nass-simscape/active_damping/index.org

#+TITLE: Active Damping applied on the Simscape Model
:DRAWER:
#+STARTUP: overview

#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

#+HTML_LINK_HOME: ../index.html
#+HTML_LINK_UP: ../index.html

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#+PROPERTY: header-args:matlab  :session *MATLAB*
#+PROPERTY: header-args:matlab+ :comments org
#+PROPERTY: header-args:matlab+ :results none
#+PROPERTY: header-args:matlab+ :exports both
#+PROPERTY: header-args:matlab+ :eval no-export
#+PROPERTY: header-args:matlab+ :output-dir figs
#+PROPERTY: header-args:matlab+ :tangle no
#+PROPERTY: header-args:matlab+ :mkdirp yes

#+PROPERTY: header-args:shell  :eval no-export

#+PROPERTY: header-args:latex  :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/}{config.tex}")
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
#+PROPERTY: header-args:latex+ :results raw replace :buffer no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports both
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
:END:

* Introduction                                                       :ignore:
First, in section [[sec:undamped_system]], we will looked at the undamped system.

Then, we will compare three active damping techniques:
- In section [[sec:iff]]: the integral force feedback is used
- In section [[sec:dvf]]: the direct velocity feedback is used
- In section [[sec:ine]]: inertial control is used

For each of the active damping technique, we will:
- Look at the damped plant
- Simulate tomography experiments
- Compare the sensitivity from disturbances

The disturbances are:
- Ground motion
- Motion errors of all the stages

* Undamped System
:PROPERTIES:
:header-args:matlab+: :tangle matlab/undamped_system.m
:header-args:matlab+: :comments none :mkdirp yes
:END:
<<sec:undamped_system>>

** ZIP file containing the data and matlab files                     :ignore:
#+begin_src bash :exports none :results none
  if [ matlab/undamped_system.m -nt data/undamped_system.zip ]; then
    cp matlab/undamped_system.m undamped_system.m;
    zip data/undamped_system \
        undamped_system.m
    rm undamped_system.m;
  fi
#+end_src

#+begin_note
  All the files (data and Matlab scripts) are accessible [[file:data/undamped_system.zip][here]].
#+end_note

** Introduction                                                      :ignore:
We first look at the undamped system.
The performance of this undamped system will be compared with the damped system using various techniques.

** 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
  addpath('active_damping/src/');
#+end_src

#+begin_src matlab
  open('active_damping/matlab/sim_nass_active_damping.slx')
#+end_src

** Identification of the dynamics for Active Damping
*** Initialize the Simulation
We initialize all the stages with the default parameters.
#+begin_src matlab
  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();
#+end_src

The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
#+begin_src matlab
  initializeNanoHexapod('actuator', 'piezo');
  initializeSample('mass', 50);
#+end_src

We set the references to zero.
#+begin_src matlab
  initializeReferences();
#+end_src

And all the controllers are set to 0.
#+begin_src matlab
  K = tf(zeros(6));
  save('./mat/controllers.mat', 'K', '-append');
  K_ine = tf(zeros(6));
  save('./mat/controllers.mat', 'K_ine', '-append');
  K_iff = tf(zeros(6));
  save('./mat/controllers.mat', 'K_iff', '-append');
  K_dvf = tf(zeros(6));
  save('./mat/controllers.mat', 'K_dvf', '-append');
#+end_src

*** Identification
First, we identify the dynamics of the system using the =linearize= function.
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'sim_nass_active_damping';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fnl'],           1, 'openinput');              io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Vlm');  io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
  G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ...
                  'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6', ...
                  'Vnlm1', 'Vnlm2', 'Vnlm3', 'Vnlm4', 'Vnlm5', 'Vnlm6'};
#+end_src

We then create transfer functions corresponding to the active damping plants.
#+begin_src matlab
  G_iff = minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}));
  G_dvf = minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}));
  G_ine = minreal(G({'Vnlm1', 'Vnlm2', 'Vnlm3', 'Vnlm4', 'Vnlm5', 'Vnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}));
#+end_src

And we save them for further analysis.
#+begin_src matlab
  save('./active_damping/mat/undamped_plants.mat', 'G_iff', 'G_dvf', 'G_ine');
#+end_src

*** Obtained Plants for Active Damping
#+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_iff(['Fnlm', num2str(i)], ['Fnl', 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_iff(['Fnlm', num2str(i)], ['Fnl', 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/nass_active_damping_iff_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:nass_active_damping_iff_plant
#+CAPTION: =G_iff=: IFF Plant ([[./figs/nass_active_damping_iff_plant.png][png]], [[./figs/nass_active_damping_iff_plant.pdf][pdf]])
[[file:figs/nass_active_damping_iff_plant.png]]

#+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_dvf(['Dnlm', num2str(i)], ['Fnl', 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_dvf(['Dnlm', num2str(i)], ['Fnl', 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/nass_active_damping_dvf_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:nass_active_damping_dvf_plant
#+CAPTION: =G_dvf=: Plant for Direct Velocity Feedback ([[./figs/nass_active_damping_dvf_plant.png][png]], [[./figs/nass_active_damping_dvf_plant.pdf][pdf]])
[[file:figs/nass_active_damping_ine_plant.png]]

#+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_ine(['Vnlm', num2str(i)], ['Fnl', num2str(i)]), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [$\frac{m/s}{N}$]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(G_ine(['Vnlm', num2str(i)], ['Fnl', 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/nass_active_damping_inertial_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:nass_active_damping_inertial_plant
#+CAPTION: Inertial Feedback Plant ([[./figs/nass_active_damping_inertial_plant.png][png]], [[./figs/nass_active_damping_inertial_plant.pdf][pdf]])
[[file:figs/nass_active_damping_inertial_plant.png]]

** Tomography Experiment
*** Simulation
We initialize elements for the tomography experiment.
#+begin_src matlab
  prepareTomographyExperiment();
#+end_src

We change the simulation stop time.
#+begin_src matlab
  load('mat/conf_simscape.mat');
  set_param(conf_simscape, 'StopTime', '3');
#+end_src

And we simulate the system.
#+begin_src matlab
  sim('sim_nass_active_damping');
#+end_src

Finally, we save the simulation results for further analysis
#+begin_src matlab
  save('./active_damping/mat/tomo_exp.mat', 'En', 'Eg', '-append');
#+end_src

*** Results
We load the results of tomography experiments.
#+begin_src matlab
  load('./active_damping/mat/tomo_exp.mat', 'En');
  t = linspace(0, 3, length(En(:,1)));
#+end_src

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(t, En(:,1), 'DisplayName', '$\epsilon_{x}$')
  plot(t, En(:,2), 'DisplayName', '$\epsilon_{y}$')
  plot(t, En(:,3), 'DisplayName', '$\epsilon_{z}$')
  hold off;
  legend();
  xlabel('Time [s]'); ylabel('Position Error [m]');
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/nass_act_damp_undamped_sim_tomo_trans.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:nass_act_damp_undamped_sim_tomo_trans
#+CAPTION: Position Error during tomography experiment - Translations ([[./figs/nass_act_damp_undamped_sim_tomo_trans.png][png]], [[./figs/nass_act_damp_undamped_sim_tomo_trans.pdf][pdf]])
[[file:figs/nass_act_damp_undamped_sim_tomo_trans.png]]

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(t, En(:,4), 'DisplayName', '$\epsilon_{\theta_x}$')
  plot(t, En(:,5), 'DisplayName', '$\epsilon_{\theta_y}$')
  plot(t, En(:,6), 'DisplayName', '$\epsilon_{\theta_z}$')
  hold off;
  xlim([0.5,inf]);
  legend();
  xlabel('Time [s]'); ylabel('Position Error [rad]');
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/nass_act_damp_undamped_sim_tomo_rot.pdf" :var figsize="wide-normal" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:nass_act_damp_undamped_sim_tomo_rot
#+CAPTION: Position Error during tomography experiment - Rotations ([[./figs/nass_act_damp_undamped_sim_tomo_rot.png][png]], [[./figs/nass_act_damp_undamped_sim_tomo_rot.pdf][pdf]])
[[file:figs/nass_act_damp_undamped_sim_tomo_rot.png]]

* Integral Force Feedback
:PROPERTIES:
:header-args:matlab+: :tangle matlab/iff.m
:header-args:matlab+: :comments none :mkdirp yes
:END:
<<sec:iff>>

** ZIP file containing the data and matlab files                     :ignore:
#+begin_src bash :exports none :results none
  if [ matlab/iff.m -nt data/iff.zip ]; then
    cp matlab/iff.m iff.m;
    zip data/iff \
        mat/plant.mat \
        iff.m
    rm iff.m;
  fi
#+end_src

#+begin_note
  All the files (data and Matlab scripts) are accessible [[file:data/iff.zip][here]].
#+end_note

** Introduction                                                      :ignore:
Integral Force Feedback is applied on the simscape model.

** 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
  addpath('active_damping/src/');
#+end_src

#+begin_src matlab
  open('active_damping/matlab/sim_nass_active_damping.slx')
#+end_src

** Control Design
*** Plant
Let's load the previously indentified undamped plant:
#+begin_src matlab
  load('./active_damping/mat/undamped_plants.mat', 'G_iff');
#+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_iff(['Fnlm', num2str(i)], ['Fnl', 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_iff(['Fnlm', num2str(i)], ['Fnl', 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]]

*** Control Design
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_iff(['Fnlm', num2str(i)], ['Fnl', 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_iff(['Fnlm', num2str(i)], ['Fnl', 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]]

*** Diagonal Controller
We create the diagonal controller and we add a minus sign as we have a positive
feedback architecture.
#+begin_src matlab
  K_iff = -K_iff*eye(6);
#+end_src

We save the controller for further analysis.
#+begin_src matlab
  save('./active_damping/mat/K_iff.mat', 'K_iff');
#+end_src

*** IFF with High Pass Filter
#+begin_src matlab
  w_hpf = 2*pi*10; % Cut-off frequency for the high pass filter [rad/s]
  w_lpf = 2*pi*200; % Cut-off frequency for the low pass filter [rad/s]

  K_iff = 2*pi*200/s * (s/w_hpf)/(s/w_hpf + 1) * 1/(s/w_lpf + 1);
#+end_src

The corresponding loop gains are shown in figure [[fig:iff_hpf_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_iff(['Fnlm', num2str(i)], ['Fnl', 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_iff(['Fnlm', num2str(i)], ['Fnl', 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_hpf_open_loop.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:iff_hpf_open_loop
#+CAPTION: Loop Gain for the Integral Force Feedback with an High pass filter ([[./figs/iff_hpf_open_loop.png][png]], [[./figs/iff_hpf_open_loop.pdf][pdf]])
[[file:figs/iff_hpf_open_loop.png]]

We create the diagonal controller and we add a minus sign as we have a positive
feedback architecture.
#+begin_src matlab
  K_iff = -K_iff*eye(6);
#+end_src

We save the controller for further analysis.
#+begin_src matlab
  save('./active_damping/mat/K_iff_hpf.mat', 'K_iff');
#+end_src

** TODO Identification of the damped plant                         :noexport:
*** Initialize the Simulation
We initialize all the stages with the default parameters.
#+begin_src matlab
    initializeGround();
    initializeGranite();
    initializeTy();
    initializeRy();
    initializeRz();
    initializeMicroHexapod();
    initializeAxisc();
    initializeMirror();
#+end_src

The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
#+begin_src matlab
  initializeNanoHexapod('actuator', 'piezo');
  initializeSample('mass', 50);
#+end_src

We set the references to zero.
#+begin_src matlab
  initializeReferences();
#+end_src

And all the controllers are set to 0 except for the IFF.
#+begin_src matlab
  K = tf(zeros(6));
  save('./mat/controllers.mat', 'K', '-append');
  K_ine = tf(zeros(6));
  save('./mat/controllers.mat', 'K_ine', '-append');
  K_iff = K_iff;
  save('./mat/controllers.mat', 'K_iff', '-append');
  K_dvf = tf(zeros(6));
  save('./mat/controllers.mat', 'K_dvf', '-append');
#+end_src

*** Identification
First, we identify the dynamics of the system using the =linearize= function.
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'sim_nass_active_damping';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fnl'],           1, 'openinput');              io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
  G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ...
                  'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'};
#+end_src

We then create transfer functions corresponding to the active damping plants.
#+begin_src matlab
  G_iff = minreal(G({'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}));
  % G_rmc = minreal(G({'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'}, {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'}));
#+end_src

And we save them for further analysis.
#+begin_src matlab
  save('./active_damping/mat/plants.mat', 'G_iff', '-append');
#+end_src

*** TODO 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]]

*** TODO 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]]

** Tomography Experiment
*** Simulation with IFF Controller
We initialize elements for the tomography experiment.
#+begin_src matlab
  prepareTomographyExperiment();
#+end_src

We set the IFF controller.
#+begin_src matlab
  load('./active_damping/mat/K_iff.mat', 'K_iff');
  save('./mat/controllers.mat', 'K_iff', '-append');
#+end_src

We change the simulation stop time.
#+begin_src matlab
  load('mat/conf_simscape.mat');
  set_param(conf_simscape, 'StopTime', '3');
#+end_src

And we simulate the system.
#+begin_src matlab
  sim('sim_nass_active_damping');
#+end_src

Finally, we save the simulation results for further analysis
#+begin_src matlab
  En_iff = En;
  Eg_iff = Eg;
  save('./active_damping/mat/tomo_exp.mat', 'En_iff', 'Eg_iff', '-append');
#+end_src

*** Simulation with IFF Controller with added High Pass Filter
We initialize elements for the tomography experiment.
#+begin_src matlab
  prepareTomographyExperiment();
#+end_src

We set the IFF controller with the High Pass Filter.
#+begin_src matlab
  load('./active_damping/mat/K_iff_hpf.mat', 'K_iff');
  save('./mat/controllers.mat', 'K_iff', '-append');
#+end_src

We change the simulation stop time.
#+begin_src matlab
  load('mat/conf_simscape.mat');
  set_param(conf_simscape, 'StopTime', '3');
#+end_src

And we simulate the system.
#+begin_src matlab
  sim('sim_nass_active_damping');
#+end_src

Finally, we save the simulation results for further analysis
#+begin_src matlab
  En_iff_hpf = En;
  Eg_iff_hpf = Eg;
  save('./active_damping/mat/tomo_exp.mat', 'En_iff_hpf', 'Eg_iff_hpf', '-append');
#+end_src

*** Compare with Undamped system
We load the results of tomography experiments.
#+begin_src matlab
  load('./active_damping/mat/tomo_exp.mat', 'En', 'En_iff', 'En_iff_hpf');
  t = linspace(0, 3, length(En(:,1)));
#+end_src

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(En(:,1),         En(:,2),         'DisplayName', '$\epsilon_{x,y}$ - OL')
  plot(En_iff(:,1),     En_iff(:,2),     'DisplayName', '$\epsilon_{x,y}$ - IFF')
  plot(En_iff_hpf(:,1), En_iff_hpf(:,2), 'DisplayName', '$\epsilon_{x,y}$ - IFF + HPF')
  xlabel('X Motion [m]'); ylabel('Y Motion [m]');
  legend('location', 'northwest');
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/nass_act_damp_iff_sim_tomo_xy.pdf" :var figsize="small-normal" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:nass_act_damp_iff_sim_tomo_xy
#+CAPTION: Position Error during tomography experiment - XY Motion ([[./figs/nass_act_damp_iff_sim_tomo_xy.png][png]], [[./figs/nass_act_damp_iff_sim_tomo_xy.pdf][pdf]])
[[file:figs/nass_act_damp_iff_sim_tomo_xy.png]]

#+begin_src matlab :exports none
  figure;
  ax1 = subplot(3, 1, 1);
  hold on;
  plot(t, En(:,1), 'DisplayName', '$\epsilon_{x}$')
  plot(t, En_iff(:,1), 'DisplayName', '$\epsilon_{x}$ - IFF')
  plot(t, En_iff_hpf(:,1), 'DisplayName', '$\epsilon_{x}$ - IFF + HPF')
  legend('location', 'southwest');

  ax2 = subplot(3, 1, 2);
  hold on;
  plot(t, En(:,2), 'DisplayName', '$\epsilon_{y}$')
  plot(t, En_iff(:,2), 'DisplayName', '$\epsilon_{y}$ - IFF')
  plot(t, En_iff_hpf(:,2), 'DisplayName', '$\epsilon_{y}$ - IFF + HPF')
  legend('location', 'southwest');
  ylabel('Position Error [m]');

  ax3 = subplot(3, 1, 3);
  hold on;
  plot(t, En(:,3), 'DisplayName', '$\epsilon_{z}$')
  plot(t, En_iff(:,3), 'DisplayName', '$\epsilon_{z}$ - IFF')
  plot(t, En_iff_hpf(:,3), 'DisplayName', '$\epsilon_{z}$ - IFF + HPF')
  legend('location', 'northwest');
  xlabel('Time [s]');
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/nass_act_damp_iff_sim_tomo_trans.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:nass_act_damp_iff_sim_tomo_trans
#+CAPTION: Position Error during tomography experiment - Translations ([[./figs/nass_act_damp_iff_sim_tomo_trans.png][png]], [[./figs/nass_act_damp_iff_sim_tomo_trans.pdf][pdf]])
[[file:figs/nass_act_damp_iff_sim_tomo_trans.png]]

#+begin_src matlab :exports none
  figure;
  ax1 = subplot(3, 1, 1);
  hold on;
  plot(t, En(:,4), 'DisplayName', '$\epsilon_{\theta_x}$')
  plot(t, En_iff(:,4), 'DisplayName', '$\epsilon_{\theta_x}$ - IFF')
  plot(t, En_iff_hpf(:,4), 'DisplayName', '$\epsilon_{\theta_x}$ - IFF + HPF')
  legend('location', 'northwest');

  ax2 = subplot(3, 1, 2);
  hold on;
  plot(t, En(:,5), 'DisplayName', '$\epsilon_{\theta_y}$')
  plot(t, En_iff(:,5), 'DisplayName', '$\epsilon_{\theta_y}$ - IFF')
  plot(t, En_iff_hpf(:,5), 'DisplayName', '$\epsilon_{\theta_y}$ - IFF + HPF')
  legend('location', 'southwest');
  ylabel('Position Error [rad]');

  ax3 = subplot(3, 1, 3);
  hold on;
  plot(t, En(:,6), 'DisplayName', '$\epsilon_{\theta_z}$')
  plot(t, En_iff(:,6), 'DisplayName', '$\epsilon_{\theta_z}$ - IFF')
  plot(t, En_iff_hpf(:,6), 'DisplayName', '$\epsilon_{\theta_z}$ - IFF + HPF')
  legend();
  xlabel('Time [s]');

  linkaxes([ax1,ax2,ax3],'x');
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/nass_act_damp_iff_sim_tomo_rot.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:nass_act_damp_iff_sim_tomo_rot
#+CAPTION: Position Error during tomography experiment - Rotations ([[./figs/nass_act_damp_iff_sim_tomo_rot.png][png]], [[./figs/nass_act_damp_iff_sim_tomo_rot.pdf][pdf]])
[[file:figs/nass_act_damp_iff_sim_tomo_rot.png]]

** Conclusion
#+begin_important
Integral Force Feedback:
- Robust (guaranteed stability)
- Acceptable Damping
- Increase the sensitivity to disturbances at low frequencies
#+end_important

* Direct Velocity Feedback
:PROPERTIES:
:header-args:matlab+: :tangle matlab/dvf.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 Direct Velocity Feedback (DVF), a derivative feedback is applied between the measured actuator displacement to the actuator force input.
The actuator displacement can be measured with a capacitive sensor for instance.

** 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
  addpath('active_damping/src/');
#+end_src

#+begin_src matlab
  open('active_damping/matlab/sim_nass_active_damping.slx')
#+end_src

** Control Design
*** Plant
Let's load the undamped plant:
#+begin_src matlab
  load('./active_damping/mat/undamped_plants.mat', 'G_dvf');
#+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: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_dvf(['Dnlm', num2str(i)], ['Fnl', 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_dvf(['Dnlm', num2str(i)], ['Fnl', 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 displacement sensor ([[./figs/dvf_plant.png][png]], [[./figs/dvf_plant.pdf][pdf]])
[[file:figs/dvf_plant.png]]

*** Control Design
The Direct Velocity Feedback is defined below.
A Low pass Filter is added to make the controller transfer function proper.
#+begin_src matlab
  K_dvf = s*20000/(1 + s/2/pi/10000);
#+end_src

The obtained loop gains are shown in figure [[fig:dvf_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_dvf*G_dvf(['Dnlm', num2str(i)], ['Fnl', 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_dvf(['Dnlm', num2str(i)], ['Fnl', 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.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:dvf_open_loop
#+CAPTION: Loop Gain for the Integral Force Feedback ([[./figs/dvf_open_loop.png][png]], [[./figs/dvf_open_loop.pdf][pdf]])
[[file:figs/dvf_open_loop.png]]

*** Diagonal Controller
We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.
#+begin_src matlab
  K_dvf = -K_dvf*eye(6);
#+end_src

We save the controller for further analysis.
#+begin_src matlab
  save('./active_damping/mat/K_dvf.mat', 'K_dvf');
#+end_src

** TODO Identification of the damped plant                         :noexport:
*** Initialize the Simulation
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));
  K_ine = tf(zeros(6));
  save('./mat/controllers.mat', 'K_ine', '-append');
  save('./mat/controllers.mat', 'K', '-append');
  K_iff = tf(zeros(6));
  save('./mat/controllers.mat', 'K_iff', '-append');
  K_dvf = K_dvf;
  save('./mat/controllers.mat', 'K_dvf', '-append');
#+end_src

*** Identification
We identify the system dynamics now that the DVF controller is ON.
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'sim_nass_active_damping';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fnl'], 1, 'openinput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Fnlm'); io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
  G.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6', ...
                  'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'};
#+end_src

And we save the damped plant for further analysis.
#+begin_src matlab
  save('./active_damping/mat/plants.mat', 'G_dvf', '-append');
#+end_src

*** Sensitivity to disturbances
As shown in figure [[fig:sensitivity_dist_dvf]], DVF 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_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', '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_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]]

** Tomography Experiment
*** Initialize the Simulation
We initialize elements for the tomography experiment.
#+begin_src matlab
  prepareTomographyExperiment();
#+end_src

We set the DVF controller.
#+begin_src matlab
  load('./active_damping/mat/K_dvf.mat', 'K_dvf');
  save('./mat/controllers.mat', 'K_dvf', '-append');
#+end_src

*** Simulation
We change the simulation stop time.
#+begin_src matlab
  load('mat/conf_simscape.mat');
  set_param(conf_simscape, 'StopTime', '3');
#+end_src

And we simulate the system.
#+begin_src matlab
  sim('sim_nass_active_damping');
#+end_src

Finally, we save the simulation results for further analysis
#+begin_src matlab
  En_dvf = En;
  Eg_dvf = Eg;
  save('./active_damping/mat/tomo_exp.mat', 'En_dvf', 'Eg_dvf', '-append');
#+end_src

*** Compare with Undamped system
We load the results of tomography experiments.
#+begin_src matlab
  load('./active_damping/mat/tomo_exp.mat', 'En', 'En_dvf');
  t = linspace(0, 3, length(En(:,1)));
#+end_src

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(En(:,1),     En(:,2),     'DisplayName', '$\epsilon_{x,y}$ - OL')
  plot(En_dvf(:,1), En_dvf(:,2), 'DisplayName', '$\epsilon_{x,y}$ - DVF')
  xlabel('X Motion [m]'); ylabel('Y Motion [m]');
  legend();
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/nass_act_damp_dvf_sim_tomo_xy.pdf" :var figsize="small-normal" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:nass_act_damp_dvf_sim_tomo_xy
#+CAPTION: Position Error during tomography experiment - XY Motion ([[./figs/nass_act_damp_dvf_sim_tomo_xy.png][png]], [[./figs/nass_act_damp_dvf_sim_tomo_xy.pdf][pdf]])
[[file:figs/nass_act_damp_dvf_sim_tomo_xy.png]]

#+begin_src matlab :exports none
  figure;
  ax1 = subplot(3, 1, 1);
  hold on;
  plot(t, En(:,1), 'DisplayName', '$\epsilon_{x}$')
  plot(t, En_dvf(:,1), 'DisplayName', '$\epsilon_{x}$ - DVF')
  legend();

  ax2 = subplot(3, 1, 2);
  hold on;
  plot(t, En(:,2), 'DisplayName', '$\epsilon_{y}$')
  plot(t, En_dvf(:,2), 'DisplayName', '$\epsilon_{y}$ - DVF')
  legend();
  ylabel('Position Error [m]');

  ax3 = subplot(3, 1, 3);
  hold on;
  plot(t, En(:,3), 'DisplayName', '$\epsilon_{z}$')
  plot(t, En_dvf(:,3), 'DisplayName', '$\epsilon_{z}$ - DVF')
  legend();
  xlabel('Time [s]');
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/nass_act_damp_dvf_sim_tomo_trans.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:nass_act_damp_dvf_sim_tomo_trans
#+CAPTION: Position Error during tomography experiment - Translations ([[./figs/nass_act_damp_dvf_sim_tomo_trans.png][png]], [[./figs/nass_act_damp_dvf_sim_tomo_trans.pdf][pdf]])
[[file:figs/nass_act_damp_dvf_sim_tomo_trans.png]]

#+begin_src matlab :exports none
  figure;
  ax1 = subplot(3, 1, 1);
  hold on;
  plot(t, En(:,4), 'DisplayName', '$\epsilon_{\theta_x}$')
  plot(t, En_dvf(:,4), 'DisplayName', '$\epsilon_{\theta_x}$ - DVF')
  legend();

  ax2 = subplot(3, 1, 2);
  hold on;
  plot(t, En(:,5), 'DisplayName', '$\epsilon_{\theta_y}$')
  plot(t, En_dvf(:,5), 'DisplayName', '$\epsilon_{\theta_y}$ - DVF')
  legend();
  ylabel('Position Error [rad]');

  ax3 = subplot(3, 1, 3);
  hold on;
  plot(t, En(:,6), 'DisplayName', '$\epsilon_{\theta_z}$')
  plot(t, En_dvf(:,6), 'DisplayName', '$\epsilon_{\theta_z}$ - DVF')
  legend();
  xlabel('Time [s]');

  linkaxes([ax1,ax2,ax3],'x');
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/nass_act_damp_dvf_sim_tomo_rot.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:nass_act_damp_dvf_sim_tomo_rot
#+CAPTION: Position Error during tomography experiment - Rotations ([[./figs/nass_act_damp_dvf_sim_tomo_rot.png][png]], [[./figs/nass_act_damp_dvf_sim_tomo_rot.pdf][pdf]])
[[file:figs/nass_act_damp_dvf_sim_tomo_rot.png]]

** Conclusion
#+begin_important
Direct Velocity Feedback:
-
#+end_important

* Inertial Control
:PROPERTIES:
:header-args:matlab+: :tangle matlab/ine.m
:header-args:matlab+: :comments none :mkdirp yes
:END:
<<sec:ine>>

** ZIP file containing the data and matlab files                     :ignore:
#+begin_src bash :exports none :results none
  if [ matlab/ine.m -nt data/ine.zip ]; then
    cp matlab/ine.m ine.m;
    zip data/ine \
        mat/plant.mat \
        ine.m
    rm ine.m;
  fi
#+end_src

#+begin_note
  All the files (data and Matlab scripts) are accessible [[file:data/ine.zip][here]].
#+end_note

** Introduction                                                      :ignore:
In Inertial Control, a feedback is applied between the measured *absolute* motion (velocity or acceleration) of the platform to the actuator force input.

** 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
  addpath('active_damping/src/');
#+end_src

#+begin_src matlab
  open('active_damping/matlab/sim_nass_active_damping.slx')
#+end_src

** Control Design
*** Plant
Let's load the undamped plant:
#+begin_src matlab
  load('./active_damping/mat/undamped_plants.mat', 'G_ine');
#+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:ine_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_ine(['Vnlm', num2str(i)], ['Fnl', num2str(i)]), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [$\frac{m/s^2}{N}$]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i=1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(G_ine(['Vnlm', num2str(i)], ['Fnl', 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/ine_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:ine_plant
#+CAPTION: Transfer function from forces applied in the legs to leg velocity sensor ([[./figs/ine_plant.png][png]], [[./figs/ine_plant.pdf][pdf]])
[[file:figs/ine_plant.png]]

*** Control Design
The controller is defined below and the obtained loop gain is shown in figure [[fig:ine_open_loop_gain]].

#+begin_src matlab
  K_ine = 1e4/(1+s/(2*pi*100));
#+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_ine*G_ine(['Vnlm', num2str(i)], ['Fnl', 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_ine*G_ine(['Vnlm', num2str(i)], ['Fnl', 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/ine_open_loop_gain.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:ine_open_loop_gain
#+CAPTION: Loop Gain for Inertial Control ([[./figs/ine_open_loop_gain.png][png]], [[./figs/ine_open_loop_gain.pdf][pdf]])
[[file:figs/ine_open_loop_gain.png]]

*** Diagonal Controller
We create the diagonal controller and we add a minus sign as we have a positive feedback architecture.
#+begin_src matlab
  K_ine = -K_ine*eye(6);
#+end_src

We save the controller for further analysis.
#+begin_src matlab
  save('./active_damping/mat/K_ine.mat', 'K_ine');
#+end_src

** TODO Identification of the damped plant                         :noexport:
*** Initialize the Simulation
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.mat', 'K', '-append');
  K_ine = -K_ine*eye(6);
  save('./mat/controllers.mat', 'K_ine', '-append');
  K_iff = tf(zeros(6));
  save('./mat/controllers.mat', 'K_iff', '-append');
  K_dvf = tf(zeros(6));
  save('./mat/controllers.mat', 'K_dvf', '-append');
#+end_src

*** Identification
We identify the system dynamics now that the Inertial controller is ON.
#+begin_src matlab
  G_ine = identifyPlant();
#+end_src

And we save the damped plant for further analysis.
#+begin_src matlab
  save('./active_damping/mat/plants.mat', 'G_ine', '-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_ine.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G_ine.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G_ine.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_ine.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G_ine.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G_ine.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_ine.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:sensitivity_dist_ine
#+CAPTION: Sensitivity to disturbance once the INE controller is applied to the system ([[./figs/sensitivity_dist_ine.png][png]], [[./figs/sensitivity_dist_ine.pdf][pdf]])
[[file:figs/sensitivity_dist_ine.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_ine.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G_ine.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G_ine.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_ine.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:sensitivity_dist_stages_ine
#+CAPTION: Sensitivity to force disturbances in various stages when INE is applied ([[./figs/sensitivity_dist_stages_ine.png][png]], [[./figs/sensitivity_dist_stages_ine.pdf][pdf]])
[[file:figs/sensitivity_dist_stages_ine.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_ine.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--');
  plot(freqs, abs(squeeze(freqresp(G_ine.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--');
  plot(freqs, abs(squeeze(freqresp(G_ine.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_ine.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--');
  plot(freqs, abs(squeeze(freqresp(G_ine.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--');
  plot(freqs, abs(squeeze(freqresp(G_ine.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_ine.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_ine.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_ine.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_ine.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_ine.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_ine.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_ine_damped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:plant_ine_damped
#+CAPTION: Damped Plant after INE is applied ([[./figs/plant_ine_damped.png][png]], [[./figs/plant_ine_damped.pdf][pdf]])
[[file:figs/plant_ine_damped.png]]

** Tomography Experiment
*** Initialize the Simulation
We initialize elements for the tomography experiment.
#+begin_src matlab
  prepareTomographyExperiment();
#+end_src

We set the Inertial controller.
#+begin_src matlab
  load('./active_damping/mat/K_ine.mat', 'K_ine');
  save('./mat/controllers.mat', 'K_ine', '-append');
#+end_src

*** Simulation
We change the simulation stop time.
#+begin_src matlab
  load('mat/conf_simscape.mat');
  set_param(conf_simscape, 'StopTime', '3');
#+end_src

And we simulate the system.
#+begin_src matlab
  sim('sim_nass_active_damping');
#+end_src

Finally, we save the simulation results for further analysis
#+begin_src matlab
  En_ine = En;
  Eg_ine = Eg;
  save('./active_damping/mat/tomo_exp.mat', 'En_ine', 'Eg_ine', '-append');
#+end_src

*** Compare with Undamped system
We load the results of tomography experiments.
#+begin_src matlab
  load('./active_damping/mat/tomo_exp.mat', 'En', 'En_ine');
  t = linspace(0, 3, length(En_ine(:,1)));
#+end_src

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(En(:,1),     En(:,2),     'DisplayName', '$\epsilon_{x,y}$ - OL')
  plot(En_ine(:,1), En_ine(:,2), 'DisplayName', '$\epsilon_{x,y}$ - Inertial')
  xlabel('X Motion [m]'); ylabel('Y Motion [m]');
  legend();
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/nass_act_damp_ine_sim_tomo_xy.pdf" :var figsize="small-normal" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:nass_act_damp_ine_sim_tomo_xy
#+CAPTION: Position Error during tomography experiment - XY Motion ([[./figs/nass_act_damp_ine_sim_tomo_xy.png][png]], [[./figs/nass_act_damp_ine_sim_tomo_xy.pdf][pdf]])
[[file:figs/nass_act_damp_ine_sim_tomo_xy.png]]

#+begin_src matlab :exports none
  figure;
  ax1 = subplot(3, 1, 1);
  hold on;
  plot(t, En(:,1), 'DisplayName', '$\epsilon_{x}$')
  plot(t, En_ine(:,1), 'DisplayName', '$\epsilon_{x}$ - Inertial')
  legend();

  ax2 = subplot(3, 1, 2);
  hold on;
  plot(t, En(:,2), 'DisplayName', '$\epsilon_{y}$')
  plot(t, En_ine(:,2), 'DisplayName', '$\epsilon_{y}$ - Inertial')
  legend();
  ylabel('Position Error [m]');

  ax3 = subplot(3, 1, 3);
  hold on;
  plot(t, En(:,3), 'DisplayName', '$\epsilon_{z}$')
  plot(t, En_ine(:,3), 'DisplayName', '$\epsilon_{z}$ - Inertial')
  legend();
  xlabel('Time [s]');
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/nass_act_damp_ine_sim_tomo_trans.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:nass_act_damp_ine_sim_tomo_trans
#+CAPTION: Position Error during tomography experiment - Translations ([[./figs/nass_act_damp_ine_sim_tomo_trans.png][png]], [[./figs/nass_act_damp_ine_sim_tomo_trans.pdf][pdf]])
[[file:figs/nass_act_damp_ine_sim_tomo_trans.png]]

#+begin_src matlab :exports none
  figure;
  ax1 = subplot(3, 1, 1);
  hold on;
  plot(t, En(:,4), 'DisplayName', '$\epsilon_{\theta_x}$')
  plot(t, En_ine(:,4), 'DisplayName', '$\epsilon_{\theta_x}$ - Inertial')
  legend();

  ax2 = subplot(3, 1, 2);
  hold on;
  plot(t, En(:,5), 'DisplayName', '$\epsilon_{\theta_y}$')
  plot(t, En_ine(:,5), 'DisplayName', '$\epsilon_{\theta_y}$ - Inertial')
  legend();
  ylabel('Position Error [rad]');

  ax3 = subplot(3, 1, 3);
  hold on;
  plot(t, En(:,6), 'DisplayName', '$\epsilon_{\theta_z}$')
  plot(t, En_ine(:,6), 'DisplayName', '$\epsilon_{\theta_z}$ - Inertial')
  legend();
  xlabel('Time [s]');

  linkaxes([ax1,ax2,ax3],'x');
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/nass_act_damp_ine_sim_tomo_rot.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:nass_act_damp_ine_sim_tomo_rot
#+CAPTION: Position Error during tomography experiment - Rotations ([[./figs/nass_act_damp_ine_sim_tomo_rot.png][png]], [[./figs/nass_act_damp_ine_sim_tomo_rot.pdf][pdf]])
[[file:figs/nass_act_damp_ine_sim_tomo_rot.png]]

** Conclusion
#+begin_important
Inertial Control:
#+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('./active_damping/mat/plants.mat', 'G', 'G_iff', 'G_ine', '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_ine.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'INE');
  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_ine.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'INE');
  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_ine.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'INE');
  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_ine.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'INE');
  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_ine.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'k--', 'DisplayName', 'INE');
  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_ine.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'INE');
  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_ine.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_ine.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'k--', 'DisplayName', 'INE');
  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_ine.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_ine.G_cart('Rz', 'Fnx'), freqs, 'Hz'))), 'k--', 'DisplayName', 'INE');
  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_ine.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]]
** Tomography Experiment
*** Load the Simulation Data
#+begin_src matlab
  load('./active_damping/mat/tomo_exp.mat', 'En', 'En_iff_hpf', 'En_dvf', 'En_ine');
  En_iff = En_iff_hpf;
  t = linspace(0, 3, length(En(:,1)));
#+end_src

*** Frequency Domain Analysis
Window used for =pwelch= function.
#+begin_src matlab
  n_av = 8;
  han_win = hanning(ceil(length(En(:, 1))/n_av));
#+end_src

#+begin_src matlab :exports none
  Ts = t(2)-t(1); % Sample Time for the Data [s]

  [pxx,     f] = pwelch(En(:,     1), han_win, [], [], 1/Ts);
  [pxx_ine, ~] = pwelch(En_ine(:, 1), han_win, [], [], 1/Ts);
  [pxx_dvf, ~] = pwelch(En_dvf(:, 1), han_win, [], [], 1/Ts);
  [pxx_iff, ~] = pwelch(En_iff(:, 1), han_win, [], [], 1/Ts);

  [pyy,     ~] = pwelch(En(:,     2), han_win, [], [], 1/Ts);
  [pyy_ine, ~] = pwelch(En_ine(:, 2), han_win, [], [], 1/Ts);
  [pyy_dvf, ~] = pwelch(En_dvf(:, 2), han_win, [], [], 1/Ts);
  [pyy_iff, ~] = pwelch(En_iff(:, 2), han_win, [], [], 1/Ts);

  [pzz,     ~] = pwelch(En(:,     3), han_win, [], [], 1/Ts);
  [pzz_ine, ~] = pwelch(En_ine(:, 3), han_win, [], [], 1/Ts);
  [pzz_dvf, ~] = pwelch(En_dvf(:, 3), han_win, [], [], 1/Ts);
  [pzz_iff, ~] = pwelch(En_iff(:, 3), han_win, [], [], 1/Ts);

  [prx,     ~] = pwelch(En(:,     4), han_win, [], [], 1/Ts);
  [prx_ine, ~] = pwelch(En_ine(:, 4), han_win, [], [], 1/Ts);
  [prx_dvf, ~] = pwelch(En_dvf(:, 4), han_win, [], [], 1/Ts);
  [prx_iff, ~] = pwelch(En_iff(:, 4), han_win, [], [], 1/Ts);

  [pry,     ~] = pwelch(En(:,     5), han_win, [], [], 1/Ts);
  [pry_ine, ~] = pwelch(En_ine(:, 5), han_win, [], [], 1/Ts);
  [pry_dvf, ~] = pwelch(En_dvf(:, 5), han_win, [], [], 1/Ts);
  [pry_iff, ~] = pwelch(En_iff(:, 5), han_win, [], [], 1/Ts);

  [prz,     ~] = pwelch(En(:,     6), han_win, [], [], 1/Ts);
  [prz_ine, ~] = pwelch(En_ine(:, 6), han_win, [], [], 1/Ts);
  [prz_dvf, ~] = pwelch(En_dvf(:, 6), han_win, [], [], 1/Ts);
  [prz_iff, ~] = pwelch(En_iff(:, 6), han_win, [], [], 1/Ts);
#+end_src

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(f, pxx_ine,    'DisplayName', 'Inertial')
  plot(f, pxx_dvf,    'DisplayName', 'DVF')
  plot(f, pxx_iff,    'DisplayName', 'IFF')
  plot(f, pxx, 'k--', 'DisplayName', 'Undamped')
  hold off;
  xlabel('Frequency [Hz]');
  ylabel('Power Spectral Density [$m^2/Hz$]');
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  legend('location', 'northeast');
  xlim([2, 500]);
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/act_damp_tomo_exp_comp_psd_trans.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:act_damp_tomo_exp_comp_psd_trans
#+CAPTION: PSD of the translation errors for applied Active Damping techniques ([[./figs/act_damp_tomo_exp_comp_psd_trans.png][png]], [[./figs/act_damp_tomo_exp_comp_psd_trans.pdf][pdf]])
[[file:figs/act_damp_tomo_exp_comp_psd_trans.png]]

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(f, prx_ine,    'DisplayName', 'Inertial')
  plot(f, prx_dvf,    'DisplayName', 'DVF')
  plot(f, prx_iff,    'DisplayName', 'IFF')
  plot(f, prx, 'k--', 'DisplayName', 'Undamped')
  hold off;
  xlabel('Frequency [Hz]');
  ylabel('Power Spectral Density [$m^2/Hz$]');
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  legend('location', 'northeast');
  xlim([2, 500]);
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/act_damp_tomo_exp_comp_psd_rot.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:act_damp_tomo_exp_comp_psd_rot
#+CAPTION: PSD of the rotation errors for applied Active Damping techniques ([[./figs/act_damp_tomo_exp_comp_psd_rot.png][png]], [[./figs/act_damp_tomo_exp_comp_psd_rot.pdf][pdf]])
[[file:figs/act_damp_tomo_exp_comp_psd_rot.png]]

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(f, flip(-cumtrapz(flip(f), flip(pxx_ine))),    'DisplayName', 'Inertial')
  plot(f, flip(-cumtrapz(flip(f), flip(pxx_dvf))),    'DisplayName', 'DVF')
  plot(f, flip(-cumtrapz(flip(f), flip(pxx_iff))),    'DisplayName', 'IFF')
  plot(f, flip(-cumtrapz(flip(f), flip(pxx))), 'k--', 'DisplayName', 'Undamped')
  hold off;
  xlabel('Frequency [Hz]');
  ylabel('Power Spectral Density [$m^2/Hz$]');
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  legend('location', 'northeast');
  xlim([2, 500]);
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/act_damp_tomo_exp_comp_cps_trans.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:act_damp_tomo_exp_comp_cps_trans
#+CAPTION: CPS of the translation errors for applied Active Damping techniques ([[./figs/act_damp_tomo_exp_comp_cps_trans.png][png]], [[./figs/act_damp_tomo_exp_comp_cps_trans.pdf][pdf]])
[[file:figs/act_damp_tomo_exp_comp_cps_trans.png]]

#+begin_src matlab :exports none
  figure;
  hold on;
  plot(f, flip(-cumtrapz(flip(f), flip(prx_ine))),    'DisplayName', 'Inertial')
  plot(f, flip(-cumtrapz(flip(f), flip(prx_dvf))),    'DisplayName', 'DVF')
  plot(f, flip(-cumtrapz(flip(f), flip(prx_iff))),    'DisplayName', 'IFF')
  plot(f, flip(-cumtrapz(flip(f), flip(prx))), 'k--', 'DisplayName', 'Undamped')
  hold off;
  xlabel('Frequency [Hz]');
  ylabel('Power Spectral Density [$m^2/Hz$]');
  set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
  legend('location', 'northeast');
  xlim([2, 500]);
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/act_damp_tomo_exp_comp_cps_rot.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src

#+NAME: fig:act_damp_tomo_exp_comp_cps_rot
#+CAPTION: CPS of the rotation errors for applied Active Damping techniques ([[./figs/act_damp_tomo_exp_comp_cps_rot.png][png]], [[./figs/act_damp_tomo_exp_comp_cps_rot.pdf][pdf]])
[[file:figs/act_damp_tomo_exp_comp_cps_rot.png]]

* Useful Functions
** prepareTomographyExperiment
:PROPERTIES:
:header-args:matlab+: :tangle src/prepareTomographyExperiment.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:prepareTomographyExperiment>>

This Matlab function is accessible [[file:src/prepareTomographyExperiment.m][here]].

*** Function Description
:PROPERTIES:
:UNNUMBERED: t
:END:

#+begin_src matlab
  function [] = prepareTomographyExperiment(args)
#+end_src

*** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
  arguments
      args.nass_actuator       char   {mustBeMember(args.nass_actuator,{'piezo', 'lorentz'})} = 'piezo'
      args.sample_mass   (1,1) double {mustBeNumeric, mustBePositive} = 50
      args.Ry_period     (1,1) double {mustBeNumeric, mustBePositive} = 1
  end
#+end_src

*** Initialize the Simulation
:PROPERTIES:
:UNNUMBERED: t
:END:
We initialize all the stages with the default parameters.
#+begin_src matlab
  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();
#+end_src

The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
#+begin_src matlab
  initializeNanoHexapod('actuator', args.nass_actuator);
  initializeSample('mass', args.sample_mass);
#+end_src

We set the references to zero.
#+begin_src matlab
  initializeReferences('Rz_type', 'rotating', 'Rz_period', args.Ry_period);
#+end_src

And all the controllers are set to 0.
#+begin_src matlab
  K = tf(zeros(6));
  save('./mat/controllers.mat', 'K', '-append');
  K_ine = tf(zeros(6));
  save('./mat/controllers.mat', 'K_ine', '-append');
  K_iff = tf(zeros(6));
  save('./mat/controllers.mat', 'K_iff', '-append');
  K_dvf = tf(zeros(6));
  save('./mat/controllers.mat', 'K_dvf', '-append');
#+end_src

* TODO Order                                                        :noexport:
** Undamped
*** Identification of the transfer function from disturbance to position error
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'sim_nass_active_damping';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwx'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwy'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Disturbances'], 1, 'openinput', [], 'Dwz'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Compute Error in NASS base'], 2, 'openoutput'); io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'Dwx', 'Dwy', 'Dwz'};
  G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src

*** Identification of the plant
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'sim_nass_active_damping';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fnl'], 1, 'openinput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Compute Error in NASS base'], 2, 'openoutput'); io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
  G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src

#+begin_src matlab
  lzoad('mat/stages.mat', 'nano_hexapod');
  G_cart = G*inv(nano_hexapod.J');
  G_cart.InputName = {'Fnx', 'Fny', 'Fnz', 'Mnx', 'Mny', 'Mnz'};
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G_cart('Edx', 'Fnx'), freqs, 'Hz'))), 'DisplayName', '$T_{x}$');
  plot(freqs, abs(squeeze(freqresp(G_cart('Edy', 'Fny'), freqs, 'Hz'))), 'DisplayName', '$T_{y}$');
  plot(freqs, abs(squeeze(freqresp(G_cart('Edz', 'Fnz'), freqs, 'Hz'))), 'DisplayName', '$T_{z}$');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'southwest')

  ax2 = subplot(2, 1, 2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_cart('Edx', 'Fnx'), freqs, 'Hz'))));
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_cart('Edy', 'Fny'), freqs, 'Hz'))));
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_cart('Edz', 'Fnz'), freqs, 'Hz'))));
  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

#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G_cart('Erx', 'Mnx'), freqs, 'Hz'))), 'DisplayName', '$R_{x}$');
  plot(freqs, abs(squeeze(freqresp(G_cart('Ery', 'Mny'), freqs, 'Hz'))), 'DisplayName', '$R_{y}$');
  plot(freqs, abs(squeeze(freqresp(G_cart('Erz', 'Mnz'), freqs, 'Hz'))), 'DisplayName', '$R_{z}$');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'southwest')

  ax2 = subplot(2, 1, 2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_cart('Erx', 'Mnx'), freqs, 'Hz'))));
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_cart('Ery', 'Mny'), freqs, 'Hz'))));
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_cart('Erz', 'Mnz'), freqs, 'Hz'))));
  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

*** TODO test
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'sim_nass_active_damping';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Micro-Station/Dy'], 1, 'openinput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/Compute Error in NASS base'], 2, 'openoutput'); io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'Dy'};
  G.OutputName = {'Edx', 'Edy', 'Edz', 'Erx', 'Ery', 'Erz'};
#+end_src

#+begin_important
Why is the transfer function from Ty displacement to position error is equal to
1 at all frequencies?
Why don't we see any resonance?
#+end_important

#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G('Edy', 'Dy(1)'), freqs, 'Hz'))), 'DisplayName', '$T_{x}$');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'southwest')

  ax2 = subplot(2, 1, 2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G('Edy', 'Dy(1)'), freqs, 'Hz'))));
  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

*** TODO test on hexapod
#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'test_nano_hexapod';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fnl'], 1, 'openinput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/x'],  1, 'openoutput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/y'],  1, 'openoutput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/z'],  1, 'openoutput'); io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
  G.OutputName = {'x', 'y', 'z'};
#+end_src

#+begin_src matlab
  %% Options for Linearized
  options = linearizeOptions;
  options.SampleTime = 0;

  %% Name of the Simulink File
  mdl = 'test_nano_hexapod';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Fx'], 1, 'openinput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/x'], 1, 'openoutput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/y'], 1, 'openoutput'); io_i = io_i + 1;
  io(io_i) = linio([mdl, '/z'], 1, 'openoutput'); io_i = io_i + 1;

  %% Run the linearization
  G = linearize(mdl, io, options);
  G.InputName  = {'Fx'};
  G.OutputName = {'x', 'y', 'z'};
#+end_src

#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G('Edy', 'Dy(1)'), freqs, 'Hz'))), 'DisplayName', '$T_{x}$');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'southwest')

  ax2 = subplot(2, 1, 2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G('Edy', 'Dy(1)'), freqs, 'Hz'))));
  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

#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G_cart('Erx', 'Mnx'), freqs, 'Hz'))), 'DisplayName', '$R_{x}$');
  plot(freqs, abs(squeeze(freqresp(G_cart('Ery', 'Mny'), freqs, 'Hz'))), 'DisplayName', '$R_{y}$');
  plot(freqs, abs(squeeze(freqresp(G_cart('Erz', 'Mnz'), freqs, 'Hz'))), 'DisplayName', '$R_{z}$');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  legend('location', 'southwest')

  ax2 = subplot(2, 1, 2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_cart('Erx', 'Mnx'), freqs, 'Hz'))));
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_cart('Ery', 'Mny'), freqs, 'Hz'))));
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_cart('Erz', 'Mnz'), freqs, 'Hz'))));
  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

*** Sensitivity to disturbances
The sensitivity to disturbances are shown on figure [[fig:sensitivity_dist_undamped]].

#+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, '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, '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_undamped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:sensitivity_dist_undamped
#+CAPTION: Undamped sensitivity to disturbances ([[./figs/sensitivity_dist_undamped.png][png]], [[./figs/sensitivity_dist_undamped.pdf][pdf]])
[[file:figs/sensitivity_dist_undamped.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, '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.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:sensitivity_dist_stages
#+CAPTION: Sensitivity to force disturbances in various stages ([[./figs/sensitivity_dist_stages.png][png]], [[./figs/sensitivity_dist_stages.pdf][pdf]])
[[file:figs/sensitivity_dist_stages.png]]
*** Undamped Plant
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]].

#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;

  ax1 = subplot(2, 1, 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'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'DisplayName', '$D_x / F_x$');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))), 'DisplayName', '$D_y / F_y$');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'DisplayName', '$D_z / F_z$');
  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', 'southwest');

  linkaxes([ax1,ax2],'x');
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/plant_undamped.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:plant_undamped
#+CAPTION: Transfer Function from cartesian forces to displacement for the undamped plant ([[./figs/plant_undamped.png][png]], [[./figs/plant_undamped.pdf][pdf]])
[[file:figs/plant_undamped.png]]


** Direct Velocity Feedback
*** 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]]
** Inertial Control
*** One degree-of-freedom example
:PROPERTIES:
:header-args:matlab+: :tangle no
:END:
<<sec:ine_1dof>>
**** Equations
#+begin_src latex :file ine_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] (Kine) {$K$};
    \draw[->] (Kine.west) -- (F) node[above right]{$F$};
    \draw[<-] (Kine.east) -- ++(0.5, 0) |- (velg.east);
  \end{tikzpicture}
#+end_src

#+name: fig:ine_1dof
#+caption: Direct Velocity Feedback applied to a 1dof system
#+RESULTS:
[[file:figs/ine_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:ine_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 $K_\text{INE}$ is:
#+begin_src matlab
  Kine = tf(-(sqrt(k*m)-c));
  Kine.InputName = {'xdot'};
  Kine.OutputName = {'F'};
#+end_src

And the closed loop system is computed below.
#+begin_src matlab
  sys_ine = feedback(sys, Kine, 'name', +1);
#+end_src

The obtained sensitivity to disturbances is shown in figure [[fig:ine_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_ine('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_ine('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_ine('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_ine('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/ine_1dof_sensitivitiy.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:ine_1dof_sensitivitiy
#+CAPTION: Sensitivity to disturbance when INE is applied on the 1dof system ([[./figs/ine_1dof_sensitivitiy.png][png]], [[./figs/ine_1dof_sensitivitiy.pdf][pdf]])
[[file:figs/ine_1dof_sensitivitiy.png]]