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Add decentralized control file
2020-03-23 09:55:26 +01:00
#+TITLE: Control in the Frame of the Legs applied on the Simscape Model
Add org-setup-file that all org file links to
2020-04-17 10:25:44 +02:00
#+SETUPFILE: ./setup/org-setup-file.org
Add decentralized control file
2020-03-23 09:55:26 +01:00
* Introduction :ignore:
In this document, we apply some decentralized control to the NASS and see what level of performance can be obtained.
* Decentralized Control
** Matlab Init :noexport:
#+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
simulinkproject('../');
#+end_src
#+begin_src matlab
open('nass_model.slx');
#+end_src
** Control Schematic
The control architecture is shown in Figure [[fig:decentralized_reference_tracking_L]].
The signals are:
- $\bm{r}_\mathcal{X}$: wanted position of the sample with respect to the granite
- $\bm{r}_{\mathcal{X}_n}$: wanted position of the sample with respect to the nano-hexapod
- $\bm{r}_\mathcal{L}$: wanted length of each of the nano-hexapod's legs
- $\bm{\tau}$: forces applied in each actuator
- $\bm{\mathcal{L}}$: measured displacement of each leg
- $\bm{\mathcal{X}}$: measured position of the sample with respect to the granite
#+begin_src latex :file decentralized_reference_tracking_L.pdf
\begin{tikzpicture}
% Blocs
\node[block={2.0cm}{2.0cm}] (P) {};
\coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$);
\coordinate[] (outputX) at ($(P.south east)!0.7!(P.north east)$);
\coordinate[] (outputL) at ($(P.south east)!0.3!(P.north east)$);
\node[block, left= of inputF] (K) {$\bm{K}_\mathcal{L}$};
\node[addb={+}{}{}{}{-}, left= of K] (subr) {};
\node[block, align=center, left= of subr] (J) {Inverse\\Kinematics};
\node[block, align=center, left= of J] (Ex) {Compute\\Pos. Error};
% Connections and labels
\draw[->] (outputL) -- ++(1, 0) node[above left]{$\bm{\mathcal{L}}$};
\draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, -1) -| (subr.south);
\draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_\mathcal{L}$};
\draw[->] (K.east) -- (inputF) node[above left]{$\bm{\tau}$};
\draw[->] (outputX) -- ++(1.8, 0) node[above left]{$\bm{\mathcal{X}}$};
\draw[->] ($(outputX) + (1.4, 0)$)node[branch]{} -- ++(0, -2.5) -| (Ex.south);
\draw[->] (Ex.east) -- (J.west) node[above left]{$\bm{r}_{\mathcal{X}_n}$};
\draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{\mathcal{L}}$};
\draw[<-] (Ex.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-1, 0);
\end{tikzpicture}
#+end_src
#+name: fig:decentralized_reference_tracking_L
#+caption: Decentralized control for reference tracking
#+RESULTS:
[[file:figs/decentralized_reference_tracking_L.png]]
** Initialize the Simscape Model
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', 'ref-track-L');
Kl = tf(zeros(6));
#+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
** Identification of the plant
Let's identify the transfer function from $\bm{\tau}$ to $\bm{\mathcal{L}}$.
#+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, '/Controller/Reference-Tracking-L/Sum'], 1, 'openoutput'); io_i = io_i + 1; % Leg length error
%% Run the linearization
G = linearize(mdl, io, 0);
G.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
G.OutputName = {'El1', 'El2', 'El3', 'El4', 'El5', 'El6'};
#+end_src
** Plant Analysis
The diagonal and off-diagonal terms of the plant are shown in Figure [[fig:decentralized_control_plant_L]].
We can see that:
- the diagonal terms have similar dynamics
- the plant is decoupled at low frequency
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
ax1 = subplot(2, 2, 1);
hold on;
for i = 1:6
plot(freqs, abs(squeeze(freqresp(G(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(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(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(G(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(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, 180/pi*angle(squeeze(freqresp(G(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
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/decentralized_control_plant_L.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:decentralized_control_plant_L
#+caption: Transfer Functions from forces applied in each actuator $\tau_i$ to the relative motion of each leg $d\mathcal{L}_i$ ([[./figs/decentralized_control_plant_L.png][png]], [[./figs/decentralized_control_plant_L.pdf][pdf]])
[[file:figs/decentralized_control_plant_L.png]]
** Controller Design
The controller consists of:
- A pure integrator
- An integrator up to little before the crossover
- A lead around the crossover
- A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin
The obtained loop gains corresponding to the diagonal elements are shown in Figure [[fig:decentralized_control_L_loop_gain]].
#+begin_src matlab
wc = 2*pi*20;
h = 1.5;
Kl = diag(1./diag(abs(freqresp(G, wc)))) * ...
wc/s * ... % Pure Integrator
((s/wc*2 + 1)/(s/wc*2)) * ... % Integrator up to wc/2
1/h * (1 + s/wc*h)/(1 + s/wc/h) * ... % Lead
1/(1 + s/3/wc) * ... % Low pass Filter
1/(1 + s/3/wc);
#+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(Kl(i, i)*G(i, i), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(i, i)*G(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
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/decentralized_control_L_loop_gain.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:decentralized_control_L_loop_gain
#+caption: Obtained Loop Gain ([[./figs/decentralized_control_L_loop_gain.png][png]], [[./figs/decentralized_control_L_loop_gain.pdf][pdf]])
[[file:figs/decentralized_control_L_loop_gain.png]]
#+begin_src matlab :exports none :tangle no
isstable(feedback(G*Kl, eye(6), -1))
#+end_src
We add a minus sign to the controller as it is not included in the Simscape model.
#+begin_src matlab
Kl = -Kl;
#+end_src
** Simulation
#+begin_src matlab
initializeController('type', 'ref-track-L');
#+end_src
#+begin_src matlab
load('mat/conf_simulink.mat');
set_param(conf_simulink, 'StopTime', '2');
#+end_src
#+begin_src matlab
sim('nass_model');
#+end_src
#+begin_src matlab
decentralized_L = simout;
save('./mat/tomo_exp_decentalized.mat', 'decentralized_L', '-append');
#+end_src
** Results
The reference path and the position of the mobile platform are shown in Figure [[fig:decentralized_L_position_errors]].
#+begin_src matlab
load('./mat/experiment_tomography.mat', 'tomo_align_dist');
load('./mat/tomo_exp_decentalized.mat', 'decentralized_L');
#+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(decentralized_L.Em.En.Time, decentralized_L.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(decentralized_L.Em.En.Time, decentralized_L.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(decentralized_L.Em.En.Time, decentralized_L.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(decentralized_L.Em.En.Time, decentralized_L.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(decentralized_L.Em.En.Time, decentralized_L.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(decentralized_L.Em.En.Time, decentralized_L.Em.En.Data(:, 6), 'DisplayName', 'HAC-DVF')
hold off;
xlabel('Time [s]');
ylabel('Rz [rad]');
legend();
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
xlim([0.5, inf]);
#+end_src
#+header: :tangle no :exports results :results none :noweb yes
#+begin_src matlab :var filepath="figs/decentralized_L_position_errors.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
<<plt-matlab>>
#+end_src
#+name: fig:decentralized_L_position_errors
#+caption: Position Errors when using the Decentralized Control Architecture ([[./figs/decentralized_L_position_errors.png][png]], [[./figs/decentralized_L_position_errors.pdf][pdf]])
[[file:figs/decentralized_L_position_errors.png]]
* HAC-LAC (IFF) Decentralized Control
** Introduction :ignore:
We here add an Active Damping Loop (Integral Force Feedback) prior to using the Decentralized control architecture using $\bm{\mathcal{L}}$.
** Control Schematic
The control architecture is shown in Figure [[fig:decentralized_reference_tracking_L]].
The signals are:
- $\bm{r}_\mathcal{X}$: wanted position of the sample with respect to the granite
- $\bm{r}_{\mathcal{X}_n}$: wanted position of the sample with respect to the nano-hexapod
- $\bm{r}_\mathcal{L}$: wanted length of each of the nano-hexapod's legs
- $\bm{\tau}$: forces applied in each actuator
- $\bm{\mathcal{L}}$: measured displacement of each leg
- $\bm{\mathcal{X}}$: measured position of the sample with respect to the granite
#+begin_src latex :file decentralized_reference_tracking_iff_L.pdf
\begin{tikzpicture}
% Blocs
\node[block={3.0cm}{3.0cm}] (P) {};
\coordinate[] (inputF) at ($(P.south west)!0.5!(P.north west)$);
\coordinate[] (outputF) at ($(P.south east)!0.8!(P.north east)$);
\coordinate[] (outputX) at ($(P.south east)!0.5!(P.north east)$);
\coordinate[] (outputL) at ($(P.south east)!0.2!(P.north east)$);
\node[block, above= of P] (Kiff) {$\bm{K}_\text{IFF}$};
\node[addb, left= of inputF] (addF) {};
\node[block, left= of addF] (K) {$\bm{K}_\mathcal{L}$};
\node[addb={+}{}{}{}{-}, left= of K] (subr) {};
\node[block, align=center, left= of subr] (J) {Inverse\\Kinematics};
\node[block, align=center, left= of J] (Ex) {Compute\\Pos. Error};
% Connections and labels
\draw[->] (outputF) -- ++(1, 0) node[below left]{$\bm{\tau}_m$};
\draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east);
\draw[->] (Kiff.west) -| (addF.north);
\draw[->] (addF.east) -- (inputF) node[above left]{$\bm{\tau}$};
\draw[->] (outputL) -- ++(1, 0) node[above left]{$\bm{\mathcal{L}}$};
\draw[->] ($(outputL) + (0.6, 0)$)node[branch]{} -- ++(0, -1) -| (subr.south);
\draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_\mathcal{L}$};
\draw[->] (K.east) -- (addF.west);
\draw[->] (outputX) -- ++(1.8, 0) node[above left]{$\bm{\mathcal{X}}$};
\draw[->] ($(outputX) + (1.4, 0)$)node[branch]{} -- ++(0, -2.5) -| (Ex.south);
\draw[->] (Ex.east) -- (J.west) node[above left]{$\bm{r}_{\mathcal{X}_n}$};
\draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{\mathcal{L}}$};
\draw[<-] (Ex.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-1, 0);
\end{tikzpicture}
#+end_src
#+name: fig:decentralized_reference_tracking_L
#+caption: Decentralized control for reference tracking
#+RESULTS:
[[file:figs/decentralized_reference_tracking_L.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
simulinkproject('../');
#+end_src
#+begin_src matlab
open('nass_model.slx');
#+end_src
** Initialize the Simscape Model
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', 'ref-track-L');
Kl = tf(zeros(6));
#+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
** Initialization
#+begin_src matlab
initializeController('type', 'ref-track-iff-L');
K_iff = tf(zeros(6));
Kl = tf(zeros(6));
#+end_src
** Identification for IFF
#+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', [], 'Fnlm'); io_i = io_i + 1; % Force Sensors
%% Run the linearization
G_iff = linearize(mdl, io, 0);
G_iff.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
G_iff.OutputName = {'Fnlm1', 'Fnlm2', 'Fnlm3', 'Fnlm4', 'Fnlm5', 'Fnlm6'};
#+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(G_iff(i, i), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff(i, i), freqs, 'Hz'))), 'DisplayName', sprintf('$\\tau_{m_%i}/\\tau_%i$', 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', 'southwest');
linkaxes([ax1,ax2],'x');
#+end_src
** Integral Force Feedback Controller
#+begin_src matlab
w0 = 2*pi*50;
K_iff = -5000/s * (s/w0)/(1 + s/w0) * eye(6);
#+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_iff(i,i)*G_iff(i, i), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(K_iff(i,i)*G_iff(i, i), freqs, 'Hz'))), 'DisplayName', sprintf('$L_{\\tau,%i}$', 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', 'southwest');
linkaxes([ax1,ax2],'x');
#+end_src
#+begin_src matlab
K_iff = -K_iff;
#+end_src
** Identification of the damped plant
#+begin_src matlab
%% Name of the Simulink DehaezeFile
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, '/Controller/Reference-Tracking-IFF-L/Sum'], 1, 'openoutput'); io_i = io_i + 1; % Leg length error
%% Run the linearization
Gd = linearize(mdl, io, 0);
Gd.InputName = {'Fnl1', 'Fnl2', 'Fnl3', 'Fnl4', 'Fnl5', 'Fnl6'};
Gd.OutputName = {'El1', 'El2', 'El3', 'El4', 'El5', 'El6'};
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
ax1 = subplot(2, 2, 1);
hold on;
for i = 1:6
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(G( i, i), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',i);
plot(freqs, abs(squeeze(freqresp(Gd(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
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*angle(squeeze(freqresp(G( i, i), freqs, 'Hz'))), 'DisplayName', sprintf('$d\\mathcal{L}_%i/\\tau_%i$', i, i));
set(gca,'ColorOrderIndex',i);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gd(i, i), 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', 'northeast');
ax3 = subplot(2, 2, 2);
hold on;
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
plot(freqs, abs(squeeze(freqresp(Gd(i, j), freqs, 'Hz'))), '--', 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(Gd(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(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gd(i, j), freqs, 'Hz'))), '--', 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',1);
plot(freqs, 180/pi*angle(squeeze(freqresp(Gd(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
#+begin_src matlab :exports none
freqs = logspace(1, 4, 1000);
figure;
ax1 = subplot(2, 2, 1);
hold on;
for i = 1:6
plot(freqs, abs(squeeze(freqresp(G(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(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(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, abs(squeeze(freqresp(G(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(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]);
end
end
set(gca,'ColorOrderIndex',1);
plot(freqs, 180/pi*angle(squeeze(freqresp(G(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
#+begin_src matlab
wc = 2*pi*300;
h = 3;
Kl = diag(1./diag(abs(freqresp(Gd, wc)))) * ...
((s/(2*pi*20) + 1)/(s/(2*pi*20))) * ... % Pure Integrator
((s/(2*pi*50) + 1)/(s/(2*pi*50))) * ... % Integrator up to wc/2
1/h * (1 + s/wc*h)/(1 + s/wc/h) * ...
1/(1 + s/(2*wc)) * ...
1/(1 + s/(3*wc));
#+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(Kl(i, i)*Gd(i, i), freqs, 'Hz'))));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Loop Gain'); set(gca, 'XTickLabel',[]);
ax2 = subplot(2, 1, 2);
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(i, i)*Gd(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
isstable(feedback(Gd*Kl, eye(6), -1))
#+end_src
#+begin_src matlab
Kl = -Kl;
#+end_src
** Simulation
#+begin_src matlab
initializeController('type', 'ref-track-iff-L');
#+end_src
#+begin_src matlab
load('mat/conf_simulink.mat');
set_param(conf_simulink, 'StopTime', '2');
#+end_src
#+begin_src matlab
sim('nass_model');
#+end_src
#+begin_src matlab
decentralized_iff_L = simout;
save('./mat/tomo_exp_decentalized.mat', 'decentralized_iff_L', '-append');
#+end_src
** Results
#+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(decentralized_iff_L.Em.En.Time, decentralized_iff_L.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(decentralized_iff_L.Em.En.Time, decentralized_iff_L.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(decentralized_iff_L.Em.En.Time, decentralized_iff_L.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(decentralized_iff_L.Em.En.Time, decentralized_iff_L.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(decentralized_iff_L.Em.En.Time, decentralized_iff_L.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(decentralized_iff_L.Em.En.Time, decentralized_iff_L.Em.En.Data(:, 6), 'DisplayName', 'IFF + Decentralized')
hold off;
xlabel('Time [s]');
ylabel('Rz [rad]');
legend();
linkaxes([ax1,ax2,ax3,ax4,ax5,ax6],'x');
xlim([0.5, inf]);
#+end_src
* Conclusion
Reference in New Issue Copy Permalink