nass-simscape/org/control_hac_lac.org

#+TITLE: HAC-LAC applied on the Simscape Model
#+SETUPFILE: ./setup/org-setup-file.org

* Introduction                                                        :ignore:
The position $\bm{\mathcal{X}}$ of the Sample with respect to the granite is measured.

It is then compare to the wanted position of the Sample $\bm{r}_\mathcal{X}$ in order to obtain the position error $\bm{\epsilon}_\mathcal{X}$ of the Sample with respect to a frame attached to the Stewart top platform.

#+begin_src latex :file hac_lac_control_schematic.pdf
  \begin{tikzpicture}
    \node[block={3.0cm}{3.0cm}] (G) {Plant};

    % Input and outputs coordinates
    \coordinate[] (outputX) at ($(G.south east)!0.25!(G.north east)$);
    \coordinate[] (outputL) at ($(G.south east)!0.75!(G.north east)$);

    \draw[->] (outputX) -- ++(1.8, 0) node[above left]{$\bm{\mathcal{X}}$};
    \draw[->] (outputL) -- ++(1.8, 0) node[above left]{$\bm{\mathcal{L}}$};

    % Blocs
    \node[addb, left= of G] (addF) {};
    \node[block, left=1.2 of addF] (Kx) {$\bm{K}_\mathcal{X}$};
    \node[block={2cm}{2cm}, align=center, left=1.2 of Kx] (subx) {Computes\\Position\\Error};

    \node[block, above= of addF] (Kl) {$\bm{K}_\mathcal{L}$};
    \node[addb={+}{}{}{-}{}, above= of Kl] (subl) {};

    \node[block, align=center, left=0.8 of subl] (invK) {Inverse\\Kinematics};

    % Connections and labels
    \draw[<-] (subx.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-0.8, 0);
    \draw[->] ($(subx.east) + (0.2, 0)$)node[branch]{} |- (invK.west);
    \draw[->] (invK.east) -- (subl.west) node[above left]{$\bm{r}_\mathcal{L}$};
    \draw[->] (subl.south) -- (Kl.north) node[above right]{$\bm{\epsilon}_\mathcal{L}$};
    \draw[->] (Kl.south) -- (addF.north);

    \draw[->] (subx.east) -- (Kx.west) node[above left]{$\bm{\epsilon}_\mathcal{X}$};
    \draw[->] (Kx.east) node[above right]{$\bm{\tau}^\prime$} -- (addF.west);
    \draw[->] (addF.east) -- (G.west) node[above left]{$\bm{\tau}$};

    \draw[->] ($(outputL.east) + (0.4, 0)$)node[branch](L){} |- (subl.east);
    \draw[->] ($(outputX.east) + (1.2, 0)$)node[branch]{} -- ++(0, -1.6) -| (subx.south);

    \begin{scope}[on background layer]
      \node[fit={(G.south-|Kl.west) (L|-subl.north)}, fill=black!20!white, draw, dashed, inner sep=8pt] (Ktot) {};
    \end{scope}
  \end{tikzpicture}
#+end_src

#+name: fig:hac_lac_control_schematic
#+caption: HAC-LAC Control Architecture used for the Control of the NASS
#+RESULTS:
[[file:figs/hac_lac_control_schematic.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('nass_model.slx')
#+end_src

* Initialization
We initialize all the stages with the default parameters.
#+begin_src matlab
  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();
#+end_src

The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.
#+begin_src matlab
  initializeNanoHexapod('actuator', 'piezo');
  initializeSample('mass', 1);
#+end_src

We set the references that corresponds to a tomography experiment.
#+begin_src matlab
  initializeReferences('Rz_type', 'rotating', 'Rz_period', 1);
#+end_src

#+begin_src matlab
  initializeDisturbances();
#+end_src

Open Loop.
#+begin_src matlab
  initializeController('type', 'open-loop');
#+end_src

And we put some gravity.
#+begin_src matlab
  initializeSimscapeConfiguration('gravity', true);
#+end_src

We log the signals.
#+begin_src matlab
  initializeLoggingConfiguration('log', 'all');
#+end_src

* Low Authority Control - Direct Velocity Feedback $\bm{K}_\mathcal{L}$
** Introduction                                                      :ignore:
The first loop closed corresponds to a direct velocity feedback loop.

The design of the associated decentralized controller is explained in [[file:control_active_damping.org][this]] file.

** Identification
#+begin_src matlab
  %% Name of the Simulink File
  mdl = 'nass_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],    1, 'openinput');               io_i = io_i + 1; % Actuator Inputs
  io(io_i) = linio([mdl, '/Micro-Station'], 3, 'openoutput', [], 'Dnlm');  io_i = io_i + 1; % Relative Motion Outputs

  %% Run the linearization
  G_dvf = linearize(mdl, io, 0);
  G_dvf.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
  G_dvf.OutputName = {'Dnlm1', 'Dnlm2', 'Dnlm3', 'Dnlm4', 'Dnlm5', 'Dnlm6'};
#+end_src

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

  figure;

  ax1 = subplot(2, 2, 1);
  hold on;
  for i = 1:6
    plot(freqs, abs(squeeze(freqresp(G_dvf(i, i), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title('Diagonal elements of the Plant');

  ax2 = subplot(2, 2, 3);
  hold on;
  for i = 1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf(i, i), freqs, 'Hz'))), 'DisplayName', sprintf('$d\\mathcal{L}_%i/\\tau_%i$', i, i));
  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]);
  legend('location', 'northwest');

  ax3 = subplot(2, 2, 2);
  hold on;
  for i = 1:5
    for j = i+1:6
      plot(freqs, abs(squeeze(freqresp(G_dvf(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
    end
  end
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(G_dvf(1, 1), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title('Off-Diagonal elements of the Plant');

  ax4 = subplot(2, 2, 4);
  hold on;
  for i = 1:5
    for j = i+1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
    end
  end
  set(gca,'ColorOrderIndex',1);
  plot(freqs, 180/pi*angle(squeeze(freqresp(G_dvf(1, 1), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

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

** Root Locus
#+begin_src matlab :exports none
  gains = logspace(0, 5, 500);

  figure;
  hold on;
  plot(real(pole(G_dvf)),  imag(pole(G_dvf)),  'x');
  set(gca,'ColorOrderIndex',1);
  plot(real(tzero(G_dvf)),  imag(tzero(G_dvf)),  'o');
  for i = 1:length(gains)
    set(gca,'ColorOrderIndex',1);
    cl_poles = pole(feedback(G_dvf, (gains(i)*s)*eye(6)));
    plot(real(cl_poles), imag(cl_poles), '.');
  end
  ylim([0, 2*pi*500]);
  xlim([-2*pi*500,0]);
  xlabel('Real Part')
  ylabel('Imaginary Part')
  axis square
#+end_src

** Controller and Loop Gain
#+begin_src matlab
  K_dvf = s*15000/(1 + s/2/pi/10000);
#+end_src

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

  figure;

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

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(K_dvf*G_dvf(i,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

#+begin_src matlab
  K_dvf = -K_dvf*eye(6);
#+end_src

* Uncertainty Improvements thanks to the LAC control
#+begin_src matlab
  K_dvf_backup = K_dvf;
  initializeController('type', 'hac-dvf');
#+end_src

#+begin_src matlab
  masses = [1, 10, 50]; % [kg]
#+end_src

#+begin_src matlab
  %% Name of the Simulink File
  mdl = 'nass_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],     1, 'input');            io_i = io_i + 1; % Actuator Inputs
  io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror
#+end_src

#+begin_src matlab :exports none
  Gm = {zeros(length(masses))};

  K_dvf = tf(zeros(6));
  Kx = tf(zeros(6));

  for i = 1:length(masses)
    initializeSample('mass', masses(i));

    %% Run the linearization
    G = linearize(mdl, io, 0);
    G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
    G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};

    Gm(i) = {G};
  end
#+end_src

#+begin_src matlab :exports none
  Gm_iff = {zeros(length(masses))};

  K_dvf = K_dvf_backup;
  Kx = tf(zeros(6));

  for i = 1:length(masses)
    initializeSample('mass', masses(i));

    %% Run the linearization
    G = linearize(mdl, io, 0);
    G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
    G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};

    Gm_iff(i) = {G};
  end
#+end_src

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

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:length(Gm_iff)
    set(gca,'ColorOrderIndex',i);
    plot(freqs, abs(squeeze(freqresp(Gm{i}(1, 1), freqs, 'Hz'))), '-');
    set(gca,'ColorOrderIndex',i);
    plot(freqs, abs(squeeze(freqresp(Gm_iff{i}(1, 1), 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:length(Gm_iff)
    set(gca,'ColorOrderIndex',i);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gm{i}(1, 1), freqs, 'Hz'))), '-', ...
         'DisplayName', sprintf('$M = %.0f$ [kg]', masses(i)));
    set(gca,'ColorOrderIndex',i);
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gm_iff{i}(1, 1), freqs, 'Hz'))), '--', ...
         'HandleVisibility', 'off');
  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]);
  legend('location', 'southwest');

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

* High Authority Control - $\bm{K}_\mathcal{X}$
** Identification of the damped plant
#+begin_src matlab
  Kx = tf(zeros(6));
#+end_src

#+begin_src matlab
  initializeController('type', 'hac-dvf');
#+end_src

#+begin_src matlab
  %% Name of the Simulink File
  mdl = 'nass_model';

  %% Input/Output definition
  clear io; io_i = 1;
  io(io_i) = linio([mdl, '/Controller'],     1, 'input');            io_i = io_i + 1; % Actuator Inputs
  io(io_i) = linio([mdl, '/Tracking Error'], 1, 'output', [], 'En'); io_i = io_i + 1; % Position Errror

  %% Run the linearization
  G = linearize(mdl, io, 0);
  G.InputName  = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
  G.OutputName = {'Ex', 'Ey', 'Ez', 'Erx', 'Ery', 'Erz'};
#+end_src

The minus sine is put here because there is already a minus sign included due to the computation of the position error.
#+begin_src matlab
  load('mat/stages.mat', 'nano_hexapod');

  Gx = -G*inv(nano_hexapod.J');
  Gx.InputName  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
#+end_src

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

  labels = {'$D_x/\mathcal{F}_x$', '$D_y/\mathcal{F}_y$', '$D_z/\mathcal{F}_z$', '$R_x/\mathcal{M}_x$', '$R_y/\mathcal{M}_y$', '$R_z/\mathcal{M}_z$'};

  figure;

  ax1 = subplot(2, 2, 1);
  hold on;
  for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Gx(i, i), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title('Diagonal elements of the Plant');

  ax2 = subplot(2, 2, 3);
  hold on;
  for i = 1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(i, i), freqs, 'Hz'))), 'DisplayName', labels{i});
  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]);
  legend();

  ax3 = subplot(2, 2, 2);
  hold on;
  for i = 1:5
    for j = i+1:6
      plot(freqs, abs(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
    end
  end
  set(gca,'ColorOrderIndex',1);
  plot(freqs, abs(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title('Off-Diagonal elements of the Plant');

  ax4 = subplot(2, 2, 4);
  hold on;
  for i = 1:5
    for j = i+1:6
      plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
    end
  end
  set(gca,'ColorOrderIndex',1);
  plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))));
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
  ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
  ylim([-180, 180]);
  yticks([-180, -90, 0, 90, 180]);

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

** Controller Design
The controller consists of:
- A pure integrator
- A Second integrator up to half the wanted bandwidth
- A Lead around the cross-over frequency
- A low pass filter with a cut-off equal to two times the wanted bandwidth

#+begin_src matlab
  wc = 2*pi*15; % Bandwidth Bandwidth [rad/s]

  h = 1.5; % Lead parameter

  Kx = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * wc/s * ((s/wc*2 + 1)/(s/wc*2)) * (1/(1 + s/wc/2));

  % Normalization of the gain of have a loop gain of 1 at frequency wc
  Kx = Kx.*diag(1./diag(abs(freqresp(Gx*Kx, wc))));
#+end_src

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

  labels = {'$L_x$', '$L_y$', '$L_z$', '$L_{R_x}$', '$L_{R_y}$', '$L_{R_z}$'};

  figure;

  ax1 = subplot(2, 1, 1);
  hold on;
  for i = 1:6
    plot(freqs, abs(squeeze(freqresp(Gx(i, i)*Kx(i,i), freqs, 'Hz'))));
  end
  hold off;
  set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
  ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
  title('Diagonal elements of the Plant');

  ax2 = subplot(2, 1, 2);
  hold on;
  for i = 1:6
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(i, i)*Kx(i,i), freqs, 'Hz'))), 'DisplayName', labels{i});
  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]);
  legend();

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

#+begin_src matlab
  isstable(feedback(Gx*Kx, eye(6), -1))
#+end_src

#+begin_src matlab
  Kx = inv(nano_hexapod.J')*Kx;
#+end_src

#+begin_src matlab
  isstable(feedback(G*Kx, eye(6), 1))
#+end_src

* Simulation
#+begin_src matlab
  load('mat/conf_simulink.mat');
  set_param(conf_simulink, 'StopTime', '2');
#+end_src

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

#+begin_src matlab
  hac_dvf = simout;
  save('./mat/tomo_exp_hac_lac.mat', 'hac_dvf');
#+end_src

* Results
Let's load the simulation when no control is applied.
#+begin_src matlab
  load('./mat/experiment_tomography.mat', 'tomo_align_dist');
  load('./mat/tomo_exp_hac_lac.mat', 'hac_dvf');
#+end_src

#+begin_src matlab :exports none
  figure;
  ax1 = subplot(2, 3, 1);
  hold on;
  plot(tomo_align_dist.Em.En.Time, tomo_align_dist.Em.En.Data(:, 1))
  plot(hac_dvf.Em.En.Time, hac_dvf.Em.En.Data(:, 1))
  hold off;
  xlabel('Time [s]');
  ylabel('Dx [m]');

  ax2 = subplot(2, 3, 2);
  hold on;
  plot(tomo_align_dist.Em.En.Time, tomo_align_dist.Em.En.Data(:, 2))
  plot(hac_dvf.Em.En.Time, hac_dvf.Em.En.Data(:, 2))
  hold off;
  xlabel('Time [s]');
  ylabel('Dy [m]');

  ax3 = subplot(2, 3, 3);
  hold on;
  plot(tomo_align_dist.Em.En.Time, tomo_align_dist.Em.En.Data(:, 3))
  plot(hac_dvf.Em.En.Time, hac_dvf.Em.En.Data(:, 3))
  hold off;
  xlabel('Time [s]');
  ylabel('Dz [m]');

  ax4 = subplot(2, 3, 4);
  hold on;
  plot(tomo_align_dist.Em.En.Time, tomo_align_dist.Em.En.Data(:, 4))
  plot(hac_dvf.Em.En.Time, hac_dvf.Em.En.Data(:, 4))
  hold off;
  xlabel('Time [s]');
  ylabel('Rx [rad]');

  ax5 = subplot(2, 3, 5);
  hold on;
  plot(tomo_align_dist.Em.En.Time, tomo_align_dist.Em.En.Data(:, 5))
  plot(hac_dvf.Em.En.Time, hac_dvf.Em.En.Data(:, 5))
  hold off;
  xlabel('Time [s]');
  ylabel('Ry [rad]');

  ax6 = subplot(2, 3, 6);
  hold on;
  plot(tomo_align_dist.Em.En.Time, tomo_align_dist.Em.En.Data(:, 6), 'DisplayName', '$\mu$-Station')
  plot(hac_dvf.Em.En.Time, hac_dvf.Em.En.Data(:, 6), 'DisplayName', 'HAC-DVF')
  hold off;
  xlabel('Time [s]');
  ylabel('Rz [rad]');
  legend();

  linkaxes([ax1,ax2,ax3,ax4],'x');
  xlim([0.5, inf]);
#+end_src