Add figure about RGA
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figs/simscape_model_rga.pdf
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figs/simscape_model_rga.png
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@ -281,7 +281,6 @@ for in_i = 2:6
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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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plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
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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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@ -442,7 +442,6 @@ This is computed over the following frequencies.
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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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plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
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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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@ -1382,6 +1381,10 @@ The analysis of the SVD control applied to the Stewart platform is performed in
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addpath('STEP');
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#+end_src
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#+begin_src matlab
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freqs = logspace(-1, 2, 1000);
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#+end_src
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** Jacobian :noexport:
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First, the position of the "joints" (points of force application) are estimated and the Jacobian computed.
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#+begin_src matlab :tangle no
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@ -1413,7 +1416,7 @@ First, the position of the "joints" (points of force application) are estimated
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J = [As' , cross(Ab, As)'];
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save('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
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save('stewart_platform/jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
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#+end_src
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** Simscape Model - Parameters
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@ -1433,6 +1436,11 @@ Definition of spring parameters:
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cz = 0.025;
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#+end_src
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We suppose the sensor is perfectly positioned.
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#+begin_src matlab
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sens_pos_error = zeros(3,1);
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#+end_src
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Gravity:
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#+begin_src matlab
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g = 0;
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@ -1440,7 +1448,7 @@ Gravity:
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We load the Jacobian (previously computed from the geometry):
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#+begin_src matlab
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load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
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load('jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
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#+end_src
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We initialize other parameters:
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@ -1524,8 +1532,6 @@ The elements of the transfer matrix $\bm{G}$ corresponding to the transfer funct
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One can easily see that the system is strongly coupled.
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#+begin_src matlab :exports none
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freqs = logspace(-1, 2, 1000);
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figure;
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% Magnitude
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@ -1563,6 +1569,21 @@ One can easily see that the system is strongly coupled.
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Consider the control architecture shown in Figure [[fig:plant_decouple_jacobian]].
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The Jacobian matrix is used to transform forces/torques applied on the top platform to the equivalent forces applied by each actuator.
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The Jacobian matrix is computed from the geometry of the platform (position and orientation of the actuators).
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(J, {}, {}, ' %.3f ');
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#+end_src
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#+caption: Computed Jacobian Matrix
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#+RESULTS:
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| 0.811 | 0.0 | 0.584 | -0.018 | -0.008 | 0.025 |
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| -0.406 | -0.703 | 0.584 | -0.016 | -0.012 | -0.025 |
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| -0.406 | 0.703 | 0.584 | 0.016 | -0.012 | 0.025 |
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| 0.811 | 0.0 | 0.584 | 0.018 | -0.008 | -0.025 |
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| -0.406 | -0.703 | 0.584 | 0.002 | 0.019 | 0.025 |
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| -0.406 | 0.703 | 0.584 | -0.002 | 0.019 | -0.025 |
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#+begin_src latex :file plant_decouple_jacobian.pdf :tangle no :exports results
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\begin{tikzpicture}
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\node[block] (G) {$G_u$};
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@ -1633,6 +1654,7 @@ This can be verified below where only the real value of $G_u(\omega_c)$ is shown
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data2orgtable(real(evalfr(Gu, j*wc)), {}, {}, ' %.1f ');
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#+end_src
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#+caption: Real part of $G$ at the decoupling frequency $\omega_c$
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#+RESULTS:
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| 4.4 | -2.1 | -2.1 | 4.4 | -2.4 | -2.4 |
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| -0.2 | -3.9 | 3.9 | 0.2 | -3.8 | 3.8 |
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@ -1651,6 +1673,32 @@ First, the Singular Value Decomposition of $H_1$ is performed:
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[U,~,V] = svd(H1);
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(U, {}, {}, ' %.1g ');
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#+end_src
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#+caption: Obtained matrix $U$
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#+RESULTS:
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| -0.005 | 7e-06 | 6e-11 | -3e-06 | -1 | 0.1 |
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| -7e-06 | -0.005 | -9e-09 | -5e-09 | -0.1 | -1 |
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| 4e-08 | -2e-10 | -6e-11 | -1 | 3e-06 | -3e-07 |
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| -0.002 | -1 | -5e-06 | 2e-10 | 0.0006 | 0.005 |
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| 1 | -0.002 | -1e-08 | 2e-08 | -0.005 | 0.0006 |
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| -4e-09 | 5e-06 | -1 | 6e-11 | -2e-09 | -1e-08 |
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(V, {}, {}, ' %.1g ');
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#+end_src
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#+caption: Obtained matrix $V$
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#+RESULTS:
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| -0.2 | 0.5 | -0.4 | -0.4 | -0.6 | -0.2 |
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| -0.3 | 0.5 | 0.4 | -0.4 | 0.5 | 0.3 |
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| -0.3 | -0.5 | -0.4 | -0.4 | 0.4 | -0.4 |
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| -0.2 | -0.5 | 0.4 | -0.4 | -0.5 | 0.3 |
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| 0.6 | -0.06 | -0.4 | -0.4 | 0.1 | 0.6 |
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| 0.6 | 0.06 | 0.4 | -0.4 | -0.006 | -0.6 |
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The obtained matrices $U$ and $V$ are used to decouple the system as shown in Figure [[fig:plant_decouple_svd]].
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#+begin_src latex :file plant_decouple_svd.pdf :tangle no :exports results
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@ -1694,10 +1742,6 @@ 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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#+begin_src matlab
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freqs = logspace(-2, 2, 1000); % [Hz]
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#+end_src
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#+begin_src matlab :exports none
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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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@ -1735,7 +1779,6 @@ This is computed over the following frequencies.
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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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plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
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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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@ -1752,14 +1795,105 @@ This is computed over the following frequencies.
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#+RESULTS:
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[[file:figs/simscape_model_gershgorin_radii.png]]
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** Verification of the decoupling using the "Relative Gain Array"
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The relative gain array (RGA) is defined as:
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\begin{equation}
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\Lambda\big(G(s)\big) = G(s) \times \big( G(s)^{-1} \big)^T
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\end{equation}
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where $\times$ denotes an element by element multiplication and $G(s)$ is an $n \times n$ square transfer matrix.
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The obtained RGA elements are shown in Figure [[fig:simscape_model_rga]].
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#+begin_src matlab :exports none
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% Relative Gain Array for the coupled plant:
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RGA_coupled = zeros(length(freqs), size(Gu,1), size(Gu,2));
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Gu_inv = inv(Gu);
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for f_i = 1:length(freqs)
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RGA_coupled(f_i, :, :) = abs(evalfr(Gu, j*2*pi*freqs(f_i)).*evalfr(Gu_inv, j*2*pi*freqs(f_i))');
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end
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% Relative Gain Array for the decoupled plant using SVD:
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RGA_svd = zeros(length(freqs), size(Gsvd,1), size(Gsvd,2));
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Gsvd_inv = inv(Gsvd);
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for f_i = 1:length(freqs)
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RGA_svd(f_i, :, :) = abs(evalfr(Gsvd, j*2*pi*freqs(f_i)).*evalfr(Gsvd_inv, j*2*pi*freqs(f_i))');
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end
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% Relative Gain Array for the decoupled plant using the Jacobian:
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RGA_x = zeros(length(freqs), size(Gx,1), size(Gx,2));
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Gx_inv = inv(Gx);
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for f_i = 1:length(freqs)
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RGA_x(f_i, :, :) = abs(evalfr(Gx, j*2*pi*freqs(f_i)).*evalfr(Gx_inv, j*2*pi*freqs(f_i))');
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end
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#+end_src
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#+begin_src matlab :exports none
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figure;
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tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
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ax1 = nexttile;
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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, RGA_svd(:, i_out, i_in), '--', '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, RGA_svd(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
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'DisplayName', '$RGA_{SVD}(i,j),\ i \neq j$');
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plot(freqs, RGA_svd(:, 1, 1), 'k-', ...
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'DisplayName', '$RGA_{SVD}(i,i)$');
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for ch_i = 1:6
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plot(freqs, RGA_svd(:, ch_i, ch_i), 'k-', ...
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'HandleVisibility', 'off');
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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'); xlabel('Frequency [Hz]');
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legend('location', 'southwest');
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ax2 = nexttile;
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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, RGA_x(:, i_out, i_in), '--', '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, RGA_x(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
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'DisplayName', '$RGA_{X}(i,j),\ i \neq j$');
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plot(freqs, RGA_x(:, 1, 1), 'k-', ...
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'DisplayName', '$RGA_{X}(i,i)$');
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for ch_i = 1:6
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plot(freqs, RGA_x(:, ch_i, ch_i), 'k-', ...
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'HandleVisibility', 'off');
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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]'); set(gca, 'YTickLabel',[]);
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legend('location', 'southwest');
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linkaxes([ax1,ax2],'y');
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ylim([1e-5, 1e1]);
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/simscape_model_rga.pdf', 'width', 'wide', 'height', 'tall');
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#+end_src
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#+name: fig:simscape_model_rga
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#+caption: Obtained norm of RGA elements for the SVD decoupled plant and the Jacobian decoupled plant
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#+RESULTS:
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[[file:figs/simscape_model_rga.png]]
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** Obtained Decoupled Plants
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<<sec:stewart_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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#+begin_src matlab :exports none
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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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@ -1812,8 +1946,6 @@ The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown i
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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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#+begin_src matlab :exports none
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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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@ -1956,8 +2088,6 @@ $G_0$ is tuned such that the crossover frequency corresponding to the diagonal t
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The obtained diagonal elements of the loop gains are shown in Figure [[fig:stewart_comp_loop_gain_diagonal]].
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#+begin_src matlab :exports none
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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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@ -2038,8 +2168,6 @@ Let's first verify the stability of the closed-loop systems:
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The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_platform_simscape_cl_transmissibility]].
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#+begin_src matlab :exports results
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freqs = logspace(-2, 2, 1000);
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figure;
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tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');
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@ -2102,6 +2230,159 @@ The obtained transmissibility in Open-loop, for the centralized control as well
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#+RESULTS:
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[[file:figs/stewart_platform_simscape_cl_transmissibility.png]]
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** Small error on the sensor location :no_export:
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Let's now consider a small position error of the sensor:
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#+begin_src matlab
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sens_pos_error = [105 5 -1]*1e-3; % [m]
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#+end_src
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The system is identified again:
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#+begin_src matlab :exports none
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%% Name of the Simulink File
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mdl = 'drone_platform';
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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, '/Dw'], 1, 'openinput'); io_i = io_i + 1; % Ground Motion
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io(io_i) = linio([mdl, '/V-T'], 1, 'openinput'); io_i = io_i + 1; % Actuator Forces
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io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration
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G = linearize(mdl, io);
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G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
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'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
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G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
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% Plant
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Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
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% Disturbance dynamics
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Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'});
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#+end_src
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#+begin_src matlab
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Gx = Gu*inv(J');
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Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
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#+end_src
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#+begin_src matlab
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Gsvd = inv(U)*Gu*inv(V');
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#+end_src
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#+begin_src matlab :exports none
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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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#+end_src
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#+begin_src matlab :exports results
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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)
|
||||
plot(freqs, Gr_jacobian(:,in_i), 'HandleVisibility', 'off');
|
||||
end
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
hold off;
|
||||
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
legend('location', 'northwest');
|
||||
ylim([1e-3, 1e3]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
L_cen = K_cen*Gx;
|
||||
G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
L_svd = K_svd*Gsvd;
|
||||
G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results output replace text
|
||||
isstable(G_cen)
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :results output replace text
|
||||
isstable(G_svd)
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports results
|
||||
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]);
|
||||
#+end_src
|
||||
|
||||
* Stewart Platform - Analytical Model :noexport:
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle stewart_platform/analytical_model.m
|
||||
|
||||
Binary file not shown.
BIN
stewart_platform/jacobian.mat
Normal file
BIN
stewart_platform/jacobian.mat
Normal file
Binary file not shown.
@ -6,6 +6,8 @@ s = zpk('s');
|
||||
|
||||
addpath('STEP');
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
% Simscape Model - Parameters
|
||||
% <<sec:stewart_simscape>>
|
||||
|
||||
@ -25,6 +27,12 @@ cz = 0.025;
|
||||
|
||||
|
||||
|
||||
% We suppose the sensor is perfectly positioned.
|
||||
|
||||
sens_pos_error = zeros(3,1);
|
||||
|
||||
|
||||
|
||||
% Gravity:
|
||||
|
||||
g = 0;
|
||||
@ -33,7 +41,7 @@ g = 0;
|
||||
|
||||
% We load the Jacobian (previously computed from the geometry):
|
||||
|
||||
load('./jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
|
||||
load('jacobian.mat', 'Aa', 'Ab', 'As', 'l', 'J');
|
||||
|
||||
|
||||
|
||||
@ -86,8 +94,6 @@ size(G)
|
||||
% One can easily see that the system is strongly coupled.
|
||||
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
|
||||
% Magnitude
|
||||
@ -174,8 +180,6 @@ Gsvd = inv(U)*Gu*inv(V');
|
||||
|
||||
% This is computed over the following frequencies.
|
||||
|
||||
freqs = logspace(-2, 2, 1000); % [Hz]
|
||||
|
||||
% Gershgorin Radii for the coupled plant:
|
||||
Gr_coupled = zeros(length(freqs), size(Gu,2));
|
||||
H = abs(squeeze(freqresp(Gu, freqs, 'Hz')));
|
||||
@ -210,21 +214,99 @@ for in_i = 2:6
|
||||
set(gca,'ColorOrderIndex',3)
|
||||
plot(freqs, Gr_jacobian(:,in_i), 'HandleVisibility', 'off');
|
||||
end
|
||||
plot(freqs, 0.5*ones(size(freqs)), 'k--', 'DisplayName', 'Limit')
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
hold off;
|
||||
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
legend('location', 'northwest');
|
||||
ylim([1e-3, 1e3]);
|
||||
|
||||
% Verification of the decoupling using the "Relative Gain Array"
|
||||
% The relative gain array (RGA) is defined as:
|
||||
% \begin{equation}
|
||||
% \Lambda\big(G(s)\big) = G(s) \times \big( G(s)^{-1} \big)^T
|
||||
% \end{equation}
|
||||
% where $\times$ denotes an element by element multiplication and $G(s)$ is an $n \times n$ square transfer matrix.
|
||||
|
||||
% The obtained RGA elements are shown in Figure [[fig:simscape_model_rga]].
|
||||
|
||||
|
||||
% Relative Gain Array for the coupled plant:
|
||||
RGA_coupled = zeros(length(freqs), size(Gu,1), size(Gu,2));
|
||||
Gu_inv = inv(Gu);
|
||||
for f_i = 1:length(freqs)
|
||||
RGA_coupled(f_i, :, :) = abs(evalfr(Gu, j*2*pi*freqs(f_i)).*evalfr(Gu_inv, j*2*pi*freqs(f_i))');
|
||||
end
|
||||
|
||||
% Relative Gain Array for the decoupled plant using SVD:
|
||||
RGA_svd = zeros(length(freqs), size(Gsvd,1), size(Gsvd,2));
|
||||
Gsvd_inv = inv(Gsvd);
|
||||
for f_i = 1:length(freqs)
|
||||
RGA_svd(f_i, :, :) = abs(evalfr(Gsvd, j*2*pi*freqs(f_i)).*evalfr(Gsvd_inv, j*2*pi*freqs(f_i))');
|
||||
end
|
||||
|
||||
% Relative Gain Array for the decoupled plant using the Jacobian:
|
||||
RGA_x = zeros(length(freqs), size(Gx,1), size(Gx,2));
|
||||
Gx_inv = inv(Gx);
|
||||
for f_i = 1:length(freqs)
|
||||
RGA_x(f_i, :, :) = abs(evalfr(Gx, j*2*pi*freqs(f_i)).*evalfr(Gx_inv, j*2*pi*freqs(f_i))');
|
||||
end
|
||||
|
||||
figure;
|
||||
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
ax1 = nexttile;
|
||||
hold on;
|
||||
for i_in = 1:6
|
||||
for i_out = [1:i_in-1, i_in+1:6]
|
||||
plot(freqs, RGA_svd(:, i_out, i_in), '--', 'color', [0 0 0 0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
plot(freqs, RGA_svd(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
|
||||
'DisplayName', '$RGA_{SVD}(i,j),\ i \neq j$');
|
||||
|
||||
plot(freqs, RGA_svd(:, 1, 1), 'k-', ...
|
||||
'DisplayName', '$RGA_{SVD}(i,i)$');
|
||||
for ch_i = 1:6
|
||||
plot(freqs, RGA_svd(:, ch_i, ch_i), 'k-', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Magnitude'); xlabel('Frequency [Hz]');
|
||||
legend('location', 'southwest');
|
||||
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
for i_in = 1:6
|
||||
for i_out = [1:i_in-1, i_in+1:6]
|
||||
plot(freqs, RGA_x(:, i_out, i_in), '--', 'color', [0 0 0 0.2], ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
end
|
||||
plot(freqs, RGA_x(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
|
||||
'DisplayName', '$RGA_{X}(i,j),\ i \neq j$');
|
||||
|
||||
plot(freqs, RGA_x(:, 1, 1), 'k-', ...
|
||||
'DisplayName', '$RGA_{X}(i,i)$');
|
||||
for ch_i = 1:6
|
||||
plot(freqs, RGA_x(:, ch_i, ch_i), 'k-', ...
|
||||
'HandleVisibility', 'off');
|
||||
end
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
|
||||
legend('location', 'southwest');
|
||||
|
||||
linkaxes([ax1,ax2],'y');
|
||||
ylim([1e-5, 1e1]);
|
||||
|
||||
% Obtained Decoupled Plants
|
||||
% <<sec:stewart_decoupled_plant>>
|
||||
|
||||
% The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown in Figure [[fig:simscape_model_decoupled_plant_svd]].
|
||||
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
@ -274,8 +356,6 @@ linkaxes([ax1,ax2],'x');
|
||||
% 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]].
|
||||
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
@ -350,8 +430,6 @@ G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
|
||||
% The obtained diagonal elements of the loop gains are shown in Figure [[fig:stewart_comp_loop_gain_diagonal]].
|
||||
|
||||
|
||||
freqs = logspace(-1, 2, 1000);
|
||||
|
||||
figure;
|
||||
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
@ -424,7 +502,140 @@ isstable(G_svd)
|
||||
% The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure [[fig:stewart_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]);
|
||||
|
||||
% Small error on the sensor location :no_export:
|
||||
% Let's now consider a small position error of the sensor:
|
||||
|
||||
sens_pos_error = [105 5 -1]*1e-3; % [m]
|
||||
|
||||
|
||||
|
||||
% The system is identified again:
|
||||
|
||||
%% Name of the Simulink File
|
||||
mdl = 'drone_platform';
|
||||
|
||||
%% Input/Output definition
|
||||
clear io; io_i = 1;
|
||||
io(io_i) = linio([mdl, '/Dw'], 1, 'openinput'); io_i = io_i + 1; % Ground Motion
|
||||
io(io_i) = linio([mdl, '/V-T'], 1, 'openinput'); io_i = io_i + 1; % Actuator Forces
|
||||
io(io_i) = linio([mdl, '/Inertial Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Top platform acceleration
|
||||
|
||||
G = linearize(mdl, io);
|
||||
G.InputName = {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz', ...
|
||||
'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
|
||||
G.OutputName = {'Ax', 'Ay', 'Az', 'Arx', 'Ary', 'Arz'};
|
||||
|
||||
% Plant
|
||||
Gu = G(:, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
|
||||
% Disturbance dynamics
|
||||
Gd = G(:, {'Dwx', 'Dwy', 'Dwz', 'Rwx', 'Rwy', 'Rwz'});
|
||||
|
||||
Gx = Gu*inv(J');
|
||||
Gx.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
|
||||
|
||||
Gsvd = inv(U)*Gu*inv(V');
|
||||
|
||||
% Gershgorin Radii for the coupled plant:
|
||||
Gr_coupled = zeros(length(freqs), size(Gu,2));
|
||||
H = abs(squeeze(freqresp(Gu, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gu,2)
|
||||
Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
end
|
||||
|
||||
% Gershgorin Radii for the decoupled plant using SVD:
|
||||
Gr_decoupled = zeros(length(freqs), size(Gsvd,2));
|
||||
H = abs(squeeze(freqresp(Gsvd, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gsvd,2)
|
||||
Gr_decoupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
end
|
||||
|
||||
% Gershgorin Radii for the decoupled plant using the Jacobian:
|
||||
Gr_jacobian = zeros(length(freqs), size(Gx,2));
|
||||
H = abs(squeeze(freqresp(Gx, freqs, 'Hz')));
|
||||
for out_i = 1:size(Gx,2)
|
||||
Gr_jacobian(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
|
||||
end
|
||||
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, Gr_coupled(:,1), 'DisplayName', 'Coupled');
|
||||
plot(freqs, Gr_decoupled(:,1), 'DisplayName', 'SVD');
|
||||
plot(freqs, Gr_jacobian(:,1), 'DisplayName', 'Jacobian');
|
||||
for in_i = 2:6
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, Gr_coupled(:,in_i), 'HandleVisibility', 'off');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, Gr_decoupled(:,in_i), 'HandleVisibility', 'off');
|
||||
set(gca,'ColorOrderIndex',3)
|
||||
plot(freqs, Gr_jacobian(:,in_i), 'HandleVisibility', 'off');
|
||||
end
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
hold off;
|
||||
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
|
||||
legend('location', 'northwest');
|
||||
ylim([1e-3, 1e3]);
|
||||
|
||||
L_cen = K_cen*Gx;
|
||||
G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
|
||||
|
||||
L_svd = K_svd*Gsvd;
|
||||
G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
|
||||
|
||||
isstable(G_cen)
|
||||
|
||||
isstable(G_svd)
|
||||
|
||||
figure;
|
||||
tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');
|
||||
|
||||
Loading…
Reference in New Issue
Block a user