7.2 KiB
Robust and Optimal Sensor Fusion
\bibliographystyle{IEEEtran}
Abstract ignore
Abstract text to be done
Keywords ignore
Complementary Filters, Sensor Fusion, H-Infinity Synthesis
Introduction
<<sec:introduction>>
Optimal Super Sensor Noise: $\mathcal{H}_2$ Synthesis
<<sec:optimal_fusion>>
Sensor Model
Sensor Fusion Architecture
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
Super Sensor Noise
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}$\mathcal{H}_2$ 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_{\hat{n}}(\omega) d\omega} \end{equation}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} \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 \end{equation}The $\mathcal{H}_2$ synthesis of the complementary filters thus minimized the RMS value of the super sensor noise.
Example
Robustness Problem
Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
<<sec:robust_fusion>>
Representation of Sensor Dynamical Uncertainty
Super Sensor Dynamical Uncertainty
$\mathcal{H_\infty}$ Synthesis of Complementary Filters
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
\bibliography{ref}