550 lines
17 KiB
Matlab
550 lines
17 KiB
Matlab
%% 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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% Simscape Model - Parameters
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open('gravimeter.slx')
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% Parameters
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l = 1.0; % Length of the mass [m]
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h = 1.7; % Height of the mass [m]
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la = l/2; % Position of Act. [m]
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ha = h/2; % Position of Act. [m]
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m = 400; % Mass [kg]
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I = 115; % Inertia [kg m^2]
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k = 15e3; % Actuator Stiffness [N/m]
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c = 0.03; % Actuator Damping [N/(m/s)]
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deq = 0.2; % Length of the actuators [m]
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g = 0; % Gravity [m/s2]
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% System Identification - Without Gravity
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%% Name of the Simulink File
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mdl = 'gravimeter';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
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io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
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G = linearize(mdl, io);
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G.InputName = {'F1', 'F2', 'F3'};
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G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
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% #+name: fig:gravimeter_plant_schematic
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% #+caption: Schematic of the gravimeter plant
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% #+RESULTS:
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% [[file:figs/gravimeter_plant_schematic.png]]
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% \begin{equation}
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% \bm{a} = \begin{bmatrix} a_{1x} \\ a_{1z} \\ a_{2x} \\ a_{2z} \end{bmatrix}
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% \end{equation}
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% \begin{equation}
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% \bm{\tau} = \begin{bmatrix}\tau_1 \\ \tau_2 \\ \tau_2 \end{bmatrix}
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% \end{equation}
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% We can check the poles of the plant:
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pole(G)
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% #+RESULTS:
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% #+begin_example
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% -0.000183495485977108 + 13.546056874877i
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% -0.000183495485977108 - 13.546056874877i
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% -7.49842878906757e-05 + 8.65934902322567i
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% -7.49842878906757e-05 - 8.65934902322567i
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% -1.33171230256362e-05 + 3.64924169037897i
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% -1.33171230256362e-05 - 3.64924169037897i
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% #+end_example
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% The plant as 6 states as expected (2 translations + 1 rotation)
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size(G)
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% #+RESULTS:
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% : State-space model with 4 outputs, 3 inputs, and 6 states.
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% The bode plot of all elements of the plant are shown in Figure [[fig:open_loop_tf]].
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freqs = logspace(-1, 2, 1000);
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figure;
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tiledlayout(4, 3, 'TileSpacing', 'None', 'Padding', 'None');
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for out_i = 1:4
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for in_i = 1:3
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nexttile;
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plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), '-');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlim([1e-1, 2e1]); ylim([1e-4, 1e0]);
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if in_i == 1
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ylabel('Amplitude [m/N]')
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else
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set(gca, 'YTickLabel',[]);
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end
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if out_i == 4
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xlabel('Frequency [Hz]')
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else
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set(gca, 'XTickLabel',[]);
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end
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end
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end
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% #+name: fig:gravimeter_decouple_jacobian
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% #+caption: Decoupled plant $\bm{G}_x$ using the Jacobian matrix $J$
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% #+RESULTS:
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% [[file:figs/gravimeter_decouple_jacobian.png]]
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% The jacobian corresponding to the sensors and actuators are defined below.
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Ja = [1 0 h/2
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0 1 -l/2
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1 0 -h/2
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0 1 0];
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Jt = [1 0 ha
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0 1 -la
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0 1 la];
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Gx = pinv(Ja)*G*pinv(Jt');
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Gx.InputName = {'Fx', 'Fz', 'My'};
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Gx.OutputName = {'Dx', 'Dz', 'Ry'};
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% The diagonal and off-diagonal elements of $G_x$ are shown in Figure [[fig:gravimeter_jacobian_plant]].
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freqs = logspace(-1, 2, 1000);
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figure;
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% Magnitude
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hold on;
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for i_in = 1:3
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for i_out = [1:i_in-1, i_in+1:3]
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plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
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'HandleVisibility', 'off');
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end
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end
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plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
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'DisplayName', '$G_x(i,j)\ i \neq j$');
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set(gca,'ColorOrderIndex',1)
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for i_in_out = 1:3
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plot(freqs, abs(squeeze(freqresp(Gx(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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legend('location', 'southeast');
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ylim([1e-8, 1e0]);
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% Real Approximation of $G$ at the decoupling frequency
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% <<sec:gravimeter_real_approx>>
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% Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_u(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
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wc = 2*pi*10; % Decoupling frequency [rad/s]
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H1 = evalfr(G, j*wc);
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% The real approximation is computed as follows:
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D = pinv(real(H1'*H1));
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H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
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% SVD Decoupling
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% <<sec:gravimeter_svd_decoupling>>
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% First, the Singular Value Decomposition of $H_1$ is performed:
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% \[ H_1 = U \Sigma V^H \]
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[U,~,V] = svd(H1);
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% #+name: fig:gravimeter_decouple_svd
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% #+caption: Decoupled plant $\bm{G}_{SVD}$ using the Singular Value Decomposition
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% #+RESULTS:
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% [[file:figs/gravimeter_decouple_svd.png]]
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% The decoupled plant is then:
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% \[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \]
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Gsvd = inv(U)*G*inv(V');
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% The diagonal and off-diagonal elements of the "SVD" plant are shown in Figure [[fig:gravimeter_svd_plant]].
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freqs = logspace(-1, 2, 1000);
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figure;
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% Magnitude
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hold on;
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for i_in = 1:3
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for i_out = [1:i_in-1, i_in+1:3]
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plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
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'HandleVisibility', 'off');
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end
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end
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plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
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'DisplayName', '$G_x(i,j)\ i \neq j$');
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set(gca,'ColorOrderIndex',1)
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for i_in_out = 1:3
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plot(freqs, abs(squeeze(freqresp(Gsvd(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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legend('location', 'southeast', 'FontSize', 8);
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ylim([1e-8, 1e0]);
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% TODO Verification of the decoupling using the "Gershgorin Radii"
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% <<sec:comp_decoupling>>
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% The "Gershgorin Radii" is computed for the coupled plant $G(s)$, for the "Jacobian plant" $G_x(s)$ and the "SVD Decoupled Plant" $G_{SVD}(s)$:
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% The "Gershgorin Radii" of a matrix $S$ is defined by:
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% \[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \]
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% This is computed over the following frequencies.
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freqs = logspace(-2, 2, 1000); % [Hz]
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% Gershgorin Radii for the coupled plant:
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Gr_coupled = zeros(length(freqs), size(Gu,2));
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H = abs(squeeze(freqresp(Gu, freqs, 'Hz')));
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for out_i = 1:size(Gu,2)
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Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
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end
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% Gershgorin Radii for the decoupled plant using SVD:
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Gr_decoupled = zeros(length(freqs), size(Gsvd,2));
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H = abs(squeeze(freqresp(Gsvd, freqs, 'Hz')));
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for out_i = 1:size(Gsvd,2)
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Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
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end
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% Gershgorin Radii for the decoupled plant using the Jacobian:
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Gr_jacobian = zeros(length(freqs), size(Gx,2));
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H = abs(squeeze(freqresp(Gx, freqs, 'Hz')));
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for out_i = 1:size(Gx,2)
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Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
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end
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figure;
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hold on;
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plot(freqs, Gr_coupled(:,1), 'DisplayName', 'Coupled');
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plot(freqs, Gr_decoupled(:,1), 'DisplayName', 'SVD');
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plot(freqs, Gr_jacobian(:,1), 'DisplayName', 'Jacobian');
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for in_i = 2:6
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set(gca,'ColorOrderIndex',1)
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plot(freqs, Gr_coupled(:,in_i), 'HandleVisibility', 'off');
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set(gca,'ColorOrderIndex',2)
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plot(freqs, Gr_decoupled(:,in_i), 'HandleVisibility', 'off');
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set(gca,'ColorOrderIndex',3)
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plot(freqs, Gr_jacobian(:,in_i), 'HandleVisibility', 'off');
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end
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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hold off;
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xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
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legend('location', 'northwest');
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ylim([1e-3, 1e3]);
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% TODO Obtained Decoupled Plants
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% <<sec:gravimeter_decoupled_plant>>
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% The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown in Figure [[fig:simscape_model_decoupled_plant_svd]].
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freqs = logspace(-1, 2, 1000);
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', '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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for i_in = 1:6
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for i_out = [1:i_in-1, i_in+1:6]
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plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
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'HandleVisibility', 'off');
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end
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end
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plot(freqs, abs(squeeze(freqresp(Gsvd(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
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'DisplayName', '$G_{SVD}(i,j),\ i \neq j$');
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set(gca,'ColorOrderIndex',1)
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for ch_i = 1:6
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plot(freqs, abs(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))), ...
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'DisplayName', sprintf('$G_{SVD}(%i,%i)$', ch_i, ch_i));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
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legend('location', 'northwest');
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ylim([1e-1, 1e5])
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% Phase
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ax2 = nexttile;
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hold on;
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for ch_i = 1:6
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))));
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end
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([-180, 180]);
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yticks([-180:90:360]);
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linkaxes([ax1,ax2],'x');
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% #+name: fig:simscape_model_decoupled_plant_svd
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% #+caption: Decoupled Plant using SVD
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% #+RESULTS:
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% [[file:figs/simscape_model_decoupled_plant_svd.png]]
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% Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant $G_x(s)$ using the Jacobian are shown in Figure [[fig:simscape_model_decoupled_plant_jacobian]].
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freqs = logspace(-1, 2, 1000);
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figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', '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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for i_in = 1:6
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for i_out = [1:i_in-1, i_in+1:6]
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plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
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'HandleVisibility', 'off');
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end
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end
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plot(freqs, abs(squeeze(freqresp(Gx(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
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'DisplayName', '$G_x(i,j),\ i \neq j$');
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set(gca,'ColorOrderIndex',1)
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plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$G_x(1,1) = A_x/F_x$');
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plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$G_x(2,2) = A_y/F_y$');
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plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = A_z/F_z$');
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plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$G_x(4,4) = A_{R_x}/M_x$');
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plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$G_x(5,5) = A_{R_y}/M_y$');
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plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$G_x(6,6) = A_{R_z}/M_z$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
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legend('location', 'northwest');
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ylim([1e-2, 2e6])
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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(Gx('Ax', 'Fx'), freqs, 'Hz'))));
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))));
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))));
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
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plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))));
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
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ylim([0, 180]);
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yticks([0:45:360]);
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linkaxes([ax1,ax2],'x');
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% #+name: fig:svd_control
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% #+caption: Control Diagram for the SVD control
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% #+RESULTS:
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% [[file:figs/svd_control.png]]
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% We choose the controller to be a low pass filter:
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% \[ K_c(s) = \frac{G_0}{1 + \frac{s}{\omega_0}} \]
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% $G_0$ is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to $\omega_c$
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wc = 2*pi*80; % Crossover Frequency [rad/s]
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w0 = 2*pi*0.1; % Controller Pole [rad/s]
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K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
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L_cen = K_cen*Gx;
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G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
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K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
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L_svd = K_svd*Gsvd;
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G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
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% The obtained diagonal elements of the loop gains are shown in Figure [[fig:gravimeter_comp_loop_gain_diagonal]].
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freqs = logspace(-1, 2, 1000);
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|
|
|
figure;
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tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
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|
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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(L_svd(1, 1), freqs, 'Hz'))), 'DisplayName', '$L_{SVD}(i,i)$');
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for i_in_out = 2:6
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|
set(gca,'ColorOrderIndex',1)
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|
plot(freqs, abs(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
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|
end
|
|
|
|
set(gca,'ColorOrderIndex',2)
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|
plot(freqs, abs(squeeze(freqresp(L_cen(1, 1), freqs, 'Hz'))), ...
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|
'DisplayName', '$L_{J}(i,i)$');
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|
for i_in_out = 2:6
|
|
set(gca,'ColorOrderIndex',2)
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|
plot(freqs, abs(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
|
|
legend('location', 'northwest');
|
|
ylim([5e-2, 2e3])
|
|
|
|
% Phase
|
|
ax2 = nexttile;
|
|
hold on;
|
|
for i_in_out = 1:6
|
|
set(gca,'ColorOrderIndex',1)
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))));
|
|
end
|
|
set(gca,'ColorOrderIndex',2)
|
|
for i_in_out = 1:6
|
|
set(gca,'ColorOrderIndex',2)
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
|
|
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
|
|
ylim([-180, 180]);
|
|
yticks([-180:90:360]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
|
|
% TODO Closed-Loop system Performances
|
|
% <<sec:gravimeter_closed_loop_results>>
|
|
|
|
% Let's first verify the stability of the closed-loop systems:
|
|
|
|
isstable(G_cen)
|
|
|
|
|
|
|
|
% #+RESULTS:
|
|
% : ans =
|
|
% : logical
|
|
% : 1
|
|
|
|
|
|
isstable(G_svd)
|
|
|
|
|
|
|
|
% #+RESULTS:
|
|
% : ans =
|
|
% : logical
|
|
% : 1
|
|
|
|
% The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:gravimeter_platform_simscape_cl_transmissibility]].
|
|
|
|
|
|
freqs = logspace(-2, 2, 1000);
|
|
|
|
figure;
|
|
tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');
|
|
|
|
ax1 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
|
|
plot(freqs, abs(squeeze(freqresp(G_cen('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
|
|
plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), '--', 'DisplayName', 'SVD');
|
|
set(gca,'ColorOrderIndex',1)
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
|
|
plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('$D_x/D_{w,x}$, $D_y/D_{w, y}$'); set(gca, 'XTickLabel',[]);
|
|
legend('location', 'southwest');
|
|
|
|
ax2 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Az', 'Dwz')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_cen('Az', 'Dwz')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_svd('Az', 'Dwz')/s^2, freqs, 'Hz'))), '--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('$D_z/D_{w,z}$'); set(gca, 'XTickLabel',[]);
|
|
|
|
ax3 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Arx', 'Rwx')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_cen('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz'))), '--');
|
|
set(gca,'ColorOrderIndex',1)
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Ary', 'Rwy')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))), '--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('$R_x/R_{w,x}$, $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
|
|
|
|
ax4 = nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G( 'Arz', 'Rwz')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_cen('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
|
|
plot(freqs, abs(squeeze(freqresp(G_svd('Arz', 'Rwz')/s^2, freqs, 'Hz'))), '--');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('$R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
|
|
|
|
linkaxes([ax1,ax2,ax3,ax4],'xy');
|
|
xlim([freqs(1), freqs(end)]);
|
|
ylim([1e-3, 1e2]);
|