Reworked all the analysis about optimal sensor fusion
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matlab/index.org
@ -86,61 +86,65 @@ The complementary filters have to be designed in order to minimize the effect no
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#+end_src
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** Architecture
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Let's consider the sensor fusion architecture shown on figure [[fig:fusion_two_noisy_sensors_with_dyn]] where two sensors 1 and 2 are measuring the same quantity $x$ with different noise characteristics determined by $W_1$ and $W_2$.
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Let's consider the sensor fusion architecture shown on figure [[fig:fusion_two_noisy_sensors_weights]] where two sensors (sensor 1 and sensor 2) are measuring the same quantity $x$ with different noise characteristics determined by $N_1(s)$ and $N_2(s)$.
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$n_1$ and $n_2$ are white noise (constant power spectral density over all frequencies).
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$\tilde{n}_1$ and $\tilde{n}_2$ are normalized white noise:
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#+name: eq:normalized_noise
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\begin{equation}
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\Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_1}(\omega) = 1
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\end{equation}
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#+name: fig:fusion_two_noisy_sensors_with_dyn
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#+name: fig:fusion_two_noisy_sensors_weights
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#+caption: Fusion of two sensors
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[[file:figs-tikz/fusion_two_noisy_sensors_with_dyn.png]]
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[[file:figs-tikz/fusion_two_noisy_sensors_weights.png]]
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We consider that the two sensor dynamics $G_1$ and $G_2$ are ideal ($G_1 = G_2 = 1$). We obtain the architecture of figure [[fig:fusion_two_noisy_sensors]].
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We consider that the two sensor dynamics $G_1(s)$ and $G_2(s)$ are ideal:
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#+name: eq:idea_dynamics
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\begin{equation}
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G_1(s) = G_2(s) = 1
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\end{equation}
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#+name: fig:fusion_two_noisy_sensors
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We obtain the architecture of figure [[fig:sensor_fusion_noisy_perfect_dyn]].
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#+name: fig:sensor_fusion_noisy_perfect_dyn
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#+caption: Fusion of two sensors with ideal dynamics
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[[file:figs-tikz/fusion_two_noisy_sensors.png]]
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[[file:figs-tikz/sensor_fusion_noisy_perfect_dyn.png]]
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$H_1$ and $H_2$ are complementary filters ($H_1 + H_2 = 1$). The goal is to design $H_1$ and $H_2$ such that the effect of the noise sources $n_1$ and $n_2$ has the smallest possible effect on the estimation $\hat{x}$.
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$H_1(s)$ and $H_2(s)$ are complementary filters:
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#+name: eq:comp_filters_property
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\begin{equation}
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H_1(s) + H_2(s) = 1
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\end{equation}
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The goal is to design $H_1(s)$ and $H_2(s)$ such that the effect of the noise sources $\tilde{n}_1$ and $\tilde{n}_2$ has the smallest possible effect on the estimation $\hat{x}$.
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We have that the Power Spectral Density (PSD) of $\hat{x}$ is:
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\[ \Gamma_{\hat{x}} = |H_1 W_1|^2 \Gamma_{n_1} + |H_2 W_2|^2 \Gamma_{n_2} \]
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\[ \Phi_{\hat{x}}(\omega) = |H_1(j\omega) N_1(j\omega)|^2 \Phi_{\tilde{n}_1}(\omega) + |H_2(j\omega) N_2(j\omega)|^2 \Phi_{\tilde{n}_2}(\omega), \quad \forall \omega \]
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And the goal is the minimize the Root Mean Square (RMS) value of $\hat{x}$:
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\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty \Gamma_{\hat{x}}(\omega) d\omega} \]
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As $n_1$ and $n_2$ are white noise: $\Gamma_{n_1} = \Gamma_{n_2} = 1$ and we have:
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\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty |H_1 W_1|^2(\omega) + |H_2 W_2|^2(\omega) d\omega} = \left\| \begin{matrix} H_1 W_1 \\ H_2 W_2 \end{matrix} \right\|_2 \]
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Thus, the goal is to design $H_1$ and $H_2$ such that $H_1 + H_2 = 1$ and such that $\left\| \begin{matrix} H_1 W_1 \\ H_2 W_2 \end{matrix} \right\|_2$ is minimized.
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For that, we will use the $\mathcal{H}_2$ Synthesis.
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#+name: eq:rms_value_estimation
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\begin{equation}
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\sigma_{\hat{x}} = \sqrt{\int_0^\infty \Phi_{\hat{x}}(\omega) d\omega}
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\end{equation}
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** Noise of the sensors
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Let's define the noise characteristics of the two sensors by choosing $W_1$ and $W_2$:
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- Sensor 1 characterized by $W_1$ has low noise at low frequency (for instance a geophone)
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- Sensor 2 characterized by $W_2$ has low noise at high frequency (for instance an accelerometer)
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#+begin_src matlab :exports none
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omegac = 2*pi; G0 = 1e-2; Ginf = 1e-6;
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W1 = ((sqrt(G0))/(s/omegac + 1))^2;
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omegac = 100*2*pi; G0 = 1e-6; Ginf = 1e-2;
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W2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
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#+end_src
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Let's define the noise characteristics of the two sensors by choosing $N_1$ and $N_2$:
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- Sensor 1 characterized by $N_1(s)$ has low noise at low frequency (for instance a geophone)
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- Sensor 2 characterized by $N_2(s)$ has low noise at high frequency (for instance an accelerometer)
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#+begin_src matlab
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omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
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W1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000);
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N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/100);
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omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
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W2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
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N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
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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, abs(squeeze(freqresp(W1, freqs, 'Hz'))), '-', 'DisplayName', '$W_1$');
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plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))), '-', 'DisplayName', '$W_2$');
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plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))), '-', 'DisplayName', '$N_1$');
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plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))), '-', 'DisplayName', '$N_2$');
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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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@ -149,15 +153,21 @@ Let's define the noise characteristics of the two sensors by choosing $W_1$ and
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#+end_src
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#+HEADER: :tangle no :exports results :results none :noweb yes
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#+begin_src matlab :var filepath="figs/nosie_characteristics_sensors.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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#+begin_src matlab :var filepath="figs/noise_characteristics_sensors.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
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<<plt-matlab>>
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#+end_src
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#+NAME: fig:nosie_characteristics_sensors
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#+CAPTION: Noise Characteristics of the two sensors ([[./figs/nosie_characteristics_sensors.png][png]], [[./figs/nosie_characteristics_sensors.pdf][pdf]])
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[[file:figs/nosie_characteristics_sensors.png]]
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#+NAME: fig:noise_characteristics_sensors
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#+CAPTION: Noise Characteristics of the two sensors ([[./figs/noise_characteristics_sensors.png][png]], [[./figs/noise_characteristics_sensors.pdf][pdf]])
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[[file:figs/noise_characteristics_sensors.png]]
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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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For that, we use the $\mathcal{H}_2$ Synthesis.
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We use the generalized plant architecture shown on figure [[fig:h_infinity_optimal_comp_filters]].
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#+name: fig:h_infinity_optimal_comp_filters
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@ -165,16 +175,16 @@ We use the generalized plant architecture shown on figure [[fig:h_infinity_optim
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[[file:figs-tikz/h_infinity_optimal_comp_filters.png]]
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The transfer function from $[n_1, n_2]$ to $\hat{x}$ is:
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\[ \begin{bmatrix} W_1 H_1 \\ W_2 (1 - H_1) \end{bmatrix} \]
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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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\[ \begin{bmatrix} W_1 H_1 \\ W_2 H_2 \end{bmatrix} \]
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\[ \begin{bmatrix} N_1 H_1 \\ N_2 H_2 \end{bmatrix} \]
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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 define the generalized plant $P$ on matlab as shown on figure [[fig:h_infinity_optimal_comp_filters]].
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#+begin_src matlab
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P = [0 W2 1;
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W1 -W2 0];
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P = [0 N2 1;
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N1 -N2 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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@ -182,17 +192,18 @@ And we do the $\mathcal{H}_2$ synthesis using the =h2syn= command.
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[H1, ~, gamma] = h2syn(P, 1, 1);
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#+end_src
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What is minimized is =norm([W1*H1,W2*H2], 2)=.
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Finally, we define $H_2 = 1 - H_1$.
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Finally, we define $H_2(s) = 1 - H_1(s)$.
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#+begin_src matlab
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H2 = 1 - H1;
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#+end_src
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** Analysis
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The complementary filters obtained are shown on figure [[fig:htwo_comp_filters]]. The PSD of the [[fig:psd_sensors_htwo_synthesis]].
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Finally, the RMS value of $\hat{x}$ is shown on table [[tab:rms_results]].
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The optimal sensor fusion has permitted to reduced the RMS value of the estimation error by a factor 8 compare to when using only one sensor.
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The complementary filters obtained are shown on figure [[fig:htwo_comp_filters]].
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The PSD of the noise of the individual sensor and of the super sensor are shown in Fig. [[fig:psd_sensors_htwo_synthesis]].
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The Cumulative Power Spectrum (CPS) is shown on Fig. [[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 :exports none
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figure;
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@ -215,14 +226,20 @@ The optimal sensor fusion has permitted to reduced the RMS value of the estimati
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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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[[file:figs/htwo_comp_filters.png]]
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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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#+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, abs(squeeze(freqresp(W1, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|W_1|^2$');
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plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|W_2|^2$');
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plot(freqs, abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(W2*H2, freqs, 'Hz'))).^2, 'k-', 'DisplayName', '$|W_1H_1|^2+|W_2H_2|^2$');
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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('Magnitude');
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xlabel('Frequency [Hz]'); ylabel('Power Spectral Density');
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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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@ -237,77 +254,76 @@ The optimal sensor fusion has permitted to reduced the RMS value of the estimati
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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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[[file:figs/psd_sensors_htwo_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([norm([W1], 2);norm([W2], 2);norm([W1*H1 + W2*H2], 2)], {'Sensor 1', 'Sensor 2', 'Optimal Sensor Fusion'}, {'rms value'}, ' %.1e');
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#+end_src
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#+name: tab:rms_results
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#+caption: RMS value of the estimation error when using the sensor individually and when using the two sensor merged using the optimal complementary filters
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#+RESULTS:
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| | rms value |
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|-----------------------+-----------|
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| Sensor 1 | 1.1e-02 |
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| Sensor 2 | 1.3e-03 |
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| Optimal Sensor Fusion | 1.5e-04 |
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** H-Infinity Synthesis
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*** First idea
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Another objective that we may have is that the noise of the super sensor $n_{SS}$ is following the minimum of the noise of the two sensors $n_1$ and $n_2$:
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\[ \Gamma_{n_{ss}}(\omega) = \min(\Gamma_{n_1}(\omega),\ \Gamma_{n_2}(\omega)) \]
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In order to obtain that ideal case, we need that the complementary filters be designed such that:
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\begin{align*}
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& H_1(j\omega) = 1 \text{ and } H_2(j\omega) = 0 \text{ at frequencies where } \Gamma_{n_1}(\omega) < \Gamma_{n_2}(\omega) \\
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& H_1(j\omega) = 0 \text{ and } H_2(j\omega) = 1 \text{ at frequencies where } \Gamma_{n_1}(\omega) > \Gamma_{n_2}(\omega)
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\end{align*}
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Which is indeed *impossible in practice*.
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We could try to use filters with the highest possible order to approximate that.
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However, we may have robustness and implementation problems.
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#+begin_src matlab
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omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
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N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000);
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omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
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N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
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#+end_src
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#+begin_src matlab
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n = 5; w0 = 2*pi*10; G0 = 1/10; G1 = 10000; Gc = 1/2;
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W1 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
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n = 5; w0 = 2*pi*8; G0 = 1000; G1 = 0.1; Gc = 1/2;
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W2 = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
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CPS_S1 = 1/pi*cumtrapz(2*pi*freqs, PSD_S1);
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CPS_S2 = 1/pi*cumtrapz(2*pi*freqs, PSD_S2);
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CPS_H2 = 1/pi*cumtrapz(2*pi*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, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$W_1 = \frac{N_1}{N_2}$');
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plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$W_2 = \frac{N_2}{N_1}$');
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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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plot(freqs, CPS_S1, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_1} = %.1e$', sqrt(CPS_S1(end))));
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plot(freqs, CPS_S2, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_2} = %.1e$', sqrt(CPS_S2(end))));
|
||||
plot(freqs, CPS_H2, 'k-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_2}} = %.1e$', sqrt(CPS_H2(end))));
|
||||
set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
legend('location', 'northeast');
|
||||
xlim([2e-1, freqs(end)]);
|
||||
ylim([1e-10 1e-5]);
|
||||
legend('location', 'southeast');
|
||||
#+end_src
|
||||
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/cps_h2_synthesis.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:cps_h2_synthesis
|
||||
#+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]])
|
||||
[[file:figs/cps_h2_synthesis.png]]
|
||||
|
||||
** H-Infinity Synthesis - method A
|
||||
Another objective that we may have is that the noise of the super sensor $n_{SS}$ is following the minimum of the noise of the two sensors $n_1$ and $n_2$:
|
||||
\[ \Gamma_{n_{ss}}(\omega) = \min(\Gamma_{n_1}(\omega),\ \Gamma_{n_2}(\omega)) \]
|
||||
|
||||
In order to obtain that ideal case, we need that the complementary filters be designed such that:
|
||||
\begin{align*}
|
||||
& |H_1(j\omega)| = 1 \text{ and } |H_2(j\omega)| = 0 \text{ at frequencies where } \Gamma_{n_1}(\omega) < \Gamma_{n_2}(\omega) \\
|
||||
& |H_1(j\omega)| = 0 \text{ and } |H_2(j\omega)| = 1 \text{ at frequencies where } \Gamma_{n_1}(\omega) > \Gamma_{n_2}(\omega)
|
||||
\end{align*}
|
||||
|
||||
Which is indeed impossible in practice.
|
||||
|
||||
We could try to approach that with the $\mathcal{H}_\infty$ synthesis by using high order filters.
|
||||
|
||||
As shown on Fig. [[fig:noise_characteristics_sensors]], the frequency where the two sensors have the same noise level is around 9Hz.
|
||||
We will thus choose weighting functions such that the merging frequency is around 9Hz.
|
||||
|
||||
The weighting functions used as well as the obtained complementary filters are shown in Fig. [[fig:weights_comp_filters_Hinfa]].
|
||||
|
||||
#+begin_src matlab
|
||||
n = 5; w0 = 2*pi*10; G0 = 1/10; G1 = 10000; Gc = 1/2;
|
||||
W1a = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
|
||||
|
||||
n = 5; w0 = 2*pi*8; G0 = 1000; G1 = 0.1; Gc = 1/2;
|
||||
W2a = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
P = [W1 -W1;
|
||||
0 W2;
|
||||
P = [W1a -W1a;
|
||||
0 W2a;
|
||||
1 0];
|
||||
#+end_src
|
||||
|
||||
And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
|
||||
#+begin_src matlab :results output replace :exports both
|
||||
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
[H2a, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
#+begin_example
|
||||
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
[H2a, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
Resetting value of Gamma min based on D_11, D_12, D_21 terms
|
||||
|
||||
Test bounds: 0.1000 < gamma <= 10500.0000
|
||||
@ -343,7 +359,7 @@ Test bounds: 0.1000 < gamma <= 10500.0000
|
||||
#+end_example
|
||||
|
||||
#+begin_src matlab
|
||||
H1 = 1 - H2;
|
||||
H1a = 1 - H2a;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
@ -352,14 +368,14 @@ Test bounds: 0.1000 < gamma <= 10500.0000
|
||||
ax1 = subplot(2,1,1);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1a, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2a, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
|
||||
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
|
||||
plot(freqs, abs(squeeze(freqresp(H1a, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
|
||||
plot(freqs, abs(squeeze(freqresp(H2a, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
|
||||
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
@ -371,9 +387,9 @@ Test bounds: 0.1000 < gamma <= 10500.0000
|
||||
ax2 = subplot(2,1,2);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H1a, freqs, 'Hz'))), '-');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H2a, freqs, 'Hz'))), '-');
|
||||
hold off;
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
set(gca, 'XScale', 'log');
|
||||
@ -384,135 +400,100 @@ Test bounds: 0.1000 < gamma <= 10500.0000
|
||||
xticks([0.1, 1, 10, 100, 1000]);
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|N_1|^2$');
|
||||
plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|N_2|^2$');
|
||||
plot(freqs, abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2, 'k-', 'DisplayName', '$|N_1H_1|^2+|N_2H_2|^2$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
legend('location', 'northeast');
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/weights_comp_filters_Hinfa.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
||||
data2orgtable([norm([N1], 2);norm([N2], 2);norm([N1*H1 + N2*H2], 2)], {'Sensor 1', 'Sensor 2', 'Optimal Sensor Fusion'}, {'rms value'}, ' %.1e');
|
||||
#+NAME: fig:weights_comp_filters_Hinfa
|
||||
#+CAPTION: Weights and Complementary Fitlers obtained ([[./figs/weights_comp_filters_Hinfa.png][png]], [[./figs/weights_comp_filters_Hinfa.pdf][pdf]])
|
||||
[[file:figs/weights_comp_filters_Hinfa.png]]
|
||||
|
||||
We then compute the Power Spectral Density as well as the Cumulative Power Spectrum.
|
||||
|
||||
#+begin_src matlab
|
||||
PSD_Ha = abs(squeeze(freqresp(N1*H1a, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2a, freqs, 'Hz'))).^2;
|
||||
CPS_Ha = 1/pi*cumtrapz(2*pi*freqs, PSD_Ha);
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
| | rms value |
|
||||
|-----------------------+-----------|
|
||||
| Sensor 1 | 1.1e-02 |
|
||||
| Sensor 2 | 1.3e-03 |
|
||||
| Optimal Sensor Fusion | 3.2e-04 |
|
||||
|
||||
*** Second idea
|
||||
** H-Infinity Synthesis - method B
|
||||
We have that:
|
||||
\[ \Phi_{n_{ss}} = \left|H_1\right|^2 \Phi_{n_1} + \left|H_2\right|^2 \Phi_{n_2} \]
|
||||
\[ \Phi_{\hat{x}}(\omega) = \left|H_1(j\omega) N_1(j\omega)\right|^2 + \left|H_2(j\omega) N_2(j\omega)\right|^2 \]
|
||||
|
||||
We model the Power Spectral Densities of the two sensors by weighting functions $N_1(s)$ and $N_2(s)$.
|
||||
\[ \Phi_{n_{ss}} = \left|H_1 N_1\right|^2 + \left|H_2\right N_2|^2 \]
|
||||
Then, at frequencies where $|H_1(j\omega)| < |H_2(j\omega)|$ we would like that $|N_1(j\omega)| = 1$ and $|N_2(j\omega)| = 0$ as we discussed before.
|
||||
Then $|H_1 N_1|^2 + |H_2 N_2|^2 = |N_1|^2$.
|
||||
|
||||
Then, at frequencies where $|H_1| < |H_2|$ we would like that $|N_1| = 1$ and $|N_2| = 0$ as we discussed before.
|
||||
Then $|H_1 N_1|^2 + |H_2 N_2|^2 = |H_1 N_1|^2$.
|
||||
|
||||
However, if we design $H_2$ such that when $|N_1| < |N_2|$, we have that:
|
||||
We know that this is impossible in practice. A more realistic choice is to design $H_2(s)$ such that when $|N_2(j\omega)| > |N_1(j\omega)|$, we have that:
|
||||
\[ |H_2 N_2|^2 = \epsilon |H_1 N_1|^2 \]
|
||||
|
||||
Which is equivalent to have (by supposing $|H_1| \approx 1$):
|
||||
\[ |H_2| = \sqrt{\epsilon} \frac{|N_1|}{|N_2|} \]
|
||||
|
||||
And we have:
|
||||
\begin{align*}
|
||||
\Phi_{n_{ss}} &= \left|H_1 N_1\right|^2 + |H_2 N_2|^2 \\
|
||||
&= (1 + \epsilon)\left|H_1 N_1\right|^2 \\
|
||||
&\approx \left|H_1 N_1\right|^2
|
||||
\Phi_{\hat{x}} &= \left|H_1 N_1\right|^2 + |H_2 N_2|^2 \\
|
||||
&= (1 + \epsilon) \left| H_1 N_1 \right|^2 \\
|
||||
&\approx \left|N_1\right|^2
|
||||
\end{align*}
|
||||
|
||||
We have then that:
|
||||
\[ |H_2 N_2| = \sqrt{\epsilon} |H_1 N_1| \]
|
||||
We suppose that $|H_1| = 1$:
|
||||
\[ |H_2| = \sqrt{\epsilon} \frac{|N_1|}{|N_2|} \]
|
||||
We can use that for the maximum magnitude of $|H_2|$ at frequencies where $|N_1| < |N_2|$
|
||||
|
||||
Similarly, we have that the maximum magnitude of $|H_1|$ at frequencies where $|N_1| > |N_2|$ is:
|
||||
Similarly, we design $H_1(s)$ such that at frequencies where $|N_1| > |N_2|$:
|
||||
\[ |H_1| = \sqrt{\epsilon} \frac{|N_2|}{|N_1|} \]
|
||||
|
||||
We could take $\epsilon = 1$, then we increase the noise of the super sensor just by a factor $\sqrt{2}$ over the all bandwidth from the idea case.
|
||||
For instance, is we take $\epsilon = 1$, then the PSD of $\hat{x}$ is increased by just by a factor $\sqrt{2}$ over the all frequencies from the idea case.
|
||||
|
||||
This could be used as weighting functions for the $\mathcal{H}_\infty$ synthesis of the complementary filters.
|
||||
We use this as the weighting functions for the $\mathcal{H}_\infty$ synthesis of the complementary filters.
|
||||
|
||||
The weighting function and the obtained complementary filters are shown in Fig. [[fig:weights_comp_filters_Hinfb]].
|
||||
|
||||
#+begin_src matlab
|
||||
omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
|
||||
N1 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/4000);
|
||||
epsilon = 2;
|
||||
|
||||
omegac = 1*2*pi; G0 = 1e-3; Ginf = 1e-8;
|
||||
N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
|
||||
W1b = 1/epsilon*N1/N2;
|
||||
W2b = 1/epsilon*N2/N1;
|
||||
|
||||
W1b = W1b/(1 + s/2/pi/1000); % this is added so that it is proper
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
W1 = N1/N2;
|
||||
W2 = N2/N1;
|
||||
|
||||
W1 = W1/(1 + s/2/pi/1000); % this is added so that it is proper
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$W_1 = \frac{N_1}{N_2}$');
|
||||
plot(freqs, abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$W_2 = \frac{N_2}{N_1}$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
legend('location', 'northeast');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab
|
||||
P = [W1 -W1;
|
||||
0 W2;
|
||||
P = [W1b -W1b;
|
||||
0 W2b;
|
||||
1 0];
|
||||
#+end_src
|
||||
|
||||
And we do the $\mathcal{H}_\infty$ synthesis using the =hinfsyn= command.
|
||||
#+begin_src matlab :results output replace :exports both
|
||||
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
[H2b, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
#+begin_example
|
||||
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
Test bounds: 0.0000 < gamma <= 2625.0000
|
||||
[H2b, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
Test bounds: 0.0000 < gamma <= 32.8125
|
||||
|
||||
gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f
|
||||
2.625e+03 1.8e+01 -2.3e-12 6.3e+00 -1.7e-14 0.0000 p
|
||||
1.313e+03 1.8e+01 -3.3e-12 6.3e+00 -5.9e-13 0.0000 p
|
||||
656.250 1.8e+01 -1.8e-12 6.3e+00 -1.2e-13 0.0000 p
|
||||
328.125 1.8e+01 -8.3e-13 6.3e+00 -7.5e-14 0.0000 p
|
||||
164.063 1.8e+01 -3.8e-13 6.3e+00 -6.1e-14 0.0000 p
|
||||
82.031 1.8e+01 -3.3e-22 6.3e+00 -5.4e-14 0.0000 p
|
||||
41.016 1.8e+01 -3.4e-12 6.3e+00 -1.4e-16 0.0000 p
|
||||
20.508 1.8e+01 -6.4e-14 6.3e+00 -4.4e-13 0.0000 p
|
||||
10.254 1.8e+01 -1.4e-13 6.3e+00 -4.9e-14 0.0000 p
|
||||
5.127 1.7e+01 -1.1e-12 6.3e+00 -1.6e-13 0.0000 p
|
||||
2.563 1.7e+01 -6.5e-13 6.3e+00 -7.3e-14 0.0000 p
|
||||
1.282 1.4e+01 -8.9e+04# 6.3e+00 -1.6e-14 0.0000 f
|
||||
1.923 1.6e+01 -4.6e+05# 6.3e+00 -2.1e-13 0.0000 f
|
||||
2.243 1.7e+01 -1.9e+06# 6.3e+00 -9.9e-14 0.0000 f
|
||||
2.403 1.7e+01 -3.7e-11 6.3e+00 -2.1e-12 0.0000 p
|
||||
2.323 1.7e+01 -4.6e+06# 6.3e+00 -5.8e-14 0.0000 f
|
||||
2.363 1.7e+01 -1.2e+07# 6.3e+00 -3.3e-14 0.0000 f
|
||||
2.383 1.7e+01 -6.8e+07# 6.3e+00 -1.2e-13 0.0000 f
|
||||
2.393 1.7e+01 -5.5e-11 6.3e+00 -1.8e-12 0.0000 p
|
||||
2.388 1.7e+01 1.3e-22 6.3e+00 -2.5e-15 0.0000 p
|
||||
2.386 1.7e+01 -1.5e+08# 6.3e+00 -1.4e-13 0.0000 f
|
||||
2.387 1.7e+01 -4.0e+08# 6.3e+00 -1.0e-13 0.0000 f
|
||||
2.388 1.7e+01 -2.1e+09# 6.3e+00 -4.2e-13 0.0000 f
|
||||
32.812 1.8e+01 3.4e-10 6.3e+00 -2.9e-13 0.0000 p
|
||||
16.406 1.8e+01 3.4e-10 6.3e+00 -1.2e-15 0.0000 p
|
||||
8.203 1.8e+01 3.3e-10 6.3e+00 -2.6e-13 0.0000 p
|
||||
4.102 1.8e+01 3.3e-10 6.3e+00 -2.1e-13 0.0000 p
|
||||
2.051 1.7e+01 3.4e-10 6.3e+00 -7.2e-16 0.0000 p
|
||||
1.025 1.6e+01 -1.3e+06# 6.3e+00 -8.3e-14 0.0000 f
|
||||
1.538 1.7e+01 3.4e-10 6.3e+00 -2.0e-13 0.0000 p
|
||||
1.282 1.7e+01 3.4e-10 6.3e+00 -7.9e-17 0.0000 p
|
||||
1.154 1.7e+01 3.6e-10 6.3e+00 -1.8e-13 0.0000 p
|
||||
1.089 1.7e+01 -3.4e+06# 6.3e+00 -1.7e-13 0.0000 f
|
||||
1.122 1.7e+01 -1.0e+07# 6.3e+00 -3.2e-13 0.0000 f
|
||||
1.138 1.7e+01 -1.3e+08# 6.3e+00 -1.8e-13 0.0000 f
|
||||
1.146 1.7e+01 3.2e-10 6.3e+00 -3.0e-13 0.0000 p
|
||||
1.142 1.7e+01 5.5e-10 6.3e+00 -2.8e-13 0.0000 p
|
||||
1.140 1.7e+01 -1.5e-10 6.3e+00 -2.3e-13 0.0000 p
|
||||
1.139 1.7e+01 -4.8e+08# 6.3e+00 -6.2e-14 0.0000 f
|
||||
1.139 1.7e+01 1.3e-09 6.3e+00 -8.9e-17 0.0000 p
|
||||
|
||||
Gamma value achieved: 2.3882
|
||||
Gamma value achieved: 1.1390
|
||||
#+end_example
|
||||
|
||||
#+begin_src matlab
|
||||
H1 = 1 - H2;
|
||||
H1b = 1 - H2b;
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
@ -521,14 +502,14 @@ Test bounds: 0.0000 < gamma <= 2625.0000
|
||||
ax1 = subplot(2,1,1);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W1b, freqs, 'Hz'))), '--', 'DisplayName', '$w_1$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
|
||||
plot(freqs, 1./abs(squeeze(freqresp(W2b, freqs, 'Hz'))), '--', 'DisplayName', '$w_2$');
|
||||
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
|
||||
plot(freqs, abs(squeeze(freqresp(H1b, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
|
||||
plot(freqs, abs(squeeze(freqresp(H2b, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
|
||||
|
||||
hold off;
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
@ -540,9 +521,9 @@ Test bounds: 0.0000 < gamma <= 2625.0000
|
||||
ax2 = subplot(2,1,2);
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H1b, freqs, 'Hz'))), '-');
|
||||
set(gca,'ColorOrderIndex',2)
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
|
||||
plot(freqs, 180/pi*phase(squeeze(freqresp(H2b, freqs, 'Hz'))), '-');
|
||||
hold off;
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
set(gca, 'XScale', 'log');
|
||||
@ -553,32 +534,96 @@ Test bounds: 0.0000 < gamma <= 2625.0000
|
||||
xticks([0.1, 1, 10, 100, 1000]);
|
||||
#+end_src
|
||||
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/weights_comp_filters_Hinfb.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:weights_comp_filters_Hinfb
|
||||
#+CAPTION: Weights and Complementary Fitlers obtained ([[./figs/weights_comp_filters_Hinfb.png][png]], [[./figs/weights_comp_filters_Hinfb.pdf][pdf]])
|
||||
[[file:figs/weights_comp_filters_Hinfb.png]]
|
||||
|
||||
#+begin_src matlab
|
||||
PSD_Hb = abs(squeeze(freqresp(N1*H1b, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2b, freqs, 'Hz'))).^2;
|
||||
CPS_Hb = 1/pi*cumtrapz(2*pi*freqs, PSD_Hb);
|
||||
#+end_src
|
||||
|
||||
** Comparison of the methods
|
||||
The three methods are now compared.
|
||||
|
||||
The Power Spectral Density of the super sensors obtained with the complementary filters designed using the three methods are shown in Fig. [[fig:comparison_psd_noise]].
|
||||
|
||||
The Cumulative Power Spectrum for the same sensors are shown on Fig. [[fig:comparison_cps_noise]].
|
||||
|
||||
The RMS value of the obtained super sensors are shown on table [[tab:rms_results]].
|
||||
|
||||
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
||||
data2orgtable([norm([N1], 2) ; norm([N2], 2) ; norm([N1*H1, N2*H2], 2) ; norm([N1*H1a, N2*H2a], 2) ; norm([N1*H1b, N2*H2b], 2)], {'Sensor 1', 'Sensor 2', 'H2 Fusion', 'H-Infinity a', 'H-Infinity b'}, {'rms value'}, ' %.1e');
|
||||
#+end_src
|
||||
|
||||
#+name: tab:rms_results
|
||||
#+caption: RMS value of the estimation error when using the sensor individually and when using the two sensor merged using the optimal complementary filters
|
||||
#+RESULTS:
|
||||
| | rms value |
|
||||
|--------------+-----------|
|
||||
| Sensor 1 | 1.3e-03 |
|
||||
| Sensor 2 | 1.3e-03 |
|
||||
| H2 Fusion | 1.2e-04 |
|
||||
| H-Infinity a | 2.4e-04 |
|
||||
| H-Infinity b | 1.4e-04 |
|
||||
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|N_1|^2$');
|
||||
plot(freqs, abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2, '-', 'DisplayName', '$|N_2|^2$');
|
||||
plot(freqs, abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2, 'k-', 'DisplayName', '$|N_1H_1|^2+|N_2H_2|^2$');
|
||||
plot(freqs, PSD_S1, '-', 'DisplayName', '$\Phi_{\hat{x}_1}$');
|
||||
plot(freqs, PSD_S2, '-', 'DisplayName', '$\Phi_{\hat{x}_2}$');
|
||||
plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_2}}$');
|
||||
plot(freqs, PSD_Ha, 'k--', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_\infty},a}$');
|
||||
plot(freqs, PSD_Hb, 'k-.', 'DisplayName', '$\Phi_{\hat{x}_{\mathcal{H}_\infty},b}$');
|
||||
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
||||
xlabel('Frequency [Hz]'); ylabel('Power Spectral Density');
|
||||
hold off;
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
legend('location', 'northeast');
|
||||
#+end_src
|
||||
|
||||
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
|
||||
data2orgtable([norm([N1], 2);norm([N2], 2);norm([N1*H1 + N2*H2], 2)], {'Sensor 1', 'Sensor 2', 'Optimal Sensor Fusion'}, {'rms value'}, ' %.1e');
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/comparison_psd_noise.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
| | rms value |
|
||||
|-----------------------+-----------|
|
||||
| Sensor 1 | 1.1e-02 |
|
||||
| Sensor 2 | 1.3e-03 |
|
||||
| Optimal Sensor Fusion | 1.9e-04 |
|
||||
#+NAME: fig:comparison_psd_noise
|
||||
#+CAPTION: Comparison of the obtained Power Spectral Density using the three methods ([[./figs/comparison_psd_noise.png][png]], [[./figs/comparison_psd_noise.pdf][pdf]])
|
||||
[[file:figs/comparison_psd_noise.png]]
|
||||
|
||||
*** Conclusion
|
||||
From the above two experiments with $\mathcal{H}_\infty$ 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 :exports none
|
||||
figure;
|
||||
hold on;
|
||||
plot(freqs, CPS_S1, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_1} = %.1e$', sqrt(CPS_S1(end))));
|
||||
plot(freqs, CPS_S2, '-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_2} = %.1e$', sqrt(CPS_S2(end))));
|
||||
plot(freqs, CPS_H2, 'k-', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_2}} = %.1e$', sqrt(CPS_H2(end))));
|
||||
plot(freqs, CPS_Ha, 'k--', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_\\infty, a}} = %.1e$', sqrt(CPS_Ha(end))));
|
||||
plot(freqs, CPS_Hb, 'k-.', 'DisplayName', sprintf('$\\sigma_{\\hat{x}_{\\mathcal{H}_\\infty, b}} = %.1e$', sqrt(CPS_Hb(end))));
|
||||
set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum');
|
||||
hold off;
|
||||
xlim([2e-1, freqs(end)]);
|
||||
ylim([1e-10 1e-5]);
|
||||
legend('location', 'southeast');
|
||||
#+end_src
|
||||
|
||||
#+HEADER: :tangle no :exports results :results none :noweb yes
|
||||
#+begin_src matlab :var filepath="figs/comparison_cps_noise.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png")
|
||||
<<plt-matlab>>
|
||||
#+end_src
|
||||
|
||||
#+NAME: fig:comparison_cps_noise
|
||||
#+CAPTION: Comparison of the obtained Cumulative Power Spectrum using the three methods ([[./figs/comparison_cps_noise.png][png]], [[./figs/comparison_cps_noise.pdf][pdf]])
|
||||
[[file:figs/comparison_cps_noise.png]]
|
||||
|
||||
** 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).
|
||||
|
||||
* Robustness to sensor dynamics uncertainty
|
||||
:PROPERTIES:
|
||||
|
||||
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After Width: | Height: | Size: 41 KiB |
153
tikz/index.org
153
tikz/index.org
@ -108,7 +108,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+begin_src latex :file h_infinity_optimal_comp_filters.pdf :tangle figs/h_infinity_optimal_comp_filters.tex
|
||||
<<tikz_default>>
|
||||
\begin{tikzpicture}
|
||||
\node[block={5.0cm}{3.0cm}, dashed] (P) {};
|
||||
\node[block={5.0cm}{3.0cm}, fill=black!20!white, dashed] (P) {};
|
||||
\node[above] at (P.north) {$P$};
|
||||
|
||||
\coordinate[] (inputn1) at ($(P.south west)!0.8!(P.north west) + (-\cdist, 0)$);
|
||||
@ -118,18 +118,18 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
\coordinate[] (outputx) at ($(P.south east)!0.5!(P.north east) + ( \cdist, 0)$);
|
||||
\coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( \cdist, 0)$);
|
||||
|
||||
\node[block, right=1.5 of inputn1] (W1){$W_1$};
|
||||
\node[block, right=1.5 of inputn2] (W2){$W_2$};
|
||||
\node[addb={+}{}{}{}{-}, right=of W1] (sub) {};
|
||||
\node[addb, right=2 of W2] (add) {};
|
||||
\node[block, right=1.5 of inputn1] (N1){$N_1$};
|
||||
\node[block, right=1.5 of inputn2] (N2){$N_2$};
|
||||
\node[addb={+}{}{}{}{-}, right=of N1] (sub) {};
|
||||
\node[addb, right=2 of N2] (add) {};
|
||||
|
||||
\node[block, below=of P] (H1) {$H_1$};
|
||||
|
||||
\draw[->] (inputn1)node[above right]{$n_1$} -- (W1.west);
|
||||
\draw[->] (inputn2)node[above right]{$n_2$} -- (W2.west);
|
||||
\draw[->] (W1) -- (sub.west);
|
||||
\draw[->] (W2) -| (sub.south);
|
||||
\draw[->] (W2-|sub.south) node[branch]{} -- (add.west);
|
||||
\draw[->] (inputn1)node[above right]{$\tilde{n}_1$} -- (N1.west);
|
||||
\draw[->] (inputn2)node[above right]{$\tilde{n}_2$} -- (N2.west);
|
||||
\draw[->] (N1) -- (sub.west);
|
||||
\draw[->] (N2) -| (sub.south);
|
||||
\draw[->] (N2-|sub.south) node[branch]{} -- (add.west);
|
||||
\draw[->] (sub.east) -- ++(0.5, 0) |- ($(outputv) + (-0.3, 0)$) |- (H1.east);
|
||||
\draw[->] (H1.west) -| ($(inputu) + (0.3, 0)$) -| (add.south);
|
||||
\draw[->] (add.east) -- (outputx) node[above left]{$\hat{x}$};
|
||||
@ -142,28 +142,40 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
[[file:figs/h_infinity_optimal_comp_filters.png]]
|
||||
* Fusion of two noisy sensors
|
||||
#+begin_src latex :file fusion_two_noisy_sensors.pdf :tangle figs/fusion_two_noisy_sensors.tex
|
||||
<<tikz_default>>
|
||||
|
||||
\begin{tikzpicture}
|
||||
\node[branch] (x) at (0, 0);
|
||||
\node[addb, above right=1.5 and 1 of x](add1){};
|
||||
\node[addb, below right=1.5 and 1 of x](add2){};
|
||||
\node[block, above=0.5 of add1](W1){$W_1$};
|
||||
\node[block, above=0.5 of add2](W2){$W_2$};
|
||||
\node[block, right=1 of add1](H1){$H_1$};
|
||||
\node[block, right=1 of add2](H2){$H_2$};
|
||||
\node[addb, right=4 of x](add){};
|
||||
\node[block, above right=0.5 and 0.5 of x](G1){$G_1(s)$};
|
||||
\node[block, below right=0.5 and 0.5 of x](G2){$G_2(s)$};
|
||||
\node[addb, right=0.8 of G1](add1){};
|
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\node[addb, right=0.8 of G2](add2){};
|
||||
\node[block, right=0.8 of add1](H1){$H_1(s)$};
|
||||
\node[block, right=0.8 of add2](H2){$H_2(s)$};
|
||||
\node[addb, right=5 of x](add){};
|
||||
|
||||
\draw[] ($(x)+(-1, 0)$) node[above right]{$x$} -- (x);
|
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\draw[->] (x) |- (add1.west);
|
||||
\draw[->] (x) |- (add2.west);
|
||||
\draw[->] (W1.south) -- (add1.north);
|
||||
\draw[->] (W2.south) -- (add2.north);
|
||||
\draw[<-] (W1.north) -- ++(0, 0.8)node[below right]{$n_1$};
|
||||
\draw[<-] (W2.north) -- ++(0, 0.8)node[below right]{$n_2$};
|
||||
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
|
||||
\draw[->] (x.center) |- (G1.west);
|
||||
\draw[->] (x.center) |- (G2.west);
|
||||
\draw[->] (G1.east) -- (add1.west);
|
||||
\draw[->] (G2.east) -- (add2.west);
|
||||
\draw[<-] (add1.north) -- ++(0, 0.8)node[below right](n1){$n_1$};
|
||||
\draw[<-] (add2.north) -- ++(0, 0.8)node[below right](n2){$n_2$};
|
||||
\draw[->] (add1.east) -- (H1.west);
|
||||
\draw[->] (add2.east) -- (H2.west);
|
||||
\draw[->] (H1) -| (add.north);
|
||||
\draw[->] (H2) -| (add.south);
|
||||
\draw[->] (add.east) -- ++(1, 0) node[above left]{$\hat{x}$};
|
||||
\draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={($(G2.south-|x)+(-0.2, -0.3)$) ($(n1.north east-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
|
||||
\node[below left] at (supersensor.north east) {Super Sensor};
|
||||
|
||||
\node[fit={($(G1.south west)+(-0.3, -0.1)$) ($(n1.north east)+(0.0, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
|
||||
\node[below right] at (sensor1.north west) {Sensor 1};
|
||||
\node[fit={($(G2.south west)+(-0.3, -0.1)$) ($(n2.north east)+(0.0, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
|
||||
\node[below right] at (sensor2.north west) {Sensor 2};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
@ -171,6 +183,97 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+caption: Fusion of two noisy sensors ([[./figs/fusion_two_noisy_sensors.png][png]], [[./figs/fusion_two_noisy_sensors.pdf][pdf]], [[./figs/fusion_two_noisy_sensors.tex][tex]]).
|
||||
#+RESULTS:
|
||||
[[file:figs/fusion_two_noisy_sensors.png]]
|
||||
|
||||
* Fusion of two noisy sensors with perfect dynamics
|
||||
#+begin_src latex :file sensor_fusion_noisy_perfect_dyn.pdf :tangle figs/sensor_fusion_noisy_perfect_dyn.tex
|
||||
<<tikz_default>>
|
||||
\begin{tikzpicture}
|
||||
\node[branch] (x) at (0, 0);
|
||||
\node[addb, above right=1.2 and 1.8 of x](addn1){};
|
||||
\node[addb, below right=1.2 and 1.8 of x](addn2){};
|
||||
\node[block, above=0.5 of addn1](N1) {$N_1(s)$};
|
||||
\node[block, above=0.5 of addn2](N2) {$N_2(s)$};
|
||||
\node[block, right=1.2 of addn1](H1){$H_1(s)$};
|
||||
\node[block, right=1.2 of addn2](H2){$H_2(s)$};
|
||||
\node[addb, right=4.5 of x](add){};
|
||||
|
||||
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
|
||||
\draw[->] (x.center) |- (addn1.west);
|
||||
\draw[->] (x.center) |- (addn2.west);
|
||||
\draw[->] (addn1.east) -- (H1.west)node[above left]{$\hat{x}_1$};
|
||||
\draw[->] (addn2.east) -- (H2.west)node[above left]{$\hat{x}_2$};
|
||||
\draw[->] ($(N1.north)+(0,0.7)$) node[below right](n1){$\tilde{n}_1$} -- (N1.north);
|
||||
\draw[->] ($(N2.north)+(0,0.7)$) node[below right](n2){$\tilde{n}_2$} -- (N2.north);
|
||||
\draw[->] (N1.south) -- (addn1.north)node[above right]{$n_1$};
|
||||
\draw[->] (N2.south) -- (addn2.north)node[above right]{$n_2$};
|
||||
\draw[->] (H1.east) -| (add.north);
|
||||
\draw[->] (H2.east) -| (add.south);
|
||||
\draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={($(addn2.south-|x)+(-0.2, -0.4)$) ($(n1.north-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
|
||||
\node[below left] at (supersensor.north east) {Super Sensor};
|
||||
\node[block, fit={($(x|-addn1.south)+(0.2, -0.2)$) ($(n1.north-|N1.east)+(0.2, 0.1)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {};
|
||||
\node[below right] at (sensor1.north west) {Sensor 1};
|
||||
\node[block, fit={($(x|-addn2.south)+(0.2, -0.2)$) ($(n2.north-|N2.east)+(0.2, 0.1)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor2) {};
|
||||
\node[below right] at (sensor2.north west) {Sensor 2};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_fusion_noisy_perfect_dyn
|
||||
#+caption: Sensor fusion architecture with sensor dynamics uncertainty ([[./figs/sensor_fusion_noisy_perfect_dyn.png][png]], [[./figs/sensor_fusion_noisy_perfect_dyn.pdf][pdf]], [[./figs/sensor_fusion_noisy_perfect_dyn.tex][tex]]).
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_fusion_noisy_perfect_dyn.png]]
|
||||
|
||||
* Fusion of two noisy sensors with sensors weights
|
||||
#+begin_src latex :file fusion_two_noisy_sensors_weights.pdf :tangle figs/fusion_two_noisy_sensors_weights.tex
|
||||
<<tikz_default>>
|
||||
|
||||
\begin{tikzpicture}
|
||||
\node[branch] (x) at (0, 0);
|
||||
\node[block, above right=1 and 0.5 of x](G1){$G_1(s)$};
|
||||
\node[block, below right=1 and 0.5 of x](G2){$G_2(s)$};
|
||||
\node[addb, right=0.8 of G1](add1){};
|
||||
\node[addb, right=0.8 of G2](add2){};
|
||||
\node[block, above=0.5 of add1](N1){$N_1(s)$};
|
||||
\node[block, above=0.5 of add2](N2){$N_2(s)$};
|
||||
\node[block, right=0.9 of add1](H1){$H_1(s)$};
|
||||
\node[block, right=0.9 of add2](H2){$H_2(s)$};
|
||||
\node[addb, right=5 of x](add){};
|
||||
|
||||
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
|
||||
\draw[->] (x.center) |- (G1.west);
|
||||
\draw[->] (x.center) |- (G2.west);
|
||||
\draw[->] (G1.east) -- (add1.west);
|
||||
\draw[->] (G2.east) -- (add2.west);
|
||||
\draw[<-] (N1.north) -- ++(0, 0.6)node[below right](n1){$\tilde{n}_1$};
|
||||
\draw[<-] (N2.north) -- ++(0, 0.6)node[below right](n2){$\tilde{n}_2$};
|
||||
\draw[->] (N1.south) -- (add1.north)node[above right]{$n_1$};
|
||||
\draw[->] (N2.south) -- (add2.north)node[above right]{$n_2$};
|
||||
\draw[->] (add1.east) -- (H1.west);
|
||||
\draw[->] (add2.east) -- (H2.west);
|
||||
\draw[->] (H1) -| (add.north);
|
||||
\draw[->] (H2) -| (add.south);
|
||||
\draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={($(G2.south-|x)+(-0.2, -0.3)$) ($(n1.north east-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
|
||||
\node[below left] at (supersensor.north east) {Super Sensor};
|
||||
|
||||
\node[fit={($(G1.south west)+(-0.3, -0.1)$) ($(n1.north-|N1.east)+(0.2, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
|
||||
\node[below right] at (sensor1.north west) {Sensor 1};
|
||||
\node[fit={($(G2.south west)+(-0.3, -0.1)$) ($(n2.north-|N1.east)+(0.2, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
|
||||
\node[below right] at (sensor2.north west) {Sensor 2};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:fusion_two_noisy_sensors_weights
|
||||
#+caption: Fusion of two noisy sensors ([[./figs/fusion_two_noisy_sensors_weights.png][png]], [[./figs/fusion_two_noisy_sensors_weights.pdf][pdf]], [[./figs/fusion_two_noisy_sensors_weights.tex][tex]]).
|
||||
#+RESULTS:
|
||||
[[file:figs/fusion_two_noisy_sensors_weights.png]]
|
||||
|
||||
* Fusion of two noisy signals
|
||||
#+begin_src latex :file fusion_two_signals.pdf :tangle figs/fusion_two_signals.tex
|
||||
\begin{tikzpicture}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user