-
-
- 1. Optimal Sensor Fusion - Minimize the Super Sensor Noise +
- 1. Optimal Sensor Fusion - Minimize the Super Sensor Noise
-
-
- 1.1. Architecture -
- 1.2. Noise of the sensors -
- 1.3. H-Two Synthesis -
- 1.4. Alternative H-Two Synthesis -
- 1.5. H-Infinity Synthesis - method A -
- 1.6. H-Infinity Synthesis - method B -
- 1.7. H-Infinity Synthesis - method C -
- 1.8. Comparison of the methods -
- 1.9. Obtained Super Sensor's noise uncertainty -
- 1.10. Conclusion +
- 1.1. Architecture +
- 1.2. Noise of the sensors +
- 1.3. H-Two Synthesis +
- 1.4. Alternative H-Two Synthesis +
- 1.5. H-Infinity Synthesis - method A +
- 1.6. H-Infinity Synthesis - method B +
- 1.7. H-Infinity Synthesis - method C +
- 1.8. Comparison of the methods +
- 1.9. Obtained Super Sensor's noise uncertainty +
- 1.10. Conclusion
- - 2. Optimal Sensor Fusion - Minimize the Super Sensor Dynamical Uncertainty +
- 2. Optimal Sensor Fusion - Minimize the Super Sensor Dynamical Uncertainty
-
-
- 2.1. Super Sensor Dynamical Uncertainty -
- 2.2. Dynamical uncertainty of the individual sensors -
- 2.3. Synthesis objective -
- 2.4. Requirements as an \(\mathcal{H}_\infty\) norm -
- 2.5. Weighting Function used to bound the super sensor uncertainty -
- 2.6. \(\mathcal{H}_\infty\) Synthesis -
- 2.7. Super sensor uncertainty -
- 2.8. Super sensor noise -
- 2.9. Conclusion +
- 2.1. Super Sensor Dynamical Uncertainty +
- 2.2. Dynamical uncertainty of the individual sensors +
- 2.3. Synthesis objective +
- 2.4. Requirements as an \(\mathcal{H}_\infty\) norm +
- 2.5. Weighting Function used to bound the super sensor uncertainty +
- 2.6. \(\mathcal{H}_\infty\) Synthesis +
- 2.7. Super sensor uncertainty +
- 2.8. Super sensor noise +
- 2.9. Conclusion
- - 3. Optimal Sensor Fusion - Mixed Synthesis +
- 3. Optimal Sensor Fusion - Mixed Synthesis
-
-
- 3.1. Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis - Introduction -
- 3.2. Noise characteristics and Uncertainty of the individual sensors -
- 3.3. Weighting Functions on the uncertainty of the super sensor -
- 3.4. Mixed Synthesis Architecture -
- 3.5. Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis -
- 3.6. Obtained Super Sensor's noise -
- 3.7. Obtained Super Sensor's Uncertainty -
- 3.8. Conclusion +
- 3.1. Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis - Introduction +
- 3.2. Noise characteristics and Uncertainty of the individual sensors +
- 3.3. Weighting Functions on the uncertainty of the super sensor +
- 3.4. Mixed Synthesis Architecture +
- 3.5. Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis +
- 3.6. Obtained Super Sensor's noise +
- 3.7. Obtained Super Sensor's Uncertainty +
- 3.8. Conclusion
- - 4. H-Infinity synthesis to ensure both performance and robustness +
- 4. Mixed Synthesis - LMI Optimization
-
-
- 4.1. Introduction -
- 4.2. Dynamical uncertainty and Noise level of the individual sensors -
- 4.3. Weights for uncertainty and performance -
- 4.4. H-infinity synthesis with 4 outputs corresponding to the 4 weights -
- 4.5. Conclusion +
- 4.1. Introduction +
- 4.2. Noise characteristics and Uncertainty of the individual sensors +
- 4.3. Weights +
- 4.4. LMI Optimization +
- 4.5. Result +
- 4.6. Comparison with the matlab Mixed Synthesis +
- 4.7. H-Infinity Objective +
- 4.8. Obtained Super Sensor's noise +
- 4.9. Obtained Super Sensor's Uncertainty
- - 5. Equivalent Super Sensor +
- 5. H-Infinity synthesis to ensure both performance and robustness
-
-
- 5.1. Sensor Fusion Architecture -
- 5.2. Equivalent Configuration -
- 5.3. Model the uncertainty of the super sensor -
- 5.4. Model the noise of the super sensor -
- 5.5. First guess +
- 5.1. Introduction +
- 5.2. Dynamical uncertainty and Noise level of the individual sensors +
- 5.3. Weights for uncertainty and performance +
- 5.4. H-infinity synthesis with 4 outputs corresponding to the 4 weights +
- 5.5. Conclusion
- - 6. Optimal And Robust Sensor Fusion in Practice +
- 6. Equivalent Super Sensor
-
-
- 6.1. Measurement of the noise characteristics of the sensors
-
-
-
- 6.1.1. Huddle Test -
- 6.1.2. Weights that represents the noises' PSD -
- 6.1.3. Comparison of the noises' PSD -
- 6.1.4. Computation of the coherence, power spectral density and cross spectral density of signals +
- 6.1. Sensor Fusion Architecture +
- 6.2. Equivalent Configuration +
- 6.3. Model the uncertainty of the super sensor +
- 6.4. Model the noise of the super sensor +
- 6.5. First guess
- - 6.2. Estimate the dynamic uncertainty of the sensors -
- 6.3. Optimal and Robust synthesis of the complementary filters +
- 7. Optimal And Robust Sensor Fusion in Practice
+
-
+
- 7.1. Measurement of the noise characteristics of the sensors + -
- 7. Methods of complementary filter synthesis
-
-
-
- 7.1. Complementary filters using analytical formula - -
- 7.2. H-Infinity synthesis of complementary filters +
- 8. Methods of complementary filter synthesis
-
-
- 7.2.1. Synthesis Architecture -
- 7.2.2. Weights -
- 7.2.3. H-Infinity Synthesis -
- 7.2.4. Obtained Complementary Filters +
- 8.1. Complementary filters using analytical formula + -
- 7.3. Feedback Control Architecture to generate Complementary Filters +
- 8.2. H-Infinity synthesis of complementary filters -
- 7.4. Analytical Formula found in the literature +
- 8.3. Feedback Control Architecture to generate Complementary Filters -
- 7.5. Comparison of the different methods of synthesis +
- 8.4. Analytical Formula found in the literature + + +
- 8.5. Comparison of the different methods of synthesis
-
-
- in section 1: the \(\mathcal{H}_2\) synthesis is used to design complementary filters such that the RMS value of the super sensor's noise is minimized -
- in section 2: the \(\mathcal{H}_\infty\) synthesis is used to design complementary filters such that the super sensor's uncertainty is bonded to acceptable values -
- in section 3: 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 -
- in section 4: the \(\mathcal{H}_\infty\) synthesis is used for both limiting the noise and uncertainty of the super sensor -
- in section 5: we try to find the characteristics of the super sensor from the characteristics of the individual sensors and of the complementary filters -
- in section 6: a methodology is proposed to apply optimal and robust sensor fusion in practice -
- in section 7: methods of complementary filter synthesis are proposed +
- in section 1: the \(\mathcal{H}_2\) synthesis is used to design complementary filters such that the RMS value of the super sensor's noise is minimized +
- in section 2: the \(\mathcal{H}_\infty\) synthesis is used to design complementary filters such that the super sensor's uncertainty is bonded to acceptable values +
- in section 3: 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 +
- in section 5: the \(\mathcal{H}_\infty\) synthesis is used for both limiting the noise and uncertainty of the super sensor +
- in section 6: we try to find the characteristics of the super sensor from the characteristics of the individual sensors and of the complementary filters +
- in section 7: a methodology is proposed to apply optimal and robust sensor fusion in practice +
- in section 8: methods of complementary filter synthesis are proposed
-1 Optimal Sensor Fusion - Minimize the Super Sensor Noise
++1 Optimal Sensor Fusion - Minimize the Super Sensor Noise
-The idea is to combine sensors that works in different frequency range using complementary filters. @@ -431,23 +444,23 @@ The Matlab scripts is accessible here
-1.1 Architecture
++1.1 Architecture
-Let's consider the sensor fusion architecture shown on figure 1 where two sensors (sensor 1 and sensor 2) are measuring the same quantity \(x\) with different noise characteristics determined by \(N_1(s)\) and \(N_2(s)\). +Let's consider the sensor fusion architecture shown on figure 1 where two sensors (sensor 1 and sensor 2) are measuring the same quantity \(x\) with different noise characteristics determined by \(N_1(s)\) and \(N_2(s)\).
\(\tilde{n}_1\) and \(\tilde{n}_2\) are normalized white noise:
\begin{equation} -\label{org51f2bdd} +\label{orgecfbe8a} \Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_1}(\omega) = 1 \end{equation} -+
Figure 1: Fusion of two sensors
@@ -457,16 +470,16 @@ Let's consider the sensor fusion architecture shown on figure 2. +We obtain the architecture of figure 2. -+-
Figure 2: Fusion of two sensors with ideal dynamics
@@ -476,7 +489,7 @@ We obtain the architecture of figure 2. \(H_1(s)\) and \(H_2(s)\) are complementary filters: \begin{equation} -\label{org7eb0521} +\label{orgc62aef9} H_1(s) + H_2(s) = 1 \end{equation} @@ -493,14 +506,14 @@ We have that the Power Spectral Density (PSD) of \(\hat{x}\) is: And the goal is the minimize the Root Mean Square (RMS) value of \(\hat{x}\): \begin{equation} -\label{org29e3384} +\label{orgb11ad80} \sigma_{\hat{x}} = \sqrt{\int_0^\infty \Phi_{\hat{x}}(\omega) d\omega} \end{equation}-1.2 Noise of the sensors
++1.2 Noise of the sensors
Let's define the noise characteristics of the two sensors by choosing \(N_1\) and \(N_2\): @@ -520,7 +533,7 @@ N2 = ( +
Figure 3: Noise Characteristics of the two sensors (png, pdf)
@@ -528,8 +541,8 @@ N2 = ( -1.3 H-Two Synthesis
++1.3 H-Two Synthesis
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: @@ -542,11 +555,11 @@ For that, we use the \(\mathcal{H}_2\) Synthesis.
-We use the generalized plant architecture shown on figure 4. +We use the generalized plant architecture shown on figure 4.
-+
Figure 4: \(\mathcal{H}_2\) Synthesis - Generalized plant used for the optimal generation of complementary filters
@@ -575,7 +588,7 @@ Thus, if we minimize the \(\mathcal{H}_2\) norm of this transfer function, we mi-We define the generalized plant \(P\) on matlab as shown on figure 4. +We define the generalized plant \(P\) on matlab as shown on figure 4.
P = [0 N2 1; @@ -600,15 +613,15 @@ Finally, we define \(H_2(s) = 1 - H_1(s)\).
-The complementary filters obtained are shown on figure 5. +The complementary filters obtained are shown on figure 5.
-The PSD of the noise of the individual sensor and of the super sensor are shown in Fig. 6. +The PSD of the noise of the individual sensor and of the super sensor are shown in Fig. 6.
-The Cumulative Power Spectrum (CPS) is shown on Fig. 7. +The Cumulative Power Spectrum (CPS) is shown on Fig. 7.
@@ -616,7 +629,7 @@ The obtained RMS value of the super sensor is lower than the RMS value of the in
-+
Figure 5: Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis (png, pdf)
@@ -630,7 +643,7 @@ PSD_H2 = abs(squeeze -+
Figure 6: Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal (png, pdf)
@@ -644,7 +657,7 @@ CPS_H2 = 1 +
Figure 7: Cumulative Power Spectrum of individual sensors and super sensor using the \(\mathcal{H}_2\) synthesis (png, pdf)
@@ -652,15 +665,15 @@ CPS_H2 = 1 -1.4 Alternative H-Two Synthesis
++1.4 Alternative H-Two Synthesis
-An alternative Alternative formulation of the \(\mathcal{H}_2\) synthesis is shown in Fig. 8. +An alternative Alternative formulation of the \(\mathcal{H}_2\) synthesis is shown in Fig. 8.
-+-
Figure 8: Alternative formulation of the \(\mathcal{H}_2\) synthesis
@@ -681,8 +694,8 @@ An alternative Alternative formulation of the \(\mathcal{H}_2\) synthesis is sho-1.5 H-Infinity Synthesis - method A
++1.5 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 \(n_2\): @@ -706,12 +719,12 @@ We could try to approach that with the \(\mathcal{H}_\infty\) synthesis by using
-As shown on Fig. 3, the frequency where the two sensors have the same noise level is around 9Hz. +As shown on Fig. 3, 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. 9. +The weighting functions used as well as the obtained complementary filters are shown in Fig. 9.
@@ -780,7 +793,7 @@ Test bounds: 0.1000 < gamma <= 10500.0000-+
Figure 9: Weights and Complementary Fitlers obtained (png, pdf)
@@ -798,8 +811,8 @@ CPS_Ha = 1 -1.6 H-Infinity Synthesis - method B
++1.6 H-Infinity Synthesis - method B
We have that: @@ -844,7 +857,7 @@ We use this as the weighting functions for the \(\mathcal{H}_\infty\) synthesis
-The weighting function and the obtained complementary filters are shown in Fig. 10. +The weighting function and the obtained complementary filters are shown in Fig. 10.
@@ -904,7 +917,7 @@ Test bounds: 0.0000 < gamma <= 32.8125-+
Figure 10: Weights and Complementary Fitlers obtained (png, pdf)
@@ -918,8 +931,8 @@ CPS_Hb = 1 -1.7 H-Infinity Synthesis - method C
++1.7 H-Infinity Synthesis - method C
-Wp = 0.56*(inv(N1)+inv(N2))/(1 + s/2/pi/1000); @@ -976,7 +989,7 @@ Test bounds: 0.0000 < gamma <= 36.7543
+
Figure 11: Weights and Complementary Fitlers obtained (png, pdf)
@@ -990,26 +1003,26 @@ CPS_Hc = 1 -1.8 Comparison of the methods
++1.8 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. 12. +The Power Spectral Density of the super sensors obtained with the complementary filters designed using the three methods are shown in Fig. 12.
-The Cumulative Power Spectrum for the same sensors are shown on Fig. 13. +The Cumulative Power Spectrum for the same sensors are shown on Fig. 13.
-The RMS value of the obtained super sensors are shown on table 1. +The RMS value of the obtained super sensors are shown on table 1.
-
- 6.1. Measurement of the noise characteristics of the sensors
-
+
+
+
+
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