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\journal{Mechanical Systems and Signal Processing}
\author[a1,a2]{Thomas Dehaeze\corref{cor1}}
\author[a3,a4]{Mohit Verma}
\author[a2,a4]{Christophe Collette}
\cortext[cor1]{Corresponding author. Email Address: dehaeze.thomas@gmail.com}
\address[a1]{European Synchrotron Radiation Facility, Grenoble, France}
\address[a2]{University of Li\`{e}ge, Department of Aerospace and Mechanical Engineering, 4000 Li\`{e}ge, Belgium.}
\address[a3]{CSIR --- Structural Engineering Research Centre, Taramani, Chennai --- 600113, India.}
\address[a4]{Universit\'{e} Libre de Bruxelles, Precision Mechatronics Laboratory, BEAMS Department, 1050 Brussels, Belgium.}
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\date{}
\title{A new method of designing complementary filters for sensor fusion using \(\mathcal{H}_\infty\) synthesis}
\begin{document}


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\begin{frontmatter}
\begin{abstract}
Sensor have limited bandwidth and are accurate only in a certain frequency band.
In many applications, the signals of different sensor are fused together in order to either enhance the stability or improve the operational bandwidth of the system.
The sensor signals can be fused using complementary filters.
The tuning of complementary filters is a complex task and is the subject of this paper.
The filters needs to meet design specifications while satisfying the complementary property.
This paper presents a framework to shape the norm of complementary filters using the \(\mathcal{H}_\infty\) norm minimization.
The design specifications are imposed as constraints in the optimization problem by appropriate selection of weighting functions.
The proposed method is quite general and easily extendable to cases where more than two sensors are fused.
Finally, the proposed method is applied to the design of complementary filter design for active vibration isolation of the Laser Interferometer Gravitation-wave Observatory (LIGO).
\end{abstract}

\begin{keyword}
Sensor fusion \sep{} Optimal filters \sep{} \(\mathcal{H}_\infty\) synthesis \sep{} Vibration isolation \sep{} Precision
\end{keyword}
\end{frontmatter}

\section{Introduction}
\label{sec:org188c07e}
\label{sec:introduction}
\begin{itemize}
\item \cite{bendat57_optim_filter_indep_measur_two} roots of sensor fusion
\end{itemize}
\begin{itemize}
\item Increase the bandwidth: \cite{zimmermann92_high_bandw_orien_measur_contr}
\item Increased robustness: \cite{collette15_sensor_fusion_method_high_perfor}
\item Decrease the noise:
\end{itemize}
\begin{itemize}
\item UAV: \cite{pascoal99_navig_system_desig_using_time}, \cite{jensen13_basic_uas}
\item Gravitational wave observer: \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}
\end{itemize}
\begin{itemize}
\item \cite{brown72_integ_navig_system_kalman_filter} alternate form of complementary filters => Kalman filtering
\item \cite{higgins75_compar_compl_kalman_filter} Compare Kalman Filtering with sensor fusion using complementary filters
\item \cite{robert12_introd_random_signal_applied_kalman} advantage of complementary filters over Kalman filtering
\end{itemize}
\begin{itemize}
\item Analog complementary filters: \cite{yong16_high_speed_vertic_posit_stage}, \cite{moore19_capac_instr_sensor_fusion_high_bandw_nanop}

\item Analytical methods:
\begin{itemize}
\item first order: \cite{corke04_inert_visual_sensin_system_small_auton_helic}
\item second order: \cite{baerveldt97_low_cost_low_weigh_attit}, \cite{stoten01_fusion_kinet_data_using_compos_filter}, \cite{jensen13_basic_uas}
\item higher order: \cite{shaw90_bandw_enhan_posit_measur_using_measur_accel}, \cite{zimmermann92_high_bandw_orien_measur_contr}, \cite{collette15_sensor_fusion_method_high_perfor}, \cite{matichard15_seism_isolat_advan_ligo}
\end{itemize}
\item \cite{pascoal99_navig_system_desig_using_time} use LMI to generate complementary filters
\item \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}: FIR + convex optimization
\item Similar to feedback system:
\begin{itemize}
\item \cite{plummer06_optim_compl_filter_their_applic_motion_measur} use H-Infinity to optimize complementary filters (flatten the super sensor noise spectral density)
\item \cite{jensen13_basic_uas} design of complementary filters with classical control theory, PID
\end{itemize}

\item 3 complementary filters: \cite{becker15_compl_filter_desig_three_frequen_bands}
\end{itemize}
\begin{itemize}
\item Robustness problems: \cite{zimmermann92_high_bandw_orien_measur_contr} change of phase near the merging frequency
\item Trial and error
\item Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to shape the norm of the complementary filters is available.
\end{itemize}
Most of the requirements => shape of the complementary filters
=> propose a way to shape complementary filters.

\section{Sensor Fusion and Complementary Filters Requirements}
\label{sec:org99f43ee}
\label{sec:requirements}
Complementary filters provides a framework for fusing signals from different sensors.
As the effectiveness of the fusion depends on the proper design of the complementary filters, they are expected to fulfill certain requirements.
These requirements are discussed in this section.
\subsection{Sensor Fusion Architecture}
\label{sec:orgec9e73a}
\label{sec:sensor_fusion}

A general sensor fusion architecture using complementary filters is shown in Figure \ref{fig:sensor_fusion_overview} where several sensors (here two) are measuring the same physical quantity \(x\).
The two sensors output signals are estimates \(\hat{x}_1\) and \(\hat{x}_2\) of \(x\).
Each of these estimates are then filtered out by complementary filters and combined to form a new estimate \(\hat{x}\).

The resulting sensor, termed as ``super sensor'', can have larger bandwidth and better noise characteristics in comparison to the individual sensor.
This means that the super sensor provides an estimate \(\hat{x}\) of \(x\) which can be more accurate over a larger frequency band than the outputs of the individual sensors.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/sensor_fusion_overview.pdf}
\caption{\label{fig:sensor_fusion_overview}Schematic of a sensor fusion architecture}
\end{figure}

The complementary property of filters \(H_1(s)\) and \(H_2(s)\) implies that the summation of their transfer functions is equal to unity.
That is, unity magnitude and zero phase at all frequencies.
Therefore, a pair of strict complementary filter needs to satisfy the following condition:
\begin{equation}
\label{eq:comp_filter}
  H_1(s) + H_2(s) = 1
\end{equation}

It will soon become clear why the complementary property is important.

\subsection{Sensor Models and Sensor Normalization}
\label{sec:org9538be3}
\label{sec:sensor_models}

In order to study such sensor fusion architecture, a model of the sensors is required.

Such model is shown in Figure \ref{fig:sensor_model} and consists of a linear time invariant (LTI) system \(G_i(s)\) representing the dynamics of the sensor and an additive noise input \(n_i\) representing its noise.
The model input \(x\) is the measured physical quantity and its output \(\tilde{x}_i\) is the ``raw'' output of the sensor.

Before filtering the sensor outputs \(\tilde{x}_i\) by the complementary filters, the sensors are usually normalized to simplify the fusion.
This normalization consists of first obtaining an estimate \(\hat{G}_i(s)\) of the sensor dynamics \(G_i(s)\).
It is supposed that the estimate of the sensor dynamics \(\hat{G}_i(s)\) can be inverted and that its inverse \(\hat{G}_i^{-1}(s)\) is proper and stable.
The raw output of the sensor \(\tilde{x}_i\) is then passed through \(\hat{G}_i^{-1}(s)\) as shown in Figure \ref{fig:sensor_model_calibrated}.
This way, the units of the estimates \(\hat{x}_i\) are equal to the units of the physical quantity \(x\).
The sensor dynamics estimate \(\hat{G}_1(s)\) can be a simple gain or more complex transfer functions.

\begin{figure}[htbp]
\begin{subfigure}[b]{0.49\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_model.pdf}
\caption{\label{fig:sensor_model} Basic sensor model consisting of a noise input $n_i$ and a dynamics $G_i(s)$}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_model_calibrated.pdf}
\caption{\label{fig:sensor_model_calibrated} Calibrated sensors using the inverse of an estimate $\hat{G}_1(s)$ of the sensor dynamics}
\end{subfigure}
\caption{\label{fig:figure_name}Sensor models with an without normalization}
\centering
\end{figure}

Two calibrated sensors and then combined to form a super sensor as shown in Figure \ref{fig:fusion_super_sensor}.

The two sensors are measuring the same physical quantity \(x\) with dynamics \(G_1(s)\) and \(G_2(s)\), and with \emph{uncorrelated} noises \(n_1\) and \(n_2\).
The normalized signals from both calibrated sensors are fed into two complementary filters \(H_1(s)\) and \(H_2(s)\) and then combined to yield an estimate \(\hat{x}\) of \(x\) as shown in Fig. \ref{fig:fusion_super_sensor}.

The super sensor output is therefore equal to:
\begin{equation}
\label{eq:comp_filter_estimate}
  \hat{x} = \Big( H_1(s) \hat{G}_1^{-1}(s) G_1(s) + H_2(s) \hat{G}_2^{-1}(s) G_2(s) \Big) x + H_1(s) \hat{G}_1^{-1}(s) G_1(s) n_1 + H_2(s) \hat{G}_2^{-1}(s) G_2(s) n_2
\end{equation}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/fusion_super_sensor.pdf}
\caption{\label{fig:fusion_super_sensor}Sensor fusion architecture}
\end{figure}

\subsection{Noise Sensor Filtering}
\label{sec:orgb03f925}
\label{sec:noise_filtering}

In this section, it is supposed that all the sensors are perfectly calibrated, such that:
\begin{equation}
\label{eq:perfect_dynamics}
  \frac{\hat{x}_i}{x} = \hat{G}_i(s) G_i(s) = 1
\end{equation}

The effect of a non-perfect normalization will be discussed in the next section.

The super sensor output \(\hat{x}\) is then:
\begin{equation}
\label{eq:estimate_perfect_dyn}
  \hat{x} = x + H_1(s) n_1 + H_2(s) n_2
\end{equation}

From \eqref{eq:estimate_perfect_dyn}, the complementary filters \(H_1(s)\) and \(H_2(s)\) are shown to only operate on the sensor's noises.
Thus, this sensor fusion architecture permits to filter the noise of both sensors without introducing any distortion in the physical quantity to be measured.

The estimation error \(\delta x\), defined as the difference between the sensor output \(\hat{x}\) and the measured quantity \(x\), is computed for the super sensor \eqref{eq:estimate_error}.
\begin{equation}
\label{eq:estimate_error}
  \delta x \triangleq \hat{x} - x = H_1(s) n_1 + H_2(s) n_2
\end{equation}

As shown in \eqref{eq:noise_filtering_psd}, the Power Spectral Density (PSD) of the estimation error \(\Phi_{\delta x}\) depends both on the norm of the two complementary filters and on the PSD of the noise sources \(\Phi_{n_1}\) and \(\Phi_{n_2}\).
\begin{equation}
\label{eq:noise_filtering_psd}
  \Phi_{\delta x}(\omega) = \left|H_1(j\omega)\right|^2 \Phi_{n_1}(\omega) + \left|H_2(j\omega)\right|^2 \Phi_{n_2}(\omega)
\end{equation}

If the two sensors have identical noise characteristics (\(\Phi_{n_1}(\omega) = \Phi_{n_2}(\omega)\)), a simple averaging (\(H_1(s) = H_2(s) = 0.5\)) is what would minimize the super sensor noise.
This the simplest form of sensor fusion with complementary filters.

However, the two sensors have usually high noise levels over distinct frequency regions.
In such case, to lower the noise of the super sensor, the value of the norm \(|H_1|\) has to be lowered when \(\Phi_{n_1}\) is larger than \(\Phi_{n_2}\) and that of \(|H_2|\) lowered when \(\Phi_{n_2}\) is larger than \(\Phi_{n_1}\).
Therefore, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor noise.

\subsection{Sensor Fusion Robustness}
\label{sec:orgfb0ea88}
\label{sec:fusion_robustness}

In practical systems the sensor normalization is not perfect and condition \eqref{eq:perfect_dynamics} is not verified.

In order to study such imperfection, a multiplicative input uncertainty is added to the sensor dynamics (Figure \ref{fig:sensor_model_uncertainty}), where the nominal model is taken as the estimated model for the normalization \(\hat{G}_i(s)\), \(\Delta_i\) is any stable transfer function satisfying \(|\Delta_i(j\omega)| \le 1,\ \forall\omega\), and \(w_i(s)\) is a weight representing the magnitude of the uncertainty.

The weight \(w_i(s)\) is chosen such that the real sensor dynamics is always contained in the uncertain region represented by a circle centered on \(1\) and with a radius equal to \(|w_i(j\omega)|\).

As the nominal sensor dynamics is taken as the normalized filter, the normalized sensor can be further simplified as shown in Figure \ref{fig:sensor_model_uncertainty_simplified}.

\begin{figure}[htbp]
\begin{subfigure}[b]{0.59\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_model_uncertainty.pdf}
\caption{\label{fig:sensor_model_uncertainty} Sensor with multiplicative input uncertainty}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.39\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_model_uncertainty_simplified.pdf}
\caption{\label{fig:sensor_model_uncertainty_simplified} Simplified sensor model}
\end{subfigure}
\hfill
\caption{\label{fig:sensor_models_uncertainty}Sensor models with dynamical uncertainty}
\centering
\end{figure}

A sensor fusion architecture with two sensors with dynamical uncertainty is shown in Figure \ref{fig:sensor_fusion_dynamic_uncertainty}.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/sensor_fusion_dynamic_uncertainty.pdf}
\caption{\label{fig:sensor_fusion_dynamic_uncertainty}Sensor fusion architecture with sensor dynamics uncertainty}
\end{figure}

The super sensor dynamics \eqref{eq:super_sensor_dyn_uncertainty} is no longer equal to \(1\) and now depends on the sensor dynamics uncertainty weights \(w_i(s)\) as well as on the complementary filters \(H_i(s)\).
\begin{equation}
\label{eq:super_sensor_dyn_uncertainty}
  \frac{\hat{x}}{x} = 1 + w_1(s) H_1(s) \Delta_1(s) + w_2(s) H_2(s) \Delta_2(s)
\end{equation}

The dynamical uncertainty of the super sensor can be graphically represented in the complex plane by a circle centered on \(1\) 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}.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/uncertainty_set_super_sensor.pdf}
\caption{\label{fig:uncertainty_set_super_sensor}Uncertainty region of the super sensor dynamics in the complex plane (solid circle). The contribution of both sensors 1 and 2 to the uncertainty are represented respectively by a blue circle and a red circle. The frequency dependency \(\omega\) is here omitted.}
\end{figure}

The super sensor dynamical uncertainty (i.e. the robustness of the fusion) clearly depends on the complementary filters norms.
For instance, the phase uncertainty \(\Delta\phi(\omega)\) added by the super sensor dynamics at frequency \(\omega\) can be found by drawing a tangent from the origin to the uncertainty circle of super sensor (Figure \ref{fig:uncertainty_set_super_sensor}) and is bounded by \eqref{eq:max_phase_uncertainty}.

\begin{equation}
\label{eq:max_phase_uncertainty}
    \Delta\phi(\omega) < \arcsin\big( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \big)
\end{equation}

As it is generally desired to limit the maximum phase added by the super sensor, \(H_1(s)\) and \(H_2(s)\) should be designed such that \(\Delta \phi\) is bounded to acceptable values.
Typically, the norm of the complementary filter \(|H_i(j\omega)|\) should be made small when \(|w_i(j\omega)|\) is large, i.e., at frequencies where the sensor dynamics is uncertain.

\section{Complementary Filters Shaping using \(\mathcal{H}_\infty\) Synthesis}
\label{sec:orgfccc360}
\label{sec:hinf_method}
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.
\subsection{Synthesis Objective}
\label{sec:orga79128d}
\label{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.
\begin{subequations}
\label{eq:comp_filter_problem_form}
  \begin{align}
  & H_1(s) + H_2(s) = 1 \label{eq:hinf_cond_complementarity} \\
  & |H_1(j\omega)| \le \frac{1}{|W_1(j\omega)|} \quad \forall\omega \label{eq:hinf_cond_h1} \\
  & |H_2(j\omega)| \le \frac{1}{|W_2(j\omega)|} \quad \forall\omega \label{eq:hinf_cond_h2}
  \end{align}
\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.

\subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
\label{sec:org91451ed}
\label{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}.
\begin{equation}
\label{eq:generalized_plant}
  \begin{bmatrix} z_1 \\ z_2 \\ v \end{bmatrix} = P(s) \begin{bmatrix} w\\u \end{bmatrix}; \quad P(s) = \begin{bmatrix}W_1(s) & -W_1(s) \\ 0 & W_2(s) \\  1 & 0 \end{bmatrix}
\end{equation}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/h_infinity_robust_fusion.pdf}
\caption{\label{fig:h_infinity_robust_fusion}Architecture used for \(\mathcal{H}_\infty\) synthesis of complementary filters}
\end{figure}

The \(\mathcal{H}_\infty\) filter design problem is then to find a stable filter \(H_2(s)\) which based on \(v\), generates a signal \(u\) such that the \(\mathcal{H}_\infty\) norm from \(w\) to \([z_1, \ z_2]\) is less than one \eqref{eq:hinf_syn_obj}.
\begin{equation}
\label{eq:hinf_syn_obj}
  \left\|\begin{matrix} \left[1 - H_2(s)\right] W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1
\end{equation}

This is equivalent to having \eqref{eq:hinf_problem} by defining \(H_1(s)\) as the complementary filter of \(H_2(s)\) \eqref{eq:definition_H1}.
\begin{equation}
\label{eq:hinf_problem}
  \left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1
\end{equation}

\begin{equation}
\label{eq:definition_H1}
  H_1(s) \triangleq 1 - H_2(s)
\end{equation}

The complementary condition \eqref{eq:hinf_cond_complementarity} is ensured by \eqref{eq:definition_H1}.
The conditions \eqref{eq:hinf_cond_h1} and \eqref{eq:hinf_cond_h2} on the filters shapes are satisfied by \eqref{eq:hinf_problem}.
Therefore, all the conditions \eqref{eq:comp_filter_problem_form} are satisfied using this synthesis method based on \(\mathcal{H}_\infty\) synthesis, and thus it permits to shape complementary filters as desired.

\subsection{Weighting Functions Design}
\label{sec:orge5f38aa}
\label{sec:hinf_weighting_func}
The proper design of the weighting functions is of primary importance for the success of the presented complementary filters \(\mathcal{H}_\infty\) synthesis.

First, only proper, stable and minimum phase transfer functions should be used.
Second, the order of the weights should stay reasonably small in order to reduce the computational costs associated with the solving of the optimization problem and for the physical implementation of the filters (the order of the synthesized filters being equal to the sum of the weighting functions order).
Third, one should not forget the fundamental limitations imposed by the complementary property \eqref{eq:comp_filter}.
This implies for instance that \(|H_1(j\omega)|\) and \(|H_2(j\omega)|\) cannot be made small at the same time.

When designing complementary filters, it is usually desired to specify the slope of the filter, its crossover frequency and its gain at low and high frequency.
To help with the design of the weighting functions such that the above specification can be easily expressed, the following formula is proposed.
\begin{equation}
\label{eq:weight_formula}
  W(s) = \left( \frac{
           \hfill{} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}
         }{
           \left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_0} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}}
         }\right)^n
\end{equation}
The parameters permit to specify:
\begin{itemize}
\item the low frequency gain: \(G_0 = lim_{\omega \to 0} |W(j\omega)|\)
\item the high frequency gain: \(G_\infty = lim_{\omega \to \infty} |W(j\omega)|\)
\item the absolute gain at \(\omega_0\): \(G_c = |W(j\omega_0)|\)
\item the absolute slope between high and low frequency: \(n\)
\end{itemize}

The parameters \(G_0\), \(G_c\) and \(G_\infty\) should either satisfy condition \eqref{eq:cond_formula_1} or \eqref{eq:cond_formula_2}.
\begin{subequations}
\label{eq:condition_params_formula}
  \begin{align}
    G_0 < 1 < G_\infty \text{ and } G_0 < G_c < G_\infty \label{eq:cond_formula_1}\\
    G_\infty < 1 < G_0 \text{ and } G_\infty < G_c < G_0 \label{eq:cond_formula_2}
  \end{align}
\end{subequations}

The general shape of a weighting function generated using \eqref{eq:weight_formula} is shown in Fig. \ref{fig:weight_formula}.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/weight_formula.pdf}
\caption{\label{fig:weight_formula}Magnitude of a weighting function generated using the proposed formula \eqref{eq:weight_formula}, \(G_0 = 1e^{-3}\), \(G_\infty = 10\), \(\omega_c = \SI{10}{Hz}\), \(G_c = 2\), \(n = 3\)}
\end{figure}

\subsection{Validation of the proposed synthesis method}
\label{sec:org0477d6e}
\label{sec:hinf_example}
Let's validate the proposed design method of complementary filters with a simple example where two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that:
\begin{itemize}
\item the merging frequency is around \(\SI{10}{Hz}\)
\item the slope of \(|H_1(j\omega)|\) is \(-2\) above \(\SI{10}{Hz}\)
\item the slope of \(|H_2(j\omega)|\) is \(+3\) below \(\SI{10}{Hz}\)
\item the gain of both filters is equal to \(10^{-3}\) away from the merging frequency
\end{itemize}

The weighting functions \(W_1(s)\) and \(W_2(s)\) are designed using \eqref{eq:weight_formula}.
The parameters used are summarized in table \ref{tab:weights_params} and the magnitude of the weighting functions is shown in Fig. \ref{fig:hinf_filters_results}.

\begin{table}[htbp]
\caption{\label{tab:weights_params}Parameters used for \(W_1(s)\) and \(W_2(s)\)}
\centering
\begin{tabularx}{0.5\linewidth}{Xcc}
\toprule
Parameter & \(W_1(s)\) & \(W_2(s)\)\\
\midrule
\(G_0\) & \(0.1\) & \(1000\)\\
\(G_\infty\) & \(1000\) & \(0.1\)\\
\(\omega_c\) [\(\si{Hz}\)] & \(11\) & \(10\)\\
\(G_c\) & \(0.5\) & \(0.5\)\\
\(n\) & \(2\) & \(3\)\\
\bottomrule
\end{tabularx}
\end{table}

The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:hinf_filters_results} and their transfer functions in the Laplace domain are given below.
\begin{align*}
  H_1(s) &= \frac{10^{-8} (s+6.6e^9) (s+3450)^2 (s^2 + 49s + 895)}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}\\
  H_2(s) &= \frac{(s+6.6e^4) (s+160) (s+4)^3}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}
\end{align*}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/hinf_filters_results.pdf}
\caption{\label{fig:hinf_filters_results}Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis}
\end{figure}

\section{Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO}
\label{sec:org70c1567}
\label{sec:application_ligo}
Several complementary filters are used in the active isolation system at the LIGO \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}.
The requirements on those filters are very tight and thus their design is complex.
The approach used in \cite{hua05_low_ligo} for their design is to write the synthesis of complementary FIR filters as a convex optimization problem.
The obtained FIR filters are compliant with the requirements. However they are of very high order so their implementation is quite complex.

The effectiveness of the proposed method is demonstrated by designing complementary filters with the same requirements as the one described in \cite{hua05_low_ligo}.
\subsection{Complementary Filters Specifications}
\label{sec:orgdfbd1f2}
\label{sec:ligo_specifications}
The specifications for one pair of complementary filters used at the LIGO are summarized below (for further details, refer to \cite{hua04_polyp_fir_compl_filter_contr_system}) and shown in Fig. \ref{fig:ligo_weights}:
\begin{itemize}
\item From \(0\) to \(\SI{0.008}{Hz}\), the magnitude of the filter's transfer function should be less or equal to \(8 \times 10^{-4}\)
\item Between \(\SI{0.008}{Hz}\) to \(\SI{0.04}{Hz}\), the filter should attenuate the input signal proportional to frequency cubed
\item Between \(\SI{0.04}{Hz}\) to \(\SI{0.1}{Hz}\), the magnitude of the transfer function should be less than \(3\)
\item Above \(\SI{0.1}{Hz}\), the magnitude of the complementary filter should be less than \(0.045\)
\end{itemize}

\subsection{Weighting Functions Design}
\label{sec:orgf9892b6}
\label{sec:ligo_weights}
The weighting functions should be designed such that their inverse magnitude is as close as possible to the specifications in order to not over-constrain the synthesis problem.
However, the order of each weight should stay reasonably small in order to reduce the computational costs of the optimization problem as well as for the physical implementation of the filters.

A Type I Chebyshev filter of order \(20\) is used as the weighting transfer function \(w_L(s)\) corresponding to the low pass filter.
For the one corresponding to the high pass filter \(w_H(s)\), a \(7^{\text{th}}\) order transfer function is designed.
The magnitudes of the weighting functions are shown in Fig. \ref{fig:ligo_weights}.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/ligo_weights.pdf}
\caption{\label{fig:ligo_weights}Specifications and weighting functions magnitudes}
\end{figure}

\subsection{\(\mathcal{H}_\infty\) Synthesis}
\label{sec:orge086b06}
\label{sec:ligo_results}
\(\mathcal{H}_\infty\) synthesis is performed using the architecture shown in Fig. \ref{eq:generalized_plant}.
The complementary filters obtained are of order \(27\).
In Fig. \ref{fig:comp_fir_ligo_hinf}, their bode plot is compared with the FIR filters of order 512 obtained in \cite{hua05_low_ligo}.
They are found to be very close to each other and this shows the effectiveness of the proposed synthesis method.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/comp_fir_ligo_hinf.pdf}
\caption{\label{fig:comp_fir_ligo_hinf}Comparison of the FIR filters (solid) designed in \cite{hua05_low_ligo} with the filters obtained with \(\mathcal{H}_\infty\) synthesis (dashed)}
\end{figure}

\section{Discussion}
\label{sec:org7b7d598}
\subsection{Alternative configuration}
\label{sec:org56a1607}
\begin{itemize}
\item Feedback architecture : Similar to mixed sensitivity
\item 2 inputs / 1 output
\end{itemize}

Explain differences

\subsection{Imposing zero at origin / roll-off}
\label{sec:org8da9d79}
3 methods:

Link to literature about doing that with mixed sensitivity

\subsection{Synthesis of Three Complementary Filters}
\label{sec:orgefead29}
\label{sec:hinf_three_comp_filters}
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.
The synthesis problem is then to compute \(n\) stable transfer functions \(H_i(s)\) such that \eqref{eq:hinf_problem_gen} is satisfied.
\begin{subequations}
\label{eq:hinf_problem_gen}
  \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}
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}.
\begin{equation}
\label{eq:hinf_syn_obj_three}
  \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}

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/comp_filter_three_hinf.pdf}
\caption{\label{fig:comp_filter_three_hinf}Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters}
\end{figure}

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
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:three_complementary_filters_results}.
The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:three_complementary_filters_results}.

\begin{figure}[htbp]
\centering
\includegraphics[scale=1,scale=1]{figs/three_complementary_filters_results.pdf}
\caption{\label{fig:three_complementary_filters_results}Frequency response of the weighting functions and three complementary filters obtained using \(\mathcal{H}_\infty\) synthesis}
\end{figure}

\section{Conclusion}
\label{sec:org2e6ce14}
\label{sec:conclusion}
This paper has shown how complementary filters can be used to combine multiple sensors in order to obtain a super sensor.
Typical specification on the super sensor noise and on the robustness of the sensor fusion has been shown to be linked to the norm of the complementary filters.
Therefore, a synthesis method that permits the shaping of the complementary filters norms has been proposed and has been successfully applied for the design of complex filters.
Future work will aim at further developing this synthesis method for the robust and optimal synthesis of complementary filters used in sensor fusion.

\section*{Acknowledgment}
\label{sec:orgde5a128}
This research benefited from a FRIA grant from the French Community of Belgium.

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