diff --git a/journal/journal.org b/journal/journal.org index b1a0ddf..a24bcd2 100644 --- a/journal/journal.org +++ b/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[T1]{fontenc} -#+LATEX_HEADER: \usepackage{graphicx} -#+LATEX_HEADER: \usepackage{grffile} -#+LATEX_HEADER: \usepackage{rotating} -#+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: \usepackage{amsfonts} +#+LATEX_HEADER: \usepackage{siunitx} +#+LATEX_HEADER_EXTRA: \usepackage{tabularx} +#+LATEX_HEADER_EXTRA: \usepackage{booktabs} +#+LATEX_HEADER_EXTRA: \usepackage{array} #+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 <> +** Sensor Models +<> + +- Noise + dynamical uncertainty +- Suppose we calibrate the sensors + ** Sensor Fusion Architecture <> @@ -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 +<> 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 +<> 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 -<> - -*** 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 <> ** 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 +<> + +*** 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 <> 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: