#+TITLE: Active Damping
: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 matlab/modal_frf_coh.m
#+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:

* Analysis of the nano-hexapod transfer functions                  :noexport:
** 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 :tangle no
  simulinkproject('../');
#+end_src

** Init
#+begin_src matlab
  %% Initialize Ground
  initializeGround();

  %% Initialize Granite
  initializeGranite(struct('rigid', false));

  %% Initialize Translation stage
  initializeTy(struct('rigid', false));

  %% Initialize Tilt Stage
  initializeRy(struct('rigid', false));

  %% Initialize Spindle
  initializeRz(struct('rigid', false));

  %% Initialize Hexapod Symétrie
  initializeMicroHexapod(struct('rigid', false));

  %% Initialize Center of gravity compensation
  initializeAxisc();

  %% Initialize the mirror
  initializeMirror();

  %% Initialize the Nano Hexapod
  initializeNanoHexapod(struct('actuator', 'piezo'));

  %% Initialize the Sample
  initializeSample(struct('mass', 50));
#+end_src

#+begin_src matlab
  K = tf(zeros(6));
  save('./mat/controllers.mat', 'K', '-append');
  K_iff = tf(zeros(6));
  save('./mat/controllers.mat', 'K_iff', '-append');
#+end_src

** Identification
#+begin_src matlab
  G = identifyPlant();
#+end_src

** Force Sensor
#+begin_src matlab
  figure;
  hold on;
  for i=1:6
    plot(freqs, abs(squeeze(freqresp(G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
  end
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [N/N]'); xlabel('Frequency [Hz]');
#+end_src
* Introduction                                                       :ignore:
First, in section [[sec:undamped]], 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:rmc]]: the relative motion control is used
- In section [[sec:dvf]]: the direct velocity feedback is used

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

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

* Undamped System
<<sec:undamped>>
** 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
  open 'simscape/sim_nano_station_id.slx'
#+end_src

** Init
We initialize all the stages with the default parameters.
The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.

#+begin_src matlab
  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();
  initializeNanoHexapod(struct('actuator', 'piezo'));
  initializeSample(struct('mass', 50));
#+end_src

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

** Identification
We identify the various transfer functions of the system
#+begin_src matlab
  G = identifyPlant();
#+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]]

** 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]]

** Save
#+begin_src matlab
  save('./active_damping/mat/plants.mat', 'G', '-append');
#+end_src

* Integral Force Feedback
<<sec:iff>>
** Introduction                                                     :ignore:
Integral Force Feedback is applied.
In section [[sec:iff_1dof]], IFF is applied on a uni-axial system to understand its behavior.
Then, it is applied on the simscape model.

** One degree-of-freedom example
<<sec:iff_1dof>>
*** Equations
#+begin_src latex :file iff_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, 1.4) rectangle ++(2, 0.8) node[pos=0.5]{$m$};
    \node[forcesensor={0.4}{0.4}] (fsensn) at (0, 1){};
    \draw[] (-0.8, 1) -- (0.8, 1);
    \node[left] at (fsensn.west) {$F_m$};

    % Displacement of the mass
    \draw[dashed] (-1, 2.2) -- ++(-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);

    % Displacements
    \node[block={0.8cm}{0.6cm}, right=0.6 of F] (Kiff) {$K$};
    \draw[->] (Kiff.west) -- (F) node[above right]{$F$};
    \draw[<-] (Kiff.east) -- ++(0.5, 0) |- (fsensn.east);
  \end{tikzpicture}
#+end_src

#+name: fig:iff_1dof
#+caption: Integral Force Feedback applied to a 1dof system
#+RESULTS:
[[file:figs/iff_1dof.png]]

The dynamic of the system is described by the following equation:
\begin{equation}
  ms^2x = F_d - kx - csx + kw + csw + F
\end{equation}
The measured force $F_m$ is:
\begin{align}
  F_m &= F - kx - csx + kw + csw \\
      &= ms^2 x - F_d
\end{align}
The Integral Force Feedback controller is $K = -\frac{g}{s}$, and thus the applied force by this controller is:
\begin{equation}
  F_{\text{IFF}} = -\frac{g}{s} F_m = -\frac{g}{s} (ms^2 x - F_d)
\end{equation}
Once the IFF is applied, the new dynamics of the system is:
\begin{equation}
  ms^2x = F_d + F - kx - csx + kw + csw - \frac{g}{s} (ms^2x - F_d)
\end{equation}

And finally:
\begin{equation}
  x = F_d \frac{1 + \frac{g}{s}}{ms^2 + (mg + c)s + k} + F \frac{1}{ms^2 + (mg + c)s + k} +  w \frac{k + cs}{ms^2 + (mg + c)s + k}
\end{equation}

We can see that this:
- adds damping to the system by a value $mg$
- lower the compliance as low frequency by a factor: $1 + g/s$

If we want critical damping:
\begin{equation}
  \xi = \frac{1}{2} \frac{c + gm}{\sqrt{km}} = \frac{1}{2}
\end{equation}

This is attainable if we have:
\begin{equation}
  g = \frac{\sqrt{km} - c}{m}
\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 $K_\text{IFF}$ is:
#+begin_src matlab
  Kiff = -((sqrt(k*m)-c)/m)/s;
  Kiff.InputName = {'Fm'};
  Kiff.OutputName = {'F'};
#+end_src

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

#+begin_src matlab :exports none
  freqs = logspace(0, 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_iff('d', 'Fd'), freqs, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
  xlim([freqs(1), freqs(end)]);

  subplot(2, 2, 3);
  title('Fd to x')
  hold on;
  plot(freqs, abs(squeeze(freqresp(sys('d', 'Fd'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(sys_iff('d', 'Fd'), freqs, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
  xlim([freqs(1), freqs(end)]);

  subplot(2, 2, 2);
  title('w to d')
  hold on;
  plot(freqs, abs(squeeze(freqresp(sys('d', 'wddot')*s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(sys_iff('d', 'wddot')*s^2, freqs, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
  xlim([freqs(1), freqs(end)]);

  subplot(2, 2, 4);
  title('w to x')
  hold on;
  plot(freqs, abs(squeeze(freqresp(1+sys('d', 'wddot')*s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(1+sys_iff('d', 'wddot')*s^2, freqs, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
  xlim([freqs(1), freqs(end)]);
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/iff_1dof_sensitivitiy.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:iff_1dof_sensitivitiy
#+CAPTION: Sensitivity to disturbance when IFF is applied on the 1dof system ([[./figs/iff_1dof_sensitivitiy.png][png]], [[./figs/iff_1dof_sensitivitiy.pdf][pdf]])
[[file:figs/iff_1dof_sensitivitiy.png]]

** Matlab Init                                             :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

#+begin_src matlab :tangle no
  simulinkproject('../');
#+end_src

#+begin_src matlab
  open 'simscape/sim_nano_station_id.slx'
#+end_src

** Control Design
Let's load the undamped plant:
#+begin_src matlab
  load('./active_damping/mat/plants.mat', 'G');
#+end_src

Let's look at the transfer function from actuator forces in the nano-hexapod to the force sensor in the nano-hexapod legs for all 6 pairs of actuator/sensor (figure [[fig:iff_plant]]).

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

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i=1:6
    plot(freqs, abs(squeeze(freqresp(G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i=1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

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

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

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

The controller for each pair of actuator/sensor is:
#+begin_src matlab
  K_iff = -1000/s;
#+end_src

The corresponding loop gains are shown in figure [[fig:iff_open_loop]].

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

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i=1:6
    plot(freqs, abs(squeeze(freqresp(K_iff*G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i=1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(K_iff*G.G_iff(['Fm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

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

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

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

** Sensitivity to disturbances
Let's initialize the system prior to identification.
#+begin_src matlab
  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();
  initializeNanoHexapod(struct('actuator', 'piezo'));
  initializeSample(struct('mass', 50));
#+end_src

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

We identify the system dynamics now that the IFF controller is ON.
#+begin_src matlab
  G_iff = identifyPlant();
#+end_src

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-normal" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:sensitivity_dist_stages_iff
#+CAPTION: Sensitivity to force disturbances in various stages when IFF is applied ([[./figs/sensitivity_dist_stages_iff.png][png]], [[./figs/sensitivity_dist_stages_iff.pdf][pdf]])
[[file:figs/sensitivity_dist_stages_iff.png]]

** Damped Plant
Now, look at the new damped plant to control.

It damps the plant (resonance of the nano hexapod as well as other resonances) as shown in figure [[fig:plant_iff_damped]].

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

  figure;

  ax1 = subplot(2, 2, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))));
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--');
  plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--');
  plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');

  ax2 = subplot(2, 2, 2);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))));
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--');
  plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--');
  plot(freqs, abs(squeeze(freqresp(G_iff.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [rad/(Nm)]'); xlabel('Frequency [Hz]');

  ax3 = subplot(2, 2, 3);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{n,x}\right|$');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{n,y}\right|$');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{n,z}\right|$');
  set(gca,'ColorOrderIndex',1);
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'northwest');

  ax4 = subplot(2, 2, 4);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), 'DisplayName', '$\left|R_x / M_{n,x}\right|$');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))), 'DisplayName', '$\left|R_y / M_{n,y}\right|$');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), 'DisplayName', '$\left|R_z / M_{n,z}\right|$');
  set(gca,'ColorOrderIndex',1);
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'northwest');

  linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src

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

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

However, it increases coupling at low frequency (figure [[fig:plant_iff_coupling]]).
#+begin_src matlab :exports none
  freqs = logspace(0, 3, 1000);

  figure;

  for ix = 1:6
    for iy = 1:6
      subplot(6, 6, (ix-1)*6 + iy);
      hold on;
      plot(freqs, abs(squeeze(freqresp(G.G_cart(ix, iy), freqs, 'Hz'))), 'k-');
      plot(freqs, abs(squeeze(freqresp(G_iff.G_cart(ix, iy), freqs, 'Hz'))), 'k--');
      set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
      ylim([1e-12, 1e-5]);
    end
  end
#+end_src

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

#+NAME: fig:plant_iff_coupling
#+CAPTION: Coupling induced by IFF ([[./figs/plant_iff_coupling.png][png]], [[./figs/plant_iff_coupling.pdf][pdf]])
[[file:figs/plant_iff_coupling.png]]

** Save
#+begin_src matlab
  save('./active_damping/mat/plants.mat', 'G_iff', '-append');
#+end_src

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

* Relative Motion Control
<<sec:rmc>>
** Introduction                                                     :ignore:
In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input.

** One degree-of-freedom example
<<sec:rmc_1dof>>
*** Equations
#+begin_src latex :file rmc_1dof.pdf :post pdf2svg(file=*this*, ext="png") :exports results
  \begin{tikzpicture}
    % Ground
    \draw (-1, 0) -- (1, 0);

    % Ground Displacement
    \draw[dashed] (-1, 0) -- ++(-0.5, 0) coordinate(w);
    \draw[->] (w) -- ++(0, 0.5) node[left]{$w$};

    % Mass
    \draw[fill=white] (-1, 1) rectangle ++(2, 0.8) node[pos=0.5]{$m$};

    % Displacement of the mass
    \draw[dashed] (-1, 1.8) -- ++(-0.5, 0) coordinate(x);
    \draw[->] (x) -- ++(0, 0.5) node[left]{$x$};

    % Spring, Damper, and Actuator
    \draw[spring] (-0.8, 0) -- (-0.8, 1) node[midway, left=0.1]{$k$};
    \draw[damper] (0, 0)    -- (0, 1)    node[midway, left=0.2]{$c$};
    \draw[actuator={0.4}{0.2}] (0.8, 0) -- (0.8, 1)  coordinate[midway, right=0.1](F);

    % Measured deformation
    \draw[dashed] (1, 0) -- ++(2, 0) coordinate(d_bot);
    \draw[dashed] (1, 1) -- ++(2, 0) coordinate(d_top);
    \draw[<->] (d_bot) --coordinate[midway](d) (d_top);

    % Displacements
    \node[block={0.8cm}{0.6cm}, right=0.6 of F] (Krmc) {$K$};
    \draw[->] (Krmc.west) -- (F) node[above right]{$F$};
    \draw[->] (d)node[above left]{$d$} -- (Krmc.east);
  \end{tikzpicture}
#+end_src

#+name: fig:rmc_1dof
#+caption: Relative Motion Control applied to a 1dof system
#+RESULTS:
[[file:figs/rmc_1dof.png]]

The dynamic of the system is:
\begin{equation}
  ms^2x = F_d - kx - csx + kw + csw + F
\end{equation}
In terms of the stage deformation $d = x - w$:
\begin{equation}
  (ms^2 + cs + k) d = -ms^2 w + F_d + F
\end{equation}
The relative motion control law is:
\begin{equation}
  K = -g s
\end{equation}
Thus, the applied force is:
\begin{equation}
  F = -g s d
\end{equation}
And the new dynamics will be:
\begin{equation}
  d = w \frac{-ms^2}{ms^2 + (c + g)s + k} + F_d \frac{1}{ms^2 + (c + g)s + k} + F \frac{1}{ms^2 + (c + g)s + k}
\end{equation}

And thus damping is added.

If critical damping is wanted:
\begin{equation}
  \xi = \frac{1}{2}\frac{c + g}{\sqrt{km}} = \frac{1}{2}
\end{equation}
This corresponds to a gain:
\begin{equation}
  g = \sqrt{km} - c
\end{equation}

*** Matlab Init                                           :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

*** Matlab Example
Let define the system parameters.
#+begin_src matlab
  m = 50; % [kg]
  k = 1e6; % [N/m]
  c = 1e3; % [N/(m/s)]
#+end_src

The state space model of the system is defined below.
#+begin_src matlab
  A = [-c/m -k/m;
       1     0];

  B = [1/m 1/m -1;
       0   0    0];

  C = [ 0  1;
       -c -k];

  D = [0 0 0;
       1 0 0];

  sys = ss(A, B, C, D);
  sys.InputName = {'F', 'Fd', 'wddot'};
  sys.OutputName = {'d', 'Fm'};
  sys.StateName = {'ddot', 'd'};
#+end_src

The controller $K_\text{RMC}$ is:
#+begin_src matlab
  Krmc = -(sqrt(k*m)-c)*s;
  Krmc.InputName = {'d'};
  Krmc.OutputName = {'F'};
#+end_src

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

#+begin_src matlab :exports none
  freqs = logspace(-1, 3, 1000);

  figure;

  subplot(2, 2, 1);
  title('Fd to d')
  hold on;
  plot(freqs, abs(squeeze(freqresp(sys('d', 'Fd'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(sys_rmc('d', 'Fd'), freqs, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
  xlim([freqs(1), freqs(end)]);

  subplot(2, 2, 3);
  title('Fd to x')
  hold on;
  plot(freqs, abs(squeeze(freqresp(sys('d', 'Fd'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(sys_rmc('d', 'Fd'), freqs, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
  xlim([freqs(1), freqs(end)]);

  subplot(2, 2, 2);
  title('w to d')
  hold on;
  plot(freqs, abs(squeeze(freqresp(sys('d', 'wddot')*s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(sys_rmc('d', 'wddot')*s^2, freqs, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
  xlim([freqs(1), freqs(end)]);

  subplot(2, 2, 4);
  title('w to x')
  hold on;
  plot(freqs, abs(squeeze(freqresp(1+sys('d', 'wddot')*s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(1+sys_rmc('d', 'wddot')*s^2, freqs, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
  xlim([freqs(1), freqs(end)]);
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/rmc_1dof_sensitivitiy.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:rmc_1dof_sensitivitiy
#+CAPTION: Sensitivity to disturbance when RMC is applied on the 1dof system ([[./figs/rmc_1dof_sensitivitiy.png][png]], [[./figs/rmc_1dof_sensitivitiy.pdf][pdf]])
[[file:figs/rmc_1dof_sensitivitiy.png]]

** Matlab Init                                             :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

#+begin_src matlab :tangle no
  simulinkproject('../');
#+end_src

#+begin_src matlab
  open 'simscape/sim_nano_station_id.slx'
#+end_src

** Control Design
Let's load the undamped plant:
#+begin_src matlab
  load('./active_damping/mat/plants.mat', 'G');
#+end_src

Let's look at the transfer function from actuator forces in the nano-hexapod to the measured displacement of the actuator for all 6 pairs of actuator/sensor (figure [[fig:rmc_plant]]).

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

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i=1:6
    plot(freqs, abs(squeeze(freqresp(G.G_dleg(['Dm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i=1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_dleg(['Dm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

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

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

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

The Relative Motion Controller is defined below.
A Low pass Filter is added to make the controller transfer function proper.
#+begin_src matlab
  K_rmc = s*50000/(1 + s/2/pi/10000);
#+end_src

The obtained loop gains are shown in figure [[fig:rmc_open_loop]].

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

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i=1:6
    plot(freqs, abs(squeeze(freqresp(K_rmc*G.G_dleg(['Dm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i=1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(K_rmc*G.G_dleg(['Dm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

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

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

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

** Sensitivity to disturbances
Let's initialize the system prior to identification.
#+begin_src matlab
  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();
  initializeNanoHexapod(struct('actuator', 'piezo'));
  initializeSample(struct('mass', 50));
#+end_src

And initialize the controllers.
#+begin_src matlab
  K = tf(zeros(6));
  save('./mat/controllers.mat', 'K', '-append');
  K_iff = tf(zeros(6));
  save('./mat/controllers.mat', 'K_iff', '-append');
  K_rmc = -K_rmc*eye(6);
  save('./mat/controllers.mat', 'K_rmc', '-append');
  K_dvf = tf(zeros(6));
  save('./mat/controllers.mat', 'K_dvf', '-append');
#+end_src

We identify the system dynamics now that the RMC controller is ON.
#+begin_src matlab
  G_rmc = identifyPlant();
#+end_src

As shown in figure [[fig:sensitivity_dist_rmc]], RMC control succeed in lowering the sensitivity to disturbances near resonance of the system.

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

  figure;

  subplot(2, 1, 1);
  title('$D_g$ to $D$');
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / D_{g,x}\right|$');
  plot(freqs, abs(squeeze(freqresp(G.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / D_{g,y}\right|$');
  plot(freqs, abs(squeeze(freqresp(G.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / D_{g,z}\right|$');
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_gm('Dy', 'Dgy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_gm('Dz', 'Dgz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
  legend('location', '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_rmc.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_fs('Dy', 'Fsy'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_fs('Dz', 'Fsz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
  legend('location', 'northeast');
#+end_src

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

#+NAME: fig:sensitivity_dist_rmc
#+CAPTION: Sensitivity to disturbance once the RMC controller is applied to the system ([[./figs/sensitivity_dist_rmc.png][png]], [[./figs/sensitivity_dist_rmc.pdf][pdf]])
[[file:figs/sensitivity_dist_rmc.png]]

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

  figure;
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{rz, z}\right|$');
  plot(freqs, abs(squeeze(freqresp(G.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{ty, z}\right|$');
  plot(freqs, abs(squeeze(freqresp(G.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{ty, x}\right|$');
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_dist('Dz', 'Frzz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_dist('Dz', 'Ftyz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_dist('Dx', 'Ftyx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
  legend('location', 'northeast');
#+end_src

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

#+NAME: fig:sensitivity_dist_stages_rmc
#+CAPTION: Sensitivity to force disturbances in various stages when RMC is applied ([[./figs/sensitivity_dist_stages_rmc.png][png]], [[./figs/sensitivity_dist_stages_rmc.pdf][pdf]])
[[file:figs/sensitivity_dist_stages_rmc.png]]

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

  figure;

  ax1 = subplot(2, 2, 1);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))));
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--');
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--');
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');

  ax2 = subplot(2, 2, 2);
  hold on;
  plot(freqs, abs(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))));
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--');
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--');
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [rad/(Nm)]'); xlabel('Frequency [Hz]');

  ax3 = subplot(2, 2, 3);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), 'DisplayName', '$\left|D_x / F_{n,x}\right|$');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dy', 'Fny'), freqs, 'Hz'))), 'DisplayName', '$\left|D_y / F_{n,y}\right|$');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), 'DisplayName', '$\left|D_z / F_{n,z}\right|$');
  set(gca,'ColorOrderIndex',1);
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Dx', 'Fnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Dy', 'Fny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Dz', 'Fnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'northwest');

  ax4 = subplot(2, 2, 4);
  hold on;
  plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), 'DisplayName', '$\left|R_x / M_{n,x}\right|$');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Ry', 'Mny'), freqs, 'Hz'))), 'DisplayName', '$\left|R_y / M_{n,y}\right|$');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), 'DisplayName', '$\left|R_z / M_{n,z}\right|$');
  set(gca,'ColorOrderIndex',1);
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Rx', 'Mnx'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Ry', 'Mny'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_rmc.G_cart('Rz', 'Mnz'), freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);
  legend('location', 'northwest');

  linkaxes([ax1,ax2,ax3,ax4],'x');
#+end_src

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

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

** Save
#+begin_src matlab
  save('./active_damping/mat/plants.mat', 'G_rmc', '-append');
#+end_src

** Conclusion
#+begin_important
Relative Motion Control:
-
#+end_important

* Direct Velocity Feedback
<<sec:dvf>>
** Introduction                                                     :ignore:
In the Relative Motion Control (RMC), a feedback is applied between the measured velocity of the platform to the actuator force input.

** One degree-of-freedom example
<<sec:dvf_1dof>>
*** Equations
#+begin_src latex :file dvf_1dof.pdf :post pdf2svg(file=*this*, ext="png") :exports results
  \begin{tikzpicture}
    % Ground
    \draw (-1, 0) -- (1, 0);

    % Ground Displacement
    \draw[dashed] (-1, 0) -- ++(-0.5, 0) coordinate(w);
    \draw[->] (w) -- ++(0, 0.5) node[left]{$w$};

    % Mass
    \draw[fill=white] (-1, 1) rectangle ++(2, 0.8) node[pos=0.5]{$m$};

    % Velocity Sensor
    \node[inertialsensor={0.3}] (velg) at (1, 1.8){};
    \node[above] at (velg.north) {$\dot{x}$};

    % Displacement of the mass
    \draw[dashed] (-1, 1.8) -- ++(-0.5, 0) coordinate(x);
    \draw[->] (x) -- ++(0, 0.5) node[left]{$x$};

    % Spring, Damper, and Actuator
    \draw[spring] (-0.8, 0) -- (-0.8, 1) node[midway, left=0.1]{$k$};
    \draw[damper] (0, 0)    -- (0, 1)    node[midway, left=0.2]{$c$};
    \draw[actuator={0.4}{0.2}] (0.8, 0) -- (0.8, 1)  coordinate[midway, right=0.1](F);

    % Control
    \node[block={0.8cm}{0.6cm}, right=0.6 of F] (Kdvf) {$K$};
    \draw[->] (Kdvf.west) -- (F) node[above right]{$F$};
    \draw[<-] (Kdvf.east) -- ++(0.5, 0) |- (velg.east);
  \end{tikzpicture}
#+end_src

#+name: fig:dvf_1dof
#+caption: Direct Velocity Feedback applied to a 1dof system
#+RESULTS:
[[file:figs/dvf_1dof.png]]

The dynamic of the system is:
\begin{equation}
  ms^2x = F_d - kx - csx + kw + csw + F
\end{equation}
In terms of the stage deformation $d = x - w$:
\begin{equation}
  (ms^2 + cs + k) d = -ms^2 w + F_d + F
\end{equation}
The direct velocity feedback law shown in figure [[fig:dvf_1dof]] is:
\begin{equation}
  K = -g
\end{equation}
Thus, the applied force is:
\begin{equation}
  F = -g \dot{x}
\end{equation}
And the new dynamics will be:
\begin{equation}
  d = w \frac{-ms^2 - gs}{ms^2 + (c + g)s + k} + F_d \frac{1}{ms^2 + (c + g)s + k} + F \frac{1}{ms^2 + (c + g)s + k}
\end{equation}

And thus damping is added.

If critical damping is wanted:
\begin{equation}
  \xi = \frac{1}{2}\frac{c + g}{\sqrt{km}} = \frac{1}{2}
\end{equation}
This corresponds to a gain:
\begin{equation}
  g = \sqrt{km} - c
\end{equation}

*** Matlab Init                                           :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

*** Matlab Example
Let define the system parameters.
#+begin_src matlab
  m = 50; % [kg]
  k = 1e6; % [N/m]
  c = 1e3; % [N/(m/s)]
#+end_src

The state space model of the system is defined below.
#+begin_src matlab
  A = [-c/m -k/m;
       1     0];

  B = [1/m 1/m -1;
       0   0    0];

  C = [1 0;
       0 1;
       0 0];

  D = [0 0 0;
       0 0 0;
       0 0 1];

  sys = ss(A, B, C, D);
  sys.InputName = {'F', 'Fd', 'wddot'};
  sys.OutputName = {'ddot', 'd', 'wddot'};
  sys.StateName = {'ddot', 'd'};
#+end_src

Because we need $\dot{x}$ for feedback, we compute it from the outputs
#+begin_src matlab
  G_xdot = [1, 0, 1/s;
            0, 1, 0];
  G_xdot.InputName = {'ddot', 'd', 'wddot'};
  G_xdot.OutputName = {'xdot', 'd'};
#+end_src

Finally, the system is described by =sys= as defined below.
#+begin_src matlab
  sys = series(sys, G_xdot, [1 2 3], [1 2 3]);
#+end_src

The controller $K_\text{DVF}$ is:
#+begin_src matlab
  Kdvf = tf(-(sqrt(k*m)-c));
  Kdvf.InputName = {'xdot'};
  Kdvf.OutputName = {'F'};
#+end_src

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

The obtained sensitivity to disturbances is shown in figure [[fig:dvf_1dof_sensitivitiy]].
#+begin_src matlab :exports none
  freqs = logspace(-1, 3, 1000);

  figure;

  subplot(2, 2, 1);
  title('Fd to d')
  hold on;
  plot(freqs, abs(squeeze(freqresp(sys('d', 'Fd'), freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(sys_dvf('d', 'Fd'), freqs, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
  xlim([freqs(1), freqs(end)]);

  subplot(2, 2, 3);
  title('Fd to x')
  hold on;
  plot(freqs, abs(squeeze(freqresp(sys('xdot', 'Fd')/s, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(sys_dvf('xdot', 'Fd')/s, freqs, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
  xlim([freqs(1), freqs(end)]);

  subplot(2, 2, 2);
  title('w to d')
  hold on;
  plot(freqs, abs(squeeze(freqresp(sys('d', 'wddot')*s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(sys_dvf('d', 'wddot')*s^2, freqs, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
  xlim([freqs(1), freqs(end)]);

  subplot(2, 2, 4);
  title('w to x')
  hold on;
  plot(freqs, abs(squeeze(freqresp(1+sys('d', 'wddot')*s^2, freqs, 'Hz'))));
  plot(freqs, abs(squeeze(freqresp(1+sys_dvf('d', 'wddot')*s^2, freqs, 'Hz'))));
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]');
  xlim([freqs(1), freqs(end)]);
#+end_src

#+HEADER: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/dvf_1dof_sensitivitiy.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
  <<plt-matlab>>
#+end_src

#+NAME: fig:dvf_1dof_sensitivitiy
#+CAPTION: Sensitivity to disturbance when DVF is applied on the 1dof system ([[./figs/dvf_1dof_sensitivitiy.png][png]], [[./figs/dvf_1dof_sensitivitiy.pdf][pdf]])
[[file:figs/dvf_1dof_sensitivitiy.png]]

** Matlab Init                                             :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
  <<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
  <<matlab-init>>
#+end_src

#+begin_src matlab :tangle no
  simulinkproject('../');
#+end_src

#+begin_src matlab
  open 'simscape/sim_nano_station_id.slx'
#+end_src

** Control Design
Let's load the undamped plant:
#+begin_src matlab
  load('./active_damping/mat/plants.mat', 'G');
#+end_src

Let's look at the transfer function from actuator forces in the nano-hexapod to the measured velocity of the nano-hexapod platform in the direction of the corresponding actuator for all 6 pairs of actuator/sensor (figure [[fig:dvf_plant]]).

The plant looks to complicated to be reasonably controlled.

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

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i=1:6
    plot(freqs, abs(squeeze(freqresp(G.G_geoph(['Vm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);

  ax2 = subplot(2, 1, 2);
  hold on;
  for i=1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(G.G_geoph(['Vm', num2str(i)], ['F', num2str(i)]), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

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

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

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

** Conclusion
#+begin_important
Direct Velocity Feedback:
- Not usable
#+end_important

* Comparison
<<sec:comparison>>

** 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

** Comparison
#+begin_src matlab
  load('./active_damping/mat/plants.mat', 'G', 'G_iff', 'G_rmc');
#+end_src

#+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', 'Undamped');
  plot(freqs, abs(squeeze(freqresp(G_iff.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), 'DisplayName', 'IFF');
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_gm('Dx', 'Dgx'), freqs, 'Hz'))), 'DisplayName', 'RMC');
  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', 'Undamped');
  plot(freqs, abs(squeeze(freqresp(G_iff.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), 'DisplayName', 'IFF');
  plot(freqs, abs(squeeze(freqresp(G_rmc.G_fs('Dx', 'Fsx'), freqs, 'Hz'))), 'DisplayName', 'RMC');
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]');
  legend('location', 'northeast');
#+end_src

* Conclusion
<<sec:conclusion>>