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b/matlab/figs/super_sensor_time_domain_h2_hinf.pdf differ diff --git a/matlab/figs/super_sensor_time_domain_h2_hinf.png b/matlab/figs/super_sensor_time_domain_h2_hinf.png new file mode 100644 index 0000000..9f24b85 Binary files /dev/null and b/matlab/figs/super_sensor_time_domain_h2_hinf.png differ diff --git a/matlab/index.html b/matlab/index.html index 75270b8..5154d95 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 @@ -35,51 +35,50 @@

Table of Contents

@@ -95,24 +94,24 @@ Two sensors are considered with both different noise characteristics and dynamic

    -
  • Section 3: the \(\mathcal{H}_2\) synthesis is used to design complementary filters such that the RMS value of the super sensor’s noise is minimized
    • -
  • Section 4: the \(\mathcal{H}_\infty\) synthesis is used to design complementary filters such that the super sensor’s uncertainty is bonded to acceptable values
    • -
  • Section 5: 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
    • +
  • Section 3: the \(\mathcal{H}_2\) synthesis is used to design complementary filters such that the RMS value of the super sensor’s noise is minimized
    • +
  • Section 4: the \(\mathcal{H}_\infty\) synthesis is used to design complementary filters such that the super sensor’s uncertainty is bonded to acceptable values
    • +
  • Section 5: 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
-
-

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
+@@ -161,8 +160,8 @@ 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
+@@ -206,18 +205,18 @@ 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

Figure 1: Sensor Model

-
-

1.1 Sensor Dynamics

+
+

1.1 Sensor Dynamics

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

@@ -248,14 +247,14 @@ G2 = g_pos/s/(1 + s/w_pos); % Position Sensor Plant [V/(m/s)]

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

@@ -263,12 +262,12 @@ 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).

@@ -280,7 +279,7 @@ 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) * ...
@@ -291,18 +290,18 @@ W2 = createWeight('n', 2, 'w0', 2*pi*1e2, 'G0', 0.05, 'G1', 4, 'Gc', 1);

-The bode plot of the sensors nominal dynamics as well as their defined dynamical spread are shown in Figure 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)

@@ -310,12 +309,12 @@ 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} @@ -329,7 +328,7 @@ 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;
@@ -341,7 +340,7 @@ N2 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/1e4);
-
+

sensors_noise.png

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

@@ -349,8 +348,8 @@ N2 = (Ginf*s/omegac + G0)/(s/omegac + 1)/(1 + s/2/pi/1e4);
-
-

1.4 Save Model

+
+

1.4 Save Model

All the dynamical systems representing the sensors are saved for further use. @@ -364,27 +363,27 @@ 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

Figure 6: Sensor Fusion Architecture

@@ -408,11 +407,11 @@ The super sensor estimate \(\hat{x}\) is given by \eqref{eq:super_sensor_estimat
-
-

2.2 Super Sensor Noise

+
+

2.2 Super Sensor Noise

- +

@@ -443,15 +442,15 @@ 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} @@ -463,18 +462,18 @@ If we consider some dynamical uncertainty (the true system dynamics \(G_i\) not \end{equation} -
+

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

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

@@ -483,18 +482,18 @@ 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 \(\sigma_n\).

-
+

sensor_fusion_noise_arch.png

Figure 9: Optimal Sensor Fusion Architecture

@@ -512,23 +511,23 @@ 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 10 and described by Equation \eqref{eq:H2_generalized_plant}. +Consider the generalized plant \(P_{\mathcal{H}_2}\) shown in Figure 10 and described by Equation \eqref{eq:H2_generalized_plant}.

-
+

h_two_optimal_fusion.png

Figure 10: Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters

@@ -584,10 +583,10 @@ Finally, \(H_1(s)\) is defined as follows

-The obtained complementary filters are shown in Figure 11. +The obtained complementary filters are shown in Figure 11.

-
+

htwo_comp_filters.png

Figure 11: Obtained complementary filters using the \(\mathcal{H}_2\) Synthesis

@@ -595,15 +594,15 @@ The obtained complementary filters are shown in Figure 11<
-
-

3.2 Super Sensor Noise

+
+

3.2 Super Sensor Noise

- +

-The Power Spectral Density of the individual sensors’ noise \(\Phi_{n_1}, \Phi_{n_2}\) and of the super sensor noise \(\Phi_{n_{\mathcal{H}_2}}\) are computed below and shown in Figure 12. +The Power Spectral Density of the individual sensors’ noise \(\Phi_{n_1}, \Phi_{n_2}\) and of the super sensor noise \(\Phi_{n_{\mathcal{H}_2}}\) are computed below and shown in Figure 12. 

PSD_S1 = abs(squeeze(freqresp(N1,    freqs, 'Hz'))).^2;
@@ -614,23 +613,9 @@ PSD_H2 = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2 + ...

-The corresponding Cumulative Power Spectrum \(\Gamma_{n_1}\), \(\Gamma_{n_2}\) and \(\Gamma_{n_{\mathcal{H}_2}}\) (cumulative integration of the PSD \eqref{eq:CPS_definition}) are computed below and shown in Figure 13. +The RMS value of the individual sensors and of the super sensor are listed in Table 3.

-
-
CPS_S1 = cumtrapz(freqs, PSD_S1);
-CPS_S2 = cumtrapz(freqs, PSD_S2);
-CPS_H2 = cumtrapz(freqs, PSD_H2);
-
-
- -\begin{equation} - \Gamma_n (\omega) = \int_0^\omega \Phi_n(\nu) d\nu \label{eq:CPS_definition} -\end{equation} - -

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

- +
@@ -663,51 +648,44 @@ 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 12: Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the optimally fused signal

- -
-

cps_h2_synthesis.png -

-

Figure 13: Cumulative Power Spectrum of individual sensors and super sensor using the \(\mathcal{H}_2\) synthesis

-
-

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 14. -The resulting noises are displayed in Figure 15. +The velocity estimates from the two sensors and from the super sensors are shown in Figure 13. +The resulting noises are displayed in Figure 14.

-
+

super_sensor_time_domain_h2.png

-

Figure 14: Noise of individual sensors and noise of the super sensor

+

Figure 13: Noise of individual sensors and noise of the super sensor

-
+

sensor_noise_H2_time_domain.png

-

Figure 15: Noise of the two sensors \(n_1, n_2\) and noise of the super sensor \(n\)

+

Figure 14: Noise of the two sensors \(n_1, n_2\) and noise of the super sensor \(n\)

-
-

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 16. +The super sensor dynamical uncertainty is shown in Figure 15.

@@ -715,20 +693,20 @@ 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 16: Super sensor dynamical uncertainty when using the \(\mathcal{H}_2\) Synthesis

+

Figure 15: Super sensor dynamical uncertainty when using the \(\mathcal{H}_2\) Synthesis

-
-

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. @@ -736,18 +714,18 @@ 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 17. +To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Figure 16.

-
+

sensor_fusion_arch_uncertainty.png

-

Figure 17: Sensor fusion architecture with sensor dynamics uncertainty

+

Figure 16: 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.

@@ -769,7 +747,7 @@ 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.

@@ -787,15 +765,15 @@ 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

- +

@@ -808,7 +786,7 @@ This is done using the \(\mathcal{H}_\infty\) synthesis in Section 18. +The uncertainty bounds of the two individual sensor as well as the wanted maximum uncertainty bounds of the super sensor are shown in Figure 17.

@@ -819,30 +797,30 @@ Wu = createWeight('n', 2, 'w0', 2*pi*4e2, 'G0', 1/sin(Dphi*pi/180), 'G1', 1/4, '
-
+

weight_uncertainty_bounds_Wu.png

-

Figure 18: Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)

+

Figure 17: Uncertainty region of the two sensors as well as the wanted maximum uncertainty of the super sensor (dashed lines)

-
-

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 19 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 18 and is described by Equation \eqref{eq:Hinf_generalized_plant}.

-
+

h_infinity_robust_fusion.png

-

Figure 19: Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters

+

Figure 18: Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters

\begin{equation} \label{eq:Hinf_generalized_plant} @@ -906,48 +884,48 @@ 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 20. +The obtained complementary filters as well as the wanted upper bounds are shown in Figure 19.

-
+

hinf_comp_filters.png

-

Figure 20: Obtained complementary filters using the \(\mathcal{H}_\infty\) Synthesis

+

Figure 19: Obtained complementary filters using the \(\mathcal{H}_\infty\) Synthesis

-
-

4.3 Super sensor uncertainty

+
+

4.3 Super sensor uncertainty

-The super sensor dynamical uncertainty is displayed in Figure 21. +The super sensor dynamical uncertainty is displayed in Figure 20. It is confirmed that the super sensor dynamical uncertainty is less than the maximum allowed uncertainty defined by the norm of \(W_u(s)\).

-The frequency band where the super sensor has small dynamical uncertainty is larger than for the individual sensor. +The \(\mathcal{H}_\infty\) synthesis thus allows to design filters such that the super sensor has specified bounded uncertainty.

-
+

super_sensor_dynamical_uncertainty_Hinf.png

-

Figure 21: Super sensor dynamical uncertainty (solid curve) when using the \(\mathcal{H}_\infty\) Synthesis

+

Figure 20: Super sensor dynamical uncertainty (solid curve) when using the \(\mathcal{H}_\infty\) Synthesis

-
-

4.4 Super sensor noise

+
+

4.4 Super sensor noise

-We now compute the obtain Power Spectral Density of the super sensor’s noise (Figure 22). +We now compute the obtain Power Spectral Density of the super sensor’s noise (Figure 21).

-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.

@@ -960,13 +938,13 @@ PSD_Hinf = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2 + ...
-
+

psd_sensors_hinf_synthesis.png

-

Figure 22: Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the

+

Figure 21: Power Spectral Density of the estimated \(\hat{x}\) using the two sensors alone and using the

- +
@@ -995,8 +973,8 @@ PSD_Hinf = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2 + ... -
-

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. @@ -1009,189 +987,153 @@ 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 22 (\(\hat{G}_i\) are omitted for space reasons). +

+ + +

sensor_fusion_arch_full.png

-

Figure 23: Sensor fusion architecture with sensor dynamics uncertainty

+

Figure 22: Sensor fusion architecture with sensor dynamics uncertainty

The goal is to design complementary filters such that:

    -
  • the maximum uncertainty of the super sensor is bounded
    • +
  • the maximum uncertainty of the super sensor is bounded to acceptable values (defined by \(W_u(s)\))
  • the RMS value of the super sensor noise is minimized

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

- -

-The Matlab function for that is h2hinfsyn (doc). +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 24. +

-The controller \(K\) is synthesized such that it: +The synthesis architecture that is used here is shown in Figure 23. +

+ +

+The filter \(H_2(s)\) is synthesized such that it:

    -
  • Keeps the \(\mathcal{H}_\infty\) norm \(G\) of the transfer function from \(w\) to \(z_\infty\) bellow some specified value
    • -
  • Keeps the \(\mathcal{H}_2\) norm \(H\) of the transfer function from \(w\) to \(z_2\) bellow some specified value
    • -
  • 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
    • +
  • keeps the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \(z_{\mathcal{H}_\infty}\) bellow some specified value
    • +
  • minimizes the \(\mathcal{H}_2\) norm of the transfer function from \(w\) to \(z_{\mathcal{H}_2}\)
-
+

mixed_h2_hinf_synthesis.png

-

Figure 24: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis

+

Figure 23: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis

-Here, we define \(P\) such that: +Let’s see that +with \(H_1(s)= 1 - H_2(s)\)

-\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*} +\begin{align} + \left\| \frac{z_\infty}{w} \right\|_\infty &= \left\| \begin{matrix}H_1(s) W_1(s) W_u(s)\\ H_2(s) W_2(s) W_u(s)\end{matrix} \right\|_\infty \\ + \left\| \frac{z_2}{w} \right\|_2 &= \left\| \begin{matrix}H_1(s) N_1(s) \\ H_2(s) N_2(s)\end{matrix} \right\|_2 = \sigma_n +\end{align}

-Then: -

-
    -
  • we specify the maximum value for the \(\mathcal{H}_\infty\) norm between \(w\) and \(z_\infty\) to be \(1\)
    • -
  • we don’t specify any maximum value for the \(\mathcal{H}_2\) norm between \(w\) and \(z_2\)
    • -
  • we choose \(W_1 = 0\) and \(W_2 = 1\) such that the objective is to minimize the \(\mathcal{H}_2\) norm between \(w\) and \(z_2\)
    • -
- -

-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. -

- -

-We define the generalized plant that will be used for the mixed synthesis. +The generalized plant \(P_{\mathcal{H}_2/\mathcal{H}_\infty}\) is defined below 

W1u = ss(W2*Wu); W2u = ss(W1*Wu); % Weight on the uncertainty
 W1n = ss(N2); W2n = ss(N1); % Weight on the noise
 
-P = [W1u -W1u;
-     0    W2u;
-     W1n -W1n;
-     0    W2n;
-     1    0];
+P = [Wu*W1 -Wu*W1;
+     0      Wu*W2;
+     N1    -N1;
+     0      N2;
+     1      0];

-The mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is performed below. +And the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis is performed.

-
Nmeas = 1; Ncon = 1; Nz2 = 2;
+
[H2, ~] = h2hinfsyn(ss(P), 1, 1, 2, [0, 1], 'HINFMAX', 1, 'H2MAX', Inf, 'DKMAX', 100, 'TOL', 0.01, 'DISPLAY', 'on');
+

+
-[H1, ~, normz, ~] = h2hinfsyn(P, Nmeas, Ncon, Nz2, [0, 1], 'HINFMAX', 1, 'H2MAX', Inf, 'DKMAX', 100, 'TOL', 0.01, 'DISPLAY', 'on'); - -H2 = 1 - H1; +
+
H1 = 1 - H2;

-The obtained complementary filters are shown in Figure 25. +The obtained complementary filters are shown in Figure 24.

-
+

htwo_hinf_comp_filters.png

-

Figure 25: Obtained complementary filters after mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis

+

Figure 24: Obtained complementary filters after mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis

-
-

5.2 Obtained Super Sensor’s noise

+
+

5.2 Obtained Super Sensor’s noise

-The PSD and CPS of the super sensor’s noise are shown in Figure 26 and Figure 27 respectively. +The Power Spectral Density of the super sensor’s noise is shown in Figure 25. +

+ +

+A time domain simulation is shown in Figure 26. +

+ +

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

-
PSD_S2     = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2;
-PSD_S1     = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2;
-PSD_H2Hinf = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
-
-CPS_S2     = cumtrapz(freqs, PSD_S2);
-CPS_S1     = cumtrapz(freqs, PSD_S1);
-CPS_H2Hinf = cumtrapz(freqs, PSD_H2Hinf);
+
PSD_S2     = abs(squeeze(freqresp(N2,    freqs, 'Hz'))).^2;
+PSD_S1     = abs(squeeze(freqresp(N1,    freqs, 'Hz'))).^2;
+PSD_H2Hinf = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2 + ...
+             abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2;
-
+

psd_sensors_htwo_hinf_synthesis.png

-

Figure 26: Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis

+

Figure 25: Power Spectral Density of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis

-
-

cps_h2_hinf_synthesis.png +

+

super_sensor_time_domain_h2_hinf.png

-

Figure 27: Cumulative Power Spectrum of the Super Sensor obtained with the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis

-
-
+

Figure 26: Noise of individual sensors and noise of the super sensor

-
-

5.3 Obtained Super Sensor’s Uncertainty

-
-

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

-
-
- -
-

5.4 Comparison Hinf H2 H2/Hinf

-
-
-
H2_filters      = load('./mat/H2_filters.mat',      'H2', 'H1');
-Hinf_filters    = load('./mat/Hinf_filters.mat',    'H2', 'H1');
-H2_Hinf_filters = load('./mat/H2_Hinf_filters.mat', 'H2', 'H1');
-
-
- -
-
PSD_H2 = abs(squeeze(freqresp(N2*H2_filters.H2, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N1*H2_filters.H1, freqs, 'Hz'))).^2;
-CPS_H2 = cumtrapz(freqs, PSD_H2);
-
-PSD_Hinf = abs(squeeze(freqresp(N2*Hinf_filters.H2, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N1*Hinf_filters.H1, freqs, 'Hz'))).^2;
-CPS_Hinf = cumtrapz(freqs, PSD_Hinf);
-
-PSD_H2Hinf = abs(squeeze(freqresp(N2*H2_Hinf_filters.H2, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N1*H2_Hinf_filters.H1, freqs, 'Hz'))).^2;
-CPS_H2Hinf = cumtrapz(freqs, PSD_H2Hinf);
-
-
- -
Table 4: Comparison of the obtained RMS noise of the super sensor
+
@@ -1208,7 +1150,7 @@ CPS_H2Hinf = cumtrapz(freqs, PSD_H2Hinf); - + @@ -1218,18 +1160,34 @@ CPS_H2Hinf = cumtrapz(freqs, PSD_H2Hinf); - +
Table 5: Comparison of the obtained RMS noise of the super sensor
Optimal: \(\mathcal{H}_2\)0.00120.0027
Mixed: \(\mathcal{H}_2/\mathcal{H}_\infty\)0.0110.01
-
-

5.5 Conclusion

-
+
+

5.3 Obtained Super Sensor’s Uncertainty

+

-This synthesis methods allows both to: +The uncertainty on the super sensor’s dynamics is shown in Figure 27. +

+ + +
+

super_sensor_dynamical_uncertainty_Htwo_Hinf.png +

+

Figure 27: Super sensor dynamical uncertainty (solid curve) when using the mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis

+
+
+
+ +
+

5.4 Conclusion

+
+

+The mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) synthesis of the complementary filters allows to:

  • limit the dynamical uncertainty of the super sensor
    • @@ -1239,18 +1197,18 @@ This synthesis methods allows both to:
-
-

6 Matlab Functions

+
+

6 Matlab Functions

- +

-
-

6.1 createWeight

+
+

6.1 createWeight

- +

@@ -1302,11 +1260,11 @@ This Matlab function is accessible here.

-
-

6.2 plotMagUncertainty

+
+

6.2 plotMagUncertainty

- +

@@ -1357,11 +1315,11 @@ end

-
-

6.3 plotPhaseUncertainty

+
+

6.3 plotPhaseUncertainty

- +

@@ -1425,7 +1383,7 @@ end

Author: Thomas Dehaeze

-

Created: 2020-10-02 ven. 17:57

+

Created: 2020-10-02 ven. 18:25

diff --git a/matlab/index.org b/matlab/index.org index c452b6d..d9a2349 100644 --- a/matlab/index.org +++ b/matlab/index.org @@ -332,10 +332,6 @@ All the dynamical systems representing the sensors are saved for further use. 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; - - CPS_S1 = cumtrapz(freqs, PSD_S1); - CPS_S2 = cumtrapz(freqs, PSD_S2); - CPS_H2 = cumtrapz(freqs, PSD_H2); #+end_src #+begin_src matlab @@ -603,20 +599,9 @@ The Power Spectral Density of the individual sensors' noise $\Phi_{n_1}, \Phi_{n abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2; #+end_src -The corresponding Cumulative Power Spectrum $\Gamma_{n_1}$, $\Gamma_{n_2}$ and $\Gamma_{n_{\mathcal{H}_2}}$ (cumulative integration of the PSD eqref:eq:CPS_definition) are computed below and shown in Figure [[fig:cps_h2_synthesis]]. -#+begin_src matlab - CPS_S1 = cumtrapz(freqs, PSD_S1); - CPS_S2 = cumtrapz(freqs, PSD_S2); - CPS_H2 = cumtrapz(freqs, PSD_H2); -#+end_src - -\begin{equation} - \Gamma_n (\omega) = \int_0^\omega \Phi_n(\nu) d\nu \label{eq:CPS_definition} -\end{equation} - The RMS value of the individual sensors and of the super sensor are listed in Table [[tab:rms_noise_H2]]. #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) - data2orgtable([sqrt(CPS_S1(end)); sqrt(CPS_S2(end)); sqrt(CPS_H2(end))], {'$\sigma_{n_1}$', '$\sigma_{n_2}$', '$\sigma_{n_{\mathcal{H}_2}}$'}, {'RMS value $[m/s]$'}, ' %.3f '); + data2orgtable([sqrt(trapz(freqs, PSD_S1)); sqrt(trapz(freqs, PSD_S2)); sqrt(trapz(freqs, PSD_H2))], {'$\sigma_{n_1}$', '$\sigma_{n_2}$', '$\sigma_{n_{\mathcal{H}_2}}$'}, {'RMS value $[m/s]$'}, ' %.3f '); #+end_src #+name: tab:rms_noise_H2 @@ -637,7 +622,7 @@ The RMS value of the individual sensors and of the super sensor are listed in Ta plot(freqs, PSD_S2, '-', 'DisplayName', '$\Phi_{n_2}$'); plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{n_{\mathcal{H}_2}}$'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); ylabel('Power Spectral Density [$(m/s)^2/Hz$]'); + xlabel('Frequency [Hz]'); ylabel('Power Spectral Density $\left[ \frac{(m/s)^2}{Hz} \right]$'); hold off; xlim([freqs(1), freqs(end)]); legend('location', 'northeast'); @@ -652,28 +637,6 @@ The RMS value of the individual sensors and of the super sensor are listed in Ta #+RESULTS: [[file:figs/psd_sensors_htwo_synthesis.png]] -#+begin_src matlab :exports none - figure; - hold on; - plot(freqs, CPS_S1, '-', 'DisplayName', '$\Gamma_{n_1}$'); - plot(freqs, CPS_S2, '-', 'DisplayName', '$\Gamma_{n_2}$'); - plot(freqs, CPS_H2, 'k-', 'DisplayName', '$\Gamma_{n_{\mathcal{H}_2}}$'); - set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); - xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum $[(m/s)^2]$'); - hold off; - xlim([2*freqs(1), freqs(end)]); - legend('location', 'southeast'); -#+end_src - -#+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/cps_h2_synthesis.pdf', 'width', 'wide', 'height', 'normal'); -#+end_src - -#+name: fig:cps_h2_synthesis -#+caption: Cumulative Power Spectrum of individual sensors and super sensor using the $\mathcal{H}_2$ synthesis -#+RESULTS: -[[file:figs/cps_h2_synthesis.png]] - A time domain simulation is now performed. The measured velocity $x$ is set to be a sweep sine with an amplitude of $0.1\ [m/s]$. The velocity estimates from the two sensors and from the super sensors are shown in Figure [[fig:super_sensor_time_domain_h2]]. @@ -1109,13 +1072,6 @@ As expected, the super sensor obtained from the $\mathcal{H}_\infty$ synthesis i PSD_H2 = abs(squeeze(freqresp(N1*H2_filters.H1, freqs, 'Hz'))).^2 + ... abs(squeeze(freqresp(N2*H2_filters.H2, freqs, 'Hz'))).^2; - CPS_H2 = cumtrapz(freqs, PSD_H2); -#+end_src - -#+begin_src matlab :exports none - CPS_S2 = cumtrapz(freqs, PSD_S2); - CPS_S1 = cumtrapz(freqs, PSD_S1); - CPS_Hinf = cumtrapz(freqs, PSD_Hinf); #+end_src #+begin_src matlab :exports none @@ -1126,7 +1082,7 @@ As expected, the super sensor obtained from the $\mathcal{H}_\infty$ synthesis i plot(freqs, PSD_H2, 'k-', 'DisplayName', '$\Phi_{n_{\mathcal{H}_2}}$'); plot(freqs, PSD_Hinf, 'k--', 'DisplayName', '$\Phi_{n_{\mathcal{H}_\infty}}$'); set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - xlabel('Frequency [Hz]'); ylabel('Power Spectral Density [$(m/s)^2/Hz$]'); + xlabel('Frequency [Hz]'); ylabel('Power Spectral Density $\left[ \frac{(m/s)^2}{Hz} \right]$'); hold off; xlim([freqs(1), freqs(end)]); legend('location', 'northeast'); @@ -1142,7 +1098,7 @@ As expected, the super sensor obtained from the $\mathcal{H}_\infty$ synthesis i [[file:figs/psd_sensors_hinf_synthesis.png]] #+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) - data2orgtable([sqrt(CPS_H2(end)), sqrt(CPS_Hinf(end))]', {'Optimal: $\mathcal{H}_2$', 'Robust: $\mathcal{H}_\infty$'}, {'RMS [m/s]'}, ' %.1e '); + data2orgtable([sqrt(trapz(freqs, PSD_H2)), sqrt(trapz(freqs, PSD_Hinf))]', {'Optimal: $\mathcal{H}_2$', 'Robust: $\mathcal{H}_\infty$'}, {'RMS [m/s]'}, ' %.1e '); #+end_src #+name: tab:rms_noise_comp_H2_Hinf @@ -1168,17 +1124,19 @@ However, the RMS of the super sensor noise is not optimized as it was the case w <> ** Introduction :ignore: +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 [[fig:sensor_fusion_arch_full]] ($\hat{G}_i$ are omitted for space reasons). + #+name: fig:sensor_fusion_arch_full #+caption: Sensor fusion architecture with sensor dynamics uncertainty [[file:figs-tikz/sensor_fusion_arch_full.png]] The goal is to design complementary filters such that: -- the maximum uncertainty of the super sensor is bounded +- the maximum uncertainty of the super sensor is bounded to acceptable values (defined by $W_u(s)$) - the RMS value of the super sensor noise is minimized -To do so, we can use the Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis. - -The Matlab function for that is =h2hinfsyn= ([[https://fr.mathworks.com/help/robust/ref/h2hinfsyn.html][doc]]). +To do so, we can use the Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis presented in Section [[sec:H2_Hinf_synthesis]]. ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) @@ -1195,53 +1153,44 @@ The Matlab function for that is =h2hinfsyn= ([[https://fr.mathworks.com/help/rob #+end_src ** Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis +<> + The synthesis architecture that is used here is shown in Figure [[fig:mixed_h2_hinf_synthesis]]. -The controller $K$ is synthesized such that it: -- Keeps the $\mathcal{H}_\infty$ norm $G$ of the transfer function from $w$ to $z_\infty$ bellow some specified value -- Keeps the $\mathcal{H}_2$ norm $H$ of the transfer function from $w$ to $z_2$ bellow some specified value -- 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 +The filter $H_2(s)$ is synthesized such that it: +- keeps the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $z_{\mathcal{H}_\infty}$ bellow some specified value +- minimizes the $\mathcal{H}_2$ norm of the transfer function from $w$ to $z_{\mathcal{H}_2}$ #+name: fig:mixed_h2_hinf_synthesis #+caption: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis [[file:figs-tikz/mixed_h2_hinf_synthesis.png]] -Here, we define $P$ such that: -\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*} +Let's see that +with $H_1(s)= 1 - H_2(s)$ +\begin{align} + \left\| \frac{z_\infty}{w} \right\|_\infty &= \left\| \begin{matrix}H_1(s) W_1(s) W_u(s)\\ H_2(s) W_2(s) W_u(s)\end{matrix} \right\|_\infty \\ + \left\| \frac{z_2}{w} \right\|_2 &= \left\| \begin{matrix}H_1(s) N_1(s) \\ H_2(s) N_2(s)\end{matrix} \right\|_2 = \sigma_n +\end{align} -Then: -- we specify the maximum value for the $\mathcal{H}_\infty$ norm between $w$ and $z_\infty$ to be $1$ -- we don't specify any maximum value for the $\mathcal{H}_2$ norm between $w$ and $z_2$ -- we choose $W_1 = 0$ and $W_2 = 1$ such that the objective is to minimize the $\mathcal{H}_2$ norm between $w$ and $z_2$ - -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. - -We define the generalized plant that will be used for the mixed synthesis. +The generalized plant $P_{\mathcal{H}_2/\mathcal{H}_\infty}$ is defined below #+begin_src matlab W1u = ss(W2*Wu); W2u = ss(W1*Wu); % Weight on the uncertainty W1n = ss(N2); W2n = ss(N1); % Weight on the noise - P = [W1u -W1u; - 0 W2u; - W1n -W1n; - 0 W2n; - 1 0]; + P = [Wu*W1 -Wu*W1; + 0 Wu*W2; + N1 -N1; + 0 N2; + 1 0]; #+end_src -The mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis is performed below. +And the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis is performed. #+begin_src matlab - Nmeas = 1; Ncon = 1; Nz2 = 2; + [H2, ~] = h2hinfsyn(ss(P), 1, 1, 2, [0, 1], 'HINFMAX', 1, 'H2MAX', Inf, 'DKMAX', 100, 'TOL', 0.01, 'DISPLAY', 'on'); +#+end_src - [H1, ~, normz, ~] = h2hinfsyn(P, Nmeas, Ncon, Nz2, [0, 1], 'HINFMAX', 1, 'H2MAX', Inf, 'DKMAX', 100, 'TOL', 0.01, 'DISPLAY', 'on'); - - H2 = 1 - H1; +#+begin_src matlab + H1 = 1 - H2; #+end_src #+begin_src matlab :exports none @@ -1256,15 +1205,12 @@ The obtained complementary filters are shown in Figure [[fig:htwo_hinf_comp_filt ax1 = subplot(2,1,1); hold on; - set(gca,'ColorOrderIndex',1) - plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$W_1$'); - set(gca,'ColorOrderIndex',2) - plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$W_2$'); + plot(freqs, 1./abs(squeeze(freqresp(Wu*W1, freqs, 'Hz'))), '--', 'DisplayName', '$1/|W_uW_1|$'); + plot(freqs, 1./abs(squeeze(freqresp(Wu*W2, freqs, 'Hz'))), '--', 'DisplayName', '$1/|W_uW_2|$'); set(gca,'ColorOrderIndex',1) - plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$'); - set(gca,'ColorOrderIndex',2) - plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$'); + plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$'); + plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); @@ -1275,10 +1221,8 @@ The obtained complementary filters are shown in Figure [[fig:htwo_hinf_comp_filt ax2 = subplot(2,1,2); hold on; - set(gca,'ColorOrderIndex',1) - plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-'); - set(gca,'ColorOrderIndex',2) plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-'); + plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-'); hold off; xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); set(gca, 'XScale', 'log'); @@ -1286,7 +1230,6 @@ The obtained complementary filters are shown in Figure [[fig:htwo_hinf_comp_filt linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); - xticks([0.1, 1, 10, 100, 1000]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace @@ -1299,12 +1242,17 @@ The obtained complementary filters are shown in Figure [[fig:htwo_hinf_comp_filt [[file:figs/htwo_hinf_comp_filters.png]] ** Obtained Super Sensor's noise -The PSD and CPS of the super sensor's noise are shown in Figure [[fig:psd_sensors_htwo_hinf_synthesis]] and Figure [[fig:cps_h2_hinf_synthesis]] respectively. +The Power Spectral Density of the super sensor's noise is shown in Figure [[fig:psd_sensors_htwo_hinf_synthesis]]. -#+begin_src matlab :exports none - % The filters are loaded - H2_filters = load('./mat/H2_filters.mat', 'H2', 'H1'); - Hinf_filters = load('./mat/Hinf_filters.mat', 'H2', 'H1'); +A time domain simulation is shown in Figure [[fig:super_sensor_time_domain_h2_hinf]]. + +The RMS values of the super sensor noise for the presented three synthesis are listed in Table [[tab:rms_noise_comp]]. + +#+begin_src matlab + PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2; + PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2; + PSD_H2Hinf = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2 + ... + abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2; #+end_src #+begin_src matlab :exports none @@ -1312,7 +1260,6 @@ The PSD and CPS of the super sensor's noise are shown in Figure [[fig:psd_sensor PSD_H2 = abs(squeeze(freqresp(N1*H2_filters.H1, freqs, 'Hz'))).^2 + ... abs(squeeze(freqresp(N2*H2_filters.H2, freqs, 'Hz'))).^2; - CPS_H2 = cumtrapz(freqs, PSD_H2); #+end_src #+begin_src matlab :exports none @@ -1320,17 +1267,6 @@ The PSD and CPS of the super sensor's noise are shown in Figure [[fig:psd_sensor PSD_Hinf = abs(squeeze(freqresp(N1*Hinf_filters.H1, freqs, 'Hz'))).^2 + ... abs(squeeze(freqresp(N2*Hinf_filters.H2, freqs, 'Hz'))).^2; - CPS_Hinf = cumtrapz(freqs, PSD_Hinf); -#+end_src - -#+begin_src matlab - PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2; - PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2; - PSD_H2Hinf = abs(squeeze(freqresp(N1*H1, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N2*H2, freqs, 'Hz'))).^2; - - CPS_S2 = cumtrapz(freqs, PSD_S2); - CPS_S1 = cumtrapz(freqs, PSD_S1); - CPS_H2Hinf = cumtrapz(freqs, PSD_H2Hinf); #+end_src #+begin_src matlab :exports none @@ -1357,32 +1293,63 @@ The PSD and CPS of the super sensor's noise are shown in Figure [[fig:psd_sensor #+RESULTS: [[file:figs/psd_sensors_htwo_hinf_synthesis.png]] +#+begin_src matlab :exports none + Fs = 1e4; % Sampling Frequency [Hz] + Ts = 1/Fs; % Sampling Time [s] + + t = 0:Ts:2; % Time Vector [s] + + v = 0.1*sin((10*t).*t)'; % Velocity measured [m/s] + + % Generate noises in velocity corresponding to sensor 1 and 2: + n1 = lsim(N1, sqrt(Fs/2)*randn(length(t), 1), t); + n2 = lsim(N2, sqrt(Fs/2)*randn(length(t), 1), t); +#+end_src + #+begin_src matlab :exports none figure; hold on; - plot(freqs, CPS_S1, '-', 'DisplayName', '$\Gamma_{n_1}$'); - plot(freqs, CPS_S2, '-', 'DisplayName', '$\Gamma_{n_2}$'); - plot(freqs, CPS_H2, 'k-', 'DisplayName', '$\Gamma_{n_{\mathcal{H}_2}}$'); - plot(freqs, CPS_Hinf, 'k--', 'DisplayName', '$\Gamma_{n_{\mathcal{H}_\infty}}$'); - plot(freqs, CPS_H2Hinf, 'k-.', 'DisplayName', '$\Gamma_{n_{\mathcal{H}_2/\mathcal{H}_\infty}}$'); - set(gca, 'YScale', 'log'); set(gca, 'XScale', 'log'); - xlabel('Frequency [Hz]'); ylabel('Cumulative Power Spectrum'); + set(gca,'ColorOrderIndex',2) + plot(t, v + n2, 'DisplayName', '$\hat{x}_2$'); + set(gca,'ColorOrderIndex',1) + plot(t, v + n1, 'DisplayName', '$\hat{x}_1$'); + set(gca,'ColorOrderIndex',3) + plot(t, v + (lsim(H1, n1, t) + lsim(H2, n2, t)), 'DisplayName', '$\hat{x}$'); + plot(t, v, 'k--', 'DisplayName', '$x$'); hold off; - xlim([2*freqs(1), freqs(end)]); - legend('location', 'southeast'); + xlabel('Time [s]'); ylabel('Velocity [m/s]'); + legend('location', 'southwest'); + ylim([-0.3, 0.3]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace - exportFig('figs/cps_h2_hinf_synthesis.pdf', 'width', 'wide', 'height', 'normal'); + exportFig('figs/super_sensor_time_domain_h2_hinf.pdf', 'width', 'wide', 'height', 'normal'); #+end_src -#+name: fig:cps_h2_hinf_synthesis -#+CAPTION: Cumulative Power Spectrum of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis +#+name: fig:super_sensor_time_domain_h2_hinf +#+caption: Noise of individual sensors and noise of the super sensor #+RESULTS: -[[file:figs/cps_h2_hinf_synthesis.png]] +[[file:figs/super_sensor_time_domain_h2_hinf.png]] + +#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) + data2orgtable([sqrt(trapz(freqs, PSD_H2)), sqrt(trapz(freqs, PSD_Hinf)), sqrt(trapz(freqs, PSD_H2Hinf))]', ... + {'Optimal: $\mathcal{H}_2$', 'Robust: $\mathcal{H}_\infty$', 'Mixed: $\mathcal{H}_2/\mathcal{H}_\infty$'}, ... + {'RMS [m/s]'}, ' %.1e '); +#+end_src + +#+name: tab:rms_noise_comp +#+caption: Comparison of the obtained RMS noise of the super sensor +#+attr_latex: :environment tabular :align cc +#+attr_latex: :center t :booktabs t :float t +#+RESULTS: +| | RMS [m/s] | +|-------------------------------------------+-----------| +| Optimal: $\mathcal{H}_2$ | 0.0027 | +| Robust: $\mathcal{H}_\infty$ | 0.041 | +| Mixed: $\mathcal{H}_2/\mathcal{H}_\infty$ | 0.01 | ** Obtained Super Sensor's Uncertainty -The uncertainty on the super sensor's dynamics is shown in Figure +The uncertainty on the super sensor's dynamics is shown in Figure [[fig:super_sensor_dynamical_uncertainty_Htwo_Hinf]]. #+begin_src matlab :exports none Dphi_Wu = 180/pi*asin(abs(squeeze(freqresp(inv(Wu), freqs, 'Hz')))); @@ -1395,18 +1362,21 @@ The uncertainty on the super sensor's dynamics is shown in Figure % Magnitude ax1 = subplot(2,1,1); hold on; - plotMagUncertainty(W1, freqs, 'color_i', 1); - plotMagUncertainty(W2, freqs, 'color_i', 2); - p = patch([freqs flip(freqs)], [1 + abs(squeeze(freqresp(W2*H2, freqs, 'Hz')))+abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))); flip(max(1 - abs(squeeze(freqresp(W2*H2, freqs, 'Hz')))-abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))), 0.001))], 'w'); - p.EdgeColor = 'black'; p.FaceAlpha = 0; - plot(freqs, 1 + abs(squeeze(freqresp(inv(Wu), freqs, 'Hz'))), 'r--', ... - 'DisplayName', '$W_u$') - plot(freqs, 1 - abs(squeeze(freqresp(inv(Wu), freqs, 'Hz'))), 'r--', ... + plotMagUncertainty(W1, freqs, 'color_i', 1, 'DisplayName', '$1 + W_1 \Delta_1$'); + plotMagUncertainty(W2, freqs, 'color_i', 2, 'DisplayName', '$1 + W_2 \Delta_2$'); + plot(freqs, 1 + abs(squeeze(freqresp(W2*H2, freqs, 'Hz')))+abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))), 'k-', ... + 'DisplayName', '$1 + W_1 \Delta_1 + W_2 \Delta_2$') + plot(freqs, max(1 - abs(squeeze(freqresp(W2*H2, freqs, 'Hz')))-abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))), 0.001), 'k-', ... + 'HandleVisibility', 'off'); + plot(freqs, 1 + abs(squeeze(freqresp(inv(Wu), freqs, 'Hz'))), 'k--', ... + 'DisplayName', '$1 + W_u^{-1}\Delta$') + plot(freqs, 1 - abs(squeeze(freqresp(inv(Wu), freqs, 'Hz'))), 'k--', ... 'HandleVisibility', 'off') set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); set(gca, 'XTickLabel',[]); ylabel('Magnitude'); ylim([1e-2, 1e1]); + legend('location', 'southeast'); hold off; % Phase @@ -1414,10 +1384,10 @@ The uncertainty on the super sensor's dynamics is shown in Figure hold on; plotPhaseUncertainty(W1, freqs, 'color_i', 1); plotPhaseUncertainty(W2, freqs, 'color_i', 2); - p = patch([freqs flip(freqs)], [Dphi_ss; flip(-Dphi_ss)], 'w'); - p.EdgeColor = 'black'; p.FaceAlpha = 0; - plot(freqs, Dphi_Wu, 'r--'); - plot(freqs, -Dphi_Wu, 'r--'); + plot(freqs, Dphi_ss, 'k-'); + plot(freqs, -Dphi_ss, 'k-'); + plot(freqs, Dphi_Wu, 'k--'); + plot(freqs, -Dphi_Wu, 'k--'); set(gca,'xscale','log'); yticks(-180:90:180); ylim([-180 180]); @@ -1427,41 +1397,17 @@ The uncertainty on the super sensor's dynamics is shown in Figure xlim([freqs(1), freqs(end)]); #+end_src -** Comparison Hinf H2 H2/Hinf -#+begin_src matlab - H2_filters = load('./mat/H2_filters.mat', 'H2', 'H1'); - Hinf_filters = load('./mat/Hinf_filters.mat', 'H2', 'H1'); - H2_Hinf_filters = load('./mat/H2_Hinf_filters.mat', 'H2', 'H1'); +#+begin_src matlab :tangle no :exports results :results file replace + exportFig('figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.pdf', 'width', 'full', 'height', 'full'); #+end_src -#+begin_src matlab - PSD_H2 = abs(squeeze(freqresp(N2*H2_filters.H2, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N1*H2_filters.H1, freqs, 'Hz'))).^2; - CPS_H2 = cumtrapz(freqs, PSD_H2); - - PSD_Hinf = abs(squeeze(freqresp(N2*Hinf_filters.H2, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N1*Hinf_filters.H1, freqs, 'Hz'))).^2; - CPS_Hinf = cumtrapz(freqs, PSD_Hinf); - - PSD_H2Hinf = abs(squeeze(freqresp(N2*H2_Hinf_filters.H2, freqs, 'Hz'))).^2+abs(squeeze(freqresp(N1*H2_Hinf_filters.H1, freqs, 'Hz'))).^2; - CPS_H2Hinf = cumtrapz(freqs, PSD_H2Hinf); -#+end_src - -#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*) - data2orgtable([sqrt(CPS_H2(end)), sqrt(CPS_Hinf(end)), sqrt(CPS_H2Hinf(end))]', {'Optimal: $\mathcal{H}_2$', 'Robust: $\mathcal{H}_\infty$', 'Mixed: $\mathcal{H}_2/\mathcal{H}_\infty$'}, {'RMS [m/s]'}, ' %.1e '); -#+end_src - -#+name: tab:rms_noise_comp -#+caption: Comparison of the obtained RMS noise of the super sensor -#+attr_latex: :environment tabular :align cc -#+attr_latex: :center t :booktabs t :float t +#+name: fig:super_sensor_dynamical_uncertainty_Htwo_Hinf +#+caption: Super sensor dynamical uncertainty (solid curve) when using the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis #+RESULTS: -| | RMS [m/s] | -|-------------------------------------------+-----------| -| Optimal: $\mathcal{H}_2$ | 0.0012 | -| Robust: $\mathcal{H}_\infty$ | 0.041 | -| Mixed: $\mathcal{H}_2/\mathcal{H}_\infty$ | 0.011 | +[[file:figs/super_sensor_dynamical_uncertainty_Htwo_Hinf.png]] ** Conclusion -This synthesis methods allows both to: +The mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis of the complementary filters allows to: - limit the dynamical uncertainty of the super sensor - minimize the RMS value of the estimation