Rework H2 synthesis section
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matlab/figs/sensor_noise_H2_time_domain.pdf
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matlab/figs/super_sensor_dynamical_uncertainty_H2.pdf
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matlab/figs/super_sensor_dynamical_uncertainty_H2.pdf
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matlab/index.org
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matlab/index.org
@ -12,6 +12,7 @@
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#+HTML_HEAD: <script src="../js/bootstrap.min.js"></script>
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#+HTML_HEAD: <script src="../js/jquery.stickytableheaders.min.js"></script>
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#+HTML_HEAD: <script src="../js/readtheorg.js"></script>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://cdn.rawgit.com/dreampulse/computer-modern-web-font/master/fonts.css">
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :tangle no
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@ -42,56 +43,32 @@ Two sensors are considered with both different noise characteristics and dynamic
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** Introduction :ignore:
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In Figure [[fig:sensor_model_noise_uncertainty]] is shown a schematic of a sensor model that is used in the following study.
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In this example, the measured quantity $x$ is the velocity of an object.
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#+name: tab:sensor_signals
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#+caption: Description of signals in Figure [[fig:sensor_model_noise_uncertainty]]
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| *Notation* | *Meaning* |
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|---------------+---------------------------------|
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| $x$ | Physical measured quantity |
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| $\tilde{n}_i$ | White noise with unitary PSD |
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| $n_i$ | Shaped noise |
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| $v_i$ | Sensor output measurement |
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| $\hat{x}_i$ | Estimate of $x$ from the sensor |
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| *Notation* | *Meaning* | *Unit* |
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|---------------+---------------------------------+---------|
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| $x$ | Physical measured quantity | $[m/s]$ |
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| $\tilde{n}_i$ | White noise with unitary PSD | |
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| $n_i$ | Shaped noise | $[m/s]$ |
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| $v_i$ | Sensor output measurement | $[V]$ |
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| $\hat{x}_i$ | Estimate of $x$ from the sensor | $[m/s]$ |
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#+name: tab:sensor_dynamical_blocks
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#+caption: Description of Systems in Figure [[fig:sensor_model_noise_uncertainty]]
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| *Notation* | *Meaning* |
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|-------------+------------------------------------------------------------------------------|
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| $\hat{G}_i$ | Nominal Sensor Dynamics |
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| $W_i$ | Weight representing the size of the uncertainty at each frequency |
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| $\Delta_i$ | Any complex perturbation such that $\vert\vert\Delta_i\vert\vert_\infty < 1$ |
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| $N_i$ | Weight representing the sensor noise |
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| *Notation* | *Meaning* | *Unit* |
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|-------------+------------------------------------------------------------------------------+-------------------|
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| $\hat{G}_i$ | Nominal Sensor Dynamics | $[\frac{V}{m/s}]$ |
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| $W_i$ | Weight representing the size of the uncertainty at each frequency | |
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| $\Delta_i$ | Any complex perturbation such that $\vert\vert\Delta_i\vert\vert_\infty < 1$ | |
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| $N_i$ | Weight representing the sensor noise | $[m/s]$ |
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#+name: fig:sensor_model_noise_uncertainty
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#+caption: Sensor Model
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#+RESULTS:
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[[file:figs-tikz/sensor_model_noise_uncertainty.png]]
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In this example, the measured quantity $x$ is the velocity of an object.
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The units of signals are listed in Table [[tab:signal_units]].
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The units of systems are listed in Table [[tab:dynamical_block_units]].
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#+name: tab:signal_units
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#+caption: Units of signals in Figure [[fig:sensor_model_noise_uncertainty]]
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| *Notation* | *Unit* |
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|---------------+---------|
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| $x$ | $[m/s]$ |
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| $\tilde{n}_i$ | |
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| $n_i$ | $[m/s]$ |
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| $v_i$ | $[V]$ |
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| $\hat{x}_i$ | $[m/s]$ |
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#+name: tab:dynamical_block_units
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#+caption: Units of Systems in Figure [[fig:sensor_model_noise_uncertainty]]
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| *Notation* | *Unit* |
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|------------------+-------------------|
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| $\hat{G}_i$ | $[\frac{V}{m/s}]$ |
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| $\hat{G}_i^{-1}$ | $[\frac{m/s}{V}]$ |
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| $W_i$ | |
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| $\Delta_i$ | |
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| $N_i$ | $[m/s]$ |
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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@ -269,7 +246,7 @@ The Power Spectral Density of the sensor noise $\Phi_{n_i}(\omega)$ is then comp
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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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#+begin_src matlab
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omegac = 0.05*2*pi; G0 = 1e-1; Ginf = 1e-6;
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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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@ -431,10 +408,10 @@ If we first suppose that the models of the sensors $\hat{G}_i$ are very close to
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As the noise of both sensors are considered to be uncorrelated, the PSD of the super sensor noise is computed as follow:
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\begin{equation}
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\Phi_n(\omega) = \left| H_1 N_1 \right|^2 + \left| H_2 N_2 \right|^2 \label{eq:super_sensor_psd_noise}
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\Phi_n(\omega) = \left| H_1(j\omega) N_1(j\omega) \right|^2 + \left| H_2(j\omega) N_2(j\omega) \right|^2 \label{eq:super_sensor_psd_noise}
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\end{equation}
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And the Root Mean Square (RMS) value of the super sensor noise $\sigma_n$ is given by eqref:eq:super_sensor_rms_noise.
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And the Root Mean Square (RMS) value of the super sensor noise $\sigma_n$ is given by Equation eqref:eq:super_sensor_rms_noise.
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\begin{equation}
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\sigma_n = \sqrt{\int_0^\infty \Phi_n(\omega) d\omega} \label{eq:super_sensor_rms_noise}
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\end{equation}
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@ -470,16 +447,23 @@ The uncertainty set of the transfer function from $\hat{x}$ to $x$ at frequency
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<<sec:optimal_comp_filters>>
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** Introduction :ignore:
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In this section, the two sensors are merged together using complementary
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The idea is to combine sensors that works in different frequency range using complementary filters.
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Doing so, one "super sensor" is obtained that can have better noise characteristics than the individual sensors over a large frequency range.
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The complementary filters have to be designed in order to minimize the effect noise of each sensor on the super sensor noise.
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In this section, the complementary filters $H_1(s)$ and $H_2(s)$ are designed in order to minimize the RMS value of super sensor noise $\sigma_n$.
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#+name: fig:sensor_fusion_noise_arch
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#+caption: Optimal Sensor Fusion Architecture
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[[file:figs-tikz/sensor_fusion_noise_arch.png]]
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The RMS value of the super sensor noise is (neglecting the model uncertainty):
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\begin{equation}
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\begin{aligned}
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\sigma_{n} &= \sqrt{\int_0^\infty |H_1(j\omega) N_1(j\omega)|^2 + |H_2(j\omega) N_2(j\omega)|^2 d\omega} \\
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&= \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2
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\end{aligned}
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\end{equation}
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The goal is to design $H_1(s)$ and $H_2(s)$ such that $H_1(s) + H_2(s) = 1$ (complementary property) and such that $\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2$ is minimized (minimized RMS value of the super sensor noise).
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This is done using the $\mathcal{H}_2$ synthesis in Section [[sec:H2_synthesis]].
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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@ -490,54 +474,51 @@ The complementary filters have to be designed in order to minimize the effect no
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#+end_src
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#+begin_src matlab
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addpath('src');
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load('./mat/model.mat', 'freqs', 'G1', 'G2', 'N2', 'N1', 'W2', 'W1');
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#+end_src
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** H-Two Synthesis
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As $\tilde{n}_1$ and $\tilde{n}_2$ are normalized white noise: $\Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_2}(\omega) = 1$ and we have:
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\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty |H_1 N_1|^2(\omega) + |H_2 N_2|^2(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2 \]
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Thus, the goal is to design $H_1(s)$ and $H_2(s)$ such that $H_1(s) + H_2(s) = 1$ and such that $\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2$ is minimized.
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** $\mathcal{H}_2$ Synthesis
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<<sec:H2_synthesis>>
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For that, we use the $\mathcal{H}_2$ Synthesis.
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We use the generalized plant architecture shown on Figure [[fig:h_two_optimal_fusion]].
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Consider the generalized plant $P_{\mathcal{H}_2}$ shown in Figure [[fig:h_two_optimal_fusion]] and described by Equation eqref:eq:H2_generalized_plant.
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#+name: fig:h_two_optimal_fusion
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#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
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[[file:figs-tikz/h_two_optimal_fusion.png]]
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\begin{equation*}
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\begin{equation} \label{eq:H2_generalized_plant}
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\begin{pmatrix}
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z \\ v
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\end{pmatrix} = \begin{pmatrix}
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0 & N_2 & 1 \\
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N_1 & -N_2 & 0
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\end{pmatrix} \begin{pmatrix}
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W_1 \\ W_2 \\ u
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z_1 \\ z_2 \\ v
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\end{pmatrix} = \underbrace{\begin{bmatrix}
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N_1 & -N_1 \\
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0 & N_2 \\
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1 & 0
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\end{bmatrix}}_{P_{\mathcal{H}_2}} \begin{pmatrix}
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w \\ u
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\end{pmatrix}
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\end{equation*}
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\end{equation}
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The transfer function from $[n_1, n_2]$ to $\hat{x}$ is:
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\[ \begin{bmatrix} N_1 H_1 \\ N_2 (1 - H_1) \end{bmatrix} \]
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If we define $H_2 = 1 - H_1$, we obtain:
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\[ \begin{bmatrix} N_1 H_1 \\ N_2 H_2 \end{bmatrix} \]
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Applying the $\mathcal{H}_2$ synthesis on $P_{\mathcal{H}_2}$ will generate a filter $H_2(s)$ such that the $\mathcal{H}_2$ norm from $w$ to $(z_1,z_2)$ which is actually equals to $\sigma_n$ by defining $H_1(s) = 1 - H_2(s)$:
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\begin{equation}
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\left\| \begin{matrix} z_1/w \\ z_2/w \end{matrix} \right\|_2 = \left\| \begin{matrix} N_1 (1 - H_2) \\ N_2 H_2 \end{matrix} \right\|_2 = \sigma_n \quad \text{with} \quad H_1(s) = 1 - H_2(s)
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\end{equation}
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Thus, if we minimize the $\mathcal{H}_2$ norm of this transfer function, we minimize the RMS value of $\hat{x}$.
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We then have that the $\mathcal{H}_2$ synthesis applied on $P_{\mathcal{H}_2}$ generates two complementary filters $H_1(s)$ and $H_2(s)$ such that the RMS value of super sensor noise is minimized.
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We define the generalized plant $P$ on matlab as shown on Figure
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The generalized plant $P_{\mathcal{H}_2}$ is defined below
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#+begin_src matlab
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P = [N1 -N1;
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0 N2;
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1 0];
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PH2 = [N1 -N1;
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0 N2;
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1 0];
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#+end_src
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And we do the $\mathcal{H}_2$ synthesis using the =h2syn= command.
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The $\mathcal{H}_2$ synthesis using the =h2syn= command
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#+begin_src matlab
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[H2, ~, gamma] = h2syn(P, 1, 1);
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[H2, ~, gamma] = h2syn(PH2, 1, 1);
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#+end_src
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Finally, we define $H_2(s) = 1 - H_1(s)$.
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Finally, $H_1(s)$ is defined as follows
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#+begin_src matlab
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H1 = 1 - H2;
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#+end_src
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@ -547,185 +528,26 @@ Finally, we define $H_2(s) = 1 - H_1(s)$.
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save('./mat/H2_filters.mat', 'H2', 'H1');
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#+end_src
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The complementary filters obtained are shown on Figure
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The obtained complementary filters are shown in Figure [[fig:htwo_comp_filters]].
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#+begin_src matlab :exports none
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figure;
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hold on;
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plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
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plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
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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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hold off;
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xlim([freqs(1), freqs(end)]);
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legend('location', 'northeast');
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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/htwo_comp_filters.pdf', 'width', 'full', 'height', 'tall');
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#+end_src
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#+name: fig:htwo_comp_filters
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#+caption: Obtained complementary filters using the $\mathcal{H}_2$ Synthesis ([[./figs/htwo_comp_filters.png][png]], [[./figs/htwo_comp_filters.pdf][pdf]])
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#+RESULTS:
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[[file:figs/htwo_comp_filters.png]]
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** Sensor Noise
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The PSD of the noise of the individual sensor and of the super sensor are shown in Figure [[fig:psd_sensors_htwo_synthesis]].
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The Cumulative Power Spectrum (CPS) is shown on Figure [[fig:cps_h2_synthesis]].
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The obtained RMS value of the super sensor is lower than the RMS value of the individual sensors.
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#+begin_src matlab
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PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2;
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PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2;
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PSD_H2 = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
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CPS_S1 = cumtrapz(freqs, PSD_S1);
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CPS_S2 = cumtrapz(freqs, PSD_S2);
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CPS_H2 = cumtrapz(freqs, PSD_H2);
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#+end_src
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#+begin_src matlab :exports none
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figure;
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hold on;
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plot(freqs, PSD_S1, '-', 'DisplayName', '$\Phi_{\hat{x}_1}$');
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plot(freqs, PSD_S2, '-', 'DisplayName', '$\Phi_{\hat{x}_2}$');
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plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_2}}$');
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set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Power Spectral Density [$(m/s)^2/Hz$]');
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hold off;
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xlim([freqs(1), freqs(end)]);
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legend('location', 'northeast');
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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/psd_sensors_htwo_synthesis.pdf', 'width', 'full', 'height', 'tall');
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#+end_src
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#+name: fig:psd_sensors_htwo_synthesis
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#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the optimally fused signal ([[./figs/psd_sensors_htwo_synthesis.png][png]], [[./figs/psd_sensors_htwo_synthesis.pdf][pdf]])
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#+RESULTS:
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[[file:figs/psd_sensors_htwo_synthesis.png]]
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#+begin_src matlab :exports none
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figure;
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hold on;
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plot(freqs, CPS_S1, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_1} = %.1e$ [m/s rms]', sqrt(CPS_S1(end))));
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plot(freqs, CPS_S2, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_2} = %.1e$ [m/s rms]', sqrt(CPS_S2(end))));
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plot(freqs, CPS_H2, 'k-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_2}} = %.1e$ [m/s rms]', sqrt(CPS_H2(end))));
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set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log');
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xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum');
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hold off;
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xlim([2*freqs(1), freqs(end)]);
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legend('location', 'southeast');
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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/cps_h2_synthesis.pdf', 'width', 'full', 'height', 'tall');
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#+end_src
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#+name: fig:cps_h2_synthesis
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#+caption: Cumulative Power Spectrum of individual sensors and super sensor using the $\mathcal{H}_2$ synthesis ([[./figs/cps_h2_synthesis.png][png]], [[./figs/cps_h2_synthesis.pdf][pdf]])
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#+RESULTS:
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[[file:figs/cps_h2_synthesis.png]]
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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data2orgtable([sqrt(CPS_S1(end)), sqrt(CPS_S2(end)), sqrt(CPS_H2(end))]', {'Integrated Acceleration', 'Derived Position', 'Super Sensor - $\mathcal{H}_2$'}, {'RMS [m/s]'}, ' %.1e ');
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#+end_src
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#+RESULTS:
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| | RMS [m/s] |
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|--------------------------------+-----------|
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| Integrated Acceleration | 0.005 |
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| Derived Position | 0.08 |
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| Super Sensor - $\mathcal{H}_2$ | 0.0012 |
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** Time Domain Simulation
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Parameters of the time domain simulation.
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#+begin_src matlab
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Fs = 1e4; % Sampling Frequency [Hz]
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Ts = 1/Fs; % Sampling Time [s]
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||||
|
||||
t = 0:Ts:2; % Time Vector [s]
|
||||
#+end_src
|
||||
|
||||
Time domain velocity.
|
||||
#+begin_src matlab
|
||||
v = 0.1*sin((10*t).*t)';
|
||||
#+end_src
|
||||
|
||||
Generate noises in velocity corresponding to sensor 1 and 2:
|
||||
#+begin_src matlab
|
||||
n1 = lsim(N1, sqrt(Fs/2)*randn(length(t), 1), t);
|
||||
n2 = lsim(N2, sqrt(Fs/2)*randn(length(t), 1), t);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(t, n2, 'DisplayName', 'Differentiated Position');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(t, n1, 'DisplayName', 'Integrated Acceleration');
|
||||
set(gca,'ColorOrderIndex',3)
|
||||
plot(t, (lsim(H1, n1, t)+lsim(H2, n2, t)), 'k-', 'DisplayName', 'Super Sensor');
|
||||
hold off;
|
||||
xlabel('Time [s]'); ylabel('Velocity [m/s]');
|
||||
legend();
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(t, v+n2, 'DisplayName', 'Differentiated Position');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(t, v+n1, 'DisplayName', 'Integrated Acceleration');
|
||||
set(gca,'ColorOrderIndex',3)
|
||||
plot(t, v+(lsim(H1, n1, t)+lsim(H2, n2, t)), 'DisplayName', 'Super Sensor');
|
||||
plot(t, v, 'k--', 'DisplayName', 'True Velocity');
|
||||
hold off;
|
||||
xlabel('Time [s]'); ylabel('Velocity [m/s]');
|
||||
legend();
|
||||
ylim([-0.3, 0.3]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/super_sensor_time_domain_h2.pdf', 'width', 'full', 'height', 'tall');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:super_sensor_time_domain_h2
|
||||
#+caption: Noise of individual sensors and noise of the super sensor
|
||||
#+RESULTS:
|
||||
[[file:figs/super_sensor_time_domain_h2.png]]
|
||||
|
||||
** Discrepancy between sensor dynamics and model
|
||||
#+begin_src matlab :exports none
|
||||
Dphi_ss = 180/pi*asin(abs(squeeze(freqresp(W2*H2, freqs, 'Hz'))) + abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))));
|
||||
Dphi_ss(abs(squeeze(freqresp(W2*H2, freqs, 'Hz'))) + abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))) > 1) = 360;
|
||||
|
||||
figure;
|
||||
% Magnitude
|
||||
ax1 = subplot(2,1,1);
|
||||
hold on;
|
||||
plotMagUncertainty(W1, freqs, 'color_i', 1);
|
||||
plotMagUncertainty(W2, freqs, 'color_i', 2);
|
||||
p = patch([freqs flip(freqs)], [1 + abs(squeeze(freqresp(W2*H2, freqs, 'Hz')))+abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))); flip(max(1 - abs(squeeze(freqresp(W2*H2, freqs, 'Hz')))-abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))), 0.001))], 'w');
|
||||
p.EdgeColor = 'black'; p.FaceAlpha = 0;
|
||||
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), 'DisplayName', '$H_1$');
|
||||
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), 'DisplayName', '$H_2$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylabel('Magnitude');
|
||||
ylim([1e-2, 1e1]);
|
||||
hold off;
|
||||
legend('location', 'northeast');
|
||||
|
||||
% Phase
|
||||
ax2 = subplot(2,1,2);
|
||||
hold on;
|
||||
plotPhaseUncertainty(W1, freqs, 'color_i', 1);
|
||||
plotPhaseUncertainty(W2, freqs, 'color_i', 2);
|
||||
p = patch([freqs flip(freqs)], [Dphi_ss; flip(-Dphi_ss)], 'w');
|
||||
p.EdgeColor = 'black'; p.FaceAlpha = 0;
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(H1, freqs, 'Hz'))));
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(H2, freqs, 'Hz'))));
|
||||
set(gca,'xscale','log');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
@ -735,10 +557,214 @@ Generate noises in velocity corresponding to sensor 1 and 2:
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
#+end_src
|
||||
|
||||
** Conclusion
|
||||
From the above complementary filter design with the $\mathcal{H}_2$ and $\mathcal{H}_\infty$ synthesis, it still seems that the $\mathcal{H}_2$ synthesis gives the complementary filters that permits to obtain the minimal super sensor noise (when measuring with the $\mathcal{H}_2$ norm).
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/htwo_comp_filters.pdf', 'width', 'full', 'height', 'tall');
|
||||
#+end_src
|
||||
|
||||
However, the synthesis does not take into account the robustness of the sensor fusion.
|
||||
#+name: fig:htwo_comp_filters
|
||||
#+caption: Obtained complementary filters using the $\mathcal{H}_2$ Synthesis
|
||||
#+RESULTS:
|
||||
[[file:figs/htwo_comp_filters.png]]
|
||||
|
||||
** Super Sensor Noise
|
||||
<<sec:H2_super_sensor_noise>>
|
||||
|
||||
The Power Spectral Density of the individual sensors' noise $\Phi_{n_1}, \Phi_{n_2}$ and of the super sensor noise $\Phi_{n_{\mathcal{H}_2}}$ are computed below and shown in Figure [[fig:psd_sensors_htwo_synthesis]].
|
||||
#+begin_src matlab
|
||||
PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2;
|
||||
PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2;
|
||||
PSD_H2 = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2 + ...
|
||||
abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
|
||||
#+end_src
|
||||
|
||||
The corresponding Cumulative Power Spectrum $\Gamma_{n_1}$, $\Gamma_{n_2}$ and $\Gamma_{n_{\mathcal{H}_2}}$ (cumulative integration of the PSD eqref:eq:CPS_definition) are computed below and shown in Figure [[fig:cps_h2_synthesis]].
|
||||
#+begin_src matlab
|
||||
CPS_S1 = cumtrapz(freqs, PSD_S1);
|
||||
CPS_S2 = cumtrapz(freqs, PSD_S2);
|
||||
CPS_H2 = cumtrapz(freqs, PSD_H2);
|
||||
#+end_src
|
||||
|
||||
\begin{equation}
|
||||
\Gamma_n (\omega) = \int_0^\omega \Phi_n(\nu) d\nu \label{eq:CPS_definition}
|
||||
\end{equation}
|
||||
|
||||
The RMS value of the individual sensors and of the super sensor are listed in Table [[tab:rms_noise_H2]].
|
||||
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
||||
data2orgtable([sqrt(CPS_S1(end)); sqrt(CPS_S2(end)); sqrt(CPS_H2(end))], {'$\sigma_{n_1}$', '$\sigma_{n_2}$', '$\sigma_{n_{\mathcal{H}_2}}$'}, {'RMS value $[m/s]$'}, ' %.1e ');
|
||||
#+end_src
|
||||
|
||||
#+name: tab:rms_noise_H2
|
||||
#+caption: RMS value of the individual sensor noise and of the super sensor using the $\mathcal{H}_2$ Synthesis
|
||||
#+RESULTS:
|
||||
| | RMS value $[m/s]$ |
|
||||
|------------------------------+-------------------|
|
||||
| $\sigma_{n_1}$ | 0.015 |
|
||||
| $\sigma_{n_2}$ | 0.08 |
|
||||
| $\sigma_{n_{\mathcal{H}_2}}$ | 0.0027 |
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, PSD_S1, '-', 'DisplayName', '$\Phi_{n_1}$');
|
||||
plot(freqs, PSD_S2, '-', 'DisplayName', '$\Phi_{n_2}$');
|
||||
plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{n_{\mathcal{H}_2}}$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Power Spectral Density [$(m/s)^2/Hz$]');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
legend('location', 'northeast');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/psd_sensors_htwo_synthesis.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:psd_sensors_htwo_synthesis
|
||||
#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the optimally fused signal
|
||||
#+RESULTS:
|
||||
[[file:figs/psd_sensors_htwo_synthesis.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, CPS_S1, '-', 'DisplayName', '$\Gamma_{n_1}$');
|
||||
plot(freqs, CPS_S2, '-', 'DisplayName', '$\Gamma_{n_2}$');
|
||||
plot(freqs, CPS_H2, 'k-', 'DisplayName', '$\Gamma_{n_{\mathcal{H}_2}}$');
|
||||
set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum $[(m/s)^2]$');
|
||||
hold off;
|
||||
xlim([2*freqs(1), freqs(end)]);
|
||||
legend('location', 'southeast');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/cps_h2_synthesis.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:cps_h2_synthesis
|
||||
#+caption: Cumulative Power Spectrum of individual sensors and super sensor using the $\mathcal{H}_2$ synthesis
|
||||
#+RESULTS:
|
||||
[[file:figs/cps_h2_synthesis.png]]
|
||||
|
||||
A time domain simulation is now performed.
|
||||
The measured velocity $x$ is set to be a sweep sine with an amplitude of $0.1\ [m/s]$.
|
||||
The velocity estimates from the two sensors and from the super sensors are shown in Figure [[fig:super_sensor_time_domain_h2]].
|
||||
The resulting noises are displayed in Figure [[fig:sensor_noise_H2_time_domain]].
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
Fs = 1e4; % Sampling Frequency [Hz]
|
||||
Ts = 1/Fs; % Sampling Time [s]
|
||||
|
||||
t = 0:Ts:2; % Time Vector [s]
|
||||
|
||||
v = 0.1*sin((10*t).*t)'; % Velocity measured [m/s]
|
||||
|
||||
% Generate noises in velocity corresponding to sensor 1 and 2:
|
||||
n1 = lsim(N1, sqrt(Fs/2)*randn(length(t), 1), t);
|
||||
n2 = lsim(N2, sqrt(Fs/2)*randn(length(t), 1), t);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(t, v + n2, 'DisplayName', '$\hat{x}_2$');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(t, v + n1, 'DisplayName', '$\hat{x}_1$');
|
||||
set(gca,'ColorOrderIndex',3)
|
||||
plot(t, v + (lsim(H1, n1, t) + lsim(H2, n2, t)), 'DisplayName', '$\hat{x}$');
|
||||
plot(t, v, 'k--', 'DisplayName', '$x$');
|
||||
hold off;
|
||||
xlabel('Time [s]'); ylabel('Velocity [m/s]');
|
||||
legend('location', 'southwest');
|
||||
ylim([-0.3, 0.3]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/super_sensor_time_domain_h2.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:super_sensor_time_domain_h2
|
||||
#+caption: Noise of individual sensors and noise of the super sensor
|
||||
#+RESULTS:
|
||||
[[file:figs/super_sensor_time_domain_h2.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(t, n2, 'DisplayName', '$n_2$');
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(t, n1, 'DisplayName', '$n_1$');
|
||||
set(gca,'ColorOrderIndex',3)
|
||||
plot(t, (lsim(H1, n1, t)+lsim(H2, n2, t)), '-', 'DisplayName', '$n$');
|
||||
hold off;
|
||||
xlabel('Time [s]'); ylabel('Sensor Noise [m/s]');
|
||||
legend();
|
||||
ylim([-0.2, 0.2]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/sensor_noise_H2_time_domain.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_noise_H2_time_domain
|
||||
#+caption:
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_noise_H2_time_domain.png]]
|
||||
|
||||
** Discrepancy between sensor dynamics and model
|
||||
If we consider sensor dynamical uncertainty as explained in Section [[sec:sensor_uncertainty]], we can compute what would be the super sensor dynamical uncertainty when using the complementary filters obtained using the $\mathcal{H}_2$ Synthesis.
|
||||
|
||||
The super sensor dynamical uncertainty is shown in Figure [[fig:super_sensor_dynamical_uncertainty_H2]].
|
||||
|
||||
It is shown that the phase uncertainty is not bounded between 100Hz and 200Hz.
|
||||
As a result the super sensor signal can not be used for feedback applications about 100Hz.
|
||||
#+begin_src matlab :exports none
|
||||
Dphi_ss = 180/pi*asin(abs(squeeze(freqresp(W2*H2, freqs, 'Hz'))) + abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))));
|
||||
Dphi_ss(abs(squeeze(freqresp(W2*H2, freqs, 'Hz'))) + abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))) > 1) = 360;
|
||||
|
||||
figure;
|
||||
% Magnitude
|
||||
ax1 = subplot(2,1,1);
|
||||
hold on;
|
||||
plotMagUncertainty(W1, freqs, 'color_i', 1, 'DisplayName', '$1 + W_1 \Delta_1$');
|
||||
plotMagUncertainty(W2, freqs, 'color_i', 2, 'DisplayName', '$1 + W_2 \Delta_2$');
|
||||
plot(freqs, 1 + abs(squeeze(freqresp(W2*H2, freqs, 'Hz')))+abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))), 'k-', ...
|
||||
'DisplayName', '$1 + W_1 \Delta_1 + W_2 \Delta_2$')
|
||||
plot(freqs, max(1 - abs(squeeze(freqresp(W2*H2, freqs, 'Hz')))-abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))), 0.001), 'k-', ...
|
||||
'HandleVisibility', 'off');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylabel('Magnitude');
|
||||
ylim([1e-2, 1e1]);
|
||||
legend('location', 'southeast');
|
||||
hold off;
|
||||
|
||||
% Phase
|
||||
ax2 = subplot(2,1,2);
|
||||
hold on;
|
||||
plotPhaseUncertainty(W1, freqs, 'color_i', 1);
|
||||
plotPhaseUncertainty(W2, freqs, 'color_i', 2);
|
||||
plot(freqs, Dphi_ss, 'k-');
|
||||
plot(freqs, -Dphi_ss, 'k-');
|
||||
set(gca,'xscale','log');
|
||||
yticks(-180:90:180);
|
||||
ylim([-180 180]);
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :tangle no :exports results :results file replace
|
||||
exportFig('figs/super_sensor_dynamical_uncertainty_H2.pdf', 'width', 'full', 'height', 'full');
|
||||
#+end_src
|
||||
|
||||
#+name: fig:super_sensor_dynamical_uncertainty_H2
|
||||
#+caption: Super sensor dynamical uncertainty when using the $\mathcal{H}_2$ Synthesis
|
||||
#+RESULTS:
|
||||
[[file:figs/super_sensor_dynamical_uncertainty_H2.png]]
|
||||
|
||||
* Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
|
||||
:PROPERTIES:
|
||||
@ -757,7 +783,7 @@ To do so, we model the uncertainty that we have on the sensor dynamics by multip
|
||||
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
|
||||
[[file:figs-tikz/sensor_fusion_arch_uncertainty.png]]
|
||||
|
||||
The objective here is to design complementary filters $H_1(s)$ and $H_2(s)$ in order to minimize the dynamical uncertainty of the super sensor.
|
||||
The objective here is to design complementary filters $H_1(s)$ and $H_2(s)$ in order to bound the dynamical uncertainty of the super sensor to acceptable values.
|
||||
|
||||
** Matlab Init :noexport:ignore:
|
||||
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
||||
@ -897,7 +923,7 @@ The obtained upper bounds on the complementary filters in order to limit the pha
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:upper_bounds_comp_filter_max_phase_uncertainty
|
||||
#+CAPTION: Upper bounds on the complementary filters set in order to limit the maximum phase uncertainty of the super sensor to 30 degrees until 500Hz ([[./figs/upper_bounds_comp_filter_max_phase_uncertainty.png][png]], [[./figs/upper_bounds_comp_filter_max_phase_uncertainty.pdf][pdf]])
|
||||
#+CAPTION: Upper bounds on the complementary filters set in order to limit the maximum phase uncertainty of the super sensor to 30 degrees until 500Hz
|
||||
[[file:figs/upper_bounds_comp_filter_max_phase_uncertainty.png]]
|
||||
|
||||
** $\mathcal{H}_\infty$ Synthesis
|
||||
@ -910,8 +936,8 @@ The $\mathcal{H}_\infty$ synthesis architecture used for the complementary filte
|
||||
The generalized plant is defined below.
|
||||
#+begin_src matlab
|
||||
P = [Wu*W1 -Wu*W1;
|
||||
0 Wu*W2;
|
||||
1 0];
|
||||
0 Wu*W2;
|
||||
1 0];
|
||||
#+end_src
|
||||
|
||||
And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
|
||||
@ -988,7 +1014,7 @@ The obtained complementary filters are shown in Figure [[fig:comp_filter_hinf_un
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:comp_filter_hinf_uncertainty
|
||||
#+CAPTION: Obtained complementary filters ([[./figs/comp_filter_hinf_uncertainty.png][png]], [[./figs/comp_filter_hinf_uncertainty.pdf][pdf]])
|
||||
#+CAPTION: Obtained complementary filters
|
||||
[[file:figs/comp_filter_hinf_uncertainty.png]]
|
||||
|
||||
** Super sensor uncertainty
|
||||
@ -1081,7 +1107,7 @@ The CPS of both sensor and of the super sensor is shown in Figure [[fig:cps_sens
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:psd_sensors_hinf_synthesis
|
||||
#+CAPTION: Power Spectral Density of the obtained super sensor using the $\mathcal{H}_\infty$ synthesis ([[./figs/psd_sensors_hinf_synthesis.png][png]], [[./figs/psd_sensors_hinf_synthesis.pdf][pdf]])
|
||||
#+CAPTION: Power Spectral Density of the obtained super sensor using the $\mathcal{H}_\infty$ synthesis
|
||||
[[file:figs/psd_sensors_hinf_synthesis.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
@ -1104,7 +1130,7 @@ The CPS of both sensor and of the super sensor is shown in Figure [[fig:cps_sens
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#+end_src
|
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#+NAME: fig:cps_sensors_hinf_synthesis
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#+CAPTION: Cumulative Power Spectrum of the obtained super sensor using the $\mathcal{H}_\infty$ synthesis ([[./figs/cps_sensors_hinf_synthesis.png][png]], [[./figs/cps_sensors_hinf_synthesis.cps][cps]])
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#+CAPTION: Cumulative Power Spectrum of the obtained super sensor using the $\mathcal{H}_\infty$ synthesis
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[[file:figs/cps_sensors_hinf_synthesis.png]]
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** Conclusion
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@ -1180,7 +1206,7 @@ Both dynamical uncertainty and noise characteristics of the individual sensors a
|
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#+end_src
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#+NAME: fig:mixed_synthesis_noise_uncertainty_sensors
|
||||
#+CAPTION: Noise characteristsics and Dynamical uncertainty of the individual sensors ([[./figs/mixed_synthesis_noise_uncertainty_sensors.png][png]], [[./figs/mixed_synthesis_noise_uncertainty_sensors.pdf][pdf]])
|
||||
#+CAPTION: Noise characteristsics and Dynamical uncertainty of the individual sensors
|
||||
[[file:figs/mixed_synthesis_noise_uncertainty_sensors.png]]
|
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|
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** Weighting Functions on the uncertainty of the super sensor
|
||||
@ -1287,7 +1313,7 @@ The obtained complementary filters are shown in Figure [[fig:comp_filters_mixed_
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:comp_filters_mixed_synthesis
|
||||
#+CAPTION: Obtained complementary filters after mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis ([[./figs/comp_filters_mixed_synthesis.png][png]], [[./figs/comp_filters_mixed_synthesis.pdf][pdf]])
|
||||
#+CAPTION: Obtained complementary filters after mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
|
||||
[[file:figs/comp_filters_mixed_synthesis.png]]
|
||||
|
||||
** Obtained Super Sensor's noise
|
||||
@ -1322,7 +1348,7 @@ The PSD and CPS of the super sensor's noise are shown in Figure [[fig:psd_super_
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:psd_super_sensor_mixed_syn
|
||||
#+CAPTION: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis ([[./figs/psd_super_sensor_mixed_syn.png][png]], [[./figs/psd_super_sensor_mixed_syn.pdf][pdf]])
|
||||
#+CAPTION: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
|
||||
[[file:figs/psd_super_sensor_mixed_syn.png]]
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
@ -1345,7 +1371,7 @@ The PSD and CPS of the super sensor's noise are shown in Figure [[fig:psd_super_
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:cps_super_sensor_mixed_syn
|
||||
#+CAPTION: Cumulative Power Spectrum of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis ([[./figs/cps_super_sensor_mixed_syn.png][png]], [[./figs/cps_super_sensor_mixed_syn.pdf][pdf]])
|
||||
#+CAPTION: Cumulative Power Spectrum of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
|
||||
[[file:figs/cps_super_sensor_mixed_syn.png]]
|
||||
|
||||
** Obtained Super Sensor's Uncertainty
|
||||
@ -1503,7 +1529,7 @@ This Matlab function is accessible [[file:src/plotMagUncertainty.m][here]].
|
||||
freqs double {mustBeNumeric, mustBeNonnegative}
|
||||
args.G = tf(1)
|
||||
args.color_i (1,1) double {mustBeInteger, mustBePositive} = 1
|
||||
args.opacity (1,1) double {mustBeNumeric, mustBePositive} = 0.3
|
||||
args.opacity (1,1) double {mustBeNumeric, mustBeNonnegative} = 0.3
|
||||
args.DisplayName char = ''
|
||||
end
|
||||
|
||||
|
||||
@ -19,7 +19,7 @@ arguments
|
||||
freqs double {mustBeNumeric, mustBeNonnegative}
|
||||
args.G = tf(1)
|
||||
args.color_i (1,1) double {mustBeInteger, mustBePositive} = 1
|
||||
args.opacity (1,1) double {mustBeNumeric, mustBePositive} = 0.3
|
||||
args.opacity (1,1) double {mustBeNumeric, mustBeNonnegative} = 0.3
|
||||
args.DisplayName char = ''
|
||||
end
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user