From 9a1379d4ebaf50584f4d4a7cbd0a4e57cb5863e0 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 22 Sep 2020 21:58:37 +0200 Subject: [PATCH] Add test about Hinf synthesis --- paper/paper.org | 110 ++++++++++++++++++++-------- paper/paper.pdf | Bin 310641 -> 322059 bytes paper/paper.tex | 185 ++++++++++++++++++++++++++++++++---------------- 3 files changed, 204 insertions(+), 91 deletions(-) diff --git a/paper/paper.org b/paper/paper.org index 0f49397..40ec259 100644 --- a/paper/paper.org +++ b/paper/paper.org @@ -42,11 +42,11 @@ \bibliographystyle{IEEEtran} :END: -* LaTeX Config :noexport: +* LaTeX Config :noexport: #+begin_src latex :tangle config.tex #+end_src -* Build :noexport: +* Build :noexport: #+NAME: startblock #+BEGIN_SRC emacs-lisp :results none (add-to-list 'org-latex-classes @@ -74,7 +74,7 @@ (setq org-latex-with-hyperref nil) #+END_SRC -* Abstract :ignore: +* Abstract :ignore: #+begin_abstract Abstract text to be done #+end_abstract @@ -97,17 +97,17 @@ ** Sensor Model -Let's consider a sensor measuring a physical quantity $x$ (Figure [[fig:sensor_model]]). +Let's consider a sensor measuring a physical quantity $x$ (Figure ref:fig:sensor_model_noise). 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} + \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 @@ -120,19 +120,19 @@ The output of the sensor $v_i$: 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]]): +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): \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 +#+name: fig:sensor_model_noise #+caption: Sensor Model #+attr_latex: :scale 1 -[[file:figs/sensor_model.pdf]] +[[file:figs/sensor_model_noise.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]]). +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). The noise sources $\tilde{n}_1$ and $\tilde{n}_2$ are considered to be uncorrelated. @@ -190,7 +190,7 @@ And the goal is the minimize the Root Mean Square (RMS) value of $n$: 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]]): +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): \begin{equation} \begin{pmatrix} z_1 \\ z_2 \\ v @@ -230,28 +230,38 @@ We then have that the $\mathcal{H}_2$ synthesis applied on $P_{\mathcal{H}_2}$ g ** 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} +In Section ref:sec:optimal_fusion, the model $\hat{G}_i(s)$ of the sensor was considered to be perfect. +In reality, there are always uncertainty (neglected dynamics) associated with the estimation of the sensor dynamics. +The Uncertainty on the sensor dynamics $G_i(s)$ is here modelled by (input) multiplicative uncertainty: +\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} +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 $\left\| \Delta_i \right\|_\infty < 1$. + +The sensor can then be represented as shown in Figure ref:fig:sensor_model_uncertainty. + +#+name: fig:sensor_model_uncertainty +#+caption: Sensor Model including Dynamical Uncertainty +#+attr_latex: :scale 1 +[[file:figs/sensor_model_uncertainty.pdf]] ** Sensor Fusion Architecture +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. + +The super sensor estimate is then: \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} +\begin{aligned} + \hat{x} &= \Big( H_1 \hat{G}_1^{-1} \hat{G}_1 (1 + W_1 \Delta_1) \\ + & \quad + H_2 \hat{G}_2^{-1} \hat{G}_2 (1 + W_2 \Delta_2) \Big) x \\ + &= \Big( H_1 (1 + W_1 \Delta_1) + H_2 (1 + W_2 \Delta_2) \Big) x +\end{aligned} \end{equation} with $\Delta_i$ is any transfer function satisfying $\| \Delta_i \|_\infty < 1$. -Suppose the model inversion is equal to the nominal model: +As $H_1$ and $H_2$ are complementary filters, we finally have: \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 + \hat{x} = \left( 1 + H_1 W_1 \Delta_1 + H_2 W_2 \Delta_2 \right) x, \quad \|\Delta_i\|_\infty<1 \end{equation} #+name: fig:sensor_fusion_arch_uncertainty @@ -261,16 +271,48 @@ Suppose the model inversion is equal to the nominal model: ** 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 $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$. +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. + + +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. #+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 +# Some comments on the weights +At frequencies where $\left|W_i(j\omega)\right| > 1$ the uncertainty exceeds $100\%$ and sensor fusion is impossible. + +** $\mathcal{H_\infty}$ Synthesis of Complementary Filters +In order for the fusion to be "robust", meaning no phase drop will be induced in the super sensor dynamics, + +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. + +This problem can be dealt with an $\mathcal{H}_\infty$ synthesis problem by considering the following generalized plant: +\begin{equation} +\begin{pmatrix} + z_1 \\ z_2 \\ v +\end{pmatrix} = \underbrace{\begin{bmatrix} + W_1 & W_1 \\ + 0 & W_2 \\ + 1 & 0 +\end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix} + w \\ u +\end{pmatrix} +\end{equation} + +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: +#+NAME: eq:Hinf_norm +\begin{equation} + \left\| \begin{matrix} z_1/w \\ z_2/w \end{matrix} \right\|_\infty = \left\| \begin{matrix} W_1 (1 - H_2) \\ W_2 H_2 \end{matrix} \right\|_\infty +\end{equation} + +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 $H_2(s)$: +\begin{equation} + H_1(s) = 1 - H_2(s) +\end{equation} -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 @@ -279,11 +321,19 @@ In order to minimize the super sensor dynamical uncertainty ** Example +# Comments on the choice of the weights => we cannot ask for less uncertainty than both sensors + * Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis <> ** Sensor Fusion Architecture +#+name: fig:sensor_model_noise_uncertainty +#+caption: Sensor Model including Noise and Dynamical Uncertainty +#+attr_latex: :scale 1 +[[file:figs/sensor_model_noise_uncertainty.pdf]] + + #+name: fig:sensor_fusion_arch_full #+caption: Super Sensor Fusion with both sensor noise and sensor model uncertainty #+attr_latex: :scale 1 diff --git a/paper/paper.pdf b/paper/paper.pdf index 1ef6ea3b86060db78da6adaff4a73d2c1ef6fc65..eeacf963765604f0725b21dbe3c31e23dc102f85 100644 GIT binary patch delta 105786 zcmZU(V{|6b7OfrIcHXdK+qTnT$4)v1CZ(P`Wt2@<|9$8=qa0VZ2-seCH;uj8 zwt><0X{2$K@pzFk4tyiD5UF6pay5vzD5G>zfaTn_);O8}7O#7%#v}Bt_bXfbk zWx2b}<1Kh8zj0R2(E_MDWXt7uK_HMo1Ac1Gzpw)}kFQy9QsiU@`sFt_0x9mSDdSIn zd1VYLEu>qR3XyK8fR+}o-r02hY9&0DNq}fN%T?xPi2FU=%2;350wm!S&)n~^syq?v 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``robust'', meaning no phase drop will be induced in the super sensor dynamics, + +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. + +This problem can be dealt with an \(\mathcal{H}_\infty\) synthesis problem by considering the following generalized plant: +\begin{equation} +\begin{pmatrix} + z_1 \\ z_2 \\ v +\end{pmatrix} = \underbrace{\begin{bmatrix} + W_1 & W_1 \\ + 0 & W_2 \\ + 1 & 0 +\end{bmatrix}}_{P_{\mathcal{H}_\infty}} \begin{pmatrix} + w \\ u +\end{pmatrix} +\end{equation} + +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: +\begin{equation} +\label{eq:Hinf_norm} + \left\| \begin{matrix} z_1/w \\ z_2/w \end{matrix} \right\|_\infty = \left\| \begin{matrix} W_1 (1 - H_2) \\ W_2 H_2 \end{matrix} \right\|_\infty +\end{equation} + +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 \(H_2(s)\): +\begin{equation} + H_1(s) = 1 - H_2(s) +\end{equation} -In order to minimize the super sensor dynamical uncertainty \begin{figure}[htbp] \centering @@ -251,14 +306,22 @@ In order to minimize the super sensor dynamical uncertainty \end{figure} \subsection{Example} -\label{sec:orgba594da} +\label{sec:orgfe98b6f} + \section{Optimal and Robust Sensor Fusion: Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} -\label{sec:orgc07eeab} +\label{sec:org9114fff} \label{sec:optimal_robust_fusion} \subsection{Sensor Fusion Architecture} -\label{sec:orgddd6d33} +\label{sec:org7816cc1} + +\begin{figure}[htbp] +\centering +\includegraphics[scale=1]{figs/sensor_model_noise_uncertainty.pdf} +\caption{\label{fig:sensor_model_noise_uncertainty}Sensor Model including Noise and Dynamical Uncertainty} +\end{figure} + \begin{figure}[htbp] \centering @@ -267,10 +330,10 @@ In order to minimize the super sensor dynamical uncertainty \end{figure} \subsection{Synthesis Objective} -\label{sec:org79824da} +\label{sec:org39451fc} \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} -\label{sec:org247ac1c} +\label{sec:orga8ff805} \begin{figure}[htbp] \centering @@ -279,30 +342,30 @@ In order to minimize the super sensor dynamical uncertainty \end{figure} \subsection{Example} -\label{sec:org7af2158} +\label{sec:orga353d87} \section{Experimental Validation} -\label{sec:orgb54c59b} +\label{sec:orgb00dce4} \label{sec:experimental_validation} \subsection{Experimental Setup} -\label{sec:org40eadad} +\label{sec:orgc725d26} \subsection{Sensor Noise and Dynamical Uncertainty} -\label{sec:orgb81743f} +\label{sec:org0b05001} \subsection{Mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) Synthesis} -\label{sec:orgb2232ac} +\label{sec:org9c0559a} \subsection{Super Sensor Noise and Dynamical Uncertainty} -\label{sec:orgd80a558} +\label{sec:orgc629276} \section{Conclusion} -\label{sec:org0da5eb6} +\label{sec:orgdd3a6b6} \label{sec:conclusion} \section{Acknowledgment} -\label{sec:orge5b9b80} +\label{sec:orge958f77} \bibliography{ref} \end{document}