diff --git a/matlab/figs-paper b/matlab/figs-paper new file mode 120000 index 0000000..30fa883 --- /dev/null +++ b/matlab/figs-paper @@ -0,0 +1 @@ +../paper/figs \ No newline at end of file diff --git a/matlab/figs-tikz b/matlab/figs-tikz deleted file mode 120000 index 9a9cf69..0000000 --- a/matlab/figs-tikz +++ /dev/null @@ -1 +0,0 @@ -../tikz/figs \ No newline at end of file diff --git a/matlab/index.html b/matlab/index.html index 623ef1f..004fce6 100644 --- a/matlab/index.html +++ b/matlab/index.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Robust and Optimal Sensor Fusion - Matlab Computation @@ -35,50 +35,50 @@

Table of Contents

@@ -89,27 +89,27 @@ This document is arranged as follows:

-
-

1 Sensor Description

+
+

1 Sensor Description

- +

-In Figure 1 is shown a schematic of a sensor model that is used in the following study. +In Figure 1 is shown a schematic of a sensor model that is used in the following study. In this example, the measured quantity \(x\) is the velocity of an object.

- - +
Table 1: Description of signals in Figure 1
+@@ -176,8 +176,8 @@ In this example, the measured quantity \(x\) is the velocity of an object.
Table 1: Description of signals in Figure 1
- - +
Table 2: Description of Systems in Figure 1
+@@ -221,18 +221,18 @@ In this example, the measured quantity \(x\) is the velocity of an object.
Table 2: Description of Systems in Figure 1
-
-

sensor_model_noise_uncertainty.png +

+

sensor_model_noise_uncertainty.png

Figure 1: Sensor Model

-
-

1.1 Sensor Dynamics

+
+

1.1 Sensor Dynamics

- + Let’s consider two sensors measuring the velocity of an object.

@@ -263,14 +263,14 @@ G2 = g_pos/s/(1 These nominal dynamics are also taken as the model of the sensor dynamics. -The true sensor dynamics has some uncertainty associated to it and described in section 1.2. +The true sensor dynamics has some uncertainty associated to it and described in section 1.2.

-Both sensor dynamics in \([\frac{V}{m/s}]\) are shown in Figure 2. +Both sensor dynamics in \([\frac{V}{m/s}]\) are shown in Figure 2.

-
+

sensors_nominal_dynamics.png

Figure 2: Sensor nominal dynamics from the velocity of the object to the output voltage

@@ -278,12 +278,12 @@ Both sensor dynamics in \([\frac{V}{m/s}]\) are shown in Figure -

1.2 Sensor Model Uncertainty

+
+

1.2 Sensor Model Uncertainty

- -The uncertainty on the sensor dynamics is described by multiplicative uncertainty (Figure 1). + +The uncertainty on the sensor dynamics is described by multiplicative uncertainty (Figure 1).

@@ -295,7 +295,7 @@ The true sensor dynamics \(G_i(s)\) is then described by \eqref{eq:sensor_dynami \end{equation}

-The weights \(W_i(s)\) representing the dynamical uncertainty are defined below and their magnitude is shown in Figure 3. +The weights \(W_i(s)\) representing the dynamical uncertainty are defined below and their magnitude is shown in Figure 3. 

W1 = createWeight('n', 2, 'w0', 2*pi*3,   'G0', 2, 'G1', 0.1,     'Gc', 1) * ...
@@ -306,18 +306,18 @@ W2 = createWeight('n', 2, 4.
+The bode plot of the sensors nominal dynamics as well as their defined dynamical spread are shown in Figure 4.
 

 
 
-

+

 
sensors_uncertainty_weights.png
 

 
Figure 3: Magnitude of the multiplicative uncertainty weights \(|W_i(j\omega)|\)

 

 
 
-

+

 
sensors_nominal_dynamics_and_uncertainty.png
 

 
Figure 4: Nominal Sensor Dynamics \(\hat{G}_i\) (solid lines) as well as the spread of the dynamical uncertainty (background color)

@@ -325,12 +325,12 @@ The bode plot of the sensors nominal dynamics as well as their defined dynamical
 

 

 
-

-
1.3 Sensor Noise

+

+
1.3 Sensor Noise

 

 

-
-The noise of the sensors \(n_i\) are modelled by shaping a white noise with unitary PSD \(\tilde{n}_i\) \eqref{eq:unitary_noise_psd} with a LTI transfer function \(N_i(s)\) (Figure 1).
+
+The noise of the sensors \(n_i\) are modelled by shaping a white noise with unitary PSD \(\tilde{n}_i\) \eqref{eq:unitary_noise_psd} with a LTI transfer function \(N_i(s)\) (Figure 1).
 

 \begin{equation}
   \Phi_{\tilde{n}_i}(\omega) = 1 \label{eq:unitary_noise_psd}
@@ -344,7 +344,7 @@ The Power Spectral Density of the sensor noise \(\Phi_{n_i}(\omega)\) is then co
 \end{equation}
 
 

-The weights \(N_1\) and \(N_2\) representing the amplitude spectral density of the sensor noises are defined below and shown in Figure 5.
+The weights \(N_1\) and \(N_2\) representing the amplitude spectral density of the sensor noises are defined below and shown in Figure 5.
 

 

 
omegac = 0.15*2*pi; G0 = 1e-1; Ginf = 1e-6;
@@ -356,7 +356,7 @@ N2 = (Ginf*s/omegac
 

 
 
-

+

 
sensors_noise.png
 

 
Figure 5: Amplitude spectral density of the sensors \(\sqrt{\Phi_{n_i}(\omega)} = |N_i(j\omega)|\)

@@ -364,8 +364,8 @@ N2 = (Ginf*s/omegac
 

 

 
-

-
1.4 Save Model

+

+
1.4 Save Model

 

 

 All the dynamical systems representing the sensors are saved for further use.
@@ -379,28 +379,28 @@ All the dynamical systems representing the sensors are saved for further use.
 

 

 
-

-
2 Introduction to Sensor Fusion

+

+
2 Introduction to Sensor Fusion

 

 

-
+
 

 

 
-

-
2.1 Sensor Fusion Architecture

+

+
2.1 Sensor Fusion Architecture

 

 

-
+
 

 
 

-The two sensors presented in Section 1 are now merged together using complementary filters \(H_1(s)\) and \(H_2(s)\) to form a super sensor (Figure 6).
+The two sensors presented in Section 1 are now merged together using complementary filters \(H_1(s)\) and \(H_2(s)\) to form a super sensor (Figure 6).
 

 
 
-

-
sensor_fusion_noise_arch.png
+

+
sensor_fusion_noise_arch.png
 

 
Figure 6: Sensor Fusion Architecture

 

@@ -423,11 +423,11 @@ The super sensor estimate \(\hat{x}\) is given by \eqref{eq:super_sensor_estimat
 

 

 
-

-
2.2 Super Sensor Noise

+

+
2.2 Super Sensor Noise

 

 

-
+
 

 
 

@@ -458,15 +458,15 @@ And the Root Mean Square (RMS) value of the super sensor noise \(\sigma_n\) is g
 

 

 
-

-
2.3 Super Sensor Dynamical Uncertainty

+

+
2.3 Super Sensor Dynamical Uncertainty

 

 

-
+
 

 
 

-If we consider some dynamical uncertainty (the true system dynamics \(G_i\) not being perfectly equal to our model \(\hat{G}_i\)) that we model by the use of multiplicative uncertainty (Figure 7), the super sensor dynamics is then equals to:
+If we consider some dynamical uncertainty (the true system dynamics \(G_i\) not being perfectly equal to our model \(\hat{G}_i\)) that we model by the use of multiplicative uncertainty (Figure 7), the super sensor dynamics is then equals to:
 

 
 \begin{equation}
@@ -478,19 +478,19 @@ If we consider some dynamical uncertainty (the true system dynamics \(G_i\) not
 \end{equation}
 
 
-

-
sensor_model_uncertainty.png
+

+
sensor_model_uncertainty.png
 

 
Figure 7: Sensor Model including Dynamical Uncertainty

 

 
 

-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 \(|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|\) as shown in Figure 8.
+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 \(|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|\) as shown in Figure 8.
 

 
 
-

-
uncertainty_set_super_sensor.png
+

+
uncertainty_set_super_sensor.png
 

 
Figure 8: Super Sensor model uncertainty displayed in the complex plane

 

@@ -498,11 +498,11 @@ The uncertainty set of the transfer function from \(\hat{x}\) to \(x\) at freque
 

 

 
-

-
3 Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis

+

+
3 Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis

 

 

-
+
 

 

 In this section, the complementary filters \(H_1(s)\) and \(H_2(s)\) are designed in order to minimize the RMS value of super sensor noise \(\sigma_n\).
@@ -520,24 +520,24 @@ The RMS value of the super sensor noise is (neglecting the model uncertainty):
 
 

 The goal is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s) = 1\) (complementary property) and such that \(\left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2\) is minimized (minimized RMS value of the super sensor noise).
-This is done using the \(\mathcal{H}_2\) synthesis in Section 3.1.
+This is done using the \(\mathcal{H}_2\) synthesis in Section 3.1.
 

 

 
-

-
3.1 \(\mathcal{H}_2\) Synthesis

+

+
3.1 \(\mathcal{H}_2\) Synthesis

 

 

-
+
 

 
 

-Consider the generalized plant \(P_{\mathcal{H}_2}\) shown in Figure 9 and described by Equation \eqref{eq:H2_generalized_plant}.
+Consider the generalized plant \(P_{\mathcal{H}_2}\) shown in Figure 9 and described by Equation \eqref{eq:H2_generalized_plant}.
 

 
 
-

-
h_two_optimal_fusion.png
+

+
h_two_optimal_fusion.png
 

 
Figure 9: Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters

 

@@ -592,10 +592,10 @@ Finally, \(H_1(s)\) is defined as follows
 

 
 

-The obtained complementary filters are shown in Figure 10.
+The obtained complementary filters are shown in Figure 10.
 

 
-

+

 
htwo_comp_filters.png
 

 
Figure 10: Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis

@@ -603,11 +603,11 @@ The obtained complementary filters are shown in Figure 10<
 

 

 
-

-
3.2 Super Sensor Noise

+

+
3.2 Super Sensor Noise

 

 

-
+
 

 
 

@@ -622,13 +622,13 @@ PSD_H2 = abs(squeeze(freqresp(N1*H1, freqs, 
 
 

-The obtained ASD are shown in Figure 11.
+The obtained ASD are shown in Figure 11.
 

 
 

-The RMS value of the individual sensors and of the super sensor are listed in Table 3.
+The RMS value of the individual sensors and of the super sensor are listed in Table 3.
 

-
+
@@ -661,7 +661,7 @@ The RMS value of the individual sensors and of the super sensor are listed in Ta
 

 Table 3: RMS value of the individual sensor noise and of the super sensor using the \(\mathcal{H}_2\) Synthesis
 
 


 
 
-

+

 
psd_sensors_htwo_synthesis.png
 

 
Figure 11: Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal

@@ -670,19 +670,19 @@ The RMS value of the individual sensors and of the super sensor are listed in Ta
 

 A time domain simulation is now performed.
 The measured velocity \(x\) is set to be a sweep sine with an amplitude of \(0.1\ [m/s]\).
-The velocity estimates from the two sensors and from the super sensors are shown in Figure 12.
-The resulting noises are displayed in Figure 13.
+The velocity estimates from the two sensors and from the super sensors are shown in Figure 12.
+The resulting noises are displayed in Figure 13.
 

 
 
-

+

 
super_sensor_time_domain_h2.png
 

 
Figure 12: Noise of individual sensors and noise of the super sensor

 

 
 
-

+

 
sensor_noise_H2_time_domain.png
 

 
Figure 13: Noise of the two sensors \(n_1, n_2\) and noise of the super sensor \(n\)

@@ -690,15 +690,15 @@ The resulting noises are displayed in Figure 13.
 

 

 
-

-
3.3 Discrepancy between sensor dynamics and model

+

+
3.3 Discrepancy between sensor dynamics and model

 

 

-If we consider sensor dynamical uncertainty as explained in Section 1.2, we can compute what would be the super sensor dynamical uncertainty when using the complementary filters obtained using the \(\mathcal{H}_2\) Synthesis.
+If we consider sensor dynamical uncertainty as explained in Section 1.2, we can compute what would be the super sensor dynamical uncertainty when using the complementary filters obtained using the \(\mathcal{H}_2\) Synthesis.
 

 
 

-The super sensor dynamical uncertainty is shown in Figure 14.
+The super sensor dynamical uncertainty is shown in Figure 14.
 

 
 

@@ -706,7 +706,7 @@ It is shown that the phase uncertainty is not bounded between 100Hz and 200Hz.
 As a result the super sensor signal can not be used for feedback applications about 100Hz.
 

 
-

+

 
super_sensor_dynamical_uncertainty_H2.png
 

 
Figure 14: Super sensor dynamical uncertainty when using the \(\mathcal{H}_2\) Synthesis

@@ -715,11 +715,11 @@ As a result the super sensor signal can not be used for feedback applications ab
 

 

 
-

-
4 Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis

+

+
4 Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis

 

 

-
+
 

 

 We initially considered perfectly known sensor dynamics so that it can be perfectly inverted.
@@ -727,18 +727,18 @@ We initially considered perfectly known sensor dynamics so that it can be perfec
 
 

 We now take into account the fact that the sensor dynamics is only partially known.
-To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Figure 15.
+To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Figure 15.
 

 
 
-

-
sensor_fusion_arch_uncertainty.png
+

+
sensor_fusion_arch_uncertainty.png
 

 
Figure 15: Sensor fusion architecture with sensor dynamics uncertainty

 

 
 

-As explained in Section 1.2, at each frequency \(\omega\), the dynamical uncertainty of the super sensor can be represented in the complex plane by a circle with a radius equals to \(|H_1(j\omega) W_1(j\omega)| + |H_2(j\omega) W_2(j\omega)|\) and centered on 1.
+As explained in Section 1.2, at each frequency \(\omega\), the dynamical uncertainty of the super sensor can be represented in the complex plane by a circle with a radius equals to \(|H_1(j\omega) W_1(j\omega)| + |H_2(j\omega) W_2(j\omega)|\) and centered on 1.
 

 
 

@@ -760,7 +760,7 @@ In order to specify a wanted upper bound on the dynamical uncertainty, a weight
 \end{align}
 
 

-The choice of \(W_u\) is presented in Section 4.1.
+The choice of \(W_u\) is presented in Section 4.1.
 

 
 
@@ -778,15 +778,15 @@ The objective is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s)
 

 
 

-This is done using the \(\mathcal{H}_\infty\) synthesis in Section 4.2.
+This is done using the \(\mathcal{H}_\infty\) synthesis in Section 4.2.
 

 

 
-

-
4.1 Weighting Function used to bound the super sensor uncertainty

+

+
4.1 Weighting Function used to bound the super sensor uncertainty

 

 

-
+
 

 
 

@@ -799,7 +799,7 @@ This is done using the \(\mathcal{H}_\infty\) synthesis in Section 16.
+The uncertainty bounds of the two individual sensor as well as the wanted maximum uncertainty bounds of the super sensor are shown in Figure 16.
 

 
 

@@ -810,7 +810,7 @@ Wu = createWeight('n', 2, 
+

 
weight_uncertainty_bounds_Wu.png
 

 
Figure 16: Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)

@@ -818,20 +818,20 @@ Wu = createWeight('n', 2, 
-
4.2 \(\mathcal{H}_\infty\) Synthesis

+

+
4.2 \(\mathcal{H}_\infty\) Synthesis

 

 

-
+
 

 
 

-The generalized plant \(P_{\mathcal{H}_\infty}\) used for the \(\mathcal{H}_\infty\) Synthesis of the complementary filters is shown in Figure 17 and is described by Equation \eqref{eq:Hinf_generalized_plant}.
+The generalized plant \(P_{\mathcal{H}_\infty}\) used for the \(\mathcal{H}_\infty\) Synthesis of the complementary filters is shown in Figure 17 and is described by Equation \eqref{eq:Hinf_generalized_plant}.
 

 
 
-

-
h_infinity_robust_fusion.png
+

+
h_infinity_robust_fusion.png
 

 
Figure 17: Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters

 

@@ -897,11 +897,11 @@ The \(\mathcal{H}_\infty\) is successful as the \(\mathcal{H}_\infty\) norm of t
 

 
 

-The obtained complementary filters as well as the wanted upper bounds are shown in Figure 18.
+The obtained complementary filters as well as the wanted upper bounds are shown in Figure 18.
 

 
 
-

+

 
hinf_comp_filters.png
 

 
Figure 18: Obtained complementary filters using the \(\mathcal{H}_\infty\) Synthesis

@@ -909,11 +909,11 @@ The obtained complementary filters as well as the wanted upper bounds are shown
 

 

 
-

-
4.3 Super sensor uncertainty

+

+
4.3 Super sensor uncertainty

 

 

-The super sensor dynamical uncertainty is displayed in Figure 19.
+The super sensor dynamical uncertainty is displayed in Figure 19.
 It is confirmed that the super sensor dynamical uncertainty is less than the maximum allowed uncertainty defined by the norm of \(W_u(s)\).
 

 
@@ -922,7 +922,7 @@ The \(\mathcal{H}_\infty\) synthesis thus allows to design filters such that the
 

 
 
-

+

 
super_sensor_dynamical_uncertainty_Hinf.png
 

 
Figure 19: Super sensor dynamical uncertainty (solid curve) when using the \(\mathcal{H}_\infty\) Synthesis

@@ -930,12 +930,12 @@ The \(\mathcal{H}_\infty\) synthesis thus allows to design filters such that the
 

 

 
-

-
4.4 Super sensor noise

+

+
4.4 Super sensor noise

 

 

 We now compute the obtain Power Spectral Density of the super sensor’s noise.
-The Amplitude Spectral Densities are shown in Figure 20.
+The Amplitude Spectral Densities are shown in Figure 20.
 

 
 

@@ -947,18 +947,18 @@ PSD_Hinf = abs(squeeze(freqresp(N1*H1, freqs, 
 
 

-The obtained RMS of the super sensor noise in the \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) case are shown in Table 4.
+The obtained RMS of the super sensor noise in the \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) case are shown in Table 4.
 As expected, the super sensor obtained from the \(\mathcal{H}_\infty\) synthesis is much noisier than the super sensor obtained from the \(\mathcal{H}_2\) synthesis.
 

 
 
-

+

 
psd_sensors_hinf_synthesis.png
 

 
Figure 20: Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the

 

 
-
+
@@ -987,8 +987,8 @@ As expected, the super sensor obtained from the \(\mathcal{H}_\infty\) synthesis
 
 
-

-
4.5 Conclusion

+

+
4.5 Conclusion

 

 

 Using the \(\mathcal{H}_\infty\) synthesis, the dynamical uncertainty of the super sensor can be bounded to acceptable values.
@@ -1001,23 +1001,23 @@ However, the RMS of the super sensor noise is not optimized as it was the case w
 

 

 
-

-
5 Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis

+

+
5 Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis

 

 

-
+
 

 

 The (optima) \(\mathcal{H}_2\) synthesis and the (robust) \(\mathcal{H}_\infty\) synthesis are now combined to form an Optimal and Robust synthesis of complementary filters for sensor fusion.
 

 
 

-The sensor fusion architecture is shown in Figure 21 (\(\hat{G}_i\) are omitted for space reasons).
+The sensor fusion architecture is shown in Figure 21 (\(\hat{G}_i\) are omitted for space reasons).
 

 
 
-

-
sensor_fusion_arch_full.png
+

+
sensor_fusion_arch_full.png
 

 
Figure 21: Sensor fusion architecture with sensor dynamics uncertainty

 

@@ -1031,18 +1031,18 @@ The goal is to design complementary filters such that:
 
 
 

-To do so, we can use the Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis presented in Section 5.1.
+To do so, we can use the Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis presented in Section 5.1.
 

 

-

-
5.1 Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis

+

+
5.1 Mixed \(\mathcal{H}_2\) / \(\mathcal{H}_\infty\) Synthesis

 

 

-
+
 

 
 

-The synthesis architecture that is used here is shown in Figure 22.
+The synthesis architecture that is used here is shown in Figure 22.
 

 
 

@@ -1054,8 +1054,8 @@ The filter \(H_2(s)\) is synthesized such that it:
 
 
 
-

-
mixed_h2_hinf_synthesis.png
+

+
mixed_h2_hinf_synthesis.png
 

 
Figure 22: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis

 

@@ -1098,11 +1098,11 @@ And the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is performed.
 

 
 

-The obtained complementary filters are shown in Figure 23.
+The obtained complementary filters are shown in Figure 23.
 

 
 
-

+

 
htwo_hinf_comp_filters.png
 

 
Figure 23: Obtained complementary filters after mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis

@@ -1110,19 +1110,19 @@ The obtained complementary filters are shown in Figure 23<
 

 

 
-

-
5.2 Obtained Super Sensor’s noise

+

+
5.2 Obtained Super Sensor’s noise

 

 

-The Amplitude Spectral Density of the super sensor’s noise is shown in Figure 24.
+The Amplitude Spectral Density of the super sensor’s noise is shown in Figure 24.
 

 
 

-A time domain simulation is shown in Figure 25.
+A time domain simulation is shown in Figure 25.
 

 
 

-The RMS values of the super sensor noise for the presented three synthesis are listed in Table 5.
+The RMS values of the super sensor noise for the presented three synthesis are listed in Table 5.
 

 
 

@@ -1134,20 +1134,20 @@ PSD_H2Hinf = abs(squeeze(freqresp(N1*H1, freqs, 
 
 
-

+

 
psd_sensors_htwo_hinf_synthesis.png
 

 
Figure 24: Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis

 

 
 
-

+

 
super_sensor_time_domain_h2_hinf.png
 

 
Figure 25: Noise of individual sensors and noise of the super sensor

 

 
-

 Table 4: Comparison of the obtained RMS noise of the super sensor
 
 

 

+
@@ -1181,15 +1181,15 @@ PSD_H2Hinf = abs(squeeze(freqresp(N1*H1, freqs, 
 
 
-

-
5.3 Obtained Super Sensor’s Uncertainty

+

+
5.3 Obtained Super Sensor’s Uncertainty

 

 

-The uncertainty on the super sensor’s dynamics is shown in Figure 26.
+The uncertainty on the super sensor’s dynamics is shown in Figure 26.
 

 
 
-

+

 
super_sensor_dynamical_uncertainty_Htwo_Hinf.png
 

 
Figure 26: Super sensor dynamical uncertainty (solid curve) when using the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis

@@ -1197,8 +1197,8 @@ The uncertainty on the super sensor’s dynamics is shown in Figure 
 

 
-

-
5.4 Conclusion

+

+
5.4 Conclusion

 

 

 The mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis of the complementary filters allows to:
@@ -1211,18 +1211,18 @@ The mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis of the complementary fi
 

 

 
-

-
6 Matlab Functions

+

+
6 Matlab Functions

 

 

-
+
 

 

-

-
6.1 createWeight

+

+
6.1 createWeight

 

 

-
+
 

 
 

@@ -1274,11 +1274,11 @@ This Matlab function is accessible here.
 

 

 
-

-
6.2 plotMagUncertainty

+

+
6.2 plotMagUncertainty

 

 

-
+
 

 
 

@@ -1329,11 +1329,11 @@ p.FaceAlpha = args.opacity;
 

 

 
-

-
6.3 plotPhaseUncertainty

+

+
6.3 plotPhaseUncertainty

 

 

-
+
 

 
 

@@ -1397,7 +1397,7 @@ p.FaceAlpha = args.opacity;
 

 

 
Author: Thomas Dehaeze

-
Created: 2020-10-05 lun. 11:45

+
Created: 2020-10-05 lun. 15:08

 

 
 
diff --git a/matlab/index.org b/matlab/index.org
index e298902..0f394e5 100644
--- a/matlab/index.org
+++ b/matlab/index.org
@@ -100,7 +100,7 @@ In this example, the measured quantity $x$ is the velocity of an object.
 #+name: fig:sensor_model_noise_uncertainty
 #+caption: Sensor Model
 #+RESULTS:
-[[file:figs-tikz/sensor_model_noise_uncertainty.png]]
+[[file:figs-paper/sensor_model_noise_uncertainty.png]]
 
 ** Matlab Init                                              :noexport:ignore:
 #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@@ -325,7 +325,7 @@ The two sensors presented in Section [[sec:sensor_description]] are now merged t
 
 #+name: fig:sensor_fusion_noise_arch
 #+caption: Sensor Fusion Architecture
-[[file:figs-tikz/sensor_fusion_noise_arch.png]]
+[[file:figs-paper/sensor_fusion_noise_arch.png]]
 
 The complementary property of $H_1(s)$ and $H_2(s)$ means that the sum of their transfer function is equal to $1$ eqref:eq:complementary_property.
 
@@ -377,13 +377,13 @@ If we consider some dynamical uncertainty (the true system dynamics $G_i$ not be
 
 #+name: fig:sensor_model_uncertainty
 #+caption: Sensor Model including Dynamical Uncertainty
-[[file:figs-tikz/sensor_model_uncertainty.png]]
+[[file:figs-paper/sensor_model_uncertainty.png]]
 
 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 $|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|$ as shown in Figure [[fig:uncertainty_set_super_sensor]].
 
 #+name: fig:uncertainty_set_super_sensor
 #+caption: Super Sensor model uncertainty displayed in the complex plane
-[[file:figs-tikz/uncertainty_set_super_sensor.png]]
+[[file:figs-paper/uncertainty_set_super_sensor.png]]
 
 * Optimal Super Sensor Noise: $\mathcal{H}_2$ Synthesis
 :PROPERTIES:
@@ -427,7 +427,7 @@ Consider the generalized plant $P_{\mathcal{H}_2}$ shown in Figure [[fig:h_two_o
 
 #+name: fig:h_two_optimal_fusion
 #+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
-[[file:figs-tikz/h_two_optimal_fusion.png]]
+[[file:figs-paper/h_two_optimal_fusion.png]]
 
 \begin{equation} \label{eq:H2_generalized_plant}
 \begin{pmatrix}
@@ -696,7 +696,7 @@ To do so, we model the uncertainty that we have on the sensor dynamics by multip
 
 #+name: fig:sensor_fusion_arch_uncertainty
 #+caption: Sensor fusion architecture with sensor dynamics uncertainty
-[[file:figs-tikz/sensor_fusion_arch_uncertainty.png]]
+[[file:figs-paper/sensor_fusion_arch_uncertainty.png]]
 
 As explained in Section [[sec:sensor_uncertainty]], at each frequency $\omega$, the dynamical uncertainty of the super sensor can be represented in the complex plane by a circle with a radius equals to $|H_1(j\omega) W_1(j\omega)| + |H_2(j\omega) W_2(j\omega)|$ and centered on 1.
 
@@ -818,7 +818,7 @@ The generalized plant $P_{\mathcal{H}_\infty}$ used for the $\mathcal{H}_\infty$
 
 #+name: fig:h_infinity_robust_fusion
 #+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
-[[file:figs-tikz/h_infinity_robust_fusion.png]]
+[[file:figs-paper/h_infinity_robust_fusion.png]]
 
 \begin{equation} \label{eq:Hinf_generalized_plant}
 \begin{pmatrix}
@@ -1017,7 +1017,7 @@ As expected, the super sensor obtained from the $\mathcal{H}_\infty$ synthesis i
 #+end_src
 
 #+name: fig:psd_sensors_hinf_synthesis
-#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the
+#+caption: Power Spectral Density of the estimated $\hat{x}$ using the two sensors alone and using the $\mathcal{H}_\infty$ synthesis
 #+RESULTS:
 [[file:figs/psd_sensors_hinf_synthesis.png]]
 
@@ -1054,7 +1054,7 @@ The sensor fusion architecture is shown in Figure [[fig:sensor_fusion_arch_full]
 
 #+name: fig:sensor_fusion_arch_full
 #+caption: Sensor fusion architecture with sensor dynamics uncertainty
-[[file:figs-tikz/sensor_fusion_arch_full.png]]
+[[file:figs-paper/sensor_fusion_arch_full.png]]
 
 The goal is to design complementary filters such that:
 - the maximum uncertainty of the super sensor is bounded to acceptable values (defined by $W_u(s)$)
@@ -1087,7 +1087,7 @@ The filter $H_2(s)$ is synthesized such that it:
 
 #+name: fig:mixed_h2_hinf_synthesis
 #+caption: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
-[[file:figs-tikz/mixed_h2_hinf_synthesis.png]]
+[[file:figs-paper/mixed_h2_hinf_synthesis.png]]
 
 Let's see that
 with $H_1(s)= 1 - H_2(s)$
@@ -1213,7 +1213,7 @@ The RMS values of the super sensor noise for the presented three synthesis are l
 #+end_src
 
 #+name: fig:psd_sensors_htwo_hinf_synthesis
-#+CAPTION: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
+#+caption: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis
 #+RESULTS:
 [[file:figs/psd_sensors_htwo_hinf_synthesis.png]]
 
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 Table 5: Comparison of the obtained RMS noise of the super sensor