phd-nass-rotating-3dof-model/matlab/rotating_5_rdc.m

%% Clear Workspace and Close figures
clear; close all; clc;

%% Intialize Laplace variable
s = zpk('s');

%% Path for functions, data and scripts
addpath('./mat/'); % Path for data
addpath('./src/'); % Path for Functions

%% Colors for the figures
colors = colororder;

%% Simscape model name
mdl = 'rotating_model';

%% Load "Generic" system dynamics
load('rotating_generic_plants.mat', 'Gs', 'Wzs');

% Decentralized Relative Damping Control
% The transfer functions from $[F_u,\ F_v]$ to $[d_u,\ d_v]$ is identified and shown in Figure ref:fig:rotating_rdc_plant_effect_rot for several rotating velocities.


%% Bode plot of the direct and coupling term for the "relative damping control" plant - Effect of rotation
freqs = logspace(-2, 1, 1000);

figure;
tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');

% Magnitude
ax1 = nexttile([2, 1]);
hold on;
for i = 1:length(Wzs)
    plot(freqs, abs(squeeze(freqresp(Gs{i}('du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:))
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude [N/N]');
title('Direct terms: $d_u/F_u$, $d_v/F_v$');
ylim([1e-3, 1e2]);

ax2 = nexttile([2, 1]);
hold on;
for i = 1:length(Wzs)
    plot(freqs, abs(squeeze(freqresp(Gs{i}('dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:), ...
         'DisplayName', sprintf('$\\Omega = %.1f \\omega_0$', Wzs(i)))
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); set(gca, 'YTickLabel',[]);
title('Coupling Terms: $d_u/F_v$, $d_v/F_u$');
ldg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
ldg.ItemTokenSize = [10, 1];
ylim([1e-3, 1e2]);

ax3 = nexttile;
hold on;
for i = 1:length(Wzs)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:))
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
yticks(-180:90:180);
ylim([-180 180]);
xticks([1e-2,1e-1,1,1e1])
xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})

ax4 = nexttile;
hold on;
for i = 1:length(Wzs)
    plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(i,:));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); set(gca, 'YTickLabel',[]);
yticks(-180:90:180);
ylim([-180 180]);
xticks([1e-2,1e-1,1,1e1])
xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})

linkaxes([ax1,ax2,ax3,ax4],'x');
xlim([freqs(1), freqs(end)]);

linkaxes([ax1,ax2],'y');



% #+name: fig:rotating_rdc_plant_effect_rot
% #+caption: Bode plot of the direct and coupling term for the "relative damping control" plant - Effect of rotation
% #+RESULTS:
% [[file:figs/rotating_rdc_plant_effect_rot.png]]

% In order to see if large damping can be added with Relative Damping Control, the root locus is computed (Figure ref:fig:rotating_rdc_root_locus).
% The closed-loop system is unconditionally stable and the poles can be damped as much as wanted.


%% Root Locus for Relative Damping Control
Krdc = s*eye(2); % Relative damping controller

gains = logspace(-2, 2, 300); % Tested gains
Wz_i = [1,3,4];

figure;
hold on;
for i = 1:length(Wz_i)
    plot(real(pole(Gs{Wz_i(i)}({'du', 'dv'}, {'Fu', 'Fv'})*Krdc)),  imag(pole(Gs{Wz_i(i)}({'du', 'dv'}, {'Fu', 'Fv'})*Krdc)), 'x', 'color', colors(i,:), ...
         'DisplayName', sprintf('$\\Omega = %.1f \\omega_0 $', Wzs(Wz_i(i))),'MarkerSize',8);
    plot(real(tzero(Gs{Wz_i(i)}({'du', 'dv'}, {'Fu', 'Fv'})*Krdc)),  imag(tzero(Gs{Wz_i(i)}({'du', 'dv'}, {'Fu', 'Fv'})*Krdc)), 'o', 'color', colors(i,:), ...
         'HandleVisibility', 'off','MarkerSize',8);
    for g = gains
        cl_poles = pole(feedback(Gs{Wz_i(i)}({'du', 'dv'}, {'Fu', 'Fv'}), g*Krdc, -1));
        plot(real(cl_poles), imag(cl_poles), '.', 'color', colors(i,:), ...
             'HandleVisibility', 'off','MarkerSize',4);
    end
end
hold off;
axis square;
xlim([-1.8, 0.2]); ylim([0, 2]);
xticks([-1, 0])
xticklabels({'-$\omega_0$', '$0$'})
yticks([0, 1, 2])
yticklabels({'$0$', '$\omega_0$', '$2 \omega_0$'})

xlabel('Real Part'); ylabel('Imaginary Part');
leg = legend('location', 'northwest', 'FontSize', 8);
leg.ItemTokenSize(1) = 8;

% Damped Plant
% Let's select a reasonable "Relative Damping Control" gain, and compute the closed-loop damped system.
% The open-loop and damped plants are compared in Figure ref:fig:rotating_rdc_damped_plant.

% The rotating aspect does not add any complexity for the use of Relative Damping Control.
% It does not increase the low frequency coupling as compared to Integral Force Feedback.


i = 2;

%% Relative Damping Controller
Krdc = 2*s*eye(2);
Krdc.InputName = {'Du', 'Dv'};
Krdc.OutputName = {'Fu', 'Fv'};

%% Compute the damped plant
G_cl_rdc = feedback(Gs{i}, Krdc, 'name');

%% Damped plant using Relative Damping Control
freqs = logspace(-3, 2, 1000);

figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');

% Magnitude
ax1 = nexttile([2, 1]);
hold on;
plot(freqs, abs(squeeze(freqresp(Gs{i}(   'Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', zeros(1,3), ...
    'DisplayName', 'OL - $G_d(1,1)$')
plot(freqs, abs(squeeze(freqresp(G_cl_rdc('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(1,:), ...
    'DisplayName', 'RDC - $G_d(1,1)$')
plot(freqs, abs(squeeze(freqresp(Gs{i}(   'Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [zeros(1,3), 0.5], ...
    'DisplayName', 'OL - $G_d(2,1)$')
plot(freqs, abs(squeeze(freqresp(G_cl_rdc('Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(1,:), 0.5], ...
    'DisplayName', 'RDC - $G_d(2,1)$')
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
ldg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
ldg.ItemTokenSize(1) = 20;
ylim([1e-6, 1e2]);

ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{i}('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', zeros(1,3))
plot(freqs, 180/pi*angle(squeeze(freqresp(G_cl_rdc('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', colors(1,:))
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
yticks(-180:90:180);
ylim([-180 180]);

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