nass-simscape/org/hac_lac.org

HAC-LAC applied on the Simscape Model

• Introduction
• Initialization
• Low Authority Control - Direct Velocity Feedback $\bm{K}_\mathcal{L}$
  • Introduction
  • Identification
  • Plant
  • Root Locus
  • Controller and Loop Gain
• High Authority Control - $\bm{K}_\mathcal{X}$
  • Identification of the damped plant
  • Controller Design
• Simulation
• Results

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{tikzpicture}
    \node[block={3.0cm}{3.0cm}] (G) {$G$};

    % 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]{$\bm{\tau}_\mathcal{X}$} -- (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}

Initialization

We initialize all the stages with the default parameters.

  initializeGround();
  initializeGranite();
  initializeTy();
  initializeRy();
  initializeRz();
  initializeMicroHexapod();
  initializeAxisc();
  initializeMirror();

The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.

  initializeNanoHexapod('actuator', 'piezo');
  initializeSample('mass', 1);

We set the references that corresponds to a tomography experiment.

  initializeReferences('Rz_type', 'rotating', 'Rz_period', 1);

  initializeDisturbances();

Open Loop.

  initializeController('type', 'open-loop');

And we put some gravity.

  initializeSimscapeConfiguration('gravity', true);

We log the signals.

  initializeLoggingConfiguration('log', 'all');

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 this file.

Identification

  %% 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'};

Plant

Root Locus

Controller and Loop Gain

  K_dvf = s*15000/(1 + s/2/pi/10000);

  K_dvf = -K_dvf*eye(6);

High Authority Control - $\bm{K}_\mathcal{X}$

Identification of the damped plant

  Kx = tf(zeros(6));

  initializeController('type', 'hac-dvf');

  %% 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'};

The minus sine is put here because there is already a minus sign included due to the computation of the position error.

  load('mat/stages.mat', 'nano_hexapod');

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

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

  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))));

  isstable(feedback(Gx*Kx, eye(6), -1))

  Kx = inv(nano_hexapod.J')*Kx;

  isstable(feedback(G*Kx, eye(6), 1))

Simulation

  load('mat/conf_simulink.mat');
  set_param(conf_simulink, 'StopTime', '1.5');

And we simulate the system.

  sim('nass_model');

  save('./mat/tomo_exp_hac_lac.mat', 'simout');

Results

  load('./mat/tomo_exp_hac_lac.mat', 'simout');