#+TITLE: ESRF Double Crystal Monochromator - Dynamical Multi-Body Model
:DRAWER:
#+LANGUAGE: en
#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

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

#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>

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#+LaTeX_CLASS: scrreprt
#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full]
#+LaTeX_HEADER_EXTRA: \input{preamble.tex}
#+LATEX_HEADER_EXTRA: \bibliography{ref}

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

#+PROPERTY: header-args:latex  :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
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#+PROPERTY: header-args:latex+ :results file raw replace
#+PROPERTY: header-args:latex+ :buffer no
#+PROPERTY: header-args:latex+ :tangle no
#+PROPERTY: header-args:latex+ :eval no-export
#+PROPERTY: header-args:latex+ :exports results
#+PROPERTY: header-args:latex+ :mkdirp yes
#+PROPERTY: header-args:latex+ :output-dir figs
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:

#+begin_export html
  <hr>
  <p>This report is also available as a <a href="./dcm-simscape.pdf">pdf</a>.</p>
  <hr>
#+end_export

#+latex: \clearpage

* Introduction                                                        :ignore:
In this document, a Simscape (.e.g. multi-body) model of the ESRF Double Crystal Monochromator (DCM) is presented and used to develop and optimize the control strategy.

It is structured as follow:
- Section [[sec:dcm_kinematics]]: the kinematics of the DCM is presented, and Jacobian matrices which are used to solve the inverse and forward kinematics are computed.
- Section [[sec:open_loop_identification]]: the system dynamics is identified in the absence of control.
- Section [[sec:active_damping_strain_gauges]]: it is studied whether if the strain gauges fixed to the piezoelectric actuators can be used to actively damp the plant.
- Section [[sec:active_damping_iff]]: piezoelectric force sensors are added in series with the piezoelectric actuators and are used to actively damp the plant using the Integral Force Feedback (IFF) control strategy.
- Section [[sec:hac_iff]]: the High Authority Control - Low Authority Control (HAC-LAC) strategy is tested on the Simscape model.

* System Kinematics
:PROPERTIES:
:header-args:matlab+: :tangle matlab/dcm_kinematics.m
:END:
<<sec:dcm_kinematics>>

** Introduction                                                      :ignore:

** Matlab Init                                              :noexport:ignore:
#+begin_src matlab
%% dcm_kinematics.m
% Computation of the DCM kinematics
#+end_src

#+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 :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src

** Bragg Angle
#+begin_src matlab
%% Tested bragg angles
bragg = linspace(5, 80, 1000); % Bragg angle [deg]
d_off = 10.5e-3; % Wanted offset between x-rays [m]
#+end_src

#+begin_src matlab
%% Vertical Jack motion as a function of Bragg angle
dz = d_off./(2*cos(bragg*pi/180));
#+end_src

#+begin_src matlab :exports none
%% Jack motion as a function of Bragg angle
figure;
plot(bragg, 1e3*dz)
xlabel('Bragg angle [deg]'); ylabel('Jack Motion [mm]');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/jack_motion_bragg_angle.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:jack_motion_bragg_angle
#+caption: Jack motion as a function of Bragg angle
#+RESULTS:
[[file:figs/jack_motion_bragg_angle.png]]

#+begin_src matlab :results value replace :exports both
%% Required Jack stroke
ans = 1e3*(dz(end) - dz(1))
#+end_src

#+RESULTS:
: 24.963

** Kinematics (111 Crystal)
*** Introduction                                                    :ignore:
The reference frame is taken at the center of the 111 second crystal.

*** Interferometers - 111 Crystal

Three interferometers are pointed to the bottom surface of the 111 crystal.

The position of the measurement points are shown in Figure [[fig:sensor_111_crystal_points]] as well as the origin where the motion of the crystal is computed.

#+begin_src latex :file sensor_111_crystal_points.pdf
\begin{tikzpicture}
  % Crystal
  \draw (-15/2, -3.5/2) rectangle (15/2, 3.5/2);

  % Measurement Points
  \node[branch] (a1) at (-7,  1.5){};
  \node[branch] (a2) at ( 0, -1.5){};
  \node[branch] (a3) at ( 7,  1.5){};

  % Labels
  \node[right] at (a1) {$\mathcal{O}_1 = (-0.07, -0.015)$};
  \node[right] at (a2) {$\mathcal{O}_2 = (0,      0.015)$};
  \node[left]  at (a3) {$\mathcal{O}_3 = ( 0.07, -0.015)$};

  % Origin
  \draw[->] (0, 0) node[] -- ++(1,  0) node[right]{$x$};
  \draw[->] (0, 0) -- ++(0, -1) node[below]{$y$};
  \draw[fill, color=black] (0, 0) circle (0.05);

  \node[left] at (0,0) {$\mathcal{O}_{111}$};
\end{tikzpicture}
#+end_src

#+name: fig:sensor_111_crystal_points
#+caption: Bottom view of the second crystal 111. Position of the measurement points.
#+RESULTS:
[[file:figs/sensor_111_crystal_points.png]]

The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure [[fig:schematic_sensor_jacobian_inverse_kinematics]]):
\begin{equation}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
=
\bm{J}_{s,111}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
\end{equation}

#+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics.pdf
\begin{tikzpicture}
  % Blocs
  \node[block] (Js) {$\bm{J}_{s,111}$};

  % Connections and labels
  \draw[->] ($(Js.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- (Js.west);
  \draw[->] (Js.east)  -- ++(1.5, 0) node[above left]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$};
\end{tikzpicture}
#+end_src

#+name: fig:schematic_sensor_jacobian_inverse_kinematics
#+caption: Inverse Kinematics - Interferometers
#+RESULTS:
[[file:figs/schematic_sensor_jacobian_inverse_kinematics.png]]


From the Figure [[fig:sensor_111_crystal_points]], the inverse kinematics can be solved as follow (for small motion):
\begin{equation}
\bm{J}_{s,111}
=
\begin{bmatrix}
1 &  0.07 & -0.015 \\
1 &  0    &  0.015 \\
1 & -0.07 & -0.015
\end{bmatrix}
\end{equation}

#+begin_src matlab
%% Sensor Jacobian matrix for 111 crystal
J_s_111 = [1,  0.07, -0.015
           1,  0,     0.015
           1, -0.07, -0.015];
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(J_s_111, {}, {}, ' %.3f ');
#+end_src

#+name: tab:jacobian_sensor_111
#+caption: Sensor Jacobian $\bm{J}_{s,111}$
#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
#+attr_latex: :center t :booktabs t
#+RESULTS:
| 1.0 |  0.07 | -0.015 |
| 1.0 |   0.0 |  0.015 |
| 1.0 | -0.07 | -0.015 |

The forward kinematics is solved by inverting the Jacobian matrix (see Figure [[fig:schematic_sensor_jacobian_forward_kinematics]]).
\begin{equation}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
=
\bm{J}_{s,111}^{-1}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
\end{equation}

#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics.pdf
\begin{tikzpicture}
  % Blocs
  \node[block] (Js_inv) {$\bm{J}_{s,111}^{-1}$};

  % Connections and labels
  \draw[->] ($(Js_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} -- (Js_inv.west);
  \draw[->] (Js_inv.east)  -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$};
\end{tikzpicture}
#+end_src

#+name: fig:schematic_sensor_jacobian_forward_kinematics
#+caption: Forward Kinematics - Interferometers
#+RESULTS:
[[file:figs/schematic_sensor_jacobian_forward_kinematics.png]]

#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(inv(J_s_111), {}, {}, ' %.2f ');
#+end_src

#+name: tab:inverse_jacobian_sensor_111
#+caption: Inverse of the sensor Jacobian $\bm{J}_{s,111}^{-1}$
#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
#+attr_latex: :center t :booktabs t
#+RESULTS:
|   0.25 |   0.5 |   0.25 |
|   7.14 |   0.0 |  -7.14 |
| -16.67 | 33.33 | -16.67 |

*** Piezo - 111 Crystal

The location of the actuators with respect with the center of the 111 second crystal are shown in Figure [[fig:actuator_jacobian_111_points]].

#+name: fig:actuator_jacobian_111_points
#+caption: Location of actuators with respect to the center of the 111 second crystal (bottom view)
#+attr_latex: :width \linewidth
[[file:figs/actuator_jacobian_111_points.png]]

Inverse Kinematics consist of deriving the axial (z) motion of the 3 actuators from the motion of the crystal's center.
\begin{equation}
\begin{bmatrix}
d_{u_r} \\ d_{u_h} \\ d_{d}
\end{bmatrix}
=
\bm{J}_{a,111}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
\end{equation}

#+begin_src latex :file schematic_actuator_jacobian_inverse_kinematics.pdf
\begin{tikzpicture}
  % Blocs
  \node[block] (Ja) {$\bm{J}_{a,111}$};

  % Connections and labels
  \draw[->] ($(Ja.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- (Ja.west);
  \draw[->] (Ja.east)  -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_d \end{bmatrix}$};
\end{tikzpicture}
#+end_src

#+name: fig:schematic_sensor_jacobian_inverse_kinematics
#+caption: Inverse Kinematics - Actuators
#+RESULTS:
[[file:figs/schematic_actuator_jacobian_inverse_kinematics.png]]

Based on the geometry in Figure [[fig:actuator_jacobian_111_points]], we obtain:
\begin{equation}
\bm{J}_{a,111}
=
\begin{bmatrix}
1 &  0.14 & -0.1525 \\
1 &  0.14 &  0.0675 \\
1 & -0.14 & -0.0425
\end{bmatrix}
\end{equation}

#+begin_src matlab
%% Actuator Jacobian - 111 crystal
J_a_111 = [1,  0.14, -0.1525
           1,  0.14,  0.0675
           1, -0.14, -0.0425];
#+end_src

#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(J_a_111, {}, {}, ' %.4f ');
#+end_src

#+name: tab:jacobian_actuator_111
#+caption: Actuator Jacobian $\bm{J}_{a,111}$
#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
#+attr_latex: :center t :booktabs t
#+RESULTS:
| 1.0 |  0.14 | -0.1525 |
| 1.0 |  0.14 |  0.0675 |
| 1.0 | -0.14 | -0.0425 |

The forward Kinematics is solved by inverting the Jacobian matrix:
\begin{equation}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
=
\bm{J}_{a,111}^{-1}
\begin{bmatrix}
d_{u_r} \\ d_{u_h} \\ d_{d}
\end{bmatrix}
\end{equation}

#+begin_src latex :file schematic_actuator_jacobian_forward_kinematics.pdf
\begin{tikzpicture}
  % Blocs
  \node[block] (Ja_inv) {$\bm{J}_{a,111}^{-1}$};

  % Connections and labels
  \draw[->] ($(Ja_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_d \end{bmatrix}$} -- (Ja_inv.west);
  \draw[->] (Ja_inv.east)  -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$};
\end{tikzpicture}
#+end_src

#+name: fig:schematic_actuator_jacobian_forward_kinematics
#+caption: Forward Kinematics - Actuators for 111 crystal
#+RESULTS:
[[file:figs/schematic_actuator_jacobian_forward_kinematics.png]]

#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(inv(J_a_111), {}, {}, ' %.4f ');
#+end_src

#+name: tab:inverse_jacobian_actuator_111
#+caption: Inverse of the actuator Jacobian $\bm{J}_{a,111}^{-1}$
#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
#+attr_latex: :center t :booktabs t
#+RESULTS:
|  0.0568 | 0.4432 |     0.5 |
|  1.7857 | 1.7857 | -3.5714 |
| -4.5455 | 4.5455 |     0.0 |

** TODO Inputs and Outputs                                         :noexport:

Disturbances:
- Motion errors of the stepper motor
- Vibrations from the outside
- Vibrations from the cooling system directly applied on the crystals

** Save Kinematics
#+begin_src matlab :exports none :tangle no
save('matlab/mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111')
#+end_src

#+begin_src matlab :eval no
save('mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111')
#+end_src

* Open Loop System Identification
:PROPERTIES:
:header-args:matlab+: :tangle matlab/dcm_identification.m
:END:
<<sec:open_loop_identification>>
** Introduction                                                      :ignore:
** Matlab Init                                              :noexport:ignore:
#+begin_src matlab
%% dcm_identification.m
% Extraction of system dynamics using Simscape model
#+end_src

#+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 :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-simscape>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src

** Identification
Let's considered the system $\bm{G}(s)$ with:
- 3 inputs: force applied to the 3 fast jacks
- 3 outputs: measured displacement by the 3 interferometers pointing at the 111 second crystal

It is schematically shown in Figure [[fig:schematic_system_inputs_outputs]].

#+begin_src latex :file schematic_system_inputs_outputs.pdf
\begin{tikzpicture}
  % Blocs
  \node[block] (G) {$\bm{G}(s)$};

  % Connections and labels
  \draw[->] ($(G.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} -- (G.west);
  \draw[->] (G.east)  -- ++(1.5, 0) node[above left]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$};
\end{tikzpicture}
#+end_src

#+name: fig:schematic_system_inputs_outputs
#+caption: Dynamical system with inputs and outputs
#+RESULTS:
[[file:figs/schematic_system_inputs_outputs.png]]

The system is identified from the Simscape model.

#+begin_src matlab
%% Input/Output definition
clear io; io_i = 1;

%% Inputs
% Control Input {3x1} [N]
io(io_i) = linio([mdl, '/control_system'], 1, 'openinput');  io_i = io_i + 1;

%% Outputs
% Interferometers {3x1} [m]
io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1;
#+end_src

#+begin_src matlab
%% Extraction of the dynamics
G = linearize(mdl, io);
#+end_src

#+begin_src matlab :exports none
%% Input and Output names
G.InputName  = {'u_ur',  'u_uh',  'u_d'};
G.OutputName = {'int_111_1', 'int_111_2', 'int_111_3'};
#+end_src

#+begin_src matlab :results output replace :exports both :tangle no
size(G)
#+end_src

#+RESULTS:
: size(G)
: State-space model with 3 outputs, 3 inputs, and 24 states.

** Plant in the frame of the fastjacks
#+begin_src matlab
load('dcm_kinematics.mat');
#+end_src

Using the forward and inverse kinematics, we can computed the dynamics from piezo forces to axial motion of the 3 fastjacks (see Figure [[fig:schematic_jacobian_frame_fastjack]]).

#+begin_src latex :file schematic_jacobian_frame_fastjack.pdf
\begin{tikzpicture}
  % Blocs
  \node[block] (G) {$\bm{G}(s)$};
  \node[block, right=1.5 of G] (Js) {$\bm{J}_{s}^{-1}$};
  \node[block, right=1.5 of Js] (Ja) {$\bm{J}_{a}$};

  % Connections and labels
  \draw[->] ($(G.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} -- (G.west);
  \draw[->] (G.east)  -- node[midway, above]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} (Js.west);
  \draw[->] (Js.east) -- node[midway, above]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} (Ja.west);
  \draw[->] (Ja.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_{d} \end{bmatrix}$};

  \begin{scope}[on background layer]
    \node[fit={(G.south west) ($(Ja.east)+(0, 1.4)$)}, fill=black!20!white, draw, inner sep=6pt] (system) {};
    \node[above] at (system.north) {$\bm{G}_{\text{fj}}(s)$};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:schematic_jacobian_frame_fastjack
#+caption: Use of Jacobian matrices to obtain the system in the frame of the fastjacks
#+RESULTS:
[[file:figs/schematic_jacobian_frame_fastjack.png]]


#+begin_src matlab
%% Compute the system in the frame of the fastjacks
G_pz = J_a_111*inv(J_s_111)*G;
#+end_src

The DC gain of the new system shows that the system is well decoupled at low frequency.
#+begin_src matlab :results value replace :exports both :tangle no
dcgain(G_pz)
#+end_src

#+name: tab:dc_gain_plan_fj
#+caption: DC gain of the plant in the frame of the fast jacks $\bm{G}_{\text{fj}}$
#+attr_latex: :environment tabularx :width 0.5\linewidth :align ccc
#+attr_latex: :center t :booktabs t
#+RESULTS:
| 4.4407e-09 | 2.7656e-12 | 1.0132e-12 |
| 2.7656e-12 | 4.4407e-09 | 1.0132e-12 |
| 1.0109e-12 | 1.0109e-12 | 4.4424e-09 |

The bode plot of $\bm{G}_{\text{fj}}(s)$ is shown in Figure [[fig:bode_plot_plant_fj]].

#+begin_src matlab :exports none
%% Bode plot for the plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_pz(1,1), freqs, 'Hz'))), ...
     'DisplayName', 'd');
plot(freqs, abs(squeeze(freqresp(G_pz(2,2), freqs, 'Hz'))), ...
     'DisplayName', 'uh');
plot(freqs, abs(squeeze(freqresp(G_pz(3,3), freqs, 'Hz'))), ...
     'DisplayName', 'ur');
for i = 1:2
    for j = i+1:3
        plot(freqs, abs(squeeze(freqresp(G_pz(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 3);
ylim([1e-13, 1e-6]);

ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_pz(1,1), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_pz(2,2), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_pz(3,3), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 180]);

linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/bode_plot_plant_fj.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:bode_plot_plant_fj
#+caption: Bode plot of the diagonal and off-diagonal elements of the plant in the frame of the fast jacks
#+RESULTS:
[[file:figs/bode_plot_plant_fj.png]]

#+begin_important
Computing the system in the frame of the fastjack gives good decoupling at low frequency (until the first resonance of the system).
#+end_important

** Plant in the frame of the crystal
#+begin_src latex :file schematic_jacobian_frame_crystal.pdf
\begin{tikzpicture}
  % Blocs
  \node[block] (G) {$\bm{G}(s)$};
  \node[block, left=1.5 of G] (Ja) {$\bm{J}_{a}^{-T}$};
  \node[block, right=1.5 of G] (Js) {$\bm{J}_{s}^{-1}$};

  % Connections and labels
  \draw[->] ($(Ja.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} \mathcal{F}_{z} \\ \mathcal{M}_{y} \\ \mathcal{M}_{x} \end{bmatrix}$} -- (Ja.west);
  \draw[->] (Ja.east)  -- node[midway, above]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} (G.west);
  \draw[->] (G.east) -- node[midway, above]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} (Js.west);
  \draw[->] (Js.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$};

  \begin{scope}[on background layer]
    \node[fit={(Ja.south west) ($(Js.east)+(0, 1.4)$)}, fill=black!20!white, draw, inner sep=6pt] (system) {};
    \node[above] at (system.north) {$\bm{G}_{\text{cr}}(s)$};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:schematic_jacobian_frame_crystal
#+caption: Use of Jacobian matrices to obtain the system in the frame of the crystal
#+RESULTS:
[[file:figs/schematic_jacobian_frame_crystal.png]]

#+begin_src matlab
G_mr = inv(J_s_111)*G*inv(J_a_111');
#+end_src

#+begin_src matlab :results value replace :exports both :tangle no
dcgain(G_mr)
#+end_src

#+RESULTS:
| 1.9978e-09 |  3.9657e-09 |  7.7944e-09 |
| 3.9656e-09 |  8.4979e-08 | -1.5135e-17 |
| 7.7944e-09 | -3.9252e-17 |   1.834e-07 |

#+begin_src matlab :exports none
%% Bode plot for the plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_mr(1,1), freqs, 'Hz'))), ...
     'DisplayName', 'd');
plot(freqs, abs(squeeze(freqresp(G_mr(2,2), freqs, 'Hz'))), ...
     'DisplayName', 'uh');
plot(freqs, abs(squeeze(freqresp(G_mr(3,3), freqs, 'Hz'))), ...
     'DisplayName', 'ur');
for i = 1:2
    for j = i+1:3
        plot(freqs, abs(squeeze(freqresp(G_mr(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);

ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_mr(1,1), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_mr(2,2), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_mr(3,3), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 180]);

linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src

#+begin_src matlab :exports none
%% Bode plot for the plant
fig = figure;
tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');

for i_out = 1:3
    for i_in = 1:3
        ax = nexttile;
        plot(freqs, abs(squeeze(freqresp(G_mr(i_out, i_in), freqs, 'Hz'))));
        set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
    end
end

linkaxes(findall(fig, 'type', 'axes'),'xy');
xlim([freqs(1), freqs(end)]);
#+end_src

This results in a coupled system.
The main reason is that, as we map forces to the center of the 111 crystal and not at the center of mass/stiffness of the moving part, vertical forces will induce rotation and torques will induce vertical motion.

** Plant at the center of stiffness :noexport:

Here, we map the piezo forces at the center of stiffness.

Let's first compute the Jacobian:

* Active Damping Plant (Strain gauges)
:PROPERTIES:
:header-args:matlab+: :tangle matlab/dcm_active_damping_strain_gauges.m
:END:
<<sec:active_damping_strain_gauges>>
** Introduction                                                      :ignore:
In this section, we wish to see whether if strain gauges fixed to the piezoelectric actuator can be used for active damping.

** Matlab Init                                              :noexport:ignore:
#+begin_src matlab
%% dcm_active_damping_strain_gauges.m
% Active Damping using relative motion sensors (strain gauges)
#+end_src

#+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 :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-simscape>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src

** Identification
#+begin_src matlab
%% Input/Output definition
clear io; io_i = 1;

%% Inputs
% Control Input {3x1} [N]
io(io_i) = linio([mdl, '/control_system'], 1, 'openinput');  io_i = io_i + 1;

%% Outputs
% Strain Gauges {3x1} [m]
io(io_i) = linio([mdl, '/DCM'], 2, 'openoutput'); io_i = io_i + 1;
#+end_src

#+begin_src matlab
%% Extraction of the dynamics
G_sg = linearize(mdl, io);
#+end_src

#+begin_src matlab :exports none
G_sg.InputName  = {'u_ur',  'u_uh',  'u_d'};
G_sg.OutputName = {'sg_ur', 'sg_uh', 'sg_d'};
#+end_src

#+begin_src matlab :results value replace :exports both :tangle no
dcgain(G_sg)
#+end_src

#+RESULTS:
| 4.4443e-09 | 1.0339e-13 |  3.774e-14 |
| 1.0339e-13 | 4.4443e-09 |  3.774e-14 |
| 3.7792e-14 | 3.7792e-14 | 4.4444e-09 |

#+begin_src matlab :exports none
%% Bode plot for the plant (strain gauge output)
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_sg(1,1), freqs, 'Hz'))), ...
     'DisplayName', 'd');
plot(freqs, abs(squeeze(freqresp(G_sg(2,2), freqs, 'Hz'))), ...
     'DisplayName', 'uh');
plot(freqs, abs(squeeze(freqresp(G_sg(3,3), freqs, 'Hz'))), ...
     'DisplayName', 'ur');
for i = 1:2
    for j = i+1:3
        plot(freqs, abs(squeeze(freqresp(G_sg(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
ylim([1e-14, 1e-7]);

ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_sg(1,1), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_sg(2,2), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_sg(3,3), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 0]);

linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/strain_gauge_plant_bode_plot.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:strain_gauge_plant_bode_plot
#+caption: Bode Plot of the transfer functions from piezoelectric forces to strain gauges measuremed displacements
#+RESULTS:
[[file:figs/strain_gauge_plant_bode_plot.png]]

#+begin_important
As the distance between the poles and zeros in Figure [[fig:iff_plant_bode_plot]] is very small, little damping can be actively added using the strain gauges.
This will be confirmed using a Root Locus plot.
#+end_important

** Relative Active Damping
#+begin_src matlab
Krad_g1 = eye(3)*s/(s^2/(2*pi*500)^2 + 2*s/(2*pi*500) + 1);
#+end_src

As can be seen in Figure [[fig:relative_damping_root_locus]], very little damping can be added using relative damping strategy using strain gauges.

#+begin_src matlab :exports none
%% Root Locus for IFF
gains = logspace(3, 8, 200);

figure;

hold on;
plot(real(pole(G_sg)),  imag(pole(G_sg)),  'x', 'color', colors(1,:), ...
    'DisplayName', '$g = 0$');
plot(real(tzero(G_sg)), imag(tzero(G_sg)), 'o', 'color', colors(1,:), ...
    'HandleVisibility', 'off');

for g = gains
    clpoles = pole(feedback(G_sg, g*Krad_g1, -1));
    plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
        'HandleVisibility', 'off');
end

% Optimal gain
g = 2e5;
clpoles = pole(feedback(G_sg, g*Krad_g1, -1));
plot(real(clpoles), imag(clpoles), 'x', 'color', colors(2,:), ...
    'DisplayName', sprintf('$g=%.0e$', g));
hold off;
xlim([-6, 0]); ylim([0, 2700]);
xlabel('Real Part'); ylabel('Imaginary Part');
legend('location', 'northwest');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/relative_damping_root_locus.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:relative_damping_root_locus
#+caption: Root Locus for the relative damping control
#+RESULTS:
[[file:figs/relative_damping_root_locus.png]]

#+begin_src matlab
Krad = -g*Krad_g1;
#+end_src

** Damped Plant
The controller is implemented on Simscape, and the damped plant is identified.

#+begin_src matlab
%% Input/Output definition
clear io; io_i = 1;

%% Inputs
% Control Input {3x1} [N]
io(io_i) = linio([mdl, '/control_system'], 1, 'input');  io_i = io_i + 1;

%% Outputs
% Force Sensor {3x1} [m]
io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1;
#+end_src

#+begin_src matlab
%% DCM Kinematics
load('dcm_kinematics.mat');
#+end_src

#+begin_src matlab
%% Identification of the Open Loop plant
controller.type = 0; % Open Loop
G_ol = J_a_111*inv(J_s_111)*linearize(mdl, io);
G_ol.InputName  = {'u_ur',  'u_uh',  'u_d'};
G_ol.OutputName = {'d_ur',  'd_uh',  'd_d'};
#+end_src

#+begin_src matlab
%% Identification of the damped plant with Relative Active Damping
controller.type = 2; % RAD
G_dp = J_a_111*inv(J_s_111)*linearize(mdl, io);
G_dp.InputName  = {'u_ur',  'u_uh',  'u_d'};
G_dp.OutputName = {'d_ur',  'd_uh',  'd_d'};
#+end_src

#+begin_src matlab :exports none
%% Comparison of the damped and undamped plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_ol(1,1), freqs, 'Hz'))), ...
     'DisplayName', 'd - OL');
plot(freqs, abs(squeeze(freqresp(G_ol(2,2), freqs, 'Hz'))), ...
     'DisplayName', 'uh - OL');
plot(freqs, abs(squeeze(freqresp(G_ol(3,3), freqs, 'Hz'))), ...
     'DisplayName', 'ur - OL');
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(G_dp(1,1), freqs, 'Hz'))), '--', ...
     'DisplayName', 'd - IFF');
plot(freqs, abs(squeeze(freqresp(G_dp(2,2), freqs, 'Hz'))), '--', ...
     'DisplayName', 'uh - IFF');
plot(freqs, abs(squeeze(freqresp(G_dp(3,3), freqs, 'Hz'))), '--', ...
     'DisplayName', 'ur - IFF');
for i = 1:2
    for j = i+1:3
        plot(freqs, abs(squeeze(freqresp(G_dp(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
ylim([1e-12, 1e-6]);

ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(1,1), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(2,2), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(3,3), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',1)
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(1,1), freqs, 'Hz'))), '--');
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(2,2), freqs, 'Hz'))), '--');
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(3,3), freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 0]);

linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/comp_damp_undamped_plant_rad_bode_plot.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:comp_damp_undamped_plant_rad_bode_plot
#+caption: Bode plot of both the open-loop plant and the damped plant using relative active damping
#+RESULTS:
[[file:figs/comp_damp_undamped_plant_rad_bode_plot.png]]

* Active Damping Plant (Force Sensors)
:PROPERTIES:
:header-args:matlab+: :tangle matlab/dcm_active_damping_iff.m
:END:
<<sec:active_damping_iff>>
** Introduction                                                      :ignore:
Force sensors are added above the piezoelectric actuators.
They can consists of a simple piezoelectric ceramic stack.
See for instance cite:fleming10_integ_strain_force_feedb_high.

** Matlab Init                                              :noexport:ignore:
#+begin_src matlab
%% dcm_active_damping_iff.m
% Test of Integral Force Feedback Strategy
#+end_src

#+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 :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-simscape>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src

** Identification
#+begin_src matlab
%% Input/Output definition
clear io; io_i = 1;

%% Inputs
% Control Input {3x1} [N]
io(io_i) = linio([mdl, '/control_system'], 1, 'openinput');  io_i = io_i + 1;

%% Outputs
% Force Sensor {3x1} [m]
io(io_i) = linio([mdl, '/DCM'], 3, 'openoutput'); io_i = io_i + 1;
#+end_src

#+begin_src matlab
%% Extraction of the dynamics
G_fs = linearize(mdl, io);
#+end_src

#+begin_src matlab :exports none
G_fs.InputName  = {'u_ur',  'u_uh',  'u_d'};
G_fs.OutputName = {'fs_ur', 'fs_uh', 'fs_d'};
#+end_src

The Bode plot of the identified dynamics is shown in Figure [[fig:iff_plant_bode_plot]].
At high frequency, the diagonal terms are constants while the off-diagonal terms have some roll-off.

#+begin_src matlab :exports none
%% Bode plot for the plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_fs(1,1), freqs, 'Hz'))), ...
     'DisplayName', 'd');
plot(freqs, abs(squeeze(freqresp(G_fs(2,2), freqs, 'Hz'))), ...
     'DisplayName', 'uh');
plot(freqs, abs(squeeze(freqresp(G_fs(3,3), freqs, 'Hz'))), ...
     'DisplayName', 'ur');
plot(freqs, abs(squeeze(freqresp(G_fs(1,2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
     'DisplayName', 'off-diag');
for i = 1:2
    for j = i+1:3
        plot(freqs, abs(squeeze(freqresp(G_fs(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2);
ylim([1e-13, 1e-7]);

ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_fs(1,1), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_fs(2,2), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_fs(3,3), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 180]);

linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/iff_plant_bode_plot.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:iff_plant_bode_plot
#+caption: Bode plot of IFF Plant
#+RESULTS:
[[file:figs/iff_plant_bode_plot.png]]

** Controller - Root Locus
We want to have integral action around the resonances of the system, but we do not want to integrate at low frequency.
Therefore, we can use a low pass filter.

#+begin_src matlab
%% Integral Force Feedback Controller
Kiff_g1 = eye(3)*1/(1 + s/2/pi/20);
#+end_src

#+begin_src matlab :exports none
%% Root Locus for IFF
gains = logspace(9, 12, 200);

figure;

hold on;
plot(real(pole(G_fs)),  imag(pole(G_fs)),  'x', 'color', colors(1,:), ...
    'DisplayName', '$g = 0$');
plot(real(tzero(G_fs)), imag(tzero(G_fs)), 'o', 'color', colors(1,:), ...
    'HandleVisibility', 'off');

for g = gains
    clpoles = pole(feedback(G_fs, g*Kiff_g1, +1));
    plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
        'HandleVisibility', 'off');
end

% Optimal gain
g = 8e10;
clpoles = pole(feedback(G_fs, g*Kiff_g1, +1));
plot(real(clpoles), imag(clpoles), 'x', 'color', colors(2,:), ...
    'DisplayName', sprintf('$g=%.0e$', g));
hold off;
axis square;
xlim([-2700, 0]); ylim([0, 2700]);
xlabel('Real Part'); ylabel('Imaginary Part');
legend('location', 'northwest');
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/iff_root_locus.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:iff_root_locus
#+caption: Root Locus plot for the IFF Control strategy
#+RESULTS:
[[file:figs/iff_root_locus.png]]

#+begin_src matlab
%% Integral Force Feedback Controller with optimal gain
Kiff = g*Kiff_g1;
#+end_src

#+begin_src matlab :exports none :tangle no
save('matlab/mat/Kiff.mat', 'Kiff');
#+end_src

#+begin_src matlab :eval no
%% Save the IFF controller
save('mat/Kiff.mat', 'Kiff');
#+end_src

** Damped Plant
Both the Open Loop dynamics (see Figure [[fig:schematic_jacobian_frame_fastjack]]) and the dynamics with IFF (see Figure [[fig:schematic_jacobian_frame_fastjack_iff]]) are identified.

We are here interested in the dynamics from $\bm{u}^\prime = [u_{u_r}^\prime,\ u_{u_h}^\prime,\ u_d^\prime]$ (input of the damped plant) to $\bm{d}_{\text{fj}} = [d_{u_r},\ d_{u_h},\ d_d]$ (motion of the crystal expressed in the frame of the fast-jacks).
This is schematically represented in Figure [[fig:schematic_jacobian_frame_fastjack_iff]].

#+begin_src latex :file schematic_jacobian_frame_fastjack_iff.pdf
\begin{tikzpicture}
  % Blocs
  \node[block={4.0cm}{3.0cm}] (G) {$\bm{G}(s)$};
  \coordinate[] (inputF) at ($(G.south west)!0.5!(G.north west)$);
  \coordinate[] (outputF) at ($(G.south east)!0.8!(G.north east)$);
  \coordinate[] (outputL) at ($(G.south east)!0.2!(G.north east)$);

  \node[block, right=1.5 of outputL] (Js) {$\bm{J}_{s}^{-1}$};
  \node[block, right=1.5 of Js] (Ja) {$\bm{J}_{a}$};
  \node[addb,  left= of G] (addF) {};
  \node[block, above=0.5 of G] (Kiff) {$\bm{K}_{\text{IFF}}(s)$};

  % Connections and labels
  \draw[->] ($(addF.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} u_{u_r}^\prime \\ u_{u_h}^\prime \\ u_d^\prime \end{bmatrix}$} -- (addF.west);
  \draw[->] (addF.east) -- node[miday, above]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} (inputF);
  \draw[->] (outputF) -- ++(2.0, 0) node[above left]{$\begin{bmatrix} \tau_{u_r} \\ \tau_{u_h} \\ \tau_d \end{bmatrix}$};
  \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east);
  \draw[->] (Kiff.west) -| (addF.north);
  \draw[->] (outputL) -- node[midway, above]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} (Js.west);
  \draw[->] (Js.east) -- node[midway, above]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} (Ja.west);
  \draw[->] (Ja.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_{d} \end{bmatrix}$};

  \begin{scope}[on background layer]
    \node[fit={(G.south -| addF.west) (Ja.east |- Kiff.north)}, fill=black!20!white, draw, inner sep=6pt] (system) {};
    \node[above] at (system.north) {$\bm{G}_{\text{fj,IFF}}(s)$};
  \end{scope}
\end{tikzpicture}
#+end_src

#+name: fig:schematic_jacobian_frame_fastjack_iff
#+caption: Use of Jacobian matrices to obtain the system in the frame of the fastjacks
#+RESULTS:
[[file:figs/schematic_jacobian_frame_fastjack_iff.png]]

#+begin_src matlab :exports none
%% Input/Output definition
clear io; io_i = 1;

%% Inputs
% Control Input {3x1} [N]
io(io_i) = linio([mdl, '/control_system'], 1, 'input');  io_i = io_i + 1;

%% Outputs
% Force Sensor {3x1} [m]
io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1;
#+end_src

#+begin_src matlab :exports none
%% Load DCM Kinematics
load('dcm_kinematics.mat');
#+end_src

#+begin_src matlab :exports none
%% Identification of the Open Loop plant
controller.type = 0; % Open Loop
G_ol = J_a_111*inv(J_s_111)*linearize(mdl, io);
G_ol.InputName  = {'u_ur',  'u_uh',  'u_d'};
G_ol.OutputName = {'d_ur',  'd_uh',  'd_d'};
#+end_src

#+begin_src matlab :exports none
%% Identification of the damped plant with IFF
controller.type = 1; % IFF
G_dp = J_a_111*inv(J_s_111)*linearize(mdl, io);
G_dp.InputName  = {'u_ur',  'u_uh',  'u_d'};
G_dp.OutputName = {'d_ur',  'd_uh',  'd_d'};
#+end_src

The dynamics from $\bm{u}$ to $\bm{d}_{\text{fj}}$ (open-loop dynamics) and from $\bm{u}^\prime$ to $\bm{d}_{\text{fs}}$ are compared in Figure [[fig:comp_damped_undamped_plant_iff_bode_plot]].
It is clear that the Integral Force Feedback control strategy is very effective in damping the resonances of the plant.

#+begin_src matlab :exports none
%% Comparison of the damped and undamped plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_ol(1,1), freqs, 'Hz'))), ...
     'DisplayName', 'd - OL');
plot(freqs, abs(squeeze(freqresp(G_ol(2,2), freqs, 'Hz'))), ...
     'DisplayName', 'uh - OL');
plot(freqs, abs(squeeze(freqresp(G_ol(3,3), freqs, 'Hz'))), ...
     'DisplayName', 'ur - OL');
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(G_dp(1,1), freqs, 'Hz'))), '--', ...
     'DisplayName', 'd - IFF');
plot(freqs, abs(squeeze(freqresp(G_dp(2,2), freqs, 'Hz'))), '--', ...
     'DisplayName', 'uh - IFF');
plot(freqs, abs(squeeze(freqresp(G_dp(3,3), freqs, 'Hz'))), '--', ...
     'DisplayName', 'ur - IFF');
for i = 1:2
    for j = i+1:3
        plot(freqs, abs(squeeze(freqresp(G_dp(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
ylim([1e-12, 1e-6]);

ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(1,1), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(2,2), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(3,3), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',1)
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(1,1), freqs, 'Hz'))), '--');
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(2,2), freqs, 'Hz'))), '--');
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(3,3), freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 0]);

linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/comp_damped_undamped_plant_iff_bode_plot.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:comp_damped_undamped_plant_iff_bode_plot
#+caption: Bode plot of both the open-loop plant and the damped plant using IFF
#+RESULTS:
[[file:figs/comp_damped_undamped_plant_iff_bode_plot.png]]

#+begin_important
The Integral Force Feedback control strategy is very effective in damping the modes present in the plant.
#+end_important

* HAC-LAC (IFF) architecture
:PROPERTIES:
:header-args:matlab+: :tangle matlab/dcm_hac_iff.m
:END:
<<sec:hac_iff>>
** Introduction                                                      :ignore:

The HAC-LAC architecture is shown in Figure [[fig:schematic_jacobian_frame_fastjack_hac_iff]].

#+begin_src latex :file schematic_jacobian_frame_fastjack_hac_iff.pdf
\begin{tikzpicture}
  % Blocs
  \node[block={3.0cm}{3.0cm}] (G) {$\bm{G}(s)$};
  \coordinate[] (inputF) at ($(G.south west)!0.5!(G.north west)$);
  \coordinate[] (outputF) at ($(G.south east)!0.8!(G.north east)$);
  \coordinate[] (outputL) at ($(G.south east)!0.2!(G.north east)$);

  \node[block, right=1.2 of outputL] (Js) {$\bm{J}_{s}^{-1}$};

  \node[addb,  left= of G] (addF) {};
  \node[block, above=0.5 of G] (Kiff) {$\bm{K}_{\text{IFF}}(s)$};

  \node[block, left=1.2 of addF] (Khac) {$\bm{K}_{\text{HAC}}(s)$};
  \node[block, left=1.2 of Khac] (Ja) {$\bm{J}_{a}$};
  \node[addb={+}{}{}{}{-},  left=1.0 of Ja] (subL) {};


  % Connections and labels
  \draw[->] (Khac.east) -- node[midway, above]{$\begin{bmatrix} u_{u_r}^\prime \\ u_{u_h}^\prime \\ u_d^\prime \end{bmatrix}$} (addF.west);
  \draw[->] (addF.east) -- node[midway, above]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} (inputF);
  \draw[->] (outputF) -- ++(2.0, 0) node[above left]{$\begin{bmatrix} \tau_{u_r} \\ \tau_{u_h} \\ \tau_d \end{bmatrix}$};
  \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east);
  \draw[->] (Kiff.west) -| (addF.north);
  \draw[->] (outputL) -- node[midway, above]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} (Js.west);
  \draw[->] (Js.east) -- ++(1.0, 0);


  \draw[->] ($(subL.west) + (-0.8, 0)$) -- node[midway, above]{$\begin{bmatrix} r_{d_z} \\ r_{r_y} \\ r_{r_x} \end{bmatrix}$} (subL.west);
  \draw[->] (subL.east) -- node[midway, above]{$\begin{bmatrix} \epsilon_{d_z} \\ \epsilon_{r_y} \\ \epsilon_{r_x} \end{bmatrix}$} (Ja.west);
  \draw[->] (Ja.east) -- node[midway, above]{$\begin{bmatrix} \epsilon_{d_{u_r}} \\ \epsilon_{d_{u_h}} \\ \epsilon_{d_d} \end{bmatrix}$} (Khac.west);

  \draw[->] ($(Js.east) + (0.6, 0)$)node[branch]{}node[above]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- ++(0, -1.0) -| (subL.south);
\end{tikzpicture}
#+end_src

#+name: fig:schematic_jacobian_frame_fastjack_hac_iff
#+caption: HAC-LAC architecture
#+attr_latex: :width \linewidth
#+RESULTS:
[[file:figs/schematic_jacobian_frame_fastjack_hac_iff.png]]

** Matlab Init                                              :noexport:ignore:
#+begin_src matlab
%% dcm_hac_iff.m
% Development of the HAC-IFF control strategy
#+end_src

#+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 :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-simscape>>
#+end_src

#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src

** System Identification
Let's identify the damped plant.

#+begin_src matlab :exports none
%% Input/Output definition
clear io; io_i = 1;

%% Inputs
% Control Input {3x1} [N]
io(io_i) = linio([mdl, '/control_system'], 1, 'input');  io_i = io_i + 1;

%% Outputs
% Force Sensor {3x1} [m]
io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1;
#+end_src

#+begin_src matlab :exports none
%% Load DCM Kinematics and IFF controller
load('dcm_kinematics.mat');
load('Kiff.mat');
#+end_src

#+begin_src matlab :exports none
%% Identification of the damped plant with IFF
controller.type = 1; % IFF
G_dp = J_a_111*inv(J_s_111)*linearize(mdl, io);
G_dp.InputName  = {'u_ur',  'u_uh',  'u_d'};
G_dp.OutputName = {'d_ur',  'd_uh',  'd_d'};
#+end_src

#+begin_src matlab :exports none
%% Comparison of the damped and undamped plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_dp(1,1), freqs, 'Hz'))), '-', ...
     'DisplayName', 'd - IFF');
plot(freqs, abs(squeeze(freqresp(G_dp(2,2), freqs, 'Hz'))), '-', ...
     'DisplayName', 'uh - IFF');
plot(freqs, abs(squeeze(freqresp(G_dp(3,3), freqs, 'Hz'))), '-', ...
     'DisplayName', 'ur - IFF');
for i = 1:2
    for j = i+1:3
        plot(freqs, abs(squeeze(freqresp(G_dp(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
ylim([1e-12, 1e-8]);

ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(1,1), freqs, 'Hz'))), '-');
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(2,2), freqs, 'Hz'))), '-');
plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(3,3), freqs, 'Hz'))), '-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 0]);

linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/bode_plot_hac_iff_plant.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:bode_plot_hac_iff_plant
#+caption: Bode Plot of the plant for the High Authority Controller (transfer function from $\bm{u}^\prime$ to $\bm{\epsilon}_d$)
#+RESULTS:
[[file:figs/bode_plot_hac_iff_plant.png]]

** High Authority Controller
Let's design a controller with a bandwidth of 100Hz.
As the plant is well decoupled and well approximated by a constant at low frequency, the high authority controller can easily be designed with SISO loop shaping.

#+begin_src matlab
%% Controller design
wc = 2*pi*100; % Wanted crossover frequency [rad/s]
a = 2; % Lead parameter

Khac = diag(1./diag(abs(evalfr(G_dp, 1j*wc)))) * ... % Diagonal controller
        wc/s * ... % Integrator
        1/(sqrt(a))*(1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))) * ... % Lead
        1/(s^2/(4*wc)^2 + 2*s/(4*wc) + 1); % Low pass filter
#+end_src

#+begin_src matlab :exports none :tangle no
save('matlab/mat/Khac_iff.mat', 'Khac');
#+end_src

#+begin_src matlab :eval no
%% Save the HAC controller
save('mat/Khac_iff.mat', 'Khac');
#+end_src

#+begin_src matlab
%% Loop Gain
L_hac_lac = G_dp * Khac;
#+end_src

#+begin_src matlab :exports none
%% Bode Plot of the Loop Gain
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');

ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(L_hac_lac(1,1), freqs, 'Hz'))), '-', ...
     'DisplayName', 'd');
plot(freqs, abs(squeeze(freqresp(L_hac_lac(2,2), freqs, 'Hz'))), '-', ...
     'DisplayName', 'uh');
plot(freqs, abs(squeeze(freqresp(L_hac_lac(3,3), freqs, 'Hz'))), '-', ...
     'DisplayName', 'ur');
for i = 1:2
    for j = i+1:3
        plot(freqs, abs(squeeze(freqresp(L_hac_lac(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
             'HandleVisibility', 'off');
    end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2);
ylim([1e-2, 1e1]);

ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(L_hac_lac(1,1), freqs, 'Hz'))), '-');
plot(freqs, 180/pi*angle(squeeze(freqresp(L_hac_lac(2,2), freqs, 'Hz'))), '-');
plot(freqs, 180/pi*angle(squeeze(freqresp(L_hac_lac(3,3), freqs, 'Hz'))), '-');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
hold off;
yticks(-360:90:360);
ylim([-180, 0]);

linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/hac_iff_loop_gain_bode_plot.pdf', 'width', 'wide', 'height', 'tall');
#+end_src

#+name: fig:hac_iff_loop_gain_bode_plot
#+caption: Bode Plot of the Loop gain for the High Authority Controller
#+RESULTS:
[[file:figs/hac_iff_loop_gain_bode_plot.png]]

#+begin_src matlab :exports none
%% Compute the Eigenvalues of the loop gain
Ldet = zeros(3, length(freqs));

Lmimo = squeeze(freqresp(L_hac_lac, freqs, 'Hz'));
for i_f = 1:length(freqs)
    Ldet(:, i_f) = eig(squeeze(Lmimo(:,:,i_f)));
end
#+end_src

As shown in the Root Locus plot in Figure [[fig:loci_hac_iff_fast_jack]], the closed loop system should be stable.

#+begin_src matlab :exports none
%% Plot of the eigenvalues of L in the complex plane
figure;
hold on;
% Angle used to draw the circles
theta = linspace(0, 2*pi, 100);
% Unit circle
plot(cos(theta), sin(theta), '--');
% Circle for module margin
plot(-1 + min(min(abs(Ldet + 1)))*cos(theta), min(min(abs(Ldet + 1)))*sin(theta), '--');

for i = 1:3
    plot(real(squeeze(Ldet(i,:))), imag(squeeze(Ldet(i,:))), 'k.');
    plot(real(squeeze(Ldet(i,:))), -imag(squeeze(Ldet(i,:))), 'k.');
end
% Unstable Point
plot(-1, 0, 'kx', 'HandleVisibility', 'off');
hold off;
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
xlabel('Real'); ylabel('Imag');
axis square;
xlim([-3, 1]); ylim([-2, 2]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/loci_hac_iff_fast_jack.pdf', 'width', 'normal', 'height', 'normal');
#+end_src

#+name: fig:loci_hac_iff_fast_jack
#+caption: Root Locus for the High Authority Controller
#+RESULTS:
[[file:figs/loci_hac_iff_fast_jack.png]]

** Performances
In order to estimate the performances of the HAC-IFF control strategy, the transfer function from motion errors of the stepper motors to the motion error of the crystal is identified both in open loop and with the HAC-IFF strategy.

#+begin_src matlab :exports none
%% Input/Output definition
clear io; io_i = 1;

%% Inputs
% Jack Motion Erros {3x1} [m]
io(io_i) = linio([mdl, '/stepper_errors'], 1, 'input');  io_i = io_i + 1;

%% Outputs
% Interferometer Output {3x1} [m]
io(io_i) = linio([mdl, '/DCM'], 1, 'output'); io_i = io_i + 1;
#+end_src

#+begin_src matlab :exports none
%% Identification of the transmissibility of errors in open-loop
controller.type = 0; % Open Loop
T_ol = inv(J_s_111)*linearize(mdl, io)*J_a_111;
T_ol.InputName  = {'e_dz',  'e_ry',  'e_rx'};
T_ol.OutputName = {'dx',  'ry',  'rx'};
#+end_src

#+begin_src matlab :exports none
%% Load DCM Kinematics and IFF controller
load('dcm_kinematics.mat');
load('Kiff.mat');
#+end_src

#+begin_src matlab :exports none
%% Identification of the transmissibility of errors with HAC-IFF
controller.type = 3; % IFF
T_hl = inv(J_s_111)*linearize(mdl, io)*J_a_111;
T_hl.InputName  = {'e_dz',  'e_ry',  'e_rx'};
T_hl.OutputName = {'dx',  'ry',  'rx'};
#+end_src

It is first verified that the closed-loop system is stable:

#+begin_src matlab :results value replace :exports both :tangle no
isstable(T_hl)
#+end_src

#+RESULTS:
: 1

And both transmissibilities are compared in Figure [[fig:stepper_transmissibility_comp_ol_hac_iff]].

#+begin_src matlab :exports none
%% Transmissibility of stepper errors
f = logspace(0, 3, 1000);

figure;
hold on;
plot(f, abs(squeeze(freqresp(T_ol(1,1), f, 'Hz'))), '-', ...
     'DisplayName', '$d_z$ - OL');
plot(f, abs(squeeze(freqresp(T_ol(2,2), f, 'Hz'))), '-', ...
     'DisplayName', '$r_y$ - OL');
plot(f, abs(squeeze(freqresp(T_ol(3,3), f, 'Hz'))), '-', ...
     'DisplayName', '$r_x$ - OL');
set(gca,'ColorOrderIndex',1)
plot(f, abs(squeeze(freqresp(T_hl(1,1), f, 'Hz'))), '--', ...
     'DisplayName', '$d_z$ - HAC-IFF');
plot(f, abs(squeeze(freqresp(T_hl(2,2), f, 'Hz'))), '--', ...
     'DisplayName', '$r_y$ - HAC-IFF');
plot(f, abs(squeeze(freqresp(T_hl(3,3), f, 'Hz'))), '--', ...
     'DisplayName', '$r_x$ - HAC-IFF');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Stepper transmissibility');
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
ylim([1e-2, 1e2]);
xlim([f(1), f(end)]);
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/stepper_transmissibility_comp_ol_hac_iff.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:stepper_transmissibility_comp_ol_hac_iff
#+caption: Comparison of the transmissibility of errors from vibrations of the stepper motor between the open-loop case and the hac-iff case.
#+RESULTS:
[[file:figs/stepper_transmissibility_comp_ol_hac_iff.png]]

#+begin_important
The HAC-IFF control strategy can effectively reduce the transmissibility of the motion errors of the stepper motors.
This reduction is effective inside the bandwidth of the controller.
#+end_important

* Helping Functions                                                 :noexport:
** Initialize Path
#+NAME: m-init-path
#+BEGIN_SRC matlab
%% Path for functions, data and scripts
addpath('./matlab/mat/'); % Path for data
addpath('./matlab/');     % Path for scripts

%% Simscape Model - Nano Hexapod
addpath('./matlab/STEPS/')
#+END_SRC

#+NAME: m-init-path-tangle
#+BEGIN_SRC matlab
%% Path for functions, data and scripts
addpath('./mat/'); % Path for data

%% Simscape Model - Nano Hexapod
addpath('./STEPS/')
#+END_SRC

** Initialize Simscape Model
#+NAME: m-init-simscape
#+begin_src matlab
%% Initialize Parameters for Simscape model
controller.type = 0; % Open Loop Control

%% Options for Linearization
options = linearizeOptions;
options.SampleTime = 0;

%% Open Simulink Model
mdl = 'simscape_dcm';

open(mdl)
#+end_src

** Initialize other elements
#+NAME: m-init-other
#+BEGIN_SRC matlab
%% Colors for the figures
colors = colororder;

%% Frequency Vector
freqs = logspace(1, 3, 1000);
#+END_SRC

* Bibliography                                                        :ignore:
#+latex: \printbibliography