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Optimal and Robust Sensor Fusion
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Abstract ignore
Abstract text to be done
Keywords ignore
Complementary Filters, Sensor Fusion, H-Infinity Synthesis
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} \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}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.
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 $n$:
\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:
\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.
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
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}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 $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$.
$\mathcal{H_\infty}$ Synthesis of Complementary Filters
In order to minimize the super sensor dynamical uncertainty
Example
Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
<<sec:optimal_robust_fusion>>
Sensor Fusion Architecture
Synthesis Objective
Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
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
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