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< h1 class = "title" > Robust and Optimal Sensor Fusion - Matlab Computation< / h1 >
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< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#org44d9894" > 1. Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis< / a >
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< ul >
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< li > < a href = "#org57fc74a" > 1.1. Architecture< / a > < / li >
< li > < a href = "#org4e01148" > 1.2. Noise of the sensors< / a > < / li >
< li > < a href = "#org2009eeb" > 1.3. H-Two Synthesis< / a > < / li >
< li > < a href = "#org5bce7a5" > 1.4. Obtained Super Sensor’ s noise uncertainty< / a > < / li >
< li > < a href = "#org3c6ed73" > 1.5. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#orgf8fcf68" > 2. Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis< / a >
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< ul >
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< li > < a href = "#org95e625c" > 2.1. Super Sensor Dynamical Uncertainty< / a > < / li >
< li > < a href = "#orgf77bae9" > 2.2. Dynamical uncertainty of the individual sensors< / a > < / li >
< li > < a href = "#orgb8c1069" > 2.3. Synthesis objective< / a > < / li >
< li > < a href = "#org3b2fc2b" > 2.4. Requirements as an \(\mathcal{H}_\infty\) norm< / a > < / li >
< li > < a href = "#org1559f0c" > 2.5. Weighting Function used to bound the super sensor uncertainty< / a > < / li >
< li > < a href = "#org0942905" > 2.6. \(\mathcal{H}_\infty\) Synthesis< / a > < / li >
< li > < a href = "#orgd70c5ba" > 2.7. Super sensor uncertainty< / a > < / li >
< li > < a href = "#org822276a" > 2.8. Super sensor noise< / a > < / li >
< li > < a href = "#org78697e8" > 2.9. Conclusion< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org885e445" > 3. Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis< / a >
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< ul >
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< li > < a href = "#orgbe7b403" > 3.1. Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis - Introduction< / a > < / li >
< li > < a href = "#org3013d41" > 3.2. Noise characteristics and Uncertainty of the individual sensors< / a > < / li >
< li > < a href = "#org39502ab" > 3.3. Weighting Functions on the uncertainty of the super sensor< / a > < / li >
< li > < a href = "#orgd08d347" > 3.4. Mixed Synthesis Architecture< / a > < / li >
< li > < a href = "#orgec6e5c4" > 3.5. Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis< / a > < / li >
< li > < a href = "#orgcf41bc0" > 3.6. Obtained Super Sensor’ s noise< / a > < / li >
< li > < a href = "#org39e64c2" > 3.7. Obtained Super Sensor’ s Uncertainty< / a > < / li >
< li > < a href = "#orgdf7555f" > 3.8. Conclusion< / a > < / li >
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< / ul >
< / li >
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< / ul >
< / div >
< / div >
< p >
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In this document, the optimal and robust design of complementary filters is studied.
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< / p >
< p >
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Two sensors are considered with both different noise characteristics and dynamical uncertainties represented by multiplicative input uncertainty.
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< / p >
< ul class = "org-ul" >
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< li > Section < a href = "#orgdd2b0ca" > 1< / a > : the \(\mathcal{H}_2\) synthesis is used to design complementary filters such that the RMS value of the super sensor’ s noise is minimized< / li >
< li > Section < a href = "#org9f38ce2" > 2< / a > : the \(\mathcal{H}_\infty\) synthesis is used to design complementary filters such that the super sensor’ s uncertainty is bonded to acceptable values< / li >
< li > Section < a href = "#org24cb1b0" > 3< / a > : the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is used to both limit the super sensor’ s uncertainty and to lower the RMS value of the super sensor’ s noise< / li >
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< / ul >
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< div id = "outline-container-org44d9894" class = "outline-2" >
< h2 id = "org44d9894" > < span class = "section-number-2" > 1< / span > Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis< / h2 >
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< div class = "outline-text-2" id = "text-1" >
< p >
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< a id = "orgdd2b0ca" > < / a >
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< / p >
< p >
The idea is to combine sensors that works in different frequency range using complementary filters.
< / p >
< p >
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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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< / p >
< p >
The complementary filters have to be designed in order to minimize the effect noise of each sensor on the super sensor noise.
< / p >
< div class = "note" >
< p >
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The Matlab scripts is accessible < a href = "matlab/optimal_comp_filters.m" > here< / a > .
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< / p >
< / div >
< / div >
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< div id = "outline-container-org57fc74a" class = "outline-3" >
< h3 id = "org57fc74a" > < span class = "section-number-3" > 1.1< / span > Architecture< / h3 >
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< div class = "outline-text-3" id = "text-1-1" >
< p >
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Let’ s consider the sensor fusion architecture shown on figure < a href = "#orgd6622a2" > 1< / a > 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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< / p >
< p >
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\(\tilde{n}_1\) and \(\tilde{n}_2\) are normalized white noise:
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< / p >
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\begin{equation}
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\label{orga7ad7f8}
\Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_2}(\omega) = 1
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\end{equation}
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< div id = "orgd6622a2" class = "figure" >
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< p > < img src = "figs-tikz/fusion_two_noisy_sensors_weights.png" alt = "fusion_two_noisy_sensors_weights.png" / >
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< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Fusion of two sensors< / p >
< / div >
< p >
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We consider that the two sensor dynamics \(G_1(s)\) and \(G_2(s)\) are ideal:
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< / p >
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\begin{equation}
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\label{org024c8f5}
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G_1(s) = G_2(s) = 1
\end{equation}
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< p >
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We obtain the architecture of figure < a href = "#org0581836" > 2< / a > .
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< / p >
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< div id = "org0581836" class = "figure" >
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< p > < img src = "figs-tikz/sensor_fusion_noisy_perfect_dyn.png" alt = "sensor_fusion_noisy_perfect_dyn.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 2: < / span > Fusion of two sensors with ideal dynamics< / p >
< / div >
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< p >
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\(H_1(s)\) and \(H_2(s)\) are complementary filters:
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< / p >
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\begin{equation}
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\label{orga71be68}
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H_1(s) + H_2(s) = 1
\end{equation}
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< p >
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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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< / p >
< p >
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We have that the Power Spectral Density (PSD) of \(\hat{x}\) is:
\[ \Phi_{\hat{x}}(\omega) = |H_1(j\omega) N_1(j\omega)|^2 \Phi_{\tilde{n}_1}(\omega) + |H_2(j\omega) N_2(j\omega)|^2 \Phi_{\tilde{n}_2}(\omega), \quad \forall \omega \]
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< / p >
< p >
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And the goal is the minimize the Root Mean Square (RMS) value of \(\hat{x}\):
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< / p >
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\begin{equation}
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\label{orgc926b79}
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\sigma_{\hat{x}} = \sqrt{\int_0^\infty \Phi_{\hat{x}}(\omega) d\omega}
\end{equation}
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< / div >
< / div >
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< div id = "outline-container-org4e01148" class = "outline-3" >
< h3 id = "org4e01148" > < span class = "section-number-3" > 1.2< / span > Noise of the sensors< / h3 >
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< div class = "outline-text-3" id = "text-1-2" >
< p >
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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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< / p >
< ul class = "org-ul" >
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< li > Sensor 1 characterized by \(N_1(s)\) has low noise at low frequency (for instance a geophone)< / li >
< li > Sensor 2 characterized by \(N_2(s)\) has low noise at high frequency (for instance an accelerometer)< / li >
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< / ul >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
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;
N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
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< / pre >
< / div >
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< div id = "org29561dd" class = "figure" >
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< p > < img src = "figs/noise_characteristics_sensors.png" alt = "noise_characteristics_sensors.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 3: < / span > Noise Characteristics of the two sensors (< a href = "./figs/noise_characteristics_sensors.png" > png< / a > , < a href = "./figs/noise_characteristics_sensors.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org2009eeb" class = "outline-3" >
< h3 id = "org2009eeb" > < span class = "section-number-3" > 1.3< / span > H-Two Synthesis< / h3 >
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< div class = "outline-text-3" id = "text-1-3" >
< p >
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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:
\[ \sigma_{\hat{x}} = \sqrt{\int_0^\infty |H_1 N_1|^2(\omega) + |H_2 N_2|^2(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2 \]
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.
< / p >
< p >
For that, we use the \(\mathcal{H}_2\) Synthesis.
< / p >
< p >
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We use the generalized plant architecture shown on figure < a href = "#orgc952e5a" > 4< / a > .
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< / p >
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< div id = "orgc952e5a" class = "figure" >
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< p > < img src = "figs-tikz/h_infinity_optimal_comp_filters.png" alt = "h_infinity_optimal_comp_filters.png" / >
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< / p >
< p > < span class = "figure-number" > Figure 4: < / span > \(\mathcal{H}_2\) Synthesis - Generalized plant used for the optimal generation of complementary filters< / p >
< / div >
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\begin{equation*}
\begin{pmatrix}
z \\ v
\end{pmatrix} = \begin{pmatrix}
0 & N_2 & 1 \\
N_1 & -N_2 & 0
\end{pmatrix} \begin{pmatrix}
w_1 \\ w_2 \\ u
\end{pmatrix}
\end{equation*}
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< p >
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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< / p >
< p >
Thus, if we minimize the \(\mathcal{H}_2\) norm of this transfer function, we minimize the RMS value of \(\hat{x}\).
< / p >
< p >
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We define the generalized plant \(P\) on matlab as shown on figure < a href = "#orgc952e5a" > 4< / a > .
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > P = [0 N2 1;
N1 -N2 0];
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< / pre >
< / div >
< p >
And we do the \(\mathcal{H}_2\) synthesis using the < code > h2syn< / code > command.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > [H1, ~, gamma] = h2syn(P, 1, 1);
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< / pre >
< / div >
< p >
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Finally, we define \(H_2(s) = 1 - H_1(s)\).
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > H2 = 1 - H1;
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< / pre >
< / div >
< p >
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The complementary filters obtained are shown on figure < a href = "#orgc176e1b" > 5< / a > .
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< / p >
< p >
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The PSD of the noise of the individual sensor and of the super sensor are shown in Fig. < a href = "#org95e2127" > 6< / a > .
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< / p >
< p >
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The Cumulative Power Spectrum (CPS) is shown on Fig. < a href = "#org3be7555" > 7< / a > .
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< / p >
< p >
The obtained RMS value of the super sensor is lower than the RMS value of the individual sensors.
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< / p >
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< div id = "orgc176e1b" class = "figure" >
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< p > < img src = "figs/htwo_comp_filters.png" alt = "htwo_comp_filters.png" / >
< / p >
< p > < span class = "figure-number" > Figure 5: < / span > Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis (< a href = "./figs/htwo_comp_filters.png" > png< / a > , < a href = "./figs/htwo_comp_filters.pdf" > pdf< / a > )< / p >
< / div >
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< div class = "org-src-container" >
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< pre class = "src 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;
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< / pre >
< / div >
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< div id = "org95e2127" class = "figure" >
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< p > < img src = "figs/psd_sensors_htwo_synthesis.png" alt = "psd_sensors_htwo_synthesis.png" / >
< / p >
< p > < span class = "figure-number" > Figure 6: < / span > Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal (< a href = "./figs/psd_sensors_htwo_synthesis.png" > png< / a > , < a href = "./figs/psd_sensors_htwo_synthesis.pdf" > pdf< / a > )< / p >
< / div >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > CPS_S1 = 1/pi*cumtrapz(2*pi*freqs, PSD_S1);
CPS_S2 = 1/pi*cumtrapz(2*pi*freqs, PSD_S2);
CPS_H2 = 1/pi*cumtrapz(2*pi*freqs, PSD_H2);
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< / pre >
< / div >
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< div id = "org3be7555" class = "figure" >
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< p > < img src = "figs/cps_h2_synthesis.png" alt = "cps_h2_synthesis.png" / >
< / p >
< p > < span class = "figure-number" > Figure 7: < / span > Cumulative Power Spectrum of individual sensors and super sensor using the \(\mathcal{H}_2\) synthesis (< a href = "./figs/cps_h2_synthesis.png" > png< / a > , < a href = "./figs/cps_h2_synthesis.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
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< / div >
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< div id = "outline-container-org5bce7a5" class = "outline-3" >
< h3 id = "org5bce7a5" > < span class = "section-number-3" > 1.4< / span > Obtained Super Sensor’ s noise uncertainty< / h3 >
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< div class = "outline-text-3" id = "text-1-4" >
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< p >
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We would like to verify if the obtained sensor fusion architecture is robust to change in the sensor dynamics.
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< / p >
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< p >
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To study the dynamical uncertainty on the super sensor, we defined some multiplicative uncertainty on both sensor dynamics.
Two weights \(w_1(s)\) and \(w_2(s)\) are used to described the amplitude of the dynamical uncertainty.
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< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > omegac = 100*2*pi; G0 = 0.1; Ginf = 10;
w1 = (Ginf*s/omegac + G0)/(s/omegac + 1);
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omegac = 0.2*2*pi; G0 = 5; Ginf = 0.1;
w2 = (Ginf*s/omegac + G0)/(s/omegac + 1);
omegac = 5000*2*pi; G0 = 1; Ginf = 50;
w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1);
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< / pre >
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< / div >
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< p >
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The sensor uncertain models are defined below.
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< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > G1 = 1 + w1*ultidyn('Delta',[1 1]);
G2 = 1 + w2*ultidyn('Delta',[1 1]);
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< / pre >
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< / div >
< p >
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The super sensor uncertain model is defined below using the complementary filters obtained with the \(\mathcal{H}_2\) synthesis.
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The dynamical uncertainty bounds of the super sensor is shown in Fig. < a href = "#orgfde2308" > 8< / a > .
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Right Half Plane zero might be introduced in the super sensor dynamics which will render the feedback system unstable.
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< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gss = G1*H1 + G2*H2;
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< / pre >
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< / div >
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< div id = "orgfde2308" class = "figure" >
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< p > < img src = "figs/uncertainty_super_sensor_H2_syn.png" alt = "uncertainty_super_sensor_H2_syn.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 8: < / span > Uncertianty regions of both individual sensors and of the super sensor when using the \(\mathcal{H}_2\) synthesis (< a href = "./figs/uncertainty_super_sensor_H2_syn.png" > png< / a > , < a href = "./figs/uncertainty_super_sensor_H2_syn.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org3c6ed73" class = "outline-3" >
< h3 id = "org3c6ed73" > < span class = "section-number-3" > 1.5< / span > Conclusion< / h3 >
< div class = "outline-text-3" id = "text-1-5" >
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< p >
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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).
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< / p >
< p >
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However, the synthesis does not take into account the robustness of the sensor fusion.
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< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgf8fcf68" class = "outline-2" >
< h2 id = "orgf8fcf68" > < span class = "section-number-2" > 2< / span > Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis< / h2 >
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< div class = "outline-text-2" id = "text-2" >
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< p >
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< a id = "org9f38ce2" > < / a >
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< / p >
< p >
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We initially considered perfectly known sensor dynamics so that it can be perfectly inverted.
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< / p >
< p >
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We now take into account the fact that the sensor dynamics is only partially known.
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To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Fig. < a href = "#org607f91f" > 9< / a > .
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< / p >
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< div id = "org607f91f" class = "figure" >
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< p > < img src = "figs-tikz/sensor_fusion_dynamic_uncertainty.png" alt = "sensor_fusion_dynamic_uncertainty.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 9: < / span > Sensor fusion architecture with sensor dynamics uncertainty< / p >
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< / div >
< p >
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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.
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< / p >
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< div class = "note" >
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< p >
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The Matlab scripts is accessible < a href = "matlab/comp_filter_robustness.m" > here< / a > .
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< / p >
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< / div >
< / div >
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< div id = "outline-container-org95e625c" class = "outline-3" >
< h3 id = "org95e625c" > < span class = "section-number-3" > 2.1< / span > Super Sensor Dynamical Uncertainty< / h3 >
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< div class = "outline-text-3" id = "text-2-1" >
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< p >
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In practical systems, the sensor dynamics has always some level of uncertainty.
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Let’ s represent that with multiplicative input uncertainty as shown on figure < a href = "#org607f91f" > 9< / a > .
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< / p >
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< div id = "org8eaa465" class = "figure" >
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< p > < img src = "figs-tikz/sensor_fusion_dynamic_uncertainty.png" alt = "sensor_fusion_dynamic_uncertainty.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 10: < / span > Fusion of two sensors with input multiplicative uncertainty< / p >
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< / div >
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< p >
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The dynamics of the super sensor is represented by
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< / p >
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\begin{align*}
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\frac{\hat{x}}{x} & = (1 + w_1 \Delta_1) H_1 + (1 + w_2 \Delta_2) H_2 \\
& = 1 + w_1 H_1 \Delta_1 + w_2 H_2 \Delta_2
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\end{align*}
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< p >
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with \(\Delta_i\) is any transfer function satisfying \(\| \Delta_i \|_\infty < 1 \ ) .
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< / p >
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< p >
We see that as soon as we have some uncertainty in the sensor dynamics, we have that the complementary filters have some effect on the transfer function from \(x\) to \(\hat{x}\).
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< / p >
< p >
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The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at frequency \(\omega\) is bounded in the complex plane by a circle centered on 1 and with a radius equal to \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\) (figure < a href = "#org5f4c4b1" > 11< / a > ).
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< / p >
< p >
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We then have that the angle introduced by the super sensor is bounded by \(\arcsin(\epsilon)\):
\[ \angle \frac{\hat{x}}{x}(j\omega) \le \arcsin \Big(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\Big) \]
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< / p >
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< div id = "org5f4c4b1" class = "figure" >
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< p > < img src = "figs-tikz/uncertainty_gain_phase_variation.png" alt = "uncertainty_gain_phase_variation.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 11: < / span > Maximum phase variation< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgf77bae9" class = "outline-3" >
< h3 id = "orgf77bae9" > < span class = "section-number-3" > 2.2< / span > Dynamical uncertainty of the individual sensors< / h3 >
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< div class = "outline-text-3" id = "text-2-2" >
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< p >
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Let say we want to merge two sensors:
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< / p >
< ul class = "org-ul" >
< li > sensor 1 that has unknown dynamics above 10Hz: \(|w_1(j\omega)| > 1\) for \(\omega > 10\text{ Hz}\)< / li >
< li > sensor 2 that has unknown dynamics below 1Hz and above 1kHz \(|w_2(j\omega)| > 1\) for \(\omega < 1 \ text { Hz } \ ) and \ ( \ omega > 1\text{ kHz}\)< / li >
< / ul >
< p >
We define the weights that are used to characterize the dynamic uncertainty of the sensors.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > omegac = 100*2*pi; G0 = 0.1; Ginf = 10;
w1 = (Ginf*s/omegac + G0)/(s/omegac + 1);
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omegac = 0.2*2*pi; G0 = 5; Ginf = 0.1;
w2 = (Ginf*s/omegac + G0)/(s/omegac + 1);
omegac = 5000*2*pi; G0 = 1; Ginf = 50;
w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1);
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< / pre >
< / div >
< p >
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From the weights, we define the uncertain transfer functions of the sensors. Some of the uncertain dynamics of both sensors are shown on Fig. < a href = "#orgb3ffcc1" > 12< / a > with the upper and lower bounds on the magnitude and on the phase.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > G1 = 1 + w1*ultidyn('Delta',[1 1]);
G2 = 1 + w2*ultidyn('Delta',[1 1]);
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< / pre >
< / div >
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< div id = "orgb3ffcc1" class = "figure" >
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< p > < img src = "figs/uncertainty_dynamics_sensors.png" alt = "uncertainty_dynamics_sensors.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 12: < / span > Dynamic uncertainty of the two sensors (< a href = "./figs/uncertainty_dynamics_sensors.png" > png< / a > , < a href = "./figs/uncertainty_dynamics_sensors.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgb8c1069" class = "outline-3" >
< h3 id = "orgb8c1069" > < span class = "section-number-3" > 2.3< / span > Synthesis objective< / h3 >
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< div class = "outline-text-3" id = "text-2-3" >
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< p >
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The uncertainty region of the super sensor dynamics is represented by a circle in the complex plane as shown in Fig. < a href = "#org5f4c4b1" > 11< / a > .
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< / p >
< p >
At each frequency \(\omega\), the radius of the circle is \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\).
< / p >
< p >
Thus, the phase shift \(\Delta\phi(\omega)\) due to the super sensor uncertainty is bounded by:
\[ |\Delta\phi(\omega)| \leq \arcsin\big( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \big) \]
< / p >
< p >
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Let’ s define some allowed frequency depend phase shift \(\Delta\phi_\text{max}(\omega) > 0\) such that:
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\[ |\Delta\phi(\omega)| < \Delta\phi_\text{max}(\omega), \quad \forall\omega \]
< / p >
< p >
If \(H_1(s)\) and \(H_2(s)\) are designed such that
\[ |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| < \sin\big( \Delta\phi_\text{max}(\omega) \big) \]
< / p >
< p >
The maximum phase shift due to dynamic uncertainty at frequency \(\omega\) will be \(\Delta\phi_\text{max}(\omega)\).
< / p >
< / div >
< / div >
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< div id = "outline-container-org3b2fc2b" class = "outline-3" >
< h3 id = "org3b2fc2b" > < span class = "section-number-3" > 2.4< / span > Requirements as an \(\mathcal{H}_\infty\) norm< / h3 >
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< div class = "outline-text-3" id = "text-2-4" >
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< p >
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We now try to express this requirement in terms of an \(\mathcal{H}_\infty\) norm.
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< / p >
< p >
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Let’ s define one weight \(w_\phi(s)\) that represents the maximum wanted phase uncertainty:
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\[ |w_{\phi}(j\omega)|^{-1} \approx \sin(\Delta\phi_{\text{max}}(\omega)), \quad \forall\omega \]
< / p >
< p >
Then:
< / p >
\begin{align*}
& |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| < \sin\big( \Delta\phi_\text{max}(\omega) \big), \quad \forall\omega \\
\Longleftrightarrow & |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| < |w_\phi(j\omega)|^{-1}, \quad \forall\omega \\
\Longleftrightarrow & \left| w_1(j\omega) H_1(j\omega) w_\phi(j\omega) \right| + \left| w_2(j\omega) H_2(j\omega) w_\phi(j\omega) \right| < 1 , \ quad \ forall \ omega
\end{align*}
< p >
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Which is approximately equivalent to (with an error of maximum \(\sqrt{2}\)):
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< / p >
\begin{equation}
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\label{org559e8db}
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\left\| \begin{matrix} w_1(s) w_\phi(s) H_1(s) \\ w_2(s) w_\phi(s) H_2(s) \end{matrix} \right\|_\infty < 1
\end{equation}
< p >
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One should not forget that at frequency where both sensors has unknown dynamics (\(|w_1(j\omega)| > 1\) and \(|w_2(j\omega)| > 1\)), the super sensor dynamics will also be unknown and the phase uncertainty cannot be bounded.
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Thus, at these frequencies, \(|w_\phi|\) should be smaller than \(1\).
< / p >
< / div >
< / div >
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< div id = "outline-container-org1559f0c" class = "outline-3" >
< h3 id = "org1559f0c" > < span class = "section-number-3" > 2.5< / span > Weighting Function used to bound the super sensor uncertainty< / h3 >
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< div class = "outline-text-3" id = "text-2-5" >
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< p >
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Let’ s define \(w_\phi(s)\) in order to bound the maximum allowed phase uncertainty \(\Delta\phi_\text{max}\) of the super sensor dynamics.
The magnitude \(|w_\phi(j\omega)|\) is shown in Fig. < a href = "#org90016c8" > 13< / a > and the corresponding maximum allowed phase uncertainty of the super sensor dynamics of shown in Fig. < a href = "#orgf2ea8be" > 14< / a > .
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Dphi = 20; % [deg]
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n = 4; w0 = 2*pi*900; G0 = 1/sin(Dphi*pi/180); Ginf = 1/100; Gc = 1;
wphi = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/Ginf)^(2/n)))*s + (G0/Gc)^(1/n))/((1/Ginf)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/Ginf)^(2/n)))*s + (1/Gc)^(1/n)))^n;
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W1 = w1*wphi;
W2 = w2*wphi;
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< / pre >
< / div >
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< div id = "org90016c8" class = "figure" >
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< p > < img src = "figs/magnitude_wphi.png" alt = "magnitude_wphi.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 13: < / span > Magnitude of the weght \(w_\phi(s)\) that is used to bound the uncertainty of the super sensor (< a href = "./figs/magnitude_wphi.png" > png< / a > , < a href = "./figs/magnitude_wphi.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "orgf2ea8be" class = "figure" >
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< p > < img src = "figs/maximum_wanted_phase_uncertainty.png" alt = "maximum_wanted_phase_uncertainty.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 14: < / span > Maximum wanted phase uncertainty using this weight (< a href = "./figs/maximum_wanted_phase_uncertainty.png" > png< / a > , < a href = "./figs/maximum_wanted_phase_uncertainty.pdf" > pdf< / a > )< / p >
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< / div >
< p >
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The obtained upper bounds on the complementary filters in order to limit the phase uncertainty of the super sensor are represented in Fig. < a href = "#org6bf03e5" > 15< / a > .
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< / p >
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< div id = "org6bf03e5" class = "figure" >
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< p > < img src = "figs/upper_bounds_comp_filter_max_phase_uncertainty.png" alt = "upper_bounds_comp_filter_max_phase_uncertainty.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 15: < / span > Upper bounds on the complementary filters set in order to limit the maximum phase uncertainty of the super sensor to 30 degrees until 500Hz (< a href = "./figs/upper_bounds_comp_filter_max_phase_uncertainty.png" > png< / a > , < a href = "./figs/upper_bounds_comp_filter_max_phase_uncertainty.pdf" > pdf< / a > )< / p >
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< / div >
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< / div >
< / div >
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< div id = "outline-container-org0942905" class = "outline-3" >
< h3 id = "org0942905" > < span class = "section-number-3" > 2.6< / span > \(\mathcal{H}_\infty\) Synthesis< / h3 >
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< div class = "outline-text-3" id = "text-2-6" >
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< p >
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The \(\mathcal{H}_\infty\) synthesis architecture used for the complementary filters is shown in Fig. < a href = "#orgb6417fc" > 16< / a > .
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< / p >
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< div id = "orgb6417fc" class = "figure" >
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< p > < img src = "figs-tikz/h_infinity_robust_fusion.png" alt = "h_infinity_robust_fusion.png" / >
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< / p >
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< p > < span class = "figure-number" > Figure 16: < / span > Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters< / p >
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< / div >
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< p >
The generalized plant is defined below.
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > P = [W1 -W1;
0 W2;
1 0];
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< / pre >
< / div >
< p >
And we do the \(\mathcal{H}_\infty\) synthesis using the < code > hinfsyn< / code > command.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > [H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
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< / pre >
< / div >
< pre class = "example" >
[H2, ~, 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: 0.0447 < gamma < = 1.3318
gamma hamx_eig xinf_eig hamy_eig yinf_eig nrho_xy p/f
1.332 1.3e+01 -1.0e-14 1.3e+00 -2.6e-18 0.0000 p
0.688 1.3e-11# ******** 1.3e+00 -6.7e-15 ******** f
1.010 1.1e+01 -1.5e-14 1.3e+00 -2.5e-14 0.0000 p
0.849 6.9e-11# ******** 1.3e+00 -2.3e-14 ******** f
0.930 5.2e-12# ******** 1.3e+00 -6.1e-18 ******** f
0.970 5.6e-11# ******** 1.3e+00 -2.3e-14 ******** f
0.990 5.0e-11# ******** 1.3e+00 -1.7e-17 ******** f
1.000 2.1e-10# ******** 1.3e+00 0.0e+00 ******** f
1.005 1.9e-10# ******** 1.3e+00 -3.7e-14 ******** f
1.008 1.1e+01 -9.1e-15 1.3e+00 0.0e+00 0.0000 p
1.006 1.2e-09# ******** 1.3e+00 -6.9e-16 ******** f
1.007 1.1e+01 -4.6e-15 1.3e+00 -1.8e-16 0.0000 p
Gamma value achieved: 1.0069
< / pre >
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< p >
And \(H_1(s)\) is defined as the complementary of \(H_2(s)\).
< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > H1 = 1 - H2;
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< / pre >
< / div >
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< p >
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The obtained complementary filters are shown in Fig. < a href = "#orgec29b6b" > 17< / a > .
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< / p >
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< div id = "orgec29b6b" class = "figure" >
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< p > < img src = "figs/comp_filter_hinf_uncertainty.png" alt = "comp_filter_hinf_uncertainty.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 17: < / span > Obtained complementary filters (< a href = "./figs/comp_filter_hinf_uncertainty.png" > png< / a > , < a href = "./figs/comp_filter_hinf_uncertainty.pdf" > pdf< / a > )< / p >
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< / div >
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< / div >
< / div >
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< div id = "outline-container-orgd70c5ba" class = "outline-3" >
< h3 id = "orgd70c5ba" > < span class = "section-number-3" > 2.7< / span > Super sensor uncertainty< / h3 >
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< div class = "outline-text-3" id = "text-2-7" >
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< p >
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We can now compute the uncertainty of the super sensor. The result is shown in Fig. < a href = "#org4dd04dd" > 18< / a > .
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > Gss = G1*H1 + G2*H2;
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< / pre >
< / div >
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< div id = "org4dd04dd" class = "figure" >
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< p > < img src = "figs/super_sensor_uncertainty_bode_plot.png" alt = "super_sensor_uncertainty_bode_plot.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 18: < / span > Uncertainty on the dynamics of the super sensor (< a href = "./figs/super_sensor_uncertainty_bode_plot.png" > png< / a > , < a href = "./figs/super_sensor_uncertainty_bode_plot.pdf" > pdf< / a > )< / p >
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< / div >
< p >
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The uncertainty of the super sensor cannot be made smaller than both the individual sensor. Ideally, it would follow the minimum uncertainty of both sensors.
< / p >
< p >
We here just used very wimple weights.
For instance, we could improve the dynamical uncertainty of the super sensor by making \(|w_\phi(j\omega)|\) smaller bellow 2Hz where the dynamical uncertainty of the sensor 1 is small.
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< / p >
< / div >
< / div >
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< div id = "outline-container-org822276a" class = "outline-3" >
< h3 id = "org822276a" > < span class = "section-number-3" > 2.8< / span > Super sensor noise< / h3 >
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< div class = "outline-text-3" id = "text-2-8" >
< p >
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We now compute the obtain Power Spectral Density of the super sensor’ s noise.
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The noise characteristics of both individual sensor are defined below.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
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;
N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
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< / pre >
< / div >
< p >
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The PSD of both sensor and of the super sensor is shown in Fig. < a href = "#orgbaedbae" > 19< / a > .
The CPS of both sensor and of the super sensor is shown in Fig. < a href = "#orgb734af5" > 20< / a > .
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< / p >
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< div id = "orgbaedbae" class = "figure" >
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< p > < img src = "figs/psd_sensors_hinf_synthesis.png" alt = "psd_sensors_hinf_synthesis.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 19: < / span > Power Spectral Density of the obtained super sensor using the \(\mathcal{H}_\infty\) synthesis (< a href = "./figs/psd_sensors_hinf_synthesis.png" > png< / a > , < a href = "./figs/psd_sensors_hinf_synthesis.pdf" > pdf< / a > )< / p >
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< / div >
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< div id = "orgb734af5" class = "figure" >
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< p > < img src = "figs/cps_sensors_hinf_synthesis.png" alt = "cps_sensors_hinf_synthesis.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 20: < / span > Cumulative Power Spectrum of the obtained super sensor using the \(\mathcal{H}_\infty\) synthesis (< a href = "./figs/cps_sensors_hinf_synthesis.png" > png< / a > , < a href = "./figs/cps_sensors_hinf_synthesis.cps" > cps< / a > )< / p >
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< / div >
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< / div >
< / div >
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< div id = "outline-container-org78697e8" class = "outline-3" >
< h3 id = "org78697e8" > < span class = "section-number-3" > 2.9< / span > Conclusion< / h3 >
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< div class = "outline-text-3" id = "text-2-9" >
< p >
Using the \(\mathcal{H}_\infty\) synthesis, the dynamical uncertainty of the super sensor can be bounded to acceptable values.
< / p >
< p >
However, the RMS of the super sensor noise is not optimized as it was the case with the \(\mathcal{H}_2\) synthesis
< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org885e445" class = "outline-2" >
< h2 id = "org885e445" > < span class = "section-number-2" > 3< / span > Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis< / h2 >
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< div class = "outline-text-2" id = "text-3" >
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< p >
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< a id = "org24cb1b0" > < / a >
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< / p >
< div class = "note" >
< p >
The Matlab scripts is accessible < a href = "matlab/mixed_synthesis_sensor_fusion.m" > here< / a > .
< / p >
< / div >
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< / div >
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< div id = "outline-container-orgbe7b403" class = "outline-3" >
< h3 id = "orgbe7b403" > < span class = "section-number-3" > 3.1< / span > Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis - Introduction< / h3 >
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< div class = "outline-text-3" id = "text-3-1" >
< p >
The goal is to design complementary filters such that:
< / p >
< ul class = "org-ul" >
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< li > the maximum uncertainty of the super sensor is bounded< / li >
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< li > the RMS value of the super sensor noise is minimized< / li >
< / ul >
< p >
To do so, we can use the Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis.
< / p >
< p >
The Matlab function for that is < code > h2hinfsyn< / code > (< a href = "https://fr.mathworks.com/help/robust/ref/h2hinfsyn.html" > doc< / a > ).
< / p >
< / div >
< / div >
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< div id = "outline-container-org3013d41" class = "outline-3" >
< h3 id = "org3013d41" > < span class = "section-number-3" > 3.2< / span > Noise characteristics and Uncertainty of the individual sensors< / h3 >
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< div class = "outline-text-3" id = "text-3-2" >
< p >
We define the weights that are used to characterize the dynamic uncertainty of the sensors. This will be used for the \(\mathcal{H}_\infty\) part of the synthesis.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > omegac = 100*2*pi; G0 = 0.1; Ginf = 10;
w1 = (Ginf*s/omegac + G0)/(s/omegac + 1);
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omegac = 0.2*2*pi; G0 = 5; Ginf = 0.1;
w2 = (Ginf*s/omegac + G0)/(s/omegac + 1);
omegac = 5000*2*pi; G0 = 1; Ginf = 50;
w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1);
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< / pre >
< / div >
< p >
We define the noise characteristics of the two sensors by choosing \(N_1\) and \(N_2\). This will be used for the \(\mathcal{H}_2\) part of the synthesis.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > omegac = 100*2*pi; G0 = 1e-5; Ginf = 1e-4;
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;
N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
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< / pre >
< / div >
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< p >
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Both dynamical uncertainty and noise characteristics of the individual sensors are shown in Fig. < a href = "#org1270265" > 21< / a > .
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< / p >
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< div id = "org1270265" class = "figure" >
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< p > < img src = "figs/mixed_synthesis_noise_uncertainty_sensors.png" alt = "mixed_synthesis_noise_uncertainty_sensors.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 21: < / span > Noise characteristsics and Dynamical uncertainty of the individual sensors (< a href = "./figs/mixed_synthesis_noise_uncertainty_sensors.png" > png< / a > , < a href = "./figs/mixed_synthesis_noise_uncertainty_sensors.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-org39502ab" class = "outline-3" >
< h3 id = "org39502ab" > < span class = "section-number-3" > 3.3< / span > Weighting Functions on the uncertainty of the super sensor< / h3 >
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< div class = "outline-text-3" id = "text-3-3" >
< p >
We design weights for the \(\mathcal{H}_\infty\) part of the synthesis in order to limit the dynamical uncertainty of the super sensor.
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The maximum wanted multiplicative uncertainty is shown in Fig. < a href = "#org56b5728" > 22< / a > . The idea here is that we don’ t really need low uncertainty at low frequency but only near the crossover frequency that is suppose to be around 300Hz here.
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< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > n = 4; w0 = 2*pi*900; G0 = 9; G1 = 1; Gc = 1.1;
H = (((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;
wphi = 0.2*(s+3.142e04)/(s+628.3)/H;
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< / pre >
< / div >
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< div id = "org56b5728" class = "figure" >
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< p > < img src = "figs/mixed_syn_hinf_weight.png" alt = "mixed_syn_hinf_weight.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 22: < / span > Wanted maximum module uncertainty of the super sensor (< a href = "./figs/mixed_syn_hinf_weight.png" > png< / a > , < a href = "./figs/mixed_syn_hinf_weight.pdf" > pdf< / a > )< / p >
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< / div >
< p >
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The equivalent Magnitude and Phase uncertainties are shown in Fig. < a href = "#orgaf5d6a0" > 23< / a > .
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< / p >
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< div id = "orgaf5d6a0" class = "figure" >
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< p > < img src = "figs/mixed_syn_objective_hinf.png" alt = "mixed_syn_objective_hinf.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 23: < / span > \(\mathcal{H}_\infty\) synthesis objective part of the mixed-synthesis (< a href = "./figs/mixed_syn_objective_hinf.png" > png< / a > , < a href = "./figs/mixed_syn_objective_hinf.pdf" > pdf< / a > )< / p >
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< / div >
< / div >
< / div >
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< div id = "outline-container-orgd08d347" class = "outline-3" >
< h3 id = "orgd08d347" > < span class = "section-number-3" > 3.4< / span > Mixed Synthesis Architecture< / h3 >
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< div class = "outline-text-3" id = "text-3-4" >
< p >
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The synthesis architecture that is used here is shown in Fig. < a href = "#orgc007636" > 24< / a > .
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< / p >
< p >
The controller \(K\) is synthesized such that it:
< / p >
< ul class = "org-ul" >
< li > Keeps the \(\mathcal{H}_\infty\) norm \(G\) of the transfer function from \(w\) to \(z_\infty\) bellow some specified value< / li >
< li > Keeps the \(\mathcal{H}_2\) norm \(H\) of the transfer function from \(w\) to \(z_2\) bellow some specified value< / li >
< li > Minimizes a trade-off criterion of the form \(W_1 G^2 + W_2 H^2\) where \(W_1\) and \(W_2\) are specified values< / li >
< / ul >
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< div id = "orgc007636" class = "figure" >
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< p > < img src = "figs-tikz/mixed_h2_hinf_synthesis.png" alt = "mixed_h2_hinf_synthesis.png" / >
< / p >
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< p > < span class = "figure-number" > Figure 24: < / span > Mixed H2/H-Infinity Synthesis< / p >
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< / div >
< p >
Here, we define \(P\) such that:
< / p >
\begin{align*}
\left\| \frac{z_\infty}{w} \right\|_\infty & = \left\| \begin{matrix}W_1(s) H_1(s) \\ W_2(s) H_2(s)\end{matrix} \right\|_\infty \\
\left\| \frac{z_2}{w} \right\|_2 & = \left\| \begin{matrix}N_1(s) H_1(s) \\ N_2(s) H_2(s)\end{matrix} \right\|_2
\end{align*}
< p >
Then:
< / p >
< ul class = "org-ul" >
< li > we specify the maximum value for the \(\mathcal{H}_\infty\) norm between \(w\) and \(z_\infty\) to be \(1\)< / li >
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< li > we don’ t specify any maximum value for the \(\mathcal{H}_2\) norm between \(w\) and \(z_2\)< / li >
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< li > we choose \(W_1 = 0\) and \(W_2 = 1\) such that the objective is to minimize the \(\mathcal{H}_2\) norm between \(w\) and \(z_2\)< / li >
< / ul >
< p >
The synthesis objective is to have:
\[ \left\| \frac{z_\infty}{w} \right\|_\infty = \left\| \begin{matrix}W_1(s) H_1(s) \\ W_2(s) H_2(s)\end{matrix} \right\|_\infty < 1 \ ]
and to minimize:
\[ \left\| \frac{z_2}{w} \right\|_2 = \left\| \begin{matrix}N_1(s) H_1(s) \\ N_2(s) H_2(s)\end{matrix} \right\|_2 \]
which is what we wanted.
< / p >
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< p >
We define the generalized plant that will be used for the mixed synthesis.
< / p >
< div class = "org-src-container" >
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< pre class = "src src-matlab" > W1u = ss(w1*wphi); W2u = ss(w2*wphi); % Weight on the uncertainty
W1n = ss(N1); W2n = ss(N2); % Weight on the noise
P = [W1u -W1u;
0 W2u;
W1n -W1n;
0 W2n;
1 0];
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< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgec6e5c4" class = "outline-3" >
< h3 id = "orgec6e5c4" > < span class = "section-number-3" > 3.5< / span > Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis< / h3 >
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< div class = "outline-text-3" id = "text-3-5" >
< p >
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The mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is performed below.
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< / p >
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< div class = "org-src-container" >
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< pre class = "src src-matlab" > Nmeas = 1; Ncon = 1; Nz2 = 2;
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[H2,~,normz,~] = h2hinfsyn(P, Nmeas, Ncon, Nz2, [0, 1], 'HINFMAX', 1, 'H2MAX', Inf, 'DKMAX', 100, 'TOL', 0.01, 'DISPLAY', 'on');
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H1 = 1 - H2;
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< / pre >
< / div >
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< p >
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The obtained complementary filters are shown in Fig. < a href = "#org13db4e3" > 25< / a > .
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< / p >
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< div id = "org13db4e3" class = "figure" >
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< p > < img src = "figs/comp_filters_mixed_synthesis.png" alt = "comp_filters_mixed_synthesis.png" / >
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< p > < span class = "figure-number" > Figure 25: < / span > Obtained complementary filters after mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis (< a href = "./figs/comp_filters_mixed_synthesis.png" > png< / a > , < a href = "./figs/comp_filters_mixed_synthesis.pdf" > pdf< / a > )< / p >
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< h3 id = "orgcf41bc0" > < span class = "section-number-3" > 3.6< / span > Obtained Super Sensor’ s noise< / h3 >
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The PSD and CPS of the super sensor’ s noise are shown in Fig. < a href = "#org217e5b6" > 26< / a > and Fig. < a href = "#orgfc270f8" > 27< / a > respectively.
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< p > < img src = "figs/psd_super_sensor_mixed_syn.png" alt = "psd_super_sensor_mixed_syn.png" / >
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< p > < span class = "figure-number" > Figure 26: < / span > Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis (< a href = "./figs/psd_super_sensor_mixed_syn.png" > png< / a > , < a href = "./figs/psd_super_sensor_mixed_syn.pdf" > pdf< / a > )< / p >
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< p > < img src = "figs/cps_super_sensor_mixed_syn.png" alt = "cps_super_sensor_mixed_syn.png" / >
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< p > < span class = "figure-number" > Figure 27: < / span > Cumulative Power Spectrum of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis (< a href = "./figs/cps_super_sensor_mixed_syn.png" > png< / a > , < a href = "./figs/cps_super_sensor_mixed_syn.pdf" > pdf< / a > )< / p >
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< h3 id = "org39e64c2" > < span class = "section-number-3" > 3.7< / span > Obtained Super Sensor’ s Uncertainty< / h3 >
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The uncertainty on the super sensor’ s dynamics is shown in Fig. < a href = "#org5613508" > 28< / a > .
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< p > < img src = "figs/super_sensor_dyn_uncertainty_mixed_syn.png" alt = "super_sensor_dyn_uncertainty_mixed_syn.png" / >
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< p > < span class = "figure-number" > Figure 28: < / span > Super Sensor Dynamical Uncertainty obtained with the mixed synthesis (< a href = "./figs/super_sensor_dyn_uncertainty_mixed_syn.png" > png< / a > , < a href = "./figs/super_sensor_dyn_uncertainty_mixed_syn.pdf" > pdf< / a > )< / p >
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< h3 id = "orgdf7555f" > < span class = "section-number-3" > 3.8< / span > Conclusion< / h3 >
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This synthesis methods allows both to:
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< li > limit the dynamical uncertainty of the super sensor< / li >
< li > minimize the RMS value of the estimation< / li >
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< style > . csl-entry { text-indent : -1.5 em ; margin-left : 1.5 em ; } < / style > < h2 class = 'citeproc-org-bib-h2' > Bibliography< / h2 >
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< div class = "csl-entry" > < a name = "citeproc_bib_item_1" > < / a > Barzilai, Aaron, Tom VanZandt, and Tom Kenny. 1998. “Technique for Measurement of the Noise of a Sensor in the Presence of Large Background Signals.” < i > Review of Scientific Instruments< / i > 69 (7):2767– < a href = "https://doi.org/10.1063/1.1149013" > https://doi.org/10.1063/1.1149013< / a > .< / div >
< div class = "csl-entry" > < a name = "citeproc_bib_item_2" > < / a > Moore, Steven Ian, Andrew J. Fleming, and Yuen Kuan Yong. 2019. “Capacitive Instrumentation and Sensor Fusion for High-Bandwidth Nanopositioning.” < i > IEEE Sensors Letters< / i > 3 (8):1– < a href = "https://doi.org/10.1109/lsens.2019.2933065" > https://doi.org/10.1109/lsens.2019.2933065< / a > .< / div >
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< div id = "postamble" class = "status" >
< p class = "author" > Author: Thomas Dehaeze< / p >
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< p class = "date" > Created: 2020-09-23 mer. 15:37< / p >
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