Rework figures, add colors
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@ -254,12 +254,27 @@ Finally, concluding remarks are presented in Section [[*Concluding remarks][5]].
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* Complementary Filters Requirements
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<<sec:requirements>>
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** Introduction :ignore:
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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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** Sensor Models
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<<sec:sensor_models>>
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- Noise + dynamical uncertainty
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- Noise + dynamics
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#+name: fig:sensor_model
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#+caption: Basic Sensor Model
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[[file:figs/sensor_model.pdf]]
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- Suppose we calibrate the sensors
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#+name: fig:sensor_model_calibrated
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#+caption: Calibrated Sensor
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[[file:figs/sensor_model_calibrated.pdf]]
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** Sensor Fusion Architecture
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<<sec:sensor_fusion>>
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@ -322,6 +337,15 @@ In practical systems the sensor dynamics is not perfect and eqref:eq:perfect_dyn
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In such case, one can use an inversion filter $\hat{G}_i^{-1}(s)$ to normalize the sensor dynamics, where $\hat{G}_i(s)$ is an estimate of the sensor dynamics $G_i(s)$.
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However, as there is always some level of uncertainty on the dynamics, it cannot be perfectly inverted and $\hat{G}_i^{-1}(s) G_i(s) \neq 1$.
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#+name: fig:sensor_model_uncertainty
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#+caption: Input Uncertainty
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[[file:figs/sensor_model_uncertainty.png]]
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#+name: fig:sensor_model_uncertainty_simplified
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#+caption: Input Uncertainty
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#+RESULTS:
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[[file:figs/sensor_model_uncertainty_simplified.png]]
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Let's represent the resulting dynamic uncertainty of the inverted sensors by an input multiplicative uncertainty as shown in Fig. ref:fig:sensor_fusion_dynamic_uncertainty where $\Delta_i$ is any stable transfer function satisfying $|\Delta_i(j\omega)| \le 1,\ \forall\omega$, and $|w_i(s)|$ is a weight representing the magnitude of the uncertainty.
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#+name: fig:sensor_fusion_dynamic_uncertainty
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@ -462,7 +486,7 @@ Let's validate the proposed design method of complementary filters with a simple
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- the gain of both filters is equal to $10^{-3}$ away from the merging frequency
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The weighting functions $W_1(s)$ and $W_2(s)$ are designed using eqref:eq:weight_formula.
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The parameters used are summarized in table ref:tab:weights_params and the magnitude of the weighting functions is shown in Fig. ref:fig:hinf_synthesis_results.
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The parameters used are summarized in table ref:tab:weights_params and the magnitude of the weighting functions is shown in Fig. ref:fig:hinf_filters_results.
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#+name: tab:weights_params
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#+caption: Parameters used for $W_1(s)$ and $W_2(s)$
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@ -476,16 +500,15 @@ The parameters used are summarized in table ref:tab:weights_params and the magni
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| $G_c$ | $0.5$ | $0.5$ |
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| $n$ | $2$ | $3$ |
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The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_synthesis_results and their transfer functions in the Laplace domain are given below.
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The bode plots of the obtained complementary filters are shown in Fig. ref:fig:hinf_filters_results and their transfer functions in the Laplace domain are given below.
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\begin{align*}
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H_1(s) &= \frac{10^{-8} (s+6.6e^9) (s+3450)^2 (s^2 + 49s + 895)}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}\\
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H_2(s) &= \frac{(s+6.6e^4) (s+160) (s+4)^3}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}
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\end{align*}
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#+name: fig:hinf_synthesis_results
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#+name: fig:hinf_filters_results
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#+caption: Frequency response of the weighting functions and complementary filters obtained using $\mathcal{H}_\infty$ synthesis
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#+attr_latex: :scale 1
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[[file:figs/hinf_synthesis_results.pdf]]
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[[file:figs/hinf_filters_results.pdf]]
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* Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO
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<<sec:application_ligo>>
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@ -1,28 +1,21 @@
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% Created 2021-04-28 mer. 15:56
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% Created 2021-04-30 ven. 11:16
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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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\usepackage[T1]{fontenc}
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\usepackage{graphicx}
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\usepackage{grffile}
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\usepackage{longtable}
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\usepackage{wrapfig}
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\usepackage{rotating}
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\usepackage[normalem]{ulem}
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\usepackage{amsmath}
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\usepackage{textcomp}
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\usepackage{amssymb}
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\usepackage{capt-of}
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\usepackage{hyperref}
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\usepackage{bm}
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\usepackage{array}
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\usepackage{amsmath,amssymb,amsfonts}
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\usepackage{algorithmic}
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\usepackage{textcomp}
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\usepackage{cases}
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\usepackage{tabularx,siunitx,booktabs}
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\usepackage{algorithmic}
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\usepackage{import}
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\usepackage{hyperref}
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\usepackage[hyperref]{xcolor}
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\hypersetup{colorlinks=true}
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\usepackage[top=2cm, bottom=2cm, left=2cm, right=2cm]{geometry}
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\usepackage{amsfonts}
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\usepackage{siunitx}
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\journal{Mechanical Systems and Signal Processing}
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\author[a1,a2]{Thomas Dehaeze\corref{cor1}}
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\author[a3,a4]{Mohit Verma}
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@ -32,6 +25,12 @@
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\address[a2]{University of Li\`{e}ge, Department of Aerospace and Mechanical Engineering, 4000 Li\`{e}ge, Belgium.}
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\address[a3]{CSIR --- Structural Engineering Research Centre, Taramani, Chennai --- 600113, India.}
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\address[a4]{Universit\'{e} Libre de Bruxelles, Precision Mechatronics Laboratory, BEAMS Department, 1050 Brussels, Belgium.}
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\usepackage{tabularx}
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\usepackage{booktabs}
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\usepackage{array}
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\usepackage[hyperref]{xcolor}
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\usepackage[top=2cm, bottom=2cm, left=2cm, right=2cm]{geometry}
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\hypersetup{colorlinks=true}
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\date{}
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\title{A new method of designing complementary filters for sensor fusion using \(\mathcal{H}_\infty\) synthesis}
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\begin{document}
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@ -58,7 +57,7 @@ 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:orgf244196}
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\label{sec:org341e767}
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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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@ -78,30 +77,64 @@ Sensor fusion \sep{} Optimal filters \sep{} \(\mathcal{H}_\infty\) synthesis \se
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\item \cite{robert12_introd_random_signal_applied_kalman} advantage of complementary filters over Kalman filtering
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\end{itemize}
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\begin{itemize}
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\item \cite{pascoal99_navig_system_desig_using_time} use LMI to generate complementary filters
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\item \cite{plummer06_optim_compl_filter_their_applic_motion_measur} use H-Infinity to optimize complementary filters (flatten the super sensor noise spectral density)
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\item \cite{jensen13_basic_uas} design of complementary filters with classical control theory
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\item \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}: FIR + convex optimization
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\item 3 complementary filters: \cite{becker15_compl_filter_desig_three_frequen_bands}
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\item Analog complementary filters: \cite{yong16_high_speed_vertic_posit_stage}, \cite{moore19_capac_instr_sensor_fusion_high_bandw_nanop}
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\item Analytical methods:
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\begin{itemize}
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\item first order: \cite{corke04_inert_visual_sensin_system_small_auton_helic}
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\item second order: \cite{baerveldt97_low_cost_low_weigh_attit}, \cite{stoten01_fusion_kinet_data_using_compos_filter}, \cite{jensen13_basic_uas}
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\item higher order: \cite{shaw90_bandw_enhan_posit_measur_using_measur_accel}, \cite{zimmermann92_high_bandw_orien_measur_contr}, \cite{collette15_sensor_fusion_method_high_perfor}, \cite{matichard15_seism_isolat_advan_ligo}
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\end{itemize}
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\item Analog complementary filters: \cite{yong16_high_speed_vertic_posit_stage}, \cite{moore19_capac_instr_sensor_fusion_high_bandw_nanop}
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\item \cite{pascoal99_navig_system_desig_using_time} use LMI to generate complementary filters
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\item \cite{hua05_low_ligo,hua04_polyp_fir_compl_filter_contr_system}: FIR + convex optimization
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\item Similar to feedback system:
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\begin{itemize}
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\item \cite{plummer06_optim_compl_filter_their_applic_motion_measur} use H-Infinity to optimize complementary filters (flatten the super sensor noise spectral density)
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\item \cite{jensen13_basic_uas} design of complementary filters with classical control theory
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\end{itemize}
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\item 3 complementary filters: \cite{becker15_compl_filter_desig_three_frequen_bands}
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\end{itemize}
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\begin{itemize}
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\item Robustness problems: \cite{zimmermann92_high_bandw_orien_measur_contr} change of phase near the merging frequency
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\item Trial and error
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\item 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.
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\end{itemize}
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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:org2279a0f}
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\label{sec:org77471d1}
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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 Models}
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\label{sec:org363af04}
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\label{sec:sensor_models}
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\begin{itemize}
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\item Noise + dynamics
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\end{itemize}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/sensor_model.pdf}
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\caption{\label{fig:sensor_model}Basic Sensor Model}
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\end{figure}
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\begin{itemize}
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\item Suppose we calibrate the sensors
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\end{itemize}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/sensor_model_calibrated.pdf}
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\caption{\label{fig:sensor_model_calibrated}Calibrated Sensor}
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\end{figure}
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\subsection{Sensor Fusion Architecture}
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\label{sec:org60aa99d}
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\label{sec:org240da2b}
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\label{sec:sensor_fusion}
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Let's consider two sensors measuring the same physical quantity \(x\) with dynamics \(G_1(s)\) and \(G_2(s)\), and with uncorrelated noise characteristics \(n_1\) and \(n_2\).
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@ -125,7 +158,7 @@ The complementary property of \(H_1(s)\) and \(H_2(s)\) implies that their trans
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\end{equation}
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\subsection{Noise Sensor Filtering}
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\label{sec:orgf9a3723}
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\label{sec:orgc5064da}
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\label{sec:noise_filtering}
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Let's first consider sensors with perfect dynamics
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@ -159,13 +192,25 @@ 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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\subsection{Robustness of the Fusion}
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\label{sec:org227af67}
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\label{sec:orgfc9ea9e}
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\label{sec:fusion_robustness}
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In practical systems the sensor dynamics is not perfect and \eqref{eq:perfect_dynamics} is not verified.
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In such case, one can use an inversion filter \(\hat{G}_i^{-1}(s)\) to normalize the sensor dynamics, where \(\hat{G}_i(s)\) is an estimate of the sensor dynamics \(G_i(s)\).
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However, as there is always some level of uncertainty on the dynamics, it cannot be perfectly inverted and \(\hat{G}_i^{-1}(s) G_i(s) \neq 1\).
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/sensor_model_uncertainty.png}
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\caption{\label{fig:sensor_model_uncertainty}Input Uncertainty}
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\end{figure}
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1]{figs/sensor_model_uncertainty_simplified.png}
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\caption{\label{fig:sensor_model_uncertainty_simplified}Input Uncertainty}
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\end{figure}
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Let's represent the resulting dynamic uncertainty of the inverted sensors by an input multiplicative uncertainty as shown in Fig. \ref{fig:sensor_fusion_dynamic_uncertainty} where \(\Delta_i\) is any stable transfer function satisfying \(|\Delta_i(j\omega)| \le 1,\ \forall\omega\), and \(|w_i(s)|\) is a weight representing the magnitude of the uncertainty.
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\begin{figure}[htbp]
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@ -204,13 +249,13 @@ 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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\section{Complementary Filters Shaping using \(\mathcal{H}_\infty\) Synthesis}
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\label{sec:org10fbb17}
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\label{sec:org678f099}
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\label{sec:hinf_method}
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As shown in Sec. \ref{sec:requirements}, the performance and robustness of the sensor fusion architecture depends on the complementary filters norms.
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Therefore, the development of a synthesis method of complementary filters that allows the shaping of their norm is necessary.
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\subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
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\label{sec:org933b14f}
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\label{sec:hinf_synthesis}
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\subsection{Synthesis Objective}
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\label{sec:orgf726b5b}
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\label{sec:synthesis_objective}
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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}.
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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.
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\begin{subequations}
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@ -223,6 +268,9 @@ This is equivalent as to finding stable transfer functions \(H_1(s)\) and \(H_2(
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\end{subequations}
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where \(W_1(s)\) and \(W_2(s)\) are two weighting transfer functions that are chosen to shape the norms of the corresponding filters.
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\subsection{Shaping of Complementary Filters using \(\mathcal{H}_\infty\) synthesis}
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\label{sec:orga266a36}
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\label{sec:hinf_synthesis}
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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}.
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\begin{equation}
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\label{eq:generalized_plant}
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@ -257,7 +305,7 @@ The conditions \eqref{eq:hinf_cond_h1} and \eqref{eq:hinf_cond_h2} on the filter
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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.
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\subsection{Weighting Functions Design}
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\label{sec:org3bb6eca}
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\label{sec:org911c399}
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\label{sec:hinf_weighting_func}
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The proper design of the weighting functions is of primary importance for the success of the presented complementary filters \(\mathcal{H}_\infty\) synthesis.
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@ -302,7 +350,7 @@ The general shape of a weighting function generated using \eqref{eq:weight_formu
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\end{figure}
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\subsection{Validation of the proposed synthesis method}
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\label{sec:org5517901}
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\label{sec:org6867aff}
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\label{sec:hinf_example}
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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:
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\begin{itemize}
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@ -313,7 +361,7 @@ Let's validate the proposed design method of complementary filters with a simple
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\end{itemize}
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The weighting functions \(W_1(s)\) and \(W_2(s)\) are designed using \eqref{eq:weight_formula}.
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The parameters used are summarized in table \ref{tab:weights_params} and the magnitude of the weighting functions is shown in Fig. \ref{fig:hinf_synthesis_results}.
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The parameters used are summarized in table \ref{tab:weights_params} and the magnitude of the weighting functions is shown in Fig. \ref{fig:hinf_filters_results}.
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\begin{table}[htbp]
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\caption{\label{tab:weights_params}Parameters used for \(W_1(s)\) and \(W_2(s)\)}
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@ -331,7 +379,7 @@ Parameter & \(W_1(s)\) & \(W_2(s)\)\\
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\end{tabularx}
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\end{table}
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The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:hinf_synthesis_results} and their transfer functions in the Laplace domain are given below.
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The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:hinf_filters_results} and their transfer functions in the Laplace domain are given below.
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\begin{align*}
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H_1(s) &= \frac{10^{-8} (s+6.6e^9) (s+3450)^2 (s^2 + 49s + 895)}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}\\
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H_2(s) &= \frac{(s+6.6e^4) (s+160) (s+4)^3}{(s+6.6e^4) (s^2 + 106 s + 3e^3) (s^2 + 72s + 3580)}
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@ -339,50 +387,12 @@ The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:
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\begin{figure}[htbp]
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\centering
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\includegraphics[scale=1,scale=1]{figs/hinf_synthesis_results.pdf}
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\caption{\label{fig:hinf_synthesis_results}Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis}
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\end{figure}
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\subsection{Synthesis of Three Complementary Filters}
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\label{sec:org1fd1484}
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\label{sec:hinf_three_comp_filters}
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Some applications may require to merge more than two sensors.
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In such a case, it is necessary to design as many complementary filters as the number of sensors used.
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The synthesis problem is then to compute \(n\) stable transfer functions \(H_i(s)\) such that \eqref{eq:hinf_problem_gen} is satisfied.
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\begin{subequations}
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\label{eq:hinf_problem_gen}
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\begin{align}
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& \sum_{i=0}^n H_i(s) = 1 \label{eq:hinf_cond_compl_gen} \\
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& \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}
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\end{align}
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\end{subequations}
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The synthesis method is generalized here for the synthesis of three complementary filters using the architecture shown in Fig. \ref{fig:comp_filter_three_hinf}.
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||||
The \(\mathcal{H}_\infty\) synthesis objective applied on \(P(s)\) is to design two stable filters \(H_2(s)\) and \(H_3(s)\) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_1,\ z_2, \ z_3]\) is less than one \eqref{eq:hinf_syn_obj_three}.
|
||||
\begin{equation}
|
||||
\label{eq:hinf_syn_obj_three}
|
||||
\left\| \begin{matrix} \left[1 - H_2(s) - H_3(s)\right] W_1(s) \\ H_2(s) W_2(s) \\ H_3(s) W_3(s) \end{matrix} \right\|_\infty \le 1
|
||||
\end{equation}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,scale=1]{figs/comp_filter_three_hinf.pdf}
|
||||
\caption{\label{fig:comp_filter_three_hinf}Architecture for \(\mathcal{H}_\infty\) synthesis of three complementary filters}
|
||||
\end{figure}
|
||||
|
||||
By choosing \(H_1(s) \triangleq 1 - H_2(s) - H_3(s)\), the proposed \(\mathcal{H}_\infty\) synthesis solves the design problem \eqref{eq:hinf_problem_gen}. \par
|
||||
An example is given to validate the method where three sensors are used in different frequency bands (up to \(\SI{1}{Hz}\), from \(1\) to \(\SI{10}{Hz}\) and above \(\SI{10}{Hz}\) respectively).
|
||||
Three weighting functions are designed using \eqref{eq:weight_formula} and shown by dashed curves in Fig. \ref{fig:hinf_three_synthesis_results}.
|
||||
The bode plots of the obtained complementary filters are shown in Fig. \ref{fig:hinf_three_synthesis_results}.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1,scale=1]{figs/hinf_three_synthesis_results.pdf}
|
||||
\caption{\label{fig:hinf_three_synthesis_results}Frequency response of the weighting functions and three complementary filters obtained using \(\mathcal{H}_\infty\) synthesis}
|
||||
\includegraphics[scale=1]{figs/hinf_filters_results.pdf}
|
||||
\caption{\label{fig:hinf_filters_results}Frequency response of the weighting functions and complementary filters obtained using \(\mathcal{H}_\infty\) synthesis}
|
||||
\end{figure}
|
||||
|
||||
\section{Application: Design of Complementary Filters used in the Active Vibration Isolation System at the LIGO}
|
||||
\label{sec:org925754e}
|
||||
\label{sec:org377e66e}
|
||||
\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.
|
||||
@ -391,7 +401,7 @@ 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:org4ade5f6}
|
||||
\label{sec:org75813ae}
|
||||
\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}
|
||||
@ -402,7 +412,7 @@ The specifications for one pair of complementary filters used at the LIGO are su
|
||||
\end{itemize}
|
||||
|
||||
\subsection{Weighting Functions Design}
|
||||
\label{sec:org8c85120}
|
||||
\label{sec:org7015511}
|
||||
\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.
|
||||
@ -418,7 +428,7 @@ The magnitudes of the weighting functions are shown in Fig. \ref{fig:ligo_weight
|
||||
\end{figure}
|
||||
|
||||
\subsection{\(\mathcal{H}_\infty\) Synthesis}
|
||||
\label{sec:org89d27e1}
|
||||
\label{sec:orge5b6fe6}
|
||||
\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 \(27\).
|
||||
@ -432,7 +442,7 @@ They are found to be very close to each other and this shows the effectiveness o
|
||||
\end{figure}
|
||||
|
||||
\section{Conclusion}
|
||||
\label{sec:org153df0f}
|
||||
\label{sec:org39b90d9}
|
||||
\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.
|
||||
@ -440,7 +450,7 @@ 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:org6c8e8b0}
|
||||
\label{sec:org1ece332}
|
||||
This research benefited from a FRIA grant from the French Community of Belgium.
|
||||
|
||||
\bibliographystyle{elsarticle-num}
|
||||
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<title>Complementary Filters Shaping Using \(\mathcal{H}_\infty\) Synthesis - Matlab Computation</title>
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<title>Complementary Filters Shaping Using $\mathcal{H}_\infty$ Synthesis - Matlab Computation</title>
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<meta name="author" content="Thomas Dehaeze" />
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<body>
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<div id="org-div-home-and-up">
|
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@ -30,42 +39,63 @@
|
||||
<h2>Table of Contents</h2>
|
||||
<div id="text-table-of-contents">
|
||||
<ul>
|
||||
<li><a href="#orga86008b">1. H-Infinity synthesis of complementary filters</a>
|
||||
<li><a href="#sec:h_inf_synthesis_complementary_filters">1. H-Infinity synthesis of complementary filters</a>
|
||||
<ul>
|
||||
<li><a href="#orga8d0882">1.1. Synthesis Architecture</a></li>
|
||||
<li><a href="#orgf309349">1.2. Design of Weighting Function</a></li>
|
||||
<li><a href="#org3d519e3">1.3. H-Infinity Synthesis</a></li>
|
||||
<li><a href="#org42911cd">1.4. Obtained Complementary Filters</a></li>
|
||||
<li><a href="#org38a6275">1.1. Synthesis Architecture</a></li>
|
||||
<li><a href="#org8d0a2ea">1.2. Design of Weighting Function</a></li>
|
||||
<li><a href="#orgc235246">1.3. H-Infinity Synthesis</a></li>
|
||||
<li><a href="#org326a6a1">1.4. Obtained Complementary Filters</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org1cd882b">2. Generating 3 complementary filters</a>
|
||||
<li><a href="#orgb616673">2. Generating 3 complementary filters</a>
|
||||
<ul>
|
||||
<li><a href="#org201b962">2.1. Theory</a></li>
|
||||
<li><a href="#orgbb81a3a">2.2. Weights</a></li>
|
||||
<li><a href="#orgc782a41">2.3. H-Infinity Synthesis</a></li>
|
||||
<li><a href="#orgbe6c26a">2.4. Obtained Complementary Filters</a></li>
|
||||
<li><a href="#orge13ec24">2.1. Theory</a></li>
|
||||
<li><a href="#org0043ce8">2.2. Weights</a></li>
|
||||
<li><a href="#orge8e2214">2.3. H-Infinity Synthesis</a></li>
|
||||
<li><a href="#org3e0db09">2.4. Obtained Complementary Filters</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgfb4a473">3. Implement complementary filters for LIGO</a>
|
||||
<li><a href="#orgb0c5eb8">3. Implement complementary filters for LIGO</a>
|
||||
<ul>
|
||||
<li><a href="#org0a64590">3.1. Specifications</a></li>
|
||||
<li><a href="#org5187f2d">3.2. FIR Filter</a></li>
|
||||
<li><a href="#org6e83a71">3.3. Weights</a></li>
|
||||
<li><a href="#org56349cf">3.4. H-Infinity Synthesis</a></li>
|
||||
<li><a href="#org3ef818f">3.5. Compare FIR and H-Infinity Filters</a></li>
|
||||
<li><a href="#org68b2264">3.1. Specifications</a></li>
|
||||
<li><a href="#org639758d">3.2. FIR Filter</a></li>
|
||||
<li><a href="#org3361251">3.3. Weights</a></li>
|
||||
<li><a href="#org3ea9781">3.4. H-Infinity Synthesis</a></li>
|
||||
<li><a href="#org2e38aee">3.5. Compare FIR and H-Infinity Filters</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org6fa1123">4. Alternative Synthesis</a>
|
||||
<li><a href="#org278cb17">4. Alternative Synthesis</a>
|
||||
<ul>
|
||||
<li><a href="#org1bb8ee7">4.1. Two generalized plants</a></li>
|
||||
<li><a href="#orga117463">4.2. Shaping the Low pass filter or the high pass filter?</a></li>
|
||||
<li><a href="#orgc2f7629">4.1. Two generalized plants</a></li>
|
||||
<li><a href="#org14ddd98">4.2. Shaping the Low pass filter or the high pass filter?</a></li>
|
||||
<li><a href="#orgc9d9779">4.3. Using Feedback architecture</a></li>
|
||||
<li><a href="#org0aa4270">4.4. Adding feature in the filters</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#orgf082879">5. Impose a positive slope at DC or a negative slope at infinite frequency</a>
|
||||
<li><a href="#orgb026d30">5. Impose a positive slope at DC or a negative slope at infinite frequency</a>
|
||||
<ul>
|
||||
<li><a href="#org96df1d1">5.1. Manually shift zeros to the origin after synthesis</a></li>
|
||||
<li><a href="#org6b92ce0">5.2. Imposing a positive slope at DC during the synthesis phase</a></li>
|
||||
<li><a href="#org71e3235">5.3. Imposing a negative slope at infinity frequency during the synthesis phase</a></li>
|
||||
<li><a href="#org4d4e2ad">5.1. Manually shift zeros to the origin after synthesis</a></li>
|
||||
<li><a href="#orgf1693db">5.2. Imposing a positive slope at DC during the synthesis phase</a></li>
|
||||
<li><a href="#org52c02ef">5.3. Imposing a negative slope at infinity frequency during the synthesis phase</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org29e79e7">6. Functions</a>
|
||||
<ul>
|
||||
<li><a href="#orgfbca8a7">6.1. <code>generateWF</code>: Generate Weighting Functions</a>
|
||||
<ul>
|
||||
<li><a href="#org95965a4">Function description</a></li>
|
||||
<li><a href="#orgaeb61de">Optional Parameters</a></li>
|
||||
<li><a href="#orgd91e493">Generate the Weighting function</a></li>
|
||||
<li><a href="#org84eb536">Verification of the \(G_0\), \(G_c\) and \(G_\infty\) gains</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
<li><a href="#org2f9ac50">6.2. <code>generateCF</code>: Generate Complementary Filters</a>
|
||||
<ul>
|
||||
<li><a href="#org7c373f8">Function description</a></li>
|
||||
<li><a href="#org00c9d5d">Optional Parameters</a></li>
|
||||
<li><a href="#org92a7c04">H-Infinity Synthesis</a></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
@ -90,25 +120,22 @@ To achieve this, the sensors included in the filter should complement one anothe
|
||||
This document is divided into several sections:
|
||||
</p>
|
||||
<ul class="org-ul">
|
||||
<li>in section <a href="#orge265c61">1</a>, the \(\mathcal{H}_\infty\) synthesis is used for generating two complementary filters</li>
|
||||
<li>in section <a href="#org0f5d922">2</a>, a method using the \(\mathcal{H}_\infty\) synthesis is proposed to shape three of more complementary filters</li>
|
||||
<li>in section <a href="#org2c84916">3</a>, the \(\mathcal{H}_\infty\) synthesis is used and compared with FIR complementary filters used for LIGO</li>
|
||||
<li>in section <a href="#sec:h_inf_synthesis_complementary_filters">1</a>, the \(\mathcal{H}_\infty\) synthesis is used for generating two complementary filters</li>
|
||||
<li>in section <a href="#orgd4d516e">2</a>, a method using the \(\mathcal{H}_\infty\) synthesis is proposed to shape three of more complementary filters</li>
|
||||
<li>in section <a href="#org9327342">3</a>, the \(\mathcal{H}_\infty\) synthesis is used and compared with FIR complementary filters used for LIGO</li>
|
||||
</ul>
|
||||
|
||||
<div class="note" id="org9abbfdc">
|
||||
<div class="note" id="orgad6d854">
|
||||
<p>
|
||||
Add the Matlab code use to obtain the results presented in the paper are accessible <a href="matlab.zip">here</a> and presented below.
|
||||
</p>
|
||||
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga86008b" class="outline-2">
|
||||
<h2 id="orga86008b"><span class="section-number-2">1</span> H-Infinity synthesis of complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-1">
|
||||
<p>
|
||||
<a id="orge265c61"></a>
|
||||
</p>
|
||||
<div class="note" id="org7b6c965">
|
||||
<div id="outline-container-sec:h_inf_synthesis_complementary_filters" class="outline-2">
|
||||
<h2 id="sec:h_inf_synthesis_complementary_filters"><span class="section-number-2">1</span> H-Infinity synthesis of complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-sec:h_inf_synthesis_complementary_filters">
|
||||
<div class="note" id="org65d886e">
|
||||
<p>
|
||||
The Matlab file corresponding to this section is accessible <a href="matlab/h_inf_synthesis_complementary_filters.m">here</a>.
|
||||
</p>
|
||||
@ -116,8 +143,8 @@ The Matlab file corresponding to this section is accessible <a href="matlab/h_in
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga8d0882" class="outline-3">
|
||||
<h3 id="orga8d0882"><span class="section-number-3">1.1</span> Synthesis Architecture</h3>
|
||||
<div id="outline-container-org38a6275" class="outline-3">
|
||||
<h3 id="org38a6275"><span class="section-number-3">1.1</span> Synthesis Architecture</h3>
|
||||
<div class="outline-text-3" id="text-1-1">
|
||||
<p>
|
||||
We here synthesize two complementary filters using the \(\mathcal{H}_\infty\) synthesis.
|
||||
@ -125,18 +152,18 @@ The goal is to specify upper bounds on the norms of the two complementary filter
|
||||
</p>
|
||||
|
||||
<p>
|
||||
In order to do so, we use the generalized plant shown on figure <a href="#orge741156">1</a> where \(W_1(s)\) and \(W_2(s)\) are weighting transfer functions that will be used to shape \(H_1(s)\) and \(H_2(s)\) respectively.
|
||||
In order to do so, we use the generalized plant shown on figure <a href="#org49ae644">1</a> where \(W_1(s)\) and \(W_2(s)\) are weighting transfer functions that will be used to shape \(H_1(s)\) and \(H_2(s)\) respectively.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orge741156" class="figure">
|
||||
<div id="org49ae644" class="figure">
|
||||
<p><img src="figs-tikz/h_infinity_robust_fusion.png" alt="h_infinity_robust_fusion.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 1: </span>\(\mathcal{H}_\infty\) synthesis of the complementary filters</p>
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The \(\mathcal{H}_\infty\) synthesis applied on this generalized plant will give a transfer function \(H_2\) (figure <a href="#orge741156">1</a>) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_1,\ z_2]\) is less than one:
|
||||
The \(\mathcal{H}_\infty\) synthesis applied on this generalized plant will give a transfer function \(H_2\) (figure <a href="#org49ae644">1</a>) such that the \(\mathcal{H}_\infty\) norm of the transfer function from \(w\) to \([z_1,\ z_2]\) is less than one:
|
||||
\[ \left\| \begin{array}{c} (1 - H_2(s)) W_1(s) \\ H_2(s) W_2(s) \end{array} \right\|_\infty < 1 \]
|
||||
</p>
|
||||
|
||||
@ -156,8 +183,8 @@ We then see that \(W_1(s)\) and \(W_2(s)\) can be used to shape both \(H_1(s)\)
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf309349" class="outline-3">
|
||||
<h3 id="orgf309349"><span class="section-number-3">1.2</span> Design of Weighting Function</h3>
|
||||
<div id="outline-container-org8d0a2ea" class="outline-3">
|
||||
<h3 id="org8d0a2ea"><span class="section-number-3">1.2</span> Design of Weighting Function</h3>
|
||||
<div class="outline-text-3" id="text-1-2">
|
||||
<p>
|
||||
A formula is proposed to help the design of the weighting functions:
|
||||
@ -181,11 +208,11 @@ The parameters permits to specify:
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
The general shape of a weighting function generated using the formula is shown in figure <a href="#org18f93a4">2</a>.
|
||||
The general shape of a weighting function generated using the formula is shown in figure <a href="#orgca3464c">2</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org18f93a4" class="figure">
|
||||
<div id="orgca3464c" class="figure">
|
||||
<p><img src="figs/weight_formula.png" alt="weight_formula.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 2: </span>Gain of the Weighting Function formula</p>
|
||||
@ -201,7 +228,7 @@ W2 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org54a46b0" class="figure">
|
||||
<div id="org1ce2cf7" class="figure">
|
||||
<p><img src="figs/weights_W1_W2.png" alt="weights_W1_W2.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 3: </span>Weights on the complementary filters \(W_1\) and \(W_2\) and the associated performance weights</p>
|
||||
@ -209,8 +236,8 @@ W2 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3d519e3" class="outline-3">
|
||||
<h3 id="org3d519e3"><span class="section-number-3">1.3</span> H-Infinity Synthesis</h3>
|
||||
<div id="outline-container-orgc235246" class="outline-3">
|
||||
<h3 id="orgc235246"><span class="section-number-3">1.3</span> H-Infinity Synthesis</h3>
|
||||
<div class="outline-text-3" id="text-1-3">
|
||||
<p>
|
||||
We define the generalized plant \(P\) on matlab.
|
||||
@ -230,7 +257,7 @@ And we do the \(\mathcal{H}_\infty\) synthesis using the <code>hinfsyn</code> co
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<pre class="example" id="orgb213b31">
|
||||
<pre class="example" id="org44ccac0">
|
||||
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
Resetting value of Gamma min based on D_11, D_12, D_21 terms
|
||||
|
||||
@ -264,7 +291,7 @@ Test bounds: 0.1000 < gamma <= 1050.0000
|
||||
</pre>
|
||||
|
||||
<p>
|
||||
We then define the high pass filter \(H_1 = 1 - H_2\). The bode plot of both \(H_1\) and \(H_2\) is shown on figure <a href="#orgc79ce80">4</a>.
|
||||
We then define the high pass filter \(H_1 = 1 - H_2\). The bode plot of both \(H_1\) and \(H_2\) is shown on figure <a href="#orgdd6084f">4</a>.
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
@ -274,15 +301,15 @@ We then define the high pass filter \(H_1 = 1 - H_2\). The bode plot of both \(H
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org42911cd" class="outline-3">
|
||||
<h3 id="org42911cd"><span class="section-number-3">1.4</span> Obtained Complementary Filters</h3>
|
||||
<div id="outline-container-org326a6a1" class="outline-3">
|
||||
<h3 id="org326a6a1"><span class="section-number-3">1.4</span> Obtained Complementary Filters</h3>
|
||||
<div class="outline-text-3" id="text-1-4">
|
||||
<p>
|
||||
The obtained complementary filters are shown on figure <a href="#orgc79ce80">4</a>.
|
||||
The obtained complementary filters are shown on figure <a href="#orgdd6084f">4</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgc79ce80" class="figure">
|
||||
<div id="orgdd6084f" class="figure">
|
||||
<p><img src="figs/hinf_filters_results.png" alt="hinf_filters_results.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 4: </span>Obtained complementary filters using \(\mathcal{H}_\infty\) synthesis</p>
|
||||
@ -291,13 +318,13 @@ The obtained complementary filters are shown on figure <a href="#orgc79ce80">4</
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org1cd882b" class="outline-2">
|
||||
<h2 id="org1cd882b"><span class="section-number-2">2</span> Generating 3 complementary filters</h2>
|
||||
<div id="outline-container-orgb616673" class="outline-2">
|
||||
<h2 id="orgb616673"><span class="section-number-2">2</span> Generating 3 complementary filters</h2>
|
||||
<div class="outline-text-2" id="text-2">
|
||||
<p>
|
||||
<a id="org0f5d922"></a>
|
||||
<a id="orgd4d516e"></a>
|
||||
</p>
|
||||
<div class="note" id="org7c35287">
|
||||
<div class="note" id="org1a1e099">
|
||||
<p>
|
||||
The Matlab file corresponding to this section is accessible <a href="matlab/three_comp_filters.m">here</a>.
|
||||
</p>
|
||||
@ -305,8 +332,8 @@ The Matlab file corresponding to this section is accessible <a href="matlab/thre
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org201b962" class="outline-3">
|
||||
<h3 id="org201b962"><span class="section-number-3">2.1</span> Theory</h3>
|
||||
<div id="outline-container-orge13ec24" class="outline-3">
|
||||
<h3 id="orge13ec24"><span class="section-number-3">2.1</span> Theory</h3>
|
||||
<div class="outline-text-3" id="text-2-1">
|
||||
<p>
|
||||
We want:
|
||||
@ -319,11 +346,11 @@ We want:
|
||||
\end{align*}
|
||||
|
||||
<p>
|
||||
For that, we use the \(\mathcal{H}_\infty\) synthesis with the architecture shown on figure <a href="#org86ebebf">5</a>.
|
||||
For that, we use the \(\mathcal{H}_\infty\) synthesis with the architecture shown on figure <a href="#org38edab5">5</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org86ebebf" class="figure">
|
||||
<div id="org38edab5" class="figure">
|
||||
<p><img src="figs-tikz/comp_filter_three_hinf.png" alt="comp_filter_three_hinf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 5: </span>Generalized architecture for generating 3 complementary filters</p>
|
||||
@ -344,8 +371,8 @@ And thus if we choose \(H_1 = 1 - H_2 - H_3\) we have solved the problem.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbb81a3a" class="outline-3">
|
||||
<h3 id="orgbb81a3a"><span class="section-number-3">2.2</span> Weights</h3>
|
||||
<div id="outline-container-org0043ce8" class="outline-3">
|
||||
<h3 id="org0043ce8"><span class="section-number-3">2.2</span> Weights</h3>
|
||||
<div class="outline-text-3" id="text-2-2">
|
||||
<p>
|
||||
First we define the weights.
|
||||
@ -362,7 +389,7 @@ W3 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org6ef9224" class="figure">
|
||||
<div id="orgf1d851e" class="figure">
|
||||
<p><img src="figs/three_weighting_functions.png" alt="three_weighting_functions.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 6: </span>Three weighting functions used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters</p>
|
||||
@ -370,8 +397,8 @@ W3 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc782a41" class="outline-3">
|
||||
<h3 id="orgc782a41"><span class="section-number-3">2.3</span> H-Infinity Synthesis</h3>
|
||||
<div id="outline-container-orge8e2214" class="outline-3">
|
||||
<h3 id="orge8e2214"><span class="section-number-3">2.3</span> H-Infinity Synthesis</h3>
|
||||
<div class="outline-text-3" id="text-2-3">
|
||||
<p>
|
||||
Then we create the generalized plant <code>P</code>.
|
||||
@ -392,7 +419,7 @@ And we do the \(\mathcal{H}_\infty\) synthesis.
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<pre class="example" id="orga35c0a2">
|
||||
<pre class="example" id="org65963f1">
|
||||
[H, ~, gamma, ~] = hinfsyn(P, 1, 2,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
Resetting value of Gamma min based on D_11, D_12, D_21 terms
|
||||
|
||||
@ -427,8 +454,8 @@ Test bounds: 0.1000 < gamma <= 1050.0000
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbe6c26a" class="outline-3">
|
||||
<h3 id="orgbe6c26a"><span class="section-number-3">2.4</span> Obtained Complementary Filters</h3>
|
||||
<div id="outline-container-org3e0db09" class="outline-3">
|
||||
<h3 id="org3e0db09"><span class="section-number-3">2.4</span> Obtained Complementary Filters</h3>
|
||||
<div class="outline-text-3" id="text-2-4">
|
||||
<p>
|
||||
The obtained filters are:
|
||||
@ -441,7 +468,7 @@ H1 = 1 <span class="org-type">-</span> H2 <span class="org-type">-</span> H3;
|
||||
</div>
|
||||
|
||||
|
||||
<div id="orga85736d" class="figure">
|
||||
<div id="orgd968c86" class="figure">
|
||||
<p><img src="figs/three_complementary_filters_results.png" alt="three_complementary_filters_results.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 7: </span>The three complementary filters obtained after \(\mathcal{H}_\infty\) synthesis</p>
|
||||
@ -450,13 +477,13 @@ H1 = 1 <span class="org-type">-</span> H2 <span class="org-type">-</span> H3;
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgfb4a473" class="outline-2">
|
||||
<h2 id="orgfb4a473"><span class="section-number-2">3</span> Implement complementary filters for LIGO</h2>
|
||||
<div id="outline-container-orgb0c5eb8" class="outline-2">
|
||||
<h2 id="orgb0c5eb8"><span class="section-number-2">3</span> Implement complementary filters for LIGO</h2>
|
||||
<div class="outline-text-2" id="text-3">
|
||||
<p>
|
||||
<a id="org2c84916"></a>
|
||||
<a id="org9327342"></a>
|
||||
</p>
|
||||
<div class="note" id="org4890f37">
|
||||
<div class="note" id="orgd570d71">
|
||||
<p>
|
||||
The Matlab file corresponding to this section is accessible <a href="matlab/comp_filters_ligo.m">here</a>.
|
||||
</p>
|
||||
@ -472,8 +499,8 @@ The FIR complementary filters designed in (<a href="#citeproc_bib_item_1">Hua 20
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0a64590" class="outline-3">
|
||||
<h3 id="org0a64590"><span class="section-number-3">3.1</span> Specifications</h3>
|
||||
<div id="outline-container-org68b2264" class="outline-3">
|
||||
<h3 id="org68b2264"><span class="section-number-3">3.1</span> Specifications</h3>
|
||||
<div class="outline-text-3" id="text-3-1">
|
||||
<p>
|
||||
The specifications for the filters are:
|
||||
@ -486,11 +513,11 @@ The specifications for the filters are:
|
||||
</ol>
|
||||
|
||||
<p>
|
||||
The specifications are translated in upper bounds of the complementary filters are shown on figure <a href="#org40b6368">8</a>.
|
||||
The specifications are translated in upper bounds of the complementary filters are shown on figure <a href="#orge912401">8</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org40b6368" class="figure">
|
||||
<div id="orge912401" class="figure">
|
||||
<p><img src="figs/ligo_specifications.png" alt="ligo_specifications.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 8: </span>Specification for the LIGO complementary filters</p>
|
||||
@ -498,8 +525,8 @@ The specifications are translated in upper bounds of the complementary filters a
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5187f2d" class="outline-3">
|
||||
<h3 id="org5187f2d"><span class="section-number-3">3.2</span> FIR Filter</h3>
|
||||
<div id="outline-container-org639758d" class="outline-3">
|
||||
<h3 id="org639758d"><span class="section-number-3">3.2</span> FIR Filter</h3>
|
||||
<div class="outline-text-3" id="text-3-2">
|
||||
<p>
|
||||
We here try to implement the FIR complementary filter synthesis as explained in (<a href="#citeproc_bib_item_1">Hua 2005</a>).
|
||||
@ -580,7 +607,7 @@ h = y(2<span class="org-type">:</span>end);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<pre class="example" id="org199d33a">
|
||||
<pre class="example" id="org1229f63">
|
||||
cvx_begin
|
||||
variable y(n+1,1)
|
||||
% t
|
||||
@ -657,7 +684,7 @@ h = y(2:end);
|
||||
</pre>
|
||||
|
||||
<p>
|
||||
Finally, we compute the filter response over the frequency vector defined and the result is shown on figure <a href="#org1807c4b">9</a> which is very close to the filters obtain in (<a href="#citeproc_bib_item_1">Hua 2005</a>).
|
||||
Finally, we compute the filter response over the frequency vector defined and the result is shown on figure <a href="#org13d9ffd">9</a> which is very close to the filters obtain in (<a href="#citeproc_bib_item_1">Hua 2005</a>).
|
||||
</p>
|
||||
|
||||
<div class="org-src-container">
|
||||
@ -667,7 +694,7 @@ H = [exp(<span class="org-type">-</span><span class="org-constant">j</span><span
|
||||
</div>
|
||||
|
||||
|
||||
<div id="org1807c4b" class="figure">
|
||||
<div id="org13d9ffd" class="figure">
|
||||
<p><img src="figs/fir_filter_ligo.png" alt="fir_filter_ligo.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 9: </span>FIR Complementary filters obtain after convex optimization</p>
|
||||
@ -675,8 +702,8 @@ H = [exp(<span class="org-type">-</span><span class="org-constant">j</span><span
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6e83a71" class="outline-3">
|
||||
<h3 id="org6e83a71"><span class="section-number-3">3.3</span> Weights</h3>
|
||||
<div id="outline-container-org3361251" class="outline-3">
|
||||
<h3 id="org3361251"><span class="section-number-3">3.3</span> Weights</h3>
|
||||
<div class="outline-text-3" id="text-3-3">
|
||||
<p>
|
||||
We design weights that will be used for the \(\mathcal{H}_\infty\) synthesis of the complementary filters.
|
||||
@ -690,11 +717,11 @@ Here are the requirements on the filters:
|
||||
</ul>
|
||||
|
||||
<p>
|
||||
The bode plot of the weights is shown on figure <a href="#org8999a4f">10</a>.
|
||||
The bode plot of the weights is shown on figure <a href="#org17273a4">10</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org8999a4f" class="figure">
|
||||
<div id="org17273a4" class="figure">
|
||||
<p><img src="figs/ligo_weights.png" alt="ligo_weights.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 10: </span>Weights for the \(\mathcal{H}_\infty\) synthesis</p>
|
||||
@ -702,11 +729,11 @@ The bode plot of the weights is shown on figure <a href="#org8999a4f">10</a>.
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org56349cf" class="outline-3">
|
||||
<h3 id="org56349cf"><span class="section-number-3">3.4</span> H-Infinity Synthesis</h3>
|
||||
<div id="outline-container-org3ea9781" class="outline-3">
|
||||
<h3 id="org3ea9781"><span class="section-number-3">3.4</span> H-Infinity Synthesis</h3>
|
||||
<div class="outline-text-3" id="text-3-4">
|
||||
<p>
|
||||
We define the generalized plant as shown on figure <a href="#orge741156">1</a>.
|
||||
We define the generalized plant as shown on figure <a href="#org49ae644">1</a>.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">P = [0 wL;
|
||||
@ -723,7 +750,7 @@ And we do the \(\mathcal{H}_\infty\) synthesis using the <code>hinfsyn</code> co
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<pre class="example" id="orgd8777e8">
|
||||
<pre class="example" id="org82970b4">
|
||||
[Hl, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
Resetting value of Gamma min based on D_11, D_12, D_21 terms
|
||||
|
||||
@ -758,18 +785,18 @@ The high pass filter is defined as \(H_H = 1 - H_L\).
|
||||
The size of the filters is shown below.
|
||||
</p>
|
||||
|
||||
<pre class="example" id="orgc30176a">
|
||||
<pre class="example" id="org5988dfe">
|
||||
size(Hh), size(Hl)
|
||||
State-space model with 1 outputs, 1 inputs, and 27 states.
|
||||
State-space model with 1 outputs, 1 inputs, and 27 states.
|
||||
</pre>
|
||||
|
||||
<p>
|
||||
The bode plot of the obtained filters as shown on figure <a href="#orgf3626bd">11</a>.
|
||||
The bode plot of the obtained filters as shown on figure <a href="#org1a89dc1">11</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orgf3626bd" class="figure">
|
||||
<div id="org1a89dc1" class="figure">
|
||||
<p><img src="figs/hinf_synthesis_ligo_results.png" alt="hinf_synthesis_ligo_results.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 11: </span>Obtained complementary filters using the \(\mathcal{H}_\infty\) synthesis</p>
|
||||
@ -777,15 +804,15 @@ The bode plot of the obtained filters as shown on figure <a href="#orgf3626bd">1
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org3ef818f" class="outline-3">
|
||||
<h3 id="org3ef818f"><span class="section-number-3">3.5</span> Compare FIR and H-Infinity Filters</h3>
|
||||
<div id="outline-container-org2e38aee" class="outline-3">
|
||||
<h3 id="org2e38aee"><span class="section-number-3">3.5</span> Compare FIR and H-Infinity Filters</h3>
|
||||
<div class="outline-text-3" id="text-3-5">
|
||||
<p>
|
||||
Let’s now compare the FIR filters designed in (<a href="#citeproc_bib_item_1">Hua 2005</a>) and the one obtained with the \(\mathcal{H}_\infty\) synthesis on figure <a href="#orga60b7f8">12</a>.
|
||||
Let’s now compare the FIR filters designed in (<a href="#citeproc_bib_item_1">Hua 2005</a>) and the one obtained with the \(\mathcal{H}_\infty\) synthesis on figure <a href="#orgf51384d">12</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="orga60b7f8" class="figure">
|
||||
<div id="orgf51384d" class="figure">
|
||||
<p><img src="figs/comp_fir_ligo_hinf.png" alt="comp_fir_ligo_hinf.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 12: </span>Comparison between the FIR filters developped for LIGO and the \(\mathcal{H}_\infty\) complementary filters</p>
|
||||
@ -794,15 +821,15 @@ Let’s now compare the FIR filters designed in (<a href="#citeproc_bib_item
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6fa1123" class="outline-2">
|
||||
<h2 id="org6fa1123"><span class="section-number-2">4</span> Alternative Synthesis</h2>
|
||||
<div id="outline-container-org278cb17" class="outline-2">
|
||||
<h2 id="org278cb17"><span class="section-number-2">4</span> Alternative Synthesis</h2>
|
||||
<div class="outline-text-2" id="text-4">
|
||||
</div>
|
||||
<div id="outline-container-org1bb8ee7" class="outline-3">
|
||||
<h3 id="org1bb8ee7"><span class="section-number-3">4.1</span> Two generalized plants</h3>
|
||||
<div id="outline-container-orgc2f7629" class="outline-3">
|
||||
<h3 id="orgc2f7629"><span class="section-number-3">4.1</span> Two generalized plants</h3>
|
||||
<div class="outline-text-3" id="text-4-1">
|
||||
<p>
|
||||
In order to synthesize the complementary filter using the proposed method, we can use two alternative generalized plant as shown in Figures <a href="#orgdfb88a5">13</a> and <a href="#orgb41d84e">14</a>.
|
||||
In order to synthesize the complementary filter using the proposed method, we can use two alternative generalized plant as shown in Figures <a href="#orgdc5c349">13</a> and <a href="#orgf13eaf1">14</a>.
|
||||
</p>
|
||||
|
||||
\begin{equation}
|
||||
@ -811,7 +838,7 @@ In order to synthesize the complementary filter using the proposed method, we ca
|
||||
|
||||
|
||||
|
||||
<div id="orgdfb88a5" class="figure">
|
||||
<div id="orgdc5c349" class="figure">
|
||||
<p><img src="figs/h_infinity_arch_1.png" alt="h_infinity_arch_1.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 13: </span>Complementary Filter Synthesis - Conf 1</p>
|
||||
@ -822,7 +849,7 @@ In order to synthesize the complementary filter using the proposed method, we ca
|
||||
\end{equation}
|
||||
|
||||
|
||||
<div id="orgb41d84e" class="figure">
|
||||
<div id="orgf13eaf1" class="figure">
|
||||
<p><img src="figs/h_infinity_arch_2.png" alt="h_infinity_arch_2.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 14: </span>Complementary Filter Synthesis - Conf 2</p>
|
||||
@ -852,7 +879,7 @@ W2 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<pre class="example" id="org80f6d17">
|
||||
<pre class="example" id="org14b6e96">
|
||||
[H2, ~, gamma, ~] = hinfsyn(P1, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
|
||||
Test bounds: 0.3263 <= gamma <= 1000
|
||||
@ -888,7 +915,7 @@ W2 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<pre class="example" id="org7155dd4">
|
||||
<pre class="example" id="org594f397">
|
||||
[H2b, ~, gamma, ~] = hinfsyn(P2, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
|
||||
Test bounds: 0.3263 <= gamma <= 1000
|
||||
@ -914,10 +941,10 @@ W2 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((
|
||||
</pre>
|
||||
|
||||
<p>
|
||||
And indeed, we can see that the exact same filters are obtained (Figure <a href="#org4ff7339">15</a>).
|
||||
And indeed, we can see that the exact same filters are obtained (Figure <a href="#org806650d">15</a>).
|
||||
</p>
|
||||
|
||||
<div id="org4ff7339" class="figure">
|
||||
<div id="org806650d" class="figure">
|
||||
<p><img src="figs/hinf_comp_P1_P2_syn.png" alt="hinf_comp_P1_P2_syn.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 15: </span>Comparison of \(H_2(s)\) when using \(P_1(s)\) or \(P_2(s)\)</p>
|
||||
@ -925,8 +952,8 @@ And indeed, we can see that the exact same filters are obtained (Figure <a href=
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orga117463" class="outline-3">
|
||||
<h3 id="orga117463"><span class="section-number-3">4.2</span> Shaping the Low pass filter or the high pass filter?</h3>
|
||||
<div id="outline-container-org14ddd98" class="outline-3">
|
||||
<h3 id="org14ddd98"><span class="section-number-3">4.2</span> Shaping the Low pass filter or the high pass filter?</h3>
|
||||
<div class="outline-text-3" id="text-4-2">
|
||||
<p>
|
||||
Let’s see if there is a difference by explicitly shaping \(H_1(s)\) or \(H_2(s)\).
|
||||
@ -956,7 +983,7 @@ Let’s first synthesize \(H_1(s)\):
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<pre class="example" id="org45692a7">
|
||||
<pre class="example" id="org661bf48">
|
||||
[H1, ~, gamma, ~] = hinfsyn(P1, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
|
||||
Test bounds: 0.3263 <= gamma <= 1.712
|
||||
@ -995,7 +1022,7 @@ And now \(H_2(s)\):
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<pre class="example" id="org6543cab">
|
||||
<pre class="example" id="org5a20bfb">
|
||||
[H2b, ~, gamma, ~] = hinfsyn(P2, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
||||
|
||||
Test bounds: 0.3263 <= gamma <= 1000
|
||||
@ -1021,24 +1048,104 @@ And now \(H_2(s)\):
|
||||
</pre>
|
||||
|
||||
<p>
|
||||
And compare \(H_1(s)\) with \(1 - H_2(s)\) and \(H_2(s)\) with \(1 - H_1(s)\) in Figure <a href="#org4a9724c">16</a>.
|
||||
And compare \(H_1(s)\) with \(1 - H_2(s)\) and \(H_2(s)\) with \(1 - H_1(s)\) in Figure <a href="#orgb9c9d7c">16</a>.
|
||||
</p>
|
||||
|
||||
<div id="org4a9724c" class="figure">
|
||||
<div id="orgb9c9d7c" class="figure">
|
||||
<p><img src="figs/hinf_comp_H1_H2_syn.png" alt="hinf_comp_H1_H2_syn.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 16: </span>Comparison of \(H_1(s)\) with \(1-H_2(s)\), and \(H_2(s)\) with \(1-H_1(s)\)</p>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgc9d9779" class="outline-3">
|
||||
<h3 id="orgc9d9779"><span class="section-number-3">4.3</span> Using Feedback architecture</h3>
|
||||
<div class="outline-text-3" id="text-4-3">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">n = 2; w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>11; G0 = 1<span class="org-type">/</span>10; G1 = 1000; Gc = 1<span class="org-type">/</span>2;
|
||||
W1 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n))<span class="org-type">/</span>((1<span class="org-type">/</span>G1)<span class="org-type">^</span>(1<span class="org-type">/</span>n)<span class="org-type">*</span>(1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n)))<span class="org-type">^</span>n;
|
||||
|
||||
n = 3; w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10; G0 = 1000; G1 = 0.1; Gc = 1<span class="org-type">/</span>2;
|
||||
W2 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n))<span class="org-type">/</span>((1<span class="org-type">/</span>G1)<span class="org-type">^</span>(1<span class="org-type">/</span>n)<span class="org-type">*</span>(1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n)))<span class="org-type">^</span>n;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgf082879" class="outline-2">
|
||||
<h2 id="orgf082879"><span class="section-number-2">5</span> Impose a positive slope at DC or a negative slope at infinite frequency</h2>
|
||||
<p>
|
||||
Let’s first synthesize \(H_1(s)\):
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">P = [W1 <span class="org-type">-</span>W1;
|
||||
0 W2;
|
||||
1 <span class="org-type">-</span>1];
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">[K, <span class="org-type">~</span>, gamma, <span class="org-type">~</span>] = hinfsyn(P, 1, 1,<span class="org-string">'TOLGAM'</span>, 0.001, <span class="org-string">'METHOD'</span>, <span class="org-string">'lmi'</span>, <span class="org-string">'DISPLAY'</span>, <span class="org-string">'on'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">H1 = inv(1 <span class="org-type">+</span> K);
|
||||
H2 = 1 <span class="org-type">-</span> H1;
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org0aa4270" class="outline-3">
|
||||
<h3 id="org0aa4270"><span class="section-number-3">4.4</span> Adding feature in the filters</h3>
|
||||
<div class="outline-text-3" id="text-4-4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">n = 2; w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>11; G0 = 1<span class="org-type">/</span>10; G1 = 1000; Gc = 1<span class="org-type">/</span>2;
|
||||
W1 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n))<span class="org-type">/</span>((1<span class="org-type">/</span>G1)<span class="org-type">^</span>(1<span class="org-type">/</span>n)<span class="org-type">*</span>(1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n)))<span class="org-type">^</span>n;
|
||||
|
||||
n = 3; w0 = 2<span class="org-type">*</span><span class="org-constant">pi</span><span class="org-type">*</span>10; G0 = 1000; G1 = 0.1; Gc = 1<span class="org-type">/</span>2;
|
||||
W2 = (((1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n))<span class="org-type">/</span>((1<span class="org-type">/</span>G1)<span class="org-type">^</span>(1<span class="org-type">/</span>n)<span class="org-type">*</span>(1<span class="org-type">/</span>w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(G0<span class="org-type">/</span>Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>n))<span class="org-type">/</span>(1<span class="org-type">-</span>(Gc<span class="org-type">/</span>G1)<span class="org-type">^</span>(2<span class="org-type">/</span>n)))<span class="org-type">*</span>s <span class="org-type">+</span> (1<span class="org-type">/</span>Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>n)))<span class="org-type">^</span>n;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">Wf = (1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>1)<span class="org-type">/</span>s;
|
||||
Wf = s<span class="org-type">/</span>(1 <span class="org-type">+</span> s<span class="org-type">/</span>2<span class="org-type">/</span><span class="org-constant">pi</span><span class="org-type">/</span>1e2);
|
||||
|
||||
<span class="org-comment">% </span><span class="org-comment"><span class="org-constant">W2 </span></span><span class="org-comment">= W2/Wf/(1 + s/2/pi/1e3);</span>
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">P = [W1 <span class="org-type">-</span>Wf<span class="org-type">*</span>W1;
|
||||
0 Wf<span class="org-type">*</span>W2;
|
||||
1 <span class="org-type">-</span>Wf];
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">[Ka, <span class="org-type">~</span>, gamma, <span class="org-type">~</span>] = hinfsyn(P, 1, 1,<span class="org-string">'TOLGAM'</span>, 0.001, <span class="org-string">'METHOD'</span>, <span class="org-string">'lmi'</span>, <span class="org-string">'DISPLAY'</span>, <span class="org-string">'on'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">K = Ka<span class="org-type">*</span>Wf;
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">H1 = inv(1 <span class="org-type">+</span> K);
|
||||
H2 = 1 <span class="org-type">-</span> H1;
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgb026d30" class="outline-2">
|
||||
<h2 id="orgb026d30"><span class="section-number-2">5</span> Impose a positive slope at DC or a negative slope at infinite frequency</h2>
|
||||
<div class="outline-text-2" id="text-5">
|
||||
</div>
|
||||
<div id="outline-container-org96df1d1" class="outline-3">
|
||||
<h3 id="org96df1d1"><span class="section-number-3">5.1</span> Manually shift zeros to the origin after synthesis</h3>
|
||||
<div id="outline-container-org4d4e2ad" class="outline-3">
|
||||
<h3 id="org4d4e2ad"><span class="section-number-3">5.1</span> Manually shift zeros to the origin after synthesis</h3>
|
||||
<div class="outline-text-3" id="text-5-1">
|
||||
<p>
|
||||
Suppose we want \(H_2(s)\) to be an high pass filter with a slope of +2 at low frequency (from 0Hz).
|
||||
@ -1084,7 +1191,7 @@ And we do the \(\mathcal{H}_\infty\) synthesis using the <code>hinfsyn</code> co
|
||||
<p>
|
||||
Looking at the zeros, we see two low frequency complex conjugate zeros.
|
||||
</p>
|
||||
<pre class="example" id="orge6a4f22">
|
||||
<pre class="example" id="org45e17b6">
|
||||
z{1}
|
||||
ans =
|
||||
-4690930.24283199 + 0i
|
||||
@ -1119,12 +1226,12 @@ And as usual, \(H_{1z}(s)\) is defined as the complementary of \(H_{2z}(s)\):
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The bode plots of \(H_1(s)\), \(H_2(s)\), \(H_{1z}(s)\) and \(H_{2z}(s)\) are shown in Figure <a href="#org4631977">17</a>.
|
||||
The bode plots of \(H_1(s)\), \(H_2(s)\), \(H_{1z}(s)\) and \(H_{2z}(s)\) are shown in Figure <a href="#orga18e536">17</a>.
|
||||
And we see that \(H_{1z}(s)\) is slightly modified when setting the zeros at the origin for \(H_{2z}(s)\).
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org4631977" class="figure">
|
||||
<div id="orga18e536" class="figure">
|
||||
<p><img src="figs/comp_filters_shift_zero.png" alt="comp_filters_shift_zero.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 17: </span>Bode plots of \(H_1(s)\), \(H_2(s)\), \(H_{1z}(s)\) and \(H_{2z}(s)\)</p>
|
||||
@ -1132,16 +1239,16 @@ And we see that \(H_{1z}(s)\) is slightly modified when setting the zeros at the
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org6b92ce0" class="outline-3">
|
||||
<h3 id="org6b92ce0"><span class="section-number-3">5.2</span> Imposing a positive slope at DC during the synthesis phase</h3>
|
||||
<div id="outline-container-orgf1693db" class="outline-3">
|
||||
<h3 id="orgf1693db"><span class="section-number-3">5.2</span> Imposing a positive slope at DC during the synthesis phase</h3>
|
||||
<div class="outline-text-3" id="text-5-2">
|
||||
<p>
|
||||
Suppose we want to synthesize \(H_2(s)\) such that it has a slope of +2 from DC.
|
||||
We can include this “feature” in the generalized plant as shown in Figure <a href="#org00fe83c">18</a>.
|
||||
We can include this “feature” in the generalized plant as shown in Figure <a href="#orgf2b673c">18</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org00fe83c" class="figure">
|
||||
<div id="orgf2b673c" class="figure">
|
||||
<p><img src="figs/h_infinity_arch_H2_feature.png" alt="h_infinity_arch_H2_feature.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 18: </span>Generalized plant with included wanted feature represented by \(H_{2w}(s)\)</p>
|
||||
@ -1184,7 +1291,7 @@ H2w = (s<span class="org-type">/</span>w0<span class="org-type">/</span>(s<span
|
||||
</div>
|
||||
|
||||
<p>
|
||||
We define the generalized plant as shown in Figure <a href="#org00fe83c">18</a>.
|
||||
We define the generalized plant as shown in Figure <a href="#orgf2b673c">18</a>.
|
||||
</p>
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">P = [W1 <span class="org-type">-</span>W1;
|
||||
@ -1218,11 +1325,11 @@ And we define \(H_1(s)\) to be the complementary of \(H_2(s)\):
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The obtained complementary filters are shown in Figure <a href="#org551ae15">19</a>.
|
||||
The obtained complementary filters are shown in Figure <a href="#orgf11ca9c">19</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org551ae15" class="figure">
|
||||
<div id="orgf11ca9c" class="figure">
|
||||
<p><img src="figs/comp_filters_H2_feature.png" alt="comp_filters_H2_feature.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 19: </span>Obtained complementary fitlers</p>
|
||||
@ -1230,15 +1337,15 @@ The obtained complementary filters are shown in Figure <a href="#org551ae15">19<
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org71e3235" class="outline-3">
|
||||
<h3 id="org71e3235"><span class="section-number-3">5.3</span> Imposing a negative slope at infinity frequency during the synthesis phase</h3>
|
||||
<div id="outline-container-org52c02ef" class="outline-3">
|
||||
<h3 id="org52c02ef"><span class="section-number-3">5.3</span> Imposing a negative slope at infinity frequency during the synthesis phase</h3>
|
||||
<div class="outline-text-3" id="text-5-3">
|
||||
<p>
|
||||
Let’s suppose we now want to shape a low pass filter that as a negative slope until infinite frequency.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
The used technique is the same as in the previous section, and the generalized plant is shown in Figure <a href="#org00fe83c">18</a>.
|
||||
The used technique is the same as in the previous section, and the generalized plant is shown in Figure <a href="#orgf2b673c">18</a>.
|
||||
</p>
|
||||
|
||||
<p>
|
||||
@ -1296,11 +1403,11 @@ And \(H_1(s)\) is defined as follows:
|
||||
</div>
|
||||
|
||||
<p>
|
||||
The obtained complementary filters are shown in Figure <a href="#org7f33e5d">20</a>.
|
||||
The obtained complementary filters are shown in Figure <a href="#orgab78578">20</a>.
|
||||
</p>
|
||||
|
||||
|
||||
<div id="org7f33e5d" class="figure">
|
||||
<div id="orgab78578" class="figure">
|
||||
<p><img src="figs/comp_filters_H2_feature_neg_slope.png" alt="comp_filters_H2_feature_neg_slope.png" />
|
||||
</p>
|
||||
<p><span class="figure-number">Figure 20: </span>Obtained complementary fitlers</p>
|
||||
@ -1318,10 +1425,179 @@ The obtained complementary filters are shown in Figure <a href="#org7f33e5d">20<
|
||||
<div class="csl-bib-body">
|
||||
<div class="csl-entry"><a name="citeproc_bib_item_1"></a>Hua, Wensheng. 2005. “Low Frequency Vibration Isolation and Alignment System for Advanced LIGO.” stanford university.</div>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org29e79e7" class="outline-2">
|
||||
<h2 id="org29e79e7"><span class="section-number-2">6</span> Functions</h2>
|
||||
<div class="outline-text-2" id="text-6">
|
||||
</div>
|
||||
<div id="outline-container-orgfbca8a7" class="outline-3">
|
||||
<h3 id="orgfbca8a7"><span class="section-number-3">6.1</span> <code>generateWF</code>: Generate Weighting Functions</h3>
|
||||
<div class="outline-text-3" id="text-6-1">
|
||||
<p>
|
||||
<a id="org0d68f63"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
This Matlab function is accessible <a href="matlab/src/generateWF.m">here</a>.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org95965a4" class="outline-4">
|
||||
<h4 id="org95965a4">Function description</h4>
|
||||
<div class="outline-text-4" id="text-org95965a4">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[W]</span> = <span class="org-function-name">generateWF</span>(<span class="org-variable-name">args</span>)
|
||||
<span class="org-comment">% createWeight -</span>
|
||||
<span class="org-comment">%</span>
|
||||
<span class="org-comment">% Syntax: [W] = generateWeight(args)</span>
|
||||
<span class="org-comment">%</span>
|
||||
<span class="org-comment">% Inputs:</span>
|
||||
<span class="org-comment">% - n - Weight Order (integer)</span>
|
||||
<span class="org-comment">% - G0 - Low frequency Gain</span>
|
||||
<span class="org-comment">% - G1 - High frequency Gain</span>
|
||||
<span class="org-comment">% - Gc - Gain of the weight at frequency w0</span>
|
||||
<span class="org-comment">% - w0 - Frequency at which |W(j w0)| = Gc [rad/s]</span>
|
||||
<span class="org-comment">%</span>
|
||||
<span class="org-comment">% Outputs:</span>
|
||||
<span class="org-comment">% - W - Generated Weight</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgaeb61de" class="outline-4">
|
||||
<h4 id="orgaeb61de">Optional Parameters</h4>
|
||||
<div class="outline-text-4" id="text-orgaeb61de">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-keyword">arguments</span>
|
||||
<span class="org-variable-name">args</span>.n (1,1) double {mustBeInteger, mustBePositive} = 1
|
||||
<span class="org-variable-name">args</span>.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
|
||||
<span class="org-variable-name">args</span>.Ginf (1,1) double {mustBeNumeric, mustBePositive} = 10
|
||||
<span class="org-variable-name">args</span>.Gc (1,1) double {mustBeNumeric, mustBePositive} = 1
|
||||
<span class="org-variable-name">args</span>.w0 (1,1) double {mustBeNumeric, mustBePositive} = 1
|
||||
<span class="org-keyword">end</span>
|
||||
|
||||
mustBeBetween(args.G0, args.Gc, args.Ginf);
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgd91e493" class="outline-4">
|
||||
<h4 id="orgd91e493">Generate the Weighting function</h4>
|
||||
<div class="outline-text-4" id="text-orgd91e493">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">s = zpk(<span class="org-string">'s'</span>);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">W = (((1<span class="org-type">/</span>args.w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(args.G0<span class="org-type">/</span>args.Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>args.n))<span class="org-type">/</span>(1<span class="org-type">-</span>(args.Gc<span class="org-type">/</span>args.Ginf)<span class="org-type">^</span>(2<span class="org-type">/</span>args.n)))<span class="org-type">*</span>s <span class="org-type">+</span> ...
|
||||
(args.G0<span class="org-type">/</span>args.Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>args.n))<span class="org-type">/</span>...
|
||||
((1<span class="org-type">/</span>args.Ginf)<span class="org-type">^</span>(1<span class="org-type">/</span>args.n)<span class="org-type">*</span>(1<span class="org-type">/</span>args.w0)<span class="org-type">*</span>sqrt((1<span class="org-type">-</span>(args.G0<span class="org-type">/</span>args.Gc)<span class="org-type">^</span>(2<span class="org-type">/</span>args.n))<span class="org-type">/</span>(1<span class="org-type">-</span>(args.Gc<span class="org-type">/</span>args.Ginf)<span class="org-type">^</span>(2<span class="org-type">/</span>args.n)))<span class="org-type">*</span>s <span class="org-type">+</span> ...
|
||||
(1<span class="org-type">/</span>args.Gc)<span class="org-type">^</span>(1<span class="org-type">/</span>args.n)))<span class="org-type">^</span>args.n;
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
|
||||
<div id="outline-container-org84eb536" class="outline-4">
|
||||
<h4 id="org84eb536">Verification of the \(G_0\), \(G_c\) and \(G_\infty\) gains</h4>
|
||||
<div class="outline-text-4" id="text-org84eb536">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-comment">% Custom validation function</span>
|
||||
<span class="org-keyword">function</span> <span class="org-function-name">mustBeBetween</span>(<span class="org-variable-name">a</span>,<span class="org-variable-name">b</span>,<span class="org-variable-name">c</span>)
|
||||
<span class="org-keyword">if</span> <span class="org-type">~</span>((a <span class="org-type">></span> b <span class="org-type">&&</span> b <span class="org-type">></span> c) <span class="org-type">||</span> (c <span class="org-type">></span> b <span class="org-type">&&</span> b <span class="org-type">></span> a))
|
||||
eid = <span class="org-string">'createWeight:inputError'</span>;
|
||||
msg = <span class="org-string">'Gc should be between G0 and Ginf.'</span>;
|
||||
throwAsCaller(MException(eid,msg))
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2f9ac50" class="outline-3">
|
||||
<h3 id="org2f9ac50"><span class="section-number-3">6.2</span> <code>generateCF</code>: Generate Complementary Filters</h3>
|
||||
<div class="outline-text-3" id="text-6-2">
|
||||
<p>
|
||||
<a id="orgf94493e"></a>
|
||||
</p>
|
||||
|
||||
<p>
|
||||
This Matlab function is accessible <a href="matlab/src/generateCF.m">here</a>.
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org7c373f8" class="outline-4">
|
||||
<h4 id="org7c373f8">Function description</h4>
|
||||
<div class="outline-text-4" id="text-org7c373f8">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-keyword">function</span> <span class="org-variable-name">[H1, H2]</span> = <span class="org-function-name">generateCF</span>(<span class="org-variable-name">W1</span>, <span class="org-variable-name">W2</span>, <span class="org-variable-name">args</span>)
|
||||
<span class="org-comment">% createWeight -</span>
|
||||
<span class="org-comment">%</span>
|
||||
<span class="org-comment">% Syntax: [W] = generateCF(args)</span>
|
||||
<span class="org-comment">%</span>
|
||||
<span class="org-comment">% Inputs:</span>
|
||||
<span class="org-comment">% - W1 - Weighting Function for H1</span>
|
||||
<span class="org-comment">% - W2 - Weighting Function for H2</span>
|
||||
<span class="org-comment">% - args:</span>
|
||||
<span class="org-comment">% - method - H-Infinity solver ('lmi' or 'ric')</span>
|
||||
<span class="org-comment">% - display - Display synthesis results ('on' or 'off')</span>
|
||||
<span class="org-comment">%</span>
|
||||
<span class="org-comment">% Outputs:</span>
|
||||
<span class="org-comment">% - H1 - Generated H1 Filter</span>
|
||||
<span class="org-comment">% - H2 - Generated H2 Filter</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org00c9d5d" class="outline-4">
|
||||
<h4 id="org00c9d5d">Optional Parameters</h4>
|
||||
<div class="outline-text-4" id="text-org00c9d5d">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab"><span class="org-keyword">arguments</span>
|
||||
<span class="org-variable-name">W1</span>
|
||||
<span class="org-variable-name">W2</span>
|
||||
<span class="org-variable-name">args</span>.method char {mustBeMember(args.method,{<span class="org-string">'lmi'</span>, <span class="org-string">'ric'</span>})} = <span class="org-string">'ric'</span>
|
||||
<span class="org-variable-name">args</span>.display char {mustBeMember(args.display,{<span class="org-string">'on'</span>, <span class="org-string">'off'</span>})} = <span class="org-string">'on'</span>
|
||||
<span class="org-keyword">end</span>
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org92a7c04" class="outline-4">
|
||||
<h4 id="org92a7c04">H-Infinity Synthesis</h4>
|
||||
<div class="outline-text-4" id="text-org92a7c04">
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">P = [W1 <span class="org-type">-</span>W1;
|
||||
0 W2;
|
||||
1 0];
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">[H2, <span class="org-type">~</span>, gamma, <span class="org-type">~</span>] = hinfsyn(P, 1, 1,<span class="org-string">'TOLGAM'</span>, 0.001, <span class="org-string">'METHOD'</span>, args.method, <span class="org-string">'DISPLAY'</span>, args.display);
|
||||
</pre>
|
||||
</div>
|
||||
|
||||
<div class="org-src-container">
|
||||
<pre class="src src-matlab">H1 = 1 <span class="org-type">-</span> H2;
|
||||
</pre>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Thomas Dehaeze</p>
|
||||
<p class="date">Created: 2020-12-11 ven. 14:05</p>
|
||||
<p class="date">Created: 2021-04-29 jeu. 17:25</p>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
@ -43,7 +43,7 @@ One use of complementary filter is described below:
|
||||
#+end_quote
|
||||
|
||||
This document is divided into several sections:
|
||||
- in section [[sec:h_inf_synthesis_complementary_filters]], the $\mathcal{H}_\infty$ synthesis is used for generating two complementary filters
|
||||
- in section [[#sec:h_inf_synthesis_complementary_filters]], the $\mathcal{H}_\infty$ synthesis is used for generating two complementary filters
|
||||
- in section [[sec:three_comp_filters]], a method using the $\mathcal{H}_\infty$ synthesis is proposed to shape three of more complementary filters
|
||||
- in section [[sec:comp_filters_ligo]], the $\mathcal{H}_\infty$ synthesis is used and compared with FIR complementary filters used for LIGO
|
||||
|
||||
@ -55,8 +55,8 @@ This document is divided into several sections:
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle matlab/h_inf_synthesis_complementary_filters.m
|
||||
:header-args:matlab+: :comments org :mkdirp yes
|
||||
:CUSTOM_ID: sec:h_inf_synthesis_complementary_filters
|
||||
:END:
|
||||
<<sec:h_inf_synthesis_complementary_filters>>
|
||||
|
||||
** Introduction :ignore:
|
||||
#+begin_note
|
||||
@ -93,7 +93,7 @@ In order to do so, we use the generalized plant shown on figure [[fig:h_infinity
|
||||
|
||||
#+name: fig:h_infinity_robst_fusion
|
||||
#+caption: $\mathcal{H}_\infty$ synthesis of the complementary filters
|
||||
[[file:figs-tikz/h_infinity_robust_fusion.png]]
|
||||
[[file:figs/h_infinity_robust_fusion.png]]
|
||||
|
||||
The $\mathcal{H}_\infty$ synthesis applied on this generalized plant will give a transfer function $H_2$ (figure [[fig:h_infinity_robst_fusion]]) such that the $\mathcal{H}_\infty$ norm of the transfer function from $w$ to $[z_1,\ z_2]$ is less than one:
|
||||
\[ \left\| \begin{array}{c} (1 - H_2(s)) W_1(s) \\ H_2(s) W_2(s) \end{array} \right\|_\infty < 1 \]
|
||||
@ -352,7 +352,7 @@ For that, we use the $\mathcal{H}_\infty$ synthesis with the architecture shown
|
||||
|
||||
#+name: fig:comp_filter_three_hinf
|
||||
#+caption: Generalized architecture for generating 3 complementary filters
|
||||
[[file:figs-tikz/comp_filter_three_hinf.png]]
|
||||
[[file:figs/comp_filter_three_hinf.png]]
|
||||
|
||||
The $\mathcal{H}_\infty$ objective is:
|
||||
\begin{align*}
|
||||
@ -585,7 +585,7 @@ exportFig('figs/ligo_specifications.pdf', 'width', 'wide', 'height', 'normal');
|
||||
#+RESULTS:
|
||||
[[file:figs/ligo_specifications.png]]
|
||||
|
||||
** TODO FIR Filter
|
||||
** FIR Filter
|
||||
We here try to implement the FIR complementary filter synthesis as explained in cite:hua05_low_ligo.
|
||||
For that, we use the [[http://cvxr.com/cvx/][CVX matlab Toolbox]].
|
||||
|
||||
@ -937,7 +937,7 @@ exportFig('figs/hinf_synthesis_ligo_results.pdf', 'width', 'wide', 'height', 'no
|
||||
#+RESULTS:
|
||||
[[file:figs/hinf_synthesis_ligo_results.png]]
|
||||
|
||||
** TODO Compare FIR and H-Infinity Filters
|
||||
** Compare FIR and H-Infinity Filters
|
||||
Let's now compare the FIR filters designed in cite:hua05_low_ligo and the one obtained with the $\mathcal{H}_\infty$ synthesis on figure [[fig:comp_fir_ligo_hinf]].
|
||||
|
||||
#+begin_src matlab :exports none
|
||||
@ -965,25 +965,25 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
||||
ylabel('Magnitude');
|
||||
set(gca, 'XTickLabel',[]);
|
||||
ylim([5e-3, 10]);
|
||||
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
|
||||
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
||||
leg.ItemTokenSize(1) = 16;
|
||||
|
||||
% Phase
|
||||
ax2 = nexttile;
|
||||
hold on;
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Hh, freqs, 'Hz'))), '-');
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hh, freqs, 'Hz')))), '-');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(freqs, 180/pi*angle(squeeze(freqresp(Hl, freqs, 'Hz'))), '-');
|
||||
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(Hl, freqs, 'Hz')))), '-');
|
||||
|
||||
set(gca,'ColorOrderIndex',1);
|
||||
plot(w, 180/pi*angle(H), '--');
|
||||
set(gca,' H2ColorOrderIndex',2);
|
||||
plot(w, 180/pi*angle(1-H), '--');
|
||||
plot(w, 180/pi*unwrap(angle(H)), '--');
|
||||
set(gca,'ColorOrderIndex',2);
|
||||
plot(w, 180/pi*unwrap(angle(1-H)), '--');
|
||||
set(gca, 'XScale', 'log');
|
||||
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
||||
hold off;
|
||||
yticks([-180:90:180]); ylim([-180, 180]);
|
||||
yticks([-450:90:180]); ylim([-450, 200]);
|
||||
|
||||
linkaxes([ax1,ax2],'x');
|
||||
xlim([freqs(1), freqs(end)]);
|
||||
@ -1897,7 +1897,9 @@ function [H1, H2] = generateCF(W1, W2, args)
|
||||
% Inputs:
|
||||
% - W1 - Weighting Function for H1
|
||||
% - W2 - Weighting Function for H2
|
||||
% - args -
|
||||
% - args:
|
||||
% - method - H-Infinity solver ('lmi' or 'ric')
|
||||
% - display - Display synthesis results ('on' or 'off')
|
||||
%
|
||||
% Outputs:
|
||||
% - H1 - Generated H1 Filter
|
||||
|
||||
@ -6,7 +6,9 @@ function [H1, H2] = generateCF(W1, W2, args)
|
||||
% Inputs:
|
||||
% - W1 - Weighting Function for H1
|
||||
% - W2 - Weighting Function for H2
|
||||
% - args -
|
||||
% - args:
|
||||
% - method - H-Infinity solver ('lmi' or 'ric')
|
||||
% - display - Display synthesis results ('on' or 'off')
|
||||
%
|
||||
% Outputs:
|
||||
% - H1 - Generated H1 Filter
|
||||
|
||||
480
tikz/index.org
480
tikz/index.org
@ -6,56 +6,184 @@
|
||||
#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
|
||||
#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
|
||||
|
||||
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/tikz/}{config.tex}")
|
||||
#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/papers/dehaeze21_desig_compl_filte/tikz/}{config.tex}")
|
||||
#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
|
||||
#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
|
||||
#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
|
||||
#+PROPERTY: header-args:latex+ :results raw replace :buffer no
|
||||
#+PROPERTY: header-args:latex+ :results file raw replace
|
||||
#+PROPERTY: header-args:latex+ :buffer no
|
||||
#+PROPERTY: header-args:latex+ :eval no-export
|
||||
#+PROPERTY: header-args:latex+ :exports both
|
||||
#+PROPERTY: header-args:latex+ :mkdirp yes
|
||||
#+PROPERTY: header-args:latex+ :noweb yes
|
||||
#+PROPERTY: header-args:latex+ :output-dir figs
|
||||
#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
|
||||
:END:
|
||||
|
||||
Configuration file is accessible [[file:config.org][here]].
|
||||
|
||||
* Fig 1: Sensor Fusion Architecture
|
||||
* Sensor Model
|
||||
#+begin_src latex :file sensor_model.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[addb](add1){};
|
||||
\node[block, right=0.8 of add1](G1){$G_1(s)$};
|
||||
|
||||
\draw[->] ($(add1.west)+(-0.7, 0)$) node[above right]{$x$} -- (add1.west);
|
||||
\draw[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){$n_1$};
|
||||
\draw[->] (add1.east) -- (G1.west);
|
||||
\draw[->] (G1.east) -- ++(0.7, 0) node[above left]{$\tilde{x}_1$};
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=black!20!white, draw, inner sep=3pt] (sensor1) {};
|
||||
\node[below left] at (sensor1.north east) {Sensor 1};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_model
|
||||
#+caption: Basic Sensor Model
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_model.png]]
|
||||
|
||||
* Sensor Model with calibration
|
||||
#+begin_src latex :file sensor_model_calibrated.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[addb](add1){};
|
||||
\node[block, right=0.8 of add1](G1){$G_1(s)$};
|
||||
\node[block, right=0.8 of G1](G1inv){$\hat{G}_1^{-1}(s)$};
|
||||
|
||||
\draw[->] ($(add1.west)+(-0.7, 0)$) node[above right]{$x$} -- (add1.west);
|
||||
\draw[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){$n_1$};
|
||||
\draw[->] (add1.east) -- (G1.west);
|
||||
\draw[->] (G1.east) -- (G1inv.west) node[above left]{$\tilde{x}_1$};
|
||||
\draw[->] (G1inv.east) -- ++(0.8, 0) node[above left]{$\hat{x}_1$};
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(add1.west |- G1inv.south) (n1.north -| G1inv.east)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
|
||||
\node[below left] at (sensor1cal.north east) {Calibration};
|
||||
|
||||
\node[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=black!20!white, draw, inner sep=3pt] (sensor1) {};
|
||||
\node[below left] at (sensor1.north east) {Sensor 1};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_model_calibrated
|
||||
#+caption: Calibrated Sensor
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_model_calibrated.png]]
|
||||
|
||||
* Sensor Model with Uncertainty
|
||||
#+begin_src latex :file sensor_model_uncertainty.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[branch] (input) at (0,0) {};
|
||||
\node[block, above right= 0.4 and 0.4 of input](W1){$w_1(s)$};
|
||||
\node[block, right=0.4 of W1](delta1){$\Delta_1(s)$};
|
||||
\node[addb] (addu) at ($(delta1.east|-input) + (0.4, 0)$) {};
|
||||
\node[addb, right=0.4 of addu] (addn) {};
|
||||
\node[block, right=0.4 of addn] (G1) {$\hat{G}_1(s)$};
|
||||
\node[block, right=0.8 of G1](G1inv){$\hat{G}_1^{-1}(s)$};
|
||||
|
||||
\draw[->] ($(input)+(-0.7, 0)$) node[above right]{$x$} -- (addu);
|
||||
\draw[->] (input.center) |- (W1.west);
|
||||
\draw[->] (W1.east) -- (delta1.west);
|
||||
\draw[->] (delta1.east) -| (addu.north);
|
||||
\draw[->] (addu.east) -- (addn.west);
|
||||
\draw[->] (addn.east) -- (G1.west);
|
||||
\draw[<-] (addn.north) -- ++(0, 0.7)node[below right](n1){$n_1$};
|
||||
\draw[->] (G1.east) -- (G1inv.west) node[above left]{$\tilde{x}_1$};
|
||||
\draw[->] (G1inv.east) -- ++(0.8, 0) node[above left]{$\hat{x}_1$};
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(input.west |- G1inv.south) (delta1.north -| G1inv.east)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
|
||||
\node[below left] at (sensor1cal.north east) {Calibration};
|
||||
|
||||
\node[fit={(input.west |- G1.south) (delta1.north -| G1.east)}, fill=black!20!white, draw, inner sep=3pt] (sensor1) {};
|
||||
\node[below left] at (sensor1.north east) {Sensor 1};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_model_uncertainty
|
||||
#+caption: Input Uncertainty
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_model_uncertainty.png]]
|
||||
|
||||
* Sensor Model with Uncertainty - Simplified
|
||||
#+begin_src latex :file sensor_model_uncertainty_simplified.pdf
|
||||
\begin{tikzpicture}
|
||||
\node[branch] (input) at (0,0) {};
|
||||
\node[block, above right= 0.4 and 0.4 of input](W1){$w_1(s)$};
|
||||
\node[block, right=0.4 of W1](delta1){$\Delta_1(s)$};
|
||||
\node[addb] (addu) at ($(delta1.east|-input) + (0.4, 0)$) {};
|
||||
\node[addb, right=0.4 of addu] (addn) {};
|
||||
|
||||
\draw[->] ($(input)+(-0.7, 0)$) node[above right]{$x$} -- (addu);
|
||||
\draw[->] (input.center) |- (W1.west);
|
||||
\draw[->] (W1.east) -- (delta1.west);
|
||||
\draw[->] (delta1.east) -| (addu.north);
|
||||
\draw[->] (addu.east) -- (addn.west);
|
||||
\draw[<-] (addn.north) -- ++(0, 0.7)node[below right](n1){$n_1$};
|
||||
\draw[->] (addn.east) -- ++(0.9, 0) node[above left]{$\hat{x}_1$};
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(input.west |- addu.south) ($(delta1.north -| addn.east) + (0.1, 0.3)$)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
|
||||
\node[below left] at (sensor1cal.north east) {Calibrated Sensor};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_model_uncertainty_simplified
|
||||
#+caption: Input Uncertainty
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_model_uncertainty_simplified.png]]
|
||||
|
||||
* Sensor Fusion Architecture
|
||||
#+begin_src latex :file fusion_super_sensor.pdf :tangle figs/fusion_super_sensor.tex
|
||||
\begin{tikzpicture}
|
||||
\node[branch] (x) at (0, 0);
|
||||
\node[block, above right=0.5 and 0.5 of x](G1){$G_1(s)$};
|
||||
\node[block, below right=0.5 and 0.5 of x](G2){$G_2(s)$};
|
||||
\node[addb, right=0.8 of G1](add1){};
|
||||
\node[addb, right=0.8 of G2](add2){};
|
||||
\node[block, right=0.8 of add1](H1){$H_1(s)$};
|
||||
\node[block, right=0.8 of add2](H2){$H_2(s)$};
|
||||
\node[addb, right=5 of x](add){};
|
||||
\definecolor{myblue}{rgb}{0, 0.447, 0.741}
|
||||
\definecolor{myred}{rgb}{0.8500, 0.325, 0.098}
|
||||
|
||||
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
|
||||
\draw[->] (x.center) |- (G1.west);
|
||||
\draw[->] (x.center) |- (G2.west);
|
||||
\draw[->] (G1.east) -- (add1.west);
|
||||
\draw[->] (G2.east) -- (add2.west);
|
||||
\draw[<-] (add1.north) -- ++(0, 0.8)node[below right](n1){$n_1$};
|
||||
\draw[<-] (add2.north) -- ++(0, 0.8)node[below right](n2){$n_2$};
|
||||
\draw[->] (add1.east) -- (H1.west);
|
||||
\draw[->] (add2.east) -- (H2.west);
|
||||
\draw[->] (H1) -| (add.north);
|
||||
\draw[->] (H2) -| (add.south);
|
||||
\draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
|
||||
\begin{tikzpicture}
|
||||
\node[branch] (x) at (0, 0);
|
||||
\node[addb, above right=0.8 and 0.5 of x](add1){};
|
||||
\node[addb, below right=0.8 and 0.5 of x](add2){};
|
||||
\node[block, right=0.8 of add1](G1){$G_1(s)$};
|
||||
\node[block, right=0.8 of add2](G2){$G_2(s)$};
|
||||
\node[block, right=0.8 of G1](G1inv){$\hat{G}_1^{-1}(s)$};
|
||||
\node[block, right=0.8 of G2](G2inv){$\hat{G}_2^{-2}(s)$};
|
||||
\node[block, right=0.8 of G1inv](H1){$H_1(s)$};
|
||||
\node[block, right=0.8 of G2inv](H2){$H_2(s)$};
|
||||
\node[addb, right=7 of x](add){};
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={($(G2.south-|x)+(-0.2, -0.3)$) ($(n1.north east-|add.east)+(0.2, 0.3)$)}, fill=black!10!white, draw, dashed, inner sep=0pt] (supersensor) {};
|
||||
\node[below left] at (supersensor.north east) {Super Sensor};
|
||||
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
|
||||
\draw[->] (x.center) |- (add1.west);
|
||||
\draw[->] (x.center) |- (add2.west);
|
||||
\draw[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){$n_1$};
|
||||
\draw[->] (add1.east) -- (G1.west);
|
||||
\draw[->] (G1.east) -- (G1inv.west) node[above left]{$\tilde{x}_1$};
|
||||
\draw[->] (G1inv.east) -- (H1.west) node[above left]{$\hat{x}_1$};
|
||||
\draw[<-] (add2.north) -- ++(0, 0.7)node[below right](n2){$n_2$};
|
||||
\draw[->] (add2.east) -- (G2.west);
|
||||
\draw[->] (G2.east) -- (G2inv.west) node[above left]{$\tilde{x}_2$};
|
||||
\draw[->] (G2inv.east) -- (H2.west) node[above left]{$\hat{x}_2$};
|
||||
\draw[->] (H1) -| (add.north);
|
||||
\draw[->] (H2) -| (add.south);
|
||||
\draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
|
||||
|
||||
\node[fit={($(G1.south west)+(-0.3, -0.1)$) ($(n1.north east)+(0.0, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor1) {};
|
||||
\node[below right] at (sensor1.north west) {Sensor 1};
|
||||
\node[fit={($(G2.south west)+(-0.3, -0.1)$) ($(n2.north east)+(0.0, 0.1)$)}, fill=black!20!white, draw, dashed, inner sep=0pt] (sensor2) {};
|
||||
\node[below right] at (sensor2.north west) {Sensor 2};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(G2.south-|x) (n1.north-|add.east)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
|
||||
\node[below left] at (supersensor.north east) {Super Sensor};
|
||||
|
||||
\node[fit={(add1.west |- G1inv.south) (n1.north -| G1inv.east)}, fill=myblue!20!white, draw, inner sep=6pt] (sensor1cal) {};
|
||||
\node[below left] at (sensor1cal.north east) {Calibration};
|
||||
\node[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=myblue!30!white, draw, inner sep=3pt] (sensor1) {};
|
||||
\node[below left] at (sensor1.north east) {Sensor 1};
|
||||
|
||||
\node[fit={(add2.west |- G2inv.south) (n2.north -| G2inv.east)}, fill=myred!20!white, draw, inner sep=6pt] (sensor2cal) {};
|
||||
\node[below left] at (sensor2cal.north east) {Calibration};
|
||||
\node[fit={(add2.west |- G2.south) (n2.north -| G2.east)}, fill=myred!30!white, draw, inner sep=3pt] (sensor2) {};
|
||||
\node[below left] at (sensor2.north east) {Sensor 2};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:fusion_super_sensor
|
||||
@ -63,42 +191,61 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+RESULTS:
|
||||
[[file:figs/fusion_super_sensor.png]]
|
||||
|
||||
* Fig 2: Sensor fusion architecture with sensor dynamics uncertainty
|
||||
#+begin_src latex :file sensor_fusion_dynamic_uncertainty.pdf :tangle figs/sensor_fusion_dynamic_uncertainty.tex
|
||||
\begin{tikzpicture}
|
||||
\node[branch] (x) at (0, 0);
|
||||
\node[addb, above right=0.8 and 4 of x](add1){};
|
||||
\node[addb, below right=0.8 and 4 of x](add2){};
|
||||
\node[block, above left=0.2 and 0.1 of add1](delta1){$\Delta_1(s)$};
|
||||
\node[block, above left=0.2 and 0.1 of add2](delta2){$\Delta_2(s)$};
|
||||
\node[block, left=0.5 of delta1](W1){$w_1(s)$};
|
||||
\node[block, left=0.5 of delta2](W2){$w_2(s)$};
|
||||
\node[block, right=0.5 of add1](H1){$H_1(s)$};
|
||||
\node[block, right=0.5 of add2](H2){$H_2(s)$};
|
||||
\node[addb, right=6 of x](add){};
|
||||
* Sensor fusion architecture with sensor dynamics uncertainty
|
||||
#+begin_src latex :file sensor_fusion_dynamic_uncertainty.pdf :tangle figs/fusion_super_sensor.tex
|
||||
\definecolor{myblue}{rgb}{0, 0.447, 0.741}
|
||||
\definecolor{myred}{rgb}{0.8500, 0.325, 0.098}
|
||||
|
||||
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
|
||||
\draw[->] (x.center) |- (add1.west);
|
||||
\draw[->] (x.center) |- (add2.west);
|
||||
\draw[->] ($(add1-|W1.west)+(-0.5, 0)$)node[branch](S1){} |- (W1.west);
|
||||
\draw[->] ($(add2-|W2.west)+(-0.5, 0)$)node[branch](S1){} |- (W2.west);
|
||||
\draw[->] (W1.east) -- (delta1.west);
|
||||
\draw[->] (W2.east) -- (delta2.west);
|
||||
\draw[->] (delta1.east) -| (add1.north);
|
||||
\draw[->] (delta2.east) -| (add2.north);
|
||||
\draw[->] (add1.east) -- (H1.west);
|
||||
\draw[->] (add2.east) -- (H2.west);
|
||||
\draw[->] (H1.east) -| (add.north);
|
||||
\draw[->] (H2.east) -| (add.south);
|
||||
\draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
|
||||
\begin{tikzpicture}
|
||||
\node[branch] (x) at (0, 0);
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[block, fit={($(W1.north-|S1)+(-0.2, 0.2)$) ($(add1.south east)+(0.2, -0.3)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor1) {};
|
||||
\node[above right] at (sensor1.south west) {Sensor 1};
|
||||
\node[block, fit={($(W2.north-|S1)+(-0.2, 0.2)$) ($(add2.south east)+(0.2, -0.3)$)}, fill=black!20!white, dashed, inner sep=0pt] (sensor2) {};
|
||||
\node[above right] at (sensor2.south west) {Sensor 2};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
\node[branch, above right=0.9 and 0.3 of x] (input1) {};
|
||||
\node[branch, below right=0.9 and 0.3 of x] (input2) {};
|
||||
\node[block, above right= 0.4 and 0.4 of input1](W1){$w_1(s)$};
|
||||
\node[block, above right= 0.4 and 0.4 of input2](W2){$w_2(s)$};
|
||||
\node[block, right=0.4 of W1](delta1){$\Delta_1(s)$};
|
||||
\node[block, right=0.4 of W2](delta2){$\Delta_2(s)$};
|
||||
\node[addb] (addu1) at ($(delta1.east|-input1) + (0.4, 0)$) {};
|
||||
\node[addb] (addu2) at ($(delta2.east|-input2) + (0.4, 0)$) {};
|
||||
\node[addb, right=0.4 of addu1] (addn1) {};
|
||||
\node[addb, right=0.4 of addu2] (addn2) {};
|
||||
\node[block, right=0.9 of addn1](H1){$H_1(s)$};
|
||||
\node[block, right=0.9 of addn2](H2){$H_2(s)$};
|
||||
|
||||
\node[addb, right=7 of x](add){};
|
||||
|
||||
|
||||
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
|
||||
\draw[->] (x.center) |- (addu1.west);
|
||||
\draw[->] (x.center) |- (addu2.west);
|
||||
\draw[->] (input1.center) |- (W1.west);
|
||||
\draw[->] (W1.east) -- (delta1.west);
|
||||
\draw[->] (delta1.east) -| (addu1.north);
|
||||
\draw[->] (addu1.east) -- (addn1.west);
|
||||
\draw[<-] (addn1.north) -- ++(0, 0.7)node[below right](n1){$n_1$};
|
||||
\draw[->] (input2.center) |- (W2.west);
|
||||
\draw[->] (W2.east) -- (delta2.west);
|
||||
\draw[->] (delta2.east) -| (addu2.north);
|
||||
\draw[->] (addu2.east) -- (addn2.west);
|
||||
\draw[<-] (addn2.north) -- ++(0, 0.7)node[below right](n2){$n_2$};
|
||||
|
||||
\draw[->] (addn1.east) -- (H1.west) node[above left]{$\hat{x}_1$};
|
||||
\draw[->] (addn2.east) -- (H2.west) node[above left]{$\hat{x}_2$};
|
||||
\draw[->] (H1) -| (add.north);
|
||||
\draw[->] (H2) -| (add.south);
|
||||
\draw[->] (add.east) -- ++(0.7, 0) node[above left]{$\hat{x}$};
|
||||
|
||||
\begin{scope}[on background layer]
|
||||
\node[fit={(addn2.south-|x) (delta1.north-|add.east)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
|
||||
\node[below left] at (supersensor.north east) {Super Sensor};
|
||||
|
||||
\node[fit={(input1.west |- addu1.south) ($(delta1.north -| addn1.east) + (0.1, 0.0)$)}, fill=myblue!20!white, draw, inner sep=6pt] (sensor1cal) {};
|
||||
\node[below left] at (sensor1cal.north east) {Sensor 1};
|
||||
|
||||
\node[fit={(input2.west |- addu2.south) ($(delta2.north -| addn1.east) + (0.1, 0.0)$)}, fill=myred!20!white, draw, inner sep=6pt] (sensor2cal) {};
|
||||
\node[below left] at (sensor2cal.north east) {Sensor 2};
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:sensor_fusion_dynamic_uncertainty
|
||||
@ -106,40 +253,44 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+RESULTS:
|
||||
[[file:figs/sensor_fusion_dynamic_uncertainty.png]]
|
||||
|
||||
* Fig 3: Uncertainty set of the super sensor dynamics
|
||||
* Uncertainty set of the super sensor dynamics
|
||||
#+begin_src latex :file uncertainty_set_super_sensor.pdf :tangle figs/uncertainty_set_super_sensor.tex :exports both
|
||||
\begin{tikzpicture}
|
||||
\begin{scope}[shift={(4, 0)}]
|
||||
\definecolor{myblue}{rgb}{0, 0.447, 0.741}
|
||||
\definecolor{myred}{rgb}{0.8500, 0.325, 0.098}
|
||||
|
||||
% Uncertainty Circle
|
||||
\node[draw, circle, fill=black!20!white, minimum size=3.6cm] (c) at (0, 0) {};
|
||||
\path[draw, dotted] (0, 0) circle [radius=1.0];
|
||||
\path[draw, dashed] (135:1.0) circle [radius=0.8];
|
||||
\begin{tikzpicture}
|
||||
\begin{scope}[shift={(4, 0)}]
|
||||
|
||||
% Center of Circle
|
||||
\node[below] at (0, 0){$1$};
|
||||
% Uncertainty Circle
|
||||
\node[draw, circle, fill=black!20!white, minimum size=3.6cm] (c) at (0, 0) {};
|
||||
\path[draw, fill=myblue!20!white] (0, 0) circle [radius=1.0];
|
||||
\path[draw, fill=myred!20!white] (135:1.0) circle [radius=0.8];
|
||||
\path[draw, dashed] (0, 0) circle [radius=1.0];
|
||||
|
||||
\draw[<->, dashed] (0, 0) node[branch]{} -- coordinate[midway](r1) ++(45:1.0);
|
||||
\draw[<->, dashed] (135:1.0)node[branch]{} -- coordinate[midway](r2) ++(90:0.8);
|
||||
% Center of Circle
|
||||
\node[below] at (0, 0){$1$};
|
||||
|
||||
\node[] (l1) at (2, 1.5) {$|w_1 H_1|$};
|
||||
\draw[->, dashed, out=-90, in=0] (l1.south) to (r1);
|
||||
\draw[<->] (0, 0) node[branch]{} -- coordinate[midway](r1) ++(45:1.0);
|
||||
\draw[<->] (135:1.0)node[branch]{} -- coordinate[midway](r2) ++(135:0.8);
|
||||
|
||||
\node[] (l2) at (-2.5, 1.5) {$|w_2 H_2|$};
|
||||
\draw[->, dashed, out=0, in=-180] (l2.east) to (r2);
|
||||
\node[] (l1) at (2, 1.5) {$|w_1 H_1|$};
|
||||
\draw[->, out=-90, in=0] (l1.south) to (r1);
|
||||
|
||||
\draw[<->, dashed] (0, 0) -- coordinate[near end](r3) ++(200:1.8);
|
||||
\node[] (l3) at (-2.5, -1.5) {$|w_1 H_1| + |w_2 H_2|$};
|
||||
\draw[->, dashed, out=90, in=-90] (l3.north) to (r3);
|
||||
\end{scope}
|
||||
\node[] (l2) at (-3.2, 1.2) {$|w_2 H_2|$};
|
||||
\draw[->, out=0, in=-180] (l2.east) to (r2);
|
||||
|
||||
% Real and Imaginary Axis
|
||||
\draw[->] (-0.5, 0) -- (7.0, 0) node[below left]{Re};
|
||||
\draw[->] (0, -1.7) -- (0, 1.7) node[below left]{Im};
|
||||
\draw[<->] (0, 0) -- coordinate[near end](r3) ++(200:1.8);
|
||||
\node[] (l3) at (-2.5, -1.5) {$|w_1 H_1| + |w_2 H_2|$};
|
||||
\draw[->, out=90, in=-90] (l3.north) to (r3);
|
||||
\end{scope}
|
||||
|
||||
\draw[dashed] (0, 0) -- (tangent cs:node=c,point={(0, 0)},solution=2);
|
||||
\draw[dashed] (1, 0) arc (0:28:1) node[midway, right]{$\Delta \phi$};
|
||||
\end{tikzpicture}
|
||||
% Real and Imaginary Axis
|
||||
\draw[->] (-0.5, 0) -- (7.0, 0) node[below left]{Re};
|
||||
\draw[->] (0, -1.7) -- (0, 1.7) node[below left]{Im};
|
||||
|
||||
\draw[dashed] (0, 0) -- (tangent cs:node=c,point={(0, 0)},solution=2);
|
||||
\draw[dashed] (1, 0) arc (0:28:1) node[midway, right]{$\Delta \phi$};
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:uncertainty_set_super_sensor
|
||||
@ -147,33 +298,33 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+RESULTS:
|
||||
[[file:figs/uncertainty_set_super_sensor.png]]
|
||||
|
||||
* Fig 4: Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
|
||||
* Architecture used for $\mathcal{H}_\infty$ synthesis of complementary filters
|
||||
#+begin_src latex :file h_infinity_robust_fusion.pdf :tangle figs/h_infinity_robust_fusion.tex :exports both
|
||||
\begin{tikzpicture}
|
||||
\node[block={4.0cm}{2.5cm}, fill=black!20!white, dashed] (P) {};
|
||||
\node[above] at (P.north) {$P(s)$};
|
||||
\begin{tikzpicture}
|
||||
\node[block={4.0cm}{3.0cm}, fill=black!10!white] (P) {};
|
||||
\node[above] at (P.north) {$P(s)$};
|
||||
|
||||
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
|
||||
\coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$);
|
||||
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
|
||||
\coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.7, 0)$);
|
||||
|
||||
\coordinate[] (output1) at ($(P.south east)!0.75!(P.north east) + ( 0.7, 0)$);
|
||||
\coordinate[] (output2) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$);
|
||||
\coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( 0.7, 0)$);
|
||||
\coordinate[] (output1) at ($(P.south east)!0.75!(P.north east) + ( 0.7, 0)$);
|
||||
\coordinate[] (output2) at ($(P.south east)!0.35!(P.north east) + ( 0.7, 0)$);
|
||||
\coordinate[] (outputv) at ($(P.south east)!0.1!(P.north east) + ( 0.7, 0)$);
|
||||
|
||||
\node[block, left=1.4 of output1] (W1){$W_1(s)$};
|
||||
\node[block, left=1.4 of output2] (W2){$W_2(s)$};
|
||||
\node[addb={+}{}{}{}{-}, left=of W1] (sub) {};
|
||||
\node[block, left=1.4 of output1] (W1){$W_1(s)$};
|
||||
\node[block, left=1.4 of output2] (W2){$W_2(s)$};
|
||||
\node[addb={+}{}{}{}{-}, left=of W1] (sub) {};
|
||||
|
||||
\node[block, below=0.3 of P] (H2) {$H_2(s)$};
|
||||
\node[block, below=0.3 of P] (H2) {$H_2(s)$};
|
||||
|
||||
\draw[->] (inputw) node[above right]{$w$} -- (sub.west);
|
||||
\draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (W2.west);
|
||||
\draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
|
||||
\draw[->] (sub.east) -- (W1.west);
|
||||
\draw[->] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- ($(outputv)+(-0.35, 0)$) node[above]{$v$} |- (H2.east);
|
||||
\draw[->] (W1.east) -- (output1)node[above left]{$z_1$};
|
||||
\draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
|
||||
\end{tikzpicture}
|
||||
\draw[->] (inputw) node[above right]{$w$} -- (sub.west);
|
||||
\draw[->] (H2.west) -| ($(inputu)+(0.35, 0)$) node[above]{$u$} -- (W2.west);
|
||||
\draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
|
||||
\draw[->] (sub.east) -- (W1.west);
|
||||
\draw[->] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- ($(outputv)+(-0.35, 0)$) node[above]{$v$} |- (H2.east);
|
||||
\draw[->] (W1.east) -- (output1)node[above left]{$z_1$};
|
||||
\draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:h_infinity_robust_fusion
|
||||
@ -181,98 +332,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+RESULTS:
|
||||
[[file:figs/h_infinity_robust_fusion.png]]
|
||||
|
||||
* Fig 5: Magnitude of a weighting function generated using the proposed formula
|
||||
#+begin_src matlab :exports none :results none
|
||||
s = zpk('s');
|
||||
|
||||
freqs = logspace(-1, 2, 500);
|
||||
|
||||
n = 2;
|
||||
w0 = 2*pi*10;
|
||||
G0 = 1e-3;
|
||||
G1 = 10;
|
||||
Gc = 2;
|
||||
|
||||
W = (((1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (G0/Gc)^(1/n))/((1/G1)^(1/n)*(1/w0)*sqrt((1-(G0/Gc)^(2/n))/(1-(Gc/G1)^(2/n)))*s + (1/Gc)^(1/n)))^n;
|
||||
|
||||
T = table(freqs', ...
|
||||
abs(squeeze(freqresp(W, freqs, 'Hz'))), ...
|
||||
'VariableNames', {'freqs', 'ampl'});
|
||||
writetable(T, '../matlab/mat/weight_formula.csv');
|
||||
#+end_src
|
||||
|
||||
|
||||
#+begin_src latex :file weight_formula.pdf :tangle figs/weight_formula.tex :exports both
|
||||
\setlength\fwidth{6.5cm}
|
||||
\setlength\fheight{3.5cm}
|
||||
|
||||
\begin{tikzpicture}
|
||||
\begin{axis}[%
|
||||
width=1.0\fwidth,
|
||||
height=1.0\fheight,
|
||||
at={(0.0\fwidth, 0.0\fheight)},
|
||||
scale only axis,
|
||||
xmode=log,
|
||||
xmin=0.1,
|
||||
xmax=100,
|
||||
xtick={0.1,1,10, 100},
|
||||
xminorticks=true,
|
||||
ymode=log,
|
||||
ymin=0.0005,
|
||||
ymax=20,
|
||||
ytick={0.001, 0.01, 0.1, 1, 10},
|
||||
yminorticks=true,
|
||||
ylabel={Magnitude},
|
||||
xlabel={Frequency [Hz]},
|
||||
xminorgrids,
|
||||
yminorgrids,
|
||||
]
|
||||
|
||||
\addplot [color=black, line width=1.5pt, forget plot]
|
||||
table [x=freqs, y=ampl, col sep=comma] {/home/thomas/Cloud/thesis/papers/dehaeze19_desig_compl_filte/matlab/matweight_formula.csv};
|
||||
|
||||
\addplot [color=black, dashed, line width=1.5pt]
|
||||
table[row sep=crcr]{%
|
||||
1 10\\
|
||||
100 10\\
|
||||
};
|
||||
\addplot [color=black, dashed, line width=1.5pt]
|
||||
table[row sep=crcr]{%
|
||||
0.1 0.001\\
|
||||
3 0.001\\
|
||||
};
|
||||
|
||||
\addplot [color=black, line width=1.5pt]
|
||||
table[row sep=crcr]{%
|
||||
0.1 1\\
|
||||
100 1\\
|
||||
};
|
||||
|
||||
\addplot [color=black, dashed, line width=1.5pt]
|
||||
table[row sep=crcr]{%
|
||||
10 2\\
|
||||
10 1\\
|
||||
};
|
||||
|
||||
\node[below] at (2, 10) {$G_\infty$};
|
||||
\node[above] at (2, 0.001) {$G_0$};
|
||||
|
||||
\node[branch] at (10, 2){};
|
||||
\draw[dashed, line cap=round] (7, 2) -- (20, 2) node[right]{$G_c$};
|
||||
\draw[dashed, line cap=round] (10, 2) -- (10, 1) node[below]{$\omega_c$};
|
||||
|
||||
\node[right] at (3, 0.1) {$+n$};
|
||||
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
#+end_src
|
||||
|
||||
#+name: fig:weight_formula
|
||||
#+caption: Magnitude of a weighting function generated using the proposed formula ([[./figs/weight_formula.png][png]], [[./figs/weight_formula.pdf][pdf]], [[./figs/weight_formula.tex][tex]]).
|
||||
#+RESULTS:
|
||||
[[file:figs/weight_formula.png]]
|
||||
|
||||
* Fig 6: Frequency response of the weighting functions and complementary filters obtained using $\mathcal{H}_\infty$ synthesis
|
||||
* Frequency response of the weighting functions and complementary filters obtained using $\mathcal{H}_\infty$ synthesis
|
||||
#+begin_src latex :file hinf_synthesis_results.pdf :tangle figs/hinf_synthesis_results.tex :exports both
|
||||
\setlength\fwidth{6.5cm}
|
||||
\setlength\fheight{6cm}
|
||||
@ -348,7 +408,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+RESULTS:
|
||||
[[file:figs/hinf_synthesis_results.png]]
|
||||
|
||||
* Fig 7: Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters
|
||||
* Architecture for $\mathcal{H}_\infty$ synthesis of three complementary filters
|
||||
#+begin_src latex :file comp_filter_three_hinf.pdf :tangle figs/comp_filter_three_hinf.tex
|
||||
\begin{tikzpicture}
|
||||
\node[block={5.0cm}{3.5cm}, fill=black!20!white, dashed] (P) {};
|
||||
@ -393,7 +453,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+RESULTS:
|
||||
[[file:figs/comp_filter_three_hinf.png]]
|
||||
|
||||
* Fig 8: Frequency response of the weighting functions and three complementary filters obtained using $\mathcal{H}_\infty$ synthesis
|
||||
* Frequency response of the weighting functions and three complementary filters obtained using $\mathcal{H}_\infty$ synthesis
|
||||
#+begin_src latex :file hinf_three_synthesis_results.pdf :tangle figs/hinf_three_synthesis_results.tex :exports both
|
||||
\setlength\fwidth{6.5cm}
|
||||
\setlength\fheight{6cm}
|
||||
@ -482,7 +542,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+RESULTS:
|
||||
[[file:figs/hinf_three_synthesis_results.png]]
|
||||
|
||||
* Fig 9: Specifications and weighting functions magnitude used for $\mathcal{H}_\infty$ synthesis
|
||||
* Specifications and weighting functions magnitude used for $\mathcal{H}_\infty$ synthesis
|
||||
#+begin_src latex :file ligo_weights.pdf :tangle figs/ligo_weights.tex :exports both
|
||||
\setlength\fwidth{6.5cm}
|
||||
\setlength\fheight{3.2cm}
|
||||
@ -557,7 +617,7 @@ Configuration file is accessible [[file:config.org][here]].
|
||||
#+RESULTS:
|
||||
[[file:figs/ligo_weights.png]]
|
||||
|
||||
* Fig 10: Comparison of the FIR filters (solid) with the filters obtained with $\mathcal{H}_\infty$ synthesis (dashed)
|
||||
* Comparison of the FIR filters (solid) with the filters obtained with $\mathcal{H}_\infty$ synthesis (dashed)
|
||||
#+begin_src latex :file comp_fir_ligo_hinf.pdf :tangle figs/comp_fir_ligo_hinf.tex :exports both
|
||||
\setlength\fwidth{6.5cm}
|
||||
\setlength\fheight{6.8cm}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user