Sensor have limited bandwidth and are accurate only in a certain frequency band.
In many applications, the signals of different sensor are fused together in order to either enhance the stability or improve the operational bandwidth of the system.
The sensor signals can be fused using complementary filters.
The tuning of complementary filters is a complex task and is the subject of this paper.
The filters needs to meet design specifications while satisfying the complementary property.
This paper presents a framework to shape the norm of complementary filters using the $\mathcal{H}_\infty$ norm minimization.
The design specifications are imposed as constraints in the optimization problem by appropriate selection of weighting functions.
The proposed method is quite general and easily extendable to cases where more than two sensors are fused.
Finally, the proposed method is applied to the design of complementary filter design for active vibration isolation of the Laser Interferometer Gravitation-wave Observatory (LIGO).
- cite:anderson53_instr_approac_system_steer_comput earliest application of complementary filters (A simple RC circuit was used to physically realize the complementary filters)
In some situations, it is used to increase the bandwidth of the sensor cite:shaw90_bandw_enhan_posit_measur_using_measur_accel,zimmermann92_high_bandw_orien_measur_contr,min15_compl_filter_desig_angle_estim.
For instance by increasing the high frequency bandwidth of a position sensor using an accelerometer.
Tjepkema et al. cite:tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip used sensor fusion to isolate precision equipment from the ground motion.
There are mainly two ways to perform sensor fusion: using complementary filters or using Kalman filtering cite:brown72_integ_navig_system_kalman_filter.
Advantages of complementary filtering over Kalman filtering for sensor fusion:
- Less computation cite:higgins75_compar_compl_kalman_filter
- For Kalman filtering, we are forced to make assumption about the probabilistic character of the sensor noises cite:robert12_introd_random_signal_applied_kalman
In some cases, complementary filters are implemented in an analog way such as in cite:yong16_high_speed_vertic_posit_stage,moore19_capac_instr_sensor_fusion_high_bandw_nanop, but most of the time it is implemented numerically which allows much more complex
- cite:pascoal99_navig_system_desig_using_time use LMI to generate complementary filters (convex optimization techniques), specific for navigation systems
- cite:plummer06_optim_compl_filter_their_applic_motion_measur use H-Infinity to optimize complementary filters (flatten the super sensor noise spectral density)
- Robustness problems: cite:zimmermann92_high_bandw_orien_measur_contr,plummer06_optim_compl_filter_their_applic_motion_measur change of phase near the merging frequency
- Although many design methods of complementary filters have been proposed in the literature, no simple method that allows to shape the norm of the complementary filters is available.
A general sensor fusion architecture using complementary filters is shown in Figure ref:fig:sensor_fusion_overview where several sensors (here two) are measuring the same physical quantity $x$.
The resulting sensor, termed as "super sensor", can have larger bandwidth and better noise characteristics in comparison to the individual sensor.
This means that the super sensor provides an estimate $\hat{x}$ of $x$ which can be more accurate over a larger frequency band than the outputs of the individual sensors.
Such model is shown in Figure ref:fig:sensor_model and consists of a linear time invariant (LTI) system $G_i(s)$ representing the dynamics of the sensor and an additive noise input $n_i$ representing its noise.
The model input $x$ is the measured physical quantity and its output $\tilde{x}_i$ is the "raw" output of the sensor.
The normalized signals from both calibrated sensors are fed into two complementary filters $H_1(s)$ and $H_2(s)$ and then combined to yield an estimate $\hat{x}$ of $x$ as shown in Fig. ref:fig:fusion_super_sensor.
Thus, this sensor fusion architecture permits to filter the noise of both sensors without introducing any distortion in the physical quantity to be measured.
The estimation error $\delta x$, defined as the difference between the sensor output $\hat{x}$ and the measured quantity $x$, is computed for the super sensor eqref:eq:estimate_error.
As shown in eqref:eq:noise_filtering_psd, the Power Spectral Density (PSD) of the estimation error $\Phi_{\delta x}$ depends both on the norm of the two complementary filters and on the PSD of the noise sources $\Phi_{n_1}$ and $\Phi_{n_2}$.
If the two sensors have identical noise characteristics ($\Phi_{n_1}(\omega) = \Phi_{n_2}(\omega)$), a simple averaging ($H_1(s) = H_2(s) = 0.5$) is what would minimize the super sensor noise.
However, the two sensors have usually high noise levels over distinct frequency regions.
In such case, to lower the noise of the super sensor, the value of the norm $|H_1|$ has to be lowered when $\Phi_{n_1}$ is larger than $\Phi_{n_2}$ and that of $|H_2|$ lowered when $\Phi_{n_2}$ is larger than $\Phi_{n_1}$.
Therefore, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor noise.
In order to study such imperfection, a multiplicative input uncertainty is added to the sensor dynamics (Figure ref:fig:sensor_model_uncertainty), where the nominal model is taken as the estimated model for the normalization $\hat{G}_i(s)$, $\Delta_i$ is any stable transfer function satisfying $|\Delta_i(j\omega)| \le 1,\ \forall\omega$, and $w_i(s)$ is a weight representing the magnitude of the uncertainty.
The weight $w_i(s)$ is chosen such that the real sensor dynamics is always contained in the uncertain region represented by a circle centered on $1$ and with a radius equal to $|w_i(j\omega)|$.
As the nominal sensor dynamics is taken as the normalized filter, the normalized sensor can be further simplified as shown in Figure ref:fig:sensor_model_uncertainty_simplified.
The super sensor dynamics eqref:eq:super_sensor_dyn_uncertainty is no longer equal to $1$ and now depends on the sensor dynamics uncertainty weights $w_i(s)$ as well as on the complementary filters $H_i(s)$.
The dynamical uncertainty of the super sensor can be graphically represented in the complex plane by a circle centered on $1$ with a radius equal to $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$ as shown in Figure ref:fig:uncertainty_set_super_sensor.
#+caption: Uncertainty region of the super sensor dynamics in the complex plane (solid circle). The contribution of both sensors 1 and 2 to the uncertainty are represented respectively by a blue circle and a red circle. The frequency dependency $\omega$ is here omitted.
The super sensor dynamical uncertainty (i.e. the robustness of the fusion) clearly depends on the complementary filters norms.
For instance, the phase uncertainty $\Delta\phi(\omega)$ added by the super sensor dynamics at frequency $\omega$ can be found by drawing a tangent from the origin to the uncertainty circle of super sensor (Figure ref:fig:uncertainty_set_super_sensor) and is bounded by eqref:eq:max_phase_uncertainty.
As it is generally desired to limit the maximum phase added by the super sensor, $H_1(s)$ and $H_2(s)$ should be designed such that $\Delta \phi$ is bounded to acceptable values.
Typically, the norm of the complementary filter $|H_i(j\omega)|$ should be made small when $|w_i(j\omega)|$ is large, i.e., at frequencies where the sensor dynamics is uncertain.
This is equivalent as to finding proper and stable transfer functions $H_1(s)$ and $H_2(s)$ such that conditions eqref:eq:comp_filter_problem_form are satisfied.
where $W_1(s)$ and $W_2(s)$ are two weighting transfer functions that are chosen to specify the maximum wanted norms of the complementary filters during the synthesis.
In this section, it is shown that the synthesis objective can be easily expressed as a standard $\mathcal{H}_\infty$ optimal control problem and therefore solved using convenient tools readily available.
Consider the generalized plant $P(s)$ shown in Figure ref:fig:h_infinity_robust_fusion and mathematically described by eqref:eq:generalized_plant.
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 from $w$ to $[z_1, \ z_2]$ is less than one eqref:eq:hinf_syn_obj.
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.
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.
The above optimization problem can be efficiently solved in Matlab cite:matlab20 using the Robust Control Toolbox.
Weighting functions are used during the synthesis to specify what is the maximum allowed norms of the complementary filters.
The proper design of these weighting functions is of primary importance for the success of the presented complementary filters $\mathcal{H}_\infty$ synthesis.
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 order).
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 its slope, its crossover frequency and its maximum gain at low and high frequency.
To help with the design of the weighting functions such that the above specification can be easily expressed, the formula eqref:eq:weight_formula is proposed.
#+caption: Magnitude of a weighting function generated using the proposed formula eqref:eq:weight_formula, $G_0 = 1e^{-3}$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$, $n = 3$
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 $\mathcal{H}_\infty$ synthesis is applied on the generalized plant of Figure ref:fig:h_infinity_robust_fusion using the Matlab =hinfsyn= command.
The synthesized filter $H_2(s)$ is such that $\mathcal{H}_\infty$ norm between $w$ and $[z_1,\ z_2]^T$ is minimized and here found close to one eqref:eq:hinf_synthesis_result.
The bode plots of the obtained complementary filters are shown by solid lines in Figure ref:fig:hinf_filters_results and their transfer functions in the Laplace domain are given in eqref:eq:hinf_synthesis_result_tf.
The obtained transfer functions are of order $5$ as expected (sum of the weighting functions orders), and their magnitudes are bellow the maximum specified ones as ensured by eqref:eq:hinf_synthesis_result.
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: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 cite:hua04_low_ligo.
A set of complementary filters ($L_2,H_2$) is first used to fuse a seismometer and a geophone.
Then, another set of complementary filters ($L_1,H_1$) is used to merge the output of the first "inertial super sensor" with a position sensor.
A simplified block diagram of the sensor fusion architecture is shown in Figure ref:fig:ligo_super_sensor_architecture.
#+name: fig:ligo_super_sensor_architecture
#+caption: Simplified block diagram of the sensor blending strategy for the first stage at the LIGO cite:hua04_low_ligo
#+attr_latex: :scale 1
[[file:figs/ligo_super_sensor_architecture.pdf]]
The fusion of the position sensor at low frequency with the "inertial super sensor" at high frequency using the complementary filters ($L_1,H_1$) is done for several reasons, first of which is to give the super sensor a DC sensibility and therefore allow the feedback loop to have authority at zero frequency.
The requirements on those filters are very tight and thus their design is complex and should be expressed as an optimization problem.
The approach used in cite:hua05_low_ligo is to use FIR complementary filters and to write the synthesis as a convex optimization problem.
After synthesis, the obtained FIR filters were found to be compliant with the requirements.
However they are of very high order so their implementation is quite complex.
In this section, the effectiveness of the proposed complementary filter synthesis strategy is demonstrated on the same set of requirements.
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):
- From $0$ to $\SI{0.008}{Hz}$, the magnitude $|L_1(j\omega)|$ should be less or equal to $8 \times 10^{-4}$
- Between $\SI{0.008}{Hz}$ to $\SI{0.04}{Hz}$, the filter $L_1(s)$ should attenuate the input signal proportional to frequency cubed
- Between $\SI{0.04}{Hz}$ to $\SI{0.1}{Hz}$, the magnitude $|L_1(j\omega)|$ should be less than $3$
- Above $\SI{0.1}{Hz}$, the magnitude $|H_1(j\omega)|$ should be less than $0.045$
These specifications are therefore upper bounds on the complementary filters' magnitudes.
They are physically represented in Figure ref:fig:fir_filter_ligo as well as the obtained magnitude of the FIR filters in cite:hua05_low_ligo.
The weighting functions should be designed such that their inverse magnitude is as close as possible to the specifications in order to not over-constrain the synthesis problem.
However, the order of each weight should stay reasonably small in order to reduce the computational costs of the optimization problem as well as for the physical implementation of the filters.
A Type I Chebyshev filter of order $20$ is used as the weighting transfer function $w_L(s)$ corresponding to the low pass filter.
For the one corresponding to the high pass filter $w_H(s)$, a $7^{\text{th}}$ order transfer function is designed.
The magnitudes of the weighting functions are shown in Fig. ref:fig:ligo_weights.
#+name: fig:ligo_weights
#+caption: Specifications and weighting functions magnitudes
#+attr_latex: :scale 1
[[file:figs/ligo_weights.pdf]]
** $\mathcal{H}_\infty$ Synthesis
<<sec:ligo_results>>
$\mathcal{H}_\infty$ synthesis is performed using the architecture shown in Fig. ref:eq:generalized_plant.
The complementary filters obtained are of order $27$.
In Fig. ref:fig:comp_fir_ligo_hinf, their bode plot is compared with the FIR filters of order 512 obtained in cite:hua05_low_ligo.
They are found to be very close to each other and this shows the effectiveness of the proposed synthesis method.
#+name: fig:comp_fir_ligo_hinf
#+caption: Comparison of the FIR filters (solid) designed in cite:hua05_low_ligo with the filters obtained with $\mathcal{H}_\infty$ synthesis (dashed)
- cite:plummer06_optim_compl_filter_their_applic_motion_measur use H-Infinity to optimize complementary filters (flatten the super sensor noise spectral density)
- cite:jensen13_basic_uas design of complementary filters with classical control theory, PID
*** Sensor Fusion with "closed-loop" complementary filters :ignore:
Therefore, two filters can be merged as shown in Figure ref:fig:feedback_sensor_fusion_arch.
#+name: fig:feedback_sensor_fusion_arch
#+caption: Classical feedback architecture for sensor fusion
#+attr_latex: :scale 1
[[file:figs/feedback_sensor_fusion_arch.pdf]]
One of the main advantage of this configuration is that standard tools of the linear control theory can be applied.
*** Mixed Sensitivity Synthesis :ignore:
If one want to shape both the transfer functions $\frac{\hat{x}}{\hat{x}_1}(s) = S(s)$ and $\frac{\hat{x}}{\hat{x}_2}(s) = T(s)$, this corresponds to the $\mathcal{H}_\infty$ mixed-sensitivity synthesis.
The $\mathcal{H}_\infty$ mixed-sensitivity synthesis can be perform by applying the $\mathcal{H}_\infty$ synthesis to the generalized plant $P_L(s)$ shown in Figure ref:fig:feedback_synthesis_architecture_generalized_plant and described by eqref:eq:generalized_plant_mixed_sensitivity where $W_1(s)$ and $W_2(s)$ are weighting functions used to respectively shape $S(s)$ and $T(s)$.
The sensor fusion can be implemented as shown in Figure ref:fig:feedback_sensor_fusion_arch using the feedback architecture or more classically as shown in Figure ref:fig:sensor_fusion_overview using eqref:eq:comp_filters_feedback.
For instance at the LIGO, three sensors (an LVDT, a seismometer and a geophone) are merged to form a super sensor (Figure ref:fig:ligo_super_sensor_architecture). \par
The fusion can either be done in a "sequential" way where $n-1$ sets of two complementary filters are used (Figure ref:fig:sensor_fusion_three_sequential), or in a "parallel" way where one set of $n$ complementary filters is used (Figure ref:fig:sensor_fusion_three_parallel).
In the first case, typical sensor fusion synthesis techniques can be used.
However, when a parallel architecture is used, a new synthesis method for a set of more than two complementary filters is required.
Such synthesis method is presented in this section. \par
*************** TODO Say possible advantages of parallel architecture
The synthesis objective is to compute a set of $n$ stable transfer functions $[H_1(s),\ H_2(s),\ \dots,\ H_n(s)]$ such that eqref:eq:hinf_problem_gen is satisfied.
where $[W_1(s),\ W_2(s),\ \dots,\ W_n(s)]$ are weighting transfer functions that are chosen to specify the maximum wanted norms of the complementary filters during the synthesis.
Such synthesis objective is very close to the one described in Section ref:sec:synthesis_objective, and indeed the proposed synthesis architecture is also very similar. \par
Consider the generalized plant $P_3(s)$ shown in Figure ref:fig:comp_filter_three_hinf which is also described by eqref:eq:generalized_plant_three_filters.
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 Figure ref:fig:comp_filter_three_hinf) 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.
By defining $H_1(s) \triangleq 1 - H_2(s) - H_3(s)$, the proposed $\mathcal{H}_\infty$ synthesis solves the design problem eqref:eq:hinf_problem_gen with $n=3$. \par
An example is given to validate the method where three sensors are used in different frequency bands (up to $\SI{1}{Hz}$, from $1$ to $\SI{10}{Hz}$ and above $\SI{10}{Hz}$ respectively).
This paper has shown how complementary filters can be used to combine multiple sensors in order to obtain a super sensor.
Typical specification on the super sensor noise and on the robustness of the sensor fusion has been shown to be linked to the norm of the complementary filters.
Therefore, a synthesis method that permits the shaping of the complementary filters norms has been proposed and has been successfully applied for the design of complex filters.
Future work will aim at further developing this synthesis method for the robust and optimal synthesis of complementary filters used in sensor fusion.
* Acknowledgment
:PROPERTIES:
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This research benefited from a FRIA grant from the French Community of Belgium.
