433 lines
17 KiB
Org Mode
433 lines
17 KiB
Org Mode
#+TITLE: Optimal and Robust Sensor Fusion
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:DRAWER:
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#+LATEX_CLASS: IEEEtran
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#+STARTUP: overview
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#+DATE: {{{time(%Y-%m-%d)}}}
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#+AUTHOR: @@latex:\IEEEauthorblockN{Dehaeze Thomas}@@
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#+AUTHOR: @@latex:\IEEEauthorblockA{\textit{European Synchrotron Radiation Facility} \\@@
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#+AUTHOR: @@latex:Grenoble, France\\@@
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#+AUTHOR: @@latex:\textit{Precision Mechatronics Laboratory} \\@@
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#+AUTHOR: @@latex:\textit{University of Liege}, Belgium \\@@
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#+AUTHOR: @@latex:thomas.dehaeze@esrf.fr@@
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#+AUTHOR: @@latex:}\and@@
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#+AUTHOR: @@latex:\IEEEauthorblockN{Collette Christophe}@@
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#+AUTHOR: @@latex:\IEEEauthorblockA{\textit{BEAMS Department}\\@@
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#+AUTHOR: @@latex:\textit{Free University of Brussels}, Belgium\\@@
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#+AUTHOR: @@latex:\textit{Precision Mechatronics Laboratory} \\@@
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#+AUTHOR: @@latex:\textit{University of Liege}, Belgium \\@@
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#+AUTHOR: @@latex:ccollett@ulb.ac.be@@
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#+AUTHOR: @@latex:}@@
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#+END_SRC
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* Abstract :ignore:
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#+begin_abstract
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Abstract text to be done
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#+end_abstract
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* Keywords :ignore:
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#+begin_IEEEkeywords
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Complementary Filters, Sensor Fusion, H-Infinity Synthesis
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#+end_IEEEkeywords
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* Introduction
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<<sec:introduction>>
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- Section ref:sec:optimal_fusion
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- Section ref:sec:robust_fusion
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- Section ref:sec:optimal_robust_fusion
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- Section ref:sec:experimental_validation
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* Optimal Super Sensor Noise: $\mathcal{H}_2$ Synthesis
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<<sec:optimal_fusion>>
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** Sensor Model
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Let's consider a sensor measuring a physical quantity $x$ (Figure ref:fig:sensor_model_noise).
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The sensor has an internal dynamics which is here modelled with a Linear Time Invariant (LTI) system transfer function $G_i(s)$.
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The noise of sensor can be described by the Power Spectral Density (PSD) $\Phi_{n_i}(\omega)$.
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This is approximated by shaping a white noise with unitary PSD $\tilde{n}_i$ eqref:eq:unitary_sensor_noise_psd with a LTI transfer function $N_i(s)$:
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\begin{equation}
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\begin{aligned}
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\Phi_{n_i}(\omega) &= \left| N_i(j\omega) \right|^2 \Phi_{\tilde{n}_i}(\omega) \\
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&= \left| N_i(j\omega) \right|^2
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\end{aligned}
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\end{equation}
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#+name: eq:unitary_sensor_noise_psd
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\begin{equation}
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\Phi_{\tilde{n}_i}(\omega) = 1
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\end{equation}
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The output of the sensor $v_i$:
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\begin{equation}
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v_i = \left( G_i \right) x + \left( G_i N_i \right) \tilde{n}_i
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\end{equation}
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In order to obtain an estimate $\hat{x}_i$ of $x$, a model $\hat{G}_i$ of the (true) sensor dynamics $G_i$ is inverted and applied at the output (Figure ref:fig:sensor_model_noise):
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\begin{equation}
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\hat{x}_i = \left( \hat{G}_i^{-1} G_i \right) x + \left( \hat{G}_i^{-1} G_i N_i \right) \tilde{n}_i
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\end{equation}
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#+name: fig:sensor_model_noise
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#+caption: Sensor Model
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#+attr_latex: :scale 1
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[[file:figs/sensor_model_noise.pdf]]
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** Sensor Fusion Architecture
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Let's now consider two sensors measuring the same physical quantity $x$ but with different dynamics $(G_1, G_2)$ and noise characteristics $(N_1, N_2)$ (Figure ref:fig:sensor_fusion_noise_arch).
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The noise sources $\tilde{n}_1$ and $\tilde{n}_2$ are considered to be uncorrelated.
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#+name: fig:sensor_fusion_noise_arch
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#+caption: Sensor Fusion Architecture with sensor noise
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#+attr_latex: :scale 1
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[[file:figs/sensor_fusion_noise_arch.pdf]]
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The output of both sensors $(v1,v2)$ are then passed through the inverse of the sensor model to obtained two estimates $(\hat{x}_1, \hat{x}_2)$ of $x$.
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These two estimates are then filtered out by two filters $H_1$ and $H_2$ and summed to gives the super sensor estimate $\hat{x}$.
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\begin{equation}
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\begin{split}
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\hat{x} = {}&\left( H_1 \hat{G}_1^{-1} G_1 + H_2 \hat{G}_2^{-1} G_2 \right) x \\
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&+ \left( H_1 \hat{G}_1^{-1} G_1 N_1 \right) \tilde{n}_1 + \left( H_2 \hat{G}_2^{-1} G_2 N_2 \right) \tilde{n}_2
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\end{split}
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\end{equation}
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Suppose the sensor dynamical model $\hat{G}_i$ is perfect:
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\begin{equation}
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\hat{G}_i = G_i
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\end{equation}
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We considered here complementary filters:
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\begin{equation}
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H_1(s) + H_2(s) = 1
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\end{equation}
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In such case, the super sensor estimate $\hat{x}$ is equal to $x$ plus the noise of the individual sensors filtered out by the complementary filters:
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\begin{equation}
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\hat{x} = x + \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2
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\end{equation}
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** Super Sensor Noise
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Let's note $n$ the super sensor noise.
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\begin{equation}
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n = \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2
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\end{equation}
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As the noise of both sensors are considered to be uncorrelated, the PSD of the super sensor noise is computed as follow:
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\begin{equation}
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\Phi_n(\omega) = \left| H_1 N_1 \right|^2 + \left| H_2 N_2 \right|^2
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\end{equation}
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It is clear that the PSD of the super sensor depends on the norm of the complementary filters.
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** $\mathcal{H}_2$ Synthesis of Complementary Filters
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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 $\hat{x}$.
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And the goal is the minimize the Root Mean Square (RMS) value of $n$:
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#+name: eq:rms_value_estimation
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\begin{equation}
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\sigma_{n} = \sqrt{\int_0^\infty \Phi_{n}(\omega) d\omega} = \left\| \begin{matrix} H_1 N_1 \\ H_2 N_2 \end{matrix} \right\|_2
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\end{equation}
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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 $\sigma_n$ is minimized.
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This can be cast into an $\mathcal{H}_2$ synthesis problem by considering the following generalized plant (also represented in Figure ref:fig:h_two_optimal_fusion):
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\begin{equation}
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\begin{pmatrix}
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z_1 \\ z_2 \\ v
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\end{pmatrix} = \underbrace{\begin{bmatrix}
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N_1 & N_1 \\
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0 & N_2 \\
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1 & 0
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\end{bmatrix}}_{P_{\mathcal{H}_2}} \begin{pmatrix}
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w \\ u
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\end{pmatrix}
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\end{equation}
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Applying the $\mathcal{H}_2$ synthesis on $P_{\mathcal{H}_2}$ will generate a filter $H_2(s)$ such that the $\mathcal{H}_2$ norm from $w$ to $(z_1,z_2)$ is minimized:
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#+NAME: eq:H2_norm
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\begin{equation}
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\left\| \begin{matrix} z_1/w \\ z_2/w \end{matrix} \right\|_2 = \left\| \begin{matrix} N_1 (1 - H_2) \\ N_2 H_2 \end{matrix} \right\|_2
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\end{equation}
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The $\mathcal{H}_2$ norm of Eq. eqref:eq:H2_norm is equals to $\sigma_n$ by defining $H_1(s)$ to be the complementary filter of $H_2(s)$:
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\begin{equation}
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H_1(s) = 1 - H_2(s)
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\end{equation}
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We then have that the $\mathcal{H}_2$ synthesis applied on $P_{\mathcal{H}_2}$ generates two complementary filters $H_1(s)$ and $H_2(s)$ such that the RMS value of super sensor noise is minimized.
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#+name: fig:h_two_optimal_fusion
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#+caption: Generalized plant $P_{\mathcal{H}_2}$ used for the $\mathcal{H}_2$ synthesis of complementary filters
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#+attr_latex: :scale 1
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[[file:figs/h_two_optimal_fusion.pdf]]
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** Example
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** Robustness Problem
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* Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
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<<sec:robust_fusion>>
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** Representation of Sensor Dynamical Uncertainty
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In Section ref:sec:optimal_fusion, the model $\hat{G}_i(s)$ of the sensor was considered to be perfect.
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In reality, there are always uncertainty (neglected dynamics) associated with the estimation of the sensor dynamics.
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The Uncertainty on the sensor dynamics $G_i(s)$ is here modelled by (input) multiplicative uncertainty:
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\begin{equation}
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G_i(s) = \hat{G}_i(s) \left( 1 + W_i(s) \Delta_i(s) \right); \quad |\Delta_i(j\omega)| < 1 \forall \omega
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\end{equation}
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where $\hat{G}_i(s)$ is the nominal model, $W_i$ a weight representing the size of the uncertainty at each frequency, and $\Delta_i$ is any complex perturbation such that $\left\| \Delta_i \right\|_\infty < 1$.
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The sensor can then be represented as shown in Figure ref:fig:sensor_model_uncertainty.
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#+name: fig:sensor_model_uncertainty
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#+caption: Sensor Model including Dynamical Uncertainty
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#+attr_latex: :scale 1
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[[file:figs/sensor_model_uncertainty.pdf]]
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** Sensor Fusion Architecture
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Let's consider the sensor fusion architecture shown in Figure ref:fig:sensor_fusion_arch_uncertainty where the dynamical uncertainties of both sensors are included.
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The super sensor estimate is then:
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\begin{equation}
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\begin{aligned}
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\hat{x} &= \Big( H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + W_1 \Delta_1) \\
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& \quad + H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + W_2 \Delta_2) \Big) x \\
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&= \Big( H_1 (1 + W_1 \Delta_1) + H_2 (1 + W_2 \Delta_2) \Big) x
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\end{aligned}
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\end{equation}
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with $\Delta_i$ is any transfer function satisfying $\| \Delta_i \|_\infty < 1$.
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As $H_1$ and $H_2$ are complementary filters, we finally have:
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\begin{equation}
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\hat{x} = \left( 1 + H_1 W_1 \Delta_1 + H_2 W_2 \Delta_2 \right) x, \quad \|\Delta_i\|_\infty<1
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\end{equation}
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#+name: fig:sensor_fusion_arch_uncertainty
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#+caption: Sensor Fusion Architecture with sensor model uncertainty
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#+attr_latex: :scale 1
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[[file:figs/sensor_fusion_arch_uncertainty.pdf]]
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** Super Sensor Dynamical Uncertainty
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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 ref:fig:uncertainty_set_super_sensor.
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And we can see that the dynamical uncertainty of the super sensor is equal to the sum of the individual sensor uncertainties filtered out by the complementary filters.
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#+name: fig:uncertainty_set_super_sensor
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#+caption: Super Sensor model uncertainty displayed in the complex plane
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#+attr_latex: :scale 1
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[[file:figs/uncertainty_set_super_sensor.pdf]]
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# Some comments on the weights
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At frequencies where $\left|W_i(j\omega)\right| > 1$ the uncertainty exceeds $100\%$ and sensor fusion is impossible.
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** $\mathcal{H_\infty}$ Synthesis of Complementary Filters
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In order for the fusion to be "robust", meaning no phase drop will be induced in the super sensor dynamics,
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The goal is to design two complementary filters $H_1(s)$ and $H_2(s)$ such that the super sensor noise uncertainty is kept reasonably small.
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To define what by "small" we mean, we use a weighting filter $W_u(s)$ such that the synthesis objective is:
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\begin{equation}
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\left| W_1(j\omega)H_1(j\omega) \right| + \left| W_2(j\omega)H_2(j\omega) \right| < \frac{1}{\left| W_u(j\omega) \right|}, \quad \forall \omega
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\end{equation}
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# Comments on the choice of the weights => we cannot ask for less uncertainty than both sensors
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This is actually almost equivalent as to have (within a factor $\sqrt{2}$):
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\begin{equation}
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\left\| \begin{matrix} W_u W_1 H_1 \\ W_u W_2 H_2 \end{matrix} \right\|_\infty < 1
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\end{equation}
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This problem can thus be dealt with an $\mathcal{H}_\infty$ synthesis problem by considering the following generalized plant (Figure ref:fig:h_infinity_robust_fusion):
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\begin{equation}
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\begin{pmatrix}
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z_1 \\ z_2 \\ v
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\end{pmatrix} = \underbrace{\begin{bmatrix}
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W_u W_1 & W_u W_1 \\
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0 & W_u W_2 \\
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1 & 0
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\end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix}
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w \\ u
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\end{pmatrix}
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\end{equation}
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Applying the $\mathcal{H}_\infty$ synthesis on $P_{\mathcal{H}_\infty}$ will generate a filter $H_2(s)$ such that the $\mathcal{H}_\infty$ norm from $w$ to $(z_1,z_2)$ is minimized:
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#+NAME: eq:Hinf_norm
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\begin{equation}
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\left\| \begin{matrix} z_1/w \\ z_2/w \end{matrix} \right\|_\infty = \left\| \begin{matrix} W_u W_1 (1 - H_2) \\ W_u W_2 H_2 \end{matrix} \right\|_\infty
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\end{equation}
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The $\mathcal{H}_\infty$ norm of Eq. eqref:eq:Hinf_norm is equals to $\sigma_n$ by defining $H_1(s)$ to be the complementary filter of $H_2(s)$:
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\begin{equation}
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H_1(s) = 1 - H_2(s)
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\end{equation}
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#+name: fig:h_infinity_robust_fusion
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#+caption: Generalized plant $P_{\mathcal{H}_\infty}$ used for the $\mathcal{H}_\infty$ synthesis of complementary filters
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#+attr_latex: :scale 1
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[[file:figs/h_infinity_robust_fusion.pdf]]
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** Example
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* Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
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<<sec:optimal_robust_fusion>>
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** Sensor with noise and model uncertainty
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We wish now to combine the two previous synthesis, that is to say
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The sensors are now modelled by a white noise with unitary PSD $\tilde{n}_i$ shaped by a LTI transfer function $N_i(s)$.
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The dynamical uncertainty of the sensor is modelled using multiplicative uncertainty
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\begin{equation}
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v_i = \hat{G}_i (1 + W_i \Delta_i) x + \hat{G_i} (1 + W_i \Delta_i) N_i \tilde{n}_i
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\end{equation}
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Multiplying by the inverse of the nominal model of the sensor dynamics gives an estimate $\hat{x}_i$ of $x$:
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\begin{equation}
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\hat{x} = (1 + W_i \Delta_i) x + (1 + W_i \Delta_i) N_i \tilde{n}_i
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\end{equation}
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#+name: fig:sensor_model_noise_uncertainty
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#+caption: Sensor Model including Noise and Dynamical Uncertainty
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#+attr_latex: :scale 1
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[[file:figs/sensor_model_noise_uncertainty.pdf]]
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** Sensor Fusion Architecture
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For reason of space, the blocks $\hat{G}_i$ and $\hat{G}_i^{-1}$ are omitted.
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\begin{equation}
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\begin{aligned}
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\hat{x} = &\Big( H_1 (1 + W_1 \Delta_1) + H_2 (1 + W_2 \Delta_2) \Big) x \\
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&+ \Big( H_1 (1 + W_1 \Delta_1) N_1 \Big) \tilde{n}_1 + \Big( H_2 (1 + W_2 \Delta_2) N_2 \Big) \tilde{n}_2
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\end{aligned}
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\end{equation}
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\begin{equation}
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\begin{aligned}
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\hat{x} = &\Big( 1 + H_1 W_1 \Delta_1 + H_2 W_2 \Delta_2 \Big) x \\
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&+ \Big( H_1 (1 + W_1 \Delta_1) N_1 \Big) \tilde{n}_1 + \Big( H_2 (1 + W_2 \Delta_2) N_2 \Big) \tilde{n}_2
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\end{aligned}
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\end{equation}
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The estimate $\hat{x}$ of $x$
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#+name: fig:sensor_fusion_arch_full
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#+caption: Super Sensor Fusion with both sensor noise and sensor model uncertainty
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#+attr_latex: :scale 1
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[[file:figs/sensor_fusion_arch_full.pdf]]
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** Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
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The synthesis objective is to generate two complementary filters $H_1(s)$ and $H_2(s)$ such that the uncertainty associated with the super sensor is kept reasonably small and such that the RMS value of super sensors noise is minimized.
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To specify how small we want the super sensor dynamic spread, we use a weighting filter $W_u(s)$ as was done in Section ref:sec:robust_fusion.
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This synthesis problem can be solved using the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis on the following generalized plant:
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\begin{equation}
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\begin{pmatrix}
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z_{\infty, 1} \\ z_{\infty, 2} \\ z_{2, 1} \\ z_{2, 2} \\ v
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\end{pmatrix} = \underbrace{\begin{bmatrix}
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W_u W_1 & W_u W_1 \\
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0 & W_u W_2 \\
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N_1 & N_1 \\
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0 & N_2 \\
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1 & 0
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\end{bmatrix}}_{P_{\mathcal{H}_2/\mathcal{H}_\infty}} \begin{pmatrix}
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w \\ u
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\end{pmatrix}
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\end{equation}
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The synthesis objective is to:
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- Keep the $\mathcal{H}_\infty$ norm from $w$ to $(z_{\infty,1}, z_{\infty,2})$ below $1$
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- Minimize the $\mathcal{H}_2$ norm from $w$ to $(z_{2,1}, z_{2,2})$
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#+name: fig:mixed_h2_hinf_synthesis
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#+caption: Generalized plant $P_{\mathcal{H}_2/\matlcal{H}_\infty}$ used for the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis of complementary filters
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#+attr_latex: :scale 1
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[[file:figs/mixed_h2_hinf_synthesis.pdf]]
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|
|
|
** Example
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|
|
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* Experimental Validation
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|
<<sec:experimental_validation>>
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|
|
|
** Experimental Setup
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|
|
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** Sensor Noise and Dynamical Uncertainty
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|
|
|
** Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
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|
|
|
** Super Sensor Noise and Dynamical Uncertainty
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|
|
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* Conclusion
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<<sec:conclusion>>
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|
|
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* Acknowledgment
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|
|
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* Bibliography :ignore:
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|
\bibliography{ref}
|