#+TITLE: Optimal and Robust Sensor Fusion
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#+DATE: {{{time(%Y-%m-%d)}}}

#+AUTHOR: @@latex:\IEEEauthorblockN{Dehaeze Thomas}@@
#+AUTHOR: @@latex:\IEEEauthorblockA{\textit{European Synchrotron Radiation Facility} \\@@
#+AUTHOR: @@latex:Grenoble, France\\@@
#+AUTHOR: @@latex:\textit{Precision Mechatronics Laboratory} \\@@
#+AUTHOR: @@latex:\textit{University of Liege}, Belgium \\@@
#+AUTHOR: @@latex:thomas.dehaeze@esrf.fr@@
#+AUTHOR: @@latex:}\and@@
#+AUTHOR: @@latex:\IEEEauthorblockN{Collette Christophe}@@
#+AUTHOR: @@latex:\IEEEauthorblockA{\textit{BEAMS Department}\\@@
#+AUTHOR: @@latex:\textit{Free University of Brussels}, Belgium\\@@
#+AUTHOR: @@latex:\textit{Precision Mechatronics Laboratory} \\@@
#+AUTHOR: @@latex:\textit{University of Liege}, Belgium \\@@
#+AUTHOR: @@latex:ccollett@ulb.ac.be@@
#+AUTHOR: @@latex:}@@

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* Abstract                                                           :ignore:
#+begin_abstract
  Abstract text to be done
#+end_abstract

* Keywords                                                            :ignore:
#+begin_IEEEkeywords
  Complementary Filters, Sensor Fusion, H-Infinity Synthesis
#+end_IEEEkeywords

* Introduction
<<sec:introduction>>

- Section ref:sec:optimal_fusion
- Section ref:sec:robust_fusion
- Section ref:sec:optimal_robust_fusion
- Section ref:sec:experimental_validation

* Optimal Super Sensor Noise: $\mathcal{H}_2$ Synthesis
<<sec:optimal_fusion>>

** Sensor Model

Let's consider a sensor measuring a physical quantity $x$ (Figure [[fig:sensor_model]]).
The sensor has an internal dynamics which is here modelled with a Linear Time Invariant (LTI) system transfer function $G_i(s)$.

The noise of sensor can be described by the Power Spectral Density (PSD) $\Phi_{n_i}(\omega)$.

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 $N_i(s)$:
\begin{equation}
\begin{aligned}
  \Phi_{n_i}(\omega) &= \left| N_i(j\omega) \right|^2 \Phi_{\tilde{n}_i}(\omega) \\
                     &= \left| N_i(j\omega) \right|^2
\end{aligned}
\end{equation}

#+name: eq:unitary_sensor_noise_psd
\begin{equation}
  \Phi_{\tilde{n}_i}(\omega) = 1
\end{equation}

The output of the sensor $v_i$:
\begin{equation}
  v_i = \left( G_i \right) x + \left( G_i N_i \right) \tilde{n}_i
\end{equation}

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 [[fig:sensor_model]]):
\begin{equation}
  \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
\end{equation}

#+name: fig:sensor_model
#+caption: Sensor Model
#+attr_latex: :scale 1
[[file:figs/sensor_model.pdf]]

** Sensor Fusion Architecture

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 [[fig:sensor_fusion_noise_arch]]).

The noise sources $\tilde{n}_1$ and $\tilde{n}_2$ are considered to be uncorrelated.

#+name: fig:sensor_fusion_noise_arch
#+caption: Sensor Fusion Architecture with sensor noise
#+attr_latex: :scale 1
[[file:figs/sensor_fusion_noise_arch.pdf]]

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 $x$.
These two estimates are then filtered out by two filters $H_1$ and $H_2$ and summed to gives the super sensor estimate $\hat{x}$.

\begin{equation}
  \begin{split}
    \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} G_1 N_1 \right) \tilde{n}_1 + \left( H_2 \hat{G}_2^{-1} G_2 N_2 \right) \tilde{n}_2
  \end{split}
\end{equation}

Suppose the sensor dynamical model $\hat{G}_i$ is perfect:
\begin{equation}
  \hat{G}_i = G_i
\end{equation}

We considered here complementary filters:
\begin{equation}
  H_1(s) + H_2(s) = 1
\end{equation}

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:
\begin{equation}
  \hat{x} = x + \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2
\end{equation}

** Super Sensor Noise
Let's note $n$ the super sensor noise.
\begin{equation}
  n = \left( H_1 N_1 \right) \tilde{n}_1 + \left( H_2 N_2 \right) \tilde{n}_2
\end{equation}

As the noise of both sensors are considered to be uncorrelated, the PSD of the super sensor noise is computed as follow:
\begin{equation}
  \Phi_n(\omega) = \left| H_1 N_1 \right|^2 + \left| H_2 N_2 \right|^2
\end{equation}

It is clear that the PSD of the super sensor depends on the norm of the complementary filters.

** $\mathcal{H}_2$ Synthesis of Complementary Filters
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 $n$:
#+name: eq:rms_value_estimation
\begin{equation}
  \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
\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 $\sigma_n$ is minimized.

This can be cast into an $\mathcal{H}_2$ synthesis problem by considering the following generalized plant (also represented in Figure [[fig:h_two_optimal_fusion]]):
\begin{equation}
\begin{pmatrix}
  z_1 \\ z_2 \\ v
\end{pmatrix} = \underbrace{\begin{bmatrix}
  N_1 & N_1 \\
  0   & N_2 \\
  1   &  0
\end{bmatrix}}_{P_{\mathcal{H}_2}} \begin{pmatrix}
  w \\ u
\end{pmatrix}
\end{equation}

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:
#+NAME: eq:H2_norm
\begin{equation}
   \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
\end{equation}

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 $H_2(s)$:
\begin{equation}
  H_1(s) = 1 - H_2(s)
\end{equation}

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.

#+name: fig:h_two_optimal_fusion
#+caption: Generalized plant $P_{\mathcal{H}_2}$ used for the $\mathcal{H}_2$ synthesis of complementary filters
#+attr_latex: :scale 1
[[file:figs/h_two_optimal_fusion.pdf]]

** Example

** Robustness Problem

* Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
<<sec:robust_fusion>>

** Representation of Sensor Dynamical Uncertainty

Suppose that the sensor dynamics $G_i(s)$ can be modelled by a nominal d
\begin{equation}
  G_i(s) = \hat{G}_i(s) \left( 1 + w_i(s) \Delta_i(s) \right); \quad |\Delta_i(j\omega)| < 1 \forall \omega
\end{equation}


** Sensor Fusion Architecture
\begin{equation}
\begin{split}
  \hat{x} = \Big( {} & H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + w_1 \Delta_1) \\
             + & H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + w_2 \Delta_2) \Big) x
\end{split}
\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
#+caption: Sensor Fusion Architecture with sensor model uncertainty
#+attr_latex: :scale 1
[[file:figs/sensor_fusion_arch_uncertainty.pdf]]

** 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
#+caption: Super Sensor model uncertainty displayed in the complex plane
#+attr_latex: :scale 1
[[file:figs/uncertainty_set_super_sensor.pdf]]

** $\mathcal{H_\infty}$ Synthesis of Complementary Filters

In order to minimize the super sensor dynamical uncertainty

#+name: fig:h_infinity_robust_fusion
#+caption: Generalized plant $P_{\mathcal{H}_\infty}$ used for the $\mathcal{H}_\infty$ synthesis of complementary filters
#+attr_latex: :scale 1
[[file:figs/h_infinity_robust_fusion.pdf]]

** Example

* Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
<<sec:optimal_robust_fusion>>

** Sensor Fusion Architecture

#+name: fig:sensor_fusion_arch_full
#+caption: Super Sensor Fusion with both sensor noise and sensor model uncertainty
#+attr_latex: :scale 1
[[file:figs/sensor_fusion_arch_full.pdf]]

** Synthesis Objective

** Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis

#+name: fig:mixed_h2_hinf_synthesis
#+caption: Generalized plant $P_{\mathcal{H}_2/\matlcal{H}_\infty}$ used for the mixed $\mathcal{H}_2/\mathcal{H}_\infty$ synthesis of complementary filters
#+attr_latex: :scale 1
[[file:figs/mixed_h2_hinf_synthesis.pdf]]

** Example

* Experimental Validation
<<sec:experimental_validation>>

** Experimental Setup

** Sensor Noise and Dynamical Uncertainty

** Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis

** Super Sensor Noise and Dynamical Uncertainty

* Conclusion
<<sec:conclusion>>

* Acknowledgment

* Bibliography                                                       :ignore:
\bibliography{ref}