Change packages

journal/journal.org

@@ -17,27 +17,14 @@
 #+LATEX_HEADER_EXTRA: \address[a3]{CSIR --- Structural Engineering Research Centre, Taramani, Chennai --- 600113, India.}
 #+LATEX_HEADER_EXTRA: \address[a4]{Universit\'{e} Libre de Bruxelles, Precision Mechatronics Laboratory, BEAMS Department, 1050 Brussels, Belgium.}
 
-#+LATEX_HEADER: \usepackage[utf8]{inputenc}
+#+LATEX_HEADER: \usepackage{amsfonts}
-#+LATEX_HEADER: \usepackage[T1]{fontenc}
+#+LATEX_HEADER: \usepackage{siunitx}
-#+LATEX_HEADER: \usepackage{graphicx}
+#+LATEX_HEADER_EXTRA: \usepackage{tabularx}
-#+LATEX_HEADER: \usepackage{grffile}
+#+LATEX_HEADER_EXTRA: \usepackage{booktabs}
-#+LATEX_HEADER: \usepackage{rotating}
+#+LATEX_HEADER_EXTRA: \usepackage{array}
-#+LATEX_HEADER: \usepackage[normalem]{ulem}
-#+LATEX_HEADER: \usepackage{capt-of}
-#+LATEX_HEADER: \usepackage{hyperref}
-#+LATEX_HEADER: \usepackage{bm}
-#+LATEX_HEADER: \usepackage{array}
-#+LATEX_HEADER: \usepackage{amsmath,amssymb,amsfonts}
-#+LATEX_HEADER: \usepackage{algorithmic}
-#+LATEX_HEADER: \usepackage{textcomp}
-#+LATEX_HEADER: \usepackage{cases}
-#+LATEX_HEADER: \usepackage{tabularx,siunitx,booktabs}
-#+LATEX_HEADER: \usepackage{algorithmic}
-#+LATEX_HEADER: \usepackage{import}
-#+LATEX_HEADER_EXTRA: \usepackage{hyperref}
 #+LATEX_HEADER_EXTRA: \usepackage[hyperref]{xcolor}
-#+LATEX_HEADER_EXTRA: \hypersetup{colorlinks=true}
 #+LATEX_HEADER_EXTRA: \usepackage[top=2cm, bottom=2cm, left=2cm, right=2cm]{geometry}
+#+LATEX_HEADER_EXTRA: \hypersetup{colorlinks=true}
 :END:
 
 * Build                                                            :noexport:
@@ -267,6 +254,12 @@ Finally, concluding remarks are presented in Section [[*Concluding remarks][5]].
 
 * Complementary Filters Requirements
 <<sec:requirements>>
+** Sensor Models
+<<sec:sensor_models>>
+
+- Noise + dynamical uncertainty
+- Suppose we calibrate the sensors
+
 ** Sensor Fusion Architecture
 <<sec:sensor_fusion>>
 
@@ -370,8 +363,8 @@ Thus the norm of the complementary filter $|H_i|$ should be made small at freque
 As shown in Sec. ref:sec:requirements, the performance and robustness of the sensor fusion architecture depends on the complementary filters norms.
 Therefore, the development of a synthesis method of complementary filters that allows the shaping of their norm is necessary.
 
-** Shaping of Complementary Filters using $\mathcal{H}_\infty$ synthesis
+** Synthesis Objective
-<<sec:hinf_synthesis>>
+<<sec:synthesis_objective>>
 The synthesis objective is to shape the norm of two filters $H_1(s)$ and $H_2(s)$ while ensuring their complementary property eqref:eq:comp_filter.
 This is equivalent as to finding stable transfer functions $H_1(s)$ and $H_2(s)$ such that conditions eqref:eq:comp_filter_problem_form are satisfied.
 #+name: eq:comp_filter_problem_form
@@ -384,6 +377,8 @@ This is equivalent as to finding stable transfer functions $H_1(s)$ and $H_2(s)$
 \end{subequations}
 where $W_1(s)$ and $W_2(s)$ are two weighting transfer functions that are chosen to shape the norms of the corresponding filters.
 
+** Shaping of Complementary Filters using $\mathcal{H}_\infty$ synthesis
+<<sec:hinf_synthesis>>
 In order to express this optimization problem as a standard $\mathcal{H}_\infty$ problem, the architecture shown in Fig. ref:fig:h_infinity_robust_fusion is used where the generalized plant $P$ is described by eqref:eq:generalized_plant.
 #+name: eq:generalized_plant
 \begin{equation}
@@ -492,49 +487,6 @@ The bode plots of the obtained complementary filters are shown in Fig. ref:fig:h
 #+attr_latex: :scale 1
 [[file:figs/hinf_synthesis_results.pdf]]
 
-** Synthesis of Three Complementary Filters
-<<sec:hinf_three_comp_filters>>
-
-*** Why it is used sometimes                                       :ignore:
-Some applications may require to merge more than two sensors.
-In such a case, it is necessary to design as many complementary filters as the number of sensors used.
-
-*** Mathematical Problem                                           :ignore:
-The synthesis problem is then to compute $n$ stable transfer functions $H_i(s)$ such that eqref:eq:hinf_problem_gen is satisfied.
-#+name: eq:hinf_problem_gen
-\begin{subequations}
-  \begin{align}
-  & \sum_{i=0}^n H_i(s) = 1 \label{eq:hinf_cond_compl_gen} \\
-  & \left| H_i(j\omega) \right| < \frac{1}{\left| W_i(j\omega) \right|}, \quad \forall \omega,\ i = 1 \dots n \label{eq:hinf_cond_perf_gen}
-  \end{align}
-\end{subequations}
-
-*** H-Infinity Architecture                                        :ignore:
-The synthesis method is generalized here for the synthesis of three complementary filters using the architecture shown in Fig. ref:fig:comp_filter_three_hinf.
-
-The $\mathcal{H}_\infty$ synthesis objective applied on $P(s)$ is to design two stable filters $H_2(s)$ and $H_3(s)$ such that the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $[z_1,\ z_2, \ z_3]$ is less than one eqref:eq:hinf_syn_obj_three.
-#+name: eq:hinf_syn_obj_three
-\begin{equation}
-  \left\| \begin{matrix} \left[1 - H_2(s) - H_3(s)\right] W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1
-\end{equation}
-
-#+name: fig:comp_filter_three_hinf
-#+caption: Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters
-#+attr_latex: :scale 1
-[[file:figs/comp_filter_three_hinf.pdf]]
-
-By choosing $H_1(s) \triangleq 1 - H_2(s) - H_3(s)$, the proposed $\mathcal{H}_\infty$ synthesis solves the design problem eqref:eq:hinf_problem_gen. \par
-
-*** Example of generated complementary filters                     :ignore:
-An example is given to validate the method where three sensors are used in different frequency bands (up to $\SI{1}{Hz}$, from $1$ to $\SI{10}{Hz}$ and above $\SI{10}{Hz}$ respectively).
-Three weighting functions are designed using eqref:eq:weight_formula and shown by dashed curves in Fig. ref:fig:hinf_three_synthesis_results.
-The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_three_synthesis_results.
-
-#+name: fig:hinf_three_synthesis_results
-#+caption: Frequency response of the weighting functions and three complementary filters obtained using $\mathcal{H}_\infty$ synthesis
-#+attr_latex: :scale 1
-[[file:figs/hinf_three_synthesis_results.pdf]]
-
 * Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO
 <<sec:application_ligo>>
 ** Introduction                                                     :ignore:
@@ -579,6 +531,61 @@ They are found to be very close to each other and this shows the effectiveness o
 #+attr_latex: :scale 1
 [[file:figs/comp_fir_ligo_hinf.pdf]]
 
+* Discussion                                                        :noexport:
+** Alternative configuration
+- Feedback architecture : Similar to mixed sensitivity
+- 2 inputs / 1 output
+
+Explain differences
+
+** Imposing zero at origin / roll-off
+3 methods:
+
+Link to literature about doing that with mixed sensitivity
+
+** Synthesis of Three Complementary Filters
+<<sec:hinf_three_comp_filters>>
+
+*** Why it is used sometimes                                       :ignore:
+Some applications may require to merge more than two sensors.
+In such a case, it is necessary to design as many complementary filters as the number of sensors used.
+
+*** Mathematical Problem                                           :ignore:
+The synthesis problem is then to compute $n$ stable transfer functions $H_i(s)$ such that eqref:eq:hinf_problem_gen is satisfied.
+#+name: eq:hinf_problem_gen
+\begin{subequations}
+  \begin{align}
+  & \sum_{i=0}^n H_i(s) = 1 \label{eq:hinf_cond_compl_gen} \\
+  & \left| H_i(j\omega) \right| < \frac{1}{\left| W_i(j\omega) \right|}, \quad \forall \omega,\ i = 1 \dots n \label{eq:hinf_cond_perf_gen}
+  \end{align}
+\end{subequations}
+
+*** H-Infinity Architecture                                        :ignore:
+The synthesis method is generalized here for the synthesis of three complementary filters using the architecture shown in Fig. ref:fig:comp_filter_three_hinf.
+
+The $\mathcal{H}_\infty$ synthesis objective applied on $P(s)$ is to design two stable filters $H_2(s)$ and $H_3(s)$ such that the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $[z_1,\ z_2, \ z_3]$ is less than one eqref:eq:hinf_syn_obj_three.
+#+name: eq:hinf_syn_obj_three
+\begin{equation}
+  \left\| \begin{matrix} \left[1 - H_2(s) - H_3(s)\right] W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1
+\end{equation}
+
+#+name: fig:comp_filter_three_hinf
+#+caption: Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters
+#+attr_latex: :scale 1
+[[file:figs/comp_filter_three_hinf.pdf]]
+
+By choosing $H_1(s) \triangleq 1 - H_2(s) - H_3(s)$, the proposed $\mathcal{H}_\infty$ synthesis solves the design problem eqref:eq:hinf_problem_gen. \par
+
+*** Example of generated complementary filters                     :ignore:
+An example is given to validate the method where three sensors are used in different frequency bands (up to $\SI{1}{Hz}$, from $1$ to $\SI{10}{Hz}$ and above $\SI{10}{Hz}$ respectively).
+Three weighting functions are designed using eqref:eq:weight_formula and shown by dashed curves in Fig. ref:fig:hinf_three_synthesis_results.
+The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_three_synthesis_results.
+
+#+name: fig:hinf_three_synthesis_results
+#+caption: Frequency response of the weighting functions and three complementary filters obtained using $\mathcal{H}_\infty$ synthesis
+#+attr_latex: :scale 1
+[[file:figs/hinf_three_synthesis_results.pdf]]
+
 * Conclusion
 <<sec:conclusion>>
 This paper has shown how complementary filters can be used to combine multiple sensors in order to obtain a super sensor.
@@ -595,3 +602,9 @@ This research benefited from a FRIA grant from the French Community of Belgium.
 * Bibliography                                                       :ignore:
 \bibliographystyle{elsarticle-num}
 \bibliography{ref}
+
+
+* Local Variables                                                   :noexport:
+# Local Variables:
+# org-latex-packages-alist: nil
+# End: