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@ -12,7 +12,7 @@
#+LATEX_HEADER_EXTRA: \author[a2,a4]{Christophe Collette} #+LATEX_HEADER_EXTRA: \author[a2,a4]{Christophe Collette}
#+LATEX_HEADER_EXTRA: \cortext[cor1]{Corresponding author. Email Address: thomas.dehaeze@esrf.fr} #+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[a1]{European Synchrotron Radiation Facility, 38000 Grenoble, France}
#+LATEX_HEADER_EXTRA: \address[a2]{University of Li\`{e}ge, PML, Department of Aerospace and Mechanical Engineering, 4000 Li\`{e}ge, Belgium.} #+LATEX_HEADER_EXTRA: \address[a2]{University of Li\`{e}ge, PML, 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[a3]{CSIR --- Structural Engineering Research Centre, Taramani, Chennai --- 600113, India.}
#+LATEX_HEADER_EXTRA: \address[a4]{Universit\'{e} Libre de Bruxelles, Precision Mechatronics Laboratory, BEAMS Department, 1050 Brussels, Belgium.} #+LATEX_HEADER_EXTRA: \address[a4]{Universit\'{e} Libre de Bruxelles, Precision Mechatronics Laboratory, BEAMS Department, 1050 Brussels, Belgium.}
@ -74,13 +74,13 @@
#+begin_frontmatter #+begin_frontmatter
#+begin_abstract #+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". 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. The obtained "super sensor" combines 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. 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. 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. In this study, this issue is addressed and a new method for designing complementary filters is proposed.
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. This method uses weighting functions to specify the wanted shape of the complementary filters that are then 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. 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. Such synthesis method is further extended for the shaping of a set of more than two complementary filters.
#+end_abstract #+end_abstract
#+begin_keyword #+begin_keyword
@ -93,16 +93,16 @@ Sensor fusion \sep{} Complementary filters \sep{} $\mathcal{H}_\infty$ synthesis
** Introduction to Sensor Fusion :ignore: ** Introduction to Sensor Fusion :ignore:
Measuring a physical quantity using sensors is always subject to several limitations. 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 electrical noise of the conditioning electronics being used. First, the accuracy of the measurement is affected by several noise sources, such as 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. 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. 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. Fortunately, a wide variety of sensors exists, each with different characteristics.
By carefully choosing the fused sensors, a so called "super sensor" is obtained that combines benefits of individual sensors. \par By carefully choosing the fused sensors, a so called "super sensor" is obtained that can combines benefits of the individual sensors. \par
** Advantages of Sensor Fusion :ignore: ** 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. 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 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 an 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. For other applications, sensor fusion is used to obtain an 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 obtain interesting properties for control\nbsp{}cite:collette15_sensor_fusion_method_high_perfor,yong16_high_speed_vertic_posit_stage. More recently, the fusion of sensors measuring different physical quantities has been proposed to obtain 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 for improving the stability margins of the feedback controller. \par 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 for improving the stability margins of the feedback controller. \par
@ -119,7 +119,7 @@ Emblematic examples are the isolation stages of gravitational wave detectors\nbs
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. 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. 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 with complementary filters. However, for Kalman filtering, assumptions must be made about the probabilistic character of the sensor noises\nbsp{}cite:robert12_introd_random_signal_applied_kalman whereas it is not the case with complementary filters.
Furthermore, the advantages of complementary filters over Kalman filtering for sensor fusion are their general applicability, their low computational cost\nbsp{}cite:higgins75_compar_compl_kalman_filter, and the fact that they are intuitive as their effects can be easily interpreted in the frequency domain. \par Furthermore, the advantages of complementary filters over Kalman filtering for sensor fusion are their general applicability, their low computational cost\nbsp{}cite:higgins75_compar_compl_kalman_filter, and the fact that they are intuitive as their effects can be easily interpreted in the frequency domain. \par
** Design Methods of Complementary filters :ignore: ** Design Methods of Complementary filters :ignore:
@ -132,8 +132,8 @@ Several design methods have been developed over the years to optimize complement
The easiest way to design complementary filters is to use analytical formulas. 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 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 have been developed over the years. As the characteristics of the super sensor depends on the proper design of the complementary filters\nbsp{}cite:dehaeze19_compl_filter_shapin_using_synth, several optimization techniques have been developed.
Some are based on the finding 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. Some are based on the finding of 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. 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. 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 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
@ -141,27 +141,27 @@ For instance, in\nbsp{}cite:jensen13_basic_uas the two gains of a Proportional I
** Problematic / gap in the research :ignore: ** Problematic / gap in the research :ignore:
The common objective of all these complementary filters design methods is to obtain a super sensor that has desired characteristics, usually in terms of noise and dynamics. The common objective of 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. 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. Hence, the robustness of the fusion is also of concern 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 desired super sensor characteristic while ensuring good fusion robustness has been proposed. \par Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to specify the desired super sensor characteristic while ensuring good fusion robustness has been proposed. \par
** Describe the paper itself / the problem which is addressed :ignore: ** 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. 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 Based on that, this paper introduces a new way to design complementary filters using the $\mathcal{H}_\infty$ synthesis which allows to shape the complementary filters' magnitude in an easy and intuitive way. \par
** Introduce Each part of the paper :ignore: ** 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. Section\nbsp{}ref:sec:requirements introduces the sensor fusion architecture and demonstrates how typical requirements can be linked to the complementary filters' magnitude.
In Section\nbsp{}ref:sec:hinf_method, the shaping of complementary filters is formulated as an $\mathcal{H}_\infty$ optimization problem using weighting functions, and the simplicity of the proposed method is illustrated with an example. In Section\nbsp{}ref:sec:hinf_method, the shaping of complementary filters is formulated 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. 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. Section\nbsp{}ref:sec:discussion compares the proposed synthesis method with the classical mixed-sensitivity synthesis, and extends it for the shaping of more than two complementary filters.
* Sensor Fusion and Complementary Filters Requirements * Sensor Fusion and Complementary Filters Requirements
<<sec:requirements>> <<sec:requirements>>
** Introduction :ignore: ** Introduction :ignore:
Complementary filters provides a framework for fusing signals from different sensors. Complementary filtering 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. 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. These requirements are discussed in this section.
@ -170,16 +170,16 @@ These requirements are discussed in this section.
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 $x$. 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 $x$. 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 $\hat{x}$. 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. The resulting sensor, termed as super sensor, can have larger bandwidth and better noise characteristics in comparison to the individual sensors.
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. 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 #+name: fig:sensor_fusion_overview
#+caption: Schematic of a sensor fusion architecture using complementary filters. #+caption: Schematic of a sensor fusion architecture using complementary filters.
[[file:figs/sensor_fusion_overview.pdf]] [[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. The complementary property of filters $H_1(s)$ and $H_2(s)$ implies that the sum of their transfer functions is equal to one.
That is, unity magnitude and zero phase at all frequencies. That is, unity magnitude and zero phase at all frequencies.
Therefore, a pair of complementary filter needs to satisfy the following condition: Therefore, a pair of complementary filter needs to satisfy the following condition:
#+name: eq:comp_filter #+name: eq:comp_filter
@ -200,7 +200,7 @@ Before filtering the sensor outputs $\tilde{x}_i$ by the complementary filters,
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. 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. 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 $x$. 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. The sensor dynamics estimate $\hat{G}_i(s)$ can be a simple gain or a more complex transfer function.
#+begin_export latex #+begin_export latex
\begin{figure}[htbp] \begin{figure}[htbp]
@ -245,13 +245,13 @@ In this section, it is supposed that all the sensors are perfectly normalized, s
The effect of a non-perfect normalization will be discussed in the next section. The effect of a non-perfect normalization will be discussed in the next section.
Provided\nbsp{}eqref:eq:perfect_dynamics is verified, the super sensor output $\hat{x}$ is then: Provided\nbsp{}eqref:eq:perfect_dynamics is verified, the super sensor output $\hat{x}$ is then equal to:
#+name: eq:estimate_perfect_dyn #+name: eq:estimate_perfect_dyn
\begin{equation} \begin{equation}
\hat{x} = x + H_1(s) n_1 + H_2(s) n_2 \hat{x} = x + H_1(s) n_1 + H_2(s) n_2
\end{equation} \end{equation}
From\nbsp{}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. From\nbsp{}eqref:eq:estimate_perfect_dyn, the complementary filters $H_1(s)$ and $H_2(s)$ are shown to only operate on the noise of the sensors.
Thus, this sensor fusion architecture permits to filter the noise of both sensors without introducing any distortion in the physical quantity to be measured. 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. This is why the two filters must be complementary.
@ -272,7 +272,7 @@ This is the simplest form of sensor fusion with complementary filters.
However, the two sensors have usually high noise levels over distinct frequency regions. 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 $\Phi_{n_1}(\omega)$. 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. Hence, by properly shaping the norm of the complementary filters, it is possible to reduce the noise of the super sensor.
** Sensor Fusion Robustness ** Sensor Fusion Robustness
<<sec:fusion_robustness>> <<sec:fusion_robustness>>
@ -280,9 +280,9 @@ Therefore, by properly shaping the norm of the complementary filters, it is poss
In practical systems the sensor normalization is not perfect and condition\nbsp{}eqref:eq:perfect_dynamics is not verified. In practical systems the sensor normalization is not perfect and condition\nbsp{}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). 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 nominal model is the estimated model used for the normalization $\hat{G}_i(s)$, $\Delta_i(s)$ 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 $|w_i(j\omega)|$. The weight $w_i(s)$ is chosen such that the real sensor dynamics $G_i(j\omega)$ 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. 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.
@ -305,7 +305,7 @@ As the nominal sensor dynamics is taken as the normalized filter, the normalized
\end{figure} \end{figure}
#+end_export #+end_export
The sensor fusion architecture with two sensor models including dynamical uncertainty is shown in Fig.\nbsp{}ref:fig:sensor_fusion_dynamic_uncertainty. The sensor fusion architecture with the sensor models including dynamical uncertainty is shown in Fig.\nbsp{}ref:fig:sensor_fusion_dynamic_uncertainty.
#+name: fig:sensor_fusion_dynamic_uncertainty #+name: fig:sensor_fusion_dynamic_uncertainty
#+caption: Sensor fusion architecture with sensor dynamics uncertainty. #+caption: Sensor fusion architecture with sensor dynamics uncertainty.
@ -317,15 +317,14 @@ The super sensor dynamics\nbsp{}eqref:eq:super_sensor_dyn_uncertainty is no long
\frac{\hat{x}}{x} = 1 + w_1(s) H_1(s) \Delta_1(s) + w_2(s) H_2(s) \Delta_2(s) \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} \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. 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 $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$ (Fig.\nbsp{}ref:fig:uncertainty_set_super_sensor).
#+name: 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. #+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]] [[file:figs/uncertainty_set_super_sensor.pdf]]
The super sensor dynamical uncertainty, and hence the robustness of the fusion, clearly depends on the complementary filters norms. The super sensor dynamical uncertainty, and hence the robustness of the fusion, clearly depends on the complementary filters' norm.
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). 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) and that is mathematically described by\nbsp{}eqref:eq:max_phase_uncertainty.
Therefore, the phase uncertainty of the super sensor dynamics depends on the Complementary filters norms\nbsp{}eqref:eq:max_phase_uncertainty.
#+name: eq:max_phase_uncertainty #+name: eq:max_phase_uncertainty
\begin{equation} \begin{equation}
@ -339,10 +338,10 @@ Typically, the norm of the complementary filter $|H_i(j\omega)|$ should be made
<<sec:hinf_method>> <<sec:hinf_method>>
** Introduction :ignore: ** 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. As shown in Section\nbsp{}ref:sec:requirements, the noise and robustness of the super sensor are a function of the complementary filters' norm.
Therefore, a complementary filters synthesis method that allows to shape their norms would be of great use. Therefore, a synthesis method of complementary filters that allows to shape their norm 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. 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. As weighting functions are used to represent the wanted complementary filters' shape during the synthesis, their proper design is discussed.
Finally, the synthesis method is validated on an simple example. Finally, the synthesis method is validated on an simple example.
** Synthesis Objective ** Synthesis Objective
@ -359,7 +358,7 @@ This is equivalent as to finding proper and stable transfer functions $H_1(s)$ a
& |H_2(j\omega)| \le \frac{1}{|W_2(j\omega)|} \quad \forall\omega \label{eq:hinf_cond_h2} & |H_2(j\omega)| \le \frac{1}{|W_2(j\omega)|} \quad \forall\omega \label{eq:hinf_cond_h2}
\end{align} \end{align}
\end{subequations} \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. $W_1(s)$ and $W_2(s)$ are two weighting transfer functions that are carefully chosen to specify the maximum wanted norm of the complementary filters during the synthesis.
** Shaping of Complementary Filters using $\mathcal{H}_\infty$ synthesis ** Shaping of Complementary Filters using $\mathcal{H}_\infty$ synthesis
<<sec:hinf_synthesis>> <<sec:hinf_synthesis>>
@ -393,13 +392,13 @@ Consider the generalized plant $P(s)$ shown in Fig.\nbsp{}ref:fig:h_infinity_rob
\end{figure} \end{figure}
#+end_export #+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\nbsp{}eqref:eq:hinf_syn_obj. Applying the standard $\mathcal{H}_\infty$ synthesis to 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\nbsp{}eqref:eq:hinf_syn_obj.
#+name: eq:hinf_syn_obj #+name: eq:hinf_syn_obj
\begin{equation} \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 \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} \end{equation}
By then defining $H_1(s)$ to be the complementary of $H_2(s)$\nbsp{}eqref:eq:definition_H1, the $\mathcal{H}_\infty$ synthesis objective becomes equivalent to\nbsp{}eqref:eq:hinf_problem which ensure that\nbsp{}eqref:eq:hinf_cond_h1 and\nbsp{}eqref:eq:hinf_cond_h2 are satisfied. 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\nbsp{}eqref:eq:hinf_problem which ensures that\nbsp{}eqref:eq:hinf_cond_h1 and\nbsp{}eqref:eq:hinf_cond_h2 are satisfied.
#+name: eq:definition_H1 #+name: eq:definition_H1
\begin{equation} \begin{equation}
@ -411,26 +410,25 @@ By then defining $H_1(s)$ to be the complementary of $H_2(s)$\nbsp{}eqref:eq:def
\left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1 \left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1
\end{equation} \end{equation}
Therefore, applying the $\mathcal{H}_\infty$ synthesis on the standard plant $P(s)$\nbsp{}eqref:eq:generalized_plant will generate two filters $H_2(s)$ and $H_1(s) \triangleq 1 - H_2(s)$ that are complementary\nbsp{}eqref:eq:comp_filter_problem_form and such that there norms are bellow specified bounds [[eqref:eq:hinf_cond_h1]],\nbsp{}eqref:eq:hinf_cond_h2. Therefore, applying the $\mathcal{H}_\infty$ synthesis to 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\nbsp{}eqref:eq:comp_filter_problem_form and such that there norms are bellow specified bounds [[eqref:eq:hinf_cond_h1]],\nbsp{}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. Note that there is only an implication 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 and not an equivalence.
Hence, the optimization may be a little bit conservative with respect to the "set" of filters on which it is performed. Hence, the optimization may be a little bit conservative with respect to the set of filters on which it is performed, see\nbsp{}[[cite:skogestad07_multiv_feedb_contr][Chap. 2.8.3]].
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. In practice, this is however not an found to be an issue.
** Weighting Functions Design ** Weighting Functions Design
<<sec:hinf_weighting_func>> <<sec:hinf_weighting_func>>
Weighting functions are used during the synthesis to specify the maximum allowed norms of the complementary filters. Weighting functions are used during the synthesis to specify the maximum allowed complementary filters' norm.
The proper design of these weighting functions is of primary importance for the success of the presented $\mathcal{H}_\infty$ synthesis of 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. 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). 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 synthesized filters' order being equal to the sum of the weighting functions' order).
Third, one should not forget the fundamental limitations imposed by the complementary property\nbsp{}eqref:eq:comp_filter. Third, one should not forget the fundamental limitations imposed by the complementary property\nbsp{}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. 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. 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\nbsp{}eqref:eq:weight_formula is proposed. To easily express these specifications, formula\nbsp{}eqref:eq:weight_formula is proposed to help with the design of weighting functions.
#+name: eq:weight_formula #+name: eq:weight_formula
\begin{equation} \begin{equation}
@ -447,7 +445,7 @@ The parameters in formula\nbsp{}eqref:eq:weight_formula are:
- $G_c = |W(j\omega_c)|$: the gain at a specific frequency $\omega_c$ in $\si{rad/s}$. - $G_c = |W(j\omega_c)|$: 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. - $n$: 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\nbsp{}eqref:eq:cond_formula_1 or\nbsp{}eqref:eq:cond_formula_2. The parameters $G_0$, $G_c$ and $G_\infty$ should either satisfy\nbsp{}eqref:eq:cond_formula_1 or\nbsp{}eqref:eq:cond_formula_2.
#+name: eq:condition_params_formula #+name: eq:condition_params_formula
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
@ -456,21 +454,24 @@ The parameters $G_0$, $G_c$ and $G_\infty$ should either satisfy condition\nbsp{
\end{align} \end{align}
\end{subequations} \end{subequations}
An example of the obtained magnitude of a weighting function generated using\nbsp{}eqref:eq:weight_formula is shown in Fig.\nbsp{}ref:fig:weight_formula. The typical magnitude of a weighting function generated using\nbsp{}eqref:eq:weight_formula is shown in Fig.\nbsp{}ref:fig:weight_formula.
#+name: fig:weight_formula #+name: fig:weight_formula
#+caption: Magnitude of a weighting function generated using the proposed formula\nbsp{}eqref:eq:weight_formula, $G_0 = 1e^{-3}$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$, $n = 3$. #+caption: Magnitude of a weighting function generated using formula\nbsp{}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]] [[file:figs/weight_formula.pdf]]
** Validation of the proposed synthesis method ** Validation of the proposed synthesis method
<<sec:hinf_example>> <<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 proposed methodology for the design of complementary filters is now applied on a simple example.
- the blending frequency is around $\SI{10}{Hz}$ Let's suppose two complementary filters $H_1(s)$ and $H_2(s)$ have to be designed such that:
- the slope of $|H_1(j\omega)|$ is $+2$ below $\SI{10}{Hz}$, its low frequency gain is $10^{-3}$ - the blending frequency is around $\SI{10}{Hz}$.
- the slope of $|H_2(j\omega)|$ is $-3$ above $\SI{10}{Hz}$, its high frequency gain is $10^{-3}$ - 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 first step is to translate the above requirements by properly designing the weighting functions.
The proposed formula\nbsp{}eqref:eq:weight_formula is here used for such purpose. The proposed formula\nbsp{}eqref:eq:weight_formula is here used for such purpose.
Parameters used are summarized in Table\nbsp{}ref:tab:weights_params. 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. 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.
@ -479,18 +480,18 @@ The inverse magnitudes of the designed weighting functions, which are representi
\begin{figure} \begin{figure}
\begin{minipage}[b]{0.49\linewidth} \begin{minipage}[b]{0.49\linewidth}
\centering \centering
\begin{tabularx}{0.65\linewidth}{ccc} \begin{tabularx}{0.60\linewidth}{ccc}
\toprule \toprule
Parameters & \(W_1(s)\) & \(W_2(s)\)\\ Parameters & \(W_1(s)\) & \(W_2(s)\)\\
\midrule \midrule
\(G_0\) & \(0.1\) & \(1000\)\\ \(G_0\) & \(0.1\) & \(1000\)\\
\(G_\infty\) & \(1000\) & \(0.1\)\\ \(G_\infty\) & \(1000\) & \(0.1\)\\
\(\omega_c\) & \(2\pi\cdot10\) & \(2\pi\cdot10\)\\ \(\omega_c\) & \(2\pi\cdot10\) & \(2\pi\cdot10\)\\
\(G_c\) & \(0.45\) & \(0.45\)\\ \(G_c\) & \(0.45\) & \(0.45\)\\
\(n\) & \(2\) & \(3\)\\ \(n\) & \(2\) & \(3\)\\
\bottomrule \bottomrule
\end{tabularx} \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}.} \captionof{table}{\label{tab:weights_params}Parameters used for \(W_1(s)\) and \(W_2(s)\) using~\eqref{eq:weight_formula}.}
\end{minipage} \end{minipage}
\hfill \hfill
\begin{minipage}[b]{0.49\linewidth} \begin{minipage}[b]{0.49\linewidth}
@ -501,7 +502,7 @@ Parameters & \(W_1(s)\) & \(W_2(s)\)\\
\end{figure} \end{figure}
#+end_export #+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 standard $\mathcal{H}_\infty$ synthesis is then applied to the generalized plant of Fig.\nbsp{}ref:fig:h_infinity_robust_fusion_plant and efficiently solved using Matlab\nbsp{}cite:matlab20.
The filter $H_2(s)$ that minimizes the $\mathcal{H}_\infty$ norm between $w$ and $[z_1,\ z_2]^T$ is obtained. 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\nbsp{}eqref:eq:hinf_synthesis_result which indicates that the synthesis is successful: the complementary filters norms are below the maximum specified upper bounds. The $\mathcal{H}_\infty$ norm is here found to be close to one\nbsp{}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. This is confirmed by the bode plots of the obtained complementary filters in Fig.\nbsp{}ref:fig:hinf_filters_results.
@ -511,8 +512,8 @@ This is confirmed by the bode plots of the obtained complementary filters in Fig
\left\|\begin{matrix} \left(1 - H_2(s)\right) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \approx 1 \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} \end{equation}
Their transfer functions in the Laplace domain are given in\nbsp{}eqref:eq:hinf_synthesis_result_tf. The transfer functions in the Laplace domain of the complementary filters are given in\nbsp{}eqref:eq:hinf_synthesis_result_tf.
As expected, the obtained filters are of order $5$, that is the sum of the weighting functions orders. As expected, the obtained filters are of order $5$, that is the sum of the weighting functions' order.
#+name: eq:hinf_synthesis_result_tf #+name: eq:hinf_synthesis_result_tf
\begin{subequations} \begin{subequations}
@ -534,9 +535,9 @@ A more complex real life example is taken up in the next section.
<<sec:application_ligo>> <<sec:application_ligo>>
** Introduction :ignore: ** 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]]. Sensor fusion using complementary filters are widely used in the active vibration isolation systems at gravitational wave detectors, such as 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. In the first isolation stage at the LIGO, two sets of complementary filters are used to form a super sensor that is incorporated 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. 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. 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. A simplified block diagram of the sensor fusion architecture is shown in Fig.\nbsp{}ref:fig:ligo_super_sensor_architecture.
@ -545,24 +546,24 @@ A simplified block diagram of the sensor fusion architecture is shown in Fig.\nb
#+caption: Simplified block diagram of the sensor blending strategy for the first stage at the LIGO\nbsp{}cite:hua04_low_ligo. #+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]] [[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 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 that allows the feedback loop to have authority at zero frequency.
The requirements on those filters are stringent and thus their design is complex and should be expressed as an optimization problem. The requirements on those filters are stringent 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. 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. After synthesis, the obtained FIR filters were found to be compliant with the requirements.
However they are of high order so their implementation is quite complex. However they are of 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. In this section, the effectiveness of the proposed complementary filter synthesis strategy is demonstrated by using the same set of requirements.
** Complementary Filters Specifications ** Complementary Filters Specifications
<<sec:ligo_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): 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 $8 \times 10^{-4}$ - Below $\SI{0.008}{Hz}$, the magnitude $|L_1(j\omega)|$ should be less than $8 \times 10^{-4}$.
- Between $\SI{0.008}{Hz}$ to $\SI{0.04}{Hz}$, the filter $L_1(s)$ should attenuate the input signal proportional to frequency cubed - From $\SI{0.008}{Hz}$ 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 $3$ - From $\SI{0.04}{Hz}$ 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 $0.045$ - 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. These specifications are therefore upper bounds on the complementary filters' magnitude.
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. 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 #+name: fig:fir_filter_ligo
@ -576,11 +577,11 @@ The weighting functions should be designed such that their inverse magnitude is
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. 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. 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. For the one corresponding to the high pass filter $W_H(s)$, a $7^{\text{th}}$ order transfer function is manually designed.
The magnitudes of the weighting functions are shown in Fig.\nbsp{}ref:fig:ligo_weights. The inverse magnitudes of the weighting functions are shown in Fig.\nbsp{}ref:fig:ligo_weights.
#+name: fig:ligo_weights #+name: fig:ligo_weights
#+caption: Specifications and weighting functions inverse magnitudes. #+caption: Specifications and weighting functions' inverse magnitude.
[[file:figs/ligo_weights.pdf]] [[file:figs/ligo_weights.pdf]]
** $\mathcal{H}_\infty$ Synthesis of the complementary filters ** $\mathcal{H}_\infty$ Synthesis of the complementary filters
@ -598,12 +599,11 @@ This confirms the effectiveness of the proposed synthesis method even when the c
* Discussion * Discussion
<<sec:discussion>> <<sec:discussion>>
** Introduction :ignore:
** "Closed-Loop" complementary filters ** "Closed-Loop" complementary filters
<<sec: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. 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\nbsp{}cite:mahony05_compl_filter_desig_special_orthog,plummer06_optim_compl_filter_their_applic_motion_measur,jensen13_basic_uas. This idea is discussed in\nbsp{}cite:mahony05_compl_filter_desig_special_orthog,plummer06_optim_compl_filter_their_applic_motion_measur,jensen13_basic_uas.
#+name: fig:feedback_sensor_fusion #+name: fig:feedback_sensor_fusion
#+caption: "Closed-Loop" complementary filters. #+caption: "Closed-Loop" complementary filters.
@ -617,14 +617,14 @@ The output $\hat{x}$ is linked to the inputs by\nbsp{}eqref:eq:closed_loop_compl
\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 \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} \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. As for any classical feedback architecture, we have that the sum of the sensitivity transfer function $S(s)$ and complementary sensitivity transfer function $T_(s)$ is equal to one\nbsp{}eqref:eq:sensitivity_sum.
#+name: eq:sensitivity_sum #+name: eq:sensitivity_sum
\begin{equation} \begin{equation}
T(s) + S(s) = 1 S(s) + T(s) = 1
\end{equation} \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. Therefore, provided that the the closed-loop system in Fig.\nbsp{}ref:fig:feedback_sensor_fusion is stable, it can be used as a set of two complementary filters.
Two sensors can then be merged as shown in Fig.\nbsp{}ref:fig:feedback_sensor_fusion_arch. Two sensors can then be merged as shown in Fig.\nbsp{}ref:fig:feedback_sensor_fusion_arch.
#+name: fig:feedback_sensor_fusion_arch #+name: fig:feedback_sensor_fusion_arch
@ -635,8 +635,8 @@ One of the main advantage of implementing and designing complementary filters us
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. 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). 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. Then the system is rearranged 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\nbsp{}eqref:eq:generalized_plant_mixed_sensitivity. The $\mathcal{H}_\infty$ mixed-sensitivity synthesis can finally be performed by applying the standard $\mathcal{H}_\infty$ synthesis to the generalized plant $P_L(s)$ which is described by\nbsp{}eqref:eq:generalized_plant_mixed_sensitivity.
#+name: eq:generalized_plant_mixed_sensitivity #+name: eq:generalized_plant_mixed_sensitivity
\begin{equation} \begin{equation}
@ -646,23 +646,22 @@ The $\mathcal{H}_\infty$ mixed-sensitivity synthesis can finally be performed by
\end{bmatrix} \end{bmatrix}
\end{equation} \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\nbsp{}eqref:eq:comp_filters_feedback_obj. 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 $z$ of the system in Fig.\nbsp{}ref:fig:feedback_sensor_fusion is less than one\nbsp{}eqref:eq:comp_filters_feedback_obj.
#+name: eq:comp_filters_feedback_obj #+name: eq:comp_filters_feedback_obj
\begin{equation} \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 \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} \end{equation}
If the synthesis is successful, two complementary filters are obtained with their magnitudes bounded by the inverse magnitudes of the weighting functions. If the synthesis is successful, the transfer functions from $\hat{x}_1$ to $\hat{x}$ and from $\hat{x}_2$ to $\hat{x}$ have their magnitude bounded by the inverse magnitude of the corresponding 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. The sensor fusion can then be implemented using the feedback architecture in Fig.\nbsp{}ref:fig:feedback_sensor_fusion_arch or more classically as shown in Fig.\nbsp{}ref:fig:sensor_fusion_overview by defining the two complementary filters using\nbsp{}eqref:eq:comp_filters_feedback.
The two architectures are equivalent regarding their inputs/outputs relationships.
#+name: eq:comp_filters_feedback #+name: eq:comp_filters_feedback
\begin{equation} \begin{equation}
H_1(s) = \frac{1}{1 + L(s)}; \quad H_2(s) = \frac{L(s)}{1 + L(s)} H_1(s) = \frac{1}{1 + L(s)}; \quad H_2(s) = \frac{L(s)}{1 + L(s)}
\end{equation} \end{equation}
The two architectures are equivalent regarding their inputs/outputs relationships.
#+begin_export latex #+begin_export latex
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}[t]{0.6\linewidth} \begin{subfigure}[t]{0.6\linewidth}
@ -684,7 +683,7 @@ The two architectures are equivalent regarding their inputs/outputs relationship
As an example, two "closed-loop" complementary filters are designed using the $\mathcal{H}_\infty$ mixed-sensitivity synthesis. 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. 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. After synthesis, a filter $L(s)$ is obtained whose magnitude is shown in Fig.\nbsp{}ref:fig:hinf_filters_results_mixed_sensitivity by the black 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 "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. The obtained "closed-loop" complementary filters are indeed equal to the ones obtained in Section\nbsp{}ref:sec:hinf_example.
@ -692,17 +691,17 @@ The obtained "closed-loop" complementary filters are indeed equal to the ones ob
#+caption: Bode plot of the obtained complementary filters after $\mathcal{H}_\infty$ mixed-sensitivity synthesis. #+caption: Bode plot of the obtained complementary filters after $\mathcal{H}_\infty$ mixed-sensitivity synthesis.
[[file:figs/hinf_filters_results_mixed_sensitivity.pdf]] [[file:figs/hinf_filters_results_mixed_sensitivity.pdf]]
** Synthesis of more than two Complementary Filters ** Synthesis of a set of three complementary filters
<<sec:hinf_three_comp_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. 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 For instance at the LIGO, three sensors (an LVDT, a seismometer and a geophone) are merged to form a super sensor\nbsp{}cite:matichard15_seism_isolat_advan_ligo (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. 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). 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. 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}. 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 A generalization of the proposed synthesis method of complementary filters is presented in this section. \par
#+begin_export latex #+begin_export latex
@ -727,15 +726,49 @@ The synthesis objective is to compute a set of $n$ stable transfer functions $[H
#+name: eq:hinf_problem_gen #+name: eq:hinf_problem_gen
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
& \sum_{i=0}^n H_i(s) = 1 \label{eq:hinf_cond_compl_gen} \\ & \sum_{i=1}^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} & \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{align}
\end{subequations} \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.
$[W_1(s),\ W_2(s),\ \dots,\ W_n(s)]$ are weighting transfer functions that are chosen to specify the maximum complementary filters' norm during the synthesis.
Such synthesis objective is closely related 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 Such synthesis objective is closely related 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. A set of $n$ complementary filters can be shaped by applying the standard $\mathcal{H}_\infty$ synthesis to 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}
If the synthesis if successful, a set of $n-1$ filters $[H_2(s),\ H_3(s),\ \dots,\ H_n(s)]$ are obtained such that\nbsp{}eqref:eq:hinf_syn_obj_gen is verified.
#+name: eq:hinf_syn_obj_gen
\begin{equation}
\left\|\begin{matrix} \left(1 - \left[ H_2(s) + H_3(s) + \dots + H_n(s) \right]\right) W_1(s) \\ H_2(s) W_2(s) \\ \vdots \\ H_n(s) W_n(s) \end{matrix}\right\|_\infty \le 1
\end{equation}
$H_1(s)$ is then defined using\nbsp{}eqref:eq:h1_comp_h2_hn which is ensuring the complementary property for the set of $n$ filters\nbsp{}eqref:eq:hinf_cond_compl_gen.
Condition\nbsp{}eqref:eq:hinf_cond_perf_gen is satisfied thanks to\nbsp{}eqref:eq:hinf_syn_obj_gen.
#+name: eq:h1_comp_h2_hn
\begin{equation}
H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) + \dots + H_n(s) \big]
\end{equation}
An example is given to validate the proposed method for the synthesis of a set of three complementary filters.
The sensors to be merged are 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 magnitude are shown in Fig.\nbsp{}ref:fig:three_complementary_filters_results (dashed curves).
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. 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 #+name: eq:generalized_plant_three_filters
@ -763,49 +796,32 @@ Consider the generalized plant $P_3(s)$ shown in Fig.\nbsp{}ref:fig:comp_filter_
\end{figure} \end{figure}
#+end_export #+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. The standard $\mathcal{H}_\infty$ synthesis is performed on the generalized plant $P_3(s)$.
Two filters $H_2(s)$ and $H_3(s)$ are obtained such that the $\mathcal{H}_\infty$ norm of the closed-loop transfer from $w$ to $[z_1,\ z_2,\ z_3]$ of the system in Fig.\nbsp{}ref:fig:comp_filter_three_hinf_fb is less than one.
Filter $H_1(s)$ is defined using\nbsp{}eqref:eq:h1_compl_h2_h3 thus ensuring the complementary property of the obtained set of filters.
#+name: eq:hinf_syn_obj_three #+name: eq:h1_compl_h2_h3
\begin{equation} \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 H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) \big]
\end{equation} \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\nbsp{}eqref:eq:hinf_problem_gen with $n=3$. \par Figure\nbsp{}ref:fig:three_complementary_filters_results displays the three synthesized complementary filters (solid lines) which confirms that the synthesis is successful.\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 #+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. #+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]] [[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 * Conclusion
<<sec:conclusion>> <<sec:conclusion>>
The sensor fusion robustness and the obtained super sensor noise can be linked to the magnitude of the complementary filters. A new method for designing complementary filters using the $\mathcal{H}_\infty$ synthesis has been proposed.
In this paper, a synthesis method that enables the shaping of the complementary filters norms has been proposed. It allows to shape the magnitude of the filters by the use of weighting functions during the synthesis.
This is very valuable in practice as the characteristics of the super sensor are linked to the complementary filters' magnitude.
Therefore typical sensor fusion objectives can be translated into requirements on the magnitudes of the filters.
Several examples were used to emphasize the simplicity and the effectiveness of the proposed method. Several examples were used to emphasize the simplicity and the effectiveness of the proposed method.
Links with "closed-loop" complementary filters were highlighted, and the proposed method was generalized for designing a set of more than two complementary filters.
The future work will aim at developing a complementary filter synthesis method that minimizes the super sensor noise while ensuring the robustness of the fusion. However, the shaping of the complementary filters' magnitude does not allow to directly optimize the super sensor noise and dynamical characteristics.
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 * Acknowledgment
:PROPERTIES: :PROPERTIES:
@ -813,6 +829,7 @@ The future work will aim at developing a complementary filter synthesis method t
:END: :END:
This research benefited from a FRIA grant from the French Community of Belgium. This research benefited from a FRIA grant from the French Community of Belgium.
This paper has been assigned the LIGO document number LIGO-P2100328.
* Data Availability * Data Availability
:PROPERTIES: :PROPERTIES:

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@ -1,4 +1,4 @@
% Created 2021-09-02 jeu. 10:02 % Created 2021-09-08 mer. 10:49
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[preprint, sort&compress]{elsarticle} \documentclass[preprint, sort&compress]{elsarticle}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
@ -22,7 +22,7 @@
\author[a3,a4]{Mohit Verma} \author[a3,a4]{Mohit Verma}
\author[a2,a4]{Christophe Collette} \author[a2,a4]{Christophe Collette}
\cortext[cor1]{Corresponding author. Email Address: thomas.dehaeze@esrf.fr} \cortext[cor1]{Corresponding author. Email Address: thomas.dehaeze@esrf.fr}
\address[a1]{European Synchrotron Radiation Facility, Grenoble, France} \address[a1]{European Synchrotron Radiation Facility, 38000 Grenoble, France}
\address[a2]{University of Li\`{e}ge, PML, Department of Aerospace and Mechanical Engineering, 4000 Li\`{e}ge, Belgium.} \address[a2]{University of Li\`{e}ge, PML, Department of Aerospace and Mechanical Engineering, 4000 Li\`{e}ge, Belgium.}
\address[a3]{CSIR --- Structural Engineering Research Centre, Taramani, Chennai --- 600113, India.} \address[a3]{CSIR --- Structural Engineering Research Centre, Taramani, Chennai --- 600113, India.}
\address[a4]{Universit\'{e} Libre de Bruxelles, Precision Mechatronics Laboratory, BEAMS Department, 1050 Brussels, Belgium.} \address[a4]{Universit\'{e} Libre de Bruxelles, Precision Mechatronics Laboratory, BEAMS Department, 1050 Brussels, Belgium.}
@ -43,13 +43,13 @@
\begin{frontmatter} \begin{frontmatter}
\begin{abstract} \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''. 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. The obtained ``super sensor'' combines 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. 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. 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. In this study, this issue is addressed and a new method for designing complementary filters is proposed.
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. This method uses weighting functions to specify the wanted shape of the complementary filters that are then 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. 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. Such synthesis method is further extended for the shaping of a set of more than two complementary filters.
\end{abstract} \end{abstract}
\begin{keyword} \begin{keyword}
@ -60,13 +60,13 @@ Sensor fusion \sep{} Complementary filters \sep{} \(\mathcal{H}_\infty\) synthes
\section{Introduction} \section{Introduction}
\label{sec:introduction} \label{sec:introduction}
Measuring a physical quantity using sensors is always subject to several limitations. 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 electrical noise of the conditioning electronics being used. First, the accuracy of the measurement is affected by several noise sources, such as 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. 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''~\cite{bendat57_optim_filter_indep_measur_two}. One way to overcome these limitations is to combine several sensors using a technique called ``sensor fusion''~\cite{bendat57_optim_filter_indep_measur_two}.
Fortunately, a wide variety of sensors exist, each with different characteristics. Fortunately, a wide variety of sensors exists, each with different characteristics.
By carefully choosing the fused sensors, a so called ``super sensor'' is obtained that combines benefits of individual sensors. \par By carefully choosing the fused sensors, a so called ``super sensor'' is obtained that can combines benefits of the individual sensors. \par
In some situations, sensor fusion is used to increase the bandwidth of the measurement~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,min15_compl_filter_desig_angle_estim}. In some situations, sensor fusion is used to increase the bandwidth of the measurement~\cite{shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,min15_compl_filter_desig_angle_estim}.
For instance, in~\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 instance, in~\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 an estimate of the measured quantity with lower noise~\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}. For other applications, sensor fusion is used to obtain an estimate of the measured quantity with lower noise~\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 obtain interesting properties for control~\cite{collette15_sensor_fusion_method_high_perfor,yong16_high_speed_vertic_posit_stage}. More recently, the fusion of sensors measuring different physical quantities has been proposed to obtain interesting properties for control~\cite{collette15_sensor_fusion_method_high_perfor,yong16_high_speed_vertic_posit_stage}.
In~\cite{collette15_sensor_fusion_method_high_perfor}, an inertial sensor used for active vibration isolation is fused with a sensor collocated with the actuator for improving the stability margins of the feedback controller. \par In~\cite{collette15_sensor_fusion_method_high_perfor}, an inertial sensor used for active vibration isolation is fused with a sensor collocated with the actuator for improving the stability margins of the feedback controller. \par
@ -77,7 +77,7 @@ Sensor fusion was also shown to be a key technology to improve the performance o
Emblematic examples are the isolation stages of gravitational wave detectors~\cite{collette15_sensor_fusion_method_high_perfor,heijningen18_low} such as the ones used at the LIGO~\cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system} and at the VIRGO~\cite{lucia18_low_frequen_optim_perfor_advan}. \par Emblematic examples are the isolation stages of gravitational wave detectors~\cite{collette15_sensor_fusion_method_high_perfor,heijningen18_low} such as the ones used at the LIGO~\cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system} and at the VIRGO~\cite{lucia18_low_frequen_optim_perfor_advan}. \par
There are mainly two ways to perform sensor fusion: either using a set of complementary filters~\cite{anderson53_instr_approac_system_steer_comput} or using Kalman filtering~\cite{brown72_integ_navig_system_kalman_filter,odry18_kalman_filter_mobil_robot_attit_estim}. There are mainly two ways to perform sensor fusion: either using a set of complementary filters~\cite{anderson53_instr_approac_system_steer_comput} or using Kalman filtering~\cite{brown72_integ_navig_system_kalman_filter,odry18_kalman_filter_mobil_robot_attit_estim}.
For sensor fusion applications, both methods are sharing many relationships~\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}. For sensor fusion applications, both methods are sharing many relationships~\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~\cite{robert12_introd_random_signal_applied_kalman} whereas it is not the case with complementary filters. However, for Kalman filtering, assumptions must be made about the probabilistic character of the sensor noises~\cite{robert12_introd_random_signal_applied_kalman} whereas it is not the case with complementary filters.
Furthermore, the advantages of complementary filters over Kalman filtering for sensor fusion are their general applicability, their low computational cost~\cite{higgins75_compar_compl_kalman_filter}, and the fact that they are intuitive as their effects can be easily interpreted in the frequency domain. \par Furthermore, the advantages of complementary filters over Kalman filtering for sensor fusion are their general applicability, their low computational cost~\cite{higgins75_compar_compl_kalman_filter}, and the fact that they are intuitive as their effects can be easily interpreted in the frequency domain. \par
A set of filters is said to be complementary if the sum of their transfer functions is equal to one at all frequencies. A set of filters is said to be complementary if the sum of their transfer functions is equal to one at all frequencies.
In the early days of complementary filtering, analog circuits were employed to physically realize the filters~\cite{anderson53_instr_approac_system_steer_comput}. In the early days of complementary filtering, analog circuits were employed to physically realize the filters~\cite{anderson53_instr_approac_system_steer_comput}.
@ -87,25 +87,25 @@ Several design methods have been developed over the years to optimize complement
The easiest way to design complementary filters is to use analytical formulas. The easiest way to design complementary filters is to use analytical formulas.
Depending on the application, the formulas used are of first order~\cite{corke04_inert_visual_sensin_system_small_auton_helic,yeh05_model_contr_hydraul_actuat_two,yong16_high_speed_vertic_posit_stage}, second order~\cite{baerveldt97_low_cost_low_weigh_attit,stoten01_fusion_kinet_data_using_compos_filter,jensen13_basic_uas} or even higher orders~\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 Depending on the application, the formulas used are of first order~\cite{corke04_inert_visual_sensin_system_small_auton_helic,yeh05_model_contr_hydraul_actuat_two,yong16_high_speed_vertic_posit_stage}, second order~\cite{baerveldt97_low_cost_low_weigh_attit,stoten01_fusion_kinet_data_using_compos_filter,jensen13_basic_uas} or even higher orders~\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~\cite{dehaeze19_compl_filter_shapin_using_synth}, several optimization techniques have been developed over the years. As the characteristics of the super sensor depends on the proper design of the complementary filters~\cite{dehaeze19_compl_filter_shapin_using_synth}, several optimization techniques have been developed.
Some are based on the finding optimal parameters of analytical formulas~\cite{jensen13_basic_uas,min15_compl_filter_desig_angle_estim,becker15_compl_filter_desig_three_frequen_bands}, while other are using convex optimization tools~\cite{hua04_polyp_fir_compl_filter_contr_system,hua05_low_ligo} such as linear matrix inequalities~\cite{pascoal99_navig_system_desig_using_time}. Some are based on the finding of optimal parameters of analytical formulas~\cite{jensen13_basic_uas,min15_compl_filter_desig_angle_estim,becker15_compl_filter_desig_three_frequen_bands}, while other are using convex optimization tools~\cite{hua04_polyp_fir_compl_filter_contr_system,hua05_low_ligo} such as linear matrix inequalities~\cite{pascoal99_navig_system_desig_using_time}.
As shown in~\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. As shown in~\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. Therefore, all the powerful tools developed for the classical control theory can also be used for the design of complementary filters.
For instance, in~\cite{jensen13_basic_uas} the two gains of a Proportional Integral (PI) controller are optimized to minimize the noise of the super sensor. \par For instance, in~\cite{jensen13_basic_uas} the two gains of a Proportional Integral (PI) controller are optimized to minimize the noise of the super sensor. \par
The common objective of all these complementary filters design methods is to obtain a super sensor that has desired characteristics, usually in terms of noise and dynamics. The common objective of 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~\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. Moreover, as reported in~\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. Hence, the robustness of the fusion is also of concern 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 desired super sensor characteristic while ensuring good fusion robustness has been proposed. \par Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to specify the desired super sensor characteristic while ensuring good fusion robustness has been proposed. \par
Fortunately, both the robustness of the fusion and the super sensor characteristics can be linked to the magnitude of the complementary filters~\cite{dehaeze19_compl_filter_shapin_using_synth}. Fortunately, both the robustness of the fusion and the super sensor characteristics can be linked to the magnitude of the complementary filters~\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 Based on that, this paper introduces a new way to design complementary filters using the \(\mathcal{H}_\infty\) synthesis which allows to shape the complementary filters' magnitude in an easy and intuitive way. \par
Section~\ref{sec:requirements} introduces the sensor fusion architecture and demonstrates how typical requirements can be linked to the complementary filters magnitudes. Section~\ref{sec:requirements} introduces the sensor fusion architecture and demonstrates how typical requirements can be linked to the complementary filters' magnitude.
In Section~\ref{sec:hinf_method}, the shaping of complementary filters is formulated as an \(\mathcal{H}_\infty\) optimization problem using weighting functions, and the simplicity of the proposed method is illustrated with an example. In Section~\ref{sec:hinf_method}, the shaping of complementary filters is formulated 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~\ref{sec:application_ligo} by designing complex complementary filters. The synthesis method is further validated in Section~\ref{sec:application_ligo} by designing complex complementary filters.
Section~\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. Section~\ref{sec:discussion} compares the proposed synthesis method with the classical mixed-sensitivity synthesis, and extends it for the shaping of more than two complementary filters.
\section{Sensor Fusion and Complementary Filters Requirements} \section{Sensor Fusion and Complementary Filters Requirements}
\label{sec:requirements} \label{sec:requirements}
Complementary filters provides a framework for fusing signals from different sensors. Complementary filtering 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. 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. These requirements are discussed in this section.
\subsection{Sensor Fusion Architecture} \subsection{Sensor Fusion Architecture}
@ -113,9 +113,9 @@ These requirements are discussed in this section.
A general sensor fusion architecture using complementary filters is shown in Fig.~\ref{fig:sensor_fusion_overview} where several sensors (here two) are measuring the same physical quantity \(x\). A general sensor fusion architecture using complementary filters is shown in Fig.~\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 \(x\). 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 \(\hat{x}\). 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. The resulting sensor, termed as super sensor, can have larger bandwidth and better noise characteristics in comparison to the individual sensors.
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. This means that the super sensor provides an estimate \(\hat{x}\) of \(x\) which can be more accurate over a larger frequency band than the outputs of the individual sensors.
\begin{figure}[htbp] \begin{figure}[htbp]
@ -124,7 +124,7 @@ This means that the super sensor provides an estimate \(\hat{x}\) of \(x\) which
\caption{\label{fig:sensor_fusion_overview}Schematic of a sensor fusion architecture using complementary filters.} \caption{\label{fig:sensor_fusion_overview}Schematic of a sensor fusion architecture using complementary filters.}
\end{figure} \end{figure}
The complementary property of filters \(H_1(s)\) and \(H_2(s)\) implies that the sum of their transfer functions is equal to unity. The complementary property of filters \(H_1(s)\) and \(H_2(s)\) implies that the sum of their transfer functions is equal to one.
That is, unity magnitude and zero phase at all frequencies. That is, unity magnitude and zero phase at all frequencies.
Therefore, a pair of complementary filter needs to satisfy the following condition: Therefore, a pair of complementary filter needs to satisfy the following condition:
\begin{equation} \begin{equation}
@ -145,7 +145,7 @@ Before filtering the sensor outputs \(\tilde{x}_i\) by the complementary filters
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.~\ref{fig:sensor_model_calibrated}. 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.~\ref{fig:sensor_model_calibrated}.
It is here supposed that the sensor inverse \(\hat{G}_i^{-1}(s)\) is proper and stable. 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 \(x\). 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. The sensor dynamics estimate \(\hat{G}_i(s)\) can be a simple gain or a more complex transfer function.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}[b]{0.49\linewidth} \begin{subfigure}[b]{0.49\linewidth}
@ -190,13 +190,13 @@ In this section, it is supposed that all the sensors are perfectly normalized, s
The effect of a non-perfect normalization will be discussed in the next section. 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: Provided~\eqref{eq:perfect_dynamics} is verified, the super sensor output \(\hat{x}\) is then equal to:
\begin{equation} \begin{equation}
\label{eq:estimate_perfect_dyn} \label{eq:estimate_perfect_dyn}
\hat{x} = x + H_1(s) n_1 + H_2(s) n_2 \hat{x} = x + H_1(s) n_1 + H_2(s) n_2
\end{equation} \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. From~\eqref{eq:estimate_perfect_dyn}, the complementary filters \(H_1(s)\) and \(H_2(s)\) are shown to only operate on the noise of the sensors.
Thus, this sensor fusion architecture permits to filter the noise of both sensors without introducing any distortion in the physical quantity to be measured. 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. This is why the two filters must be complementary.
@ -217,7 +217,7 @@ This is the simplest form of sensor fusion with complementary filters.
However, the two sensors have usually high noise levels over distinct frequency regions. 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 \(\Phi_{n_1}(\omega)\). 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. Hence, by properly shaping the norm of the complementary filters, it is possible to reduce the noise of the super sensor.
\subsection{Sensor Fusion Robustness} \subsection{Sensor Fusion Robustness}
\label{sec:fusion_robustness} \label{sec:fusion_robustness}
@ -225,9 +225,9 @@ Therefore, by properly shaping the norm of the complementary filters, it is poss
In practical systems the sensor normalization is not perfect and condition~\eqref{eq:perfect_dynamics} is not verified. 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.~\ref{fig:sensor_model_uncertainty}). In order to study such imperfection, a multiplicative input uncertainty is added to the sensor dynamics (Fig.~\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 nominal model is the estimated model used for the normalization \(\hat{G}_i(s)\), \(\Delta_i(s)\) 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 \(|w_i(j\omega)|\). The weight \(w_i(s)\) is chosen such that the real sensor dynamics \(G_i(j\omega)\) 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.~\ref{fig:sensor_model_uncertainty_simplified}. As the nominal sensor dynamics is taken as the normalized filter, the normalized sensor can be further simplified as shown in Fig.~\ref{fig:sensor_model_uncertainty_simplified}.
@ -248,7 +248,7 @@ As the nominal sensor dynamics is taken as the normalized filter, the normalized
\centering \centering
\end{figure} \end{figure}
The sensor fusion architecture with two sensor models including dynamical uncertainty is shown in Fig.~\ref{fig:sensor_fusion_dynamic_uncertainty}. The sensor fusion architecture with the sensor models including dynamical uncertainty is shown in Fig.~\ref{fig:sensor_fusion_dynamic_uncertainty}.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -262,7 +262,7 @@ The super sensor dynamics~\eqref{eq:super_sensor_dyn_uncertainty} is no longer e
\frac{\hat{x}}{x} = 1 + w_1(s) H_1(s) \Delta_1(s) + w_2(s) H_2(s) \Delta_2(s) \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} \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.~\ref{fig:uncertainty_set_super_sensor}. 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 \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\) (Fig.~\ref{fig:uncertainty_set_super_sensor}).
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -270,9 +270,8 @@ The dynamical uncertainty of the super sensor can be graphically represented in
\caption{\label{fig:uncertainty_set_super_sensor}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.} \caption{\label{fig:uncertainty_set_super_sensor}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.}
\end{figure} \end{figure}
The super sensor dynamical uncertainty, and hence the robustness of the fusion, clearly depends on the complementary filters norms. The super sensor dynamical uncertainty, and hence the robustness of the fusion, clearly depends on the complementary filters' norm.
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.~\ref{fig:uncertainty_set_super_sensor}). 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.~\ref{fig:uncertainty_set_super_sensor}) and that is mathematically described by~\eqref{eq:max_phase_uncertainty}.
Therefore, the phase uncertainty of the super sensor dynamics depends on the Complementary filters norms~\eqref{eq:max_phase_uncertainty}.
\begin{equation} \begin{equation}
\label{eq:max_phase_uncertainty} \label{eq:max_phase_uncertainty}
@ -284,10 +283,10 @@ Typically, the norm of the complementary filter \(|H_i(j\omega)|\) should be mad
\section{Complementary Filters Shaping} \section{Complementary Filters Shaping}
\label{sec:hinf_method} \label{sec:hinf_method}
As shown in Section~\ref{sec:requirements}, the noise and robustness of the super sensor are a function of the complementary filters norms. As shown in Section~\ref{sec:requirements}, the noise and robustness of the super sensor are a function of the complementary filters' norm.
Therefore, a complementary filters synthesis method that allows to shape their norms would be of great use. Therefore, a synthesis method of complementary filters that allows to shape their norm 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. 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. As weighting functions are used to represent the wanted complementary filters' shape during the synthesis, their proper design is discussed.
Finally, the synthesis method is validated on an simple example. Finally, the synthesis method is validated on an simple example.
\subsection{Synthesis Objective} \subsection{Synthesis Objective}
\label{sec:synthesis_objective} \label{sec:synthesis_objective}
@ -303,7 +302,7 @@ This is equivalent as to finding proper and stable transfer functions \(H_1(s)\)
& |H_2(j\omega)| \le \frac{1}{|W_2(j\omega)|} \quad \forall\omega \label{eq:hinf_cond_h2} & |H_2(j\omega)| \le \frac{1}{|W_2(j\omega)|} \quad \forall\omega \label{eq:hinf_cond_h2}
\end{align} \end{align}
\end{subequations} \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. \(W_1(s)\) and \(W_2(s)\) are two weighting transfer functions that are carefully chosen to specify the maximum wanted norm of the complementary filters during the synthesis.
\subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis} \subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
\label{sec:hinf_synthesis} \label{sec:hinf_synthesis}
@ -335,13 +334,13 @@ Consider the generalized plant \(P(s)\) shown in Fig.~\ref{fig:h_infinity_robust
\centering \centering
\end{figure} \end{figure}
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.~\ref{fig:h_infinity_robust_fusion_fb} from \(w\) to \([z_1, \ z_2]\) is less than one~\eqref{eq:hinf_syn_obj}. Applying the standard \(\mathcal{H}_\infty\) synthesis to 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.~\ref{fig:h_infinity_robust_fusion_fb} from \(w\) to \([z_1, \ z_2]\) is less than one~\eqref{eq:hinf_syn_obj}.
\begin{equation} \begin{equation}
\label{eq:hinf_syn_obj} \label{eq:hinf_syn_obj}
\left\|\begin{matrix} \left(1 - H_2(s)\right) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1 \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} \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. 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 ensures that~\eqref{eq:hinf_cond_h1} and~\eqref{eq:hinf_cond_h2} are satisfied.
\begin{equation} \begin{equation}
\label{eq:definition_H1} \label{eq:definition_H1}
@ -353,26 +352,25 @@ By then defining \(H_1(s)\) to be the complementary of \$H\_2(s)\$~\eqref{eq:def
\left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1 \left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1
\end{equation} \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}. Therefore, applying the \(\mathcal{H}_\infty\) synthesis to 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~\eqref{eq:hinf_problem} and the initial synthesis objectives~\eqref{eq:hinf_cond_h1} and~\eqref{eq:hinf_cond_h2}, but only an implication. Note that there is only an implication between the \(\mathcal{H}_\infty\) norm condition~\eqref{eq:hinf_problem} and the initial synthesis objectives~\eqref{eq:hinf_cond_h1} and~\eqref{eq:hinf_cond_h2} and not an equivalence.
Hence, the optimization may be a little bit conservative with respect to the ``set'' of filters on which it is performed. Hence, the optimization may be a little bit conservative with respect to the set of filters on which it is performed, see~\cite[Chap. 2.8.3]{skogestad07_multiv_feedb_contr}.
There might be solutions were the objectives~\eqref{eq:hinf_cond_h1} and~\eqref{eq:hinf_cond_h2} are valid but where the \(\mathcal{H}_\infty\) norm~\eqref{eq:hinf_problem} is larger than one.
In practice, this is however not an found to be an issue. In practice, this is however not an found to be an issue.
\subsection{Weighting Functions Design} \subsection{Weighting Functions Design}
\label{sec:hinf_weighting_func} \label{sec:hinf_weighting_func}
Weighting functions are used during the synthesis to specify the maximum allowed norms of the complementary filters. Weighting functions are used during the synthesis to specify the maximum allowed complementary filters' norm.
The proper design of these weighting functions is of primary importance for the success of the presented \(\mathcal{H}_\infty\) synthesis of 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. 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). 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 synthesized filters' order 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}. 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. 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. 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. To easily express these specifications, formula~\eqref{eq:weight_formula} is proposed to help with the design of weighting functions.
\begin{equation} \begin{equation}
\label{eq:weight_formula} \label{eq:weight_formula}
@ -391,7 +389,7 @@ The parameters in formula~\eqref{eq:weight_formula} are:
\item \(n\): the slope between high and low frequency. It also corresponds to the order of the weighting function. \item \(n\): the slope between high and low frequency. It also corresponds to the order of the weighting function.
\end{itemize} \end{itemize}
The parameters \(G_0\), \(G_c\) and \(G_\infty\) should either satisfy condition~\eqref{eq:cond_formula_1} or~\eqref{eq:cond_formula_2}. The parameters \(G_0\), \(G_c\) and \(G_\infty\) should either satisfy~\eqref{eq:cond_formula_1} or~\eqref{eq:cond_formula_2}.
\begin{subequations} \begin{subequations}
\label{eq:condition_params_formula} \label{eq:condition_params_formula}
\begin{align} \begin{align}
@ -400,25 +398,28 @@ The parameters \(G_0\), \(G_c\) and \(G_\infty\) should either satisfy condition
\end{align} \end{align}
\end{subequations} \end{subequations}
An example of the obtained magnitude of a weighting function generated using~\eqref{eq:weight_formula} is shown in Fig.~\ref{fig:weight_formula}. The typical magnitude of a weighting function generated using~\eqref{eq:weight_formula} is shown in Fig.~\ref{fig:weight_formula}.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[scale=1]{figs/weight_formula.pdf} \includegraphics[scale=1]{figs/weight_formula.pdf}
\caption{\label{fig:weight_formula}Magnitude of a weighting function generated using the proposed formula~\eqref{eq:weight_formula}, \(G_0 = 1e^{-3}\), \(G_\infty = 10\), \(\omega_c = \SI{10}{Hz}\), \(G_c = 2\), \(n = 3\).} \caption{\label{fig:weight_formula}Magnitude of a weighting function generated using formula~\eqref{eq:weight_formula}, \(G_0 = 1e^{-3}\), \(G_\infty = 10\), \(\omega_c = \SI{10}{Hz}\), \(G_c = 2\), \(n = 3\).}
\end{figure} \end{figure}
\subsection{Validation of the proposed synthesis method} \subsection{Validation of the proposed synthesis method}
\label{sec:hinf_example} \label{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 proposed methodology for the design of complementary filters is now applied on a simple example.
Let's suppose two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that:
\begin{itemize} \begin{itemize}
\item the blending frequency is around \(\SI{10}{Hz}\) \item the blending frequency is around \(\SI{10}{Hz}\).
\item the slope of \(|H_1(j\omega)|\) is \(+2\) below \(\SI{10}{Hz}\), its low frequency gain is \(10^{-3}\) \item the slope of \(|H_1(j\omega)|\) is \(+2\) below \(\SI{10}{Hz}\).
\item the slope of \(|H_2(j\omega)|\) is \(-3\) above \(\SI{10}{Hz}\), its high frequency gain is \(10^{-3}\) Its low frequency gain is \(10^{-3}\).
\item the slope of \(|H_2(j\omega)|\) is \(-3\) above \(\SI{10}{Hz}\).
Its high frequency gain is \(10^{-3}\).
\end{itemize} \end{itemize}
The first step is to translate the above requirements into the design of the weighting functions. The first step is to translate the above requirements by properly designing the weighting functions.
The proposed formula~\eqref{eq:weight_formula} is here used for such purpose. The proposed formula~\eqref{eq:weight_formula} is here used for such purpose.
Parameters used are summarized in Table~\ref{tab:weights_params}. Parameters used are summarized in Table~\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.~\ref{fig:weights_W1_W2}. 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.~\ref{fig:weights_W1_W2}.
@ -426,18 +427,18 @@ The inverse magnitudes of the designed weighting functions, which are representi
\begin{figure} \begin{figure}
\begin{minipage}[b]{0.49\linewidth} \begin{minipage}[b]{0.49\linewidth}
\centering \centering
\begin{tabularx}{0.65\linewidth}{ccc} \begin{tabularx}{0.60\linewidth}{ccc}
\toprule \toprule
Parameters & \(W_1(s)\) & \(W_2(s)\)\\ Parameters & \(W_1(s)\) & \(W_2(s)\)\\
\midrule \midrule
\(G_0\) & \(0.1\) & \(1000\)\\ \(G_0\) & \(0.1\) & \(1000\)\\
\(G_\infty\) & \(1000\) & \(0.1\)\\ \(G_\infty\) & \(1000\) & \(0.1\)\\
\(\omega_c\) & \(2\pi\cdot10\) & \(2\pi\cdot10\)\\ \(\omega_c\) & \(2\pi\cdot10\) & \(2\pi\cdot10\)\\
\(G_c\) & \(0.45\) & \(0.45\)\\ \(G_c\) & \(0.45\) & \(0.45\)\\
\(n\) & \(2\) & \(3\)\\ \(n\) & \(2\) & \(3\)\\
\bottomrule \bottomrule
\end{tabularx} \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}.} \captionof{table}{\label{tab:weights_params}Parameters used for \(W_1(s)\) and \(W_2(s)\) using~\eqref{eq:weight_formula}.}
\end{minipage} \end{minipage}
\hfill \hfill
\begin{minipage}[b]{0.49\linewidth} \begin{minipage}[b]{0.49\linewidth}
@ -447,7 +448,7 @@ Parameters & \(W_1(s)\) & \(W_2(s)\)\\
\end{minipage} \end{minipage}
\end{figure} \end{figure}
The \(\mathcal{H}_\infty\) synthesis is then applied to the generalized plant of Fig.~\ref{fig:h_infinity_robust_fusion_plant} on efficiently solved in Matlab~\cite{matlab20} using the Robust Control Toolbox. The standard \(\mathcal{H}_\infty\) synthesis is then applied to the generalized plant of Fig.~\ref{fig:h_infinity_robust_fusion_plant} and efficiently solved using Matlab~\cite{matlab20}.
The filter \(H_2(s)\) that minimizes the \(\mathcal{H}_\infty\) norm between \(w\) and \([z_1,\ z_2]^T\) is obtained. 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. 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.~\ref{fig:hinf_filters_results}. This is confirmed by the bode plots of the obtained complementary filters in Fig.~\ref{fig:hinf_filters_results}.
@ -457,8 +458,8 @@ This is confirmed by the bode plots of the obtained complementary filters in Fig
\left\|\begin{matrix} \left(1 - H_2(s)\right) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \approx 1 \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} \end{equation}
Their transfer functions in the Laplace domain are given in~\eqref{eq:hinf_synthesis_result_tf}. The transfer functions in the Laplace domain of the complementary filters 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. As expected, the obtained filters are of order \(5\), that is the sum of the weighting functions' order.
\begin{subequations} \begin{subequations}
\label{eq:hinf_synthesis_result_tf} \label{eq:hinf_synthesis_result_tf}
@ -480,9 +481,9 @@ A more complex real life example is taken up in the next section.
\section{Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO} \section{Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO}
\label{sec:application_ligo} \label{sec:application_ligo}
Sensor fusion using complementary filters are widely used in active vibration isolation systems in gravitational wave detectors such at the LIGO~\cite{matichard15_seism_isolat_advan_ligo,hua05_low_ligo}, the VIRGO~\cite{lucia18_low_frequen_optim_perfor_advan,heijningen18_low} and the KAGRA \cite[Chap. 5]{sekiguchi16_study_low_frequen_vibrat_isolat_system}. Sensor fusion using complementary filters are widely used in the active vibration isolation systems at gravitational wave detectors, such as at the LIGO~\cite{matichard15_seism_isolat_advan_ligo,hua05_low_ligo}, the VIRGO~\cite{lucia18_low_frequen_optim_perfor_advan,heijningen18_low} and the KAGRA \cite[Chap. 5]{sekiguchi16_study_low_frequen_vibrat_isolat_system}.
In the first isolation stage at the LIGO, two sets of complementary filters are used and included in a feedback loop~\cite{hua04_low_ligo}. In the first isolation stage at the LIGO, two sets of complementary filters are used to form a super sensor that is incorporated in a feedback loop~\cite{hua04_low_ligo}.
A set of complementary filters (\(L_2,H_2\)) is first used to fuse a seismometer and a geophone. 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. 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.~\ref{fig:ligo_super_sensor_architecture}. A simplified block diagram of the sensor fusion architecture is shown in Fig.~\ref{fig:ligo_super_sensor_architecture}.
@ -493,25 +494,25 @@ A simplified block diagram of the sensor fusion architecture is shown in Fig.~\r
\caption{\label{fig:ligo_super_sensor_architecture}Simplified block diagram of the sensor blending strategy for the first stage at the LIGO~\cite{hua04_low_ligo}.} \caption{\label{fig:ligo_super_sensor_architecture}Simplified block diagram of the sensor blending strategy for the first stage at the LIGO~\cite{hua04_low_ligo}.}
\end{figure} \end{figure}
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 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 that allows the feedback loop to have authority at zero frequency.
The requirements on those filters are stringent and thus their design is complex and should be expressed as an optimization problem. The requirements on those filters are stringent and thus their design is complex and should be expressed as an optimization problem.
The approach used in~\cite{hua05_low_ligo} is to use FIR complementary filters and to write the synthesis as a convex optimization problem. The approach used in~\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. After synthesis, the obtained FIR filters were found to be compliant with the requirements.
However they are of high order so their implementation is quite complex. However they are of 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. In this section, the effectiveness of the proposed complementary filter synthesis strategy is demonstrated by using the same set of requirements.
\subsection{Complementary Filters Specifications} \subsection{Complementary Filters Specifications}
\label{sec:ligo_specifications} \label{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~\cite{hua04_polyp_fir_compl_filter_contr_system}): The specifications for the set of complementary filters (\(L_1,H_1\)) used at the LIGO are summarized below (for further details, refer to~\cite{hua04_polyp_fir_compl_filter_contr_system}):
\begin{itemize} \begin{itemize}
\item From \(0\) to \(\SI{0.008}{Hz}\), the magnitude \(|L_1(j\omega)|\) should be less or equal to \(8 \times 10^{-4}\) \item Below \(\SI{0.008}{Hz}\), the magnitude \(|L_1(j\omega)|\) should be less than \(8 \times 10^{-4}\).
\item Between \(\SI{0.008}{Hz}\) to \(\SI{0.04}{Hz}\), the filter \(L_1(s)\) should attenuate the input signal proportional to frequency cubed \item From \(\SI{0.008}{Hz}\) to \(\SI{0.04}{Hz}\), the filter \(L_1(s)\) should attenuate the input signal proportional to frequency cubed.
\item Between \(\SI{0.04}{Hz}\) to \(\SI{0.1}{Hz}\), the magnitude \(|L_1(j\omega)|\) should be less than \(3\) \item From \(\SI{0.04}{Hz}\) to \(\SI{0.1}{Hz}\), the magnitude \(|L_1(j\omega)|\) should be less than \(3\).
\item Above \(\SI{0.1}{Hz}\), the magnitude \(|H_1(j\omega)|\) should be less than \(0.045\) \item Above \(\SI{0.1}{Hz}\), the magnitude \(|H_1(j\omega)|\) should be less than \(0.045\).
\end{itemize} \end{itemize}
These specifications are therefore upper bounds on the complementary filters' magnitudes. These specifications are therefore upper bounds on the complementary filters' magnitude.
They are physically represented in Fig.~\ref{fig:fir_filter_ligo} as well as the obtained magnitude of the FIR filters in~\cite{hua05_low_ligo}. They are physically represented in Fig.~\ref{fig:fir_filter_ligo} as well as the obtained magnitude of the FIR filters in~\cite{hua05_low_ligo}.
\begin{figure}[htbp] \begin{figure}[htbp]
@ -527,13 +528,13 @@ The weighting functions should be designed such that their inverse magnitude is
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. 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. 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. For the one corresponding to the high pass filter \(W_H(s)\), a \(7^{\text{th}}\) order transfer function is manually designed.
The magnitudes of the weighting functions are shown in Fig.~\ref{fig:ligo_weights}. The inverse magnitudes of the weighting functions are shown in Fig.~\ref{fig:ligo_weights}.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
\includegraphics[scale=1]{figs/ligo_weights.pdf} \includegraphics[scale=1]{figs/ligo_weights.pdf}
\caption{\label{fig:ligo_weights}Specifications and weighting functions inverse magnitudes.} \caption{\label{fig:ligo_weights}Specifications and weighting functions' inverse magnitude.}
\end{figure} \end{figure}
\subsection{\(\mathcal{H}_\infty\) Synthesis of the complementary filters} \subsection{\(\mathcal{H}_\infty\) Synthesis of the complementary filters}
@ -555,8 +556,9 @@ This confirms the effectiveness of the proposed synthesis method even when the c
\label{sec:discussion} \label{sec:discussion}
\subsection{``Closed-Loop'' complementary filters} \subsection{``Closed-Loop'' complementary filters}
\label{sec:closed_loop_complementary_filters} \label{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.~\ref{fig:feedback_sensor_fusion}. An alternative way to implement complementary filters is by using a fundamental property of the classical feedback architecture shown in Fig.~\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}. This idea is discussed in~\cite{mahony05_compl_filter_desig_special_orthog,plummer06_optim_compl_filter_their_applic_motion_measur,jensen13_basic_uas}.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -572,14 +574,14 @@ The output \(\hat{x}\) is linked to the inputs by~\eqref{eq:closed_loop_compleme
\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 \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} \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~\eqref{eq:sensitivity_sum}. As for any classical feedback architecture, we have that the sum of the sensitivity transfer function \(S(s)\) and complementary sensitivity transfer function \(T_(s)\) is equal to one~\eqref{eq:sensitivity_sum}.
\begin{equation} \begin{equation}
\label{eq:sensitivity_sum} \label{eq:sensitivity_sum}
T(s) + S(s) = 1 S(s) + T(s) = 1
\end{equation} \end{equation}
Therefore, provided that the closed-loop system is stable, the closed-loop system in Fig.~\ref{fig:feedback_sensor_fusion} is corresponding to two complementary filters. Therefore, provided that the the closed-loop system in Fig.~\ref{fig:feedback_sensor_fusion} is stable, it can be used as a set of two complementary filters.
Two sensors can then be merged as shown in Fig.~\ref{fig:feedback_sensor_fusion_arch}. Two sensors can then be merged as shown in Fig.~\ref{fig:feedback_sensor_fusion_arch}.
\begin{figure}[htbp] \begin{figure}[htbp]
@ -592,8 +594,8 @@ One of the main advantage of implementing and designing complementary filters us
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. 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.~\ref{fig:feedback_synthesis_architecture}). To do so, weighting functions \(W_1(s)\) and \(W_2(s)\) are added to respectively shape \(S(s)\) and \(T(s)\) (Fig.~\ref{fig:feedback_synthesis_architecture}).
Then the system is re-organized to form the generalized plant \(P_L(s)\) shown in Fig.~\ref{fig:feedback_synthesis_architecture_generalized_plant}. Then the system is rearranged to form the generalized plant \(P_L(s)\) shown in Fig.~\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}. The \(\mathcal{H}_\infty\) mixed-sensitivity synthesis can finally be performed by applying the standard \(\mathcal{H}_\infty\) synthesis to the generalized plant \(P_L(s)\) which is described by~\eqref{eq:generalized_plant_mixed_sensitivity}.
\begin{equation} \begin{equation}
\label{eq:generalized_plant_mixed_sensitivity} \label{eq:generalized_plant_mixed_sensitivity}
@ -603,23 +605,22 @@ The \(\mathcal{H}_\infty\) mixed-sensitivity synthesis can finally be performed
\end{bmatrix} \end{bmatrix}
\end{equation} \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}. 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 \(z\) of the system in Fig.~\ref{fig:feedback_sensor_fusion} is less than one~\eqref{eq:comp_filters_feedback_obj}.
\begin{equation} \begin{equation}
\label{eq:comp_filters_feedback_obj} \label{eq:comp_filters_feedback_obj}
\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 \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} \end{equation}
If the synthesis is successful, two complementary filters are obtained with their magnitudes bounded by the inverse magnitudes of the weighting functions. If the synthesis is successful, the transfer functions from \(\hat{x}_1\) to \(\hat{x}\) and from \(\hat{x}_2\) to \(\hat{x}\) have their magnitude bounded by the inverse magnitude of the corresponding weighting functions.
The sensor fusion can then be implemented as shown in Fig.~\ref{fig:feedback_sensor_fusion_arch} using the feedback architecture or more classically as shown in Fig.~\ref{fig:sensor_fusion_overview} by defining the two complementary filters as in~\eqref{eq:comp_filters_feedback}. The sensor fusion can then be implemented using the feedback architecture in Fig.~\ref{fig:feedback_sensor_fusion_arch} or more classically as shown in Fig.~\ref{fig:sensor_fusion_overview} by defining the two complementary filters using~\eqref{eq:comp_filters_feedback}.
The two architectures are equivalent regarding their inputs/outputs relationships.
\begin{equation} \begin{equation}
\label{eq:comp_filters_feedback} \label{eq:comp_filters_feedback}
H_1(s) = \frac{1}{1 + L(s)}; \quad H_2(s) = \frac{L(s)}{1 + L(s)} H_1(s) = \frac{1}{1 + L(s)}; \quad H_2(s) = \frac{L(s)}{1 + L(s)}
\end{equation} \end{equation}
The two architectures are equivalent regarding their inputs/outputs relationships.
\begin{figure}[htbp] \begin{figure}[htbp]
\begin{subfigure}[t]{0.6\linewidth} \begin{subfigure}[t]{0.6\linewidth}
\centering \centering
@ -639,7 +640,7 @@ The two architectures are equivalent regarding their inputs/outputs relationship
As an example, two ``closed-loop'' complementary filters are designed using the \(\mathcal{H}_\infty\) mixed-sensitivity synthesis. 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~\eqref{eq:weight_formula} with parameters shown in Table~\ref{tab:weights_params}. The weighting functions are designed using formula~\eqref{eq:weight_formula} with parameters shown in Table~\ref{tab:weights_params}.
After synthesis, a filter \(L(s)\) is obtained, its magnitude is shown in Fig.~\ref{fig:hinf_filters_results_mixed_sensitivity} by the dashed line. After synthesis, a filter \(L(s)\) is obtained whose magnitude is shown in Fig.~\ref{fig:hinf_filters_results_mixed_sensitivity} by the black dashed line.
The ``closed-loop'' complementary filters are compared with the inverse magnitude of the weighting functions in Fig.~\ref{fig:hinf_filters_results_mixed_sensitivity} confirming that the synthesis is successful. The ``closed-loop'' complementary filters are compared with the inverse magnitude of the weighting functions in Fig.~\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~\ref{sec:hinf_example}. The obtained ``closed-loop'' complementary filters are indeed equal to the ones obtained in Section~\ref{sec:hinf_example}.
@ -649,17 +650,17 @@ The obtained ``closed-loop'' complementary filters are indeed equal to the ones
\caption{\label{fig:hinf_filters_results_mixed_sensitivity}Bode plot of the obtained complementary filters after \(\mathcal{H}_\infty\) mixed-sensitivity synthesis.} \caption{\label{fig:hinf_filters_results_mixed_sensitivity}Bode plot of the obtained complementary filters after \(\mathcal{H}_\infty\) mixed-sensitivity synthesis.}
\end{figure} \end{figure}
\subsection{Synthesis of more than two Complementary Filters} \subsection{Synthesis of a set of three complementary filters}
\label{sec:hinf_three_comp_filters} \label{sec:hinf_three_comp_filters}
Some applications may require to merge more than two sensors~\cite{stoten01_fusion_kinet_data_using_compos_filter,becker15_compl_filter_desig_three_frequen_bands}. Some applications may require to merge more than two sensors~\cite{stoten01_fusion_kinet_data_using_compos_filter,becker15_compl_filter_desig_three_frequen_bands}.
For instance at the LIGO~\cite{matichard15_seism_isolat_advan_ligo}, three sensors (an LVDT, a seismometer and a geophone) are merged to form a super sensor (Fig.~\ref{fig:ligo_super_sensor_architecture}).\par For instance at the LIGO, three sensors (an LVDT, a seismometer and a geophone) are merged to form a super sensor~\cite{matichard15_seism_isolat_advan_ligo} (Fig.~\ref{fig:ligo_super_sensor_architecture}).\par
When merging \(n>2\) sensors using complementary filters, two architectures can be used as shown in Fig.~\ref{fig:sensor_fusion_three}. When merging \(n>2\) sensors using complementary filters, two architectures can be used as shown in Fig.~\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.~\ref{fig:sensor_fusion_three_sequential}), or in a ``parallel'' way where one set of \(n\) complementary filters is used (Fig.~\ref{fig:sensor_fusion_three_parallel}). The fusion can either be done in a ``sequential'' way where \(n-1\) sets of two complementary filters are used (Fig.~\ref{fig:sensor_fusion_three_sequential}), or in a ``parallel'' way where one set of \(n\) complementary filters is used (Fig.~\ref{fig:sensor_fusion_three_parallel}).
In the first case, typical sensor fusion synthesis techniques can be used. 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~\{\cite{stoten01_fusion_kinet_data_using_compos_filter,becker15_compl_filter_desig_three_frequen_bands}\}. 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~\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 A generalization of the proposed synthesis method of complementary filters is presented in this section. \par
\begin{figure}[htbp] \begin{figure}[htbp]
@ -682,15 +683,49 @@ The synthesis objective is to compute a set of \(n\) stable transfer functions \
\begin{subequations} \begin{subequations}
\label{eq:hinf_problem_gen} \label{eq:hinf_problem_gen}
\begin{align} \begin{align}
& \sum_{i=0}^n H_i(s) = 1 \label{eq:hinf_cond_compl_gen} \\ & \sum_{i=1}^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} & \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{align}
\end{subequations} \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.
\([W_1(s),\ W_2(s),\ \dots,\ W_n(s)]\) are weighting transfer functions that are chosen to specify the maximum complementary filters' norm during the synthesis.
Such synthesis objective is closely related to the one described in Section~\ref{sec:synthesis_objective}, and indeed the proposed synthesis method is a generalization of the one presented in Section~\ref{sec:hinf_synthesis}. \par Such synthesis objective is closely related to the one described in Section~\ref{sec:synthesis_objective}, and indeed the proposed synthesis method is a generalization of the one presented in Section~\ref{sec:hinf_synthesis}. \par
Before presenting the generalized synthesis method, the case with three sensors is presented. A set of \(n\) complementary filters can be shaped by applying the standard \(\mathcal{H}_\infty\) synthesis to the generalized plant \(P_n(s)\) described by~\eqref{eq:generalized_plant_n_filters}.
\begin{equation}
\label{eq:generalized_plant_n_filters}
\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}
If the synthesis if successful, a set of \(n-1\) filters \([H_2(s),\ H_3(s),\ \dots,\ H_n(s)]\) are obtained such that~\eqref{eq:hinf_syn_obj_gen} is verified.
\begin{equation}
\label{eq:hinf_syn_obj_gen}
\left\|\begin{matrix} \left(1 - \left[ H_2(s) + H_3(s) + \dots + H_n(s) \right]\right) W_1(s) \\ H_2(s) W_2(s) \\ \vdots \\ H_n(s) W_n(s) \end{matrix}\right\|_\infty \le 1
\end{equation}
\(H_1(s)\) is then defined using~\eqref{eq:h1_comp_h2_hn} which is ensuring the complementary property for the set of \(n\) filters~\eqref{eq:hinf_cond_compl_gen}.
Condition~\eqref{eq:hinf_cond_perf_gen} is satisfied thanks to~\eqref{eq:hinf_syn_obj_gen}.
\begin{equation}
\label{eq:h1_comp_h2_hn}
H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) + \dots + H_n(s) \big]
\end{equation}
An example is given to validate the proposed method for the synthesis of a set of three complementary filters.
The sensors to be merged are 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~\eqref{eq:weight_formula} and their inverse magnitude are shown in Fig.~\ref{fig:three_complementary_filters_results} (dashed curves).
Consider the generalized plant \(P_3(s)\) shown in Fig.~\ref{fig:comp_filter_three_hinf_gen_plant} which is also described by~\eqref{eq:generalized_plant_three_filters}. Consider the generalized plant \(P_3(s)\) shown in Fig.~\ref{fig:comp_filter_three_hinf_gen_plant} which is also described by~\eqref{eq:generalized_plant_three_filters}.
\begin{equation} \begin{equation}
@ -716,19 +751,16 @@ Consider the generalized plant \(P_3(s)\) shown in Fig.~\ref{fig:comp_filter_thr
\centering \centering
\end{figure} \end{figure}
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.~\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~\eqref{eq:hinf_syn_obj_three}. The standard \(\mathcal{H}_\infty\) synthesis is performed on the generalized plant \(P_3(s)\).
Two filters \(H_2(s)\) and \(H_3(s)\) are obtained such that the \(\mathcal{H}_\infty\) norm of the closed-loop transfer from \(w\) to \([z_1,\ z_2,\ z_3]\) of the system in Fig.~\ref{fig:comp_filter_three_hinf_fb} is less than one.
Filter \(H_1(s)\) is defined using~\eqref{eq:h1_compl_h2_h3} thus ensuring the complementary property of the obtained set of filters.
\begin{equation} \begin{equation}
\label{eq:hinf_syn_obj_three} \label{eq:h1_compl_h2_h3}
\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 H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) \big]
\end{equation} \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 Figure~\ref{fig:three_complementary_filters_results} displays the three synthesized complementary filters (solid lines) which confirms that the synthesis is successful.\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~\eqref{eq:weight_formula} and their inverse magnitudes are shown in Fig.~\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.~\ref{fig:three_complementary_filters_results} (solid lines). \par
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
@ -736,34 +768,21 @@ The \(\mathcal{H}_\infty\) synthesis is performed on the generalized plant \(P_3
\caption{\label{fig:three_complementary_filters_results}Bode plot of the inverse weighting functions and of the three complementary filters obtained using the \(\mathcal{H}_\infty\) synthesis.} \caption{\label{fig:three_complementary_filters_results}Bode plot of the inverse weighting functions and of the three complementary filters obtained using the \(\mathcal{H}_\infty\) synthesis.}
\end{figure} \end{figure}
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~\eqref{eq:generalized_plant_n_filters}.
\begin{equation}
\label{eq:generalized_plant_n_filters}
\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}
\section{Conclusion} \section{Conclusion}
\label{sec:conclusion} \label{sec:conclusion}
The sensor fusion robustness and the obtained super sensor noise can be linked to the magnitude of the complementary filters. A new method for designing complementary filters using the \(\mathcal{H}_\infty\) synthesis has been proposed.
In this paper, a synthesis method that enables the shaping of the complementary filters norms has been proposed. It allows to shape the magnitude of the filters by the use of weighting functions during the synthesis.
This is very valuable in practice as the characteristics of the super sensor are linked to the complementary filters' magnitude.
Therefore typical sensor fusion objectives can be translated into requirements on the magnitudes of the filters.
Several examples were used to emphasize the simplicity and the effectiveness of the proposed method. Several examples were used to emphasize the simplicity and the effectiveness of the proposed method.
Links with ``closed-loop'' complementary filters were highlighted, and the proposed method was generalized for designing a set of more than two complementary filters.
The future work will aim at developing a complementary filter synthesis method that minimizes the super sensor noise while ensuring the robustness of the fusion. However, the shaping of the complementary filters' magnitude does not allow to directly optimize the super sensor noise and dynamical characteristics.
Future work will aim at developing a complementary filter synthesis method that minimizes the super sensor noise while ensuring the robustness of the fusion.
\section*{Acknowledgment} \section*{Acknowledgment}
This research benefited from a FRIA grant from the French Community of Belgium. This research benefited from a FRIA grant from the French Community of Belgium.
This paper has been assigned the LIGO document number LIGO-P2100328.
\section*{Data Availability} \section*{Data Availability}
Matlab~\cite{matlab20} was used for this study. Matlab~\cite{matlab20} was used for this study.

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journal/ref.bib
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@ -386,3 +386,13 @@
Fusion Using the $\mathcal{H}_\infty$ Synthesis}}, Fusion Using the $\mathcal{H}_\infty$ Synthesis}},
year = 2021, year = 2021,
} }
@book{skogestad07_multiv_feedb_contr,
author = {Skogestad, Sigurd and Postlethwaite, Ian},
title = {Multivariable Feedback Control: Analysis and Design -
Second Edition},
year = 2007,
publisher = {John Wiley},
isbn = {978-0470011683},
note = {isbn:978-0470011683},
}