Update CSS
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.org-widget-field { /* widget-field */ background-color: #d9d9d9; }
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.org-widget-field { /* widget-field */ background-color: #d9d9d9; }
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.org-widget-inactive { /* widget-inactive */ color: #7f7f7f; }
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.org-widget-inactive { /* widget-inactive */ color: #7f7f7f; }
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.org-widget-single-line-field { /* widget-single-line-field */ background-color: #d9d9d9; }
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.org-widget-single-line-field { /* widget-single-line-field */ background-color: #d9d9d9; }
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pre {background-color:#FFFFFF;}
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pre span.org-builtin {color:#006FE0;font-weight:bold;}
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pre span.org-string {color:#008000;}
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pre span.org-doc {color:#008000;}
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pre span.org-keyword {color:#0000FF;}
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pre span.org-variable-name {color:#BA36A5;}
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pre span.org-function-name {color:#006699;}
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pre span.org-type {color:#6434A3;}
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pre span.org-preprocessor {color:#808080;font-weight:bold;}
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pre span.org-constant {color:#D0372D;}
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pre span.org-comment-delimiter {color:#8D8D84;}
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pre span.org-comment {color:#8D8D84;font-style:italic}
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pre span.org-outshine-level-1 {color:#8D8D84;font-style:italic}
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pre span.org-outshine-level-2 {color:#8D8D84;font-style:italic}
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pre span.org-outshine-level-8 {color:#8D8D84;font-style:italic}
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pre span.org-rainbow-delimiters-depth-1 {color:#707183;}
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pre span.org-rainbow-delimiters-depth-2 {color:#7388d6;}
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pre span.org-rainbow-delimiters-depth-3 {color:#909183;}
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pre span.org-rainbow-delimiters-depth-4 {color:#709870;}
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pre span.org-rainbow-delimiters-depth-5 {color:#907373;}
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pre span.org-rainbow-delimiters-depth-6 {color:#6276ba;}
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pre span.org-rainbow-delimiters-depth-8 {color:#80a880;}
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pre span.org-rainbow-delimiters-depth-9 {color:#887070;}
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pre span.org-sh-quoted-exec {color:#FF1493;}
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pre span.org-diff-added {color:#008000;}
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pre span.org-diff-changed {color:#0000FF;}
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pre span.org-diff-header {color:#800000;}
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pre span.org-diff-hunk-header {color:#990099;}
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pre span.org-diff-none {color:#545454;}
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pre span.org-diff-removed {color:#A60000;}
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@ -345,9 +345,11 @@ table{
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border-collapse:collapse;
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border-collapse:collapse;
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border-spacing:0;
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border-spacing:0;
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empty-cells:show;
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empty-cells:show;
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margin-bottom:24px;
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border-bottom:1px solid #e1e4e5;
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border-bottom:1px solid #e1e4e5;
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margin: 0 auto;
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margin-right: auto;
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margin-left: auto;
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margin-bottom:24px;
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/* margin: 0 auto; */
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}
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}
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td{
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td{
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@ -1101,3 +1103,32 @@ h2.footnotes{
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margin-bottom: 24px;
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margin-bottom: 24px;
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font-family:"Roboto Slab","ff-tisa-web-pro","Georgia",Arial,sans-serif;
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font-family:"Roboto Slab","ff-tisa-web-pro","Georgia",Arial,sans-serif;
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}
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}
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details {
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/* color: #2980B9; */
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background: #fbfbfb;
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border: 1px solid #c9c9c9;
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border-radius: 3px;
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padding: 0.25em;
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margin-bottom: 1.0em;
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}
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details pre.src {
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border: 0;
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background: none;
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margin: 0;
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}
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details pre.src-lisp::before { content: ""; }
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summary {
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outline: 0;
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color: #c9c9c9;
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}
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summary::after {
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font-size: 0.85em;
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color: #c9c9c9;
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display: inline-block;
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float: right;
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content: "Click to fold/unfold";
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padding-right: 0.5em;
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}
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174
matlab/matlab/sensor_description.m
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174
matlab/matlab/sensor_description.m
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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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addpath('src');
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freqs = logspace(0, 4, 1000);
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% Sensor Dynamics
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% <<sec:sensor_dynamics>>
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% Let's consider two sensors measuring the velocity of an object.
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% The first sensor is an accelerometer.
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% Its nominal dynamics $\hat{G}_1(s)$ is defined below.
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m_acc = 0.01; % Inertial Mass [kg]
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c_acc = 5; % Damping [N/(m/s)]
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k_acc = 1e5; % Stiffness [N/m]
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g_acc = 1e5; % Gain [V/m]
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G1 = g_acc*m_acc*s/(m_acc*s^2 + c_acc*s + k_acc); % Accelerometer Plant [V/(m/s)]
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% The second sensor is a displacement sensor, its nominal dynamics $\hat{G}_2(s)$ is defined below.
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w_pos = 2*pi*2e3; % Measurement Banwdith [rad/s]
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g_pos = 1e4; % Gain [V/m]
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G2 = g_pos/s/(1 + s/w_pos); % Position Sensor Plant [V/(m/s)]
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% These nominal dynamics are also taken as the model of the sensor dynamics.
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% The true sensor dynamics has some uncertainty associated to it and described in section [[sec:sensor_uncertainty]].
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% Both sensor dynamics in $[\frac{V}{m/s}]$ are shown in Figure [[fig:sensors_nominal_dynamics]].
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figure;
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% Magnitude
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ax1 = subplot(2,1,1);
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hold on;
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plot(freqs, abs(squeeze(freqresp(G1, freqs, 'Hz'))), '-', 'DisplayName', '$G_1(j\omega)$');
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plot(freqs, abs(squeeze(freqresp(G2, freqs, 'Hz'))), '-', 'DisplayName', '$G_2(j\omega)$');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Magnitude $\left[\frac{V}{m/s}\right]$'); set(gca, 'XTickLabel',[]);
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legend('location', 'northeast', 'FontSize', 8);
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hold off;
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% Phase
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ax2 = subplot(2,1,2);
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hold on;
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plot(freqs, 180/pi*angle(squeeze(freqresp(G1, freqs, 'Hz'))), '-');
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plot(freqs, 180/pi*angle(squeeze(freqresp(G2, freqs, 'Hz'))), '-');
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set(gca,'xscale','log');
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yticks(-180:90:180);
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ylim([-180 180]);
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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hold off;
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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% Sensor Model Uncertainty
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% <<sec:sensor_uncertainty>>
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% The uncertainty on the sensor dynamics is described by multiplicative uncertainty (Figure [[fig:sensor_model_noise_uncertainty]]).
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% The true sensor dynamics $G_i(s)$ is then described by eqref:eq:sensor_dynamics_uncertainty.
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% \begin{equation}
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% G_i(s) = \hat{G}_i(s) \left( 1 + W_i(s) \Delta_i(s) \right); \quad |\Delta_i(j\omega)| < 1 \forall \omega \label{eq:sensor_dynamics_uncertainty}
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% \end{equation}
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% The weights $W_i(s)$ representing the dynamical uncertainty are defined below and their magnitude is shown in Figure [[fig:sensors_uncertainty_weights]].
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W1 = createWeight('n', 2, 'w0', 2*pi*3, 'G0', 2, 'G1', 0.1, 'Gc', 1) * ...
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createWeight('n', 2, 'w0', 2*pi*1e3, 'G0', 1, 'G1', 4/0.1, 'Gc', 1/0.1);
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W2 = createWeight('n', 2, 'w0', 2*pi*1e2, 'G0', 0.05, 'G1', 4, 'Gc', 1);
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% The bode plot of the sensors nominal dynamics as well as their defined dynamical spread are shown in Figure [[fig:sensors_nominal_dynamics_and_uncertainty]].
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figure;
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hold on;
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plot(freqs, abs(squeeze(freqresp(W1, freqs, 'Hz'))), 'DisplayName', '$|W_1(j\omega)|$');
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plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))), 'DisplayName', '$|W_2(j\omega)|$');
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hold off;
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
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xlabel('Frequency [Hz]'); ylabel('Magnitude');
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ylim([0, 5]);
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xlim([freqs(1), freqs(end)]);
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legend('location', 'northwest', 'FontSize', 8);
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% #+name: fig:sensors_uncertainty_weights
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% #+caption: Magnitude of the multiplicative uncertainty weights $|W_i(j\omega)|$
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% #+RESULTS:
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% [[file:figs/sensors_uncertainty_weights.png]]
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figure;
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% Magnitude
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ax1 = subplot(2,1,1);
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hold on;
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plotMagUncertainty(W1, freqs, 'G', G1, 'color_i', 1, 'DisplayName', '$G_1$');
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plotMagUncertainty(W2, freqs, 'G', G2, 'color_i', 2, 'DisplayName', '$G_2$');
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set(gca,'ColorOrderIndex',1)
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plot(freqs, abs(squeeze(freqresp(G1, freqs, 'Hz'))), 'DisplayName', '$\hat{G}_1$');
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plot(freqs, abs(squeeze(freqresp(G2, freqs, 'Hz'))), 'DisplayName', '$\hat{G}_2$');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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set(gca, 'XTickLabel',[]);
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ylabel('Magnitude $[\frac{V}{m/s}]$');
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ylim([1e-2, 2e3]);
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legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2);
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hold off;
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ylim([1e-2, 1e4])
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% Phase
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ax2 = subplot(2,1,2);
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hold on;
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plotPhaseUncertainty(W1, freqs, 'G', G1, 'color_i', 1);
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plotPhaseUncertainty(W2, freqs, 'G', G2, 'color_i', 2);
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set(gca,'ColorOrderIndex',1)
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plot(freqs, 180/pi*angle(squeeze(freqresp(G1, freqs, 'Hz'))), 'DisplayName', '$\hat{G}_1$');
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plot(freqs, 180/pi*angle(squeeze(freqresp(G2, freqs, 'Hz'))), 'DisplayName', '$\hat{G}_2$');
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set(gca,'xscale','log');
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yticks(-180:90:180);
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ylim([-180 180]);
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xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
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hold off;
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linkaxes([ax1,ax2],'x');
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xlim([freqs(1), freqs(end)]);
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% Sensor Noise
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% <<sec:sensor_noise>>
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% The noise of the sensors $n_i$ are modelled by shaping a white noise with unitary PSD $\tilde{n}_i$ eqref:eq:unitary_noise_psd with a LTI transfer function $N_i(s)$ (Figure [[fig:sensor_model_noise_uncertainty]]).
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% \begin{equation}
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% \Phi_{\tilde{n}_i}(\omega) = 1 \label{eq:unitary_noise_psd}
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% \end{equation}
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% The Power Spectral Density of the sensor noise $\Phi_{n_i}(\omega)$ is then computed using eqref:eq:sensor_noise_shaping and expressed in $[\frac{(m/s)^2}{Hz}]$.
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% \begin{equation}
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% \Phi_{n_i}(\omega) = \left| N_i(j\omega) \right|^2 \Phi_{\tilde{n}_i}(\omega) \label{eq:sensor_noise_shaping}
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% \end{equation}
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% The weights $N_1$ and $N_2$ representing the amplitude spectral density of the sensor noises are defined below and shown in Figure [[fig:sensors_noise]].
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omegac = 0.15*2*pi; G0 = 1e-1; Ginf = 1e-6;
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N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/1e4);
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omegac = 1000*2*pi; G0 = 1e-6; Ginf = 1e-3;
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N2 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/1e4);
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figure;
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hold on;
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plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))), '-', 'DisplayName', '$|N_1(j\omega)|$');
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plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))), '-', 'DisplayName', '$|N_2(j\omega)|$');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('ASD $\left[ \frac{m/s}{\sqrt{Hz}} \right]$');
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hold off;
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xlim([freqs(1), freqs(end)]);
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legend('location', 'northeast', 'FontSize', 8);
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% Save Model
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% All the dynamical systems representing the sensors are saved for further use.
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save('./mat/model.mat', 'freqs', 'G1', 'G2', 'N2', 'N1', 'W2', 'W1');
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