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Thomas Dehaeze
2021-04-29 16:55:38 +02:00
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@@ -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.}
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: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: