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@ -75,7 +75,7 @@ Let's consider the sensor fusion architecture shown on figure [[fig:fusion_two_n
$\tilde{n}_1$ and $\tilde{n}_2$ are normalized white noise: $\tilde{n}_1$ and $\tilde{n}_2$ are normalized white noise:
#+name: eq:normalized_noise #+name: eq:normalized_noise
\begin{equation} \begin{equation}
\Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_1}(\omega) = 1 \Phi_{\tilde{n}_1}(\omega) = \Phi_{\tilde{n}_2}(\omega) = 1
\end{equation} \end{equation}
#+name: fig:fusion_two_noisy_sensors_weights #+name: fig:fusion_two_noisy_sensors_weights
@ -1027,7 +1027,7 @@ If $H_1(s)$ and $H_2(s)$ are designed such that
The maximum phase shift due to dynamic uncertainty at frequency $\omega$ will be $\Delta\phi_\text{max}(\omega)$. The maximum phase shift due to dynamic uncertainty at frequency $\omega$ will be $\Delta\phi_\text{max}(\omega)$.
** Requirements as an $\mathcal{H}_\infty$ norm ** Requirements as an $\mathcal{H}_\infty$ norm
We know try to express this requirement in terms of an $\mathcal{H}_\infty$ norm. We now try to express this requirement in terms of an $\mathcal{H}_\infty$ norm.
Let's define one weight $w_\phi(s)$ that represents the maximum wanted phase uncertainty: Let's define one weight $w_\phi(s)$ that represents the maximum wanted phase uncertainty:
\[ |w_{\phi}(j\omega)|^{-1} \approx \sin(\Delta\phi_{\text{max}}(\omega)), \quad \forall\omega \] \[ |w_{\phi}(j\omega)|^{-1} \approx \sin(\Delta\phi_{\text{max}}(\omega)), \quad \forall\omega \]
@ -1738,7 +1738,7 @@ We define the generalized plant that will be used for the mixed synthesis.
#+end_src #+end_src
** Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis ** Mixed $\mathcal{H}_2$ / $\mathcal{H}_\infty$ Synthesis
The mixed $\mathcal{H}_2$/$\mathcal{H}_\infty$ synthesis is performed below. The mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis is performed below.
#+begin_src matlab #+begin_src matlab
Nmeas = 1; Ncon = 1; Nz2 = 2; Nmeas = 1; Ncon = 1; Nz2 = 2;
@ -1793,7 +1793,7 @@ The obtained complementary filters are shown in Fig. [[fig:comp_filters_mixed_sy
#+end_src #+end_src
#+NAME: fig:comp_filters_mixed_synthesis #+NAME: fig:comp_filters_mixed_synthesis
#+CAPTION: Obtained complementary filters after mixed $\mathcal{H}_2$/$\mathcal{H}_\infty$ synthesis ([[./figs/comp_filters_mixed_synthesis.png][png]], [[./figs/comp_filters_mixed_synthesis.pdf][pdf]]) #+CAPTION: Obtained complementary filters after mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis ([[./figs/comp_filters_mixed_synthesis.png][png]], [[./figs/comp_filters_mixed_synthesis.pdf][pdf]])
[[file:figs/comp_filters_mixed_synthesis.png]] [[file:figs/comp_filters_mixed_synthesis.png]]
** Obtained Super Sensor's noise ** Obtained Super Sensor's noise
@ -1824,7 +1824,7 @@ The PSD and CPS of the super sensor's noise are shown in Fig. [[fig:psd_super_se
#+end_src #+end_src
#+NAME: fig:psd_super_sensor_mixed_syn #+NAME: fig:psd_super_sensor_mixed_syn
#+CAPTION: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2$/$\mathcal{H}_\infty$ synthesis ([[./figs/psd_super_sensor_mixed_syn.png][png]], [[./figs/psd_super_sensor_mixed_syn.pdf][pdf]]) #+CAPTION: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis ([[./figs/psd_super_sensor_mixed_syn.png][png]], [[./figs/psd_super_sensor_mixed_syn.pdf][pdf]])
[[file:figs/psd_super_sensor_mixed_syn.png]] [[file:figs/psd_super_sensor_mixed_syn.png]]
@ -1854,7 +1854,7 @@ The PSD and CPS of the super sensor's noise are shown in Fig. [[fig:psd_super_se
#+end_src #+end_src
#+NAME: fig:cps_super_sensor_mixed_syn #+NAME: fig:cps_super_sensor_mixed_syn
#+CAPTION: Cumulative Power Spectrum of the Super Sensor obtained with the mixed $\mathcal{H}_2$/$\mathcal{H}_\infty$ synthesis ([[./figs/cps_super_sensor_mixed_syn.png][png]], [[./figs/cps_super_sensor_mixed_syn.pdf][pdf]]) #+CAPTION: Cumulative Power Spectrum of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis ([[./figs/cps_super_sensor_mixed_syn.png][png]], [[./figs/cps_super_sensor_mixed_syn.pdf][pdf]])
[[file:figs/cps_super_sensor_mixed_syn.png]] [[file:figs/cps_super_sensor_mixed_syn.png]]
** Obtained Super Sensor's Uncertainty ** Obtained Super Sensor's Uncertainty
@ -2285,7 +2285,7 @@ The PSD and CPS of the super sensor's noise obtained with the CVX toolbox and =h
#+end_src #+end_src
#+NAME: fig:psd_compare_cvx_h2i #+NAME: fig:psd_compare_cvx_h2i
#+CAPTION: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2$/$\mathcal{H}_\infty$ synthesis ([[./figs/psd_compare_cvx_h2i.png][png]], [[./figs/psd_compare_cvx_h2i.pdf][pdf]]) #+CAPTION: Power Spectral Density of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis ([[./figs/psd_compare_cvx_h2i.png][png]], [[./figs/psd_compare_cvx_h2i.pdf][pdf]])
[[file:figs/psd_compare_cvx_h2i.png]] [[file:figs/psd_compare_cvx_h2i.png]]
@ -2317,7 +2317,7 @@ The PSD and CPS of the super sensor's noise obtained with the CVX toolbox and =h
#+end_src #+end_src
#+NAME: fig:cps_compare_cvx_h2i #+NAME: fig:cps_compare_cvx_h2i
#+CAPTION: Cumulative Power Spectrum of the Super Sensor obtained with the mixed $\mathcal{H}_2$/$\mathcal{H}_\infty$ synthesis ([[./figs/cps_compare_cvx_h2i.png][png]], [[./figs/cps_compare_cvx_h2i.pdf][pdf]]) #+CAPTION: Cumulative Power Spectrum of the Super Sensor obtained with the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis ([[./figs/cps_compare_cvx_h2i.png][png]], [[./figs/cps_compare_cvx_h2i.pdf][pdf]])
[[file:figs/cps_compare_cvx_h2i.png]] [[file:figs/cps_compare_cvx_h2i.png]]
** Obtained Super Sensor's Uncertainty ** Obtained Super Sensor's Uncertainty
@ -3509,7 +3509,7 @@ Since the input signal $x$ and the instrumental noise $n$ are incoherent:
|\Phi_{\hat{x}}(\omega)| = |\Phi_n(\omega)| + |\Phi_x(\omega)| |\Phi_{\hat{x}}(\omega)| = |\Phi_n(\omega)| + |\Phi_x(\omega)|
\end{equation} \end{equation}
From equations [[eq:coh_bis]] and [[eq:incoherent_noise]], we finally obtain From equations eqref:eq:coh_bis and eqref:eq:incoherent_noise, we finally obtain
#+begin_important #+begin_important
#+NAME: eq:noise_psd #+NAME: eq:noise_psd
\begin{equation} \begin{equation}
@ -4238,6 +4238,254 @@ The generated complementary filters using $\mathcal{H}_\infty$ and the analytica
- the analytical formula provides a very simple way to generate the complementary filters (and thus the controller), they could even be used to tune the controller online using the parameters $\alpha$ and $\omega_0$. However, these formula have the property that $|H_H|$ and $|H_L|$ are symmetrical with the frequency $\omega_0$ which may not be desirable. - the analytical formula provides a very simple way to generate the complementary filters (and thus the controller), they could even be used to tune the controller online using the parameters $\alpha$ and $\omega_0$. However, these formula have the property that $|H_H|$ and $|H_L|$ are symmetrical with the frequency $\omega_0$ which may not be desirable.
- while the $\mathcal{H}_\infty$ synthesis of the complementary filters is not as straightforward as using the analytical formula, it provides a more optimized procedure to obtain the complementary filters - while the $\mathcal{H}_\infty$ synthesis of the complementary filters is not as straightforward as using the analytical formula, it provides a more optimized procedure to obtain the complementary filters
* Bibliography :ignore: * Real World Example of optimal sensor fusion
** Introduction :ignore:
cite:moore19_capac_instr_sensor_fusion_high_bandw_nanop
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src
** Sensor Noise :noexport:
#+begin_src matlab
A1 = 19.13; % [uV2/Hz]
A2 = 0.1632; % [uV2/Hz]
A3 = 6.847; % [uV2/Hz]
wnc = 3057; % [rad]
wx = 7929; % [rad/s]
Fx = 1/(1 - s/wx)/(1 - s/wx);
[A B C D] = butter(2, 0.5, 'low');
Fx = ss(A, B, C, D);
Sq = A3*wnc/s + A3;
Sx = A1*Fx + A2;
#+end_src
#+begin_src matlab :exports none
freqs = logspace(1, 5, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(Sq, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(Sx, freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
#+end_src
** Matlab Code
Take an Accelerometer and a Geophone both measuring the absolute motion of a structure.
Parameters of the inertial sensors.
#+begin_src matlab
m_acc = 0.01;
k_acc = 1e6;
c_acc = 20;
m_geo = 1;
k_geo = 1e3;
c_geo = 10;
#+end_src
Transfer function from motion to measurement
For the accelerometer.
The measurement is the relative motion structure/inertial mass:
\[ \frac{d}{\ddot{w}} = \frac{-m}{ms^2 + cs + k} \]
For the geophone.
The measurement is the relative velocity structure/inertial mass:
\[ \frac{\dot{d}}{\dot{w}} = \frac{-ms^2}{ms^2 + cs + k} \]
#+begin_src matlab
G_acc = -m_acc/(m_acc*s^2 + c_acc*s + k_acc); % [m/(m/s^2)]
G_geo = -m_geo*s^2/(m_geo*s^2 + c_geo*s + k_geo); % [m/s/m/s]
#+end_src
Suppose the measure of the relative motion for the accelerometer (capacitive sensor for instance) has a white noise characteristic:
Suppose the measure of the relative velocity (current flowing through the coil) has a white noise characteristic:
Define the noise characteristics
#+begin_src matlab
n = 1; w0 = 2*pi*5e3; G0 = 5e-12; G1 = 1e-15; Gc = G0/2;
L_acc = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
n = 1; w0 = 2*pi*5e3; G0 = 1e-6; G1 = 1e-8; Gc = G0/2;
L_geo = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
#+end_src
Transfer function of the conversion to obtain the velocity:
#+begin_src matlab
C_acc = (-k_acc/m_acc/(2*pi + s));
C_geo = tf(-1);
#+end_src
Let's plot the noise of both sensors:
#+begin_src matlab :exports none
freqs = logspace(-1, 4, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(L_acc*C_acc, freqs, 'Hz'))), 'DisplayName', 'Acc');
plot(freqs, abs(squeeze(freqresp(L_geo*C_geo, freqs, 'Hz'))), 'DisplayName', 'Geo');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Noise ASD [$m/s/\sqrt{Hz}$]');
legend('location', 'northeast')
#+end_src
Dynamics of both sensors
#+begin_src matlab :exports none
freqs = logspace(-1, 4, 1000);
figure;
hold on;
plot(freqs, abs(squeeze(freqresp(s*G_acc*C_acc, freqs, 'Hz'))), 'DisplayName', 'Acc');
plot(freqs, abs(squeeze(freqresp(G_geo*C_geo, freqs, 'Hz'))), 'DisplayName', 'Geo');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'northeast')
#+end_src
** Time domain signals
#+begin_src matlab
Fs = 1e4; % Sampling Frequency [Hz]
Ts = 1/Fs; % Sampling Time [s]
t = 0:Ts:10; % Time Vector [s]
#+end_src
#+begin_src matlab
n_acc = lsim(L_acc*C_acc, sqrt(Fs/2)*randn(length(t), 1), t); % [m/s]
n_geo = lsim(L_geo*C_geo, sqrt(Fs/2)*randn(length(t), 1), t); % [m/s]
#+end_src
#+begin_src matlab
figure;
hold on;
plot(t, n_geo)
plot(t, n_acc)
hold off;
#+end_src
** H2 Synthesis
#+begin_src matlab
N1 = L_acc*C_acc;
N2 = L_geo*C_geo;
#+end_src
#+begin_src matlab
bodeFig({N1, N2}, logspace(-1, 5, 1000))
#+end_src
#+begin_src matlab
P = [0 N2 1;
N1 -N2 0];
#+end_src
And we do the $\mathcal{H}_2$ synthesis using the =h2syn= command.
#+begin_src matlab
[H1, ~, gamma] = h2syn(P, 1, 1);
#+end_src
Finally, we define $H_2(s) = 1 - H_1(s)$.
#+begin_src matlab
H2 = 1 - H1;
#+end_src
#+begin_src matlab
bodeFig({H1, H2}, struct('phase', true))
#+end_src
#+begin_src matlab
n_acc_filt = lsim(H1, n_acc, t);
n_geo_filt = lsim(H2, n_geo, t);
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
data2orgtable([rms(n_acc), rms(n_geo), rms(n_acc_filt + n_geo_filt)]', {'Accelerometer', 'Geophone', 'Super Sensor'}, {'RMS'}, ' %.1e ');
#+end_src
#+RESULTS:
| | RMS |
|---------------+---------|
| Accelerometer | 9.7e-05 |
| Geophone | 5.9e-05 |
| Super Sensor | 1.5e-05 |
#+begin_src matlab
figure;
hold on;
plot(t, n_geo)
plot(t, n_acc)
plot(t, n_acc_filt + n_geo_filt)
hold off;
#+end_src
** Signal and Noise
Velocity Signal:
#+begin_src matlab
v = lsim(1/(1 + s/2/pi/2), 1e-4*sqrt(Fs/2)*randn(length(t), 1), t);
v = 1e-4 * sin(2*pi*100*t);
#+end_src
#+begin_src matlab
v_acc = lsim(s*G_acc*C_acc, v, t) + n_acc;
v_geo = lsim(G_geo*C_geo, v, t) + n_geo;
#+end_src
#+begin_src matlab
v_ss = lsim(H1, v_acc, t) + lsim(H2, v_geo, t);
#+end_src
#+begin_src matlab
figure;
hold on;
plot(t, v_geo)
plot(t, v_acc)
plot(t, v_ss)
plot(t, v, 'k--')
hold off;
xlim([1, 1+0.1])
#+end_src
** PSD and CPS
#+begin_src matlab
nx = length(n_acc);
na = 16;
win = hanning(floor(nx/na));
[p_acc, f] = pwelch(n_acc, win, 0, [], Fs);
[p_geo, ~] = pwelch(n_geo, win, 0, [], Fs);
[p_ss, ~] = pwelch(n_acc_filt + n_geo_filt, win, 0, [], Fs);
#+end_src
#+begin_src matlab :exports none
figure;
hold on;
plot(f, p_acc, 'DisplayName', 'Accelerometer');
plot(f, p_geo, 'DisplayName', 'Geophone');
plot(f, p_ss, 'DisplayName', 'Super Sensor');
hold off;
set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log');
xlabel('Frequency [Hz]');
ylabel('Power Spectral Density $\left[\frac{(m/s)^2}{Hz}\right]$');
legend('location', 'southwest');
#+end_src
** Transfer function of the super sensor
#+begin_src matlab
bodeFig({s*C_acc*G_acc, C_geo*G_geo, s*C_acc*G_acc*H1+C_geo*G_geo*H2}, struct('phase', true))
#+end_src
* Bibliography :ignore:
bibliographystyle:unsrt bibliographystyle:unsrt
bibliography:ref.bib bibliography:ref.bib

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@ -182,3 +182,18 @@
doi = {10.1063/1.1149013}, doi = {10.1063/1.1149013},
url = {https://doi.org/10.1063/1.1149013}, url = {https://doi.org/10.1063/1.1149013},
} }
@article{moore19_capac_instr_sensor_fusion_high_bandw_nanop,
author = {Steven Ian Moore and Andrew J. Fleming and Yuen Kuan Yong},
title = {Capacitive Instrumentation and Sensor Fusion for
High-Bandwidth Nanopositioning},
journal = {IEEE Sensors Letters},
volume = 3,
number = 8,
pages = {1-3},
year = 2019,
doi = {10.1109/lsens.2019.2933065},
url = {https://doi.org/10.1109/lsens.2019.2933065},
tags = {complementary filters},
}

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@ -143,22 +143,21 @@ The goal is to design $H_1(s)$ and $H_2(s)$ such that the effect of the noise so
And the goal is the minimize the Root Mean Square (RMS) value of $n$: And the goal is the minimize the Root Mean Square (RMS) value of $n$:
#+name: eq:rms_value_estimation #+name: eq:rms_value_estimation
\begin{equation} \begin{equation}
\sigma_{n} = \sqrt{\int_0^\infty \Phi_{\hat{n}}(\omega) d\omega} \sigma_{n} = \sqrt{\int_0^\infty \Phi_{\hat{n}}(\omega) d\omega} = \left\| \begin{matrix} \hat{G}_1^{-1} N_1 H_1 \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2
\end{equation} \end{equation}
Thus, the goal is to design $H_1(s)$ and $H_2(s)$ such that $H_1(s) + H_2(s) = 1$ and such that $\left\| \begin{matrix} \hat{G}_1^{-1} N_1 H_1 \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2$ is minimized.
The goal is to minimize the $\mathcal{H}_2$ norm between $w$ and $[z_1\ z_2]$:
\begin{equation}
\left\| \begin{matrix} w/z_1 \\ w/z_2 \end{matrix} \right\|_2 = \left\| \begin{matrix} \hat{G}_1^{-1} N_1 (1 - H_2) \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2
\end{equation}
By defining:
\begin{equation}
H_1 = 1 - H_2
\end{equation}
\begin{equation} \begin{equation}
\left\| \begin{matrix} w/z_1 \\ w/z_2 \end{matrix} \right\|_2 = \left\| \begin{matrix} \hat{G}_1^{-1} N_1 H_1 \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2 = \sqrt{\int_0^\infty \left| H_1 \hat{G}_1^{-1} N_1 \right|^2 + \left| H_2 \hat{G}_2^{-1} N_2 \right|^2 d\omega} = \sigma_n \begin{pmatrix}
z_1 \\ z_2 \\ v
\end{pmatrix} = \begin{bmatrix}
\hat{G}_1^{-1} N_1 & -\hat{G}_1^{-1} N_1 \\
0 & \hat{G}_2^{-1} N_2 \\
1 & 0
\end{bmatrix} \begin{pmatrix}
w \\ u
\end{pmatrix}
\end{equation} \end{equation}
The $\mathcal{H}_2$ synthesis of the complementary filters thus minimized the RMS value of the super sensor noise. The $\mathcal{H}_2$ synthesis of the complementary filters thus minimized the RMS value of the super sensor noise.
@ -177,6 +176,21 @@ The $\mathcal{H}_2$ synthesis of the complementary filters thus minimized the RM
** Representation of Sensor Dynamical Uncertainty ** Representation of Sensor Dynamical Uncertainty
** Sensor Fusion Architecture
\begin{equation}
\hat{x} = \left( H_1 \hat{G}_1^{-1} (1 + w_1 \Delta_1) G_1 + H_2 \hat{G}_2^{-1} (1 + w_2 \Delta_2) G_2 \right) x
\end{equation}
with $\Delta_i$ is any transfer function satisfying $\| \Delta_i \|_\infty < 1$.
Suppose the model inversion is equal to the nominal model:
\begin{equation}
\hat{G}_i = G_i
\end{equation}
\begin{equation}
\hat{x} = \left( 1 + H_1 w_1 \Delta_1 + H_2 w_2 \Delta_2 \right) x
\end{equation}
#+name: fig:sensor_fusion_arch_uncertainty #+name: fig:sensor_fusion_arch_uncertainty
#+caption: Figure caption #+caption: Figure caption
#+attr_latex: :scale 1 #+attr_latex: :scale 1
@ -184,6 +198,8 @@ The $\mathcal{H}_2$ synthesis of the complementary filters thus minimized the RM
** Super Sensor Dynamical Uncertainty ** Super Sensor 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 $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$.
#+name: fig:uncertainty_set_super_sensor #+name: fig:uncertainty_set_super_sensor
#+caption: Figure caption #+caption: Figure caption
#+attr_latex: :scale 1 #+attr_latex: :scale 1
@ -191,6 +207,8 @@ The $\mathcal{H}_2$ synthesis of the complementary filters thus minimized the RM
** $\mathcal{H_\infty}$ Synthesis of Complementary Filters ** $\mathcal{H_\infty}$ Synthesis of Complementary Filters
In order to minimize the super sensor dynamical uncertainty
#+name: fig:h_infinity_robust_fusion #+name: fig:h_infinity_robust_fusion
#+caption: Figure caption #+caption: Figure caption
#+attr_latex: :scale 1 #+attr_latex: :scale 1

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@ -1,4 +1,4 @@
% Created 2020-08-17 lun. 18:06 % Created 2020-09-22 mar. 09:51
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[conference]{IEEEtran} \documentclass[conference]{IEEEtran}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
@ -22,7 +22,6 @@
\usepackage{siunitx} \usepackage{siunitx}
\IEEEoverridecommandlockouts \IEEEoverridecommandlockouts
\usepackage{cite} \usepackage{cite}
\usepackage{showframe}
\usepackage{amsmath,amssymb,amsfonts} \usepackage{amsmath,amssymb,amsfonts}
\usepackage{algorithmic} \usepackage{algorithmic}
\usepackage{graphicx} \usepackage{graphicx}
@ -34,8 +33,9 @@
\usepackage{import, hyperref} \usepackage{import, hyperref}
\renewcommand{\citedash}{--} \renewcommand{\citedash}{--}
\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
\usepackage{showframe}
\author{\IEEEauthorblockN{Dehaeze Thomas} \IEEEauthorblockA{\textit{European Synchrotron Radiation Facility} \\ Grenoble, France\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ thomas.dehaeze@esrf.fr }\and \IEEEauthorblockN{Collette Christophe} \IEEEauthorblockA{\textit{BEAMS Department}\\ \textit{Free University of Brussels}, Belgium\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ ccollett@ulb.ac.be }} \author{\IEEEauthorblockN{Dehaeze Thomas} \IEEEauthorblockA{\textit{European Synchrotron Radiation Facility} \\ Grenoble, France\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ thomas.dehaeze@esrf.fr }\and \IEEEauthorblockN{Collette Christophe} \IEEEauthorblockA{\textit{BEAMS Department}\\ \textit{Free University of Brussels}, Belgium\\ \textit{Precision Mechatronics Laboratory} \\ \textit{University of Liege}, Belgium \\ ccollett@ulb.ac.be }}
\date{2020-08-17} \date{2020-09-22}
\title{Robust and Optimal Sensor Fusion} \title{Robust and Optimal Sensor Fusion}
\begin{document} \begin{document}
@ -50,15 +50,18 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{IEEEkeywords} \end{IEEEkeywords}
\section{Introduction} \section{Introduction}
\label{sec:orgc7e68f1} \label{sec:org2c6d9ef}
\label{sec:introduction} \label{sec:introduction}
\section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis} \section{Optimal Super Sensor Noise: \(\mathcal{H}_2\) Synthesis}
\label{sec:org9dea05a} \label{sec:org5aa0717}
\label{sec:optimal_fusion} \label{sec:optimal_fusion}
\subsection{Sensor Model}
\label{sec:org8f1053d}
\subsection{Sensor Fusion Architecture} \subsection{Sensor Fusion Architecture}
\label{sec:org02189c8} \label{sec:orgc40deb4}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -66,11 +69,70 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\caption{\label{fig:sensor_fusion_noise_arch}Figure caption} \caption{\label{fig:sensor_fusion_noise_arch}Figure caption}
\end{figure} \end{figure}
Let note \(\Phi\) the PSD.
\(\tilde{n}_1\) and \(\tilde{n}_2\) are white noise with unitary power spectral density:
\begin{equation}
\Phi_{\tilde{n}_i}(\omega) = 1
\end{equation}
\begin{equation}
\hat{x} = \left( H_1 \hat{G}_1^{-1} G_1 + H_2 \hat{G}_2^{-1} G_2 \right) x + \left( H_1 \hat{G}_1^{-1} N_1 \right) \tilde{n}_1 + \left( H_2 \hat{G}_2^{-1} N_2 \right) \tilde{n}_2
\end{equation}
Suppose the sensor dynamical model \(\hat{G}_i\) is perfect:
\begin{equation}
\hat{G}_i = G_i
\end{equation}
Complementary Filters
\begin{equation}
H_1(s) + H_2(s) = 1
\end{equation}
\begin{equation}
\hat{x} = x + \left( H_1 \hat{G}_1^{-1} N_1 \right) \tilde{n}_1 + \left( H_2 \hat{G}_2^{-1} N_2 \right) \tilde{n}_2
\end{equation}
Perfect dynamics + filter noise
\subsection{Super Sensor Noise} \subsection{Super Sensor Noise}
\label{sec:org75bf123} \label{sec:orgf4b6ca9}
Let's note \(n\) the super sensor noise.
Its PSD is determined by:
\begin{equation}
\Phi_n(\omega) = \left| H_1 \hat{G}_1^{-1} N_1 \right|^2 + \left| H_2 \hat{G}_2^{-1} N_2 \right|^2
\end{equation}
\subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters} \subsection{\(\mathcal{H}_2\) Synthesis of Complementary Filters}
\label{sec:org2372e4d} \label{sec:org5773772}
The goal is to design \(H_1(s)\) and \(H_2(s)\) such that the effect of the noise sources \(\tilde{n}_1\) and \(\tilde{n}_2\) has the smallest possible effect on the noise \(n\) of the estimation \(\hat{x}\).
And the goal is the minimize the Root Mean Square (RMS) value of \(n\):
\begin{equation}
\label{eq:rms_value_estimation}
\sigma_{n} = \sqrt{\int_0^\infty \Phi_{\hat{n}}(\omega) d\omega} = \left\| \begin{matrix} \hat{G}_1^{-1} N_1 H_1 \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2
\end{equation}
Thus, the goal is to design \(H_1(s)\) and \(H_2(s)\) such that \(H_1(s) + H_2(s) = 1\) and such that \(\left\| \begin{matrix} \hat{G}_1^{-1} N_1 H_1 \\ \hat{G}_2^{-1} N_2 H_2 \end{matrix} \right\|_2\) is minimized.
\begin{equation}
\begin{pmatrix}
z_1 \\ z_2 \\ v
\end{pmatrix} = \begin{bmatrix}
\hat{G}_1^{-1} N_1 & -\hat{G}_1^{-1} N_1 \\
0 & \hat{G}_2^{-1} N_2 \\
1 & 0
\end{bmatrix} \begin{pmatrix}
w \\ u
\end{pmatrix}
\end{equation}
The \(\mathcal{H}_2\) synthesis of the complementary filters thus minimized the RMS value of the super sensor noise.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -79,17 +141,33 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{figure} \end{figure}
\subsection{Example} \subsection{Example}
\label{sec:orgf58d3c3} \label{sec:orged06a27}
\subsection{Robustness Problem} \subsection{Robustness Problem}
\label{sec:org25143ef} \label{sec:org62b375f}
\section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis} \section{Robust Sensor Fusion: \(\mathcal{H}_\infty\) Synthesis}
\label{sec:orge0ab922} \label{sec:orgef03e7c}
\label{sec:robust_fusion} \label{sec:robust_fusion}
\subsection{Representation of Sensor Dynamical Uncertainty} \subsection{Representation of Sensor Dynamical Uncertainty}
\label{sec:orgde1ca0e} \label{sec:org9c9762b}
\subsection{Sensor Fusion Architecture}
\label{sec:org9572e70}
\begin{equation}
\hat{x} = \left( H_1 \hat{G}_1^{-1} (1 + w_1 \Delta_1) G_1 + H_2 \hat{G}_2^{-1} (1 + w_2 \Delta_2) G_2 \right) x
\end{equation}
with \(\Delta_i\) is any transfer function satisfying \(\| \Delta_i \|_\infty < 1\).
Suppose the model inversion is equal to the nominal model:
\begin{equation}
\hat{G}_i = G_i
\end{equation}
\begin{equation}
\hat{x} = \left( 1 + H_1 w_1 \Delta_1 + H_2 w_2 \Delta_2 \right) x
\end{equation}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -98,7 +176,9 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{figure} \end{figure}
\subsection{Super Sensor Dynamical Uncertainty} \subsection{Super Sensor Dynamical Uncertainty}
\label{sec:org7f51f8d} \label{sec:orgb9ee83e}
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 \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\).
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -107,7 +187,9 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{figure} \end{figure}
\subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters} \subsection{\(\mathcal{H_\infty}\) Synthesis of Complementary Filters}
\label{sec:org04f04f3} \label{sec:orgf4e3c8e}
In order to minimize the super sensor dynamical uncertainty
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -116,14 +198,14 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{figure} \end{figure}
\subsection{Example} \subsection{Example}
\label{sec:orgbeb8891} \label{sec:org4f663bc}
\section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} \section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:orgf739e93} \label{sec:org150b612}
\label{sec:optimal_robust_fusion} \label{sec:optimal_robust_fusion}
\subsection{Sensor Fusion Architecture} \subsection{Sensor Fusion Architecture}
\label{sec:org8fde0b7} \label{sec:org9bc69b7}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -132,10 +214,10 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{figure} \end{figure}
\subsection{Synthesis Objective} \subsection{Synthesis Objective}
\label{sec:org91a1cb3} \label{sec:orgbc5ac30}
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:orgcd9ab73} \label{sec:org541ef02}
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -144,30 +226,30 @@ Complementary Filters, Sensor Fusion, H-Infinity Synthesis
\end{figure} \end{figure}
\subsection{Example} \subsection{Example}
\label{sec:orgc2f161d} \label{sec:org046c2e2}
\section{Experimental Validation} \section{Experimental Validation}
\label{sec:org9c8a2db} \label{sec:org1bb9cff}
\label{sec:experimental_validation} \label{sec:experimental_validation}
\subsection{Experimental Setup} \subsection{Experimental Setup}
\label{sec:orga1abc9b} \label{sec:org2c63393}
\subsection{Sensor Noise and Dynamical Uncertainty} \subsection{Sensor Noise and Dynamical Uncertainty}
\label{sec:org9d92c2b} \label{sec:orgb0c6496}
\subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis}
\label{sec:orgbda5f50} \label{sec:orgfb3986f}
\subsection{Super Sensor Noise and Dynamical Uncertainty} \subsection{Super Sensor Noise and Dynamical Uncertainty}
\label{sec:orgb1aa4c9} \label{sec:orgfd5c11e}
\section{Conclusion} \section{Conclusion}
\label{sec:org468c862} \label{sec:orgda418fa}
\label{sec:conclusion} \label{sec:conclusion}
\section{Acknowledgment} \section{Acknowledgment}
\label{sec:org07d8818} \label{sec:orgabdae67}
\bibliography{ref} \bibliography{ref}
\end{document} \end{document}

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@ -30,39 +30,44 @@ Configuration file is accessible [[file:config.org][here]].
#+begin_src latex :file sensor_fusion_noise_arch.pdf #+begin_src latex :file sensor_fusion_noise_arch.pdf
\begin{tikzpicture} \begin{tikzpicture}
\node[branch] (x) at (0, 0); \node[branch] (x) at (0, 0);
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\node[block, below right=1.0 and 0.5 of x](G2){$G_2(s)$}; \node[addb, below right=1.0 and 0.5 of x](add2){};
\node[addb, right=0.4 of G1](add1){}; \node[block, right=0.4 of add1](G1){$G_1$};
\node[addb, right=0.4 of G2](add2){}; \node[block, right=0.4 of add2](G2){$G_2$};
\node[block, right=1.1 of add1](H1){$H_1(s)$}; \node[block, right=0.7 of G1](Ginv1){$\hat{G}_1^{-1}$};
\node[block, right=1.1 of add2](H2){$H_2(s)$}; \node[block, right=0.7 of G2](Ginv2){$\hat{G}_2^{-1}$};
\node[block, above=0.5 of add1](N1){$N_1(s)$}; \node[block, right=0.6 of Ginv1](H1){$H_1$};
\node[block, above=0.5 of add2](N2){$N_2(s)$}; \node[block, right=0.6 of Ginv2](H2){$H_2$};
\node[addb, right=4.8 of x](add){}; \node[block, above=0.5 of add1](N1){$N_1$};
\node[block, above=0.5 of add2](N2){$N_2$};
\node[addb, right=5.2 of x](add){};
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center); \draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
\draw[->] (x.center) |- (G1.west); \draw[->] (x.center) |- (add1.west);
\draw[->] (x.center) |- (G2.west); \draw[->] (x.center) |- (add2.west);
\draw[->] (G1.east) -- (add1.west); \draw[->] (add1.east) -- (G1.west);
\draw[->] (G2.east) -- (add2.west); \draw[->] (add2.east) -- (G2.west);
\draw[->] (N1.south) -- (add1.north)node[above right]{$n_1$}; \draw[->] (N1.south) -- (add1.north)node[above left]{$n_1$};
\draw[->] (N2.south) -- (add2.north)node[above right]{$n_2$}; \draw[->] (N2.south) -- (add2.north)node[above left]{$n_2$};
\draw[<-] (N1.north) -- ++(0, 0.6)node[below right](n1){$\tilde{n}_1$}; \draw[<-] (N1.north)node[above left](n1){$\tilde{n}_1$} -- ++(0, 0.4);
\draw[<-] (N2.north) -- ++(0, 0.6)node[below right](n2){$\tilde{n}_2$}; \draw[<-] (N2.north)node[above left](n2){$\tilde{n}_2$} -- ++(0, 0.4);
\draw[->] (add1.east) -- (H1.west)node[above left]{$\hat{x}_1$}; \draw[->] (G1.east) -- (Ginv1.west)node[above left]{$v_1$};
\draw[->] (add2.east) -- (H2.west)node[above left]{$\hat{x}_2$}; \draw[->] (G2.east) -- (Ginv2.west)node[above left]{$v_2$};
\draw[->] (Ginv1.east) -- (H1.west)node[above left]{$\hat{x}_1$};
\draw[->] (Ginv2.east) -- (H2.west)node[above left]{$\hat{x}_2$};
\draw[->] (H1) -| (add.north); \draw[->] (H1) -| (add.north);
\draw[->] (H2) -| (add.south); \draw[->] (H2) -| (add.south);
\draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$}; \draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
\begin{scope}[on background layer] \begin{scope}[on background layer]
\node[fit={($(G2.south-|x)+(-0.2, -0.3)$) ($(n1.north east-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {}; \node[fit={($(G2.south-|x)+(-0.2, -0.3)$) ($(n1.north east-|add.east)+(0.2, 0.2)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
\node[below left] at (supersensor.north east) {Super Sensor}; \node[below left] at (supersensor.north east) {Super Sensor};
\node[fit={($(G1.south west)+(-0.3, -0.1)$) ($(n1.north east)+(0.1, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {}; \node[fit={($(G1.south east)+(0.15, -0.15)$) ($(n1.north west)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
\node[below right] at (sensor1.north west) {Sensor 1}; \node[below left] at (sensor1.north east) {Sensor 1};
\node[fit={($(G2.south west)+(-0.3, -0.1)$) ($(n2.north east)+(0.1, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
\node[below right] at (sensor2.north west) {Sensor 2}; \node[fit={($(G2.south east)+(0.15, -0.15)$) ($(n2.north west)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
\node[below left] at (sensor2.north east) {Sensor 2};
\end{scope} \end{scope}
\end{tikzpicture} \end{tikzpicture}
#+end_src #+end_src
@ -75,8 +80,8 @@ Configuration file is accessible [[file:config.org][here]].
* Architecture used for $\mathcal{H}_2$ synthesis of complementary filters * Architecture used for $\mathcal{H}_2$ synthesis of complementary filters
#+begin_src latex :file h_two_optimal_fusion.pdf #+begin_src latex :file h_two_optimal_fusion.pdf
\begin{tikzpicture} \begin{tikzpicture}
\node[block={4.0cm}{2.5cm}, fill=black!20!white, dashed] (P) {}; \node[block={3.5cm}{2.5cm}, fill=black!20!white, dashed] (P) {};
\node[above] at (P.north) {$P(s)$}; \node[above] at (P.north) {$P$};
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$); \coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
\coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$); \coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$);
@ -85,11 +90,11 @@ Configuration file is accessible [[file:config.org][here]].
\coordinate[] (output2) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$); \coordinate[] (output2) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$);
\coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( 0.7, 0)$); \coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( 0.7, 0)$);
\node[block, left=1.4 of output1] (N1){$N_1(s)$}; \node[block, left=0.9 of output1] (N1){$N_1$};
\node[block, left=1.4 of output2] (N2){$N_2(s)$}; \node[block, left=0.9 of output2] (N2){$N_2$};
\node[addb={+}{}{}{}{-}, left=of N1] (sub) {}; \node[addb={+}{}{}{}{-}, left=of N1] (sub) {};
\node[block, below=0.3 of P] (H2) {$H_2(s)$}; \node[block, below=0.3 of P] (H2) {$H_2$};
\draw[->] (inputw) node[above right]{$w$} -- (sub.west); \draw[->] (inputw) node[above right]{$w$} -- (sub.west);
\draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (N2.west); \draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (N2.west);
@ -110,27 +115,37 @@ Configuration file is accessible [[file:config.org][here]].
#+begin_src latex :file sensor_fusion_arch_uncertainty.pdf #+begin_src latex :file sensor_fusion_arch_uncertainty.pdf
\begin{tikzpicture} \begin{tikzpicture}
\node[branch] (x) at (0, 0); \node[branch] (x) at (0, 0);
\node[addb, above right=0.8 and 4 of x](add1){}; \node[branch, above right=0.8 and 0.2 of x](b1){};
\node[addb, below right=0.8 and 4 of x](add2){}; \node[branch, below right=0.8 and 0.2 of x](b2){};
\node[block, above left=0.2 and 0.1 of add1](delta1){$\Delta_1(s)$}; \node[block, above right=0.3 and 0.3 of b1](W1){$w_1$};
\node[block, above left=0.2 and 0.1 of add2](delta2){$\Delta_2(s)$}; \node[block, above right=0.3 and 0.3 of b2](W2){$w_2$};
\node[block, left=0.5 of delta1](W1){$w_1(s)$}; \node[block, right=0.3 of W1](delta1){$\Delta_1$};
\node[block, left=0.5 of delta2](W2){$w_2(s)$}; \node[block, right=0.3 of W2](delta2){$\Delta_2$};
\node[block, right=0.5 of add1](H1){$H_1(s)$}; \node[addb, right=0.3 of b1-|delta1](add1){};
\node[block, right=0.5 of add2](H2){$H_2(s)$}; \node[addb, right=0.3 of b2-|delta2](add2){};
\node[addb, right=6 of x](add){}; \node[block, right=0.3 of add1](G1){$\hat{G}_1$};
\node[block, right=0.3 of add2](G2){$\hat{G}_2$};
\node[block, right=0.7 of G1](Ginv1){$\hat{G}_1^{-1}$};
\node[block, right=0.7 of G2](Ginv2){$\hat{G}_2^{-1}$};
\node[block, right=0.4 of Ginv1](H1){$H_1$};
\node[block, right=0.4 of Ginv2](H2){$H_2$};
\node[addb, right=7 of x](add){};
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center); \draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
\draw[->] (x.center) |- (add1.west); \draw[->] (x.center) |- (add1.west);
\draw[->] (x.center) |- (add2.west); \draw[->] (x.center) |- (add2.west);
\draw[->] ($(add1-|W1.west)+(-0.5, 0)$)node[branch](S1){} |- (W1.west); \draw[->] (add1.east) -- (G1.west);
\draw[->] ($(add2-|W2.west)+(-0.5, 0)$)node[branch](S1){} |- (W2.west); \draw[->] (add2.east) -- (G2.west);
\draw[->] (b1) |- (W1.west);
\draw[->] (b2) |- (W2.west);
\draw[->] (W1.east) -- (delta1.west); \draw[->] (W1.east) -- (delta1.west);
\draw[->] (W2.east) -- (delta2.west); \draw[->] (W2.east) -- (delta2.west);
\draw[->] (delta1.east) -| (add1.north); \draw[->] (delta1.east) -| (add1.north);
\draw[->] (delta2.east) -| (add2.north); \draw[->] (delta2.east) -| (add2.north);
\draw[->] (add1.east) -- (H1.west); \draw[->] (G1.east) -- (Ginv1.west)node[above left]{$v_1$};
\draw[->] (add2.east) -- (H2.west); \draw[->] (G2.east) -- (Ginv2.west)node[above left]{$v_2$};
\draw[->] (Ginv1.east) -- (H1.west);
\draw[->] (Ginv2.east) -- (H2.west);
\draw[->] (H1.east) -| (add.north); \draw[->] (H1.east) -| (add.north);
\draw[->] (H2.east) -| (add.south); \draw[->] (H2.east) -| (add.south);
\draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$}; \draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
@ -139,10 +154,10 @@ Configuration file is accessible [[file:config.org][here]].
\node[fit={($(H2.south-|x)+(-0.2, -0.3)$) ($(delta1.north east-|add.east)+(0.2, 0.4)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {}; \node[fit={($(H2.south-|x)+(-0.2, -0.3)$) ($(delta1.north east-|add.east)+(0.2, 0.4)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
\node[below left] at (supersensor.north east) {Super Sensor}; \node[below left] at (supersensor.north east) {Super Sensor};
\node[block, fit={($(W1.north-|S1)+(-0.2, 0.2)$) ($(add1.south east)+(0.2, -0.3)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {}; \node[block, fit={($(b1|-W1.north) + (-0.15, 0.15)$) ($(G1.south east)+(0.15, -0.15)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {};
\node[above right] at (sensor1.south west) {Sensor 1}; \node[below left] at (sensor1.north east) {Sensor 1};
\node[block, fit={($(W2.north-|S1)+(-0.2, 0.2)$) ($(add2.south east)+(0.2, -0.3)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor2) {}; \node[block, fit={($(b2|-W2.north) + (-0.15, 0.15)$) ($(G2.south east)+(0.15, -0.15)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor2) {};
\node[above right] at (sensor2.south west) {Sensor 2}; \node[below left] at (sensor2.north east) {Sensor 2};
\end{scope} \end{scope}
\end{tikzpicture} \end{tikzpicture}
#+end_src #+end_src
@ -196,8 +211,8 @@ Configuration file is accessible [[file:config.org][here]].
* Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters * Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
#+begin_src latex :file h_infinity_robust_fusion.pdf #+begin_src latex :file h_infinity_robust_fusion.pdf
\begin{tikzpicture} \begin{tikzpicture}
\node[block={4.0cm}{2.5cm}, fill=black!20!white, dashed] (P) {}; \node[block={3.5cm}{2.5cm}, fill=black!20!white, dashed] (P) {};
\node[above] at (P.north) {$P(s)$}; \node[above] at (P.north) {$P$};
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$); \coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
\coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$); \coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$);
@ -206,11 +221,11 @@ Configuration file is accessible [[file:config.org][here]].
\coordinate[] (output2) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$); \coordinate[] (output2) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$);
\coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( 0.7, 0)$); \coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( 0.7, 0)$);
\node[block, left=1.4 of output1] (W1){$W_1(s)$}; \node[block, left=0.9 of output1] (W1){$W_1$};
\node[block, left=1.4 of output2] (W2){$W_2(s)$}; \node[block, left=0.9 of output2] (W2){$W_2$};
\node[addb={+}{}{}{}{-}, left=of W1] (sub) {}; \node[addb={+}{}{}{}{-}, left=of W1] (sub) {};
\node[block, below=0.3 of P] (H2) {$H_2(s)$}; \node[block, below=0.3 of P] (H2) {$H_2$};
\draw[->] (inputw) node[above right]{$w$} -- (sub.west); \draw[->] (inputw) node[above right]{$w$} -- (sub.west);
\draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (W2.west); \draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (W2.west);
@ -231,48 +246,50 @@ Configuration file is accessible [[file:config.org][here]].
#+begin_src latex :file sensor_fusion_arch_full.pdf #+begin_src latex :file sensor_fusion_arch_full.pdf
\begin{tikzpicture} \begin{tikzpicture}
\node[branch] (x) at (0, 0); \node[branch] (x) at (0, 0);
\node[addb, above right=1.2 and 3.5 of x](add1){}; \node[addb, above right=1.0 and 0.6 of x](addn1){};
\node[addb, below right=1.2 and 3.5 of x](add2){}; \node[addb, below right=1.0 and 0.6 of x](addn2){};
\node[block, above left=0.2 and 0 of add1](delta1){$\Delta_1(s)$}; \node[addb, right=2.6 of addn1](add1){};
\node[block, above left=0.2 and 0 of add2](delta2){$\Delta_2(s)$}; \node[addb, right=2.6 of addn2](add2){};
\node[block, left=0.4 of delta1](W1){$w_1(s)$}; \node[block, above left=0.2 and 0 of add1](delta1){$\Delta_1$};
\node[block, left=0.4 of delta2](W2){$w_2(s)$}; \node[block, above left=0.2 and 0 of add2](delta2){$\Delta_2$};
\node[addb, right=0.4 of add1](addn1){}; \node[block, left=0.4 of delta1](W1){$w_1$};
\node[addb, right=0.4 of add2](addn2){}; \node[block, left=0.4 of delta2](W2){$w_2$};
\node[block, above=0.5 of addn1](N1) {$N_1(s)$}; \node[block, above=0.5 of addn1](N1) {$N_1$};
\node[block, above=0.5 of addn2](N2) {$N_2(s)$}; \node[block, above=0.5 of addn2](N2) {$N_2$};
\node[block, right=1.1 of addn1](H1){$H_1(s)$}; \node[block, right=0.7 of add1](H1){$H_1$};
\node[block, right=1.1 of addn2](H2){$H_2(s)$}; \node[block, right=0.7 of add2](H2){$H_2$};
\node[addb, right=7.0 of x](add){}; \node[addb, right=5.8 of x](add){};
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center); \draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
\draw[->] (x.center) |- (add1.west); \draw[->] (x.center) |- (addn1.west);
\draw[->] (x.center) |- (add2.west); \draw[->] (x.center) |- (addn2.west);
\draw[->] ($(add1-|W1.west)+(-0.5, 0)$)node[branch](S1){} |- (W1.west); \draw[->] ($(addn1-|W1.west)+(-0.3, 0)$)node[branch](S1){} |- (W1.west);
\draw[->] ($(add2-|W2.west)+(-0.5, 0)$)node[branch](S2){} |- (W2.west); \draw[->] ($(addn2-|W2.west)+(-0.3, 0)$)node[branch](S2){} |- (W2.west);
\draw[->] (W1.east) -- (delta1.west); \draw[->] (W1.east) -- (delta1.west);
\draw[->] (W2.east) -- (delta2.west); \draw[->] (W2.east) -- (delta2.west);
\draw[->] (delta1.east) -| (add1.north); \draw[->] (delta1.east) -| (add1.north);
\draw[->] (delta2.east) -| (add2.north); \draw[->] (delta2.east) -| (add2.north);
\draw[->] (add1.east) -- (addn1.west); \draw[->] (addn1.east) -- (add1.west);
\draw[->] (add2.east) -- (addn2.west); \draw[->] (addn2.east) -- (add2.west);
\draw[->] (addn1.east) -- (H1.west)node[above left]{$\hat{x}_1$}; \draw[->] (add1.east) -- (H1.west)node[above left]{$\hat{x}_1$};
\draw[->] (addn2.east) -- (H2.west)node[above left]{$\hat{x}_2$}; \draw[->] (add2.east) -- (H2.west)node[above left]{$\hat{x}_2$};
\draw[->] ($(N1.north)+(0,0.7)$) node[below right](n1){$\tilde{n}_1$} -- (N1.north); \draw[<-] (N1.north)node[above left](n1){$\tilde{n}_1$} -- ++(0, 0.4);
\draw[->] ($(N2.north)+(0,0.7)$) node[below right](n2){$\tilde{n}_2$} -- (N2.north); \draw[<-] (N2.north)node[above left](n2){$\tilde{n}_2$} -- ++(0, 0.4);
\draw[->] (N1.south) -- (addn1.north)node[above right]{$n_1$}; \draw[->] (N1.south) -- (addn1.north)node[above left]{$n_1$};
\draw[->] (N2.south) -- (addn2.north)node[above right]{$n_2$}; \draw[->] (N2.south) -- (addn2.north)node[above left]{$n_2$};
\draw[->] (H1.east) -| (add.north); \draw[->] (H1.east) -| (add.north);
\draw[->] (H2.east) -| (add.south); \draw[->] (H2.east) -| (add.south);
\draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$}; \draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
\begin{scope}[on background layer] \begin{scope}[on background layer]
\node[fit={($(add2.south-|x)+(-0.2, -0.4)$) ($(n1.north-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {}; \node[fit={($(H2.south-|x)+(-0.2, -0.3)$) ($(n1.north east-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
\node[below left] at (supersensor.north east) {Super Sensor}; \node[below left] at (supersensor.north east) {Super Sensor};
\node[block, fit={($(S1|-add1.south)+(-0.2, -0.2)$) ($(n1.north-|N1.east)+(0.2, 0.1)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {};
\node[below right] at (sensor1.north west) {Sensor 1}; \node[fit={(n1.north west) ($(add1.south -| add1.east) + (0.1, -0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
\node[block, fit={($(S2|-add2.south)+(-0.2, -0.2)$) ($(n2.north-|N2.east)+(0.2, 0.1)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor2) {}; \node[below left] at (sensor1.north east) {Sensor 1};
\node[below right] at (sensor2.north west) {Sensor 2};
\node[fit={(n2.north west) ($(add2.south -| add2.east) + (0.1, -0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
\node[below left] at (sensor2.north east) {Sensor 2};
\end{scope} \end{scope}
\end{tikzpicture} \end{tikzpicture}
#+end_src #+end_src
@ -285,27 +302,27 @@ Configuration file is accessible [[file:config.org][here]].
* Mixed H2/H-Infinity Synthesis * Mixed H2/H-Infinity Synthesis
#+begin_src latex :file mixed_h2_hinf_synthesis.pdf #+begin_src latex :file mixed_h2_hinf_synthesis.pdf
\begin{tikzpicture} \begin{tikzpicture}
\node[block={5.0cm}{5.0cm}, fill=black!20!white, dashed] (P) {}; \node[block={4.0cm}{4.0cm}, fill=black!20!white, dashed] (P) {};
\node[above] at (P.north) {$P(s)$}; \node[above] at (P.north) {$P$};
\coordinate[] (inputw) at ($(P.south west)!0.85!(P.north west) + (-0.7, 0)$); \coordinate[] (inputw) at ($(P.south west)!0.85!(P.north west) + (-0.7, 0)$);
\coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$); \coordinate[] (inputu) at ($(P.south west)!0.25!(P.north west) + (-0.7, 0)$);
\coordinate[] (output1) at ($(P.south east)!0.85!(P.north east) + ( 0.7, 0)$); \coordinate[] (output1) at ($(P.south east)!0.85!(P.north east) + ( 0.7, 0)$);
\coordinate[] (output2) at ($(P.south east)!0.7!(P.north east) + ( 0.7, 0)$); \coordinate[] (output2) at ($(P.south east)!0.65!(P.north east) + ( 0.7, 0)$);
\coordinate[] (output3) at ($(P.south east)!0.5!(P.north east) + ( 0.7, 0)$); \coordinate[] (output3) at ($(P.south east)!0.45!(P.north east) + ( 0.7, 0)$);
\coordinate[] (output4) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$); \coordinate[] (output4) at ($(P.south east)!0.25!(P.north east) + ( 0.7, 0)$);
\coordinate[] (outputv) at ($(P.south east)!0.15!(P.north east) + ( 0.7, 0)$); \coordinate[] (outputv) at ($(P.south east)!0.10!(P.north east) + ( 0.7, 0)$);
\node[block, left=1.4 of output1] (W1){$W_1(s)$}; \node[block, left=1.0 of output1] (W1){$W_1$};
\node[block, left=1.4 of output2] (W2){$W_2(s)$}; \node[block, left=1.0 of output2] (W2){$W_2$};
\node[addb={+}{}{}{}{-}, left=1.6 of W1] (sub1) {}; \node[addb={+}{}{}{}{-}, left=1.2 of W1] (sub1) {};
\node[block, left=1.4 of output3] (N1){$N_1(s)$}; \node[block, left=1.0 of output3] (N1){$N_1$};
\node[block, left=1.4 of output4] (N2){$N_2(s)$}; \node[block, left=1.0 of output4] (N2){$N_2$};
\node[addb={+}{}{}{}{-}, left=1 of N1] (sub2) {}; \node[addb={+}{}{}{}{-}, left=0.6 of N1] (sub2) {};
\node[block, below=0.3 of P] (H2) {$H_2(s)$}; \node[block, below=0.3 of P] (H2) {$H_2$};
\draw[->] (inputw) node[above right]{$w$} -- (sub1.west); \draw[->] (inputw) node[above right]{$w$} -- (sub1.west);
\draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (N2.west); \draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (N2.west);
@ -316,10 +333,14 @@ Configuration file is accessible [[file:config.org][here]].
\draw[->] (sub2.east) -- (N1.west); \draw[->] (sub2.east) -- (N1.west);
\draw[->] ($(sub1.west)+(-0.6, 0)$) node[branch](w_branch){} |- ($(outputv)+(-0.35, 0)$) node[above]{$v$} |- (H2.east); \draw[->] ($(sub1.west)+(-0.6, 0)$) node[branch](w_branch){} |- ($(outputv)+(-0.35, 0)$) node[above]{$v$} |- (H2.east);
\draw[->] (w_branch|-sub2) node[branch]{} -- (sub2.west); \draw[->] (w_branch|-sub2) node[branch]{} -- (sub2.west);
\draw[->] (W1.east) -- (output1)node[above left]{$z_1$}; \draw[->] (W1.east) -- (output1)node[above left](z1){$z_1$};
\draw[->] (W2.east) -- (output2)node[above left]{$z_2$}; \draw[->] (W2.east) -- (output2)node[above left](z2){$z_2$};
\draw[->] (N1.east) -- (output3)node[above left]{$z_3$}; \draw[->] (N1.east) -- (output3)node[above left](z3){$z_3$};
\draw[->] (N2.east) -- (output4)node[above left]{$z_4$}; \draw[->] (N2.east) -- (output4)node[above left](z4){$z_4$};
\draw [decoration={brace, raise=5pt}, decorate] (z1.north east) -- node[right=6pt]{$z_{\mathcal{H}_\infty}$} (z2.south east);
\draw [decoration={brace, raise=5pt}, decorate] (z3.north east) -- node[right=6pt]{$z_{\mathcal{H}_2}$} (z4.south east);
\end{tikzpicture} \end{tikzpicture}
#+end_src #+end_src