Finish reworking section 2
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@ -253,7 +253,7 @@ The framework for the design of complementary filters is detailed in Section [[*
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This is followed by the application of the design method to complementary filter design for the active vibration isolation at LIGO in Section [[*Application: Complementary Filter Design for Active Vibration Isolation of LIGO][4]].
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Finally, concluding remarks are presented in Section [[*Concluding remarks][5]].
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* Complementary Filters Requirements
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* Sensor Fusion and Complementary Filters Requirements
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<<sec:requirements>>
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** Introduction :ignore:
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@ -264,16 +264,20 @@ These requirements are discussed in this section.
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** Sensor Fusion Architecture
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<<sec:sensor_fusion>>
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A general sensor fusion architecture is shown in Figure ref:fig:sensor_fusion_overview where several sensors (here two) are measuring the same physical quantity $x$.
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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 $x$.
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The two sensors output signals are estimates $\hat{x}_1$ and $\hat{x}_2$ of $x$.
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Each of these estimates are then filtered out by complementary filters and combined to form a new estimate $\hat{x}$.
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We further call the overall system from $x$ to $\hat{x}$ the "super sensor".
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The resulting sensor, termed as "super sensor", can have larger bandwidth and better noise characteristics in comparison to the individual sensor.
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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.
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#+name: fig:sensor_fusion_overview
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#+caption: Schematic of a sensor fusion architecture
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[[file:figs/sensor_fusion_overview.pdf]]
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The filters $H_1(s)$ and $H_2(s)$ are complementary which implies that:
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The complementary property of filters $H_1(s)$ and $H_2(s)$ implies that the summation of their transfer functions is equal to unity.
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That is, unity magnitude and zero phase at all frequencies.
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Therefore, a pair of strict complementary filter needs to satisfy the following condition:
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#+name: eq:comp_filter
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\begin{equation}
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H_1(s) + H_2(s) = 1
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@ -284,15 +288,15 @@ It will soon become clear why the complementary property is important.
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** Sensor Models and Sensor Normalization
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<<sec:sensor_models>>
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In order to study such sensor fusion architecture, a model of the sensor is required.
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In order to study such sensor fusion architecture, a model of the sensors is required.
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The sensor model is shown in Figure ref:fig:sensor_model.
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It consists of a Linear Time Invariant system (LTI) $G_i(s)$ representing the dynamics of the sensor and an additive noise input $n_i$ representing its noise.
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The model input $x$ is the measured quantity and its output $\tilde{x}_i$ is the "raw" output of the sensor.
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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.
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The model input $x$ is the measured physical quantity and its output $\tilde{x}_i$ is the "raw" output of the sensor.
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Before filtering the sensor outputs $\tilde{x}_i$ by the complementary filters, the sensors are usually normalized.
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This normalization consists of obtaining an estimate $\hat{G}_i(s)$ of the sensor dynamics $G_i(s)$.
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The raw output of the sensor $\tilde{x}_i$ is then passed through the inverse of the sensor dynamics estimate as shown in Figure ref:fig:sensor_model_calibrated.
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Before filtering the sensor outputs $\tilde{x}_i$ by the complementary filters, the sensors are usually normalized to simplify the fusion.
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This normalization consists of first obtaining an estimate $\hat{G}_i(s)$ of the sensor dynamics $G_i(s)$.
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It is supposed that the estimate of the sensor dynamics $\hat{G}_i(s)$ can be inverted and that its inverse $\hat{G}_i^{-1}(s)$ is proper and stable.
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The raw output of the sensor $\tilde{x}_i$ is then passed through $\hat{G}_i^{-1}(s)$ as shown in Figure ref:fig:sensor_model_calibrated.
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This way, the units of the estimates $\hat{x}_i$ are equal to the units of the physical quantity $x$.
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The sensor dynamics estimate $\hat{G}_1(s)$ can be a simple gain or more complex transfer functions.
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@ -314,16 +318,15 @@ The sensor dynamics estimate $\hat{G}_1(s)$ can be a simple gain or more complex
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\end{figure}
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#+end_export
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Let's now combine the two calibrated sensors models (Figure ref:fig:sensor_model_calibrated) with the sensor fusion architecture of figure ref:fig:sensor_fusion_overview.
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The result is shown in Figure ref:fig:fusion_super_sensor.
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Two calibrated sensors and then combined to form a super sensor as shown in Figure ref:fig:fusion_super_sensor.
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The two sensors are measuring the same physical quantity $x$ with dynamics $G_1(s)$ and $G_2(s)$, and with /uncorrelated/ noises $n_1$ and $n_2$.
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The 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.
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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.
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The super sensor output is therefore equal to:
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#+name: eq:comp_filter_estimate
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\begin{equation}
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\hat{x} = \Big( H_1(s) \hat{G}_1(s) G_1(s) + H_2(s) \hat{G}_2(s) G_2(s) \Big) x + H_1(s) \hat{G}_1(s) G_1(s) n_1 + H_2(s) \hat{G}_2(s) G_2(s) n_2
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\hat{x} = \Big( H_1(s) \hat{G}_1^{-1}(s) G_1(s) + H_2(s) \hat{G}_2^{-1}(s) G_2(s) \Big) x + H_1(s) \hat{G}_1^{-1}(s) G_1(s) n_1 + H_2(s) \hat{G}_2^{-1}(s) G_2(s) n_2
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\end{equation}
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#+name: fig:fusion_super_sensor
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@ -334,13 +337,13 @@ The super sensor output is therefore equal to:
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** Noise Sensor Filtering
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<<sec:noise_filtering>>
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In this section, it is suppose that all the sensors are correctly calibrated, such that:
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In this section, it is supposed that all the sensors are perfectly calibrated, such that:
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#+name: eq:perfect_dynamics
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\begin{equation}
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\frac{\hat{x}_i}{x} = \hat{G}_i(s) G_i(s) \approx 1
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\frac{\hat{x}_i}{x} = \hat{G}_i(s) G_i(s) = 1
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\end{equation}
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The effect of a non-ideal normalization will be discussed in the next section.
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The effect of a non-perfect normalization will be discussed in the next section.
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The super sensor output $\hat{x}$ is then:
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#+name: eq:estimate_perfect_dyn
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@ -348,11 +351,10 @@ The super sensor output $\hat{x}$ is then:
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\hat{x} = x + H_1(s) n_1 + H_2(s) n_2
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\end{equation}
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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.
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Thus, this sensor fusion architecture permits to filter the noise of both sensors without introducing any distortion in the physical quantity to be measured.
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Let's define the estimation error $\delta x$ by eqref:eq:estimate_error.
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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.
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#+name: eq:estimate_error
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\begin{equation}
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\delta x \triangleq \hat{x} - x = H_1(s) n_1 + H_2(s) n_2
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@ -364,18 +366,21 @@ As shown in eqref:eq:noise_filtering_psd, the Power Spectral Density (PSD) of th
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\Phi_{\delta x}(\omega) = \left|H_1(j\omega)\right|^2 \Phi_{n_1}(\omega) + \left|H_2(j\omega)\right|^2 \Phi_{n_2}(\omega)
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\end{equation}
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# TODO - Rework, tell that we can put requirements on the *norm* of the complementary filters
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Usually, the two sensors have high noise levels over distinct frequency regions.
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In order to lower the noise of the super sensor, the value of the norm $|H_1|$ has to be lowered when $\Phi_{n_1}$ is larger than $\Phi_{n_2}$ and that of $|H_2|$ lowered when $\Phi_{n_2}$ is larger than $\Phi_{n_1}$.
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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.
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This the simplest form of sensor fusion with complementary filters.
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** Robustness of the Fusion
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However, the two sensors have usually high noise levels over distinct frequency regions.
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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 $\Phi_{n_1}$.
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Therefore, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor noise.
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** Sensor Fusion Robustness
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<<sec:fusion_robustness>>
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In practical systems the sensor normalization is not perfect and eqref:eq:perfect_dynamics is not verified.
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In practical systems the sensor normalization is not perfect and condition eqref:eq:perfect_dynamics is not verified.
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In order to study such imperfection, the sensor dynamical uncertainty is modeled using multiplicative input uncertainty (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.
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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.
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# The weight $w_i(s)$ is chosen such that the real sensor dynamics is contained in the uncertain region represented by...
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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 $|w_i(j\omega)|$.
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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.
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@ -410,26 +415,22 @@ The super sensor dynamics eqref:eq:super_sensor_dyn_uncertainty is no longer equ
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\frac{\hat{x}}{x} = 1 + w_1(s) H_1(s) \Delta_1(s) + w_2(s) H_2(s) \Delta_2(s)
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\end{equation}
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The uncertainty region of the super sensor can be represented in the complex plane by a circle centered on $1$ with a radius equal to $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$ as shown in Figure ref:fig:uncertainty_set_super_sensor.
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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.
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#+name: fig:uncertainty_set_super_sensor
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#+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
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#+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.
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[[file:figs/uncertainty_set_super_sensor.pdf]]
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The maximum phase added $\Delta\phi(\omega)$ by the super sensor dynamics at frequency $\omega$ is then
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The super sensor dynamical uncertainty (i.e. the robustness of the fusion) clearly depends on the complementary filters norms.
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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.
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#+name: eq:max_phase_uncertainty
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\begin{equation}
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\Delta\phi(\omega) = \arcsin\big( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \big)
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\Delta\phi(\omega) < \arcsin\big( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \big)
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\end{equation}
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As it is generally desired to limit the maximum phase added by the super sensor, $H_1(s)$ and $H_2(s)$ should be designed such that eqref:eq:max_uncertainty_super_sensor is satisfied.
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#+name: eq:max_uncertainty_super_sensor
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\begin{equation}
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\max_\omega \big( \left|w_1 H_1\right| + \left|w_2 H_2\right|\big) < \sin\left( \Delta \phi_\text{max} \right)
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\end{equation}
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where $\Delta \phi_\text{max}$ is the maximum allowed added phase.
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Thus the norm of the complementary filter $|H_i|$ should be made small at frequencies where $|w_i|$ is large.
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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.
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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.
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* Complementary Filters Shaping using $\mathcal{H}_\infty$ Synthesis
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<<sec:hinf_method>>
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@ -1,4 +1,4 @@
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% Created 2021-05-03 lun. 17:46
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% Created 2021-05-19 mer. 11:46
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% Intended LaTeX compiler: pdflatex
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\documentclass[preprint, sort&compress]{elsarticle}
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\usepackage[utf8]{inputenc}
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@ -58,7 +58,7 @@ Sensor fusion \sep{} Optimal filters \sep{} \(\mathcal{H}_\infty\) synthesis \se
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\end{frontmatter}
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\section{Introduction}
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\label{sec:org0c85494}
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\label{sec:org188c07e}
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\label{sec:introduction}
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\begin{itemize}
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\item \cite{bendat57_optim_filter_indep_measur_two} roots of sensor fusion
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@ -104,20 +104,22 @@ Sensor fusion \sep{} Optimal filters \sep{} \(\mathcal{H}_\infty\) synthesis \se
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Most of the requirements => shape of the complementary filters
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=> propose a way to shape complementary filters.
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\section{Complementary Filters Requirements}
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\label{sec:org05c7608}
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\section{Sensor Fusion and Complementary Filters Requirements}
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\label{sec:org99f43ee}
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\label{sec:requirements}
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Complementary filters provides a framework for fusing signals from different sensors.
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As the effectiveness of the fusion depends on the proper design of the complementary filters, they are expected to fulfill certain requirements.
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These requirements are discussed in this section.
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\subsection{Sensor Fusion Architecture}
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\label{sec:orgca80a74}
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\label{sec:orgec9e73a}
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\label{sec:sensor_fusion}
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A general sensor fusion architecture is shown in Figure \ref{fig:sensor_fusion_overview} where several sensors (here two) are measuring the same physical quantity \(x\).
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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 \(x\).
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The two sensors output signals are estimates \(\hat{x}_1\) and \(\hat{x}_2\) of \(x\).
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Each of these estimates are then filtered out by complementary filters and combined to form a new estimate \(\hat{x}\).
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We further call the overall system from \(x\) to \(\hat{x}\) the ``super sensor''.
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The resulting sensor, termed as ``super sensor'', can have larger bandwidth and better noise characteristics in comparison to the individual sensor.
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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.
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\begin{figure}[htbp]
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\centering
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@ -125,7 +127,9 @@ We further call the overall system from \(x\) to \(\hat{x}\) the ``super sensor'
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\caption{\label{fig:sensor_fusion_overview}Schematic of a sensor fusion architecture}
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\end{figure}
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The filters \(H_1(s)\) and \(H_2(s)\) are complementary which implies that:
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The complementary property of filters \(H_1(s)\) and \(H_2(s)\) implies that the summation of their transfer functions is equal to unity.
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That is, unity magnitude and zero phase at all frequencies.
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Therefore, a pair of strict complementary filter needs to satisfy the following condition:
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\begin{equation}
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\label{eq:comp_filter}
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H_1(s) + H_2(s) = 1
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@ -134,18 +138,18 @@ The filters \(H_1(s)\) and \(H_2(s)\) are complementary which implies that:
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It will soon become clear why the complementary property is important.
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\subsection{Sensor Models and Sensor Normalization}
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\label{sec:orgfc7a65c}
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\label{sec:org9538be3}
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\label{sec:sensor_models}
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In order to study such sensor fusion architecture, a model of the sensor is required.
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In order to study such sensor fusion architecture, a model of the sensors is required.
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The sensor model is shown in Figure \ref{fig:sensor_model}.
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It consists of a Linear Time Invariant system (LTI) \(G_i(s)\) representing the dynamics of the sensor and an additive noise input \(n_i\) representing its noise.
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The model input \(x\) is the measured quantity and its output \(\tilde{x}_i\) is the ``raw'' output of the sensor.
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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.
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The model input \(x\) is the measured physical quantity and its output \(\tilde{x}_i\) is the ``raw'' output of the sensor.
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Before filtering the sensor outputs \(\tilde{x}_i\) by the complementary filters, the sensors are usually normalized.
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This normalization consists of obtaining an estimate \(\hat{G}_i(s)\) of the sensor dynamics \(G_i(s)\).
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The raw output of the sensor \(\tilde{x}_i\) is then passed through the inverse of the sensor dynamics estimate as shown in Figure \ref{fig:sensor_model_calibrated}.
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Before filtering the sensor outputs \(\tilde{x}_i\) by the complementary filters, the sensors are usually normalized to simplify the fusion.
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This normalization consists of first obtaining an estimate \(\hat{G}_i(s)\) of the sensor dynamics \(G_i(s)\).
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It is supposed that the estimate of the sensor dynamics \(\hat{G}_i(s)\) can be inverted and that its inverse \(\hat{G}_i^{-1}(s)\) is proper and stable.
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The raw output of the sensor \(\tilde{x}_i\) is then passed through \(\hat{G}_i^{-1}(s)\) as shown in Figure \ref{fig:sensor_model_calibrated}.
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This way, the units of the estimates \(\hat{x}_i\) are equal to the units of the physical quantity \(x\).
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The sensor dynamics estimate \(\hat{G}_1(s)\) can be a simple gain or more complex transfer functions.
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@ -165,16 +169,15 @@ The sensor dynamics estimate \(\hat{G}_1(s)\) can be a simple gain or more compl
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\centering
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\end{figure}
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Let's now combine the two calibrated sensors models (Figure \ref{fig:sensor_model_calibrated}) with the sensor fusion architecture of figure \ref{fig:sensor_fusion_overview}.
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The result is shown in Figure \ref{fig:fusion_super_sensor}.
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Two calibrated sensors and then combined to form a super sensor as shown in Figure \ref{fig:fusion_super_sensor}.
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The two sensors are measuring the same physical quantity \(x\) with dynamics \(G_1(s)\) and \(G_2(s)\), and with \emph{uncorrelated} noises \(n_1\) and \(n_2\).
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The 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}.
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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}.
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The super sensor output is therefore equal to:
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\begin{equation}
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\label{eq:comp_filter_estimate}
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\hat{x} = \Big( H_1(s) \hat{G}_1(s) G_1(s) + H_2(s) \hat{G}_2(s) G_2(s) \Big) x + H_1(s) \hat{G}_1(s) G_1(s) n_1 + H_2(s) \hat{G}_2(s) G_2(s) n_2
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\hat{x} = \Big( H_1(s) \hat{G}_1^{-1}(s) G_1(s) + H_2(s) \hat{G}_2^{-1}(s) G_2(s) \Big) x + H_1(s) \hat{G}_1^{-1}(s) G_1(s) n_1 + H_2(s) \hat{G}_2^{-1}(s) G_2(s) n_2
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\end{equation}
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\begin{figure}[htbp]
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@ -184,16 +187,16 @@ The super sensor output is therefore equal to:
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\end{figure}
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\subsection{Noise Sensor Filtering}
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\label{sec:org2a2ea67}
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\label{sec:orgb03f925}
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\label{sec:noise_filtering}
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In this section, it is suppose that all the sensors are correctly calibrated, such that:
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||||
In this section, it is supposed that all the sensors are perfectly calibrated, such that:
|
||||
\begin{equation}
|
||||
\label{eq:perfect_dynamics}
|
||||
\frac{\hat{x}_i}{x} = \hat{G}_i(s) G_i(s) \approx 1
|
||||
\frac{\hat{x}_i}{x} = \hat{G}_i(s) G_i(s) = 1
|
||||
\end{equation}
|
||||
|
||||
The effect of a non-ideal normalization will be discussed in the next section.
|
||||
The effect of a non-perfect normalization will be discussed in the next section.
|
||||
|
||||
The super sensor output \(\hat{x}\) is then:
|
||||
\begin{equation}
|
||||
@ -201,11 +204,10 @@ The super sensor output \(\hat{x}\) is then:
|
||||
\hat{x} = x + H_1(s) n_1 + H_2(s) n_2
|
||||
\end{equation}
|
||||
|
||||
|
||||
From \eqref{eq:estimate_perfect_dyn}, the complementary filters \(H_1(s)\) and \(H_2(s)\) are shown to only operate on the sensor's noises.
|
||||
Thus, this sensor fusion architecture permits to filter the noise of both sensors without introducing any distortion in the physical quantity to be measured.
|
||||
|
||||
Let's define the estimation error \(\delta x\) by \eqref{eq:estimate_error}.
|
||||
The estimation error \(\delta x\), defined as the difference between the sensor output \(\hat{x}\) and the measured quantity \(x\), is computed for the super sensor \eqref{eq:estimate_error}.
|
||||
\begin{equation}
|
||||
\label{eq:estimate_error}
|
||||
\delta x \triangleq \hat{x} - x = H_1(s) n_1 + H_2(s) n_2
|
||||
@ -217,16 +219,22 @@ As shown in \eqref{eq:noise_filtering_psd}, the Power Spectral Density (PSD) of
|
||||
\Phi_{\delta x}(\omega) = \left|H_1(j\omega)\right|^2 \Phi_{n_1}(\omega) + \left|H_2(j\omega)\right|^2 \Phi_{n_2}(\omega)
|
||||
\end{equation}
|
||||
|
||||
Usually, the two sensors have high noise levels over distinct frequency regions.
|
||||
In order to lower the noise of the super sensor, the value of the norm \(|H_1|\) has to be lowered when \(\Phi_{n_1}\) is larger than \(\Phi_{n_2}\) and that of \(|H_2|\) lowered when \(\Phi_{n_2}\) is larger than \(\Phi_{n_1}\).
|
||||
If the two sensors have identical noise characteristics (\(\Phi_{n_1}(\omega) = \Phi_{n_2}(\omega)\)), a simple averaging (\(H_1(s) = H_2(s) = 0.5\)) is what would minimize the super sensor noise.
|
||||
This the simplest form of sensor fusion with complementary filters.
|
||||
|
||||
\subsection{Robustness of the Fusion}
|
||||
\label{sec:orgca279c9}
|
||||
However, the two sensors have usually high noise levels over distinct frequency regions.
|
||||
In such case, to lower the noise of the super sensor, the value of the norm \(|H_1|\) has to be lowered when \(\Phi_{n_1}\) is larger than \(\Phi_{n_2}\) and that of \(|H_2|\) lowered when \(\Phi_{n_2}\) is larger than \(\Phi_{n_1}\).
|
||||
Therefore, by properly shaping the norm of the complementary filters, it is possible to minimize the noise of the super sensor noise.
|
||||
|
||||
\subsection{Sensor Fusion Robustness}
|
||||
\label{sec:orgfb0ea88}
|
||||
\label{sec:fusion_robustness}
|
||||
|
||||
In practical systems the sensor normalization is not perfect and \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, the sensor dynamical uncertainty is modeled using multiplicative input uncertainty (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.
|
||||
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}.
|
||||
|
||||
@ -261,36 +269,32 @@ 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)
|
||||
\end{equation}
|
||||
|
||||
The uncertainty region of the super sensor can be represented in the complex plane by a circle centered on \(1\) with a radius equal to \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\) as shown in Figure \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 \(|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|\) as shown in Figure \ref{fig:uncertainty_set_super_sensor}.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/uncertainty_set_super_sensor.pdf}
|
||||
\caption{\label{fig:uncertainty_set_super_sensor}Uncertainty region of the super sensor dynamics in the complex plane (solid circle). The contribution of both sensors 1 and 2 to the uncertainty are represented respectively by a blue circle and a red circle}
|
||||
\caption{\label{fig:uncertainty_set_super_sensor}Uncertainty region of the super sensor dynamics in the complex plane (solid circle). The contribution of both sensors 1 and 2 to the uncertainty are represented respectively by a blue circle and a red circle. The frequency dependency \(\omega\) is here omitted.}
|
||||
\end{figure}
|
||||
|
||||
The maximum phase added \(\Delta\phi(\omega)\) by the super sensor dynamics at frequency \(\omega\) is then
|
||||
The super sensor dynamical uncertainty (i.e. the robustness of the fusion) clearly depends on the complementary filters norms.
|
||||
For instance, the phase uncertainty \(\Delta\phi(\omega)\) added by the super sensor dynamics at frequency \(\omega\) can be found by drawing a tangent from the origin to the uncertainty circle of super sensor (Figure \ref{fig:uncertainty_set_super_sensor}) and is bounded by \eqref{eq:max_phase_uncertainty}.
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:max_phase_uncertainty}
|
||||
\Delta\phi(\omega) = \arcsin\big( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \big)
|
||||
\Delta\phi(\omega) < \arcsin\big( |w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)| \big)
|
||||
\end{equation}
|
||||
|
||||
As it is generally desired to limit the maximum phase added by the super sensor, \(H_1(s)\) and \(H_2(s)\) should be designed such that \eqref{eq:max_uncertainty_super_sensor} is satisfied.
|
||||
\begin{equation}
|
||||
\label{eq:max_uncertainty_super_sensor}
|
||||
\max_\omega \big( \left|w_1 H_1\right| + \left|w_2 H_2\right|\big) < \sin\left( \Delta \phi_\text{max} \right)
|
||||
\end{equation}
|
||||
where \(\Delta \phi_\text{max}\) is the maximum allowed added phase.
|
||||
|
||||
Thus the norm of the complementary filter \(|H_i|\) should be made small at frequencies where \(|w_i|\) is large.
|
||||
As it is generally desired to limit the maximum phase added by the super sensor, \(H_1(s)\) and \(H_2(s)\) should be designed such that \(\Delta \phi\) is bounded to acceptable values.
|
||||
Typically, the norm of the complementary filter \(|H_i(j\omega)|\) should be made small when \(|w_i(j\omega)|\) is large, i.e., at frequencies where the sensor dynamics is uncertain.
|
||||
|
||||
\section{Complementary Filters Shaping using \(\mathcal{H}_\infty\) Synthesis}
|
||||
\label{sec:org3d11f72}
|
||||
\label{sec:orgfccc360}
|
||||
\label{sec:hinf_method}
|
||||
As shown in Sec. \ref{sec:requirements}, the performance and robustness of the sensor fusion architecture depends on the complementary filters norms.
|
||||
Therefore, the development of a synthesis method of complementary filters that allows the shaping of their norm is necessary.
|
||||
\subsection{Synthesis Objective}
|
||||
\label{sec:org867aacd}
|
||||
\label{sec:orga79128d}
|
||||
\label{sec:synthesis_objective}
|
||||
The synthesis objective is to shape the norm of two filters \(H_1(s)\) and \(H_2(s)\) while ensuring their complementary property \eqref{eq:comp_filter}.
|
||||
This is equivalent as to finding stable transfer functions \(H_1(s)\) and \(H_2(s)\) such that conditions \eqref{eq:comp_filter_problem_form} are satisfied.
|
||||
@ -305,7 +309,7 @@ This is equivalent as to finding stable transfer functions \(H_1(s)\) and \(H_2(
|
||||
where \(W_1(s)\) and \(W_2(s)\) are two weighting transfer functions that are chosen to shape the norms of the corresponding filters.
|
||||
|
||||
\subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
|
||||
\label{sec:orgec7ca01}
|
||||
\label{sec:org91451ed}
|
||||
\label{sec:hinf_synthesis}
|
||||
In order to express this optimization problem as a standard \(\mathcal{H}_\infty\) problem, the architecture shown in Fig. \ref{fig:h_infinity_robust_fusion} is used where the generalized plant \(P\) is described by \eqref{eq:generalized_plant}.
|
||||
\begin{equation}
|
||||
@ -341,7 +345,7 @@ The conditions \eqref{eq:hinf_cond_h1} and \eqref{eq:hinf_cond_h2} on the filter
|
||||
Therefore, all the conditions \eqref{eq:comp_filter_problem_form} are satisfied using this synthesis method based on \(\mathcal{H}_\infty\) synthesis, and thus it permits to shape complementary filters as desired.
|
||||
|
||||
\subsection{Weighting Functions Design}
|
||||
\label{sec:org1b0a8b2}
|
||||
\label{sec:orge5f38aa}
|
||||
\label{sec:hinf_weighting_func}
|
||||
The proper design of the weighting functions is of primary importance for the success of the presented complementary filters \(\mathcal{H}_\infty\) synthesis.
|
||||
|
||||
@ -386,7 +390,7 @@ The general shape of a weighting function generated using \eqref{eq:weight_formu
|
||||
\end{figure}
|
||||
|
||||
\subsection{Validation of the proposed synthesis method}
|
||||
\label{sec:org9091752}
|
||||
\label{sec:org0477d6e}
|
||||
\label{sec:hinf_example}
|
||||
Let's validate the proposed design method of complementary filters with a simple example where two complementary filters \(H_1(s)\) and \(H_2(s)\) have to be designed such that:
|
||||
\begin{itemize}
|
||||
@ -428,7 +432,7 @@ The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:
|
||||
\end{figure}
|
||||
|
||||
\section{Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO}
|
||||
\label{sec:orgf547be3}
|
||||
\label{sec:org70c1567}
|
||||
\label{sec:application_ligo}
|
||||
Several complementary filters are used in the active isolation system at the LIGO \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}.
|
||||
The requirements on those filters are very tight and thus their design is complex.
|
||||
@ -437,7 +441,7 @@ The obtained FIR filters are compliant with the requirements. However they are o
|
||||
|
||||
The effectiveness of the proposed method is demonstrated by designing complementary filters with the same requirements as the one described in \cite{hua05_low_ligo}.
|
||||
\subsection{Complementary Filters Specifications}
|
||||
\label{sec:orgd0486d1}
|
||||
\label{sec:orgdfbd1f2}
|
||||
\label{sec:ligo_specifications}
|
||||
The specifications for one pair of complementary filters used at the LIGO are summarized below (for further details, refer to \cite{hua04_polyp_fir_compl_filter_contr_system}) and shown in Fig. \ref{fig:ligo_weights}:
|
||||
\begin{itemize}
|
||||
@ -448,7 +452,7 @@ The specifications for one pair of complementary filters used at the LIGO are su
|
||||
\end{itemize}
|
||||
|
||||
\subsection{Weighting Functions Design}
|
||||
\label{sec:org1a654aa}
|
||||
\label{sec:orgf9892b6}
|
||||
\label{sec:ligo_weights}
|
||||
The weighting functions should be designed such that their inverse magnitude is as close as possible to the specifications in order to not over-constrain the synthesis problem.
|
||||
However, the order of each weight should stay reasonably small in order to reduce the computational costs of the optimization problem as well as for the physical implementation of the filters.
|
||||
@ -464,7 +468,7 @@ The magnitudes of the weighting functions are shown in Fig. \ref{fig:ligo_weight
|
||||
\end{figure}
|
||||
|
||||
\subsection{\(\mathcal{H}_\infty\) Synthesis}
|
||||
\label{sec:org93cef71}
|
||||
\label{sec:orge086b06}
|
||||
\label{sec:ligo_results}
|
||||
\(\mathcal{H}_\infty\) synthesis is performed using the architecture shown in Fig. \ref{eq:generalized_plant}.
|
||||
The complementary filters obtained are of order \(27\).
|
||||
@ -478,9 +482,9 @@ They are found to be very close to each other and this shows the effectiveness o
|
||||
\end{figure}
|
||||
|
||||
\section{Discussion}
|
||||
\label{sec:org016320e}
|
||||
\label{sec:org7b7d598}
|
||||
\subsection{Alternative configuration}
|
||||
\label{sec:org69bd60e}
|
||||
\label{sec:org56a1607}
|
||||
\begin{itemize}
|
||||
\item Feedback architecture : Similar to mixed sensitivity
|
||||
\item 2 inputs / 1 output
|
||||
@ -489,13 +493,13 @@ They are found to be very close to each other and this shows the effectiveness o
|
||||
Explain differences
|
||||
|
||||
\subsection{Imposing zero at origin / roll-off}
|
||||
\label{sec:org7f88310}
|
||||
\label{sec:org8da9d79}
|
||||
3 methods:
|
||||
|
||||
Link to literature about doing that with mixed sensitivity
|
||||
|
||||
\subsection{Synthesis of Three Complementary Filters}
|
||||
\label{sec:orgd378e04}
|
||||
\label{sec:orgefead29}
|
||||
\label{sec:hinf_three_comp_filters}
|
||||
Some applications may require to merge more than two sensors.
|
||||
In such a case, it is necessary to design as many complementary filters as the number of sensors used.
|
||||
@ -533,7 +537,7 @@ The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:
|
||||
\end{figure}
|
||||
|
||||
\section{Conclusion}
|
||||
\label{sec:org46a0029}
|
||||
\label{sec:org2e6ce14}
|
||||
\label{sec:conclusion}
|
||||
This paper has shown how complementary filters can be used to combine multiple sensors in order to obtain a super sensor.
|
||||
Typical specification on the super sensor noise and on the robustness of the sensor fusion has been shown to be linked to the norm of the complementary filters.
|
||||
@ -541,7 +545,7 @@ Therefore, a synthesis method that permits the shaping of the complementary filt
|
||||
Future work will aim at further developing this synthesis method for the robust and optimal synthesis of complementary filters used in sensor fusion.
|
||||
|
||||
\section*{Acknowledgment}
|
||||
\label{sec:orgc8d6b1f}
|
||||
\label{sec:orgde5a128}
|
||||
This research benefited from a FRIA grant from the French Community of Belgium.
|
||||
|
||||
\bibliographystyle{elsarticle-num}
|
||||
|
||||
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