#+TITLE: A new method of designing complementary filters for sensor fusion using the $\mathcal{H}_\infty$ synthesis
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#+LATEX_HEADER_EXTRA: \journal{Mechanical Systems and Signal Processing}

#+LATEX_HEADER_EXTRA: \author[a1,a2]{Thomas Dehaeze\corref{cor1}}
#+LATEX_HEADER_EXTRA: \author[a3,a4]{Mohit Verma}
#+LATEX_HEADER_EXTRA: \author[a2,a4]{Christophe Collette}
#+LATEX_HEADER_EXTRA: \cortext[cor1]{Corresponding author. Email Address: thomas.dehaeze@esrf.fr}

#+LATEX_HEADER_EXTRA: \address[a1]{European Synchrotron Radiation Facility, Grenoble, France}
#+LATEX_HEADER_EXTRA: \address[a2]{University of Li\`{e}ge, Department of Aerospace and Mechanical Engineering, 4000 Li\`{e}ge, Belgium.}
#+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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* =hypersetup=                                                        :ignore:

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* Abstract and Keywords                                               :ignore:
#+begin_frontmatter
#+begin_abstract
In order to obtain a better estimate of a quantity being measured, several sensors having different characteristics can be merged with a technique called "sensor fusion".
The obtained "super sensor" can combine the benefits of the individual sensors provided that the complementary filters used in the fusion are well designed.
Indeed, properties of the super sensor are linked to the magnitude of the complementary filters.
Properly shaping the magnitude of complementary filters is a difficult and time-consuming task.
In this study, we address this issue and propose a new method for designing complementary filters.
This method uses weighting functions to specify the wanted shape of the complementary filter that are then easily obtained using the standard $\mathcal{H}_\infty$ synthesis.
The proper choice of the weighting functions is discussed, and the effectiveness and simplicity of the design method is highlighted using several examples.
Such synthesis method is further extended for the shaping of more than two complementary filters.
#+end_abstract

#+begin_keyword
Sensor fusion \sep{} Complementary filters \sep{} $\mathcal{H}_\infty$ synthesis \sep{} Vibration isolation \sep{} Motion control
#+end_keyword
#+end_frontmatter

* Introduction
<<sec:introduction>>
** Introduction to Sensor Fusion                                     :ignore:

Measuring a physical quantity using sensors is always subject to several limitations.
First, the accuracy of the measurement will be affected by several noise sources, such as the electrical noise of the conditioning electronics being used.
Second, the frequency range in which the measurement is relevant is bounded by the bandwidth of the sensor.
One way to overcome these limitations is to combine several sensors using a technique called "sensor fusion"\nbsp{}cite:bendat57_optim_filter_indep_measur_two.
Fortunately, a wide variety of sensors exist, each with different characteristics.
By carefully choosing the fused sensors, a so called "super sensor" is obtain that combines benefits of individual sensors and yields a better estimate of the measured physical quantity. \par

** Advantages of Sensor Fusion                                       :ignore:

In some situations, sensor fusion is used to increase the bandwidth of the measurement\nbsp{}cite:shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,min15_compl_filter_desig_angle_estim.
For instance, in\nbsp{}cite:shaw90_bandw_enhan_posit_measur_using_measur_accel, the bandwidth of a position sensor is increased by fusing it with an accelerometer providing the high frequency motion information.
For other applications, sensor fusion is used to obtain a estimate of the measured quantity with lower noise\nbsp{}cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system,plummer06_optim_compl_filter_their_applic_motion_measur,robert12_introd_random_signal_applied_kalman.
More recently, the fusion of sensors measuring different physical quantities has been proposed to obtained interesting properties for control\nbsp{}cite:collette15_sensor_fusion_method_high_perfor,yong16_high_speed_vertic_posit_stage.
In\nbsp{}cite:collette15_sensor_fusion_method_high_perfor, an inertial sensor used for active vibration isolation is fused with a sensor collocated with the actuator to improve the stability margins of the feedback controller. \par

** Applications                                                      :ignore:

Practical applications of sensor fusion are numerous.
It is widely used for the attitude estimation of several autonomous vehicles such as unmanned aerial vehicle\nbsp{}cite:baerveldt97_low_cost_low_weigh_attit,corke04_inert_visual_sensin_system_small_auton_helic,jensen13_basic_uas and underwater vehicles\nbsp{}cite:pascoal99_navig_system_desig_using_time,batista10_optim_posit_veloc_navig_filter_auton_vehic.
Naturally, it is of great benefits for high performance positioning control as shown in\nbsp{}cite:shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,min15_compl_filter_desig_angle_estim,yong16_high_speed_vertic_posit_stage.
Sensor fusion was also shown to be a key technology to improve the performances of active vibration isolation systems\nbsp{}cite:tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip.
This is particularly apparent for the isolation stages of gravitational wave observer\nbsp{}cite:collette15_sensor_fusion_method_high_perfor,heijningen18_low such as the ones used at the LIGO\nbsp{}cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system and at the VIRGO\nbsp{}cite:lucia18_low_frequen_optim_perfor_advan. \par

** Kalman Filtering / Complementary filters                          :ignore:

There are mainly two ways to perform sensor fusion: either using a set of complementary filters\nbsp{}cite:anderson53_instr_approac_system_steer_comput or using Kalman filtering\nbsp{}cite:brown72_integ_navig_system_kalman_filter,odry18_kalman_filter_mobil_robot_attit_estim.
For sensor fusion applications, both methods are sharing many relationships\nbsp{}cite:brown72_integ_navig_system_kalman_filter,higgins75_compar_compl_kalman_filter,robert12_introd_random_signal_applied_kalman,becker15_compl_filter_desig_three_frequen_bands.
However, for Kalman filtering, we are forced to make assumption about the probabilistic character of the sensor noises\nbsp{}cite:robert12_introd_random_signal_applied_kalman whereas it is not the case for complementary filters.
Furthermore, the main advantages of complementary filters over Kalman filtering for sensor fusion are their very general applicability, their low computational cost\nbsp{}cite:higgins75_compar_compl_kalman_filter, and the fact that they are very intuitive as their effects can be easily interpreted in the frequency domain. \par

** Design Methods of Complementary filters                           :ignore:

A set of filters is said to be complementary if the sum of their transfer functions is equal to one at all frequencies.
For the earliest used of complementary filtering for sensor fusion, analog circuits were used to physically realize the filters\nbsp{}cite:anderson53_instr_approac_system_steer_comput.
Analog complementary filters are still used today\nbsp{}cite:yong16_high_speed_vertic_posit_stage,moore19_capac_instr_sensor_fusion_high_bandw_nanop, but most of the time they are now implemented numerically as it allows for much more flexibility. \par

Several design methods have been developed over the years to optimize complementary filters.
The easiest way to design complementary filters is to use analytical formulas.
Depending on the application, the formulas used are of first order\nbsp{}cite:corke04_inert_visual_sensin_system_small_auton_helic,yeh05_model_contr_hydraul_actuat_two,yong16_high_speed_vertic_posit_stage, second order\nbsp{}cite:baerveldt97_low_cost_low_weigh_attit,stoten01_fusion_kinet_data_using_compos_filter,jensen13_basic_uas or even higher orders\nbsp{}cite:shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,stoten01_fusion_kinet_data_using_compos_filter,collette15_sensor_fusion_method_high_perfor,matichard15_seism_isolat_advan_ligo. \par

As the characteristics of the "super sensor" depends on the design of the complementary filters\nbsp{}cite:dehaeze19_compl_filter_shapin_using_synth, several optimization techniques were developed over the years.
Some are based on the finding the optimal parameters of analytical formulas\nbsp{}cite:jensen13_basic_uas,min15_compl_filter_desig_angle_estim,becker15_compl_filter_desig_three_frequen_bands, while other are using convex optimization tools\nbsp{}cite:hua04_polyp_fir_compl_filter_contr_system,hua05_low_ligo such as linear matrix inequalities\nbsp{}cite:pascoal99_navig_system_desig_using_time.
As shown in\nbsp{}cite:plummer06_optim_compl_filter_their_applic_motion_measur, the design of complementary filters can also be linked to the standard mixed-sensitivity control problem.
Therefore, all the powerful tools developed for the classical control theory can also be used for the design of complementary filters.
For instance, in\nbsp{}cite:jensen13_basic_uas the two gains of a Proportional Integral (PI) controller are optimized to minimize the noise of the super sensor. \par

** Problematic / gap in the research                                 :ignore:

The common objective to all these complementary filters design methods is to obtain a super sensor that has desired characteristics, usually in terms of noise and dynamics.
Moreover, as reported in\nbsp{}cite:zimmermann92_high_bandw_orien_measur_contr,plummer06_optim_compl_filter_their_applic_motion_measur, phase shifts and magnitude bumps of the "super sensors" dynamics can be observed if either the complementary filters are poorly designed or if the sensors are not well calibrated.
Therefore, the robustness of the fusion is also of concerned when designing the complementary filters.
Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to specify the wanted super sensor characteristic while ensuring good fusion robustness has been proposed. \par

** Describe the paper itself / the problem which is addressed        :ignore:

Fortunately, both the robustness of the fusion and the super sensor characteristics can be linked to the magnitude of the complementary filters\nbsp{}cite:dehaeze19_compl_filter_shapin_using_synth.
Based on that, this paper introduces a new way to design complementary filters using the $\mathcal{H}_\infty$ synthesis which allows to shape the magnitude of the complementary filters in an easy and intuitive way. \par

** Introduce Each part of the paper                                  :ignore:

Section\nbsp{}ref:sec:requirements introduces the sensor fusion architecture and demonstrates how typical requirements can be linked to the complementary filters magnitudes.
In section\nbsp{}ref:sec:hinf_method, the shaping of complementary filters is written as an $\mathcal{H}_\infty$ optimization problem using weighting functions, and the simplicity of the proposed method is illustrated with an example.
The synthesis method is further validated in Section\nbsp{}ref:sec:application_ligo by designing complex complementary filters.
Section\nbsp{}ref:sec:discussion compares the proposed synthesis method with the classical mixed-sensitivity synthesis, and extends it to the shaping of more than two complementary filters.

* Sensor Fusion and Complementary Filters Requirements
<<sec:requirements>>
** Introduction                                                      :ignore:

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.

** Sensor Fusion Architecture
<<sec:sensor_fusion>>

A general sensor fusion architecture using complementary filters is shown in Fig.\nbsp{}ref:fig:sensor_fusion_overview where several sensors (here two) are measuring the same physical quantity $x$.
The two sensors output signals $\hat{x}_1$ and $\hat{x}_2$ are estimates 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.

#+name: fig:sensor_fusion_overview
#+caption: Schematic of a sensor fusion architecture using complementary filters
[[file:figs/sensor_fusion_overview.pdf]]

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

It will soon become clear why the complementary property is important for the sensor fusion architecture.

** Sensor Models and Sensor Normalization
<<sec:sensor_models>>

In order to study such sensor fusion architecture, a model for the sensors is required.
Such model is shown in Fig.\nbsp{}ref:fig:sensor_model and consists of a linear time invariant (LTI) system $G_i(s)$ representing the sensor dynamics and an input $n_i$ representing the sensor 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 using an estimate $\hat{G}_i(s)$ of the sensor dynamics $G_i(s)$, and filtering the sensor output by the inverse of this estimate $\hat{G}_i^{-1}(s)$ as shown in Fig.\nbsp{}ref:fig:sensor_model_calibrated.
It is here supposed that the sensor inverse $\hat{G}_i^{-1}(s)$ is proper and stable.
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 a more complex transfer function.

#+begin_export latex
\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 linear time invariant transfer function $G_i(s)$}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_model_calibrated.pdf}
\caption{\label{fig:sensor_model_calibrated} Normalized sensors using the inverse of an estimate $\hat{G}_i(s)$ of the sensor dynamics}
\end{subfigure}
\caption{\label{fig:sensor_models}Sensor models with and without normalization}
\centering
\end{figure}
#+end_export

Two normalized sensors are then combined to form a super sensor as shown in Fig.\nbsp{}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 /uncorrelated/ noises $n_1$ and $n_2$.
The signals from both normalized sensors are fed into two complementary filters $H_1(s)$ and $H_2(s)$ and then combined to yield an estimate $\hat{x}$ of $x$.

The super sensor output is therefore equal to:
#+name: eq:comp_filter_estimate
\begin{equation}
  \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}

#+name: fig:fusion_super_sensor
#+caption: Sensor fusion architecture with two normalized sensors
[[file:figs/fusion_super_sensor.pdf]]

** Noise Sensor Filtering
<<sec:noise_filtering>>

In this section, it is supposed that all the sensors are perfectly normalized, such that:
#+name: eq:perfect_dynamics
\begin{equation}
  \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.

Provided eqref:eq:perfect_dynamics is verified, the super sensor output $\hat{x}$ is then:
#+name: eq:estimate_perfect_dyn
\begin{equation}
  \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.
This is why the two filters must be complementary.

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.
#+name: eq:estimate_error
\begin{equation}
  \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 $\Phi_{n_2}$.
#+name: eq:noise_filtering_psd
\begin{equation}
  \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 is 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 norm $|H_1(j\omega)|$ has to be small when $\Phi_{n_1}(\omega)$ is larger than $\Phi_{n_2}(\omega)$ and the norm $|H_2(j\omega)|$ has to be small when $\Phi_{n_2}(\omega)$ is larger than $\Phi_{n_1}(\omega)$.
Therefore, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor noise.

** Sensor Fusion Robustness
<<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 (Fig.\nbsp{}ref:fig:sensor_model_uncertainty).
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 weighting transfer function representing the magnitude of the uncertainty.

The weight $w_i(s)$ is chosen such that the real sensor dynamics $G(j\omega)$ is contained in the uncertain region represented by a circle in the complex plane, 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 Fig.\nbsp{}ref:fig:sensor_model_uncertainty_simplified.

#+begin_export latex
\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}
#+end_export

The sensor fusion architecture with two sensor models including dynamical uncertainty is shown in Fig.\nbsp{}ref:fig:sensor_fusion_dynamic_uncertainty.

#+name: fig:sensor_fusion_dynamic_uncertainty
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
[[file:figs/sensor_fusion_dynamic_uncertainty.pdf]]

The super sensor dynamics eqref:eq:super_sensor_dyn_uncertainty is no longer equal to $1$ and now depends on the sensor dynamical uncertainty weights $w_i(s)$ as well as on the complementary filters $H_i(s)$.
#+name: eq:super_sensor_dyn_uncertainty
\begin{equation}
  \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 Fig.\nbsp{}ref:fig:uncertainty_set_super_sensor.

#+name: fig:uncertainty_set_super_sensor
#+caption: Uncertainty region of the super sensor dynamics in the complex plane (grey circle). The contribution of both sensors 1 and 2 to the total uncertainty are represented respectively by a blue circle and a red circle. The frequency dependency $\omega$ is here omitted.
[[file:figs/uncertainty_set_super_sensor.pdf]]

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

#+name: eq:max_phase_uncertainty
\begin{equation}
    \Delta\phi_\text{max}(\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.

* Complementary Filters Shaping
<<sec:hinf_method>>
** Introduction                                                     :ignore:

As shown in Section\nbsp{}ref:sec:requirements, the noise and robustness of the super sensor are a function of the complementary filters norms.
Therefore, a complementary filters synthesis method that allows to shape their norms would be of great use.
In this section, such synthesis is proposed by writing the synthesis objective as a standard $\mathcal{H}_\infty$ optimization problem.
As weighting functions are used to represent the wanted complementary filters shapes during the synthesis, the proper design of weighting functions is discussed.
Finally, the synthesis method is validated on an simple example.

** 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 proper and stable transfer functions $H_1(s)$ and $H_2(s)$ such that conditions eqref:eq:hinf_cond_complementarity, eqref:eq:hinf_cond_h1 and eqref:eq:hinf_cond_h2 are satisfied.

#+name: eq:comp_filter_problem_form
\begin{subequations}
  \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}
$W_1(s)$ and $W_2(s)$ are two weighting transfer functions that are carefully chosen to specify the maximum wanted norms of the complementary filters during the synthesis.

** Shaping of Complementary Filters using $\mathcal{H}_\infty$ synthesis
<<sec:hinf_synthesis>>

In this section, it is shown that the synthesis objective can be easily expressed as a standard $\mathcal{H}_\infty$ optimization problem and therefore solved using convenient tools readily available.

Consider the generalized plant $P(s)$ shown in Fig.\nbsp{}ref:fig:h_infinity_robust_fusion_plant and mathematically described by eqref:eq:generalized_plant.

#+name: eq:generalized_plant
\begin{equation}
  \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 & \phantom{+}W_2(s) \\  1 & 0 \end{bmatrix}
\end{equation}

#+begin_export latex
\begin{figure}[htbp]
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/h_infinity_robust_fusion_plant.pdf}
\caption{\label{fig:h_infinity_robust_fusion_plant} Generalized plant}
\vfill
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/h_infinity_robust_fusion_fb.pdf}
\caption{\label{fig:h_infinity_robust_fusion_fb} Generalized plant with the synthesized filter}
\end{subfigure}
\hfill
\caption{\label{fig:h_infinity_robust_fusion} Architecture for the $\mathcal{H}_\infty$ synthesis of complementary filters}
\centering
\end{figure}
#+end_export

Applying the standard $\mathcal{H}_\infty$ synthesis on the generalized plant $P(s)$ is then equivalent as finding a stable filter $H_2(s)$ which based on $v$, generates a signal $u$ such that the $\mathcal{H}_\infty$ norm of the system in Fig.\nbsp{}ref:fig:h_infinity_robust_fusion_fb from $w$ to $[z_1, \ z_2]$ is less than one eqref:eq:hinf_syn_obj.
#+name: eq:hinf_syn_obj
\begin{equation}
  \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}

By then defining $H_1(s)$ to be the complementary of $H_2(s)$ eqref:eq:definition_H1, the $\mathcal{H}_\infty$ synthesis objective becomes equivalent to eqref:eq:hinf_problem which ensure that eqref:eq:hinf_cond_h1 and eqref:eq:hinf_cond_h2 are satisfied.

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

#+name: eq:hinf_problem
\begin{equation}
  \left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1
\end{equation}

Therefore, applying the $\mathcal{H}_\infty$ synthesis on the standard plant $P(s)$ eqref:eq:generalized_plant will generate two filters $H_2(s)$ and $H_1(s) \triangleq 1 - H_2(s)$ that are complementary eqref:eq:comp_filter_problem_form and such that there norms are bellow specified bounds [[eqref:eq:hinf_cond_h1]],eqref:eq:hinf_cond_h2.

Note that there is not an equivalence between the $\mathcal{H}_\infty$ norm condition\nbsp{}eqref:eq:hinf_problem and the initial synthesis objectives\nbsp{}eqref:eq:hinf_cond_h1 and\nbsp{}eqref:eq:hinf_cond_h2, but only an implication.
Hence, the optimization may be a little bit conservative with respect to the "set" of filters on which it is performed.
There might be solutions were the objectives\nbsp{}eqref:eq:hinf_cond_h1 and\nbsp{}eqref:eq:hinf_cond_h2 are valid but where the $\mathcal{H}_\infty$ norm\nbsp{}eqref:eq:hinf_problem is larger than one.
In practice, this is however not an found to be an issue.

** Weighting Functions Design
<<sec:hinf_weighting_func>>

Weighting functions are used during the synthesis to specify the maximum allowed norms of the complementary filters.
The proper design of these weighting functions is of primary importance for the success of the presented $\mathcal{H}_\infty$ synthesis of complementary filters.

First, only proper and stable transfer functions should be used.
Second, the order of the weighting functions 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 orders).
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 frequency.

When designing complementary filters, it is usually desired to specify their slopes, their "blending" frequency and their maximum gains at low and high frequency.
To help with the design of the weighting functions such that the above specification can be easily expressed, the formula eqref:eq:weight_formula is proposed.

#+name: eq:weight_formula
\begin{equation}
  W(s) = \left( \frac{
           \hfill{} \frac{1}{\omega_c} \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_c} \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 in formula eqref:eq:weight_formula are:
- $G_0 = lim_{\omega \to 0} |W(j\omega)|$: the low frequency gain
- $G_\infty = lim_{\omega \to \infty} |W(j\omega)|$: the high frequency gain
- $G_c = |W(j\omega_c)|$: the gain at a specific frequency $\omega_c$ in $\si{rad/s}$.
- $n$: the slope between high and low frequency. It also corresponds to the order of the weighting function.

The parameters $G_0$, $G_c$ and $G_\infty$ should either satisfy condition eqref:eq:cond_formula_1 or eqref:eq:cond_formula_2.
#+name: eq:condition_params_formula
\begin{subequations}
  \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}

An example of the obtained magnitude of a weighting function generated using eqref:eq:weight_formula is shown in Fig.\nbsp{}ref:fig:weight_formula.

#+name: fig:weight_formula
#+caption: Magnitude of a weighting function generated using the proposed formula eqref:eq:weight_formula, $G_0 = 1e^{-3}$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$, $n = 3$
[[file:figs/weight_formula.pdf]]

** Validation of the proposed synthesis method
<<sec:hinf_example>>

The proposed methodology for the design of complementary filters is now applied on a simple example where two complementary filters $H_1(s)$ and $H_2(s)$ have to be designed such that:
- the blending frequency is around $\SI{10}{Hz}$
- the slope of $|H_1(j\omega)|$ is $+2$ below $\SI{10}{Hz}$, its low frequency gain is $10^{-3}$
- the slope of $|H_2(j\omega)|$ is $-3$ above $\SI{10}{Hz}$, its high frequency gain is $10^{-3}$

The first step is to translate the above requirements into the design of the weighting functions.
The proposed formula eqref:eq:weight_formula is here used for such purpose.
Parameters used are summarized in Table\nbsp{}ref:tab:weights_params.
The inverse magnitudes of the designed weighting functions, which are representing the maximum allowed norms of the complementary filters, are shown by the dashed lines in Fig.\nbsp{}ref:fig:weights_W1_W2.

#+begin_export latex
\begin{figure}
\begin{minipage}[b]{0.49\linewidth}
\centering
\begin{tabularx}{0.65\linewidth}{ccc}
\toprule
Parameters & \(W_1(s)\) & \(W_2(s)\)\\
\midrule
\(G_0\) & \(0.1\) & \(1000\)\\
\(G_\infty\) & \(1000\) & \(0.1\)\\
\(\omega_c\) & \(2\pi\cdot10\) & \(2\pi\cdot10\)\\
\(G_c\) & \(0.45\) & \(0.45\)\\
\(n\) & \(2\) & \(3\)\\
\bottomrule
\end{tabularx}
\captionof{table}{\label{tab:weights_params}Parameters used for weighting functions \(W_1(s)\) and \(W_2(s)\) using \eqref{eq:weight_formula}}
\end{minipage}
\hfill
\begin{minipage}[b]{0.49\linewidth}
\centering
\includegraphics[scale=1]{figs/weights_W1_W2.pdf}
\caption{\label{fig:weights_W1_W2}Inverse magnitude of the weighting functions}
\end{minipage}
\end{figure}
#+end_export

The $\mathcal{H}_\infty$ synthesis is then applied to the generalized plant of Fig.\nbsp{}ref:fig:h_infinity_robust_fusion_plant on efficiently solved in Matlab\nbsp{}cite:matlab20 using the Robust Control Toolbox.
The filter $H_2(s)$ that minimizes the $\mathcal{H}_\infty$ norm between $w$ and $[z_1,\ z_2]^T$ is obtained.
The $\mathcal{H}_\infty$ norm is here found to be close to one eqref:eq:hinf_synthesis_result which indicates that the synthesis is successful: the complementary filters norms are below the maximum specified upper bounds.
This is confirmed by the bode plots of the obtained complementary filters in Fig.\nbsp{}ref:fig:hinf_filters_results.

#+name: eq:hinf_synthesis_result
\begin{equation}
  \left\|\begin{matrix} \left(1 - H_2(s)\right) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \approx 1
\end{equation}

Their transfer functions in the Laplace domain are given in eqref:eq:hinf_synthesis_result_tf.
As expected, the obtained filters are of order $5$, that is the sum of the weighting functions orders.

#+name: eq:hinf_synthesis_result_tf
\begin{subequations}
  \begin{align}
    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)} \\
    H_1(s) &\triangleq H_2(s) - 1 = \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)}
  \end{align}
\end{subequations}


#+name: fig:hinf_filters_results
#+caption: Bode plot of the obtained complementary filters
[[file:figs/hinf_filters_results.pdf]]

This simple example illustrates the fact that the proposed methodology for complementary filters shaping is easy to use and effective.
A more complex real life example is taken up in the next section.

* Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO
<<sec:application_ligo>>
** Introduction                                                     :ignore:

Sensor fusion using complementary filters are widely used in active vibration isolation systems in gravitational wave detectors such at the LIGO\nbsp{}cite:matichard15_seism_isolat_advan_ligo,hua05_low_ligo, the VIRGO\nbsp{}cite:lucia18_low_frequen_optim_perfor_advan,heijningen18_low and the KAGRA [[cite:sekiguchi16_study_low_frequen_vibrat_isolat_system][Chap. 5]].

In the first isolation stage at the LIGO, two sets of complementary filters are used and included in a feedback loop\nbsp{}cite:hua04_low_ligo.
A set of complementary filters ($L_2,H_2$) is first used to fuse a seismometer and a geophone.
Then, another set of complementary filters ($L_1,H_1$) is used to merge the output of the first "inertial super sensor" with a position sensor.
A simplified block diagram of the sensor fusion architecture is shown in Fig.\nbsp{}ref:fig:ligo_super_sensor_architecture.

#+name: fig:ligo_super_sensor_architecture
#+caption: Simplified block diagram of the sensor blending strategy for the first stage at the LIGO\nbsp{}cite:hua04_low_ligo
[[file:figs/ligo_super_sensor_architecture.pdf]]

The fusion of the position sensor at low frequency with the "inertial super sensor" at high frequency using the complementary filters ($L_1,H_1$) is done for several reasons, first of which is to give the super sensor a DC sensibility and therefore allow the feedback loop to have authority at zero frequency.
The requirements on those filters are very tight and thus their design is complex and should be expressed as an optimization problem.

The approach used in\nbsp{}cite:hua05_low_ligo is to use FIR complementary filters and to write the synthesis as a convex optimization problem.
After synthesis, the obtained FIR filters were found to be compliant with the requirements.
However they are of very high order so their implementation is quite complex.
In this section, the effectiveness of the proposed complementary filter synthesis strategy is demonstrated on the same set of requirements.

** Complementary Filters Specifications
<<sec:ligo_specifications>>

The specifications for the set of complementary filters ($L_1,H_1$) used at the LIGO are summarized below (for further details, refer to\nbsp{}cite:hua04_polyp_fir_compl_filter_contr_system):
- From $0$ to $\SI{0.008}{Hz}$, the magnitude $|L_1(j\omega)|$ should be less or equal to $8 \times 10^{-4}$
- Between $\SI{0.008}{Hz}$ to $\SI{0.04}{Hz}$, the filter $L_1(s)$ should attenuate the input signal proportional to frequency cubed
- Between $\SI{0.04}{Hz}$ to $\SI{0.1}{Hz}$, the magnitude $|L_1(j\omega)|$ should be less than $3$
- Above $\SI{0.1}{Hz}$, the magnitude $|H_1(j\omega)|$ should be less than $0.045$

These specifications are therefore upper bounds on the complementary filters' magnitudes.
They are physically represented in Fig.\nbsp{}ref:fig:fir_filter_ligo as well as the obtained magnitude of the FIR filters in\nbsp{}cite:hua05_low_ligo.

#+name: fig:fir_filter_ligo
#+caption: Specifications and Bode plot of the obtained FIR complementary filters in\nbsp{}cite:hua05_low_ligo. The filters are here obtained using the SeDuMi Matlab toolbox\nbsp{}cite:sturm99_using_sedum
[[file:figs/fir_filter_ligo.pdf]]

** Weighting Functions Design
<<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 for 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.\nbsp{}ref:fig:ligo_weights.

#+name: fig:ligo_weights
#+caption: Specifications and weighting functions inverse magnitudes
[[file:figs/ligo_weights.pdf]]

** $\mathcal{H}_\infty$ Synthesis of the complementary filters
<<sec:ligo_results>>

The proposed $\mathcal{H}_\infty$ synthesis is performed on the generalized plant shown in Fig.\nbsp{}ref:fig:h_infinity_robust_fusion_plant.
After optimization, the $\mathcal{H}_\infty$ norm from $w$ to $[z_1,\ z_2]^T$ is found close to one indication successful synthesis.
In Fig.\nbsp{}ref:fig:comp_fir_ligo_hinf, the bode plot of the obtained complementary filters are compared with the FIR filters of order 512 obtained in\nbsp{}cite:hua05_low_ligo.
Even though the complementary filters using the $\mathcal{H}_\infty$ synthesis are of much lower order (order 27), they are found to be very close to the FIR filters.
This confirms the effectiveness of the proposed synthesis method even when the complementary filters are subject to complex requirements.

#+name: fig:comp_fir_ligo_hinf
#+caption: Comparison of the FIR filters (dashed) designed in\nbsp{}cite:hua05_low_ligo with the filters obtained with $\mathcal{H}_\infty$ synthesis (solid)
[[file:figs/comp_fir_ligo_hinf.pdf]]

* Discussion
<<sec:discussion>>
** Introduction                                                      :ignore:

** "Closed-Loop" complementary filters
<<sec:closed_loop_complementary_filters>>
An alternative way to implement complementary filters is by using a fundamental property of the classical feedback architecture shown in Fig.\nbsp{}ref:fig:feedback_sensor_fusion.
This is for instance presented in cite:mahony05_compl_filter_desig_special_orthog,plummer06_optim_compl_filter_their_applic_motion_measur,jensen13_basic_uas.

#+name: fig:feedback_sensor_fusion
#+caption: "Closed-Loop" complementary filters
[[file:figs/feedback_sensor_fusion.pdf]]

Consider the feedback architecture of Fig.\nbsp{}ref:fig:feedback_sensor_fusion, with two inputs $\hat{x}_1$ and $\hat{x}_2$, and one output $\hat{x}$.
The output $\hat{x}$ is linked to the inputs by eqref:eq:closed_loop_complementary_filters.

#+name: eq:closed_loop_complementary_filters
\begin{equation}
\hat{x} = \underbrace{\frac{1}{1 + L(s)}}_{S(s)} \hat{x}_1 + \underbrace{\frac{L(s)}{1 + L(s)}}_{T(s)} \hat{x}_2
\end{equation}

As for any classical feedback architecture, we have that the sum of the sensitivity and complementary sensitivity transfer function is equal to one\nbsp{}eqref:eq:sensitivity_sum.

#+name: eq:sensitivity_sum
\begin{equation}
T(s) + S(s) = 1
\end{equation}

Therefore, provided that the closed-loop system is stable, the closed-loop system in Fig.\nbsp{}ref:fig:feedback_sensor_fusion is corresponding to two complementary filters.
Two sensors can then be merged as shown in Fig.\nbsp{}ref:fig:feedback_sensor_fusion_arch.

#+name: fig:feedback_sensor_fusion_arch
#+caption: Classical feedback architecture used for sensor fusion
[[file:figs/feedback_sensor_fusion_arch.pdf]]

One of the main advantage of implementing and designing complementary filters using the feedback architecture of Fig.\nbsp{}ref:fig:feedback_sensor_fusion is that all the tools of the linear control theory can be applied for the design of the filters.
If one want to shape both $\frac{\hat{x}}{\hat{x}_1}(s) = S(s)$ and $\frac{\hat{x}}{\hat{x}_2}(s) = T(s)$, the $\mathcal{H}_\infty$ mixed-sensitivity synthesis can be easily applied.

To do so, weighting functions $W_1(s)$ and $W_2(s)$ are added to respectively shape $S(s)$ and $T(s)$ (Fig.\nbsp{}ref:fig:feedback_synthesis_architecture).
Then the system is re-organized to form the generalized plant $P_L(s)$ shown in Fig.\nbsp{}ref:fig:feedback_synthesis_architecture_generalized_plant.
The $\mathcal{H}_\infty$ mixed-sensitivity synthesis can finally be performed by applying the $\mathcal{H}_\infty$ synthesis to the generalized plant $P_L(s)$ which is described by eqref:eq:generalized_plant_mixed_sensitivity.

#+name: eq:generalized_plant_mixed_sensitivity
\begin{equation}
  \begin{bmatrix} z \\ v \end{bmatrix} = P_L(s) \begin{bmatrix} w_1 \\ w_2 \\ u \end{bmatrix}; \quad P_L(s) = \begin{bmatrix}
    \phantom{+}W_1(s) & 0      & \phantom{+}1 \\
              -W_1(s) & W_2(s) &           -1
  \end{bmatrix}
\end{equation}

The output of the synthesis is a filter $L(s)$ such that the "closed-loop" $\mathcal{H}_\infty$ norm from $[w_1,\ w_2]$ to $z$ is less than one eqref:eq:comp_filters_feedback_obj.

#+name: eq:comp_filters_feedback_obj
\begin{equation}
  \left\| \begin{matrix} \frac{z}{w_1} \\ \frac{z}{w_2} \end{matrix} \right\|_\infty = \left\| \begin{matrix} \frac{1}{1 + L(s)} W_1(s) \\ \frac{L(s)}{1 + L(s)} W_2(s) \end{matrix} \right\|_\infty \le 1
\end{equation}

If the synthesis is successful, two complementary filters are obtained with their magnitudes bounded by the inverse magnitudes of the weighting functions.
The sensor fusion can then be implemented as shown in Fig.\nbsp{}ref:fig:feedback_sensor_fusion_arch using the feedback architecture or more classically as shown in Fig.\nbsp{}ref:fig:sensor_fusion_overview by defining the two complementary filters as in\nbsp{}eqref:eq:comp_filters_feedback.

#+name: eq:comp_filters_feedback
\begin{equation}
H_1(s) = \frac{1}{1 + L(s)}; \quad H_2(s) = \frac{L(s)}{1 + L(s)}
\end{equation}

The two architecture are equivalent regarding their inputs/outputs relationships.

#+begin_export latex
\begin{figure}[htbp]
\begin{subfigure}[t]{0.6\linewidth}
\centering
\includegraphics[scale=1]{figs/feedback_synthesis_architecture.pdf}
\caption{\label{fig:feedback_synthesis_architecture} Feedback architecture with included weights}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.4\linewidth}
\centering
\includegraphics[scale=1]{figs/feedback_synthesis_architecture_generalized_plant.pdf}
\caption{\label{fig:feedback_synthesis_architecture_generalized_plant} Generalized plant}
\end{subfigure}
\hfill
\caption{\label{fig:h_inf_mixed_sensitivity_synthesis} $\mathcal{H}_\infty$ mixed-sensitivity synthesis}
\centering
\end{figure}
#+end_export

As an example, two "closed-loop" complementary filters are designed using the $\mathcal{H}_\infty$ mixed-sensitivity synthesis.
The weighting functions are designed using formula\nbsp{}eqref:eq:weight_formula with parameters shown in Table\nbsp{}ref:tab:weights_params.
After synthesis, a filter $L(s)$ is obtained, its magnitude is shown in Fig.\nbsp{}ref:fig:hinf_filters_results_mixed_sensitivity by the dashed line.
The "closed-loop" complementary filters are compared with the inverse magnitude of the weighting functions in Fig.\nbsp{}ref:fig:hinf_filters_results_mixed_sensitivity confirming that the synthesis is successful.
The obtained "closed-loop" complementary filters are indeed equal to the ones obtained in Section\nbsp{}ref:sec:hinf_example.

#+name: fig:hinf_filters_results_mixed_sensitivity
#+caption: Bode plot of the obtained complementary filters after $\mathcal{H}_\infty$ mixed-sensitivity synthesis
[[file:figs/hinf_filters_results_mixed_sensitivity.pdf]]

** Synthesis of more than two Complementary Filters
<<sec:hinf_three_comp_filters>>

Some applications may require to merge more than two sensors\nbsp{}cite:stoten01_fusion_kinet_data_using_compos_filter,becker15_compl_filter_desig_three_frequen_bands.
For instance at the LIGO\nbsp{}cite:matichard15_seism_isolat_advan_ligo, three sensors (an LVDT, a seismometer and a geophone) are merged to form a super sensor (Fig.\nbsp{}ref:fig:ligo_super_sensor_architecture).\par

When merging $n>2$ sensors using complementary filters, two architectures can be used as shown in Fig.\nbsp{}ref:fig:sensor_fusion_three.
The fusion can either be done in a "sequential" way where $n-1$ sets of two complementary filters are used (Fig.\nbsp{}ref:fig:sensor_fusion_three_sequential), or in a "parallel" way where one set of $n$ complementary filters is used (Fig.\nbsp{}ref:fig:sensor_fusion_three_parallel).

In the first case, typical sensor fusion synthesis techniques can be used.
However, when a parallel architecture is used, a new synthesis method for a set of more than two complementary filters is required as only simple analytical formulas have been proposed in the literature\nbsp{cite:stoten01_fusion_kinet_data_using_compos_filter,becker15_compl_filter_desig_three_frequen_bands}.
A generalization of the proposed synthesis method of complementary filters is presented in this section. \par

#+begin_export latex
\begin{figure}[htbp]
\begin{subfigure}[b]{0.59\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_fusion_three_sequential.pdf}
\caption{\label{fig:sensor_fusion_three_sequential}Sequential fusion}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.39\linewidth}
\centering
\includegraphics[scale=1]{figs/sensor_fusion_three_parallel.pdf}
\caption{\label{fig:sensor_fusion_three_parallel}Parallel fusion}
\end{subfigure}
\caption{\label{fig:sensor_fusion_three}Possible sensor fusion architecture when more than two sensors are to be merged}
\centering
\end{figure}
#+end_export

The synthesis objective is to compute a set of $n$ stable transfer functions $[H_1(s),\ H_2(s),\ \dots,\ H_n(s)]$ such that conditions\nbsp{}eqref:eq:hinf_cond_compl_gen and\nbsp{}eqref:eq:hinf_cond_perf_gen are 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}
where $[W_1(s),\ W_2(s),\ \dots,\ W_n(s)]$ are weighting transfer functions that are chosen to specify the maximum wanted norms of the complementary filters during the synthesis.

Such synthesis objective is very close to the one described in Section\nbsp{}ref:sec:synthesis_objective, and indeed the proposed synthesis method is a generalization of the one presented in Section\nbsp{}ref:sec:hinf_synthesis. \par

Before presenting the generalized synthesis method, the case with three sensors is presented.
Consider the generalized plant $P_3(s)$ shown in Fig.\nbsp{}ref:fig:comp_filter_three_hinf_gen_plant which is also described by\nbsp{}eqref:eq:generalized_plant_three_filters.

#+name: eq:generalized_plant_three_filters
\begin{equation}
  \begin{bmatrix} z_1 \\ z_2 \\ z_3 \\ v \end{bmatrix} = P_3(s) \begin{bmatrix} w \\ u_1 \\ u_2 \end{bmatrix}; \quad P_3(s) = \begin{bmatrix}W_1(s) & -W_1(s) & -W_1(s) \\ 0 & \phantom{+}W_2(s) & 0 \\ 0 & 0 & \phantom{+}W_3(s) \\ 1 & 0 & 0 \end{bmatrix}
\end{equation}

#+begin_export latex
\begin{figure}[htbp]
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/comp_filter_three_hinf_gen_plant.pdf}
\caption{\label{fig:comp_filter_three_hinf_gen_plant} Generalized plant}
\vfill
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.5\linewidth}
\centering
\includegraphics[scale=1]{figs/comp_filter_three_hinf_fb.pdf}
\caption{\label{fig:comp_filter_three_hinf_fb} Generalized plant with the synthesized filter}
\end{subfigure}
\hfill
\caption{\label{fig:comp_filter_three_hinf} Architecture for the $\mathcal{H}_\infty$ synthesis of three complementary filters}
\centering
\end{figure}
#+end_export

Applying the $\mathcal{H}_\infty$ synthesis on the generalized plant $P_3(s)$ is equivalent as to find two stable filters $[H_2(s),\ H_3(s)]$ (shown in Fig.\nbsp{}ref:fig:comp_filter_three_hinf_fb) such that the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $[z_1,\ z_2, \ z_3]$ is less than one\nbsp{}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}

By defining $H_1(s) \triangleq 1 - H_2(s) - H_3(s)$, the proposed $\mathcal{H}_\infty$ synthesis solves the design problem eqref:eq:hinf_problem_gen with $n=3$. \par

An example is given to validate the method where three sensors are used in different frequency bands.
For instance a displacement sensor from DC up to $\SI{1}{Hz}$, a geophone from $1$ to $\SI{10}{Hz}$ and an accelerometer above $\SI{10}{Hz}$.
Three weighting functions are designed using formula\nbsp{}eqref:eq:weight_formula and their inverse magnitudes are shown in Fig.\nbsp{}ref:fig:three_complementary_filters_results (dashed curves).
The $\mathcal{H}_\infty$ synthesis is performed on the generalized plant $P_3(s)$ and the bode plot of the obtained complementary filters are shown in Fig.\nbsp{}ref:fig:three_complementary_filters_results (solid lines). \par

#+name: fig:three_complementary_filters_results
#+caption: Bode plot of the inverse weighting functions and of the three complementary filters obtained using the $\mathcal{H}_\infty$ synthesis
[[file:figs/three_complementary_filters_results.pdf]]

Even though there might not be any practical application for a set of more than 3 complementary filters, it can still be designed using the same procedure.
A set of $n$ complementary filters can be shaped using the generalized plant $P_n(s)$ described by\nbsp{}eqref:eq:generalized_plant_n_filters.

#+name: eq:generalized_plant_n_filters
\begin{equation}
  \begin{bmatrix} z_1 \\ \vdots \\ z_n \\ v \end{bmatrix} = P_n(s) \begin{bmatrix} w \\ u_1 \\ \vdots \\ u_{n-1} \end{bmatrix}; \quad
  P_n(s) = \begin{bmatrix}
    W_1    & -W_1   & \dots  & \dots  & -W_1 \\
    0      &  W_2   & 0      & \dots  & 0 \\
    \vdots & \ddots & \ddots & \ddots & \vdots \\
    \vdots &        & \ddots & \ddots & 0 \\
    0      & \dots  & \dots  & 0      & W_n \\
    1      & 0      & \dots  & \dots  & 0
    \end{bmatrix}
\end{equation}

* Conclusion
<<sec:conclusion>>

Sensors measuring a physical quantities are always subject to limitations both in terms of bandwidth or accuracy.
Complementary filters can be used to fuse multiple sensors with different characteristics in order to combine their benefits and yield a better estimate of the measured physical quantity.

The sensor fusion robustness and the obtained super sensor noise can be linked to the magnitude of the complementary filters.
In this paper, a synthesis method that enables the shaping of the complementary filters norms has been proposed.
Several example were used to emphasize the simplicity and the effectiveness of the proposed method.
Links with "closed-loop" complementary filters where highlighted, and the proposed method was generalized for the design of a set of more than two complementary filters.

Future work will aim at developing a complementary filter synthesis method that minimizes the super sensor noise while ensuring the robustness of the fusion.

* Acknowledgment
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This research benefited from a FRIA grant from the French Community of Belgium.
This paper is based on a paper previously presented at the ICCMA conference\nbsp{}cite:dehaeze19_compl_filter_shapin_using_synth.

* Data Availability
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Matlab cite:matlab20 was used for this study.
The source code is available under a MIT License and archived in Zenodo\nbsp{}cite:dehaeze21_new_method_desig_compl_filter_code.

* Bibliography                                                       :ignore:
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