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@ -396,6 +396,10 @@ All the dynamical systems representing the sensors are saved for further use.
** Introduction :ignore:
The idea is to combine sensors that works in different frequency range using complementary filters.
#+name: fig:sensor_fusion_noise_arch
#+caption: Optimal Sensor Fusion Architecture
[[file:figs-tikz/sensor_fusion_noise_arch.png]]
Doing so, one "super sensor" is obtained that can have better noise characteristics than the individual sensors over a large frequency range.
The complementary filters have to be designed in order to minimize the effect noise of each sensor on the super sensor noise.
@ -425,11 +429,12 @@ Thus, the goal is to design $H_1(s)$ and $H_2(s)$ such that $H_1(s) + H_2(s) = 1
For that, we use the $\mathcal{H}_2$ Synthesis.
We use the generalized plant architecture shown on figure [[fig:h_infinity_optimal_comp_filters]].
We use the generalized plant architecture shown on Figure [[fig:h_two_optimal_fusion]].
#+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]]
#+name: fig:h_infinity_optimal_comp_filters
#+caption: $\mathcal{H}_2$ Synthesis - Generalized plant used for the optimal generation of complementary filters
[[file:figs-tikz/h_infinity_optimal_comp_filters.png]]
\begin{equation*}
\begin{pmatrix}
@ -494,9 +499,9 @@ The complementary filters obtained are shown on figure [[fig:htwo_comp_filters]]
[[file:figs/htwo_comp_filters.png]]
** Sensor Noise
The PSD of the noise of the individual sensor and of the super sensor are shown in Fig. [[fig:psd_sensors_htwo_synthesis]].
The PSD of the noise of the individual sensor and of the super sensor are shown in Figure [[fig:psd_sensors_htwo_synthesis]].
The Cumulative Power Spectrum (CPS) is shown on Fig. [[fig:cps_h2_synthesis]].
The Cumulative Power Spectrum (CPS) is shown on Figure [[fig:cps_h2_synthesis]].
The obtained RMS value of the super sensor is lower than the RMS value of the individual sensors.
@ -675,11 +680,11 @@ However, the synthesis does not take into account the robustness of the sensor f
We initially considered perfectly known sensor dynamics so that it can be perfectly inverted.
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 Fig. [[fig:sensor_fusion_dynamic_uncertainty]].
To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Figure [[fig:sensor_fusion_arch_uncertainty]].
#+name: fig:sensor_fusion_dynamic_uncertainty
#+name: fig:sensor_fusion_arch_uncertainty
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
[[file:figs-tikz/sensor_fusion_dynamic_uncertainty.png]]
[[file:figs-tikz/sensor_fusion_arch_uncertainty.png]]
The objective here is to design complementary filters $H_1(s)$ and $H_2(s)$ in order to minimize the dynamical uncertainty of the super sensor.
@ -704,11 +709,6 @@ The objective here is to design complementary filters $H_1(s)$ and $H_2(s)$ in o
** Super Sensor Dynamical Uncertainty
In practical systems, the sensor dynamics has always some level of uncertainty.
Let's represent that with multiplicative input uncertainty as shown on figure [[fig:sensor_fusion_dynamic_uncertainty]].
#+name: fig:sensor_fusion_dynamic_uncertainty
#+caption: Fusion of two sensors with input multiplicative uncertainty
[[file:figs-tikz/sensor_fusion_dynamic_uncertainty.png]]
The dynamics of the super sensor is represented by
\begin{align*}
@ -729,7 +729,7 @@ We then have that the angle introduced by the super sensor is bounded by $\arcsi
[[file:figs-tikz/uncertainty_gain_phase_variation.png]]
** Synthesis objective
The uncertainty region of the super sensor dynamics is represented by a circle in the complex plane as shown in Fig. [[fig:uncertainty_gain_phase_variation]].
The uncertainty region of the super sensor dynamics is represented by a circle in the complex plane as shown in Figure [[fig:uncertainty_gain_phase_variation]].
At each frequency $\omega$, the radius of the circle is $|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|$.
@ -811,7 +811,7 @@ Let's define $W_\phi(s)$ in order to bound the maximum allowed phase uncertainty
xlim([freqs(1), freqs(end)]);
#+end_src
The obtained upper bounds on the complementary filters in order to limit the phase uncertainty of the super sensor are represented in Fig. [[fig:upper_bounds_comp_filter_max_phase_uncertainty]].
The obtained upper bounds on the complementary filters in order to limit the phase uncertainty of the super sensor are represented in Figure [[fig:upper_bounds_comp_filter_max_phase_uncertainty]].
#+begin_src matlab :exports none
figure;
@ -835,7 +835,7 @@ The obtained upper bounds on the complementary filters in order to limit the pha
[[file:figs/upper_bounds_comp_filter_max_phase_uncertainty.png]]
** $\mathcal{H}_\infty$ Synthesis
The $\mathcal{H}_\infty$ synthesis architecture used for the complementary filters is shown in Fig. [[fig:h_infinity_robust_fusion]].
The $\mathcal{H}_\infty$ synthesis architecture used for the complementary filters is shown in Figure [[fig:h_infinity_robust_fusion]].
#+name: fig:h_infinity_robust_fusion
#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
@ -883,7 +883,7 @@ And $H_1(s)$ is defined as the complementary of $H_2(s)$.
save('./mat/Hinf_filters.mat', 'H2', 'H1');
#+end_src
The obtained complementary filters are shown in Fig. [[fig:comp_filter_hinf_uncertainty]].
The obtained complementary filters are shown in Figure [[fig:comp_filter_hinf_uncertainty]].
#+begin_src matlab :exports none
figure;
@ -980,8 +980,8 @@ For instance, we could improve the dynamical uncertainty of the super sensor by
We now compute the obtain Power Spectral Density of the super sensor's noise.
The noise characteristics of both individual sensor are defined below.
The PSD of both sensor and of the super sensor is shown in Fig. [[fig:psd_sensors_hinf_synthesis]].
The CPS of both sensor and of the super sensor is shown in Fig. [[fig:cps_sensors_hinf_synthesis]].
The PSD of both sensor and of the super sensor is shown in Figure [[fig:psd_sensors_hinf_synthesis]].
The CPS of both sensor and of the super sensor is shown in Figure [[fig:cps_sensors_hinf_synthesis]].
#+begin_src matlab
PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2;
@ -1052,6 +1052,12 @@ However, the RMS of the super sensor noise is not optimized as it was the case w
:header-args:matlab+: :comments org :mkdirp yes
:END:
<<sec:mixed_synthesis_sensor_fusion>>
** Introduction :ignore:
#+name: fig:sensor_fusion_arch_full
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
[[file:figs-tikz/sensor_fusion_arch_full.png]]
** ZIP file containing the data and matlab files :ignore:
#+begin_note
The Matlab scripts is accessible [[file:matlab/mixed_synthesis_sensor_fusion.m][here]].
@ -1081,7 +1087,7 @@ The Matlab function for that is =h2hinfsyn= ([[https://fr.mathworks.com/help/rob
#+end_src
** Noise characteristics and Uncertainty of the individual sensors
Both dynamical uncertainty and noise characteristics of the individual sensors are shown in Fig. [[fig:mixed_synthesis_noise_uncertainty_sensors]].
Both dynamical uncertainty and noise characteristics of the individual sensors are shown in Figure [[fig:mixed_synthesis_noise_uncertainty_sensors]].
#+begin_src matlab :exports none
figure;
@ -1118,10 +1124,10 @@ Both dynamical uncertainty and noise characteristics of the individual sensors a
** Weighting Functions on the uncertainty of the super sensor
We design weights for the $\mathcal{H}_\infty$ part of the synthesis in order to limit the dynamical uncertainty of the super sensor.
The maximum wanted multiplicative uncertainty is shown in Fig. .The idea here is that we don't really need low uncertainty at low frequency but only near the crossover frequency that is suppose to be around 300Hz here.
The maximum wanted multiplicative uncertainty is shown in Figure .The idea here is that we don't really need low uncertainty at low frequency but only near the crossover frequency that is suppose to be around 300Hz here.
** Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis
The synthesis architecture that is used here is shown in Fig. [[fig:mixed_h2_hinf_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
@ -1129,7 +1135,7 @@ The controller $K$ is synthesized such that it:
- 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
#+name: fig:mixed_h2_hinf_synthesis
#+caption: Mixed H2/H-Infinity Synthesis
#+caption: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
[[file:figs-tikz/mixed_h2_hinf_synthesis.png]]
Here, we define $P$ such that:
@ -1174,7 +1180,7 @@ The mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis is performed below.
save('./mat/H2_Hinf_filters.mat', 'H2', 'H1');
#+end_src
The obtained complementary filters are shown in Fig. [[fig:comp_filters_mixed_synthesis]].
The obtained complementary filters are shown in Figure [[fig:comp_filters_mixed_synthesis]].
#+begin_src matlab :exports none
figure;
@ -1224,7 +1230,7 @@ The obtained complementary filters are shown in Fig. [[fig:comp_filters_mixed_sy
[[file:figs/comp_filters_mixed_synthesis.png]]
** Obtained Super Sensor's noise
The PSD and CPS of the super sensor's noise are shown in Fig. [[fig:psd_super_sensor_mixed_syn]] and Fig. [[fig:cps_super_sensor_mixed_syn]] respectively.
The PSD and CPS of the super sensor's noise are shown in Figure [[fig:psd_super_sensor_mixed_syn]] and Figure [[fig:cps_super_sensor_mixed_syn]] respectively.
#+begin_src matlab
PSD_S2 = abs(squeeze(freqresp(N2, freqs, 'Hz'))).^2;
@ -1282,7 +1288,7 @@ The PSD and CPS of the super sensor's noise are shown in Fig. [[fig:psd_super_se
[[file:figs/cps_super_sensor_mixed_syn.png]]
** Obtained Super Sensor's Uncertainty
The uncertainty on the super sensor's dynamics is shown in Fig.
The uncertainty on the super sensor's dynamics is shown in Figure
#+begin_src matlab :exports none
Dphi_ss = 180/pi*asin(abs(squeeze(freqresp(W2*H2, freqs, 'Hz'))) + abs(squeeze(freqresp(W1*H1, freqs, 'Hz'))));
@ -1510,9 +1516,9 @@ Finally, we define $H_2(s) = 1 - H_1(s)$.
The complementary filters obtained are shown on figure [[fig:htwo_comp_filters]].
The PSD of the noise of the individual sensor and of the super sensor are shown in Fig. [[fig:psd_sensors_htwo_synthesis]].
The PSD of the noise of the individual sensor and of the super sensor are shown in Figure [[fig:psd_sensors_htwo_synthesis]].
The Cumulative Power Spectrum (CPS) is shown on Fig. [[fig:cps_h2_synthesis]].
The Cumulative Power Spectrum (CPS) is shown on Figure [[fig:cps_h2_synthesis]].
The obtained RMS value of the super sensor is lower than the RMS value of the individual sensors.
@ -1698,7 +1704,7 @@ The sensor uncertain models are defined below.
#+end_src
The super sensor uncertain model is defined below using the complementary filters obtained with the $\mathcal{H}_2$ synthesis.
The dynamical uncertainty bounds of the super sensor is shown in Fig. [[fig:uncertainty_super_sensor_H2_syn]].
The dynamical uncertainty bounds of the super sensor is shown in Figure [[fig:uncertainty_super_sensor_H2_syn]].
Right Half Plane zero might be introduced in the super sensor dynamics which will render the feedback system unstable.
#+begin_src matlab
@ -1790,7 +1796,7 @@ However, the synthesis does not take into account the robustness of the sensor f
We initially considered perfectly known sensor dynamics so that it can be perfectly inverted.
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 Fig. [[fig:sensor_fusion_dynamic_uncertainty]].
To do so, we model the uncertainty that we have on the sensor dynamics by multiplicative input uncertainty as shown in Figure [[fig:sensor_fusion_dynamic_uncertainty]].
#+name: fig:sensor_fusion_dynamic_uncertainty
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
@ -1859,7 +1865,7 @@ We define the weights that are used to characterize the dynamic uncertainty of t
w2 = w2*(Ginf*s/omegac + G0)/(s/omegac + 1);
#+end_src
From the weights, we define the uncertain transfer functions of the sensors. Some of the uncertain dynamics of both sensors are shown on Fig. [[fig:uncertainty_dynamics_sensors]] with the upper and lower bounds on the magnitude and on the phase.
From the weights, we define the uncertain transfer functions of the sensors. Some of the uncertain dynamics of both sensors are shown on Figure [[fig:uncertainty_dynamics_sensors]] with the upper and lower bounds on the magnitude and on the phase.
#+begin_src matlab
G1 = 1 + w1*ultidyn('Delta',[1 1]);
G2 = 1 + w2*ultidyn('Delta',[1 1]);
@ -1935,7 +1941,7 @@ From the weights, we define the uncertain transfer functions of the sensors. Som
[[file:figs/uncertainty_dynamics_sensors.png]]
*** Synthesis objective
The uncertainty region of the super sensor dynamics is represented by a circle in the complex plane as shown in Fig. [[fig:uncertainty_gain_phase_variation]].
The uncertainty region of the super sensor dynamics is represented by a circle in the complex plane as shown in Figure [[fig:uncertainty_gain_phase_variation]].
At each frequency $\omega$, the radius of the circle is $|W_1(j\omega) H_1(j\omega)| + |W_2(j\omega) H_2(j\omega)|$.
@ -1975,7 +1981,7 @@ Thus, at these frequencies, $|W_\phi|$ should be smaller than $1$.
*** Weighting Function used to bound the super sensor uncertainty
Let's define $W_\phi(s)$ in order to bound the maximum allowed phase uncertainty $\Delta\phi_\text{max}$ of the super sensor dynamics.
The magnitude $|W_\phi(j\omega)|$ is shown in Fig. [[fig:magnitude_wphi]] and the corresponding maximum allowed phase uncertainty of the super sensor dynamics of shown in Fig. [[fig:maximum_wanted_phase_uncertainty]].
The magnitude $|W_\phi(j\omega)|$ is shown in Figure [[fig:magnitude_wphi]] and the corresponding maximum allowed phase uncertainty of the super sensor dynamics of shown in Figure [[fig:maximum_wanted_phase_uncertainty]].
#+begin_src matlab
Dphi = 20; % [deg]
@ -2035,7 +2041,7 @@ The magnitude $|W_\phi(j\omega)|$ is shown in Fig. [[fig:magnitude_wphi]] and th
#+CAPTION: Maximum wanted phase uncertainty using this weight ([[./figs/maximum_wanted_phase_uncertainty.png][png]], [[./figs/maximum_wanted_phase_uncertainty.pdf][pdf]])
[[file:figs/maximum_wanted_phase_uncertainty.png]]
The obtained upper bounds on the complementary filters in order to limit the phase uncertainty of the super sensor are represented in Fig. [[fig:upper_bounds_comp_filter_max_phase_uncertainty]].
The obtained upper bounds on the complementary filters in order to limit the phase uncertainty of the super sensor are represented in Figure [[fig:upper_bounds_comp_filter_max_phase_uncertainty]].
#+begin_src matlab :exports none
figure;
@ -2059,7 +2065,7 @@ The obtained upper bounds on the complementary filters in order to limit the pha
[[file:figs/upper_bounds_comp_filter_max_phase_uncertainty.png]]
*** $\mathcal{H}_\infty$ Synthesis
The $\mathcal{H}_\infty$ synthesis architecture used for the complementary filters is shown in Fig. [[fig:h_infinity_robust_fusion]].
The $\mathcal{H}_\infty$ synthesis architecture used for the complementary filters is shown in Figure [[fig:h_infinity_robust_fusion]].
#+name: fig:h_infinity_robust_fusion
#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
@ -2106,7 +2112,7 @@ And $H_1(s)$ is defined as the complementary of $H_2(s)$.
H1 = 1 - H2;
#+end_src
The obtained complementary filters are shown in Fig. [[fig:comp_filter_hinf_uncertainty]].
The obtained complementary filters are shown in Figure [[fig:comp_filter_hinf_uncertainty]].
#+begin_src matlab :exports none
figure;
@ -2154,7 +2160,7 @@ The obtained complementary filters are shown in Fig. [[fig:comp_filter_hinf_unce
[[file:figs/comp_filter_hinf_uncertainty.png]]
*** Super sensor uncertainty
We can now compute the uncertainty of the super sensor. The result is shown in Fig. [[fig:super_sensor_uncertainty_bode_plot]].
We can now compute the uncertainty of the super sensor. The result is shown in Figure [[fig:super_sensor_uncertainty_bode_plot]].
#+begin_src matlab
Gss = G1*H1 + G2*H2;
@ -2246,8 +2252,8 @@ The noise characteristics of both individual sensor are defined below.
N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
#+end_src
The PSD of both sensor and of the super sensor is shown in Fig. [[fig:psd_sensors_hinf_synthesis]].
The CPS of both sensor and of the super sensor is shown in Fig. [[fig:cps_sensors_hinf_synthesis]].
The PSD of both sensor and of the super sensor is shown in Figure [[fig:psd_sensors_hinf_synthesis]].
The CPS of both sensor and of the super sensor is shown in Figure [[fig:cps_sensors_hinf_synthesis]].
#+begin_src matlab :exports none
PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2;
@ -2475,7 +2481,7 @@ We define the noise characteristics of the two sensors by choosing $N_1$ and $N_
N2 = ((sqrt(Ginf)*s/omegac + sqrt(G0))/(s/omegac + 1))^2/(1 + s/2/pi/4000)^2;
#+end_src
Both dynamical uncertainty and noise characteristics of the individual sensors are shown in Fig. [[fig:mixed_synthesis_noise_uncertainty_sensors]].
Both dynamical uncertainty and noise characteristics of the individual sensors are shown in Figure [[fig:mixed_synthesis_noise_uncertainty_sensors]].
#+begin_src matlab :exports none
figure;
@ -2512,7 +2518,7 @@ Both dynamical uncertainty and noise characteristics of the individual sensors a
*** Weighting Functions on the uncertainty of the super sensor
We design weights for the $\mathcal{H}_\infty$ part of the synthesis in order to limit the dynamical uncertainty of the super sensor.
The maximum wanted multiplicative uncertainty is shown in Fig. [[fig:mixed_syn_hinf_weight]]. The idea here is that we don't really need low uncertainty at low frequency but only near the crossover frequency that is suppose to be around 300Hz here.
The maximum wanted multiplicative uncertainty is shown in Figure [[fig:mixed_syn_hinf_weight]]. The idea here is that we don't really need low uncertainty at low frequency but only near the crossover frequency that is suppose to be around 300Hz here.
#+begin_src matlab
n = 4; w0 = 2*pi*900; G0 = 9; G1 = 1; Gc = 1.1;
@ -2542,7 +2548,7 @@ The maximum wanted multiplicative uncertainty is shown in Fig. [[fig:mixed_syn_h
#+CAPTION: Wanted maximum module uncertainty of the super sensor ([[./figs/mixed_syn_hinf_weight.png][png]], [[./figs/mixed_syn_hinf_weight.pdf][pdf]])
[[file:figs/mixed_syn_hinf_weight.png]]
The equivalent Magnitude and Phase uncertainties are shown in Fig. [[fig:mixed_syn_objective_hinf]].
The equivalent Magnitude and Phase uncertainties are shown in Figure [[fig:mixed_syn_objective_hinf]].
#+begin_src matlab :exports none
G1 = 1 + w1*ultidyn('Delta',[1 1]);
@ -2622,7 +2628,7 @@ The equivalent Magnitude and Phase uncertainties are shown in Fig. [[fig:mixed_s
[[file:figs/mixed_syn_objective_hinf.png]]
*** Mixed Synthesis Architecture
The synthesis architecture that is used here is shown in Fig. [[fig:mixed_h2_hinf_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
@ -2672,7 +2678,7 @@ The mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis is performed below.
H1 = 1 - H2;
#+end_src
The obtained complementary filters are shown in Fig. [[fig:comp_filters_mixed_synthesis]].
The obtained complementary filters are shown in Figure [[fig:comp_filters_mixed_synthesis]].
#+begin_src matlab :exports none
figure;
@ -2722,7 +2728,7 @@ The obtained complementary filters are shown in Fig. [[fig:comp_filters_mixed_sy
[[file:figs/comp_filters_mixed_synthesis.png]]
*** Obtained Super Sensor's noise
The PSD and CPS of the super sensor's noise are shown in Fig. [[fig:psd_super_sensor_mixed_syn]] and Fig. [[fig:cps_super_sensor_mixed_syn]] respectively.
The PSD and CPS of the super sensor's noise are shown in Figure [[fig:psd_super_sensor_mixed_syn]] and Figure [[fig:cps_super_sensor_mixed_syn]] respectively.
#+begin_src matlab :exports none
PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2;
@ -2783,7 +2789,7 @@ The PSD and CPS of the super sensor's noise are shown in Fig. [[fig:psd_super_se
[[file:figs/cps_super_sensor_mixed_syn.png]]
*** Obtained Super Sensor's Uncertainty
The uncertainty on the super sensor's dynamics is shown in Fig.
The uncertainty on the super sensor's dynamics is shown in Figure
#+begin_src matlab :exports none
G1 = 1 + w1*ultidyn('Delta',[1 1]);
@ -3025,7 +3031,7 @@ Finally, we can compute the obtained complementary filters.
#+end_src
** Result
The obtained complementary filters are compared with the required upper bounds on Fig. [[fig:LMI_obtained_comp_filters]].
The obtained complementary filters are compared with the required upper bounds on Figure [[fig:LMI_obtained_comp_filters]].
#+begin_src matlab :exports none
figure;
@ -3085,7 +3091,7 @@ The Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis is performed below.
H1m = 1 - H2m;
#+end_src
The obtained filters are compare with the one obtained using the CVX toolbox in Fig. [[]].
The obtained filters are compare with the one obtained using the CVX toolbox in Figure [[]].
#+begin_src matlab :exports none
figure;
@ -3141,7 +3147,7 @@ The obtained filters are compare with the one obtained using the CVX toolbox in
[[file:figs/compare_cvx_h2hinf_comp_filters.png]]
** H-Infinity Objective
In terms of the $\mathcal{H}_\infty$ objective, both synthesis method are satisfying the requirements as shown in Fig. [[fig:comp_cvx_h2i_hinf_norm]].
In terms of the $\mathcal{H}_\infty$ objective, both synthesis method are satisfying the requirements as shown in Figure [[fig:comp_cvx_h2i_hinf_norm]].
#+begin_src matlab :exports none
figure;
@ -3181,7 +3187,7 @@ In terms of the $\mathcal{H}_\infty$ objective, both synthesis method are satisf
[[file:figs/comp_cvx_h2i_hinf_norm.png]]
** Obtained Super Sensor's noise
The PSD and CPS of the super sensor's noise obtained with the CVX toolbox and =h2hinfsyn= command are compared in Fig. [[fig:psd_compare_cvx_h2i]] and [[fig:cps_compare_cvx_h2i]].
The PSD and CPS of the super sensor's noise obtained with the CVX toolbox and =h2hinfsyn= command are compared in Figure [[fig:psd_compare_cvx_h2i]] and [[fig:cps_compare_cvx_h2i]].
#+begin_src matlab :exports none
PSD_S1 = abs(squeeze(freqresp(N1, freqs, 'Hz'))).^2;
@ -3246,7 +3252,7 @@ The PSD and CPS of the super sensor's noise obtained with the CVX toolbox and =h
[[file:figs/cps_compare_cvx_h2i.png]]
** Obtained Super Sensor's Uncertainty
The uncertainty on the super sensor's dynamics is shown in Fig. [[]].
The uncertainty on the super sensor's dynamics is shown in Figure [[]].
#+begin_src matlab :exports none
G1 = 1 + w1*ultidyn('Delta',[1 1]);