_stages_dvf.png]]
|
|
|
|
** Damped Plant
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
ax1 = subplot(2, 2, 1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
|
|
ax2 = subplot(2, 2, 2);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))));
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [rad/(Nm)]'); xlabel('Frequency [Hz]');
|
|
|
|
ax3 = subplot(2, 2, 3);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{n,x}\right|$');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{n,y}\right|$');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{n,z}\right|$');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
legend('location', 'northwest');
|
|
|
|
ax4 = subplot(2, 2, 4);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), 'DisplayName', '$\left|R_x / M_{n,x}\right|$');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))), 'DisplayName', '$\left|R_y / M_{n,y}\right|$');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), 'DisplayName', '$\left|R_z / M_{n,z}\right|$');
|
|
set(gca,'ColorOrderIndex',1);
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
legend('location', 'northwest');
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/plant_dvf_damped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:plant_dvf_damped
|
|
#+CAPTION: Damped Plant after DVF is applied ([[./figs/plant_dvf_damped.png][png]], [[./figs/plant_dvf_damped.pdf][pdf]])
|
|
[[file:figs/plant_dvf_damped.png]]
|
|
|
|
** Conclusion
|
|
#+begin_important
|
|
Direct Velocity Feedback:
|
|
#+end_important
|
|
|
|
* Comparison
|
|
<<sec:comparison>>
|
|
** Introduction :ignore:
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
cd('../');
|
|
#+end_src
|
|
|
|
** Load the plants
|
|
#+begin_src matlab
|
|
load('./mat/active_damping_uniaxial_plants.mat', 'G', 'G_iff', 'G_rmc', 'G_dvf');
|
|
#+end_src
|
|
|
|
** Sensitivity to Disturbance
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
title('$D_{g,z}$ to $D_z$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_gm( 'Dz', 'Dgz'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/sensitivity_comp_ground_motion_z.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:sensitivity_comp_ground_motion_z
|
|
#+CAPTION: caption ([[./figs/sensitivity_comp_ground_motion_z.png][png]], [[./figs/sensitivity_comp_ground_motion_z.pdf][pdf]])
|
|
[[file:figs/sensitivity_comp_ground_motion_z.png]]
|
|
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
title('$F_{s,z}$ to $D_z$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_fs( 'Dz', 'Fsz'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/sensitivity_comp_direct_forces_z.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:sensitivity_comp_direct_forces_z
|
|
#+CAPTION: caption ([[./figs/sensitivity_comp_direct_forces_z.png][png]], [[./figs/sensitivity_comp_direct_forces_z.pdf][pdf]])
|
|
[[file:figs/sensitivity_comp_direct_forces_z.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
title('$F_{rz,z}$ to $D_z$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_dist( 'Dz', 'Frzz'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/sensitivity_comp_spindle_z.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:sensitivity_comp_spindle_z
|
|
#+CAPTION: caption ([[./figs/sensitivity_comp_spindle_z.png][png]], [[./figs/sensitivity_comp_spindle_z.pdf][pdf]])
|
|
[[file:figs/sensitivity_comp_spindle_z.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
title('$F_{ty,z}$ to $D_z$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_dist( 'Dz', 'Ftyz'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/sensitivity_comp_ty_z.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:sensitivity_comp_ty_z
|
|
#+CAPTION: caption ([[./figs/sensitivity_comp_ty_z.png][png]], [[./figs/sensitivity_comp_ty_z.pdf][pdf]])
|
|
[[file:figs/sensitivity_comp_ty_z.png]]
|
|
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
title('$F_{ty,x}$ to $D_x$');
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_dist( 'Dx', 'Ftyx'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/sensitivity_comp_ty_x.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:sensitivity_comp_ty_x
|
|
#+CAPTION: caption ([[./figs/sensitivity_comp_ty_x.png][png]], [[./figs/sensitivity_comp_ty_x.pdf][pdf]])
|
|
[[file:figs/sensitivity_comp_ty_x.png]]
|
|
|
|
** Damped Plant
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
title('$F_{n,z}$ to $D_z$');
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart( 'Dz', 'Fnz'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
legend('location', 'northeast');
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart ('Dz', 'Fnz'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'k:');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'k--');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'k-.');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/plant_comp_damping_z.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:plant_comp_damping_z
|
|
#+CAPTION: Plant for the $z$ direction for different active damping technique used ([[./figs/plant_comp_damping_z.png][png]], [[./figs/plant_comp_damping_z.pdf][pdf]])
|
|
[[file:figs/plant_comp_damping_z.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
title('$F_{n,z}$ to $D_z$');
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart( 'Dx', 'Fnx'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
legend('location', 'northeast');
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart ('Dx', 'Fnx'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'k:');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'k--');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'k-.');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/plant_comp_damping_x.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:plant_comp_damping_x
|
|
#+CAPTION: Plant for the $x$ direction for different active damping technique used ([[./figs/plant_comp_damping_x.png][png]], [[./figs/plant_comp_damping_x.pdf][pdf]])
|
|
[[file:figs/plant_comp_damping_x.png]]
|
|
|
|
#+begin_src matlab :exports none
|
|
freqs = logspace(0, 3, 1000);
|
|
|
|
figure;
|
|
|
|
title('$F_{n,x}$ to $R_z$');
|
|
ax1 = subplot(2, 1, 1);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G.G_cart( 'Rz', 'Fnx'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'Undamped');
|
|
plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Rz', 'Fnx'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF');
|
|
plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Rz', 'Fnx'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC');
|
|
plot(freqs, abs(squeeze(freqresp(G_dvf.G_cart('Rz', 'Fnx'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
legend('location', 'northeast');
|
|
|
|
ax2 = subplot(2, 1, 2);
|
|
hold on;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart ('Ry', 'Fnx'), freqs, 'Hz'))), 'k-');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Ry', 'Fnx'), freqs, 'Hz'))), 'k:');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Ry', 'Fnx'), freqs, 'Hz'))), 'k--');
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf.G_cart('Ry', 'Fnx'), freqs, 'Hz'))), 'k-.');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180, -90, 0, 90, 180]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
#+end_src
|
|
|
|
#+HEADER: :tangle no :exports results :results none :noweb yes
|
|
#+begin_src matlab :var filepath="figs/plant_comp_damping_coupling.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
|
<<plt-matlab>>
|
|
#+end_src
|
|
|
|
#+NAME: fig:plant_comp_damping_coupling
|
|
#+CAPTION: Comparison of one off-diagonal plant for different damping technique applied ([[./figs/plant_comp_damping_coupling.png][png]], [[./figs/plant_comp_damping_coupling.pdf][pdf]])
|
|
[[file:figs/plant_comp_damping_coupling.png]]
|
|
|
|
* Conclusion
|
|
<<sec:conclusion>>
|