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#+end_IEEEkeywords
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* Introduction
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<<sec:introduction>>
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The basic idea of a complementary filter involves taking two or more sensors, filtering out unreliable frequencies for each sensor and combining the filtered outputs to get a better estimate throughout the entire bandwidth of the system.
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To achieve this, the sensors included in the filter should complement one another by performing better over specific parts of the system bandwidth.
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<<sec:introduction>>
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** Establish the importance of the research topic :ignore:
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# What are Complementary Filters
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A set of filters is said to be complementary if the sum of their transfer functions is equal to one at all frequencies, (i.e.) its magnitude is one and its phase is zero.
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* Optimal Super Sensor Noise: $\mathcal{H}_2$ Synthesis
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<<sec:optimal_fusion>>
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# Why Design of Complementary Filter is important
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The proper design of this particular kind of filter is of primary importance in a wide range of applications.
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** Sensor Fusion Architecture
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** Descriptions of the applications of complementary filtering :ignore:
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Often, multiple sensors with different noise or dynamical properties are used to measure the same physical quantity.
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In such case, complementary filters can be used to merge the sensors and forms a "super sensor" that has gives a better estimate of the physical quantity over a wider bandwidth.
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This is called sensor blending or sensor fusion.
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** Super Sensor Noise
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# List of all the applications
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** $\mathcal{H}_2$ Synthesis of Complementary Filters
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# Increasing Sensor Bandwidth
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This is widely used for the attitude estimation of unmanned aerial vehicles using various kind of sensors (accelerometers, gyroscopes, vision sensors, inclinometer) cite:zimmermann92_high_bandw_orien_measur_contr,corke04_inert_visual_sensin_system_small_auton_helic,min15_compl_filter_desig_angle_estim.
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** Example
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cite:shaw90_bandw_enhan_posit_measur_using_measur_accel Fast position measurement of flexible structure
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** Robustness Problem
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# Merging of different sensor types
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cite:matichard15_seism_isolat_advan_ligo (relative displacement measurement at low frequencies with inertial at high frequencies)
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* Robust Sensor Fusion: $\mathcal{H}_\infty$ Synthesis
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<<sec:robust_fusion>>
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# Reducing sensor noise
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cite:hua04_polyp_fir_compl_filter_contr_system
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** Representation of Sensor Dynamical Uncertainty
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# Improving the control robustness
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cite:collette15_sensor_fusion_method_high_perfor
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** Super Sensor Dynamical Uncertainty
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** Current research focus of the field / Literature review :ignore:
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# Discuss the different approach to complementary filter design
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The design methods for such filters goes from simple analytical formulas
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** $\mathcal{H_\infty}$ Synthesis of Complementary Filters
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# Analytical Formulas with limited number of parameter that let the designer choose the parameter value
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# First Order
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cite:corke04_inert_visual_sensin_system_small_auton_helic
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** Example
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# Second Order
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cite:min15_compl_filter_desig_angle_estim
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cite:jensen13_basic_uas
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* Optimal and Robust Sensor Fusion: Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
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<<sec:optimal_robust_fusion>>
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# Third Order and Higher orders
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cite:shaw90_bandw_enhan_posit_measur_using_measur_accel
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cite:zimmermann92_high_bandw_orien_measur_contr
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cite:matichard15_seism_isolat_advan_ligo
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cite:collette15_sensor_fusion_method_high_perfor
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** Sensor Fusion Architecture
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# FIR Filters
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cite:hua05_low_ligo
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cite:hua04_polyp_fir_compl_filter_contr_system
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cite:matichard15_seism_isolat_advan_ligo
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** Synthesis Objective
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# Non-linear methods
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cite:mahony08_nonlin_compl_filter_special_orthog_group
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** Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
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# Time Varying
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cite:pascoal99_navig_system_desig_using_time
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** Example
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# Alternate Formulation
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cite:jensen13_basic_uas (feedback system, P, PI, classical control theory for filter design)
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cite:brown72_integ_navig_system_kalman_filter
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* Experimental Validation
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<<sec:experimental_validation>>
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# LMI / convex Optimization
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cite:pascoal99_navig_system_desig_using_time
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** Experimental Setup
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# Least Square method for finding the optimal filter coefficients
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cite:min15_compl_filter_desig_angle_estim
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** Sensor Noise and Dynamical Uncertainty
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** Describe a gap in the research :ignore:
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# There is a need to design optimal complementary filters
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Although
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** Mixed $\mathcal{H}_2/\mathcal{H}_\infty$ Synthesis
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** Describe the paper itself / the problem which is addressed :ignore:
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In this paper, we propose
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** Super Sensor Noise and Dynamical Uncertainty
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** Introduce Each part of the paper :ignore:
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The body of the paper consists of five parts followed by a conclusion.
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* H-Infinity synthesis of complementary filters
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<<sec:hinf_filters>>
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First order complementary filters are easy to synthesize. For instance, one can use the following filters
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\begin{equation}
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H_H(s) = \frac{s/\w_0}{1 + s/\w_0}; \quad H_L(s) = \frac{1}{1 + s/\w_0}
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\end{equation}
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with $\w_0$ is the tuning parameter corresponding to the crossover frequency of the filters.
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However, the manual design of higher order complementary filters is far more complex and we have to use an automatic synthesis technique.
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As shown in Sec. ref:sec:trans_perf, most of the performance requirements can be expressed as upper bounds on the magnitude of the complementary filters.
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Thus, the $\mathcal{H}_\infty$ framework seems adapted and we here propose a technique to synthesis complementary filters while specifying uppers bounds on their magnitudes.
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** $\hinf$ problem formulation
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<<sec:hinf_conf>>
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In this section, we formulate the $\hinf$ problem for the synthesis of complementary filters.
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The synthesis objective is to shape an high pass filter $H_H$ and a low pass filter $H_L$ while ensuring their complementary property ($H_H + H_L = 1$).
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To do so, we define two weighting functions $w_L$ and $w_H$ that will respectively used to shape $H_L$ and $H_H$.
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The synthesis problem is then
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\begin{subnumcases}{\text{Find } H_L, H_H \text{ such that}}
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H_L \text{ and } H_H \text{ are stable} \label{eq:hinf_cond_stability}\\
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H_L + H_H = 1 \label{eq:hinf_cond_complementarity} \\
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|w_L H_L| \le 1 \quad \forall\omega \label{eq:hinf_cond_hl} \\
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|w_H H_H| \le 1 \quad \forall\omega \label{eq:hinf_cond_hh}
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\end{subnumcases}
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To express this synthesis problem into an $\hinf$ synthesis problem, we define the following generalized plant $P$ (also shown on Fig. ref:fig:sf_hinf_filters_plant_b):
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#+NAME: eq:generalized_plant
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\begin{equation}
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\colvec{w\\u} = P \colvec{z_H \\ z_L \\ v}; \quad P = \begin{bmatrix} w_H & -w_H \\ 0 & w_L \\ 1 & 0 \end{bmatrix}
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\end{equation}
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#+name: fig:sf_hinf_filters_plant_b
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#+caption: Generalized plant for the synthesis of the complementary filters
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#+attr_latex: :float t :scale 1
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file:./figs/sf_hinf_filters_plant_b.pdf
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The $\hinf$ synthesis objective is then to design a stable filter $H_L$ (Fig. ref:fig:sf_hinf_filters_b) such that the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $[z_H, \ z_L]$ is less than $1$:
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\begin{equation}
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\hnorm{\begin{matrix} (1 - H_L) w_H \\ H_L w_L \end{matrix}} \le 1
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\end{equation}
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Which is equivalent to
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#+NAME: eq:hinf_problem
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\begin{equation}
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\hnorm{\begin{matrix} H_H w_H \\ H_L w_L \end{matrix}} < 1 \text{ by choosing } H_H = 1 - H_L
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\end{equation}
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#+name: fig:sf_hinf_filters_b
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#+caption: $\hinf\text{-synthesis}$ of complementary filters
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#+attr_latex: :float t :scale 1
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file:./figs/sf_hinf_filters_b.pdf
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Performance conditions eqref:eq:hinf_cond_hl and eqref:eq:hinf_cond_hl are satisfied by eqref:eq:hinf_problem.
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Complementary condition eqref:eq:hinf_cond_complementarity is satisfied by design: $H_H = 1 - H_L$ and thus $H_L + H_H = 1$.
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The stability condition eqref:eq:hinf_cond_stability is guaranteed by the $H_\infty$ synthesis (*reference*).
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Using this synthesis method, we are then able to shape at the same time the high pass and low pass filters while ensuring their complementary.
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** Control requirements as $\mathcal{H}_\infty$ norm of complementary filters
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As presented in Sec. ref:sec:trans_perf, almost all the requirements can be specified with upper bounds on the complementary filters.
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However, robust performance condition eqref:eq:robust_perf_a is not.
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With the $\mathcal{H}_\infty$ synthesis the condition eqref:eq:hinf_problem only ensure
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\begin{align*}
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\hnorm{\begin{matrix} H_H w_H \\ H_L w_L \end{matrix}} \le 1 &\Leftrightarrow \max_\omega \sqrt{|w_L H_L|^2 + |w_H H_H|^2} \le 1\\
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&\Rightarrow |w_L H_L| + |w_H H_H| \le \sqrt{2} \quad \forall\omega
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\end{align*}
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And thus we have almost robust stability.
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** Choice of the weighting functions
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<<sec:hinf_weighting_func>>
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We here give some advice on the choice of the weighting functions used for the synthesis of the complementary filters.
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The shape should be such that the performance requirements are met as explain in Sec. ref:sec:trans_perf.
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However, one should be careful when designing the complementary filters, and should only use stable and minimum phase transfer functions.
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The order of the weights should stay reasonably small as this will increase the complexity of the optimization problem.
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One should not forget the fundamental limitations of feedback control such that $S + T = 1$.
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Similarly, we here have that $H_L + H_H = 1$ which implies that $H_L$ and $H_H$ cannot be made small at the same time.
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** Trade-off between performance and robustness
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** Analytical formula of complementary filters
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<<sec:analytical_complementary_filters>>
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To simplify the synthesis, one can use already synthesized filters
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\begin{align}
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H_L(s) = \frac{1}{1 + \frac{s}{\omega_0}}\\
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H_H(s) = \frac{\frac{s}{\omega_0}}{1 + \frac{s}{\omega_0}}
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\end{align}
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\begin{align}
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H_L(s) &= \frac{(1+\alpha) (\frac{s}{\omega_0})+1}{\left((\frac{s}{\omega_0})+1\right) \left((\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1\right)}\\
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H_H(s) &= \frac{(\frac{s}{\omega_0})^2 \left((\frac{s}{\omega_0})+1+\alpha\right)}{\left((\frac{s}{\omega_0})+1\right) \left((\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1\right)}
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\end{align}
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\begin{align}
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H_L(s) &= \frac{\left(1+(\alpha+1)(\beta+1)\right) (\frac{s}{\omega_0})^2 + (1+\alpha+\beta)(\frac{s}{\omega_0}) + 1}{\left(\frac{s}{\omega_0} + 1\right) \left( (\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1 \right) \left( (\frac{s}{\omega_0})^2 + \beta (\frac{s}{\omega_0}) + 1 \right)}\\
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H_H(s) &= \frac{(\frac{s}{\omega_0})^3 \left( (\frac{s}{\omega_0})^2 + (1+\alpha+\beta) (\frac{s}{\omega_0}) + (1+(\alpha+1)(\beta+1)) \right)}{\left(\frac{s}{\omega_0} + 1\right) \left( (\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1 \right) \left( (\frac{s}{\omega_0})^2 + \beta (\frac{s}{\omega_0}) + 1 \right)}
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\end{align}
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* Discussion
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* Conclusion
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<<sec:conclusion>>
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<<sec:conclusion>>
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* Acknowledgment
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