#+TITLE: A new method of designing complementary filters for sensor fusion using $\mathcal{H}_\infty$ 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: dehaeze.thomas@gmail.com}

#+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
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

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

# Basic explainations of sensor fusion

- cite:bendat57_optim_filter_indep_measur_two roots of sensor fusion

*** Advantages of Sensor Fusion                                     :ignore:

# Sensor Fusion can have many advantages / can be applied for various purposes

- Increase the bandwidth: cite:zimmermann92_high_bandw_orien_measur_contr
- Increased robustness: cite:collette15_sensor_fusion_method_high_perfor
- Decrease the noise:

*** Applications                                                    :ignore:

# The applications of sensor fusion are numerous

- UAV: cite:pascoal99_navig_system_desig_using_time, cite:jensen13_basic_uas
- Gravitational wave observer: cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system

*** Kalman Filtering or Complementary filters                       :ignore:

# There are mainly two ways to perform sensor fusion: using complementary filters or using Kalman filtering

- cite:brown72_integ_navig_system_kalman_filter alternate form of complementary filters => Kalman filtering
- cite:higgins75_compar_compl_kalman_filter Compare Kalman Filtering with sensor fusion using complementary filters
- cite:robert12_introd_random_signal_applied_kalman advantage of complementary filters over Kalman filtering

*** Design Methods of Complementary filters                         :ignore:

# In some cases, complementary filters are implemented in an analog way such as in [...], but most of the time it is implemented numerically which allows much more complex

- Analog complementary filters: cite:yong16_high_speed_vertic_posit_stage, cite:moore19_capac_instr_sensor_fusion_high_bandw_nanop

# Multiple design methods have been used for complementary filters

- Analytical methods:
  - first order: cite:corke04_inert_visual_sensin_system_small_auton_helic
  - second order: cite:baerveldt97_low_cost_low_weigh_attit, cite:stoten01_fusion_kinet_data_using_compos_filter, cite:jensen13_basic_uas
  - 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
- cite:pascoal99_navig_system_desig_using_time use LMI to generate complementary filters
- cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system: FIR + convex optimization
- Similar to feedback system:
  - cite:plummer06_optim_compl_filter_their_applic_motion_measur use H-Infinity to optimize complementary filters (flatten the super sensor noise spectral density)
  - cite:jensen13_basic_uas design of complementary filters with classical control theory

- 3 complementary filters: cite:becker15_compl_filter_desig_three_frequen_bands

*** Problematics / gap in the research                              :ignore:

- Robustness problems: cite:zimmermann92_high_bandw_orien_measur_contr change of phase near the merging frequency
- Trial and error
- 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.

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

Most of the requirements => shape of the complementary filters
=> propose a way to shape complementary filters.

*** Introduce Each part of the paper                                :ignore:

** Old Introduction                                         :ignore:noexport:
*** Establish the importance of the research topic                 :ignore:
# What are Complementary Filters
A set of filters is said to be complementary if the sum of their transfer functions is equal to one at all frequencies.
These filters are used when two or more sensors are measuring the same physical quantity with different noise characteristics.
Unreliable frequencies of each sensor are filtered out by the complementary filters and then combined to form a super sensor giving a better estimate of the physical quantity over a wider bandwidth.
This technique is called sensor fusion and is used in many applications.\par

*** Applications of complementary filtering                        :ignore:
# Improve bandwidth for UAV
In cite:zimmermann92_high_bandw_orien_measur_contr,corke04_inert_visual_sensin_system_small_auton_helic, various sensors (accelerometers, gyroscopes, vision sensors, etc.) are merged using complementary filters for the attitude estimation of Unmanned Aerial Vehicles (UAV).
# Improving the control robustness
In cite:collette15_sensor_fusion_method_high_perfor, several sensor fusion configurations using different types of sensors are discussed in order to increase the control bandwidth of active vibration isolation systems.
# Merging of different sensor types
Furthermore, sensor fusion is used in the isolation systems of the Laser Interferometer Gravitational-Wave Observator (LIGO) to merge inertial sensors with relative sensors
cite:matichard15_seism_isolat_advan_ligo,hua04_polyp_fir_compl_filter_contr_system. \par

*** Current design methods for complementary filters               :ignore:
# Why Design of Complementary Filter is important
As the super sensor noise characteristics largely depend on the complementary filter norms, their proper design is of primary importance for sensor fusion.
# Discuss the different approach to complementary filter design
In cite:corke04_inert_visual_sensin_system_small_auton_helic,jensen13_basic_uas, first and second order analytical formulas of complementary filters have been presented.
# Third Order and Higher orders
Higher order complementary filters have been used in
cite:shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,collette15_sensor_fusion_method_high_perfor.
# Alternate Formulation
In cite:jensen13_basic_uas, the sensitivity and complementary sensitivity transfer functions of a feedback architecture have been proposed to be used as complementary filters. The design of such filters can then benefit from the classical control theory developments.
# LMI / convex Optimization
Linear Matrix Inequalities (LMIs) are used in cite:pascoal99_navig_system_desig_using_time for the synthesis of complementary filters satisfying some frequency-like performance.
# FIR Filters
Finally, a synthesis method of high order Finite Impulse Response (FIR) complementary filters using convex optimization has been developed in cite:hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system. \par

*** Describe a gap in the research                                 :ignore:
# There is a need for easy synthesis methods for complementary filters
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.

*** Describe the paper itself / the problem which is addressed     :ignore:
# In this paper, we propose a synthesis method for the shaping of complementary filters using the $\mathcal{H}_\infty$ norm.\par
This paper presents a new design method of complementary filters based on $\mathcal{H}_\infty$ synthesis.
This design method permits to easily shape the norms of the generated filters.\par

*** Introduce Each part of the paper                               :ignore:
The section ref:sec:requirements gives a brief overview of sensor fusion using complementary filters and explains how the typical requirements for such fusion can be expressed as upper bounds on the filters norms.
In section ref:sec:hinf_method, a new design method for the shaping of complementary filters using $\mathcal{H}_\infty$ synthesis is proposed.
In section ref:sec:application_ligo, the method is used to design complex complementary filters that are used for sensor fusion at the LIGO.
Our conclusions are drawn in the final section.

** Mohit's Introduction                                            :noexport:
The sensors used for measuring physical quantity often works well within a limited frequency range called as the bandwidth of the sensor.
The signals recorded by the sensor beyond its bandwidth are often corrupt with noise and are not reliable.
Many dynamical systems require measurements over a wide frequency range.
Very often a variety of sensors are utilized to sense the same quantity.
These sensors have different operational bandwidth and are reliable only in a particular frequency range.
The signals from the different sensors are fused together in order to get the reliable measurement of the physical quantity over wider frequency band.
The combining of signals from various sensor is called sensor fusion cite:hua04_polyp_fir_compl_filter_contr_system.
The resulting sensor is referred as "super sensor" since it can have better noise characteristics and can operate over a wider frequency band as compared to the individual sensor used for merging cite:shaw90_bandw_enhan_posit_measur_using_measur_accel.

Sensor fusion is most commonly employed in the navigation systems to accurately measure the position of a vehicle.
The GPS sensors, which are accurate in low frequency band, are merged with the high-frequency accelerometers.
Zimmermann and Sulzer cite:zimmermann92_high_bandw_orien_measur_contr used sensor fusion to measure the orientation of a robot.
They merged inclinometer and accelerometers for accurate angular measurements over large frequency band.
Corke cite:corke04_inert_visual_sensin_system_small_auton_helic merged inertial measurement unit with the stereo vision system for measurement of attitude, height and velocity of an unmanned helicopter.
Baerveldt and Klang cite:baerveldt97 used an inclinometer and a gyroscope to measure the orientation of the autonomous helicopter.
The measurement of the 3D orientation using a gyroscope and an accelerometer was demonstrated by Roberts et al. cite:roberts03_low.
Cao et al. cite:cao20_adapt_compl_filter_based_post used sensor fusion to obtain the lateral and longitudinal velocities of the autonomous vehicle.

Sensor fusion is also used for enhancing the working range of the active isolation system.
For example, the active vibration isolation system at the Laser Interferometer Gravitational-Wave Observatory (LIGO) cite:matichard15_seism_isolat_advan_ligo utilizes sensor fusion.
The position sensors, seismometer and geophones are used for measuring the motion of the LIGO platform in different frequency bands cite:hua05_low_ligo.
Tjepkema et al. cite:tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip used sensor fusion to isolate precision equipment from the ground motion.
The feedback from the accelerometer was used for active isolation at low frequency while force sensor was used at high frequency.
Various configurations of sensor fusion for active vibration isolation systems are discussed by Collette and Matichard cite:collette15_sensor_fusion_method_high_perfor.
Ma and Ghasemi-Nejhad cite:ma04_frequen_weigh_adapt_contr_simul used laser sensor and piezoelectric patches for simultaneous tracking and vibration control in smart structures.
Recently, Verma et al. cite:verma21_virtual_sensor_fusion_high_precis_contr presented virtual sensor fusion for high precision control where the signals from a physical sensor are fused with a sensor simulated virtually.

Fusing signals from different sensors can typically be done using Kalman filtering cite:odry18_kalman_filter_mobil_robot_attit_estim, ren19_integ_gnss_hub_motion_estim, faria19_sensor_fusion_rotat_motion_recon, liu18_innov_infor_fusion_method_with, abdel15_const_low_cost_gps_filter, biondi17_attit_recov_from_featur_track or complementary filters cite:brown72_integ_navig_system_kalman_filter.
A set of filters is said to be complementary if the sum of their transfer functions is equal to one at all frequencies.
When two filters are complementary, usually one is a low pass filter while the other is an high pass filter.
The complementary filters are designed in such a way that their magnitude is close to one in the bandwidth of the sensor they are combined with.
This enables to measure the physical quantity over larger bandwidth.
There are two different categories of complementary filters --- frequency domain complementary filters and state space complementary filters.
Earliest application of the the frequency domain complementary filters was seen in Anderson and Fritze cite:anderson53_instr_approac_system_steer_comput.
A simple RC circuit was used to physically realize the complementary filters.
Frequency domain complementary filters were also used in cite:shaw90_bandw_enhan_posit_measur_using_measur_accel, zimmermann92_high_bandw_orien_measur_contr, baerveldt97, roberts03_low.
State space complementary filter finds application in tracking orientation of the flexible links in a robot cite:bachmann03_desig_marg_dof, salcudean91_global_conver_angul_veloc_obser, mahony08_nonlin_compl_filter_special_orthog_group and are particularly useful for multi-input multi-output systems.
Pascoal et al. cite:pascoal00_navig_system_desig_using_time presented complementary filters which can adapt with time for navigation system capable of estimating position and velocity using GPS and SONAR sensors.

The noise characteristics of the super sensor are governed by the norms of the complementary filters.
Therefore, the proper design of the complementary filters for sensor fusion is of immense importance.
The design of complementary filters is a complex task as they need to tuned as per the specification of the sensor.
In many applications, analytical formulas of first and second order complementary filters are used cite:corke04_inert_visual_sensin_system_small_auton_helic,jensen13_basic_uas.
However, these low order complementary filters are not optimal, and high order complementary filters can lead to better fusion cite:jensen13_basic_uas,shaw90_bandw_enhan_posit_measur_using_measur_accel.

Several design techniques have been proposed to design higher order complementary filters.
Pascoal cite:pascoal00_navig_system_desig_using_time used linear matrix inequalities (LMIs) cite:boyd94_linear for the design of time varying complementary filters.
LMIs were also used by Hua et al. cite:hua04_polyp_fir_compl_filter_contr_system to design finite impulse response (FIR) filters for the active vibration isolation system at LIGO.
Plummer cite:plummer06_optim_compl_filter_their_applic_motion_measur proposed an optimal design method using the $\mathcal{H}_{\infty}$ synthesis and weighting functions representing the measurement noise of the sensors.

Although various methods have been presented in the literature for the design of complementary filters, there is a lack of general and simple framework that allows to shape the norm of complementary filters.
Such a method would prove to be very useful as the noise of the "supper sensor" and its dynamical characteristics depend on the norm of the filters.
This paper presents such a framework based on the $\mathcal{H}_\infty$ norm minimization.
The proposed method is quite general and can be easily extended to a case where more than two complementary filters needs to be designed.
The organization of this paper is as follows.
Section [[*Complementary filters requirements][2]] presents the design requirements of ideal complementary filters.
It also demonstrates how the noise and robustness characteristics of the "super sensor" can be transformed into upper bounds on the norm of the complementary filters.
The framework for the design of complementary filters is detailed in Section [[*Design formulation using $\mathcal{H}_\infty$ synthesis][3]].
This is followed by the application of the design method to complementary filter design for the active vibration isolation at LIGO in Section [[*Application: Complementary Filter Design for Active Vibration Isolation of LIGO][4]].
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>>

Let's consider two sensors measuring the same physical quantity $x$ with dynamics $G_1(s)$ and $G_2(s)$, and with uncorrelated noise characteristics $n_1$ and $n_2$.

The signals from both 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.
#+name: eq:comp_filter_estimate
\begin{equation}
  \hat{x} = \left(G_1 H_1 + G_2 H_2\right) x + H_1 n_1 + H_2 n_2
\end{equation}

#+name: fig:fusion_super_sensor
#+caption: Sensor fusion architecture
#+attr_latex: :scale 1
[[file:figs/fusion_super_sensor.pdf]]

The complementary property of $H_1(s)$ and $H_2(s)$ implies that their transfer function sum is equal to one at all frequencies eqref:eq:comp_filter.
#+name: eq:comp_filter
\begin{equation}
  H_1(s) + H_2(s) = 1
\end{equation}

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

Let's first consider sensors with perfect dynamics
#+name: eq:perfect_dynamics
\begin{equation}
  G_1(s) = G_2(s) = 1
\end{equation}

The estimate $\hat{x}$ is then described by
#+name: eq:estimate_perfect_dyn
\begin{equation}
  \hat{x} = x + H_1 n_1 + H_2 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 noise.
Thus, this sensor fusion architecture permits to filter the noise of both sensors without introducing any distortion in the physical quantity to be measured.

Let's define the estimation error $\delta x$ by eqref:eq:estimate_error.
#+name: eq:estimate_error
\begin{equation}
  \delta x \triangleq \hat{x} - x = H_1 n_1 + H_2 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} = \left|H_1\right|^2 \Phi_{n_1} + \left|H_2\right|^2 \Phi_{n_2}
\end{equation}

Usually, the two sensors have high noise levels over distinct frequency regions.
In order 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 $\Phi_{n_1}$.

** Robustness of the Fusion
<<sec:fusion_robustness>>

In practical systems the sensor dynamics is not perfect and eqref:eq:perfect_dynamics is not verified.
In such case, one can use an inversion filter $\hat{G}_i^{-1}(s)$ to normalize the sensor dynamics, where $\hat{G}_i(s)$ is an estimate of the sensor dynamics $G_i(s)$.
However, as there is always some level of uncertainty on the dynamics, it cannot be perfectly inverted and $\hat{G}_i^{-1}(s) G_i(s) \neq 1$.

Let's represent the resulting dynamic uncertainty of the inverted sensors by an input multiplicative uncertainty as shown in Fig. ref:fig:sensor_fusion_dynamic_uncertainty where $\Delta_i$ 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.

#+name: fig:sensor_fusion_dynamic_uncertainty
#+caption: Sensor fusion architecture with sensor dynamics uncertainty
#+attr_latex: :scale 1
[[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 dynamics 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 uncertainty region of the super sensor can be 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. ref:fig:uncertainty_set_super_sensor.

#+name: fig:uncertainty_set_super_sensor
#+caption: 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 dotted and a dashed circle
#+attr_latex: :scale 1
[[file:figs/uncertainty_set_super_sensor.pdf]]

The maximum phase added $\Delta\phi(\omega)$ by the super sensor dynamics at frequency $\omega$ is then
#+name: eq:max_phase_uncertainty
\begin{equation}
    \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 eqref:eq:max_uncertainty_super_sensor is satisfied.
#+name: eq:max_uncertainty_super_sensor
\begin{equation}
   \max_\omega \big( \left|w_1 H_1\right| + \left|w_2 H_2\right|\big) < \sin\left( \Delta \phi_\text{max} \right)
\end{equation}
where $\Delta \phi_\text{max}$ is the maximum allowed added phase.

Thus the norm of the complementary filter $|H_i|$ should be made small at frequencies where $|w_i|$ is large.

* Complementary Filters Shaping using $\mathcal{H}_\infty$ Synthesis
<<sec:hinf_method>>
** Introduction                                                     :ignore:
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.

** 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
\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}
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}
  \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}

#+name: fig:h_infinity_robust_fusion
#+caption: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
#+attr_latex: :scale 1
[[file:figs/h_infinity_robust_fusion.pdf]]

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.
#+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}

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.
#+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}

#+name: eq:definition_H1
\begin{equation}
  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.

** Weighting Functions Design
<<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.

# Explain why we propose such weighting function
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.
#+name: eq:weight_formula
\begin{equation}
  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:
- the low frequency gain: $G_0 = lim_{\omega \to 0} |W(j\omega)|$
- the high frequency gain: $G_\infty = lim_{\omega \to \infty} |W(j\omega)|$
- the absolute gain at $\omega_0$: $G_c = |W(j\omega_0)|$
- the absolute slope between high and low frequency: $n$

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}

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

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

** Validation of the proposed synthesis method
<<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:
- the merging frequency is around $\SI{10}{Hz}$
- the slope of $|H_1(j\omega)|$ is $-2$ above $\SI{10}{Hz}$
- the slope of $|H_2(j\omega)|$ is $+3$ below $\SI{10}{Hz}$
- the gain of both filters is equal to $10^{-3}$ away from the merging frequency

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_synthesis_results.

#+name: tab:weights_params
#+caption: Parameters used for $W_1(s)$ and $W_2(s)$
#+ATTR_LATEX: :environment tabularx :width 0.5\linewidth :align Xcc
#+ATTR_LATEX: :center t :booktabs t :float t
| Parameter              | $W_1(s)$ | $W_2(s)$ |
|------------------------+----------+----------|
| $G_0$                  | $0.1$    | $1000$   |
| $G_\infty$             | $1000$   | $0.1$    |
| $\omega_c$ [$\si{Hz}$] | $11$     | $10$     |
| $G_c$                  | $0.5$    | $0.5$    |
| $n$                    | $2$      | $3$      |

The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_synthesis_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*}

#+name: fig:hinf_synthesis_results
#+caption: Frequency response of the weighting functions and complementary filters obtained using $\mathcal{H}_\infty$ synthesis
#+attr_latex: :scale 1
[[file:figs/hinf_synthesis_results.pdf]]

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

** Complementary Filters Specifications
<<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:
- From $0$ to $\SI{0.008}{Hz}$, the magnitude of the filter's transfer function should be less or equal to $8 \times 10^{-4}$
- Between $\SI{0.008}{Hz}$ to $\SI{0.04}{Hz}$, the filter should attenuate the input signal proportional to frequency cubed
- Between $\SI{0.04}{Hz}$ to $\SI{0.1}{Hz}$, the magnitude of the transfer function should be less than $3$
- Above $\SI{0.1}{Hz}$, the magnitude of the complementary filter should be less than $0.045$

** 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 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.

#+name: fig:ligo_weights
#+caption: Specifications and weighting functions magnitudes
#+attr_latex: :scale 1
[[file:figs/ligo_weights.pdf]]

** $\mathcal{H}_\infty$ Synthesis
<<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.

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

* Acknowledgment
:PROPERTIES:
:UNNUMBERED: t
:END:
This research benefited from a FRIA grant from the French Community of Belgium.

* Bibliography                                                       :ignore:
\bibliographystyle{elsarticle-num}
\bibliography{ref}


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