475 lines
18 KiB
Mathematica
475 lines
18 KiB
Mathematica
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%% Clear Workspace and Close figures
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clear; close all; clc;
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%% Intialize Laplace variable
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s = zpk('s');
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%% Path for functions, data and scripts
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addpath('./mat/'); % Path for data
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addpath('./src/'); % Path for Functions
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%% Colors for the figures
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colors = colororder;
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%% Simscape model name
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mdl = 'rotating_model';
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%% Load "Generic" system dynamics
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load('rotating_generic_plants.mat', 'Gs', 'Wzs');
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% Modified Integral Force Feedback Controller
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% The Integral Force Feedback Controller is modified such that instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used:
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% \begin{equation}
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% K_{\text{IFF}}(s) = g\frac{1}{\omega_i + s} \begin{bmatrix}
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% 1 & 0 \\
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% 0 & 1
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% \end{bmatrix}
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% \end{equation}
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% where $\omega_i$ characterize down to which frequency the signal is integrated.
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% Let's arbitrary choose the following control parameters:
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%% Modified IFF - parameters
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g = 2; % Controller gain
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wi = 0.1; % HPF Cut-Off frequency [rad/s]
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Kiff = (g/s)*eye(2); % Pure IFF
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Kiff_hpf = (g/(wi+s))*eye(2); % IFF with added HPF
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% The loop gains ($K_F(s)$ times the direct dynamics $f_u/F_u$) with and without the added HPF are shown in Figure ref:fig:rotating_iff_modified_loop_gain.
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% The effect of the added HPF limits the low frequency gain to finite values as expected.
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%% Loop gain for the IFF with pure integrator and modified IFF with added high pass filter
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freqs = logspace(-2, 1, 1000);
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Wz_i = 2;
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
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% Magnitude
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ax1 = nexttile([2, 1]);
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hold on;
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plot(freqs, abs(squeeze(freqresp(Gs{Wz_i}('fu', 'Fu')*Kiff(1,1), freqs, 'rad/s'))))
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plot(freqs, abs(squeeze(freqresp(Gs{Wz_i}('fu', 'Fu')*Kiff_hpf(1,1), freqs, 'rad/s'))))
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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set(gca, 'XTickLabel',[]); ylabel('Loop Gain');
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% Phase
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ax2 = nexttile;
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Wz_i}('fu', 'Fu')*Kiff(1,1), freqs, 'rad/s'))), 'DisplayName', 'IFF')
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{Wz_i}('fu', 'Fu')*Kiff_hpf(1,1), freqs, 'rad/s'))), 'DisplayName', 'IFF + HPF')
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [rad/s]'); ylabel('Phase [deg]');
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yticks(-180:90:180);
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ylim([-180 180]);
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xticks([1e-2,1e-1,1,1e1])
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xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
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leg = legend('location', 'southwest', 'FontSize', 8);
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leg.ItemTokenSize(1) = 20;
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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% #+name: fig:rotating_iff_modified_loop_gain
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% #+caption: Loop gain for the IFF with pure integrator and modified IFF with added high pass filter ($\Omega = 0.1\omega_0$)
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% #+RESULTS:
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% [[file:figs/rotating_iff_modified_loop_gain.png]]
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% The Root Locus plots for the decentralized IFF with and without the HPF are displayed in Figure ref:fig:rotating_iff_root_locus_hpf.
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% With the added HPF, the poles of the closed loop system are shown to be *stable up to some value of the gain* $g_\text{max}$ given by equation eqref:eq:gmax_iff_hpf.
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% #+name: eq:gmax_iff_hpf
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% \begin{equation}
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% \boxed{g_{\text{max}} = \omega_i \left( \frac{{\omega_0}^2}{\Omega^2} - 1 \right)}
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% \end{equation}
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% It is interesting to note that $g_{\text{max}}$ also corresponds to the controller gain at which the low frequency loop gain (Figure ref:fig:rotating_iff_modified_loop_gain) reaches one.
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%% Root Locus for the initial IFF and the modified IFF
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gains = logspace(-2, 4, 200);
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figure;
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tiledlayout(1, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
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ax1 = nexttile([1,2]);
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hold on;
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for g = gains
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clpoles = pole(feedback(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff));
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plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), 'HandleVisibility', 'off','MarkerSize',4);
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end
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for g = gains
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clpoles = pole(feedback(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf));
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plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:), 'HandleVisibility', 'off','MarkerSize',4);
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end
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% Pure Integrator
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plot(real(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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imag(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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'x', 'color', colors(1,:), 'DisplayName', 'IFF','MarkerSize',8);
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plot(real(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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imag(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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'o', 'color', colors(1,:), 'HandleVisibility', 'off','MarkerSize',8);
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% Modified IFF
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plot(real(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
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imag(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
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'x', 'color', colors(2,:), 'DisplayName', 'IFF + HPF','MarkerSize',8);
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plot(real(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
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imag(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
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'o', 'color', colors(2,:), 'HandleVisibility', 'off','MarkerSize',8);
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hold off;
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axis square;
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leg = legend('location', 'northwest', 'FontSize', 8);
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leg.ItemTokenSize(1) = 8;
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xlabel('Real Part'); ylabel('Imaginary Part');
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xlim([-2.25, 0.25]); ylim([-1.25, 1.25]);
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xticks([-2, -1, 0])
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xticklabels({'$-2\omega_0$', '$-\omega_0$', '$0$'})
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yticks([-1, 0, 1])
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yticklabels({'$-\omega_0$', '$0$', '$\omega_0$'})
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ax2 = nexttile();
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hold on;
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for g = gains
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clpoles = pole(feedback(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff));
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plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:),'MarkerSize',4);
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end
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for g = gains
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clpoles = pole(feedback(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff_hpf));
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plot(real(clpoles), imag(clpoles), '.', 'color', colors(2,:),'MarkerSize',4);
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end
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% Pure Integrator
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plot(real(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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imag(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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'x', 'color', colors(1,:),'MarkerSize',8);
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plot(real(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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imag(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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'o', 'color', colors(1,:),'MarkerSize',8);
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% Modified IFF
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plot(real(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
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imag(pole(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
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'x', 'color', colors(2,:),'MarkerSize',8);
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plot(real(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
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imag(tzero(Gs{Wz_i}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff_hpf)), ...
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'o', 'color', colors(2,:),'MarkerSize',8);
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hold off;
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axis square;
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xlabel('Real Part'); ylabel('Imaginary Part');
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title('Zoom near the origin');
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xlim([-0.15, 0.1]); ylim([-0.15, 0.15]);
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xticks([-0.1, 0, 0.1])
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xticklabels({'$-0.1\omega_0$', '$0$', '$0.1\omega_0$'})
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yticks([-0.1, 0, 0.1])
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yticklabels({'$-0.1\omega_0$', '$0$', '$0.1\omega_0$'})
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% Optimal IFF with HPF parameters $\omega_i$ and $g$
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% Two parameters can be tuned for the modified controller in equation eqref:eq:iff_lhf: the gain $g$ and the pole's location $\omega_i$.
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% The optimal values of $\omega_i$ and $g$ are here considered as the values for which the damping of all the closed-loop poles are simultaneously maximized.
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% In order to visualize how $\omega_i$ does affect the attainable damping, the Root Locus plots for several $\omega_i$ are displayed in Figure ref:fig:rotating_root_locus_iff_modified_effect_wi.
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% It is shown that even though small $\omega_i$ seem to allow more damping to be added to the suspension modes, the control gain $g$ may be limited to small values due to equation eqref:eq:gmax_iff_hpf.
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%% High Pass Filter Cut-Off Frequency
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wis = [0.01, 0.1, 0.5, 1]*Wzs(2); % [rad/s]
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%% Root Locus for the initial IFF and the modified IFF
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gains = logspace(-2, 4, 200);
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figure;
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tiledlayout(1, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
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ax1 = nexttile();
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hold on;
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for wi_i = 1:length(wis)
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wi = wis(wi_i);
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Kiff = (1/(wi+s))*eye(2);
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L(wi_i) = plot(nan, nan, '.', 'color', colors(wi_i,:), 'DisplayName', sprintf('$\\omega_i = %.2f \\omega_0$', wi./Wzs(2)));
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for g = gains
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clpoles = pole(feedback(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff));
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plot(real(clpoles), imag(clpoles), '.', 'color', colors(wi_i,:),'MarkerSize',4);
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end
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plot(real(pole(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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imag(pole(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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'x', 'color', colors(wi_i,:),'MarkerSize',8);
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plot(real(tzero(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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imag(tzero(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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'o', 'color', colors(wi_i,:),'MarkerSize',8);
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end
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hold off;
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axis square;
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xlim([-2.3, 0.1]); ylim([-1.2, 1.2]);
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xticks([-2:1:2]); yticks([-2:1:2]);
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leg = legend(L, 'location', 'northwest', 'FontSize', 8);
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leg.ItemTokenSize(1) = 8;
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xlabel('Real Part'); ylabel('Imaginary Part');
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clear L
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xlim([-2.25, 0.25]); ylim([-1.25, 1.25]);
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xticks([-2, -1, 0])
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xticklabels({'$-2\omega_0$', '$-\omega_0$', '$0$'})
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yticks([-1, 0, 1])
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yticklabels({'$-\omega_0$', '$0$', '$\omega_0$'})
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ax2 = nexttile();
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hold on;
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for wi_i = 1:length(wis)
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wi = wis(wi_i);
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Kiff = (1/(wi+s))*eye(2);
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L(wi_i) = plot(nan, nan, '.', 'color', colors(wi_i,:));
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for g = gains
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clpoles = pole(feedback(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'}), g*Kiff));
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plot(real(clpoles), imag(clpoles), '.', 'color', colors(wi_i,:),'MarkerSize',4);
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end
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plot(real(pole(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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imag(pole(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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'x', 'color', colors(wi_i,:),'MarkerSize',8);
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plot(real(tzero(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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imag(tzero(Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'})*Kiff)), ...
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'o', 'color', colors(wi_i,:),'MarkerSize',8);
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end
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hold off;
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axis square;
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xlim([-2.3, 0.1]); ylim([-1.2, 1.2]);
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xticks([-2:1:2]); yticks([-2:1:2]);
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xlabel('Real Part'); ylabel('Imaginary Part');
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title('Zoom near the origin');
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clear L
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xlim([-0.15, 0.1]); ylim([-0.15, 0.15]);
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xticks([-0.1, 0, 0.1])
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xticklabels({'$-0.1\omega_0$', '$0$', '$0.1\omega_0$'})
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yticks([-0.1, 0, 0.1])
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yticklabels({'$-0.1\omega_0$', '$0$', '$0.1\omega_0$'})
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% #+name: fig:rotating_root_locus_iff_modified_effect_wi
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% #+caption: Root Locus for several high pass filter cut-off frequency
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% #+RESULTS:
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% [[file:figs/rotating_root_locus_iff_modified_effect_wi.png]]
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% In order to study this trade off, the attainable closed-loop damping ratio $\xi_{\text{cl}}$ is computed as a function of $\omega_i/\omega_0$.
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% The gain $g_{\text{opt}}$ at which this maximum damping is obtained is also displayed and compared with the gain $g_{\text{max}}$ at which the system becomes unstable (Figure ref:fig:rotating_iff_hpf_optimal_gain).
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% Three regions can be observed:
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% - $\omega_i/\omega_0 < 0.02$: the added damping is limited by the maximum allowed control gain $g_{\text{max}}$
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% - $0.02 < \omega_i/\omega_0 < 0.2$: the attainable damping ratio is maximized and is reached for $g \approx 2$
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% - $0.2 < \omega_i/\omega_0$: the added damping decreases as $\omega_i/\omega_0$ increases.
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%% Compute the optimal control gain
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wis = logspace(-2, 1, 100); % [rad/s]
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opt_xi = zeros(1, length(wis)); % Optimal simultaneous damping
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opt_gain = zeros(1, length(wis)); % Corresponding optimal gain
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for wi_i = 1:length(wis)
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wi = wis(wi_i);
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Kiff = 1/(s + wi)*eye(2);
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fun = @(g)computeSimultaneousDamping(g, Gs{2}({'fu', 'fv'}, {'Fu', 'Fv'}), Kiff);
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[g_opt, xi_opt] = fminsearch(fun, 0.5*wi*((1/Wzs(2))^2 - 1));
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opt_xi(wi_i) = 1/xi_opt;
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opt_gain(wi_i) = g_opt;
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end
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|
|
||
|
|
%% Attainable damping ratio as a function of wi/w0. Corresponding control gain g_opt and g_max are also shown
|
||
|
|
figure;
|
||
|
|
yyaxis left
|
||
|
|
plot(wis, opt_xi, '-', 'DisplayName', '$\xi_{cl}$');
|
||
|
|
set(gca, 'YScale', 'lin');
|
||
|
|
ylim([0,1]);
|
||
|
|
ylabel('Damping Ratio $\xi$');
|
||
|
|
|
||
|
|
yyaxis right
|
||
|
|
hold on;
|
||
|
|
plot(wis, opt_gain, '-', 'DisplayName', '$g_{opt}$');
|
||
|
|
plot(wis, wis*((1/Wzs(2))^2 - 1), '--', 'DisplayName', '$g_{max}$');
|
||
|
|
set(gca, 'YScale', 'lin');
|
||
|
|
ylim([0,10]);
|
||
|
|
ylabel('Controller gain $g$');
|
||
|
|
|
||
|
|
xlabel('$\omega_i/\omega_0$');
|
||
|
|
set(gca, 'XScale', 'log');
|
||
|
|
legend('location', 'northeast', 'FontSize', 8);
|
||
|
|
|
||
|
|
% Obtained Damped Plant
|
||
|
|
|
||
|
|
% Let's choose $\omega_i = 0.1 \cdot \omega_0$ and compute the damped plant.
|
||
|
|
% The undamped and damped plants are compared in Figure ref:fig:rotating_iff_hpf_damped_plant in blue and red respectively.
|
||
|
|
% A well damped plant is indeed obtained.
|
||
|
|
|
||
|
|
% However, the magnitude of the coupling term ($d_v/F_u$) is larger then IFF is applied.
|
||
|
|
|
||
|
|
|
||
|
|
%% Compute damped plant
|
||
|
|
wi = 0.1; % [rad/s]
|
||
|
|
g = 2; % Gain
|
||
|
|
Kiff_hpf = (g/(wi+s))*eye(2); % IFF with added HPF
|
||
|
|
Kiff_hpf.InputName = {'fu', 'fv'};
|
||
|
|
Kiff_hpf.OutputName = {'Fu', 'Fv'};
|
||
|
|
|
||
|
|
G_iff_hpf = feedback(Gs{2}, Kiff_hpf, 'name');
|
||
|
|
|
||
|
|
isstable(G_iff_hpf) % Verify stability
|
||
|
|
|
||
|
|
%% Damped plant with IFF and added HPF - Transfer function from $F_u$ to $d_u$
|
||
|
|
freqs = logspace(-2, 1, 1000);
|
||
|
|
|
||
|
|
figure;
|
||
|
|
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||
|
|
|
||
|
|
% Magnitude
|
||
|
|
ax1 = nexttile([2, 1]);
|
||
|
|
hold on;
|
||
|
|
plot(freqs, abs(squeeze(freqresp(Gs{2}('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', [zeros(1,3)], ...
|
||
|
|
'DisplayName', '$d_u/F_u$, OL')
|
||
|
|
plot(freqs, abs(squeeze(freqresp(G_iff_hpf('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(1,:)], ...
|
||
|
|
'DisplayName', '$d_u/F_u$, IFF')
|
||
|
|
plot(freqs, abs(squeeze(freqresp(Gs{2}('Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [zeros(1,3), 0.5], ...
|
||
|
|
'DisplayName', '$d_v/F_u$, OL')
|
||
|
|
plot(freqs, abs(squeeze(freqresp(G_iff_hpf('Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(1,:), 0.5], ...
|
||
|
|
'DisplayName', '$d_v/F_u$, IFF')
|
||
|
|
hold off;
|
||
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||
|
|
set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
|
||
|
|
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
|
||
|
|
leg.ItemTokenSize(1) = 20;
|
||
|
|
|
||
|
|
ax2 = nexttile;
|
||
|
|
hold on;
|
||
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs{2}('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', zeros(1,3))
|
||
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(G_iff_hpf('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]);
|
||
|
|
xticks([1e-2,1e-1,1,1e1])
|
||
|
|
xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
|
||
|
|
|
||
|
|
linkaxes([ax1,ax2],'x');
|
||
|
|
xlim([freqs(1), freqs(end)]);
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
% #+name: fig:rotating_iff_hpf_damped_plant
|
||
|
|
% #+caption: Damped plant with IFF and added HPF - Transfer function from $F_u$ to $d_u$, $\omega_i = 0.1 \cdot \omega_0$, $\Omega = 0.1 \cdot \omega_0$
|
||
|
|
% #+RESULTS:
|
||
|
|
% [[file:figs/rotating_iff_hpf_damped_plant.png]]
|
||
|
|
|
||
|
|
% In order to study how $\omega_i$ affects the coupling of the damped plant, the closed-loop plant is identified for several $\omega_i$.
|
||
|
|
% The direct and coupling terms of the plants are shown in Figure ref:fig:rotating_iff_hpf_damped_plant_effect_wi_coupling (left) and the ratio between the two (i.e. the coupling ratio) is shown in Figure ref:fig:rotating_iff_hpf_damped_plant_effect_wi_coupling (right).
|
||
|
|
|
||
|
|
% The coupling ratio is decreasing as $\omega_i$ increases.
|
||
|
|
% There is therefore a *trade-off between achievable damping and coupling ratio* for the choice of $\omega_i$.
|
||
|
|
% The same trade-off can be seen between achievable damping and loss of compliance at low frequency (see Figure ref:fig:rotating_iff_hpf_effect_wi_compliance).
|
||
|
|
|
||
|
|
|
||
|
|
%% Compute damped plant
|
||
|
|
wis = [0.03, 0.1, 0.5]; % [rad/s]
|
||
|
|
g = 2; % Gain
|
||
|
|
|
||
|
|
Gs_iff_hpf = {};
|
||
|
|
|
||
|
|
for i = 1:length(wis)
|
||
|
|
Kiff_hpf = (g/(wis(i)+s))*eye(2); % IFF with added HPF
|
||
|
|
Kiff_hpf.InputName = {'fu', 'fv'};
|
||
|
|
Kiff_hpf.OutputName = {'Fu', 'Fv'};
|
||
|
|
|
||
|
|
Gs_iff_hpf(i) = {feedback(Gs{2}, Kiff_hpf, 'name')};
|
||
|
|
end
|
||
|
|
|
||
|
|
%% Effect of $\omega_i$ on the damped plant coupling
|
||
|
|
freqs = logspace(-2, 1, 1000);
|
||
|
|
|
||
|
|
figure;
|
||
|
|
tiledlayout(3, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||
|
|
|
||
|
|
% Magnitude
|
||
|
|
ax1 = nexttile([2, 1]);
|
||
|
|
hold on;
|
||
|
|
for i = 1:length(wis)
|
||
|
|
plot(freqs, abs(squeeze(freqresp(Gs_iff_hpf{i}('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(i,:)], ...
|
||
|
|
'DisplayName', sprintf('$d_u/F_u$, $\\omega_i = %.2f \\omega_0$', wis(i)))
|
||
|
|
plot(freqs, abs(squeeze(freqresp(Gs_iff_hpf{i}('Dv', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(i,:), 0.5], ...
|
||
|
|
'DisplayName', sprintf('$d_v/F_u$, $\\omega_i = %.2f \\omega_0$', wis(i)))
|
||
|
|
end
|
||
|
|
hold off;
|
||
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||
|
|
set(gca, 'XTickLabel',[]); ylabel('Magnitude [m/N]');
|
||
|
|
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
|
||
|
|
leg.ItemTokenSize(1) = 20;
|
||
|
|
|
||
|
|
ax3 = nexttile([3,1]);
|
||
|
|
hold on;
|
||
|
|
for i = 1:length(wis)
|
||
|
|
plot(freqs, abs(squeeze(freqresp(Gs_iff_hpf{i}('Dv', 'Fu')/Gs_iff_hpf{i}('Du', 'Fu'), freqs, 'rad/s'))), '-', 'color', [colors(i,:)], ...
|
||
|
|
'DisplayName', sprintf('$\\omega_i = %.2f \\omega_0$', wis(i)))
|
||
|
|
end
|
||
|
|
hold off;
|
||
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||
|
|
xlabel('Frequency [rad/s]'); ylabel('Coupling Ratio');
|
||
|
|
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
|
||
|
|
leg.ItemTokenSize(1) = 20;
|
||
|
|
xticks([1e-2,1e-1,1,1e1])
|
||
|
|
xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
|
||
|
|
|
||
|
|
ax2 = nexttile;
|
||
|
|
hold on;
|
||
|
|
for i = 1:length(wis)
|
||
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(Gs_iff_hpf{i}('Du', 'Fu'), freqs, 'rad/s'))), '-')
|
||
|
|
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({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
|
||
|
|
|
||
|
|
linkaxes([ax1,ax2],'x');
|
||
|
|
xlim([freqs(1), freqs(end)]);
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
% #+name: fig:rotating_iff_hpf_damped_plant_effect_wi_coupling
|
||
|
|
% #+caption: Effect of $\omega_i$ on the damped plant coupling
|
||
|
|
% #+RESULTS:
|
||
|
|
% [[file:figs/rotating_iff_hpf_damped_plant_effect_wi_coupling.png]]
|
||
|
|
|
||
|
|
|
||
|
|
%% Effect of $\omega_i$ on the obtained compliance
|
||
|
|
freqs = logspace(-2, 1, 1000);
|
||
|
|
|
||
|
|
figure;
|
||
|
|
tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
||
|
|
|
||
|
|
% Magnitude
|
||
|
|
ax1 = nexttile();
|
||
|
|
hold on;
|
||
|
|
for i = 1:length(wis)
|
||
|
|
plot(freqs, abs(squeeze(freqresp(Gs_iff_hpf{i}('Du', 'Fdx'), freqs, 'rad/s'))), '-', 'color', [colors(i,:)], ...
|
||
|
|
'DisplayName', sprintf('$d_{x}/F_{dx}$, $\\omega_i = %.2f \\omega_0$', wis(i)))
|
||
|
|
end
|
||
|
|
plot(freqs, abs(squeeze(freqresp(Gs{2}('Du', 'Fdx'), freqs, 'rad/s'))), 'k--', ...
|
||
|
|
'DisplayName', '$d_{x}/F_{dx}$, OL')
|
||
|
|
hold off;
|
||
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||
|
|
xlabel('Frequency [rad/s]'); ylabel('Compliance [m/N]');
|
||
|
|
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
|
||
|
|
leg.ItemTokenSize(1) = 20;
|
||
|
|
xticks([1e-2,1e-1,1,1e1])
|
||
|
|
xticklabels({'$0.01 \omega_0$', '$0.1 \omega_0$', '$\omega_0$', '$10 \omega_0$'})
|