Change packages
This commit is contained in:
parent
d9c3c3d2dd
commit
4e72ca326d
@ -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
|
||||
<<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
|
||||
<<sec:hinf_synthesis>>
|
||||
** Synthesis Objective
|
||||
<<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:
|
||||
|
||||
Loading…
Reference in New Issue
Block a user