4396 lines
239 KiB
Org Mode
4396 lines
239 KiB
Org Mode
#+TITLE: Control of Parallel Manipulators
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:DRAWER:
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#+LANGUAGE: en
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#+EMAIL: dehaeze.thomas@gmail.com
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#+AUTHOR: Dehaeze Thomas
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#+HTML_LINK_HOME: ../index.html
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#+HTML_LINK_UP: ../index.html
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
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#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
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#+BIND: org-latex-image-default-option "scale=1"
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#+BIND: org-latex-image-default-width ""
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#+LaTeX_CLASS: scrreprt
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#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]
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#+LATEX_HEADER: \input{preamble.tex}
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#+LATEX_HEADER_EXTRA: \input{preamble_extra.tex}
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#+LATEX_HEADER_EXTRA: \bibliography{nass-control.bib}
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#+BIND: org-latex-bib-compiler "biber"
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments no
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#+PROPERTY: header-args:matlab+ :exports none
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :noweb yes
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:matlab+ :output-dir figs
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#+PROPERTY: header-args:matlab+ :tangle no
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#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
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#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
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#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
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#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
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#+PROPERTY: header-args:latex+ :results file raw replace
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#+PROPERTY: header-args:latex+ :buffer no
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#+PROPERTY: header-args:latex+ :tangle no
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#+PROPERTY: header-args:latex+ :eval no-export
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#+PROPERTY: header-args:latex+ :exports results
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#+PROPERTY: header-args:latex+ :mkdirp yes
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#+PROPERTY: header-args:latex+ :output-dir figs
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#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
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:END:
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#+latex: \clearpage
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* Build :noexport:
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#+NAME: startblock
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#+BEGIN_SRC emacs-lisp :results none :tangle no
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(add-to-list 'org-latex-classes
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'("scrreprt"
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"\\documentclass{scrreprt}"
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("\\chapter{%s}" . "\\chapter*{%s}")
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("\\section{%s}" . "\\section*{%s}")
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("\\subsection{%s}" . "\\subsection*{%s}")
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("\\paragraph{%s}" . "\\paragraph*{%s}")
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))
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;; Remove automatic org heading labels
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(defun my-latex-filter-removeOrgAutoLabels (text backend info)
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"Org-mode automatically generates labels for headings despite explicit use of `#+LABEL`. This filter forcibly removes all automatically generated org-labels in headings."
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(when (org-export-derived-backend-p backend 'latex)
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(replace-regexp-in-string "\\\\label{sec:org[a-f0-9]+}\n" "" text)))
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(add-to-list 'org-export-filter-headline-functions
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'my-latex-filter-removeOrgAutoLabels)
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;; Remove all org comments in the output LaTeX file
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(defun delete-org-comments (backend)
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(loop for comment in (reverse (org-element-map (org-element-parse-buffer)
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'comment 'identity))
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do
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(setf (buffer-substring (org-element-property :begin comment)
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(org-element-property :end comment))
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"")))
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(add-hook 'org-export-before-processing-hook 'delete-org-comments)
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;; Use no package by default
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(setq org-latex-packages-alist nil)
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(setq org-latex-default-packages-alist nil)
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;; Do not include the subtitle inside the title
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(setq org-latex-subtitle-separate t)
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(setq org-latex-subtitle-format "\\subtitle{%s}")
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(setq org-export-before-parsing-hook '(org-ref-glossary-before-parsing
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org-ref-acronyms-before-parsing))
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#+END_SRC
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* Notes :noexport:
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** Notes
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Prefix is =detail_control=
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Up to now, basic control strategy ("decentralized" control in the frame of the struts), HAC-LAC
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Discussion about:
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- *Use multiple sensors*: Discussion about two sensor control VS HAC-LAC VS Sensor Fusion ?
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- file:~/Cloud/research/papers/published/dehaeze19_compl_filter_shapin_using_synth/index.org
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- [[file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/journal/dehaeze21_desig_compl_filte.org][file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/journal/dehaeze21_desig_compl_filte.org]]
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- [[file:~/Cloud/research/papers/dehaeze20_optim_robus_compl_filte/matlab/dehaeze22_optim_robus_compl_filte_matlab.org]]
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- *Decoupling strategies*
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- file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org
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- *Control optimization*
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- file:~/Cloud/research/papers/published/verma19_virtu_senso_fusio_high_preci_contr/elsarticle-template-harv.pdf
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- file:~/Cloud/research/papers/dehaeze20_virtu_senso_fusio/index.org
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** Unused
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*** Table with multi sensor control
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#+name: tab:detail_control_sensor_review
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#+caption: Review of Stewart platforms integrating multiple sensors
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#+attr_latex: :environment tabularx :width 0.9\linewidth :align Xcccc
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#+attr_latex: :center t :booktabs t :font \scriptsize
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| | *Actuators* | *Sensors* | *Control* | Main Object | *Reference* |
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|---+------------------+---------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
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| X | Magnetostrictive | Force (collocated), Accelerometers | Two layers: Decentralized IFF, Robust Adaptive Control | Two layer control for active damping and vibration isolation | [[cite:&geng95_intel_contr_system_multip_degree]] |
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| X | Voice Coil | Accelerometer (collocated), ext. Rx/Ry sensors | Cartesian acceleration feedback (isolation) + 2DoF pointing control (external sensor) | Decoupling, both vibration + pointing control | [[cite:&li01_simul_vibrat_isolat_point_contr]] |
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| | Voice Coil | Geophone + Eddy Current (Struts, collocated) | Decentralized (Sky Hook) + Centralized (modal) Control | | [[cite:&pu11_six_degree_of_freed_activ]] |
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| X | Piezoelectric | Force Sensor + Accelerometer | HAC-LAC (IFF + FxLMS) | Dynamic Model + Vibration Control | [[cite:&wang16_inves_activ_vibrat_isolat_stewar]] |
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| X | Voice Coil | Force sensors (strus) + accelerometer (cartesian) | Decentralized Force Feedback + Centralized H2 control based on accelerometers | | [[cite:&xie17_model_contr_hybrid_passiv_activ]] |
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|---+------------------+---------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
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| X | Voice Coil | Force (HF) and Inertial (LF) | *Sensor Fusion*, *Two Sensor Control* | | [[cite:&tjepkema12_activ_ph;&tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip]] |
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| X | Voice Coil | Force (HF) and Inertial (LF) | *Sensor Fusion*, LQG, Decentralized | Combine force/inertial sensors. Comparison of force sensor and inertial sensors. Issue on non-minimum phase zero | [[cite:&hauge04_sensor_contr_space_based_six]] |
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| X | Voice Coil | Force, LVDT, Geophones | *two sensor control* LQG, Force + geophones for vibration, LVDT for pointing | Centralized control is no better than decentralized. Geophone + Force MISO control is good | cite:&thayer98_stewar;&thayer02_six_axis_vibrat_isolat_system |
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|---+------------------+---------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
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| X | Piezoelectric | Force, Position | Vibration isolation, Model-Based, Modal control: 6x PI controllers | Stiffness of flexible joints is compensated using feedback, then the system is decoupled in the modal space | [[cite:&yang19_dynam_model_decoup_contr_flexib]] |
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| | Voice Coil | Force | IFF, centralized (decouple) + decentralized (coupled) | Specific geometry: decoupled force plant. Better perf with centralized IFF | [[cite:&mcinroy99_dynam;&mcinroy99_precis_fault_toler_point_using_stewar_platf;&mcinroy00_desig_contr_flexur_joint_hexap]] |
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*** Table with decoupling strategies
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#+name: tab:detail_control_decoupling_review
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#+caption: Litterature review about decoupling strategy for Stewart platform control
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#+attr_latex: :environment tabularx :width 0.9\linewidth :align Xccc
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#+attr_latex: :center t :booktabs t :font \scriptsize
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| | *Actuators* | *Sensors* | *Control* | *Reference* |
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|---+---------------+----------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
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| X | APA | Eddy current displacement | *Decentralized* (struts) PI + LPF control | [[cite:&furutani04_nanom_cuttin_machin_using_stewar]] |
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| X | PZT Piezo | Strain Gauge | Decentralized position feedback | [[cite:&du14_piezo_actuat_high_precis_flexib]] |
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|---+---------------+----------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
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| | Voice Coil | Force | *Cartesian frame* decoupling | [[cite:&obrien98_lesson]] |
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| X | Voice Coil | Force | Cartesian Frame, Jacobians, IFF | [[cite:&mcinroy99_dynam;&mcinroy99_precis_fault_toler_point_using_stewar_platf;&mcinroy00_desig_contr_flexur_joint_hexap]] |
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| X | Hydraulic | LVDT | Decentralized (strut) vs Centralized (cartesian) | [[cite:&kim00_robus_track_contr_desig_dof_paral_manip]] |
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| X | Voice Coil | Accelerometer (collocated), ext. Rx/Ry sensors | Cartesian acceleration feedback (isolation) + 2DoF pointing control (external sensor) | [[cite:&li01_simul_vibrat_isolat_point_contr]] |
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| | Voice Coil | Accelerometer in each leg | Centralized Vibration Control, PI, Skyhook | [[cite:&abbas14_vibrat_stewar_platf]] |
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|---+---------------+----------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
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| X | Voice Coil | Geophone + Eddy Current (Struts, collocated) | Decentralized (Sky Hook) + Centralized (*modal*) Control | [[cite:&pu11_six_degree_of_freed_activ]] |
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| X | Piezoelectric | Force, Position | Vibration isolation, Model-Based, *Modal control*: 6x PI controllers | [[cite:&yang19_dynam_model_decoup_contr_flexib]] |
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|---+---------------+----------------------------------------------------+---------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------------------------------|
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| | PZT | Geophone (struts) | *H-Infinity* and mu-synthesis | [[cite:&lei08_multi_objec_robus_activ_vibrat]] |
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| | Voice Coil | Force sensors (struts) + accelerometer (cartesian) | Decentralized Force Feedback + Centralized H2 control based on accelerometers | [[cite:&xie17_model_contr_hybrid_passiv_activ]] |
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| | Voice Coil | Accelerometers | MIMO H-Infinity, active damping | [[cite:&jiao18_dynam_model_exper_analy_stewar]] |
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*** "Closed-Loop" complementary filters
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<<ssec:detail_control_closed_loop_complementary_filters>>
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An alternative way to implement complementary filters is by using a fundamental property of the classical feedback architecture shown in Figure ref:fig:detail_control_feedback_sensor_fusion.
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This idea is discussed in [[cite:&mahony05_compl_filter_desig_special_orthog;&plummer06_optim_compl_filter_their_applic_motion_measur;&jensen13_basic_uas]].
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#+begin_src latex :file detail_control_feedback_sensor_fusion.pdf
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\tikzset{block/.default={0.8cm}{0.8cm}}
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\tikzset{addb/.append style={scale=0.7}}
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\tikzset{node distance=0.6}
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\begin{tikzpicture}
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\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
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\node[block, right=1 of addfb] (L){$L(s)$};
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\node[addb={+}{}{}{}{}, right=1 of L] (adddy){};
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\draw[<-] (addfb.west) -- ++(-1, 0) node[above right]{$\hat{x}_2$};
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\draw[->] (addfb.east) -- (L.west);
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\draw[->] (L.east) -- (adddy.west);
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\draw[->] (adddy.east) -- ++(1.4, 0) node[above left]{$\hat{x}$};
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\draw[->] ($(adddy.east) + (0.5, 0)$) node[branch]{} -- ++(0, -0.8) coordinate(botc) -| (addfb.south);
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\draw[<-] (adddy.north) -- ++(0, 1) node[below right]{$\hat{x}_1$};
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\begin{scope}[on background layer]
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\node[fit={(L.north-|addfb.west) (botc)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
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\end{scope}
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\end{tikzpicture}
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#+end_src
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#+name: fig:detail_control_feedback_sensor_fusion
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#+caption: "Closed-Loop" complementary filters.
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#+RESULTS:
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[[file:figs/detail_control_feedback_sensor_fusion.png]]
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Consider the feedback architecture of Figure ref:fig:detail_control_feedback_sensor_fusion, with two inputs $\hat{x}_1$ and $\hat{x}_2$, and one output $\hat{x}$.
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The output $\hat{x}$ is linked to the inputs by eqref:eq:detail_control_closed_loop_complementary_filters.
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\begin{equation}\label{eq:detail_control_closed_loop_complementary_filters}
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\hat{x} = \underbrace{\frac{1}{1 + L(s)}}_{S(s)} \hat{x}_1 + \underbrace{\frac{L(s)}{1 + L(s)}}_{T(s)} \hat{x}_2
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\end{equation}
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As for any classical feedback architecture, we have that the sum of the sensitivity transfer function $S(s)$ and complementary sensitivity transfer function $T_(s)$ is equal to one eqref:eq:detail_control_sensitivity_sum.
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\begin{equation}\label{eq:detail_control_sensitivity_sum}
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S(s) + T(s) = 1
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\end{equation}
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Therefore, provided that the the closed-loop system in Figure ref:fig:detail_control_feedback_sensor_fusion is stable, it can be used as a set of two complementary filters.
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Two sensors can then be merged as shown in Figure ref:fig:detail_control_feedback_sensor_fusion_arch.
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#+begin_src latex :file detail_control_feedback_sensor_fusion_arch.pdf
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\tikzset{block/.default={0.8cm}{0.8cm}}
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\tikzset{addb/.append style={scale=0.7}}
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\tikzset{node distance=0.6}
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\begin{tikzpicture}
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\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
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\node[block, right=1 of addfb] (L){$L(s)$};
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\node[addb={+}{}{}{}{}, right=1 of L] (adddy){};
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\node[block, left=1.2 of addfb] (sensor2){Sensor 2};
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\node[block, above=0.4 of sensor2] (sensor1){Sensor 1};
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\node[branch, left=0.6 of sensor2] (x){};
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\draw[->] (addfb.east) -- (L.west);
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\draw[->] (L.east) -- (adddy.west);
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\draw[->] (adddy.east) -- ++(1.4, 0) node[above left]{$\hat{x}$};
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\draw[->] ($(adddy.east) + (0.5, 0)$) node[branch]{} -- ++(0, -0.8) coordinate(botc) -| (addfb.south);
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\draw[->] (x.center) |- (sensor1.west);
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\draw[->] ($(x)-(0.8,0)$) node[above right]{$x$} -- (sensor2.west);
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\draw[->] (sensor2.east)node[above right=0 and 0.25]{$\hat{x}_2$} -- (addfb.west);
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\draw[->] (sensor1.east)node[above right=0 and 0.25]{$\hat{x}_1$} -| (adddy.north);
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\begin{scope}[on background layer]
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\node[fit={(x|-sensor1.north) (botc)}, fill=black!10!white, draw, inner sep=9pt] (supersensor) {};
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\node[fit={(sensor1.north-|addfb.west) (botc)}, fill=black!20!white, draw, inner sep=6pt] (feedbackfilter) {};
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\node[fit={(sensor2.west|-botc) (sensor1.north east)}, fill=black!20!white, draw, inner sep=6pt] (sensors) {};
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\node[above, align=center] at (sensors.south) {{\tiny Normalized}\\[-0.5em]{\tiny sensors}};
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\node[below, align=center] at (feedbackfilter.north) {{\tiny "Closed-Loop"}\\[-0.5em]{\tiny complementary filters}};
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\end{scope}
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\end{tikzpicture}
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#+end_src
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#+name: fig:detail_control_feedback_sensor_fusion_arch
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#+caption: Classical feedback architecture used for sensor fusion.
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#+RESULTS:
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[[file:figs/detail_control_feedback_sensor_fusion_arch.png]]
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One of the main advantage of implementing and designing complementary filters using the feedback architecture of Figure ref:fig:detail_control_feedback_sensor_fusion is that all the tools of the linear control theory can be applied for the design of the filters.
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If one want to shape both $\frac{\hat{x}}{\hat{x}_1}(s) = S(s)$ and $\frac{\hat{x}}{\hat{x}_2}(s) = T(s)$, the $\mathcal{H}_\infty$ mixed-sensitivity synthesis can be easily applied.
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To do so, weighting functions $W_1(s)$ and $W_2(s)$ are added to respectively shape $S(s)$ and $T(s)$ (Figure ref:fig:detail_control_feedback_synthesis_architecture).
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Then the system is rearranged to form the generalized plant $P_L(s)$ shown in Figure ref:fig:detail_control_feedback_synthesis_architecture_generalized_plant.
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The $\mathcal{H}_\infty$ mixed-sensitivity synthesis can finally be performed by applying the standard $\mathcal{H}_\infty$ synthesis to the generalized plant $P_L(s)$ which is described by eqref:eq:detail_control_generalized_plant_mixed_sensitivity.
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\begin{equation}\label{eq:detail_control_generalized_plant_mixed_sensitivity}
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\begin{bmatrix} z \\ v \end{bmatrix} = P_L(s) \begin{bmatrix} w_1 \\ w_2 \\ u \end{bmatrix}; \quad P_L(s) = \begin{bmatrix}
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\phantom{+}W_1(s) & 0 & \phantom{+}1 \\
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-W_1(s) & W_2(s) & -1
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\end{bmatrix}
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\end{equation}
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The output of the synthesis is a filter $L(s)$ such that the "closed-loop" $\mathcal{H}_\infty$ norm from $[w_1,\ w_2]$ to $z$ of the system in Figure ref:fig:detail_control_feedback_sensor_fusion is less than one eqref:eq:detail_control_comp_filters_feedback_obj.
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\begin{equation}\label{eq:detail_control_comp_filters_feedback_obj}
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\left\| \begin{matrix} \frac{z}{w_1} \\ \frac{z}{w_2} \end{matrix} \right\|_\infty = \left\| \begin{matrix} \frac{1}{1 + L(s)} W_1(s) \\ \frac{L(s)}{1 + L(s)} W_2(s) \end{matrix} \right\|_\infty \le 1
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\end{equation}
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If the synthesis is succescful, the transfer functions from $\hat{x}_1$ to $\hat{x}$ and from $\hat{x}_2$ to $\hat{x}$ have their magnitude bounded by the inverse magnitude of the corresponding weighting functions.
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The sensor fusion can then be implemented using the feedback architecture in Figure ref:fig:detail_control_feedback_sensor_fusion_arch or more classically as shown in Figure ref:fig:detail_control_sensor_fusion_overview by defining the two complementary filters using eqref:eq:detail_control_comp_filters_feedback.
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The two architectures are equivalent regarding their inputs/outputs relationships.
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\begin{equation}\label{eq:detail_control_comp_filters_feedback}
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H_1(s) = \frac{1}{1 + L(s)}; \quad H_2(s) = \frac{L(s)}{1 + L(s)}
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\end{equation}
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#+begin_src latex :file detail_control_feedback_synthesis_architecture.pdf
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\tikzset{block/.default={0.8cm}{0.8cm}}
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\tikzset{addb/.append style={scale=0.7}}
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\tikzset{node distance=0.6}
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\begin{tikzpicture}
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\node[block] (W2) at (0,0) {$W_2(s)$};
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\node[addb={+}{}{}{}{-}, right=0.8 of W2] (addfb){};
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\node[addb={+}{}{}{}{}, right=4.5 of W2] (adddy){};
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\node[block, above=0.8 of adddy] (W1){$W_1(s)$};
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\draw[<-] (W2.west) -- ++(-0.8, 0) node[above right]{$w_2$};
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\draw[->] (W2.east) -- (addfb.west) node[above left]{$\tilde{w}_2$};
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\draw[->] (addfb.east) -- ++(1, 0) node[above left]{$v$};
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\draw[<-] (adddy.west) -- ++(-1, 0) node[above right]{$u$};
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\draw[->] (adddy.east) -- ++(1.4, 0) node[above left]{$z$};
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\draw[->] (W1.south) -- (adddy.north) node[above right]{$\tilde{w}_1$};
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\draw[<-] (W1.north) -- ++(0, 0.8) node[below right]{$w_1$};
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\draw[->] ($(adddy.east) + (0.5, 0)$) node[branch]{} -- ++(0, -0.8) -| (addfb.south);
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\end{tikzpicture}
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#+end_src
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|
#+begin_src latex :file detail_control_feedback_synthesis_architecture_generalized_plant.pdf
|
|
\tikzset{block/.default={0.8cm}{0.8cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\begin{tikzpicture}
|
|
\node[block={4.5cm}{3.0cm}, fill=black!10!white] (P) {};
|
|
\node[above] at (P.north) {$P_L(s)$};
|
|
|
|
\coordinate[] (inputw1) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
|
|
\coordinate[] (inputw2) at ($(P.south west)!0.40!(P.north west) + (-0.7, 0)$);
|
|
\coordinate[] (inputu) at ($(P.south west)!0.15!(P.north west) + (-0.7, 0)$);
|
|
|
|
\coordinate[] (outputz) at ($(P.south east)!0.75!(P.north east) + ( 0.7, 0)$);
|
|
\coordinate[] (outputv) at ($(P.south east)!0.40!(P.north east) + ( 0.7, 0)$);
|
|
|
|
\node[block, right=1.2 of inputw2] (W2){$W_2(s)$};
|
|
\node[block, right=1.2 of inputw1] (W1){$W_1(s)$};
|
|
\node[addb={+}{}{}{}{}, right=0.8 of W1] (add) {};
|
|
\node[addb={+}{}{-}{}{}, right=1.8 of W2] (sub) {};
|
|
|
|
\draw[->] (inputw2) node[above right]{$w_2$} -- (W2.west);
|
|
\draw[->] (inputw1) node[above right]{$w_1$} -- (W1.west);
|
|
\draw[->] (inputu) node[above right]{$u$} -| (add.south);
|
|
\draw[->] (W2.east) -- (sub.west);
|
|
\draw[->] (W1.east) -- (add.west);
|
|
\draw[->] (add.east) -- (outputz)node[above left]{$z$};
|
|
\draw[->] (sub.east) -- (outputv)node[above left]{$v$};
|
|
\draw[->] (add-|sub) node[branch]{} -- (sub.north);
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_h_inf_mixed_sensitivity_synthesis
|
|
#+caption: $\mathcal{H}_\infty$ mixed-sensitivity synthesis
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_feedback_synthesis_architecture}Feedback architecture with included weights}
|
|
#+attr_latex: :options {0.58\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_feedback_synthesis_architecture.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_feedback_synthesis_architecture_generalized_plant}Generalized plant}
|
|
#+attr_latex: :options {0.38\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_feedback_synthesis_architecture_generalized_plant.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
As an example, two "closed-loop" complementary filters are designed using the $\mathcal{H}_\infty$ mixed-sensitivity synthesis.
|
|
The weighting functions are designed using formula eqref:eq:detail_control_weight_formula with parameters shown in Table ref:tab:detail_control_weights_params.
|
|
After synthesis, a filter $L(s)$ is obtained whose magnitude is shown in Figure ref:fig:detail_control_hinf_filters_results_mixed_sensitivity by the black dashed line.
|
|
The "closed-loop" complementary filters are compared with the inverse magnitude of the weighting functions in Figure ref:fig:detail_control_hinf_filters_results_mixed_sensitivity confirming that the synthesis is succescful.
|
|
The obtained "closed-loop" complementary filters are indeed equal to the ones obtained in Section ref:ssec:detail_control_hinf_method.
|
|
|
|
#+begin_src matlab
|
|
%% Design of "Closed-loop" complementary filters
|
|
% Design of the Weighting Functions
|
|
W1 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.45);
|
|
W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.45);
|
|
|
|
% Generalized plant for "closed-loop" complementary filter synthesis
|
|
P = [ W1 0 1;
|
|
-W1 W2 -1];
|
|
|
|
% Standard H-Infinity Synthesis
|
|
[L, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
|
|
|
% Complementary filters
|
|
H1 = inv(1 + L);
|
|
H2 = 1 - H1;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results none
|
|
%% Bode plot of the obtained complementary filters after H-infinity mixed-sensitivity synthesis
|
|
figure;
|
|
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
% Magnitude
|
|
ax1 = nexttile([2, 1]);
|
|
hold on;
|
|
set(gca,'ColorOrderIndex',1)
|
|
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'DisplayName', '$|W_1|^{-1}$');
|
|
set(gca,'ColorOrderIndex',2)
|
|
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'DisplayName', '$|W_2|^{-1}$');
|
|
|
|
set(gca,'ColorOrderIndex',1)
|
|
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'DisplayName', '$H_1$');
|
|
set(gca,'ColorOrderIndex',2)
|
|
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'DisplayName', '$H_2$');
|
|
|
|
plot(freqs, abs(squeeze(freqresp(L, freqs, 'Hz'))), 'k--', 'DisplayName', '$|L|$');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
set(gca, 'XTickLabel',[]); ylabel('Magnitude');
|
|
ylim([1e-3, 1e3]);
|
|
yticks([1e-3, 1e-2, 1e-1, 1, 1e1, 1e2, 1e3]);
|
|
yticklabels({'', '$10^{-2}$', '', '$10^0$', '', '$10^2$', ''});
|
|
leg = legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 3);
|
|
leg.ItemTokenSize(1) = 18;
|
|
|
|
% Phase
|
|
ax2 = nexttile;
|
|
hold on;
|
|
set(gca,'ColorOrderIndex',1)
|
|
plot(freqs, 180/pi*phase(squeeze(freqresp(H1, freqs, 'Hz'))), '-');
|
|
set(gca,'ColorOrderIndex',2)
|
|
plot(freqs, 180/pi*phase(squeeze(freqresp(H2, freqs, 'Hz'))), '-');
|
|
hold off;
|
|
set(gca, 'XScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
yticks([-180:90:180]);
|
|
ylim([-180, 200])
|
|
yticklabels({'-180', '', '0', '', '180'})
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([freqs(1), freqs(end)]);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_hinf_filters_results_mixed_sensitivity.pdf', 'width', 'wide', 'height', 600);
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_hinf_filters_results_mixed_sensitivity
|
|
#+caption: Bode plot of the obtained complementary filters after $\mathcal{H}_\infty$ mixed-sensitivity synthesis
|
|
#+RESULTS:
|
|
[[file:figs/detail_control_hinf_filters_results_mixed_sensitivity.png]]
|
|
|
|
*** Analytical formulas for test model
|
|
|
|
#+begin_src matlab
|
|
%% Analytical Formula for the Jacobian Matrix
|
|
% Create symbolic variables for all parameters
|
|
syms l h la ha m I real
|
|
syms c1 c2 c3 real
|
|
syms k1 k2 k3 real
|
|
|
|
% Unit vectors of the actuators (symbolic)
|
|
s1 = [1; 0]; % Actuator 1 direction (horizontal)
|
|
s2 = [0; 1]; % Actuator 2 direction (vertical)
|
|
s3 = [0; 1]; % Actuator 3 direction (vertical)
|
|
|
|
% Location of the joints with respect to the center of mass (symbolic)
|
|
Mb1 = [-l/2; -ha]; % Joint 1 position vector
|
|
Mb2 = [-la; -h/2]; % Joint 2 position vector
|
|
Mb3 = [ la; -h/2]; % Joint 3 position vector
|
|
|
|
% Jacobian matrix (Center of Mass)
|
|
J_CoM = [s1', Mb1(1)*s1(2) - Mb1(2)*s1(1);
|
|
s2', Mb2(1)*s2(2) - Mb2(2)*s2(1);
|
|
s3', Mb3(1)*s3(2) - Mb3(2)*s3(1)];
|
|
|
|
% Display the symbolic Jacobian matrix
|
|
disp('Symbolic Jacobian Matrix (J_CoM):');
|
|
disp(J_CoM);
|
|
|
|
% Jacobian at the Center of stiffness {K}
|
|
Mb1 = [-l/2; 0];
|
|
Mb2 = [-la; -h/2+ha];
|
|
Mb3 = [ la; -h/2+ha];
|
|
|
|
J_CoK = [s1', Mb1(1)*s1(2) - Mb1(2)*s1(1);
|
|
s2', Mb2(1)*s2(2) - Mb2(2)*s2(1);
|
|
s3', Mb3(1)*s3(2) - Mb3(2)*s3(1)];
|
|
|
|
% Display the symbolic Jacobian matrix
|
|
disp('Symbolic Jacobian Matrix (J_CoK):');
|
|
disp(J_CoK);
|
|
#+end_src
|
|
|
|
*** Analytical formulas for modal decoupling
|
|
|
|
#+begin_src matlab
|
|
%% Analytical Formula for the Modal Decoupling
|
|
syms l h la ha m I k c s real
|
|
|
|
% Unit vectors of the actuators
|
|
s1 = [1; 0]; % Actuator 1 direction (horizontal)
|
|
s2 = [0; 1]; % Actuator 2 direction (vertical)
|
|
s3 = [0; 1]; % Actuator 3 direction (vertical)
|
|
|
|
% Location of the joints with respect to the center of mass (symbolic)
|
|
Mb1 = [-l/2; -ha]; % Joint 1 position vector
|
|
Mb2 = [-la; -h/2]; % Joint 2 position vector
|
|
Mb3 = [ la; -h/2]; % Joint 3 position vector
|
|
|
|
% Calculate the Jacobian matrix (Center of Mass) symbolically
|
|
J_CoM = [s1', Mb1(1)*s1(2) - Mb1(2)*s1(1);
|
|
s2', Mb2(1)*s2(2) - Mb2(2)*s2(1);
|
|
s3', Mb3(1)*s3(2) - Mb3(2)*s3(1)];
|
|
|
|
disp('Symbolic Jacobian Matrix (J_CoM):');
|
|
disp(J_CoM);
|
|
|
|
% Define system matrices
|
|
M = diag([m, m, I]);
|
|
K_struts = diag([k, k, k]);
|
|
|
|
% Transform stiffness and damping to Cartesian space
|
|
K = J_CoM' * K_struts * J_CoM;
|
|
|
|
disp('Mass Matrix (M):');
|
|
disp(M);
|
|
|
|
disp('Stiffness Matrix (K):');
|
|
disp(K);
|
|
|
|
MK = simplify(M\K);
|
|
disp('M\K Matrix:');
|
|
disp(MK);
|
|
|
|
[V, ~] = eig(MK);
|
|
disp(simplify(V));
|
|
disp('1st eigen vector:');
|
|
disp(simplify(V(:,1)));
|
|
disp('2nd eigen vector:');
|
|
disp(simplify(V(:,2)));
|
|
disp('3rd eigen vector:');
|
|
disp(simplify(V(:,3)));
|
|
|
|
Vinv = inv(V);
|
|
disp(simplify(Vinv));
|
|
#+end_src
|
|
|
|
*** Simscape plant
|
|
|
|
Verify that the collocated plant is the same as the analytical plant.
|
|
*Yes they are the same*
|
|
|
|
#+begin_src matlab
|
|
%% Collocated plant
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Actuator_1'], 1, 'openoutput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Actuator_2'], 1, 'openoutput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Actuator_3'], 1, 'openoutput'); io_i = io_i + 1;
|
|
|
|
G_L_sim = linearize(mdl, io);
|
|
G_L_sim.InputName = {'F1', 'F2', 'F3'};
|
|
G_L_sim.OutputName = {'L1', 'L2', 'L3'};
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Accelerometer outputs
|
|
% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
|
|
% io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
|
|
|
|
G_acc = linearize(mdl, io);
|
|
G_acc.InputName = {'F1', 'F2', 'F3'};
|
|
G_acc.OutputName = {'Ax1', 'Ay1', 'Ax2'};
|
|
#+end_src
|
|
|
|
*** Robustness of the decoupling strategies? :noexport:
|
|
**** Introduction :ignore:
|
|
What happens if we add an additional resonance in the system (Figure ref:fig:model_test_decoupling_spurious_res).
|
|
|
|
Having less actuator than DoF (under-actuated system):
|
|
- modal decoupling: can still control first $n$ modes?
|
|
- SVD decoupling: does not matter
|
|
- Jacobian decoupling: could give poor decoupling?
|
|
|
|
#+name: fig:model_test_decoupling_spurious_res
|
|
#+caption: Plant with spurious resonance (additional DoF)
|
|
[[file:figs/model_test_decoupling_spurious_res.png]]
|
|
|
|
**** Plant
|
|
A multi body model of the system in Figure ref:fig:model_test_decoupling_spurious_res has been made using Simscape.
|
|
|
|
Its parameters are defined below:
|
|
#+begin_src matlab
|
|
leq = 20e-3; % Equilibrium length of struts [m]
|
|
mr = 5; % [kg]
|
|
kr = (2*pi*10)^2*mr; % Stiffness [N/m]
|
|
cr = 1e1; % Damping [N/(m/s)]
|
|
|
|
m = 400 - mr; % Mass [kg]
|
|
#+end_src
|
|
|
|
The plant is then identified and shown in Figure ref:fig:detail_control_coupled_plant_bode_spurious.
|
|
The added resonance only slightly modifies the plant around 10Hz.
|
|
#+begin_src matlab :exports none
|
|
%% Name of the Simulink File
|
|
mdl = 'suspended_mass';
|
|
|
|
%% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/dL'], 1, 'openoutput'); io_i = io_i + 1;
|
|
|
|
Gr = linearize(mdl, io);
|
|
Gr.InputName = {'F1', 'F2', 'F3'};
|
|
Gr.OutputName = {'L1', 'L2', 'L3'};
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
for out_i = 1:3
|
|
for in_i = 1:3
|
|
nexttile;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G(out_i,in_i), freqs, 'Hz'))), 'k-', ...
|
|
'DisplayName', sprintf('$\\mathcal{L}_%i/\\tau_%i$', out_i, in_i));
|
|
plot(freqs, abs(squeeze(freqresp(Gr(out_i,in_i), freqs, 'Hz'))), 'k--', ...
|
|
'HandleVisibility', 'off');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlim([1e-1, 2e1]); ylim([1e-6, 1e-2]);
|
|
legend('location', 'northeast', 'FontSize', 8);
|
|
|
|
if in_i == 1
|
|
ylabel('Mag. [m/N]')
|
|
else
|
|
set(gca, 'YTickLabel',[]);
|
|
end
|
|
|
|
if out_i == 3
|
|
xlabel('Frequency [Hz]')
|
|
else
|
|
set(gca, 'XTickLabel',[]);
|
|
end
|
|
end
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_coupled_plant_bode_spurious.pdf', 'width', 'full', 'height', 'tall');
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_coupled_plant_bode_spurious
|
|
#+caption: Magnitude of the coupled plant without additional mode (solid) and with the additional mode (dashed).
|
|
#+RESULTS:
|
|
[[file:figs/detail_control_coupled_plant_bode_spurious.png]]
|
|
|
|
**** Jacobian Decoupling
|
|
The obtained plant is decoupled using the Jacobian matrix.
|
|
|
|
#+begin_src matlab
|
|
Gxr = pinv(J)*Gr*pinv(J');
|
|
Gxr.InputName = {'Fx', 'Fy', 'Mz'};
|
|
Gxr.OutputName = {'Dx', 'Dy', 'Rz'};
|
|
#+end_src
|
|
|
|
The obtained plant is shown in Figure ref:fig:detail_control_jacobian_plant_spurious and is not much different than for the plant without the spurious resonance.
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
|
|
% Magnitude
|
|
hold on;
|
|
for i_in = 1:3
|
|
for i_out = [i_in+1:3]
|
|
plot(freqs, abs(squeeze(freqresp(Gxr(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
end
|
|
plot(freqs, abs(squeeze(freqresp(Gxr(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
|
'DisplayName', '$G_{x,r}(i,j)\ i \neq j$');
|
|
set(gca,'ColorOrderIndex',1)
|
|
for i_in_out = 1:3
|
|
plot(freqs, abs(squeeze(freqresp(Gxr(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{x,r}(%d,%d)$', i_in_out, i_in_out));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
|
ylim([1e-7, 1e-1]);
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_jacobian_plant_spurious.pdf', 'width', 'wide', 'height', 'normal');
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_jacobian_plant_spurious
|
|
#+caption: Plant with spurious resonance decoupled using the Jacobian matrices $G_{x,r}(s)$
|
|
#+RESULTS:
|
|
[[file:figs/detail_control_jacobian_plant_spurious.png]]
|
|
|
|
**** Modal Decoupling
|
|
The obtained plant is now decoupled using the modal matrices obtained with the plant not including the added resonance.
|
|
#+begin_src matlab
|
|
Gmr = inv(Cm)*Gr*inv(Bm);
|
|
#+end_src
|
|
|
|
The obtained decoupled plant is shown in Figure ref:fig:detail_control_modal_plant_spurious.
|
|
Compare to the decoupled plant in Figure ref:fig:detail_control_modal_plant, the added resonance induces some coupling, especially around the frequency of the added spurious resonance.
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
|
|
% Magnitude
|
|
hold on;
|
|
for i_in = 1:3
|
|
for i_out = [i_in+1:3]
|
|
plot(freqs, abs(squeeze(freqresp(Gmr(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
end
|
|
plot(freqs, abs(squeeze(freqresp(Gmr(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
|
'DisplayName', '$G_{m,r}(i,j)\ i \neq j$');
|
|
set(gca,'ColorOrderIndex',1)
|
|
for i_in_out = 1:3
|
|
plot(freqs, abs(squeeze(freqresp(Gmr(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{m,r}(%d,%d)$', i_in_out, i_in_out));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
|
ylim([1e-6, 1e2]);
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_modal_plant_spurious.pdf', 'width', 'wide', 'height', 'normal');
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_modal_plant_spurious
|
|
#+caption: Modal plant including spurious resonance $G_{m,r}(s)$
|
|
#+RESULTS:
|
|
[[file:figs/detail_control_modal_plant_spurious.png]]
|
|
|
|
**** SVD Decoupling
|
|
The SVD decoupling is performed on the new obtained plant.
|
|
The decoupling frequency is slightly shifted in order not to interfere too much with the spurious resonance.
|
|
|
|
#+begin_src matlab
|
|
%% Decoupling frequency [rad/s]
|
|
wc = 2*pi*7;
|
|
|
|
%% System's response at the decoupling frequency
|
|
H1 = evalfr(Gr, j*wc);
|
|
|
|
%% Real approximation of G(j.wc)
|
|
D = pinv(real(H1'*H1));
|
|
H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
|
|
|
|
[U,S,V] = svd(H1);
|
|
|
|
Gsvdr = inv(U)*Gr*inv(V');
|
|
#+end_src
|
|
|
|
The obtained plant is shown in Figure ref:fig:detail_control_svd_plant_spurious.
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
|
|
% Magnitude
|
|
hold on;
|
|
for i_in = 1:3
|
|
for i_out = [i_in+1:3]
|
|
plot(freqs, abs(squeeze(freqresp(Gsvdr(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
end
|
|
plot(freqs, abs(squeeze(freqresp(Gsvdr(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
|
'DisplayName', '$G_{svd,r}(i,j)\ i \neq j$');
|
|
set(gca,'ColorOrderIndex',1)
|
|
for i_in_out = 1:3
|
|
plot(freqs, abs(squeeze(freqresp(Gsvdr(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{svd,r}(%d,%d)$', i_in_out, i_in_out));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
|
ylim([1e-8, 1e-2]);
|
|
legend('location', 'northeast');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_svd_plant_spurious.pdf', 'width', 'wide', 'height', 'normal');
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_svd_plant_spurious
|
|
#+caption: SVD decoupled plant including a spurious resonance $G_{svd,r}(s)$
|
|
#+RESULTS:
|
|
[[file:figs/detail_control_svd_plant_spurious.png]]
|
|
|
|
*** Analytical formula for complementary filters
|
|
#+begin_src matlab
|
|
%% 1st order complementary filters
|
|
% Declaration of the symbolic variables
|
|
syms w0 Ts s z
|
|
|
|
% Bilinear Transform
|
|
s = 2/Ts*(z - 1)/(z + 1);
|
|
|
|
% First Order Low Pass Filter
|
|
Hl = 1/(1 + s/w0);
|
|
|
|
% Get numerator and denominator
|
|
[N,D] = numden(Hl);
|
|
|
|
% Extract coefficients (from z^0 to z^n)
|
|
num = coeffs(N, z);
|
|
den = coeffs(D, z);
|
|
|
|
sprintf('Hl: num = (%s) + (%s) * z^-1', num(2), num(1))
|
|
sprintf('Hl: den = (%s) + (%s) * z^-1', den(2), den(1))
|
|
|
|
% First Order Low Pass Filter
|
|
Hh = (s/w0)/(1 + s/w0);
|
|
|
|
% Get numerator and denominator
|
|
[N,D] = numden(Hh);
|
|
|
|
% Extract coefficients (from z^0 to z^n)
|
|
num = coeffs(N, z);
|
|
den = coeffs(D, z);
|
|
|
|
sprintf('Hh: num = (%s) + (%s) * z^-1', num(2), num(1))
|
|
sprintf('Hh: den = (%s) + (%s) * z^-1', den(2), den(1))
|
|
#+end_src
|
|
|
|
*** Response time to change of reference signal
|
|
For the nominal system, the model is accurate and the transfer function from reference signal $r$ to output $y$ is $1$ eqref:eq:detail_control_cf_cl_performance_y and does not depends of the complementary filters.
|
|
|
|
However, one can add a pre-filter as shown in Figure ref:fig:detail_control_cf_arch_class_prefilter.
|
|
|
|
#+begin_src latex :file detail_control_cf_arch_class_prefilter.pdf
|
|
\tikzset{block/.default={0.8cm}{0.6cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\def\cdist{0.7}
|
|
|
|
\begin{tikzpicture}
|
|
\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
|
|
\node[block, left=of addfb] (Kr){$K_r$};
|
|
\node[block, right=of addfb] (K){$K$};
|
|
\node[block, right=of K] (G){$G^\prime$};
|
|
\node[addb={+}{}{}{}{}, right=of G] (adddy){};
|
|
\node[addb={+}{}{}{}{}, below right=0.7 and 0.3 of adddy] (addn) {};
|
|
\node[block, left=of addn] (Hl) {$H_L$};
|
|
|
|
\draw[->] (addfb.east) -- (K.west) node[above left]{};
|
|
\draw[->] (K.east) -- (G.west) node[above left]{$u$};
|
|
\draw[->] (G.east) -- (adddy.west);
|
|
\draw[<-] (addn.east) -- ++(\cdist, 0) coordinate[](endpos) node[above left]{$n$};
|
|
\draw[->] (G-|addn)node[branch]{} -- (addn.north);
|
|
\draw[->] (adddy.east) -- (G-|endpos) node[above left]{$y$};
|
|
\draw[<-] (Kr.west) -- ++(-\cdist, 0) node[above right]{$r$};
|
|
\draw[->] (Kr.east) -- (addfb.west);
|
|
\draw[->] (addn.west) -- (Hl.east);
|
|
\draw[->] (Hl.west) -| (addfb.south);
|
|
\draw[<-] (adddy.north) -- ++(0, \cdist) node[below right]{$d_y$};
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_cf_arch_class_prefilter
|
|
#+caption: Prefilter used to limit input usage
|
|
#+RESULTS:
|
|
[[file:figs/detail_control_cf_arch_class_prefilter.png]]
|
|
|
|
The transfer function from $y$ to $r$ becomes $\frac{y}{r} = K_r$ and $K_r$ can we chosen to obtain acceptable response to change of the reference signal.
|
|
Typically, $K_r$ is a low pass filter of the form
|
|
\begin{equation}
|
|
K_r(s) = \frac{1}{1 + \tau s}
|
|
\end{equation}
|
|
with $\tau$ corresponding to the desired response time.
|
|
|
|
*** Input usage
|
|
Input usage due to disturbances $d_y$ and measurement noise $n$ is determined by $\big|\frac{u}{d_y}\big| = \big|\frac{u}{n}\big| = \big|G^{-1}H_L\big|$.
|
|
Thus it can be limited by setting an upper bound on $|H_L|$.
|
|
|
|
|
|
Input usage due to reference signal $r$ is determined by $\big|\frac{u}{r}\big| = \big|G^{-1} K_r\big|$ when using a pre-filter (Figure ref:fig:detail_control_cf_arch_class_prefilter) and $\big|\frac{u}{r}\big| = \big|G^{-1}\big|$ otherwise.
|
|
|
|
Proper choice of $|K_r|$ is then useful to limit input usage due to change of reference signal.
|
|
|
|
** DONE [#A] Fix the outline
|
|
CLOSED: [2025-04-03 Thu 12:01]
|
|
|
|
Lot is already discussed in:
|
|
- A5 [[file:~/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A5-simscape-nano-hexapod/simscape-nano-hexapod.org::*Control of Stewart Platforms][Control of Stewart Platforms]]
|
|
- Some will be discussed in the geometry optimization (cubic configuration / decoupling / ...)
|
|
|
|
- [ ] Maybe do not write this section at all?
|
|
|
|
** DONE [#B] Papers to be included here
|
|
CLOSED: [2025-04-03 Thu 12:01]
|
|
|
|
- [X] H-infinity synthesis of complementary filters:
|
|
[[file:~/Cloud/research/papers/published/dehaeze19_compl_filter_shapin_using_synth/index.org][file:~/Cloud/research/papers/published/dehaeze19_compl_filter_shapin_using_synth/index.org]]
|
|
Or newer version: [[file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/journal/dehaeze21_desig_compl_filte.org][file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/journal/dehaeze21_desig_compl_filte.org]]
|
|
- [X] Virtual sensor fusion
|
|
[[file:~/Cloud/research/papers/published/verma19_virtu_senso_fusio_high_preci_contr/elsarticle-template-harv.tex][file:~/Cloud/research/papers/published/verma19_virtu_senso_fusio_high_preci_contr/elsarticle-template-harv.tex]]
|
|
Maybe not this one.
|
|
- [X] Feedback control based on complementary filters:
|
|
[[file:~/Cloud/research/papers/dehaeze20_virtu_senso_fusio/index.org][file:~/Cloud/research/papers/dehaeze20_virtu_senso_fusio/index.org]]
|
|
|
|
** DONE [#A] Copy Paper about Complementary Filter Design
|
|
CLOSED: [2025-04-03 Thu 15:25] SCHEDULED: <2025-04-03 Thu>
|
|
|
|
[[file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/journal/dehaeze21_desig_compl_filte.org][file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/journal/dehaeze21_desig_compl_filte.org]]
|
|
|
|
- [X] Copy Matlab Code: [[file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/matlab/dehaeze21_desig_compl_filte_matlab.org]]
|
|
- [X] Copy Tikz Code: [[file:~/Cloud/research/papers/dehaeze21_desig_compl_filte/tikz/dehaeze21_desig_compl_filte_tikz.org]]
|
|
- [X] Rework all labels:
|
|
- [X] sections
|
|
- [X] equations
|
|
- [X] tables
|
|
|
|
** DONE [#A] Copy paper about closed-loop control with complementary filters
|
|
CLOSED: [2025-04-03 Thu 16:30] SCHEDULED: <2025-04-03 Thu>
|
|
|
|
file:~/Cloud/research/papers/dehaeze20_virtu_senso_fusio/index.org
|
|
|
|
- [X] Copy Content
|
|
- [X] Change citation format
|
|
- [X] Copy Tikz figures
|
|
- [X] Copy Matlab Code
|
|
- [X] Rework all labels
|
|
- [X] sections
|
|
- [X] equations
|
|
- [X] tables
|
|
|
|
** DONE [#A] Copy paper about Decoupling Control
|
|
CLOSED: [2025-04-03 Thu 17:48] SCHEDULED: <2025-04-03 Thu>
|
|
|
|
file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org
|
|
Especially [[file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org::*SVD / Jacobian / Model decoupling comparison][SVD / Jacobian / Model decoupling comparison]]
|
|
|
|
- [X] Copy Content
|
|
- [X] Copy Tikz figures
|
|
- [X] Copy Matlab Code
|
|
- [X] Rework all labels
|
|
- [X] sections
|
|
- [X] equations
|
|
- [X] tables
|
|
|
|
** DONE [#A] Verify why SVD decomposition on the proposed example gives such good performance
|
|
CLOSED: [2025-04-05 Sat 17:43]
|
|
|
|
Could be due to symmetry in the system.
|
|
|
|
** DONE [#B] Rename figures/sections/equations to have more clear separation between the three sections
|
|
CLOSED: [2025-04-08 Tue 10:46]
|
|
|
|
Prefixes:
|
|
- =detail_control_sensor=
|
|
- =detail_control_decoupling=
|
|
- =detail_control_cf=
|
|
|
|
** DONE [#A] Finish writing "multiple sensor" control section
|
|
CLOSED: [2025-04-09 Wed 13:55] SCHEDULED: <2025-04-08 Tue>
|
|
|
|
** TODO [#B] Review of control for Stewart platforms?
|
|
|
|
[[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org::*Control][Control]]
|
|
|
|
Or html version: https://research.tdehaeze.xyz/stewart-simscape/docs/bibliography.html
|
|
|
|
** TODO [#C] Discuss different strategies?
|
|
|
|
- Robust control
|
|
- Adaptive control
|
|
- etc...
|
|
|
|
* Introduction :ignore:
|
|
|
|
When controlling a MIMO system (specifically parallel manipulator such as the Stewart platform?)
|
|
|
|
- [ ] *Should the quick review of Stewart platform control be here?*
|
|
In that case it should be possible to highlight three areas:
|
|
- use of multiple sensors
|
|
- decoupling strategy
|
|
- control optimization
|
|
|
|
Several considerations:
|
|
- Section ref:sec:detail_control_sensor: How to most effectively use/combine multiple sensors
|
|
- Section ref:sec:detail_control_decoupling: How to decouple a system
|
|
- Section ref:sec:detail_control_cf: How to design the controller
|
|
|
|
* Multiple Sensor Control
|
|
:PROPERTIES:
|
|
:HEADER-ARGS:matlab+: :tangle matlab/detail_control_1_complementary_filtering.m
|
|
:END:
|
|
<<sec:detail_control_sensor>>
|
|
|
|
** Introduction :ignore:
|
|
|
|
The literature review of Stewart platforms revealed a wide diversity of designs with various sensor and actuator configurations.
|
|
Control objectives (such as active damping, vibration isolation, or precise positioning) dictate specific sensor configurations.
|
|
The selection between inertial sensors, force sensors, or relative position sensors is primarily determined by the system's control requirements.
|
|
|
|
In cases where multiple control objectives must be achieved simultaneously, as is the case for the Nano Active Stabilization System (NASS) where the Stewart platform must both position the sample and provide isolation from micro-station vibrations, combining multiple sensors within the control architecture has been demonstrated to yield significant performance benefits.
|
|
From the literature, three principal approaches for combining sensors have been identified: High Authority Control-Low Authority Control (HAC-LAC), sensor fusion, and two-sensor control architectures.
|
|
|
|
#+begin_src latex :file detail_control_sensor_arch_hac_lac.pdf
|
|
\begin{tikzpicture}
|
|
% Blocs
|
|
\node[block={2.0cm}{2.0cm}] (P) {Plant};
|
|
\coordinate[] (input) at ($(P.south west)!0.5!(P.north west)$);
|
|
\coordinate[] (outputH) at ($(P.south east)!0.2!(P.north east)$);
|
|
\coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$);
|
|
|
|
\node[block, above=0.2 of P] (Klac) {$K_\text{LAC}$};
|
|
\node[addb, left=0.5 of input] (addF) {};
|
|
\node[block, left=0.6 of addF] (Khac) {$K_\text{HAC}$};
|
|
% \node[addb={+}{}{}{}{-}, left=0.5 of Khac] (subr) {};
|
|
|
|
% Connections and labels
|
|
\draw[->] (outputL) -- ++(0.5, 0) coordinate(eastlac) |- (Klac.east);
|
|
\node[above right] at (outputL){$x^\prime$};
|
|
\draw[->] (Klac.west) -| (addF.north);
|
|
\draw[->] (addF.east) -- (input) node[above left]{$u$};
|
|
|
|
% \draw[<-] (subr.west)node[above left]{$r$} -- ++(-0.5, 0);
|
|
% \draw[->] (outputH) -- ++(0.5, 0) -- ++(0, -1.0) -| (subr.south);
|
|
\draw[->] (outputH) -- ++(0.5, 0) -- ++(0, -0.7) -| ($(Khac.west)+(-0.5, 0)$) -- (Khac.west);
|
|
\node[above right] at (outputH){$x$};
|
|
% \draw[->] (subr.east) -- (Khac.west) node[above left]{$\epsilon$};
|
|
\draw[->] (Khac.east) node[above right]{$u^\prime$} -- (addF.west);
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={(Klac.north-|eastlac) (addF.west|-P.south)}, fill=black!20!white, draw, dashed, inner sep=4pt] (Pi) {};
|
|
% \node[anchor={north west}, align=left] at (Pi.north west){\scriptsize{Damped}\\\scriptsize{Plant}};
|
|
\node[above=0 of Pi]{\scriptsize{Damped Plant}};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+begin_src latex :file detail_control_sensor_arch_sensor_fusion.pdf
|
|
\begin{tikzpicture}
|
|
% Blocs
|
|
\node[block={2.0cm}{2.0cm}] (P) {Plant};
|
|
\coordinate[] (input) at ($(P.south west)!0.5!(P.north west)$);
|
|
\coordinate[] (outputH) at ($(P.south east)!0.2!(P.north east)$);
|
|
\coordinate[] (outputL) at ($(P.south east)!0.8!(P.north east)$);
|
|
\coordinate[] (outputSS) at ($(P.south east)!0.5!(P.north east)$);
|
|
|
|
\node[block, right=0.8 of outputH] (KH) {$H_\text{HPF}$};
|
|
\node[block, right=0.8 of outputL] (KL) {$H_\text{LPF}$};
|
|
\node[addb={+}{}{}{}{}, right=2.2 of outputSS] (addss) {};
|
|
\node[block, left=0.6 of input] (K) {$K_{ss}$};
|
|
|
|
% Connections and labels
|
|
\draw[->] (outputL) -- (KL.west);
|
|
\draw[->] (outputH) -- (KH.west);
|
|
\node[above right] at (outputL){$x_L$};
|
|
\node[above right] at (outputH){$x_H$};
|
|
\draw[->] (KL.east) -| (addss.north);
|
|
\draw[->] (KH.east) -| (addss.south);
|
|
\draw[->] (addss.east) -- ++(0.9, 0);
|
|
\draw[->] ($(addss.east) + (0.5, 0)$)node[branch]{}node[above]{$x_{ss}$} -- ++(0, -1.6) -| ($(K.west)+(-0.5, 0)$) -- (K.west);
|
|
\draw[->] (K.east) -- (input) node[above left]{$u$};
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={(KL.north west) (KH.south-|addss.east)}, fill=black!20!white, draw, dashed, inner sep=4pt] (Pss) {};
|
|
\node[above=0 of Pss]{\scriptsize{Sensor Fusion}};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+begin_src latex :file detail_control_sensor_arch_two_sensor_control.pdf
|
|
\begin{tikzpicture}
|
|
% Blocs
|
|
\node[block={2.0cm}{2.0cm}] (P) {Plant};
|
|
\coordinate[] (input) at ($(P.south west)!0.5!(P.north west)$);
|
|
\coordinate[] (output1) at ($(P.south east)!0.8!(P.north east)$);
|
|
\coordinate[] (output2) at ($(P.south east)!0.2!(P.north east)$);
|
|
|
|
\node[addb={+}{}{}{}{}, left=0.8 of input] (addF) {};
|
|
\coordinate[left= 1.0 of addF] (Ks);
|
|
\node[block] (K1) at (Ks|-output1) {$K_\text{1}$};
|
|
\node[block] (K2) at (Ks|-output2) {$K_\text{2}$};
|
|
|
|
\draw[->] (output1) -| ++(0.6, 1.2) -| ($(K1.west)+(-0.5, 0)$) -- (K1.west);
|
|
\draw[->] (output2) -| ++(0.6,-1.0) -| ($(K2.west)+(-0.5, 0)$) -- (K2.west);
|
|
|
|
\draw[->] (K1.east)node[above right]{$u_1$} -| (addF.north);
|
|
\draw[->] (K2.east)node[above right]{$u_2$} -| (addF.south);
|
|
\draw[->] (addF.east) -- (input) node[above left=0 and 0.2]{$u$};
|
|
|
|
\node[above right] at (output1){$x_1$};
|
|
\node[above right] at (output2){$x_2$};
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={(K1.north west) (K2.south-|addF.east)}, fill=black!20!white, draw, dashed, inner sep=4pt] (Pss) {};
|
|
\node[above=0 of Pss]{\scriptsize{Two-Sensor Control}};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_control_multiple_sensors
|
|
#+caption: Different control strategies when using multiple sensors. High Authority Control / Low Authority Control (\subref{fig:detail_control_sensor_arch_hac_lac}). Sensor Fusion (\subref{fig:detail_control_sensor_arch_sensor_fusion}). Two-Sensor Control (\subref{fig:detail_control_sensor_arch_two_sensor_control})
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_arch_hac_lac}HAC LAC}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_sensor_arch_hac_lac.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_arch_two_sensor_control}Two Sensor Control}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_sensor_arch_two_sensor_control.png]]
|
|
#+end_subfigure
|
|
|
|
\bigskip
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_arch_sensor_fusion}Sensor Fusion}
|
|
#+attr_latex: :options {0.95\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_sensor_arch_sensor_fusion.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
The HAC-LAC approach, implemented during the conceptual phase, employs a dual-loop control strategy in which two control loops utilize different sensors for distinct purposes (Figure ref:fig:detail_control_sensor_arch_hac_lac).
|
|
In [[cite:&li01_simul_vibrat_isolat_point_contr]], vibration isolation is provided by accelerometers collocated with the voice coil actuators, while external rotational sensors are utilized to achieve pointing control.
|
|
In [[cite:&geng95_intel_contr_system_multip_degree]], force sensors collocated with the magnetostrictive actuators are used for active damping using decentralized IFF, and subsequently accelerometers are employed for adaptive vibration isolation.
|
|
Similarly, in [[cite:&wang16_inves_activ_vibrat_isolat_stewar]], piezoelectric actuators with collocated force sensors are used in a decentralized manner to provide active damping while accelerometers are implemented in an adaptive feedback loop to suppress periodic vibrations.
|
|
In [[cite:&xie17_model_contr_hybrid_passiv_activ]], force sensors are integrated in the struts for decentralized force feedback while accelerometers fixed to the top platform are employed for centralized control.
|
|
|
|
The second approach, sensor fusion (illustrated in Figure ref:fig:detail_control_sensor_arch_sensor_fusion), involves filtering signals from two sensors using complementary filters[fn:detail_control_1] and summing them to create an improved sensor signal.
|
|
In [[cite:&hauge04_sensor_contr_space_based_six]], geophones (used at low frequency) are merged with force sensors (used at high frequency).
|
|
It is demonstrated that combining both sensors using sensor fusion can improve performance compared to using the individual sensors independently.
|
|
In [[cite:&tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip]], sensor fusion architecture is implemented with an accelerometer and a force sensor.
|
|
This implementation is shown to simultaneously achieve high damping of structural modes (through the force sensors) while maintaining very low vibration transmissibility (through the accelerometers).
|
|
|
|
In [[cite:&beijen14_two_sensor_contr_activ_vibrat]], the performance of sensor fusion is compared with the more general case of "two-sensor control" (illustrated in Figure ref:fig:detail_control_sensor_arch_two_sensor_control).
|
|
It is highlighted that "two-sensor control" provides greater control freedom, potentially enhancing performance.
|
|
In [[cite:&thayer02_six_axis_vibrat_isolat_system]], the use of force sensors and geophones is compared for vibration isolation purposes.
|
|
Geophones are shown to provide better isolation performance than load cells but suffer from poor robustness.
|
|
Conversely, the controller based on force sensors exhibited inferior performance (due to the presence of a pair of low frequency zeros), but demonstrated better robustness properties.
|
|
A "two-sensor control" approach was proven to perform better than controllers based on individual sensors while maintaining better robustness.
|
|
A Linear Quadratic Regulator (LQG) was employed to optimize the two-input/one-output controller.
|
|
|
|
Beyond these three main approaches, other control architectures have been proposed for different purposes.
|
|
In [[cite:&yang19_dynam_model_decoup_contr_flexib]], a first control loop utilizes force sensors and relative motion sensors to compensate for parasitic stiffness of the flexible joints.
|
|
Subsequently, the system is decoupled in the modal space (facilitated by the removal of parasitic stiffness) and accelerometers are employed for vibration isolation.
|
|
|
|
The HAC-LAC architecture was previously investigated during the conceptual phase and successfully implemented to validate the NASS concept, demonstrating excellent performance.
|
|
At the other end of the spectrum, the two-sensor approach yields greater control design freedom but introduces increased complexity in tuning, and thus was not pursued in this study.
|
|
This work instead focuses on sensor fusion, which represents a promising middle ground between the proven HAC-LAC approach and the more complex two-sensor control strategy.
|
|
|
|
A review of sensor fusion is first presented (Section ref:ssec:detail_control_sensor_review), followed by an examination of the fundamental theoretical concepts (Section ref:ssec:detail_control_sensor_fusion_requirements).
|
|
In this section, both the robustness of the fusion and the noise characteristics of the resulting "super sensor" are derived and expressed as functions of the complementary filters' norms.
|
|
A synthesis method for designing complementary filters that allow to shape their norms is proposed (Section ref:ssec:detail_control_sensor_hinf_method).
|
|
The investigation is then extended beyond the conventional two-sensor scenario, demonstrating how the proposed complementary filter synthesis can be generalized for applications requiring the fusion of three or more sensors (Section ref:ssec:detail_control_sensor_hinf_three_comp_filters).
|
|
|
|
** Review of Sensor Fusion
|
|
<<ssec:detail_control_sensor_review>>
|
|
|
|
Sensors used to measure physical quantities have two primary limitations: measurement accuracy which is compromised by various noise sources (including electrical noise from conditioning electronics), and limited measurement bandwidth.
|
|
Sensor fusion offers a solution to these limitations by combining multiple sensors [[cite:&bendat57_optim_filter_indep_measur_two]].
|
|
By strategically selecting sensors with complementary characteristics, a "super sensor" can be created that combines the advantages of each individual sensor.
|
|
|
|
|
|
Measuring a physical quantity using sensors is always subject to several limitations.
|
|
First, the accuracy of the measurement is affected by various noise sources, such as electrical noise from the conditioning electronics.
|
|
Second, the frequency range in which the measurement is relevant is bounded by the bandwidth of the sensor.
|
|
One way to overcome these limitations is to combine several sensors using a technique called "sensor fusion" [[cite:&bendat57_optim_filter_indep_measur_two]].
|
|
Fortunately, a wide variety of sensors exists, each with different characteristics.
|
|
By carefully selecting the sensors to be fused, a "super sensor" is obtained that combines the benefits of the individual sensors.
|
|
|
|
In some applications, sensor fusion is employed to increase measurement bandwidth [[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel;&zimmermann92_high_bandw_orien_measur_contr;&min15_compl_filter_desig_angle_estim]].
|
|
For instance, in [[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel]], the bandwidth of a position sensor is extended by fusing it with an accelerometer that provides high-frequency motion information.
|
|
In other applications, sensor fusion is utilized to obtain an estimate of the measured quantity with reduced noise [[cite:&hua05_low_ligo;&hua04_polyp_fir_compl_filter_contr_system;&plummer06_optim_compl_filter_their_applic_motion_measur;&robert12_introd_random_signal_applied_kalman]].
|
|
More recently, the fusion of sensors measuring different physical quantities has been proposed to enhance control properties [[cite:&collette15_sensor_fusion_method_high_perfor;&yong16_high_speed_vertic_posit_stage]].
|
|
In [[cite:&collette15_sensor_fusion_method_high_perfor]], an inertial sensor used for active vibration isolation is fused with a sensor collocated with the actuator to improve the stability margins of the feedback controller.
|
|
|
|
On top of Stewart platforms, practical applications of sensor fusion are numerous.
|
|
It is widely implemented for attitude estimation in autonomous vehicles such as unmanned aerial vehicles [[cite:&baerveldt97_low_cost_low_weigh_attit;&corke04_inert_visual_sensin_system_small_auton_helic;&jensen13_basic_uas]] and underwater vehicles [[cite:&pascoal99_navig_system_desig_using_time;&batista10_optim_posit_veloc_navig_filter_auton_vehic]].
|
|
Sensor fusion offers significant benefits for high-performance positioning control as demonstrated in [[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel;&zimmermann92_high_bandw_orien_measur_contr;&min15_compl_filter_desig_angle_estim;&yong16_high_speed_vertic_posit_stage]].
|
|
It has also been identified as a key technology for improving the performance of active vibration isolation systems [[cite:&tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip]].
|
|
Emblematic examples include the isolation stages of gravitational wave detectors [[cite:&collette15_sensor_fusion_method_high_perfor;&heijningen18_low]] such as those employed at LIGO [[cite:&hua05_low_ligo;&hua04_polyp_fir_compl_filter_contr_system]] and Virgo [[cite:&lucia18_low_frequen_optim_perfor_advan]].
|
|
|
|
Two principal methods are employed to perform sensor fusion: using complementary filters [[cite:&anderson53_instr_approac_system_steer_comput]] or using Kalman filtering [[cite:&brown72_integ_navig_system_kalman_filter]].
|
|
For sensor fusion applications, these methods share many relationships [[cite:&brown72_integ_navig_system_kalman_filter;&higgins75_compar_compl_kalman_filter;&robert12_introd_random_signal_applied_kalman;&fonseca15_compl]].
|
|
However, Kalman filtering requires assumptions about the probabilistic characteristics of sensor noise [[cite:&robert12_introd_random_signal_applied_kalman]], whereas complementary filters do not impose such requirements.
|
|
Furthermore, complementary filters offer advantages over Kalman filtering for sensor fusion through their general applicability, low computational cost [[cite:&higgins75_compar_compl_kalman_filter]], and intuitive nature, as their effects can be readily interpreted in the frequency domain.
|
|
|
|
A set of filters is considered complementary if the sum of their transfer functions equals one at all frequencies.
|
|
In early implementations of complementary filtering, analog circuits were used to physically realize the filters [[cite:&anderson53_instr_approac_system_steer_comput]].
|
|
While analog complementary filters remain in use today [[cite:&yong16_high_speed_vertic_posit_stage;&moore19_capac_instr_sensor_fusion_high_bandw_nanop]], digital implementation is now more common as it provides greater flexibility.
|
|
|
|
Various design methods have been developed to optimize complementary filters.
|
|
The most straightforward approach utilizes analytical formulas.
|
|
Depending on the application, these formulas may be of first order [[cite:&corke04_inert_visual_sensin_system_small_auton_helic;&yeh05_model_contr_hydraul_actuat_two;&yong16_high_speed_vertic_posit_stage]], second order [[cite:&baerveldt97_low_cost_low_weigh_attit;&stoten01_fusion_kinet_data_using_compos_filter;&jensen13_basic_uas]], or higher orders [[cite:&shaw90_bandw_enhan_posit_measur_using_measur_accel;&zimmermann92_high_bandw_orien_measur_contr;&stoten01_fusion_kinet_data_using_compos_filter;&collette15_sensor_fusion_method_high_perfor;&matichard15_seism_isolat_advan_ligo]].
|
|
|
|
Since the characteristics of the super sensor depend on the proper design of complementary filters [[cite:&dehaeze19_compl_filter_shapin_using_synth]], several optimization techniques have been developed.
|
|
Some approaches focus on finding optimal parameters for analytical formulas [[cite:&jensen13_basic_uas;&min15_compl_filter_desig_angle_estim;&fonseca15_compl]], while others employ convex optimization tools [[cite:&hua04_polyp_fir_compl_filter_contr_system;&hua05_low_ligo]] such as linear matrix inequalities [[cite:&pascoal99_navig_system_desig_using_time]].
|
|
As demonstrated in [[cite:&plummer06_optim_compl_filter_their_applic_motion_measur]], complementary filter design can be linked to the standard mixed-sensitivity control problem.
|
|
Consequently, the powerful tools developed for classical control theory can be applied to complementary filter design.
|
|
For example, in [[cite:&jensen13_basic_uas]], the two gains of a Proportional Integral (PI) controller are optimized to minimize super sensor noise.
|
|
|
|
All these complementary filter design methods share the common objective of creating a super sensor with desired characteristics, typically in terms of noise and dynamics.
|
|
As reported in [[cite:&zimmermann92_high_bandw_orien_measur_contr;&plummer06_optim_compl_filter_their_applic_motion_measur]], phase shifts and magnitude bumps in the super sensor dynamics may occur if complementary filters are poorly designed or if sensors are improperly calibrated.
|
|
Therefore, the robustness of the fusion must be considered when designing complementary filters.
|
|
Despite the numerous design methods proposed in the literature, a simple approach that specifies desired super sensor characteristics while ensuring good fusion robustness has been lacking.
|
|
|
|
Fortunately, both fusion robustness and super sensor characteristics can be linked to complementary filter magnitude [[cite:&dehaeze19_compl_filter_shapin_using_synth]].
|
|
Based on this relationship, the present work introduces an approach to designing complementary filters using $\mathcal{H}_\infty$ synthesis, which enables intuitive shaping of complementary filter magnitude in a straightforward manner.
|
|
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :noweb yes
|
|
<<m-init-path>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :eval no :noweb yes
|
|
<<m-init-path-tangle>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :noweb yes
|
|
<<m-init-other>>
|
|
|
|
%% Initialize Frequency Vector
|
|
freqs = logspace(-1, 3, 1000);
|
|
#+end_src
|
|
|
|
** Sensor Fusion and Complementary Filters Requirements
|
|
<<ssec:detail_control_sensor_fusion_requirements>>
|
|
**** Sensor Fusion Architecture :ignore:
|
|
|
|
A general sensor fusion architecture using complementary filters is shown in Figure ref:fig:detail_control_sensor_fusion_overview, where multiple sensors (in this case two) measure the same physical quantity $x$.
|
|
The sensor output signals $\hat{x}_1$ and $\hat{x}_2$ represent estimates of $x$.
|
|
These estimates are filtered by complementary filters and combined to form a new estimate $\hat{x}$.
|
|
|
|
#+begin_src latex :file detail_control_sensor_fusion_overview.pdf
|
|
\tikzset{block/.default={0.8cm}{0.8cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\begin{tikzpicture}
|
|
\node[branch] (x) at (0, 0);
|
|
\node[block, above right=0.3 and 0.5 of x](sensor1){Sensor 1};
|
|
\node[block, below right=0.3 and 0.5 of x](sensor2){Sensor 2};
|
|
|
|
\node[block, right=1.1 of sensor1](H1){$H_1(s)$};
|
|
\node[block, right=1.1 of sensor2](H2){$H_2(s)$};
|
|
|
|
\node[addb, right=5.0 of x](add){};
|
|
|
|
\draw[] ($(x)+(-0.7, 0)$) node[above right]{$x$} -- (x.center);
|
|
\draw[->] (x.center) |- (sensor1.west);
|
|
\draw[->] (x.center) |- (sensor2.west);
|
|
\draw[->] (sensor1.east) -- node[midway, above]{$\hat{x}_1$} (H1.west);
|
|
\draw[->] (sensor2.east) -- node[midway, above]{$\hat{x}_2$} (H2.west);
|
|
\draw[->] (H1) -| (add.north);
|
|
\draw[->] (H2) -| (add.south);
|
|
\draw[->] (add.east) -- ++(0.9, 0) node[above left]{$\hat{x}$};
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={($(H2.south-|x) + (0, -0.2)$) ($(H1.north-|add.east) + (0.2, 0.6)$)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
|
\node[below] at (supersensor.north) {Super Sensor};
|
|
|
|
\node[fit={(sensor2.south west) (sensor1.north east)}, fill=black!20!white, draw, inner sep=6pt] (sensors) {};
|
|
\node[align=center] at (sensors.center) {{\tiny Normalized}\\[-0.5em]{\tiny Sensors}};
|
|
|
|
\node[fit={(H2.south west) (H1.north-|add.east)}, fill=black!20!white, draw, inner sep=6pt] (filters) {};
|
|
\node[align=center] at ($(filters.center) + (-0.3, 0)$) {{\tiny Complementary}\\[-0.5em]{\tiny Filters}};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_sensor_fusion_overview
|
|
#+caption: Schematic of a sensor fusion architecture using complementary filters.
|
|
#+RESULTS:
|
|
[[file:figs/detail_control_sensor_fusion_overview.png]]
|
|
|
|
The complementary property of filters $H_1(s)$ and $H_2(s)$ requires that the sum of their transfer functions equals one at all frequencies eqref:eq:detail_control_sensor_comp_filter.
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_comp_filter}
|
|
H_1(s) + H_2(s) = 1
|
|
\end{equation}
|
|
|
|
**** Sensor Models and Sensor Normalization
|
|
|
|
To analyze sensor fusion architectures, appropriate sensor models are required.
|
|
The model shown in Figure ref:fig:detail_control_sensor_model consists of a linear time invariant (LTI) system $G_i(s)$ representing the sensor dynamics and an input $n_i$ representing sensor noise.
|
|
The model input $x$ is the measured physical quantity, and its output $\tilde{x}_i$ is the "raw" output of the sensor.
|
|
|
|
Prior to filtering the sensor outputs $\tilde{x}_i$ with complementary filters, the sensors are typically normalized to simplify the fusion process.
|
|
This normalization involves using an estimate $\hat{G}_i(s)$ of the sensor dynamics $G_i(s)$, and filtering the sensor output by the inverse of this estimate $\hat{G}_i^{-1}(s)$, as shown in Figure ref:fig:detail_control_sensor_model_calibrated.
|
|
It is assumed that the sensor inverse $\hat{G}_i^{-1}(s)$ is proper and stable.
|
|
This approach ensures that the units of the estimates $\hat{x}_i$ match the units of the physical quantity $x$.
|
|
The sensor dynamics estimate $\hat{G}_i(s)$ may be a simple gain or a more complex transfer function.
|
|
|
|
#+begin_src latex :file detail_control_sensor_model.pdf
|
|
\tikzset{block/.default={0.8cm}{0.8cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\begin{tikzpicture}
|
|
\node[addb](add1){};
|
|
\node[block, right=0.8 of add1](G1){$G_i(s)$};
|
|
|
|
\draw[->] ($(add1.west)+(-0.7, 0)$) node[above right]{$x$} -- (add1.west);
|
|
\draw[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){$n_i$};
|
|
\draw[->] (add1.east) -- (G1.west);
|
|
\draw[->] (G1.east) -- ++(0.7, 0) node[above left]{$\tilde{x}_i$};
|
|
|
|
\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};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+begin_src latex :file detail_control_sensor_model_calibrated.pdf
|
|
\tikzset{block/.default={0.8cm}{0.8cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\begin{tikzpicture}
|
|
\node[addb](add1){};
|
|
\node[block, right=0.8 of add1](G1){$G_i(s)$};
|
|
\node[block, right=0.8 of G1](G1inv){$\hat{G}_i^{-1}(s)$};
|
|
|
|
\draw[->] ($(add1.west)+(-0.7, 0)$) node[above right]{$x$} -- (add1.west);
|
|
\draw[<-] (add1.north) -- ++(0, 0.7)node[below right](n1){$n_i$};
|
|
\draw[->] (add1.east) -- (G1.west);
|
|
\draw[->] (G1.east) -- (G1inv.west) node[above left]{$\tilde{x}_i$};
|
|
\draw[->] (G1inv.east) -- ++(0.8, 0) node[above left]{$\hat{x}_i$};
|
|
|
|
\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, align=right] at (sensor1cal.north east) {{\tiny Normalized}\\[-0.5em]{\tiny sensor}};
|
|
|
|
\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};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_sensor_models
|
|
#+caption: Sensor models with and without normalization.
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_model}Basic sensor model consisting of a noise input $n_i$ and a linear time invariant transfer function $G_i(s)$}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_sensor_model.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_model_calibrated}Normalized sensors using the inverse of an estimate $\hat{G}}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_sensor_model_calibrated.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
Two normalized sensors are then combined to form a super sensor as shown in Figure ref:fig:detail_control_sensor_fusion_super_sensor.
|
|
The two sensors measure the same physical quantity $x$ with dynamics $G_1(s)$ and $G_2(s)$, and with uncorrelated noises $n_1$ and $n_2$.
|
|
The signals from both normalized sensors are fed into two complementary filters $H_1(s)$ and $H_2(s)$ and then combined to yield an estimate $\hat{x}$ of $x$.
|
|
The super sensor output $\hat{x}$ is therefore described by eqref:eq:detail_control_sensor_comp_filter_estimate.
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_comp_filter_estimate}
|
|
\hat{x} = \Big( H_1(s) \hat{G}_1^{-1}(s) G_1(s) + H_2(s) \hat{G}_2^{-1}(s) G_2(s) \Big) x + H_1(s) \hat{G}_1^{-1}(s) G_1(s) n_1 + H_2(s) \hat{G}_2^{-1}(s) G_2(s) n_2
|
|
\end{equation}
|
|
|
|
#+begin_src latex :file detail_control_sensor_fusion_super_sensor.pdf
|
|
\tikzset{block/.default={0.8cm}{0.8cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\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){};
|
|
|
|
\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]{$\hat{x}$};
|
|
|
|
\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=colorblue!20!white, draw, inner sep=6pt] (sensor1cal) {};
|
|
\node[below left, align=right] at (sensor1cal.north east) {{\tiny Normalized}\\[-0.5em]{\tiny sensor}};
|
|
\node[fit={(add1.west |- G1.south) (n1.north -| G1.east)}, fill=colorblue!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=colorred!20!white, draw, inner sep=6pt] (sensor2cal) {};
|
|
\node[below left, align=right] at (sensor2cal.north east) {{\tiny Normalized}\\[-0.5em]{\tiny sensor}};
|
|
\node[fit={(add2.west |- G2.south) (n2.north -| G2.east)}, fill=colorred!30!white, draw, inner sep=3pt] (sensor2) {};
|
|
\node[below left] at (sensor2.north east) {Sensor 2};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_sensor_fusion_super_sensor
|
|
#+caption: Sensor fusion architecture with two normalized sensors.
|
|
#+RESULTS:
|
|
[[file:figs/detail_control_sensor_fusion_super_sensor.png]]
|
|
|
|
**** Noise Sensor Filtering
|
|
|
|
First, consider the case where all sensors are perfectly normalized eqref:eq:detail_control_sensor_perfect_dynamics.
|
|
The effects of imperfect normalization will be addressed subsequently.
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_perfect_dynamics}
|
|
\frac{\hat{x}_i}{x} = \hat{G}_i(s) G_i(s) = 1
|
|
\end{equation}
|
|
|
|
In that case, the super sensor output $\hat{x}$ equals $x$ plus the filtered noise from both sensors eqref:eq:detail_control_sensor_estimate_perfect_dyn.
|
|
From this equation, it is evident that the complementary filters $H_1(s)$ and $H_2(s)$ operate solely on the sensor noise.
|
|
Thus, this sensor fusion architecture allows filtering of sensor noise without introducing distortion in the measured physical quantity.
|
|
This fundamental property necessitates that the two filters must be complementary.
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_estimate_perfect_dyn}
|
|
\hat{x} = x + H_1(s) n_1 + H_2(s) n_2
|
|
\end{equation}
|
|
|
|
The estimation error $\epsilon_x$, defined as the difference between the sensor output $\hat{x}$ and the measured quantity $x$, is computed for the super sensor eqref:eq:detail_control_sensor_estimate_error.
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_estimate_error}
|
|
\epsilon_x \triangleq \hat{x} - x = H_1(s) n_1 + H_2(s) n_2
|
|
\end{equation}
|
|
|
|
As shown in eqref:eq:detail_control_sensor_noise_filtering_psd, the Power Spectral Density (PSD) of the estimation error $\Phi_{\epsilon_x}$ depends both on the norm of the two complementary filters and on the PSD of the noise sources $\Phi_{n_1}$ and $\Phi_{n_2}$.
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_noise_filtering_psd}
|
|
\Phi_{\epsilon_x}(\omega) = \left|H_1(j\omega)\right|^2 \Phi_{n_1}(\omega) + \left|H_2(j\omega)\right|^2 \Phi_{n_2}(\omega)
|
|
\end{equation}
|
|
|
|
If the two sensors have identical noise characteristics ($\Phi_{n_1}(\omega) = \Phi_{n_2}(\omega)$), simple averaging ($H_1(s) = H_2(s) = 0.5$) would minimize the super sensor noise.
|
|
This represents the simplest form of sensor fusion using complementary filters.
|
|
|
|
However, sensors typically exhibit high noise levels in different frequency regions.
|
|
In such cases, to reduce the noise of the super sensor, the norm $|H_1(j\omega)|$ should be minimized when $\Phi_{n_1}(\omega)$ exceeds $\Phi_{n_2}(\omega)$, and the norm $|H_2(j\omega)|$ should be minimized when $\Phi_{n_2}(\omega)$ exceeds $\Phi_{n_1}(\omega)$.
|
|
Therefore, by appropriately shaping the norm of the complementary filters, the noise of the super sensor can be minimized.
|
|
|
|
**** Sensor Fusion Robustness
|
|
|
|
In practical systems, sensor normalization is rarely perfect, and condition eqref:eq:detail_control_sensor_perfect_dynamics is not fully satisfied.
|
|
|
|
To analyze such imperfections, a multiplicative input uncertainty is incorporated into the sensor dynamics (Figure ref:fig:detail_control_sensor_model_uncertainty).
|
|
The nominal model is the estimated model used for normalization $\hat{G}_i(s)$, $\Delta_i(s)$ is any stable transfer function satisfying $|\Delta_i(j\omega)| \le 1,\ \forall\omega$, and $w_i(s)$ is a weighting transfer function representing the magnitude of uncertainty.
|
|
The weight $w_i(s)$ is selected such that the actual sensor dynamics $G_i(j\omega)$ remains within the uncertain region represented by a circle in the complex plane, centered on $1$ with a radius equal to $|w_i(j\omega)|$.
|
|
|
|
Since the nominal sensor dynamics is taken as the normalized filter, the normalized sensor model can be further simplified as shown in Figure ref:fig:detail_control_sensor_model_uncertainty_simplified.
|
|
|
|
#+begin_src latex :file detail_control_sensor_model_uncertainty.pdf
|
|
\tikzset{block/.default={0.8cm}{0.8cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\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, align=right] at (sensor1cal.north east) {{\tiny Normalized}\\[-0.5em]{\tiny sensor}};
|
|
|
|
\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};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+begin_src latex :file detail_control_sensor_model_uncertainty_simplified.pdf
|
|
\tikzset{block/.default={0.8cm}{0.8cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\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[->] ($(input)+(-0.8, 0)$) 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, 0.6)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) ($(delta1.north -| addn.east) + (0.1, 0)$)}, fill=black!10!white, draw, inner sep=6pt] (sensor1cal) {};
|
|
\node[below left, align=right] at (sensor1cal.north east) {{\tiny Normalized}\\[-0.5em]{\tiny sensor}};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_sensor_models_uncertainty
|
|
#+caption: Sensor models with dynamical uncertainty
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_model_uncertainty}Sensor with multiplicative input uncertainty}
|
|
#+attr_latex: :options {0.58\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :width 0.95\linewidth
|
|
[[file:figs/detail_control_sensor_model_uncertainty.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_model_uncertainty_simplified}Simplified sensor model}
|
|
#+attr_latex: :options {0.38\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :width 0.95\linewidth
|
|
[[file:figs/detail_control_sensor_model_uncertainty_simplified.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
The sensor fusion architecture incorporating sensor models with dynamical uncertainty is illustrated in Figure ref:fig:detail_control_sensor_fusion_dynamic_uncertainty.
|
|
The super sensor dynamics eqref:eq:detail_control_sensor_super_sensor_dyn_uncertainty is no longer unity but depends on the sensor dynamical uncertainty weights $w_i(s)$ and the complementary filters $H_i(s)$.
|
|
The dynamical uncertainty of the super sensor can be graphically represented in the complex plane by a circle centered on $1$ with a radius equal to $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$ (Figure ref:fig:detail_control_sensor_uncertainty_set_super_sensor).
|
|
|
|
|
|
The sensor fusion architecture with the sensor models including dynamical uncertainty is shown in Figure ref:fig:detail_control_sensor_fusion_dynamic_uncertainty.
|
|
The super sensor dynamics eqref:eq:detail_control_sensor_super_sensor_dyn_uncertainty is no longer equal to $1$ and now depends on the sensor dynamical uncertainty weights $w_i(s)$ as well as on the complementary filters $H_i(s)$.
|
|
The dynamical uncertainty of the super sensor can be graphically represented in the complex plane by a circle centered on $1$ with a radius equal to $|w_1(j\omega) H_1(j\omega)| + |w_2(j\omega) H_2(j\omega)|$ (Figure ref:fig:detail_control_sensor_uncertainty_set_super_sensor).
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_super_sensor_dyn_uncertainty}
|
|
\frac{\hat{x}}{x} = 1 + w_1(s) H_1(s) \Delta_1(s) + w_2(s) H_2(s) \Delta_2(s)
|
|
\end{equation}
|
|
|
|
#+begin_src latex :file detail_control_sensor_fusion_dynamic_uncertainty.pdf
|
|
\tikzset{block/.default={0.8cm}{0.8cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\begin{tikzpicture}
|
|
\node[branch] (x) at (0, 0);
|
|
|
|
\node[branch, above right=1.0 and 0.3 of x] (input1) {};
|
|
\node[branch, below right=1.0 and 0.3 of x] (input2) {};
|
|
\node[block, above right= 0.4 and 0.3 of input1](W1){$w_1(s)$};
|
|
\node[block, above right= 0.4 and 0.3 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, 0.6)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, 0.6)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=colorblue!20!white, draw, inner sep=6pt] (sensor1cal) {};
|
|
\node[below left, align=right] at (sensor1cal.north east) {{\tiny Normalized}\\[-0.5em]{\tiny sensor 1}};
|
|
|
|
\node[fit={(input2.west |- addu2.south) ($(delta2.north -| addn1.east) + (0.1, 0.0)$)}, fill=colorred!20!white, draw, inner sep=6pt] (sensor2cal) {};
|
|
\node[below left, align=right] at (sensor2cal.north east) {{\tiny Normalized}\\[-0.5em]{\tiny sensor 2}};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+begin_src latex :file detail_control_sensor_uncertainty_set_super_sensor.pdf
|
|
\tikzset{block/.default={0.8cm}{0.8cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\begin{tikzpicture}
|
|
\begin{scope}[shift={(4, 0)}]
|
|
|
|
% Uncertainty Circle
|
|
\node[draw, circle, fill=black!20!white, minimum size=3.6cm] (c) at (0, 0) {};
|
|
\path[draw, fill=colorblue!20!white] (0, 0) circle [radius=1.0];
|
|
\path[draw, fill=colorred!20!white] (135:1.0) circle [radius=0.8];
|
|
\path[draw, dashed] (0, 0) circle [radius=1.0];
|
|
|
|
% Center of Circle
|
|
\node[below] at (0, 0){$1$};
|
|
|
|
\draw[<->] (0, 0) node[branch]{} -- coordinate[midway](r1) ++(45:1.0);
|
|
\draw[<->] (135:1.0)node[branch]{} -- coordinate[midway](r2) ++(135:0.8);
|
|
|
|
\node[] (l1) at (2, 1.5) {$|w_1 H_1|$};
|
|
\draw[->, out=-90, in=0] (l1.south) to (r1);
|
|
|
|
\node[] (l2) at (-3.2, 1.2) {$|w_2 H_2|$};
|
|
\draw[->, out=0, in=-180] (l2.east) to (r2);
|
|
|
|
\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}
|
|
|
|
% 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]{$\Delta \phi_\text{max}$};
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_sensor_uncertainty
|
|
#+caption: Sensor fusion architecture with sensor dynamics uncertainty (\subref{fig:detail_control_sensor_fusion_dynamic_uncertainty}). Uncertainty region (\subref{fig:detail_control_sensor_uncertainty_set_super_sensor}) of the super sensor dynamics in the complex plane (grey circle). The contribution of both sensors 1 and 2 to the total uncertainty are represented respectively by a blue circle and a red circle. The frequency dependency $\omega$ is here omitted.
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_fusion_dynamic_uncertainty}Sensor Fusion Architecture}
|
|
#+attr_latex: :options {0.49\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :width 0.95\linewidth
|
|
[[file:figs/detail_control_sensor_fusion_dynamic_uncertainty.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_uncertainty_set_super_sensor}Uncertainty regions}
|
|
#+attr_latex: :options {0.49\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :width 0.95\linewidth
|
|
[[file:figs/detail_control_sensor_uncertainty_set_super_sensor.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
The super sensor dynamical uncertainty, and consequently the robustness of the fusion, clearly depends on the complementary filters' norm.
|
|
As it is generally desired to limit the dynamical uncertainty of the super sensor, the norm of the complementary filter $|H_i(j\omega)|$ should be made small when $|w_i(j\omega)|$ is large, i.e., at frequencies where the sensor dynamics is uncertain.
|
|
|
|
** Complementary Filters Shaping
|
|
<<ssec:detail_control_sensor_hinf_method>>
|
|
**** Introduction :ignore:
|
|
|
|
As established in Section ref:ssec:detail_control_sensor_fusion_requirements, the super sensor's noise characteristics and robustness are directly dependent on the complementary filters' norm.
|
|
A synthesis method enabling precise shaping of these norms would therefore offer substantial practical benefits.
|
|
This section develops such an approach by formulating the design objective as a standard $\mathcal{H}_\infty$ optimization problem.
|
|
The methodology for designing appropriate weighting functions (which specify desired complementary filter shapes during synthesis) is examined in detail, and the efficacy of the proposed method is validated with a simple example.
|
|
|
|
**** Synthesis Objective
|
|
|
|
The primary objective is to shape the norms of two filters $H_1(s)$ and $H_2(s)$ while ensuring they maintain their complementary property as defined in eqref:eq:detail_control_sensor_comp_filter.
|
|
This is equivalent to finding proper and stable transfer functions $H_1(s)$ and $H_2(s)$ that satisfy conditions eqref:eq:detail_control_sensor_hinf_cond_complementarity, eqref:eq:detail_control_sensor_hinf_cond_h1, and eqref:eq:detail_control_sensor_hinf_cond_h2.
|
|
Weighting transfer functions $W_1(s)$ and $W_2(s)$ are strategically selected to define the maximum desired norm of the complementary filters during the synthesis process.
|
|
|
|
\begin{subequations}\label{eq:detail_control_sensor_comp_filter_problem_form}
|
|
\begin{align}
|
|
& H_1(s) + H_2(s) = 1 \label{eq:detail_control_sensor_hinf_cond_complementarity} \\
|
|
& |H_1(j\omega)| \le \frac{1}{|W_1(j\omega)|} \quad \forall\omega \label{eq:detail_control_sensor_hinf_cond_h1} \\
|
|
& |H_2(j\omega)| \le \frac{1}{|W_2(j\omega)|} \quad \forall\omega \label{eq:detail_control_sensor_hinf_cond_h2}
|
|
\end{align}
|
|
\end{subequations}
|
|
|
|
**** Shaping of Complementary Filters using $\mathcal{H}_\infty$ synthesis
|
|
|
|
The synthesis objective can be readily expressed as a standard $\mathcal{H}_\infty$ optimization problem and solved using widely available computational tools.
|
|
Consider the generalized plant $P(s)$ illustrated in Figure ref:fig:detail_control_sensor_h_infinity_robust_fusion_plant and mathematically described by eqref:eq:detail_control_sensor_generalized_plant.
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_generalized_plant}
|
|
\begin{bmatrix} z_1 \\ z_2 \\ v \end{bmatrix} = P(s) \begin{bmatrix} w\\u \end{bmatrix}; \quad P(s) = \begin{bmatrix}W_1(s) & -W_1(s) \\ 0 & \phantom{+}W_2(s) \\ 1 & 0 \end{bmatrix}
|
|
\end{equation}
|
|
|
|
#+begin_src latex :file detail_control_sensor_h_infinity_robust_fusion_plant.pdf
|
|
\tikzset{block/.default={0.8cm}{0.8cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\begin{tikzpicture}
|
|
\node[block={4.0cm}{3.0cm}, fill=black!10!white] (P) {};
|
|
\node[above] at (P.north) {$P(s)$};
|
|
|
|
\node[block, below=0.2 of P, opacity=0] (H2) {$H_2(s)$};
|
|
|
|
\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[] (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) {};
|
|
|
|
\draw[->] (inputw) node[above right]{$w$} -- (sub.west);
|
|
\draw[->] (inputu) node[above right]{$u$} -- (W2.west);
|
|
\draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
|
|
\draw[->] (sub.east) -- (W1.west);
|
|
\draw[->] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- (outputv) node[above left]{$v$};
|
|
\draw[->] (W1.east) -- (output1)node[above left]{$z_1$};
|
|
\draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+begin_src latex :file detail_control_sensor_h_infinity_robust_fusion_fb.pdf
|
|
\tikzset{block/.default={0.8cm}{0.8cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\begin{tikzpicture}
|
|
\node[block={4.0cm}{3.0cm}, fill=black!10!white] (P) {};
|
|
\node[above] at (P.north) {$P(s)$};
|
|
|
|
\node[block, below=0.2 of P] (H2) {$H_2(s)$};
|
|
|
|
\coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west) + (-0.7, 0)$);
|
|
\coordinate[] (inputu) at ($(P.south west)!0.35!(P.north west) + (-0.4, 0)$);
|
|
|
|
\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 ($(P.south east)!0.1!(P.north east) + ( 0.4, 0)$);
|
|
|
|
\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) {};
|
|
|
|
\draw[->] (inputw) node[above right]{$w$} -- (sub.west);
|
|
\draw[->] (inputu-|sub) node[branch]{} -- (sub.south);
|
|
\draw[->] (sub.east) -- (W1.west);
|
|
\draw[->] ($(sub.west)+(-0.6, 0)$) node[branch]{} |- (outputv) |- (H2.east);
|
|
\draw[->] (H2.west) -| (inputu) -- (W2.west);
|
|
\draw[->] (W1.east) -- (output1)node[above left]{$z_1$};
|
|
\draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_sensor_h_infinity_robust_fusion
|
|
#+caption: Architecture for the $\mathcal{H}_\infty$ synthesis of complementary filters
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_h_infinity_robust_fusion_plant}Generalized plant}
|
|
#+attr_latex: :options {0.49\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_sensor_h_infinity_robust_fusion_plant.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_h_infinity_robust_fusion_fb}Generalized plant with the synthesized filter}
|
|
#+attr_latex: :options {0.49\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_sensor_h_infinity_robust_fusion_fb.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
Applying standard $\mathcal{H}_\infty$ synthesis to the generalized plant $P(s)$ is equivalent to finding a stable filter $H_2(s)$ that, based on input $v$, generates an output signal $u$ such that the $\mathcal{H}_\infty$ norm of the system shown in Figure ref:fig:detail_control_sensor_h_infinity_robust_fusion_fb from $w$ to $[z_1, \ z_2]$ does not exceed unity, as expressed in eqref:eq:detail_control_sensor_hinf_syn_obj.
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_hinf_syn_obj}
|
|
\left\|\begin{matrix} \left(1 - H_2(s)\right) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1
|
|
\end{equation}
|
|
|
|
By defining $H_1(s)$ as the complement of $H_2(s)$ (eqref:eq:detail_control_sensor_definition_H1), the $\mathcal{H}_\infty$ synthesis objective becomes equivalent to eqref:eq:detail_control_sensor_hinf_problem, ensuring that conditions eqref:eq:detail_control_sensor_hinf_cond_h1 and eqref:eq:detail_control_sensor_hinf_cond_h2 are satisfied.
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_definition_H1}
|
|
H_1(s) \triangleq 1 - H_2(s)
|
|
\end{equation}
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_hinf_problem}
|
|
\left\|\begin{matrix} H_1(s) W_1(s) \\ H_2(s) W_2(s) \end{matrix}\right\|_\infty \le 1
|
|
\end{equation}
|
|
|
|
Therefore, applying $\mathcal{H}_\infty$ synthesis to the standard plant $P(s)$ (eqref:eq:detail_control_sensor_generalized_plant) generates two filters, $H_2(s)$ and $H_1(s) \triangleq 1 - H_2(s)$, that are complementary as required by eqref:eq:detail_control_sensor_comp_filter_problem_form, with norms bounded by the specified constraints in eqref:eq:detail_control_sensor_hinf_cond_h1 and eqref:eq:detail_control_sensor_hinf_cond_h2.
|
|
|
|
It should be noted that there exists only an implication (not an equivalence) between the $\mathcal{H}_\infty$ norm condition in eqref:eq:detail_control_sensor_hinf_problem and the initial synthesis objectives in eqref:eq:detail_control_sensor_hinf_cond_h1 and eqref:eq:detail_control_sensor_hinf_cond_h2.
|
|
Consequently, the optimization may be somewhat conservative with respect to the set of filters on which it operates (see [[cite:&skogestad07_multiv_feedb_contr,Chap. 2.8.3]]).
|
|
|
|
**** Weighting Functions Design
|
|
|
|
Weighting functions play a crucial role during synthesis by specifying the maximum allowable norms for the complementary filters.
|
|
The proper design of these weighting functions is essential for the successful implementation of the proposed $\mathcal{H}_\infty$ synthesis approach.
|
|
|
|
Three key considerations should guide the design of weighting functions.
|
|
First, only proper and stable transfer functions should be employed.
|
|
Second, the order of the weighting functions should remain reasonably small to minimize computational costs associated with solving the optimization problem and to facilitate practical implementation of the filters (as the order of the synthesized filters equals the sum of the weighting functions' orders).
|
|
Third, the fundamental limitations imposed by the complementary property (eqref:eq:detail_control_sensor_comp_filter) must be respected, which implies that $|H_1(j\omega)|$ and $|H_2(j\omega)|$ cannot both be made small at the same frequency.
|
|
|
|
When designing complementary filters, it is typically desirable to specify their slopes, "blending" frequency, and maximum gains at low and high frequencies.
|
|
To facilitate the expression of these specifications, formula eqref:eq:detail_control_sensor_weight_formula is proposed for the design of weighting functions.
|
|
The parameters in this formula are $G_0 = \lim_{\omega \to 0} |W(j\omega)|$ (the low-frequency gain), $G_\infty = \lim_{\omega \to \infty} |W(j\omega)|$ (the high-frequency gain), $G_c = |W(j\omega_c)|$ (the gain at a specific frequency $\omega_c$ in $\si{rad/s}$), and $n$ (the slope between high and low frequency, which also corresponds to the order of the weighting function).
|
|
The typical magnitude response of a weighting function generated using eqref:eq:detail_control_sensor_weight_formula is illustrated in Figure ref:fig:detail_control_sensor_weight_formula.
|
|
|
|
#+begin_src matlab :exports none
|
|
%% Weighting Function Design
|
|
% Parameters
|
|
n = 3; w0 = 2*pi*10; G0 = 1e-3; G1 = 1e1; Gc = 2;
|
|
|
|
% Formulas
|
|
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;
|
|
|
|
% Function generateWF can be used to easily design the weighting filters
|
|
% W = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1e-3, 'Ginf', 10, 'Gc', 2);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results none
|
|
%% Magnitude of the weighting function with parameters
|
|
figure;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(W, freqs, 'Hz'))), 'k-');
|
|
|
|
plot([1e-3 1e0], [G0 G0], 'k--', 'LineWidth', 1)
|
|
text(1e0, G0, '$\quad G_0$')
|
|
|
|
plot([1e1 1e3], [G1 G1], 'k--', 'LineWidth', 1)
|
|
text(1e1,G1,'$G_{\infty}\quad$','HorizontalAlignment', 'right')
|
|
|
|
plot([w0/2/pi w0/2/pi], [1 2*Gc], 'k--', 'LineWidth', 1)
|
|
text(w0/2/pi,1,'$\omega_c$','VerticalAlignment', 'top', 'HorizontalAlignment', 'center')
|
|
|
|
plot([w0/2/pi/2 2*w0/2/pi], [Gc Gc], 'k--', 'LineWidth', 1)
|
|
text(w0/2/pi/2, Gc, '$G_c \quad$','HorizontalAlignment', 'right')
|
|
|
|
text(w0/5/pi/2, abs(evalfr(W, j*w0/5)), 'Slope: $n \quad$', 'HorizontalAlignment', 'right')
|
|
|
|
text(w0/2/pi, abs(evalfr(W, j*w0)), '$\bullet$', 'HorizontalAlignment', 'center')
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
|
hold off;
|
|
xlim([freqs(1), freqs(end)]);
|
|
ylim([5e-4, 20]);
|
|
yticks([1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1]);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_sensor_weight_formula.pdf', 'width', 'half', 'height', 'normal');
|
|
#+end_src
|
|
|
|
#+attr_latex: :options []{0.49\linewidth}
|
|
#+begin_minipage
|
|
#+name: fig:detail_control_sensor_weight_formula
|
|
#+caption: Magnitude of a weighting function generated using eqref:eq:detail_control_sensor_weight_formula, $G_0 = 10^{-3}$, $G_\infty = 10$, $\omega_c = \SI{10}{Hz}$, $G_c = 2$, $n = 3$.
|
|
#+attr_latex: :width 0.95\linewidth :float nil
|
|
[[file:figs/detail_control_sensor_weight_formula.png]]
|
|
#+end_minipage
|
|
\hfill
|
|
#+attr_latex: :options []{0.49\linewidth}
|
|
#+begin_minipage
|
|
\begin{equation}\label{eq:detail_control_sensor_weight_formula}
|
|
W(s) = \left( \frac{
|
|
\hfill{} \frac{1}{\omega_c} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{G_0}{G_c}\right)^{\frac{1}{n}}
|
|
}{
|
|
\left(\frac{1}{G_\infty}\right)^{\frac{1}{n}} \frac{1}{\omega_c} \sqrt{\frac{1 - \left(\frac{G_0}{G_c}\right)^{\frac{2}{n}}}{1 - \left(\frac{G_c}{G_\infty}\right)^{\frac{2}{n}}}} s + \left(\frac{1}{G_c}\right)^{\frac{1}{n}}
|
|
}\right)^n
|
|
\end{equation}
|
|
#+end_minipage
|
|
|
|
**** Validation of the proposed synthesis method
|
|
|
|
The proposed methodology for designing complementary filters is now applied to a simple example.
|
|
Consider the design of two complementary filters $H_1(s)$ and $H_2(s)$ with the following requirements:
|
|
- The blending frequency should be approximately $\SI{10}{Hz}$
|
|
- The slope of $|H_1(j\omega)|$ should be $+2$ below $\SI{10}{Hz}$, with a low-frequency gain of $10^{-3}$
|
|
- The slope of $|H_2(j\omega)|$ should be $-3$ above $\SI{10}{Hz}$, with a high-frequency gain of $10^{-3}$
|
|
|
|
The first step involves translating these requirements by appropriately designing the weighting functions.
|
|
The formula proposed in eqref:eq:detail_control_sensor_weight_formula is employed for this purpose.
|
|
The parameters used are summarized in Table ref:tab:detail_control_sensor_weights_params.
|
|
The inverse magnitudes of the designed weighting functions, which represent the maximum allowable norms of the complementary filters, are depicted by the dashed lines in Figure ref:fig:detail_control_sensor_hinf_filters_results.
|
|
|
|
#+begin_src matlab
|
|
%% Synthesis of Complementary Filters using H-infinity synthesis
|
|
% Design of the Weighting Functions
|
|
W1 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.45);
|
|
W2 = generateWF('n', 2, 'w0', 2*pi*10, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.45);
|
|
|
|
% Generalized Plant
|
|
P = [W1 -W1;
|
|
0 W2;
|
|
1 0];
|
|
|
|
% H-Infinity Synthesis
|
|
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
|
|
|
% Define H1 to be the complementary of H2
|
|
H1 = 1 - H2;
|
|
|
|
% The function generateCF can also be used to synthesize the complementary filters.
|
|
% [H1, H2] = generateCF(W1, W2);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results none
|
|
%% Bode plot of the Weighting filters and Obtained complementary filters
|
|
figure;
|
|
hold on;
|
|
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'color', colors(1,:),'DisplayName', '$|W_1|^{-1}$');
|
|
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'color', colors(2,:),'DisplayName', '$|W_2|^{-1}$');
|
|
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'color', [colors(1,:), 0.5], 'linewidth', 2.5,'DisplayName', '$H_1$');
|
|
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'color', [colors(2,:), 0.5], 'linewidth', 2.5,'DisplayName', '$H_2$');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
|
ylim([8e-4, 20]);
|
|
xlim([freqs(1), freqs(end)]);
|
|
leg = legend('location', 'south', 'FontSize', 8, 'NumColumns', 2);
|
|
leg.ItemTokenSize(1) = 18;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_sensor_hinf_filters_results.pdf', 'width', 'half', 'height', 400);
|
|
#+end_src
|
|
|
|
#+attr_latex: :options [b]{0.44\linewidth}
|
|
#+begin_minipage
|
|
#+attr_latex: :environment tabularx :width 0.7\linewidth :placement [b] :align ccc
|
|
#+attr_latex: :booktabs t :float nil
|
|
| Parameter | $W_1(s)$ | $W_2(s)$ |
|
|
|-------------+------------------+------------------|
|
|
| $G_0$ | $0.1$ | $1000$ |
|
|
| $G_{\infty}$ | $1000$ | $0.1$ |
|
|
| $\omega_c$ | $2 \pi \cdot 10$ | $2 \pi \cdot 10$ |
|
|
| $G_c$ | $0.45$ | $0.45$ |
|
|
| $n$ | $2$ | $3$ |
|
|
#+latex: \captionof{table}{\label{tab:detail_control_sensor_weights_params}Parameters for \(W_1(s)\) and \(W_2(s)\)}
|
|
#+end_minipage
|
|
\hfill
|
|
#+attr_latex: :options [b]{0.52\linewidth}
|
|
#+begin_minipage
|
|
#+name: fig:detail_control_sensor_hinf_filters_results
|
|
#+attr_latex: :scale 1 :float nil
|
|
#+caption: Weights and obtained filters
|
|
[[file:figs/detail_control_sensor_hinf_filters_results.png]]
|
|
#+end_minipage
|
|
|
|
Standard $\mathcal{H}_\infty$ synthesis is then applied to the generalized plant shown in Figure ref:fig:detail_control_sensor_h_infinity_robust_fusion_plant.
|
|
This yields the filter $H_2(s)$ that minimizes the $\mathcal{H}_\infty$ norm from input $w$ to outputs $[z_1,\ z_2]^{\intercal}$.
|
|
The resulting $\mathcal{H}_\infty$ norm is found to be close to unity, indicating successful synthesis: the norms of the complementary filters remain below the specified upper bounds.
|
|
This is confirmed by the Bode plots of the obtained complementary filters in Figure ref:fig:detail_control_sensor_hinf_filters_results.
|
|
This straightforward example demonstrates that the proposed methodology for shaping complementary filters is both simple and effective.
|
|
|
|
** Synthesis of a set of three complementary filters
|
|
<<ssec:detail_control_sensor_hinf_three_comp_filters>>
|
|
|
|
Certain applications necessitate the fusion of more than two sensors [[cite:&stoten01_fusion_kinet_data_using_compos_filter;&fonseca15_compl]].
|
|
At LIGO, for example, a super sensor is formed by merging three distinct sensors: an LVDT, a seismometer, and a geophone [[cite:&matichard15_seism_isolat_advan_ligo]].
|
|
|
|
For merging $n>2$ sensors with complementary filters, two architectural approaches are possible, as illustrated in Figure ref:fig:detail_control_sensor_fusion_three.
|
|
Fusion can be implemented either "sequentially," utilizing $n-1$ sets of two complementary filters (Figure ref:fig:detail_control_sensor_fusion_three_sequential), or "in parallel," employing a single set of $n$ complementary filters (Figure ref:fig:detail_control_sensor_fusion_three_parallel).
|
|
|
|
While conventional sensor fusion synthesis techniques can be applied to the sequential approach, parallel architecture implementation requires a novel synthesis method for multiple complementary filters.
|
|
Previous literature has offered only simple analytical formulas for this purpose [[cite:&stoten01_fusion_kinet_data_using_compos_filter;&fonseca15_compl]].
|
|
This section presents a generalization of the proposed complementary filter synthesis method to address this gap.
|
|
|
|
#+begin_src latex :file detail_control_sensor_fusion_three_sequential.pdf
|
|
\tikzset{block/.default={0.8cm}{0.8cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\begin{tikzpicture}
|
|
\node[branch] (x) at (0, 0);
|
|
|
|
\node[block, right=0.4 of x] (sensor2) {Sensor 2};
|
|
\node[block, above=0.4 of sensor2] (sensor1) {Sensor 1};
|
|
\node[block, below=0.4 of sensor2] (sensor3) {Sensor 3};
|
|
|
|
\node[block, right=1.1 of sensor1](H1){$H_1(s)$};
|
|
\node[block, right=1.1 of sensor2](H2){$H_2(s)$};
|
|
\node[addb] (add) at ($0.5*(H1.east)+0.5*(H2.east)+(0.6, 0)$){};
|
|
|
|
\node[block, right=0.8 of add](H1p) {$H_1^\prime(s)$};
|
|
\node[block] (H2p) at (H1p|-sensor3) {$H_2^\prime(s)$};
|
|
|
|
\node[addb] (addp) at ($0.5*(H1p.east)+0.5*(H2p.east)+(0.6, 0)$){};
|
|
|
|
\draw[->] ($(x)+(-0.8, 0)$) node[above right]{$x$} -- (sensor2.west);
|
|
\draw[->] (x.center) |- (sensor1.west);
|
|
\draw[->] (x.center) |- (sensor3.west);
|
|
\draw[->] (sensor1.east) -- (H1.west) node[above left]{$\hat{x}_1$};
|
|
\draw[->] (sensor2.east) -- (H2.west) node[above left]{$\hat{x}_2$};
|
|
\draw[->] (sensor3.east) -- (H2p.west) node[above left]{$\hat{x}_3$};
|
|
\draw[->] (H1) -| (add.north);
|
|
\draw[->] (H2) -| (add.south);
|
|
\draw[->] (add.east) -- (H1p.west) node[above left]{$\hat{x}_{12}$};
|
|
\draw[->] (H1p) -| (addp.north);
|
|
\draw[->] (H2p) -| (addp.south);
|
|
\draw[->] (addp.east) -- ++(0.8, 0) node[above left]{$\hat{x}$};
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={(x.west|-sensor3.south) (sensor1.north-|addp.east)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
|
|
|
\node[fit={(x.west|-sensor1.north) (add.east|-sensor2.south)}, fill=black!20!white, draw, inner sep=3pt] (superinertialsensor) {};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+begin_src latex :file detail_control_sensor_fusion_three_parallel.pdf
|
|
\tikzset{block/.default={0.8cm}{0.8cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\begin{tikzpicture}
|
|
\node[branch] (x) at (0, 0);
|
|
\node[block, right=0.4 of x] (sensor2) {Sensor 2};
|
|
\node[block, above=0.3 of sensor2] (sensor1) {Sensor 1};
|
|
\node[block, below=0.3 of sensor2] (sensor3) {Sensor 3};
|
|
|
|
\node[block, right=1.1 of sensor1](H1){$H_1(s)$};
|
|
\node[block, right=1.1 of sensor2](H2){$H_2(s)$};
|
|
\node[block, right=1.1 of sensor3](H3){$H_3(s)$};
|
|
|
|
\node[addb, right=0.6 of H2](add){};
|
|
|
|
\draw[->] (x.center) |- (sensor1.west);
|
|
\draw[] ($(x)+(-0.8, 0)$) node[above right]{$x$} -- (sensor2.west);
|
|
\draw[->] (x.center) |- (sensor3.west);
|
|
|
|
\draw[->] (sensor1.east) -- (H1.west) node[above left]{$\hat{x}_1$};
|
|
\draw[->] (sensor2.east) -- (H2.west) node[above left]{$\hat{x}_2$};
|
|
\draw[->] (sensor3.east) -- (H3.west) node[above left]{$\hat{x}_3$};
|
|
|
|
\draw[->] (H1) -| (add.north);
|
|
\draw[->] (H2) -- (add.west);
|
|
\draw[->] (H3) -| (add.south);
|
|
|
|
\draw[->] (add.east) -- ++(0.8, 0) node[above left]{$\hat{x}$};
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={(H3.south-|x) (H1.north-|add.east)}, fill=black!10!white, draw, inner sep=6pt] (supersensor) {};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_sensor_fusion_three
|
|
#+caption: Possible sensor fusion architecture when more than two sensors are to be merged
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_fusion_three_sequential}Sequential fusion}
|
|
#+attr_latex: :options {0.58\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 0.9
|
|
[[file:figs/detail_control_sensor_fusion_three_sequential.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_fusion_three_parallel}Parallel fusion}
|
|
#+attr_latex: :options {0.38\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 0.9
|
|
[[file:figs/detail_control_sensor_fusion_three_parallel.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
The synthesis objective is to compute a set of $n$ stable transfer functions $[H_1(s),\ H_2(s),\ \dots,\ H_n(s)]$ that satisfy conditions eqref:eq:detail_control_sensor_hinf_cond_compl_gen and eqref:eq:detail_control_sensor_hinf_cond_perf_gen.
|
|
|
|
\begin{subequations}\label{eq:detail_control_sensor_hinf_problem_gen}
|
|
\begin{align}
|
|
& \sum_{i=1}^n H_i(s) = 1 \label{eq:detail_control_sensor_hinf_cond_compl_gen} \\
|
|
& \left| H_i(j\omega) \right| < \frac{1}{\left| W_i(j\omega) \right|}, \quad \forall \omega,\ i = 1 \dots n \label{eq:detail_control_sensor_hinf_cond_perf_gen}
|
|
\end{align}
|
|
\end{subequations}
|
|
|
|
The transfer functions $[W_1(s),\ W_2(s),\ \dots,\ W_n(s)]$ are weights selected to specify the maximum complementary filters' norm during synthesis.
|
|
|
|
This synthesis objective is closely related to the one described in Section ref:ssec:detail_control_sensor_hinf_method, and the proposed synthesis method represents a generalization of the approach previously presented.
|
|
A set of $n$ complementary filters can be shaped by applying standard $\mathcal{H}_\infty$ synthesis to the generalized plant $P_n(s)$ described by eqref:eq:detail_control_sensor_generalized_plant_n_filters.
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_generalized_plant_n_filters}
|
|
\begin{bmatrix} z_1 \\ \vdots \\ z_n \\ v \end{bmatrix} = P_n(s) \begin{bmatrix} w \\ u_1 \\ \vdots \\ u_{n-1} \end{bmatrix}; \quad
|
|
P_n(s) = \begin{bmatrix}
|
|
W_1 & -W_1 & \dots & \dots & -W_1 \\
|
|
0 & W_2 & 0 & \dots & 0 \\
|
|
\vdots & \ddots & \ddots & \ddots & \vdots \\
|
|
\vdots & & \ddots & \ddots & 0 \\
|
|
0 & \dots & \dots & 0 & W_n \\
|
|
1 & 0 & \dots & \dots & 0
|
|
\end{bmatrix}
|
|
\end{equation}
|
|
|
|
If the synthesis is successful, a set of $n-1$ filters $[H_2(s),\ H_3(s),\ \dots,\ H_n(s)]$ is obtained such that eqref:eq:detail_control_sensor_hinf_syn_obj_gen is satisfied.
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_hinf_syn_obj_gen}
|
|
\left\|\begin{matrix} \left(1 - \left[ H_2(s) + H_3(s) + \dots + H_n(s) \right]\right) W_1(s) \\ H_2(s) W_2(s) \\ \vdots \\ H_n(s) W_n(s) \end{matrix}\right\|_\infty \le 1
|
|
\end{equation}
|
|
|
|
$H_1(s)$ is then defined using eqref:eq:detail_control_sensor_h1_comp_h2_hn, which ensures the complementary property for the set of $n$ filters eqref:eq:detail_control_sensor_hinf_cond_compl_gen.
|
|
Condition eqref:eq:detail_control_sensor_hinf_cond_perf_gen is satisfied through eqref:eq:detail_control_sensor_hinf_syn_obj_gen.
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_h1_comp_h2_hn}
|
|
H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) + \dots + H_n(s) \big]
|
|
\end{equation}
|
|
|
|
To validate the proposed method for synthesizing a set of three complementary filters, an example is provided.
|
|
The sensors to be merged are a displacement sensor (effective from DC up to $\SI{1}{Hz}$), a geophone (effective from $1$ to $\SI{10}{Hz}$), and an accelerometer (effective above $\SI{10}{Hz}$).
|
|
Three weighting functions are designed using formula eqref:eq:detail_control_sensor_weight_formula, and their inverse magnitudes are shown in Figure ref:fig:detail_control_sensor_three_complementary_filters_results (dashed curves).
|
|
|
|
Consider the generalized plant $P_3(s)$ shown in Figure ref:fig:detail_control_sensor_comp_filter_three_hinf_fb, which is also described by eqref:eq:detail_control_sensor_generalized_plant_three_filters.
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_generalized_plant_three_filters}
|
|
\begin{bmatrix} z_1 \\ z_2 \\ z_3 \\ v \end{bmatrix} = P_3(s) \begin{bmatrix} w \\ u_1 \\ u_2 \end{bmatrix}; \quad P_3(s) = \begin{bmatrix}W_1(s) & -W_1(s) & -W_1(s) \\ 0 & \phantom{+}W_2(s) & 0 \\ 0 & 0 & \phantom{+}W_3(s) \\ 1 & 0 & 0 \end{bmatrix}
|
|
\end{equation}
|
|
|
|
#+begin_src latex :file detail_control_sensor_comp_filter_three_hinf_gen_plant.pdf
|
|
\tikzset{block/.default={0.8cm}{0.8cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\begin{tikzpicture}
|
|
\node[block={5.0cm}{4.5cm}, fill=black!10!white] (P) {};
|
|
\node[above] at (P.north) {$P_3(s)$};
|
|
|
|
\coordinate[] (inputw) at ($(P.south west)!0.8!(P.north west) + (-0.7, 0)$);
|
|
\coordinate[] (inputu) at ($(P.south west)!0.4!(P.north west) + (-0.7, 0)$);
|
|
|
|
\coordinate[] (output1) at ($(P.south east)!0.8!(P.north east) + (0.7, 0)$);
|
|
\coordinate[] (output2) at ($(P.south east)!0.55!(P.north east) + (0.7, 0)$);
|
|
\coordinate[] (output3) at ($(P.south east)!0.3!(P.north east) + (0.7, 0)$);
|
|
\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[block, left=1.4 of output3] (W3){$W_3(s)$};
|
|
\node[addb={+}{}{}{}{-}, left=of W1] (sub1) {};
|
|
\node[addb={+}{}{}{}{-}, left=of sub1] (sub2) {};
|
|
|
|
\node[block, below=0.3 of P, opacity=0] (H) {$\begin{bmatrix}H_2(s) \\ H_3(s)\end{bmatrix}$};
|
|
|
|
\draw[->] (inputw) node[above right](w){$w$} -- (sub2.west);
|
|
\draw[->] (W3-|sub1)node[branch]{} -- (sub1.south);
|
|
\draw[->] (W2-|sub2)node[branch]{} -- (sub2.south);
|
|
\draw[->] ($(sub2.west)+(-0.5, 0)$) node[branch]{} |- (outputv);
|
|
\draw[->] (inputu|-W2) -- (W2.west);
|
|
\draw[->] (inputu|-W3) -- (W3.west);
|
|
|
|
\draw[->] (sub2.east) -- (sub1.west);
|
|
\draw[->] (sub1.east) -- (W1.west);
|
|
\draw[->] (W1.east) -- (output1)node[above left](z){$z_1$};
|
|
\draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
|
|
\draw[->] (W3.east) -- (output3)node[above left]{$z_3$};
|
|
\node[above] at (W2-|w){$u_1$};
|
|
\node[above] at (W3-|w){$u_2$};
|
|
\node[above] at (outputv-|z){$v$};
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+begin_src latex :file detail_control_sensor_comp_filter_three_hinf_fb.pdf
|
|
\tikzset{block/.default={0.8cm}{0.8cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\begin{tikzpicture}
|
|
\node[block={5.0cm}{4.5cm}, fill=black!10!white] (P) {};
|
|
\node[above] at (P.north) {$P_3(s)$};
|
|
|
|
\coordinate[] (inputw) at ($(P.south west)!0.8!(P.north west) + (-0.7, 0)$);
|
|
\coordinate[] (inputu) at ($(P.south west)!0.4!(P.north west) + (-0.7, 0)$);
|
|
|
|
\coordinate[] (output1) at ($(P.south east)!0.8!(P.north east) + (0.7, 0)$);
|
|
\coordinate[] (output2) at ($(P.south east)!0.55!(P.north east) + (0.7, 0)$);
|
|
\coordinate[] (output3) at ($(P.south east)!0.3!(P.north east) + (0.7, 0)$);
|
|
\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[block, left=1.4 of output3] (W3){$W_3(s)$};
|
|
\node[addb={+}{}{}{}{-}, left=of W1] (sub1) {};
|
|
\node[addb={+}{}{}{}{-}, left=of sub1] (sub2) {};
|
|
|
|
\node[block, below=0.3 of P] (H) {$\begin{bmatrix}H_2(s) \\ H_3(s)\end{bmatrix}$};
|
|
|
|
\draw[->] (inputw) node[above right](w){$w$} -- (sub2.west);
|
|
\draw[->] (W3-|sub1)node[branch]{} -- (sub1.south);
|
|
\draw[->] (W2-|sub2)node[branch]{} -- (sub2.south);
|
|
\draw[->] ($(sub2.west)+(-0.5, 0)$) node[branch]{} |- (outputv) |- (H.east);
|
|
\draw[->] ($(H.south west)!0.7!(H.north west)$) -| (inputu|-W2) -- (W2.west);
|
|
\draw[->] ($(H.south west)!0.3!(H.north west)$) -| ($(inputu|-W3)+(0.4, 0)$) -- (W3.west);
|
|
|
|
\draw[->] (sub2.east) -- (sub1.west);
|
|
\draw[->] (sub1.east) -- (W1.west);
|
|
\draw[->] (W1.east) -- (output1)node[above left](z){$z_1$};
|
|
\draw[->] (W2.east) -- (output2)node[above left]{$z_2$};
|
|
\draw[->] (W3.east) -- (output3)node[above left]{$z_3$};
|
|
\node[above] at (W2-|w){$u_1$};
|
|
\node[above] at (W3-|w){$u_2$};
|
|
\node[above] at (outputv-|z){$v$};
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_sensor_comp_filter_three_hinf
|
|
#+caption: Architecture for the $\mathcal{H}_\infty$ synthesis of three complementary filters (\subref{fig:detail_control_sensor_comp_filter_three_hinf_fb}). Bode plot of the inverse weighting functions and of the three obtained complementary filters (\subref{fig:detail_control_sensor_three_complementary_filters_results})
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_comp_filter_three_hinf_fb}Generalized plant with the synthesized filter}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :width 0.95\linewidth
|
|
[[file:figs/detail_control_sensor_comp_filter_three_hinf_fb.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_sensor_three_complementary_filters_results}Weights and obtained filters}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :width 0.95\linewidth
|
|
[[file:figs/detail_control_sensor_three_complementary_filters_results.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
Standard $\mathcal{H}_\infty$ synthesis is performed on the generalized plant $P_3(s)$.
|
|
Two filters, $H_2(s)$ and $H_3(s)$, are obtained such that the $\mathcal{H}_\infty$ norm of the closed-loop transfer from $w$ to $[z_1,\ z_2,\ z_3]$ of the system in Figure ref:fig:detail_control_sensor_comp_filter_three_hinf_fb is less than one.
|
|
Filter $H_1(s)$ is defined using eqref:eq:detail_control_sensor_h1_compl_h2_h3, thus ensuring the complementary property of the obtained set of filters.
|
|
|
|
\begin{equation}\label{eq:detail_control_sensor_h1_compl_h2_h3}
|
|
H_1(s) \triangleq 1 - \big[ H_2(s) + H_3(s) \big]
|
|
\end{equation}
|
|
|
|
Figure ref:fig:detail_control_sensor_three_complementary_filters_results displays the three synthesized complementary filters (solid lines), confirming the successful synthesis.
|
|
|
|
#+begin_src matlab
|
|
%% Synthesis of a set of three complementary filters
|
|
% Design of the Weighting Functions
|
|
W1 = generateWF('n', 2, 'w0', 2*pi*1, 'G0', 1/10, 'Ginf', 1000, 'Gc', 0.5);
|
|
W2 = 0.22*(1 + s/2/pi/1)^2/(sqrt(1e-4) + s/2/pi/1)^2*(1 + s/2/pi/10)^2/(1 + s/2/pi/1000)^2;
|
|
W3 = generateWF('n', 3, 'w0', 2*pi*10, 'G0', 1000, 'Ginf', 1/10, 'Gc', 0.5);
|
|
|
|
% Generalized plant for the synthesis of 3 complementary filters
|
|
P = [W1 -W1 -W1;
|
|
0 W2 0 ;
|
|
0 0 W3;
|
|
1 0 0];
|
|
|
|
% Standard H-Infinity Synthesis
|
|
[H, ~, gamma, ~] = hinfsyn(P, 1, 2,'TOLGAM', 0.001, 'METHOD', 'ric', 'DISPLAY', 'on');
|
|
|
|
% Synthesized H2 and H3 filters
|
|
H2 = tf(H(1));
|
|
H3 = tf(H(2));
|
|
|
|
% H1 is defined as the complementary filter of H2 and H3
|
|
H1 = 1 - H2 - H3;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results none
|
|
%% Bode plot of the inverse weighting functions and of the three complementary filters obtained using the H-infinity synthesis
|
|
figure;
|
|
hold on;
|
|
plot(freqs, 1./abs(squeeze(freqresp(W1, freqs, 'Hz'))), '--', 'color', colors(1,:),'DisplayName', '$|W_1|^{-1}$');
|
|
plot(freqs, 1./abs(squeeze(freqresp(W2, freqs, 'Hz'))), '--', 'color', colors(2,:),'DisplayName', '$|W_2|^{-1}$');
|
|
plot(freqs, 1./abs(squeeze(freqresp(W3, freqs, 'Hz'))), '--', 'color', colors(3,:),'DisplayName', '$|W_3|^{-1}$');
|
|
plot(freqs, abs(squeeze(freqresp(H1, freqs, 'Hz'))), '-', 'color', [colors(1,:), 0.5], 'linewidth', 2.5,'DisplayName', '$H_1$');
|
|
plot(freqs, abs(squeeze(freqresp(H2, freqs, 'Hz'))), '-', 'color', [colors(2,:), 0.5], 'linewidth', 2.5,'DisplayName', '$H_2$');
|
|
plot(freqs, abs(squeeze(freqresp(H3, freqs, 'Hz'))), '-', 'color', [colors(3,:), 0.5], 'linewidth', 2.5,'DisplayName', '$H_3$');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
|
xlim([freqs(1), freqs(end)]); ylim([1e-4, 20]);
|
|
leg = legend('location', 'southeast', 'FontSize', 8);
|
|
leg.ItemTokenSize(1) = 18;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_sensor_three_complementary_filters_results.pdf', 'width', 'half', 'height', 'normal');
|
|
#+end_src
|
|
|
|
** Conclusion
|
|
:PROPERTIES:
|
|
:UNNUMBERED: t
|
|
:END:
|
|
|
|
A new method for designing complementary filters using the $\mathcal{H}_\infty$ synthesis has been proposed.
|
|
This approach allows shaping of the filter magnitudes through the use of weighting functions during synthesis.
|
|
This capability is particularly valuable in practice since the characteristics of the super sensor are directly linked to the complementary filters' magnitude.
|
|
Consequently, typical sensor fusion objectives can be effectively translated into requirements on the magnitudes of the filters.
|
|
|
|
For the Nano Active Stabilization System (NASS), the High Authority Control-Integral Force Feedback (HAC-IFF) strategy was found to perform well and to offer the advantages of being both intuitive to understand and straightforward to tune.
|
|
Looking forward, it would be interesting to investigate how sensor fusion (particularly between the force sensor and external metrology) compares to the HAC-IFF approach in terms of performance and robustness.
|
|
|
|
* Decoupling
|
|
:PROPERTIES:
|
|
:HEADER-ARGS:matlab+: :tangle matlab/detail_control_2_decoupling.m
|
|
:END:
|
|
<<sec:detail_control_decoupling>>
|
|
|
|
** Introduction :ignore:
|
|
|
|
The control of parallel manipulators (and any MIMO system in general) typically involves a two-step approach: first decoupling the plant dynamics using various strategies, which will be discussed in this section, followed by the application of SISO control for the decoupled plant (discussed in section ref:sec:detail_control_cf).
|
|
|
|
When sensors are integrated within the struts, decentralized control may be applied, as the system is already well decoupled at low frequency.
|
|
For instance, [[cite:&furutani04_nanom_cuttin_machin_using_stewar]] implemented a system where each strut consists of piezoelectric stack actuators and eddy current displacement sensors, with separate PI controllers for each strut.
|
|
A similar control architecture was proposed in [[cite:&du14_piezo_actuat_high_precis_flexib]] using strain gauge sensors integrated in each strut.
|
|
|
|
An alternative strategy involves decoupling the system in the Cartesian frame using Jacobian matrices.
|
|
As demonstrated during the study of Stewart platform kinematics, Jacobian matrices can be utilized to map actuator forces to forces and torques applied on the top platform.
|
|
This approach enables the implementation of controllers in a defined frame.
|
|
It has been applied with various sensor types including force sensors [[cite:&mcinroy00_desig_contr_flexur_joint_hexap]], relative displacement sensors [[cite:&kim00_robus_track_contr_desig_dof_paral_manip]], and inertial sensors [[cite:&li01_simul_vibrat_isolat_point_contr;&abbas14_vibrat_stewar_platf]].
|
|
The Cartesian frame in which the system is decoupled is typically chosen at the point of interest (i.e., where the motion is of interest) or at the center of mass.
|
|
|
|
Modal control represents another noteworthy decoupling strategy, wherein the "local" plant inputs and outputs are mapped to the modal space.
|
|
In this approach, multiple SISO plants, each corresponding to a single mode, can be controlled independently.
|
|
This decoupling strategy has been implemented for active damping applications [[cite:&holterman05_activ_dampin_based_decoup_colloc_contr]], which is logical as it is often desirable to dampen specific modes.
|
|
The strategy has also been employed in [[cite:&pu11_six_degree_of_freed_activ]] for vibration isolation purposes using geophones, and in [[cite:&yang19_dynam_model_decoup_contr_flexib]] using force sensors.
|
|
|
|
Another completely different strategy, is to use implement a multivariable control directly on the coupled system.
|
|
$\mathcal{H}_\infty$ and $\mu\text{-synthesis}$ were applied to a Stewart platform model in [[cite:&lei08_multi_objec_robus_activ_vibrat]].
|
|
In [[cite:&xie17_model_contr_hybrid_passiv_activ]], decentralized force feedback was first applied, followed by $\mathcal{H}_2$ synthesis for vibration isolation based on accelerometers.
|
|
$\mathcal{H}_\infty$ synthesis was also employed in [[cite:&jiao18_dynam_model_exper_analy_stewar]] for active damping based on accelerometers.
|
|
[[cite:&thayer02_six_axis_vibrat_isolat_system]] compared $\mathcal{H}_\infty$ synthesis with decentralized control in the frame of the struts.
|
|
Their experimental closed-loop results indicated that the $\mathcal{H}_\infty$ controller did not outperform the decentralized controller in the frame of the struts.
|
|
These limitations were attributed to the model's poor ability to predict off-diagonal dynamics, which is crucial for $\mathcal{H}_\infty$ synthesis.
|
|
|
|
The purpose of this section is to compare several methods for the decoupling of parallel manipulators, an analysis that appears to be lacking in the literature.
|
|
The analysis begins in Section ref:ssec:detail_control_decoupling_model with the introduction of a simplified parallel manipulator model that serves as the foundation for evaluating various decoupling strategies.
|
|
Sections ref:ssec:detail_control_decoupling_jacobian through ref:ssec:detail_control_decoupling_svd systematically examine three distinct approaches: Jacobian matrix decoupling, modal decoupling, and Singular Value Decomposition (SVD) decoupling, respectively.
|
|
The comparative assessment of these three methodologies, along with concluding observations, is provided in Section ref:ssec:detail_control_decoupling_comp.
|
|
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :noweb yes
|
|
<<m-init-path>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :eval no :noweb yes
|
|
<<m-init-path-tangle>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :noweb yes
|
|
<<m-init-other>>
|
|
|
|
%% Initialize Frequency Vector
|
|
freqs = logspace(0, 3, 1000);
|
|
#+end_src
|
|
|
|
** Test Model
|
|
<<ssec:detail_control_decoupling_model>>
|
|
|
|
Instead of utilizing the Stewart platform for comparing decoupling strategies, a simplified parallel manipulator is employed to facilitate a more straightforward analysis.
|
|
The system illustrated in Figure ref:fig:detail_control_decoupling_model_test is used for this purpose.
|
|
It possesses three degrees of freedom (DoF) and incorporates three parallel struts.
|
|
Being a fully parallel manipulator, it is therefore quite similar to the Stewart platform.
|
|
|
|
Two reference frames are defined within this model: frame $\{M\}$ with origin $O_M$ at the center of mass of the solid body, and frame $\{K\}$ with origin $O_K$ at the center of stiffness of the parallel manipulator.
|
|
|
|
#+attr_latex: :options [b]{0.60\linewidth}
|
|
#+begin_minipage
|
|
#+name: fig:detail_control_decoupling_model_test
|
|
#+caption: Model used to compare decoupling strategies
|
|
#+attr_latex: :float nil :scale 1
|
|
[[file:figs/detail_control_decoupling_model_test.png]]
|
|
#+end_minipage
|
|
\hfill
|
|
#+attr_latex: :options [b]{0.36\linewidth}
|
|
#+begin_minipage
|
|
#+begin_scriptsize
|
|
#+latex: \centering
|
|
#+attr_latex: :environment tabularx :width \linewidth :placement [b] :align cXc
|
|
#+attr_latex: :booktabs t :float nil :center nil
|
|
| | *Description* | *Value* |
|
|
|-------+-----------------------+-------------------|
|
|
| $l_a$ | | $0.5\,m$ |
|
|
| $h_a$ | | $0.2\,m$ |
|
|
| $k$ | Actuator stiffness | $10\,N/\mu m$ |
|
|
| $c$ | Actuator damping | $200\,Ns/m$ |
|
|
| $m$ | Payload mass | $40\,\text{kg}$ |
|
|
| $I$ | Payload $R_z$ inertia | $5\,\text{kg}m^2$ |
|
|
#+latex: \captionof{table}{\label{tab:detail_control_decoupling_test_model_params}Model parameters}
|
|
#+end_scriptsize
|
|
#+end_minipage
|
|
|
|
The equations of motion are derived by applying Newton's second law to the suspended mass, expressed at its center of mass eqref:eq:detail_control_decoupling_model_eom, where $\bm{\mathcal{X}}_{\{M\}}$ represents the two translations and one rotation with respect to the center of mass, and $\bm{\mathcal{F}}_{\{M\}}$ denotes the forces and torque applied at the center of mass.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_model_eom}
|
|
\bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) = \sum \bm{\mathcal{F}}_{\{M\}}(t), \quad
|
|
\bm{\mathcal{X}}_{\{M\}} = \begin{bmatrix}
|
|
x \\
|
|
y \\
|
|
R_z
|
|
\end{bmatrix}, \quad \bm{\mathcal{F}}_{\{M\}} = \begin{bmatrix}
|
|
F_x \\
|
|
F_y \\
|
|
M_z
|
|
\end{bmatrix}
|
|
\end{equation}
|
|
|
|
The Jacobian matrix $\bm{J}_{\{M\}}$ is employed to map the spring, damping, and actuator forces to XY forces and Z torque expressed at the center of mass eqref:eq:detail_control_decoupling_jacobian_CoM.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM}
|
|
\bm{J}_{\{M\}} = \begin{bmatrix}
|
|
1 & 0 & h_a \\
|
|
0 & 1 & -l_a \\
|
|
0 & 1 & l_a \\
|
|
\end{bmatrix}
|
|
\end{equation}
|
|
|
|
Subsequently, the equation of motion relating the actuator forces $\tau$ to the motion of the mass $\bm{\mathcal{X}}_{\{M\}}$ is derived eqref:eq:detail_control_decoupling_plant_cartesian.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_plant_cartesian}
|
|
\bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{M\}} \bm{\mathcal{X}}_{\{M\}}(t) = \bm{J}_{\{M\}}^{\intercal} \bm{\tau}(t)
|
|
\end{equation}
|
|
|
|
The matrices representing the payload inertia, actuator stiffness, and damping are shown in eqref:eq:detail_control_decoupling_system_matrices.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_system_matrices}
|
|
\bm{M}_{\{M\}} = \begin{bmatrix}
|
|
m & 0 & 0 \\
|
|
0 & m & 0 \\
|
|
0 & 0 & I
|
|
\end{bmatrix}, \quad
|
|
\bm{\mathcal{K}} = \begin{bmatrix}
|
|
k & 0 & 0 \\
|
|
0 & k & 0 \\
|
|
0 & 0 & k
|
|
\end{bmatrix}, \quad
|
|
\bm{\mathcal{C}} = \begin{bmatrix}
|
|
c & 0 & 0 \\
|
|
0 & c & 0 \\
|
|
0 & 0 & c
|
|
\end{bmatrix}
|
|
\end{equation}
|
|
|
|
The parameters employed for the subsequent analysis are summarized in Table ref:tab:detail_control_decoupling_test_model_params, which includes values for geometric parameters ($l_a$, $h_a$), mechanical properties (actuator stiffness $k$ and damping $c$), and inertial characteristics (payload mass $m$ and rotational inertia $I$).
|
|
|
|
** Control in the frame of the struts
|
|
<<ssec:detail_control_decoupling_decentralized>>
|
|
|
|
The dynamics in the frame of the struts are first examined.
|
|
The equation of motion relating actuator forces $\bm{\mathcal{\tau}}$ to strut relative motion $\bm{\mathcal{L}}$ is derived from equation eqref:eq:detail_control_decoupling_plant_cartesian by mapping the Cartesian motion of the mass to the relative motion of the struts using the Jacobian matrix $\bm{J}_{\{M\}}$ defined in eqref:eq:detail_control_decoupling_jacobian_CoM.
|
|
The obtained transfer function from $\bm{\mathcal{\tau}}$ to $\bm{\mathcal{L}}$ is shown in eqref:eq:detail_control_decoupling_plant_decentralized.
|
|
|
|
#+begin_src latex :file detail_control_decoupling_control_struts.pdf
|
|
\begin{tikzpicture}
|
|
\node[block] (Gl) {$\bm{G}_{\{\mathcal{L}\}}$};
|
|
|
|
% Connections and labels
|
|
\draw[<-] (Gl.west) -- ++(-0.8, 0) node[above right]{$\bm{\tau}$};
|
|
\draw[->] (Gl.east) -- ++( 0.8, 0) node[above left]{$\bm{\mathcal{L}}$};
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
# [[file:figs/detail_control_decoupling_control_struts.png]]
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_plant_decentralized}
|
|
\frac{\bm{\mathcal{L}}}{\bm{\mathcal{\tau}}}(s) = \bm{G}_{\mathcal{L}}(s) = \left( \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} s^2 + \bm{\mathcal{C}} s + \bm{\mathcal{K}} \right)^{-1}
|
|
\end{equation}
|
|
|
|
At low frequencies, the plant converges to a diagonal constant matrix whose diagonal elements are related to the actuator stiffnesses eqref:eq:detail_control_decoupling_plant_decentralized_low_freq.
|
|
At high frequencies, the plant converges to the mass matrix mapped in the frame of the struts, which is generally highly non-diagonal.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_plant_decentralized_low_freq}
|
|
\bm{G}_{\mathcal{L}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{\mathcal{K}^{-1}}
|
|
\end{equation}
|
|
|
|
#+begin_src matlab
|
|
%% Compute Equation of motion
|
|
l = 1; h=2;
|
|
la = 0.5; % Horizontal position of actuators [m]
|
|
ha = 0.2; % Vertical of actuators [m]
|
|
|
|
m = 40; % Payload mass [kg]
|
|
I = 5; % Payload rotational inertia [kg m^2]
|
|
|
|
c = 2e2; % Actuator Damping [N/(m/s)]
|
|
k = 1e6; % Actuator Stiffness [N/m]
|
|
|
|
% Unit vectors of the actuators
|
|
s1 = [1;0];
|
|
s2 = [0;1];
|
|
s3 = [0;1];
|
|
|
|
% Stiffnesss and Damping matrices of the struts
|
|
Kr = diag([k,k,k]);
|
|
Cr = diag([c,c,c]);
|
|
|
|
% Location of the joints with respect to the center of mass
|
|
Mb1 = [-l/2;-ha];
|
|
Mb2 = [-la; -h/2];
|
|
Mb3 = [ la; -h/2];
|
|
|
|
% Jacobian matrix (Center of Mass)
|
|
J_CoM = [s1', Mb1(1)*s1(2)-Mb1(2)*s1(1);
|
|
s2', Mb2(1)*s2(2)-Mb2(2)*s2(1);
|
|
s3', Mb3(1)*s3(2)-Mb3(2)*s3(1)];
|
|
|
|
% Mass Matrix in frame {M}
|
|
M = diag([m,m,I]);
|
|
|
|
% Stiffness Matrix in frame {M}
|
|
K = J_CoM'*Kr*J_CoM;
|
|
|
|
% Damping Matrix in frame {M}
|
|
C = J_CoM'*Cr*J_CoM;
|
|
|
|
% Plant in the frame of the struts
|
|
G_L = J_CoM*inv(M*s^2 + C*s + K)*J_CoM';
|
|
#+end_src
|
|
|
|
The magnitude of the coupled plant $\bm{G}_{\mathcal{L}}$ is illustrated in Figure ref:fig:detail_control_decoupling_coupled_plant_bode.
|
|
This representation confirms that at low frequencies (below the first suspension mode), the plant is well decoupled.
|
|
Depending on the symmetry present in the system, certain diagonal elements may exhibit identical values, as demonstrated for struts 2 and 3 in this example.
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
for out_i = 1:3
|
|
for in_i = 1:3
|
|
nexttile;
|
|
plot(freqs, abs(squeeze(freqresp(G_L(out_i,in_i), freqs, 'Hz'))), 'k-', ...
|
|
'DisplayName', sprintf('$\\mathcal{L}_%i/\\tau_%i$', out_i, in_i));
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlim([freqs(1), freqs(end)]); ylim([2e-8, 4e-5]);
|
|
xticks([1e0, 1e1, 1e2])
|
|
yticks([1e-7, 1e-6, 1e-5])
|
|
leg = legend('location', 'southwest', 'FontSize', 8);
|
|
leg.ItemTokenSize(1) = 18;
|
|
|
|
if in_i == 1
|
|
ylabel('Mag. [m/N]')
|
|
else
|
|
set(gca, 'YTickLabel',[]);
|
|
end
|
|
|
|
if out_i == 3
|
|
xlabel('Frequency [Hz]')
|
|
else
|
|
set(gca, 'XTickLabel',[]);
|
|
end
|
|
end
|
|
end
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_decoupling_coupled_plant_bode.pdf', 'width', 'full', 'height', 600);
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_decoupling_coupled_plant_bode
|
|
#+caption: Model dynamics from actuator forces to relative displacement sensor of each strut.
|
|
#+RESULTS:
|
|
[[file:figs/detail_control_decoupling_coupled_plant_bode.png]]
|
|
|
|
** Jacobian Decoupling
|
|
<<ssec:detail_control_decoupling_jacobian>>
|
|
**** Jacobian Matrix
|
|
|
|
The Jacobian matrix serves a dual purpose in the decoupling process: it converts strut velocity $\dot{\mathcal{L}}$ to payload velocity and angular velocity $\dot{\bm{\mathcal{X}}}_{\{O\}}$, and it transforms actuator forces $\bm{\tau}$ to forces/torque applied on the payload $\bm{\mathcal{F}}_{\{O\}}$, as expressed in equation eqref:eq:detail_control_decoupling_jacobian.
|
|
|
|
\begin{subequations}\label{eq:detail_control_decoupling_jacobian}
|
|
\begin{align}
|
|
\dot{\bm{\mathcal{X}}}_{\{O\}} &= \bm{J}_{\{O\}} \dot{\bm{\mathcal{L}}}, \quad \dot{\bm{\mathcal{L}}} = \bm{J}_{\{O\}}^{-1} \dot{\bm{\mathcal{X}}}_{\{O\}} \\
|
|
\bm{\mathcal{F}}_{\{O\}} &= \bm{J}_{\{O\}}^{\intercal} \bm{\tau}, \quad \bm{\tau} = \bm{J}_{\{O\}}^{-\intercal} \bm{\mathcal{F}}_{\{O\}}
|
|
\end{align}
|
|
\end{subequations}
|
|
|
|
The resulting plant (Figure ref:fig:detail_control_jacobian_decoupling_arch) have inputs and outputs with clear physical interpretations:
|
|
- $\bm{\mathcal{F}}_{\{O\}}$ represents forces/torques applied on the payload at the origin of frame $\{O\}$
|
|
- $\bm{\mathcal{X}}_{\{O\}}$ represents translations/rotation of the payload expressed in frame $\{O\}$
|
|
|
|
#+begin_src latex :file detail_control_decoupling_control_jacobian.pdf
|
|
\begin{tikzpicture}
|
|
\node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
|
|
\node[block, left=0.6 of G] (Jt) {$\bm{J}_{\{O\}}^{-\intercal}$};
|
|
\node[block, right=0.6 of G] (Ja) {$\bm{J}_{\{O\}}^{-1}$};
|
|
|
|
% Connections and labels
|
|
\draw[<-] (Jt.west) -- ++(-1.4, 0) node[above right]{$\bm{\mathcal{F}}_{\{O\}}$};
|
|
\draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$};
|
|
\draw[->] (G.east) -- (Ja.west) node[above left]{$\bm{\mathcal{L}}$};
|
|
\draw[->] (Ja.east) -- ++( 1.4, 0) node[above left]{$\bm{\mathcal{X}}_{\{O\}}$};
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gx) {};
|
|
\node[above] at (Gx.north) {$\bm{G}_{\{O\}}$};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_jacobian_decoupling_arch
|
|
#+caption: Block diagram of the transfer function from $\bm{\mathcal{F}}_{\{O\}}$ to $\bm{\mathcal{X}}_{\{O\}}$
|
|
#+RESULTS:
|
|
[[file:figs/detail_control_decoupling_control_jacobian.png]]
|
|
|
|
The transfer function from $\bm{\mathcal{F}}_{\{O\}$ to $\bm{\mathcal{X}}_{\{O\}}$, denoted $\bm{G}_{\{O\}}(s)$ can be computed using eqref:eq:detail_control_decoupling_plant_jacobian.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_plant_jacobian}
|
|
\frac{\bm{\mathcal{X}}_{\{O\}}}{\bm{\mathcal{F}}_{\{O\}}}(s) = \bm{G}_{\{O\}}(s) = \left( \bm{J}_{\{O\}}^{\intercal} \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{O\}} s^2 + \bm{J}_{\{O\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{O\}} s + \bm{J}_{\{O\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{O\}} \right)^{-1}
|
|
\end{equation}
|
|
|
|
The frame $\{O\}$ can be selected according to specific requirements, but the decoupling properties are significantly influenced by this choice.
|
|
Two natural reference frames are particularly relevant: the center of mass and the center of stiffness.
|
|
|
|
**** Center Of Mass
|
|
|
|
When the decoupling frame is located at the center of mass (frame $\{M\}$ in Figure ref:fig:detail_control_decoupling_model_test), the Jacobian matrix and its inverse are expressed as in eqref:eq:detail_control_decoupling_jacobian_CoM_inverse.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_jacobian_CoM_inverse}
|
|
\bm{J}_{\{M\}} = \begin{bmatrix}
|
|
1 & 0 & h_a \\
|
|
0 & 1 & -l_a \\
|
|
0 & 1 & l_a \\
|
|
\end{bmatrix}, \quad \bm{J}_{\{M\}}^{-1} = \begin{bmatrix}
|
|
1 & \frac{h_a}{2 l_a} & \frac{-h_a}{2 l_a} \\
|
|
0 & \frac{1}{2} & \frac{1}{2} \\
|
|
0 & \frac{-1}{2 l_a} & \frac{1}{2 l_a} \\
|
|
\end{bmatrix}
|
|
\end{equation}
|
|
|
|
#+begin_src latex :file detail_control_decoupling_control_jacobian_CoM.pdf
|
|
\begin{tikzpicture}
|
|
\node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
|
|
\node[block, left=0.6 of G] (Jt) {$\bm{J}_{\{M\}}^{-\intercal}$};
|
|
\node[block, right=0.6 of G] (Ja) {$\bm{J}_{\{M\}}^{-1}$};
|
|
|
|
% Connections and labels
|
|
\draw[<-] (Jt.west) -- ++(-1.4, 0) node[above right]{$\bm{\mathcal{F}}_{\{M\}}$};
|
|
\draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$};
|
|
\draw[->] (G.east) -- (Ja.west) node[above left]{$\bm{\mathcal{L}}$};
|
|
\draw[->] (Ja.east) -- ++( 1.4, 0) node[above left]{$\bm{\mathcal{X}}_{\{M\}}$};
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gx) {};
|
|
\node[above] at (Gx.north) {$\bm{G}_{\{M\}}$};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
# [[file:figs/detail_control_decoupling_control_jacobian_CoM.png]]
|
|
|
|
Analytical formula of the plant $\bm{G}_{\{M\}}(s)$ is derived eqref:eq:detail_control_decoupling_plant_CoM.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_plant_CoM}
|
|
\frac{\bm{\mathcal{X}}_{\{M\}}}{\bm{\mathcal{F}}_{\{M\}}}(s) = \bm{G}_{\{M\}}(s) = \left( \bm{M}_{\{M\}} s^2 + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{M\}} s + \bm{J}_{\{M\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{M\}} \right)^{-1}
|
|
\end{equation}
|
|
|
|
At high frequencies, the plant converges to the inverse of the mass matrix, which is a diagonal matrix eqref:eq:detail_control_decoupling_plant_CoM_high_freq.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_plant_CoM_high_freq}
|
|
\bm{G}_{\{M\}}(j\omega) \xrightarrow[\omega \to \infty]{} -\omega^2 \bm{M}_{\{M\}}^{-1} = -\omega^2 \begin{bmatrix}
|
|
1/m & 0 & 0 \\
|
|
0 & 1/m & 0 \\
|
|
0 & 0 & 1/I
|
|
\end{bmatrix}
|
|
\end{equation}
|
|
|
|
Consequently, the plant exhibits effective decoupling at frequencies above the highest suspension mode as shown in Figure ref:fig:detail_control_decoupling_jacobian_plant_CoM.
|
|
This strategy is typically employed in systems with low-frequency suspension modes [[cite:&butler11_posit_contr_lithog_equip]], where the plant approximates decoupled mass lines.
|
|
|
|
The low-frequency coupling observed in this configuration has a clear physical interpretation.
|
|
When a static force is applied at the center of mass, the suspended mass rotates around the center of stiffness.
|
|
This rotation is due to torque induced by the stiffness of the first actuator (i.e. the one on the left side), which is not aligned with the force application point.
|
|
This phenomenon is illustrated in Figure ref:fig:detail_control_decoupling_model_test_CoM.
|
|
|
|
#+begin_src matlab
|
|
%% Jacobian Decoupling - Center of Mass
|
|
G_CoM = pinv(J_CoM)*G_L*pinv(J_CoM');
|
|
G_CoM.InputName = {'Fx', 'Fy', 'Mz'};
|
|
G_CoM.OutputName = {'Dx', 'Dy', 'Rz'};
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G_CoM(1, 3), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
|
'DisplayName', '$D_{x,\{M\}}/M_{z,\{M\}}$');
|
|
plot(freqs, abs(squeeze(freqresp(G_CoM(3, 1), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
|
'DisplayName', '$R_{z,\{M\}}/F_{x,\{M\}}$');
|
|
plot(freqs, abs(squeeze(freqresp(G_CoM(1, 1), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', '$D_{x,\{M\}}/F_{x,\{M\}}$');
|
|
plot(freqs, abs(squeeze(freqresp(G_CoM(2, 2), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', '$D_{y,\{M\}}/F_{y,\{M\}}$');
|
|
plot(freqs, abs(squeeze(freqresp(G_CoM(3, 3), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', '$R_{z,\{M\}}/M_{z,\{M\}}$');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
|
ylim([1e-10, 1e-3]);
|
|
leg = legend('location', 'southwest', 'FontSize', 8);
|
|
leg.ItemTokenSize(1) = 18;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_decoupling_jacobian_plant_CoM.pdf', 'width', 'half', 'height', 'normal');
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_jacobian_decoupling_plant_CoM_results
|
|
#+caption: Plant decoupled using the Jacobian matrix expresssed at the center of mass (\subref{fig:detail_control_decoupling_jacobian_plant_CoM}). The physical reason for low frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoM}).
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_jacobian_plant_CoM}Dynamics at the CoM}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :width 0.95\linewidth
|
|
[[file:figs/detail_control_decoupling_jacobian_plant_CoM.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_CoM}Static force applied at the CoM}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_decoupling_model_test_CoM.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
**** Center Of Stiffness
|
|
|
|
#+begin_src latex :file detail_control_decoupling_control_jacobian_CoK.pdf
|
|
\begin{tikzpicture}
|
|
\node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
|
|
\node[block, left=0.6 of G] (Jt) {$\bm{J}_{\{K\}}^{-\intercal}$};
|
|
\node[block, right=0.6 of G] (Ja) {$\bm{J}_{\{K\}}^{-1}$};
|
|
|
|
% Connections and labels
|
|
\draw[<-] (Jt.west) -- ++(-1.4, 0) node[above right]{$\bm{\mathcal{F}}_{\{K\}}$};
|
|
\draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$};
|
|
\draw[->] (G.east) -- (Ja.west) node[above left]{$\bm{\mathcal{L}}$};
|
|
\draw[->] (Ja.east) -- ++( 1.4, 0) node[above left]{$\bm{\mathcal{X}}_{\{K\}}$};
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gx) {};
|
|
\node[above] at (Gx.north) {$\bm{G}_{\{K\}}$};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+RESULTS:
|
|
# [[file:figs/detail_control_decoupling_control_jacobian_CoK.png]]
|
|
|
|
When the decoupling frame is located at the center of stiffness, the Jacobian matrix and its inverse are expressed as in eqref:eq:detail_control_decoupling_jacobian_CoK_inverse.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_jacobian_CoK_inverse}
|
|
\bm{J}_{\{K\}} = \begin{bmatrix}
|
|
1 & 0 & 0 \\
|
|
0 & 1 & -l_a \\
|
|
0 & 1 & l_a
|
|
\end{bmatrix}, \quad \bm{J}_{\{K\}}^{-1} = \begin{bmatrix}
|
|
1 & 0 & 0 \\
|
|
0 & \frac{1}{2} & \frac{1}{2} \\
|
|
0 & \frac{-1}{2 l_a} & \frac{1}{2 l_a}
|
|
\end{bmatrix}
|
|
\end{equation}
|
|
|
|
The frame $\{K\}$ was selected based on physical reasoning, positioned in line with the side strut and equidistant between the two vertical struts.
|
|
However, it could alternatively be determined through analytical methods to ensure that $\bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}}$ forms a diagonal matrix.
|
|
It should be noted that the existence of such a center of stiffness (i.e. a frame $\{K\}$ for which $\bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}}$ is diagonal) is not guaranteed for arbitrary systems.
|
|
This property is typically achievable only in systems exhibiting specific symmetrical characteristics, as is the case in the present example.
|
|
|
|
The analytical expression for the plant in this configuration was then computed ref:eq:detail_control_decoupling_plant_CoK.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_plant_CoK}
|
|
\frac{\bm{\mathcal{X}}_{\{K\}}}{\bm{\mathcal{F}}_{\{K\}}}(s) = \bm{G}_{\{K\}}(s) = \left( \bm{J}_{\{K\}}^{\intercal} \bm{J}_{\{M\}}^{-\intercal} \bm{M}_{\{M\}} \bm{J}_{\{M\}}^{-1} \bm{J}_{\{K\}} s^2 + \bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{C}} \bm{J}_{\{K\}} s + \bm{J}_{\{K\}}^{\intercal} \bm{\mathcal{K}} \bm{J}_{\{K\}} \right)^{-1}
|
|
\end{equation}
|
|
|
|
|
|
Figure ref:fig:detail_control_decoupling_jacobian_plant_CoK_results presents the dynamics of the plant when decoupled using the Jacobian matrix expressed at the center of stiffness.
|
|
The plant is well decoupled below the suspension mode with the lowest frequency eqref:eq:detail_control_decoupling_plant_CoK_low_freq, making it particularly suitable for systems with high stiffness.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_plant_CoK_low_freq}
|
|
\bm{G}_{\{K\}}(j\omega) \xrightarrow[\omega \to 0]{} \bm{J}_{\{K\}}^{-1} \bm{\mathcal{K}}^{-1} \bm{J}_{\{K\}}^{-\intercal}
|
|
\end{equation}
|
|
|
|
The physical reason for high-frequency coupling is illustrated in Figure ref:fig:detail_control_decoupling_model_test_CoK.
|
|
When a high-frequency force is applied at a point not aligned with the center of mass, it induces rotation around the center of mass.
|
|
This phenomenon explains the coupling observed between different degrees of freedom at higher frequencies.
|
|
|
|
#+begin_src matlab
|
|
%% Jacobian Decoupling - Center of Mass
|
|
% Location of the joints with respect to the center of stiffness
|
|
Mb1 = [-l/2; 0];
|
|
Mb2 = [-la; -h/2+ha];
|
|
Mb3 = [ la; -h/2+ha];
|
|
|
|
% Jacobian matrix (Center of Stiffness)
|
|
J_CoK = [s1', Mb1(1)*s1(2)-Mb1(2)*s1(1);
|
|
s2', Mb2(1)*s2(2)-Mb2(2)*s2(1);
|
|
s3', Mb3(1)*s3(2)-Mb3(2)*s3(1)];
|
|
|
|
G_CoK = pinv(J_CoK)*G_L*pinv(J_CoK');
|
|
G_CoK.InputName = {'Fx', 'Fy', 'Mz'};
|
|
G_CoK.OutputName = {'Dx', 'Dy', 'Rz'};
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G_CoK(1, 1), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', '$D_{x,\{K\}}/F_{x,\{K\}}$');
|
|
plot(freqs, abs(squeeze(freqresp(G_CoK(2, 2), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', '$D_{y,\{K\}}/F_{y,\{K\}}$');
|
|
plot(freqs, abs(squeeze(freqresp(G_CoK(3, 3), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', '$R_{z,\{K\}}/M_{z,\{K\}}$');
|
|
plot(freqs, abs(squeeze(freqresp(G_CoK(1, 3), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
|
'DisplayName', '$D_{x,\{K\}}/M_{z,\{K\}}$');
|
|
plot(freqs, abs(squeeze(freqresp(G_CoK(3, 1), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
|
'DisplayName', '$R_{z,\{K\}}/F_{x,\{K\}}$');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Kagnitude');
|
|
ylim([1e-10, 1e-3]);
|
|
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
|
leg.ItemTokenSize(1) = 18;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_decoupling_jacobian_plant_CoK.pdf', 'width', 'half', 'height', 'normal');
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_decoupling_jacobian_plant_CoK_results
|
|
#+caption: Plant decoupled using the Jacobian matrix expresssed at the center of stiffness (\subref{fig:detail_control_decoupling_jacobian_plant_CoK}). The physical reason for high frequency coupling is illustrated in (\subref{fig:detail_control_decoupling_model_test_CoK}).
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_jacobian_plant_CoK}Dynamics at the CoK}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :width 0.95\linewidth
|
|
[[file:figs/detail_control_decoupling_jacobian_plant_CoK.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_CoK}High frequency force applied at the CoK}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_decoupling_model_test_CoK.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
** Modal Decoupling
|
|
<<ssec:detail_control_decoupling_modal>>
|
|
**** Theory :ignore:
|
|
|
|
Modal decoupling represents an approach based on the principle that a mechanical system's behavior can be understood as a combination of contributions from various modes [[cite:&rankers98_machin]].
|
|
|
|
To convert the dynamics in the modal space, the equation of motion are first written with respect to the center of mass eqref:eq:detail_control_decoupling_equation_motion_CoM.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_equation_motion_CoM}
|
|
\bm{M}_{\{M\}} \ddot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{C}_{\{M\}} \dot{\bm{\mathcal{X}}}_{\{M\}}(t) + \bm{K}_{\{M\}} \bm{\mathcal{X}}_{\{M\}}(t) = \bm{J}_{\{M\}}^{\intercal} \bm{\tau}(t)
|
|
\end{equation}
|
|
|
|
For modal decoupling, a change of variables is introduced eqref:eq:detail_control_decoupling_modal_coordinates where $\bm{\mathcal{X}}_{m}$ represents the modal amplitudes and $\bm{\Phi}$ is a $n \times n$[fn:detail_control_2] matrix whose columns correspond to the mode shapes of the system, computed from $\bm{M}_{\{M\}}$ and $\bm{K}_{\{M\}}$.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_modal_coordinates}
|
|
\bm{\mathcal{X}}_{\{M\}} = \bm{\Phi} \bm{\mathcal{X}}_{m}
|
|
\end{equation}
|
|
|
|
By pre-multiplying equation eqref:eq:detail_control_decoupling_equation_motion_CoM by $\bm{\Phi}^{\intercal}$ and applying the change of variable eqref:eq:detail_control_decoupling_modal_coordinates, a new set of equations of motion is obtained eqref:eq:detail_control_decoupling_equation_modal_coordinates where $\bm{\tau}_m$ represents the modal input, while $\bm{M}_m$, $\bm{C}_m$, and $\bm{K}_m$ denote the modal mass, damping, and stiffness matrices respectively.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_equation_modal_coordinates}
|
|
\underbrace{\bm{\Phi}^{\intercal} \bm{M} \bm{\Phi}}_{\bm{M}_m} \bm{\ddot{\mathcal{X}}}_m(t) + \underbrace{\bm{\Phi}^{\intercal} \bm{C} \bm{\Phi}}_{\bm{C}_m} \bm{\dot{\mathcal{X}}}_m(t) + \underbrace{\bm{\Phi}^{\intercal} \bm{K} \bm{\Phi}}_{\bm{K}_m} \bm{\mathcal{X}}_m(t) = \underbrace{\bm{\Phi}^{\intercal} \bm{J}^{\intercal} \bm{\tau}(t)}_{\bm{\tau}_m(t)}
|
|
\end{equation}
|
|
The inherent mathematical structure of the mass, damping, and stiffness matrices [[cite:&lang17_under, chapt. 8]] ensures that modal matrices are diagonal [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition, chapt. 2.3]].
|
|
This diagonalization transforms equation eqref:eq:detail_control_decoupling_equation_modal_coordinates into a set of $n$ decoupled equations, enabling independent control of each mode without cross-interaction.
|
|
|
|
To implement this approach from a decentralized plant, the architecture shown in Figure ref:fig:detail_control_decoupling_modal is employed.
|
|
Inputs of the decoupling plant are the modal modal inputs $\bm{\tau}_m$ and the outputs are the modal amplitudes $\bm{\mathcal{X}}_m$.
|
|
This implementation requires knowledge of the system's equations of motion, from which the mode shapes matrix $\bm{\Phi}$ is derived.
|
|
The resulting decoupled system features diagonal elements each representing second-order resonant systems that are straightforward to control individually.
|
|
|
|
#+begin_src latex :file detail_control_decoupling_modal.pdf
|
|
\begin{tikzpicture}
|
|
\node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
|
|
\node[block, left=0.6 of G] (Jt) {$\bm{J}_{\{M\}}^{-\intercal}$};
|
|
\node[block, left=1.2 of Jt] (Bm) {$\bm{\Phi}^{-\intercal}$};
|
|
\node[block, right=0.6 of G] (J) {$\bm{J}_{\{M\}}^{-1}$};
|
|
\node[block, right=1.2 of J] (Cm) {$\bm{\Phi}^{-1}$};
|
|
|
|
% Connections and labels
|
|
\draw[<-] (Bm.west) -- ++(-1.0, 0) node[above right]{$\bm{\tau}_m$};
|
|
\draw[->] (Bm.east) -- (Jt.west) node[above left]{$\bm{\mathcal{F}}_{\{M\}}$};
|
|
\draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$};
|
|
\draw[->] (G.east) -- (J.west) node[above left]{$\bm{\mathcal{L}}$};
|
|
\draw[->] (J.east) -- (Cm.west) node[above left]{$\bm{\mathcal{X}}_{\{M\}}$};
|
|
\draw[->] (Cm.east) -- ++( 1.0, 0) node[above left]{$\bm{\mathcal{X}}_m$};
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={(Bm.south west) (Cm.north east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gm) {};
|
|
\node[above] at (Gm.north) {$\bm{G}_m$};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_decoupling_modal
|
|
#+caption: Modal Decoupling Architecture
|
|
#+RESULTS:
|
|
[[file:figs/detail_control_decoupling_modal.png]]
|
|
|
|
**** Example :ignore:
|
|
|
|
Modal decoupling was then applied to the test model.
|
|
First, the eigenvectors $\bm{\Phi}$ of $\bm{M}_{\{M\}}^{-1}\bm{K}_{\{M\}}$ were computed eqref:eq:detail_control_decoupling_modal_eigenvectors.
|
|
While analytical derivation of eigenvectors could be obtained for such a simple system, they are typically computed numerically for practical applications.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_modal_eigenvectors}
|
|
\bm{\Phi} = \begin{bmatrix}
|
|
\frac{I - h_a^2 m - 2 l_a^2 m - \alpha}{2 h_a m} & 0 & \frac{I - h_a^2 m - 2 l_a^2 m + \alpha}{2 h_a m} \\
|
|
0 & 1 & 0 \\
|
|
1 & 0 & 1
|
|
\end{bmatrix},\ \alpha = \sqrt{\left( I + m (h_a^2 - 2 l_a^2) \right)^2 + 8 m^2 h_a^2 l_a^2}
|
|
\end{equation}
|
|
|
|
The numerical values for the eigenvector matrix and its inverse are shown in eqref:eq:detail_control_decoupling_modal_eigenvectors_matrices.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_modal_eigenvectors_matrices}
|
|
\bm{\Phi} = \begin{bmatrix}
|
|
-0.905 & 0 & -0.058 \\
|
|
0 & 1 & 0 \\
|
|
0.424 & 0 & -0.998
|
|
\end{bmatrix}, \quad
|
|
\bm{\Phi}^{-1} = \begin{bmatrix}
|
|
-1.075 & 0 & 0.063 \\
|
|
0 & 1 & 0 \\
|
|
-0.457 & 0 & -0.975
|
|
\end{bmatrix}
|
|
\end{equation}
|
|
|
|
The two computed matrices were implemented in the control architecture of Figure ref:fig:detail_control_decoupling_modal, resulting in three distinct second order plants as depicted in Figure ref:fig:detail_control_decoupling_modal_plant.
|
|
Each of these diagonal elements corresponds to a specific mode, as shown in Figure ref:fig:detail_control_decoupling_model_test_modal, resulting in a perfectly decoupled system.
|
|
|
|
#+begin_src matlab
|
|
%% Modal decoupling
|
|
% Compute the eigen vectors
|
|
[phi, wi] = eig(M\K);
|
|
% Sort the eigen vectors by increasing associated frequency
|
|
[~, i] = sort(diag(wi));
|
|
phi = phi(:, i);
|
|
|
|
% Plant in the modal space
|
|
Gm = inv(phi)*inv(J_CoM)*G_L*inv(J_CoM')*inv(phi');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
%% Modal decoupled plant
|
|
figure;
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(Gm(1,1), freqs, 'Hz'))), 'color', colors(1,:), 'DisplayName', '$\mathcal{X}_{m,1}/\tau_{m,1}$');
|
|
plot(freqs, abs(squeeze(freqresp(Gm(2,2), freqs, 'Hz'))), 'color', colors(2,:), 'DisplayName', '$\mathcal{X}_{m,2}/\tau_{m,2}$');
|
|
plot(freqs, abs(squeeze(freqresp(Gm(3,3), freqs, 'Hz'))), 'color', colors(3,:), 'DisplayName', '$\mathcal{X}_{m,3}/\tau_{m,3}$');
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
|
ylim([1e-8, 1e-4]);
|
|
leg = legend('location', 'northeast', 'FontSize', 8);
|
|
leg.ItemTokenSize(1) = 18;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_decoupling_modal_plant.pdf', 'width', 'half', 'height', 'normal');
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_decoupling_modal_plant_modes
|
|
#+caption: Plant using modal decoupling consists of second order plants (\subref{fig:detail_control_decoupling_modal_plant}) which can be used to invidiually address different modes illustrated in (\subref{fig:detail_control_decoupling_model_test_modal})
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_modal_plant}Decoupled plant in modal space}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :width 0.95\linewidth
|
|
[[file:figs/detail_control_decoupling_modal_plant.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_modal}Individually controlled modes}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :width 0.95\linewidth
|
|
[[file:figs/detail_control_decoupling_model_test_modal.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
** SVD Decoupling
|
|
<<ssec:detail_control_decoupling_svd>>
|
|
**** Singular Value Decomposition
|
|
|
|
Singular Value Decomposition (SVD) represents a powerful mathematical tool with extensive applications in data analysis [[cite:&brunton22_data, chapt. 1]] and multivariable control systems [[cite:&skogestad07_multiv_feedb_contr]], where it is particularly valuable for analyzing directional properties in multivariable systems.
|
|
|
|
The SVD constitutes a unique matrix decomposition applicable to any complex matrix $\bm{X} \in \mathbb{C}^{n \times m}$, expressed as:
|
|
|
|
\begin{equation}\label{eq:detail_control_svd}
|
|
\bm{X} = \bm{U} \bm{\Sigma} \bm{V}^H
|
|
\end{equation}
|
|
|
|
where $\bm{U} \in \mathbb{C}^{n \times n}$ and $\bm{V} \in \mathbb{C}^{m \times m}$ are unitary matrices with orthonormal columns, and $\bm{\Sigma} \in \mathbb{R}^{n \times n}$ is a diagonal matrix with real, non-negative entries.
|
|
For real matrices $\bm{X}$, the resulting $\bm{U}$ and $\bm{V}$ matrices are also real, making them suitable for decoupling applications.
|
|
|
|
**** Decoupling using the SVD
|
|
|
|
The procedure for SVD-based decoupling begins with identifying the system dynamics from inputs to outputs, typically represented as a Frequency Response Function (FRF), which yields a complex matrix $\bm{G}(\omega_i)$ for multiple frequency points $\omega_i$.
|
|
A specific frequency is then selected for optimal decoupling, with the targeted crossover frequency $\omega_c$ often serving as an appropriate choice.
|
|
|
|
Since real matrices are required for the decoupling transformation, a real approximation of the complex measured response at the selected frequency must be computed.
|
|
In this work, the method proposed in [[cite:&kouvaritakis79_theor_pract_charac_locus_desig_method]] was used as it preserves maximal orthogonality in the directional properties of the input complex matrix.
|
|
|
|
Following this approximation, a real matrix $\tilde{\bm{G}}(\omega_c)$ is obtained, and SVD is performed on this matrix.
|
|
The resulting (real) unitary matrices $\bm{U}$ and $\bm{V}$ are structured such that $\bm{V}^{-\intercal} \tilde{\bm{G}}(\omega_c) \bm{U}^{-1}$ forms a diagonal matrix.
|
|
These singular input and output matrices are then applied to decouple the system as illustrated in Figure ref:fig:detail_control_decoupling_svd, and the decoupled plant is described by eqref:eq:detail_control_decoupling_plant_svd.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_plant_svd}
|
|
\bm{G}_{\text{SVD}}(s) = \bm{U}^{-1} \bm{G}_{\{\mathcal{L}\}}(s) \bm{V}^{-\intercal}
|
|
\end{equation}
|
|
|
|
#+begin_src latex :file detail_control_decoupling_svd.pdf
|
|
\begin{tikzpicture}
|
|
\node[block] (G) {$\bm{G}_{\{\mathcal{L}\}}$};
|
|
\node[block, left=0.6 of G.west] (V) {$\bm{V}^{-\intercal}$};
|
|
\node[block, right=0.6 of G.east] (U) {$\bm{U}^{-1}$};
|
|
|
|
% Connections and labels
|
|
\draw[<-] (V.west) -- ++(-0.8, 0) node[above right]{$\bm{u}$};
|
|
\draw[->] (V.east) -- (G.west) node[above left]{$\bm{\tau}$};
|
|
\draw[->] (G.east) -- (U.west) node[above left]{$\bm{\mathcal{L}}$};
|
|
\draw[->] (U.east) -- ++( 0.8, 0) node[above left]{$\bm{y}$};
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={(V.south west) (G.north-|U.east)}, fill=black!10!white, draw, dashed, inner sep=4pt] (Gsvd) {};
|
|
\node[above] at (Gsvd.north) {$\bm{G}_{\text{SVD}}$};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_decoupling_svd
|
|
#+caption: Decoupled plant $\bm{G}_{\text{SVD}}$ using the Singular Value Decomposition
|
|
#+RESULTS:
|
|
[[file:figs/detail_control_decoupling_svd.png]]
|
|
|
|
Implementation of SVD decoupling requires access to the system's FRF, at least in the vicinity of the desired decoupling frequency.
|
|
This information can be obtained either experimentally or derived from a model.
|
|
While this approach ensures effective decoupling near the chosen frequency, it provides no guarantees regarding decoupling performance away from this frequency.
|
|
Furthermore, the quality of decoupling depends significantly on the accuracy of the real approximation, potentially limiting its effectiveness for plants with high damping.
|
|
|
|
**** Example
|
|
|
|
Plant decoupling using the Singular Value Decomposition was then applied on the test model.
|
|
A decoupling frequency of $\SI{100}{Hz}$ was used.
|
|
The plant response at that frequency, as well as its real approximation and the obtained $\bm{U}$ and $\bm{V}$ matrices are shown in eqref:eq:detail_control_decoupling_svd_example.
|
|
|
|
\begin{equation}\label{eq:detail_control_decoupling_svd_example}
|
|
\begin{align}
|
|
& \bm{G}_{\{\mathcal{L}\}}(\omega_c = 2\pi \cdot 100) = 10^{-9} \begin{bmatrix}
|
|
-99 - j 2.6 & 74 + j 4.2 & -74 - j 4.2 \\
|
|
74 + j 4.2 & -247 - j 9.7 & 102 + j 7.0 \\
|
|
-74 - j 4.2 & 102 + j 7.0 & -247 - j 9.7
|
|
\end{bmatrix} \\
|
|
& \xrightarrow[\text{approximation}]{\text{real}} \tilde{\bm{G}}_{\{\mathcal{L}\}}(\omega_c) = 10^{-9} \begin{bmatrix}
|
|
-99 & 74 & -74 \\
|
|
74 & -247 & 102 \\
|
|
-74 & 102 & -247
|
|
\end{bmatrix} \\
|
|
& \xrightarrow[\text{SVD}]{\phantom{\text{approximation}}} \bm{U} = \begin{bmatrix}
|
|
0.34 & 0 & 0.94 \\
|
|
-0.66 & 0.71 & 0.24 \\
|
|
0.66 & 0.71 & -0.24
|
|
\end{bmatrix}, \ \bm{V} = \begin{bmatrix}
|
|
-0.34 & 0 & -0.94 \\
|
|
0.66 & -0.71 & -0.24 \\
|
|
-0.66 & -0.71 & 0.24
|
|
\end{bmatrix}
|
|
\end{align}
|
|
\end{equation}
|
|
|
|
Using these $\bm{U}$ and $\bm{V}$ matrices, the decoupled plant is computed according to equation eqref:eq:detail_control_decoupling_plant_svd.
|
|
The resulting plant, depicted in Figure ref:fig:detail_control_decoupling_svd_plant, exhibits remarkable decoupling across a broad frequency range, extending well beyond the vicinity of $\omega_c$.
|
|
Additionally, the diagonal terms manifest as second-order dynamic systems, facilitating straightforward controller design.
|
|
|
|
#+begin_src matlab
|
|
%% SVD Decoupling
|
|
wc = 2*pi*100; % Decoupling frequency [rad/s]
|
|
% System's response at the decoupling frequency
|
|
H1 = evalfr(G_L, j*wc);
|
|
|
|
% Real approximation of G(j.wc)
|
|
D = pinv(real(H1'*H1));
|
|
H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
|
|
|
|
[U,S,V] = svd(H1);
|
|
|
|
Gsvd = inv(U)*G_L*inv(V');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
figure;
|
|
hold on;
|
|
for i_in = 1:3
|
|
for i_out = [i_in+1:3]
|
|
plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
end
|
|
plot(freqs, abs(squeeze(freqresp(Gsvd(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
|
'DisplayName', '$G_{SVD}(i,j)\ i \neq j$');
|
|
set(gca,'ColorOrderIndex',1)
|
|
for i_in_out = 1:3
|
|
plot(freqs, abs(squeeze(freqresp(Gsvd(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{SVD}(%d,%d)$', i_in_out, i_in_out));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
|
ylim([1e-10, 2e-4]);
|
|
leg = legend('location', 'northeast', 'FontSize', 8);
|
|
leg.ItemTokenSize(1) = 18;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_decoupling_svd_plant.pdf', 'width', 'wide', 'height', 'normal');
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_decoupling_svd_plant
|
|
#+caption: Plant dynamics $\bm{G}_{\text{SVD}}(s)$ obtained after decoupling using Singular Value Decomposition
|
|
#+RESULTS:
|
|
[[file:figs/detail_control_decoupling_svd_plant.png]]
|
|
|
|
As it was surprising to obtain such a good decoupling at all frequencies, a variant system with identical dynamics but different sensor configurations was examined.
|
|
Instead of using relative motion sensors aligned with the struts, three relative motion sensors were positioned as shown in Figure ref:fig:detail_control_decoupling_model_test_alt.
|
|
Although Jacobian matrices could theoretically map between these different sensor arrangements, application of the same SVD decoupling procedure yielded the plant response shown in Figure ref:fig:detail_control_decoupling_svd_alt_plant, which exhibits significantly greater coupling.
|
|
Notably, the coupling demonstrates local minima near the decoupling frequency, consistent with the fact that the decoupling matrices were derived specifically for that frequency point.
|
|
|
|
#+begin_src matlab
|
|
%% Simscape model with relative motion sensor at alternative positions
|
|
mdl = 'detail_control_decoupling_test_model';
|
|
open(mdl)
|
|
|
|
deq = 0.2; % Length of the actuators [m]
|
|
|
|
% Input/Output definition
|
|
clear io; io_i = 1;
|
|
io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Payload'], 1, 'openoutput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Payload'], 2, 'openoutput'); io_i = io_i + 1;
|
|
io(io_i) = linio([mdl, '/Payload'], 3, 'openoutput'); io_i = io_i + 1;
|
|
|
|
G_L_alt = linearize(mdl, io);
|
|
G_L_alt.InputName = {'F1', 'F2', 'F3'};
|
|
G_L_alt.OutputName = {'d1', 'd2', 'd32'};
|
|
|
|
% SVD Decoupling with the new plant
|
|
wc = 2*pi*100; % Decoupling frequency [rad/s]
|
|
% System's response at the decoupling frequency
|
|
H1 = evalfr(G_L_alt, j*wc);
|
|
|
|
% Real approximation of G(j.wc)
|
|
D = pinv(real(H1'*H1));
|
|
H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
|
|
|
|
[U,S,V] = svd(H1);
|
|
|
|
Gsvd_alt = inv(U)*G_L_alt*inv(V');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results none
|
|
%% Obtained plant after SVD decoupling - Relative motion sensors are not collocated with the actuators
|
|
figure;
|
|
hold on;
|
|
for i_in = 1:3
|
|
for i_out = [i_in+1:3]
|
|
plot(freqs, abs(squeeze(freqresp(Gsvd_alt(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
|
'HandleVisibility', 'off');
|
|
end
|
|
end
|
|
plot(freqs, abs(squeeze(freqresp(Gsvd_alt(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
|
|
'DisplayName', '$G_{SVD}(i,j)\ i \neq j$');
|
|
set(gca,'ColorOrderIndex',1)
|
|
for i_in_out = 1:3
|
|
plot(freqs, abs(squeeze(freqresp(Gsvd_alt(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_{SVD}(%d,%d)$', i_in_out, i_in_out));
|
|
end
|
|
hold off;
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
|
ylim([5e-11, 7e-5]);
|
|
leg = legend('location', 'southwest', 'FontSize', 8);
|
|
leg.ItemTokenSize(1) = 18;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_decoupling_svd_alt_plant.pdf', 'width', 'half', 'height', 'normal');
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_svd_decoupling_not_symmetrical
|
|
#+caption: Application of SVD decoupling on a system schematically shown in (\subref{fig:detail_control_decoupling_model_test_alt}). The obtained decoupled plant is shown in (\subref{fig:detail_control_decoupling_svd_alt_plant}).
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_model_test_alt}Alternative location of sensors}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_decoupling_model_test_alt.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_decoupling_svd_alt_plant}Obtained decoupled plant}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :width 0.95\linewidth
|
|
[[file:figs/detail_control_decoupling_svd_alt_plant.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
The exceptional performance of SVD decoupling on the plant with collocated sensors warrants further investigation.
|
|
This effectiveness may be attributed to the symmetrical properties of the plant, as evidenced in the Bode plots of the decentralized plant shown in Figure ref:fig:detail_control_decoupling_coupled_plant_bode.
|
|
The phenomenon potentially relates to previous research on SVD controllers applied to systems with specific symmetrical characteristics [[cite:&hovd97_svd_contr_contr]].
|
|
|
|
** Comparison of decoupling strategies
|
|
<<ssec:detail_control_decoupling_comp>>
|
|
|
|
While the three proposed decoupling methods may appear similar in their mathematical implementation (each involving pre-multiplication and post-multiplication of the plant with constant matrices), they differ significantly in their underlying approaches and practical implications, as summarized in Table ref:tab:detail_control_decoupling_strategies_comp.
|
|
|
|
Each method employs a distinct conceptual framework: Jacobian decoupling is "topology-driven", relying on the geometric configuration of the system; modal decoupling is "physics-driven", based on the system's dynamical equations; and SVD decoupling is "data-driven", utilizing measured frequency response functions.
|
|
|
|
The physical interpretation of decoupled plant inputs and outputs varies considerably among these methods.
|
|
With Jacobian decoupling, inputs and outputs retain clear physical meaning, corresponding to forces/torques and translations/rotations in a specified reference frame.
|
|
Modal decoupling arranges inputs to excite individual modes, with outputs combined to measure these modes separately.
|
|
For SVD decoupling, inputs and outputs represent special directions ordered by decreasing controllability and observability at the chosen frequency, though physical interpretation becomes challenging for parallel manipulators.
|
|
|
|
This difference in interpretation relates directly to the "control space" in which the controllers operate.
|
|
When these "control spaces" meaningfully relate to the control objectives, controllers can be tuned to directly match specific requirements.
|
|
For Jacobian decoupling, the controller typically operates in a frame positioned at the point where motion needs to be controlled, for instance where the light is focused in the NASS application.
|
|
Modal decoupling provides a natural framework when specific vibrational modes require targeted control.
|
|
SVD decoupling generally results in a loss of physical meaning for the "control space", potentially complicating the process of relating controller design to practical system requirements.
|
|
|
|
The quality of decoupling achieved through these methods also exhibits distinct characteristics.
|
|
Jacobian decoupling performance depends on the chosen reference frame, with optimal decoupling at low frequencies when aligned at the center of stiffness, or at high frequencies when aligned with the center of mass.
|
|
Systems designed with coincident centers of mass and stiffness may achieve excellent decoupling using this approach.
|
|
Modal decoupling offers good decoupling across all frequencies, though its effectiveness relies on the accuracy of the system model, with discrepancies potentially resulting in significant off-diagonal elements.
|
|
The diagonal elements typically manifest as second-order low-pass filters, facilitating straightforward control design.
|
|
SVD decoupling can be implemented using measured data without requiring a model, with optimal performance near the chosen decoupling frequency, though its effectiveness may diminish at other frequencies and depends on the quality of the real approximation of the response at the selected frequency point.
|
|
|
|
#+name: tab:detail_control_decoupling_strategies_comp
|
|
#+caption: Comparison of decoupling strategies
|
|
#+attr_latex: :environment tabularx :width \linewidth :align lXXX
|
|
#+attr_latex: :center t :booktabs t :font \scriptsize
|
|
| | *Jacobian* | *Modal* | *SVD* |
|
|
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
|
|
| *Philosophy* | Topology Driven | Physics Driven | Data Driven |
|
|
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
|
|
| *Requirements* | Known geometry | Known equations of motion | Identified FRF |
|
|
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
|
|
| *Decoupling Matrices* | Decoupling using $\bm{J}_{\{O\}}$ obtained from geometry | Decoupling using $\bm{\Phi}$ obtained from modal decomposition | Decoupling using $\bm{U}$ and $\bm{V}$ obtained from SVD |
|
|
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
|
|
| *Decoupled Plant* | $\bm{G}_{\{O\}}(s) = \bm{J}_{\{O\}}^{-1} \bm{G}_{\mathcal{L}}(s) \bm{J}_{\{O\}}^{-\intercal}$ | $\bm{G}_m(s) = \bm{\Phi}^{-1} \bm{G}_{\mathcal{X}}(s) \bm{\Phi}^{-\intercal}$ | $\bm{G}_{\text{SVD}}(s) = \bm{U}^{-1} \bm{G}(s) \bm{V}^{-\intercal}$ |
|
|
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
|
|
| *Controller* | $\bm{K}_{\{O\}}(s) = \bm{J}_{\{O\}}^{-\intercal} \bm{K}_{d}(s) \bm{J}_{\{O\}}^{-1}$ | $\bm{K}_m(s) = \bm{\Phi}^{-\intercal} \bm{K}_{d}(s) \bm{\Phi}^{-1}$ | $\bm{K}_{\text{SVD}}(s) = \bm{V}^{-\intercal} \bm{K}_{d}(s) \bm{U}^{-1}$ |
|
|
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
|
|
| *Interpretation* | Forces/Torques to Displacement/Rotation in chosen frame | Inputs to excite individual modes | Directions of max to min controllability/observability |
|
|
| | | Output to sense individual modes | |
|
|
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
|
|
| *Properties* | Decoupling at low or high frequency depending on the chosen frame | Good decoupling at all frequencies | Good decoupling near the chosen frequency |
|
|
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
|
|
| *Pros* | Physical inputs / outputs | Target specific modes | Good Decoupling near the crossover |
|
|
| | Good decoupling at High frequency (diagonal mass matrix if Jacobian taken at the CoM) | 2nd order diagonal plant | Very General |
|
|
| | Good decoupling at Low frequency (if Jacobian taken at specific point) | | |
|
|
| | Easy integration of meaningful reference inputs | | |
|
|
| | | | |
|
|
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
|
|
| *Cons* | Coupling between force/rotation may be high at low frequency (non diagonal terms in K) | Need analytical equations | Loose the physical meaning of inputs /outputs |
|
|
| | Limited to parallel mechanisms (?) | | Decoupling depends on the real approximation validity |
|
|
| | If good decoupling at all frequencies => requires specific mechanical architecture | | Diagonal plants may not be easy to control |
|
|
|-----------------------+----------------------------------------------------------------------------------------+-----------------------------------------------------------------------+------------------------------------------------------------------|
|
|
| *Applicability* | Parallel Mechanisms | Systems whose dynamics that can be expressed with M and K matrices | Very general |
|
|
| | Only small motion for the Jacobian matrix to stay constant | | Need FRF data (either experimentally or analytically) |
|
|
|
|
* Closed-Loop Shaping using Complementary Filters
|
|
:PROPERTIES:
|
|
:HEADER-ARGS:matlab+: :tangle matlab/detail_control_3_close_loop_shaping.m
|
|
:END:
|
|
<<sec:detail_control_cf>>
|
|
|
|
** Introduction :ignore:
|
|
|
|
Once the system is properly decoupled using one of the approaches described in Section ref:sec:detail_control_decoupling, SISO controllers can be individually tuned for each decoupled "directions".
|
|
Several ways to design a controller to obtain a given performance while ensuring good robustness properties can be implemented.
|
|
|
|
# Add reference
|
|
In some cases, "fixed" controller structures are utilized, such as PI and PID controllers [[cite:&furutani04_nanom_cuttin_machin_using_stewar;&du14_piezo_actuat_high_precis_flexib;&yang19_dynam_model_decoup_contr_flexib]].
|
|
In such cases, the controller coefficients are manually tuned to obtain acceptable performance and robustness.
|
|
|
|
Another popular method is Open-Loop shaping, that was used during the conceptual phase after the plan was decoupled in the frame of the struts.
|
|
The idea of open-loop shaping is to tune the controller (using a series of standard leads, lags, notches, low pass filters) such that the open-loop transfer function $G(s)K(s)$ is made according to specification (i.e.
|
|
bandwidth, gain and phase margins, gain at a specific frequency, etc...) [[cite:&schmidt20_desig_high_perfor_mechat_third_revis_edition, chapt. 4.4.7]].
|
|
Open-Loop shaping is very popular because the open-loop transfer function depends linearly on the controller, making it relatively straightforward to tune the controller to achieve desired open-loop characteristics.
|
|
Another key advantage is that controllers can be tuned directly from measured frequency response functions without requiring an explicit plant model.
|
|
|
|
However, the behavior (i.e. performance) of a feedback system is a function of closed-loop transfer functions [[cite:&skogestad07_multiv_feedb_contr, chapt. 3]].
|
|
Specifications can therefore be expressed in terms of the magnitude of closed-loop transfer functions, such as the sensitivity, plant sensitivity, and complementary sensitivity transfer functions.
|
|
With open-loop shaping, closed-loop transfer functions are changed only indirectly, which may make it difficult to directly address the specifications that are in terms of the closed-loop transfer functions.
|
|
|
|
In order to synthesize a controller that directly shapes the closed-loop transfer functions (and therefore the performance metric), $\mathcal{H}_\infty$ loop-shaping may be used [[cite:&skogestad07_multiv_feedb_contr]].
|
|
This approach requires a good model of the plant and expertise in selecting weighting functions that will define the wanted shape of different closed-loop transfer functions [[cite:&bibel92_guidel_h]].
|
|
$\mathcal{H}_{\infty}$ synthesis has been applied for the Stewart platform [[cite:&jiao18_dynam_model_exper_analy_stewar]], but comparative studies with more simple decentralized controllers did not show large improvements [[cite:&thayer02_six_axis_vibrat_isolat_system;&hauge04_sensor_contr_space_based_six]].
|
|
|
|
In this section, an alternative controller synthesis scheme is proposed in which complementary filters are used for directly shaping the closed-loop transfer functions (i.e., directly addressing the closed-loop performances).
|
|
|
|
In Section ref:ssec:detail_control_cf_control_arch, the proposed control architecture including the complementary filters is presented.
|
|
In Section ref:ssec:detail_control_cf_trans_perf, typical performance requirements are translated into the shape of the complementary filters.
|
|
The design of the complementary filters is briefly discussed in Section ref:ssec:detail_control_cf_analytical_complementary_filters, and analytical formulas are proposed such that it is possible to change the closed-loop behavior of the system in real time.
|
|
Finally, in Section ref:ssec:detail_control_cf_simulations, a numerical example is used to show how the proposed control architecture can be implemented in practice.
|
|
|
|
** Matlab Init :noexport:ignore:
|
|
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
|
|
<<matlab-dir>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results silent :noweb yes
|
|
<<matlab-init>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :noweb yes
|
|
<<m-init-path>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :eval no :noweb yes
|
|
<<m-init-path-tangle>>
|
|
#+end_src
|
|
|
|
#+begin_src matlab :noweb yes
|
|
<<m-init-other>>
|
|
|
|
%% Initialize Frequency Vector
|
|
freqs = logspace(-1, 3, 1000);
|
|
#+end_src
|
|
|
|
** Control Architecture
|
|
<<ssec:detail_control_cf_control_arch>>
|
|
**** Virtual Sensor Fusion
|
|
|
|
The concept of using complementary filters in control architecture originates from sensor fusion techniques [[cite:&collette15_sensor_fusion_method_high_perfor]], where two sensors are combined using complementary filters.
|
|
Building upon this concept, "virtual sensor fusion" [[cite:&verma20_virtual_sensor_fusion_high_precis_contr]] replaces one physical sensor with a model $G$ of the plant.
|
|
|
|
The control architecture is illustrated in Figure ref:fig:detail_control_cf_arch, where $G^\prime$ represents the physical plant to be controlled, $G$ is a model of the plant, $k$ is the controller, and $H_L$ and $H_H$ are complementary filters satisfying $H_L(s) + H_H(s) = 1$.
|
|
In this arrangement, the physical plant is controlled at low frequencies, while the plant model is utilized at high frequencies to enhance robustness.
|
|
|
|
#+begin_src latex :file detail_control_cf_arch.pdf
|
|
\tikzset{block/.default={0.8cm}{0.6cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\def\cdist{0.7}
|
|
|
|
\begin{tikzpicture}
|
|
\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
|
|
\node[block, right=0.3 of addfb] (K){$k$};
|
|
\node[block, right=2.2 of K] (G){$G^\prime$};
|
|
\node[addb={+}{}{}{}{}, right=0.3 of G] (adddy){};
|
|
\coordinate[] (KG) at ($(K.east)+(0.3, 0)$);
|
|
\node[block, below=of KG] (Gm){$G$};
|
|
\node[block, below=0.4 of Gm] (Hh){$H_H$};
|
|
\node[addb={+}{}{}{}{}, below=0.4 of Hh] (addcf){};
|
|
\node[block, right=0.3 of addcf] (Hl) {$H_L$};
|
|
\node[addb={+}{}{}{}{}, right=2.1 of Hl] (addn) {};
|
|
|
|
|
|
\draw[->] (addfb.east) -- (K.west) node[above left]{};
|
|
\draw[->] (K.east) -- (G.west) node[above left]{$u$};
|
|
\draw[->] (KG) node[branch]{} -- (Gm.north);
|
|
\draw[->] (Gm.south) -- (Hh.north);
|
|
\draw[->] (Hh.south) -- (addcf.north) node[above left]{};
|
|
\draw[->] (Hl.west) -- (addcf.east);
|
|
\draw[->] (addcf.west) -| (addfb.south) node[below right]{};
|
|
\draw[->] (G.east) -- (adddy.west);
|
|
\draw[<-] (addn.east) -- ++(0.5, 0) coordinate[](endpos) node[above left]{$n$};
|
|
\draw[->] (adddy.east) -- (G-|endpos) node[above left]{$y$};
|
|
\draw[->] (adddy-|addn) node[branch]{} -- (addn.north);
|
|
\draw[<-] (addfb.west) -- ++(-0.5, 0) node[above right](r){$r$};
|
|
\draw[->] (addn.west) -- (Hl.east) node[above right]{$y_m$};
|
|
\draw[<-] (adddy.north) -- ++(0, 0.5) node[below right]{$d_y$};
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={(Hl.south east) (r.north west)}, inner sep=4pt, draw, fill=black!20!white, dashed, label={RT controller}] (Kfb) {};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+begin_src latex :file detail_control_cf_arch_eq.pdf
|
|
\tikzset{block/.default={0.8cm}{0.6cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\def\cdist{0.7}
|
|
|
|
\begin{tikzpicture}
|
|
\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
|
|
\node[addb={+}{}{}{}{-}, right=0.3 of addfb] (addK){};
|
|
\node[block, right=0.6 of addK] (K){$k$};
|
|
\node[block, right=1.5 of K] (G){$G^\prime$};
|
|
\node[addb={+}{}{}{}{}, right=0.3 of G] (adddy){};
|
|
\node[block, below right=0.5 and -0.15 of K] (Gm){$G$};
|
|
\node[block, below left =0.5 and -0.15 of K] (Hh){$H_H$};
|
|
\node[block, below=1.5 of K] (Hl) {$H_L$};
|
|
\node[addb={+}{}{}{}{}, right=3.0 of Hl] (addn) {};
|
|
|
|
\draw[->] (addfb.east) -- (addK.west);
|
|
\draw[->] (addK.east) -- (K.west);
|
|
\draw[->] (K.east) -- (G.west) node[above left]{$u$};
|
|
\draw[->] (G.east) -- (adddy.west);
|
|
\draw[->] ($(G.west)+(-0.5, 0)$) node[branch](cffb){} |- (Gm.east);
|
|
\draw[->] (Gm.west) -- (Hh.east);
|
|
\draw[->] (Hh.west) -| (addK.south);
|
|
\draw[<-] (addn.east) -- ++(0.5, 0) coordinate[](endpos) node[above left]{$n$};
|
|
\draw[->] (adddy.east) -- (G-|endpos) node[above left]{$y$};
|
|
\draw[->] (adddy-|addn) node[branch]{} -- (addn.north);
|
|
\draw[<-] (addfb.west) -- ++(-0.5, 0) node[above right](r){$r$};
|
|
\draw[->] (addn.west) -- (Hl.east) node[above right]{$y_m$};
|
|
\draw[<-] (adddy.north) -- ++(0, 0.5) node[below right]{$d_y$};
|
|
\draw[->] (Hl.west) -| (addfb.south) node[below right]{};
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={(Hl.south -| cffb) (r.north west)}, inner sep=4pt, draw, fill=black!20!white, dashed, label={RT controller}] (Kfb) {};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_cf_arch_and_eq
|
|
#+caption: Control architecture for virtual sensor fusion (\subref{fig:detail_control_cf_arch}). An equivalent architecture is shown in (\subref{fig:detail_control_cf_arch_eq}). The signals are the reference signal $r$, the output perturbation $d_y$, the measurement noise $n$ and the control input $u$.
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_arch}Virtual Sensor Fusion}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 0.9
|
|
[[file:figs/detail_control_cf_arch.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_arch_eq}Equivalent Architecture}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 0.9
|
|
[[file:figs/detail_control_cf_arch_eq.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
Although the control architecture shown in Figure ref:fig:detail_control_cf_arch appears to be a multi-loop system, it should be noted that no non-linear saturation-type elements are present in the inner loop (containing $k$, $G$, and $H_H$, all numerically implemented).
|
|
Consequently, this structure is mathematically equivalent to the single-loop architecture illustrated in Figure ref:fig:detail_control_cf_arch_eq.
|
|
|
|
**** Asymptotic behavior
|
|
|
|
When considering the extreme case of very high values for $k$, the effective controller $K(s)$ converges to the inverse of the plant model multiplied by the inverse of the high-pass filter, as expressed in eqref:eq:detail_control_cf_high_k.
|
|
|
|
\begin{equation}\label{eq:detail_control_cf_high_k}
|
|
\lim_{k\to\infty} K(s) = \lim_{k\to\infty} \frac{k}{1+H_H(s) G(s) k} = \left( H_H(s) G(s) \right)^{-1}
|
|
\end{equation}
|
|
|
|
If the resulting $K$ is improper, a low-pass filter with sufficiently high corner frequency can be added to ensure its causal realization.
|
|
Furthermore, for $K$ to be stable, both $G$ and $H_H$ must be minimum phase transfer functions.
|
|
|
|
With these assumptions, the resulting control architecture is illustrated in Figure ref:fig:detail_control_cf_arch_class, where the complementary filters $H_L$ and $H_H$ remain the only tuning parameters.
|
|
The dynamics of this closed-loop system are described by eqref:eq:detail_control_cf_sf_cl_tf_K_inf.
|
|
|
|
#+begin_src latex :file detail_control_cf_arch_class.pdf
|
|
\tikzset{block/.default={0.8cm}{0.6cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\def\cdist{0.7}
|
|
|
|
\begin{tikzpicture}
|
|
\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
|
|
\node[block, right=of addfb] (K){$K$};
|
|
\node[block, right=of K] (G){$G^\prime$};
|
|
\node[addb={+}{}{}{}{}, right=of G] (adddy){};
|
|
\node[addb={+}{}{}{}{}, below right=0.7 and 0.3 of adddy] (addn) {};
|
|
\node[block] (Hl) at (K|-addn) {$H_L$};
|
|
|
|
\draw[->] (addfb.east) -- (K.west) node[above left]{};
|
|
\draw[->] (K.east) -- (G.west) node[above left]{$u$};
|
|
\draw[->] (G.east) -- (adddy.west);
|
|
\draw[<-] (addn.east) -- ++(\cdist, 0) coordinate[](endpos) node[above left]{$n$};
|
|
\draw[->] (G-|addn)node[branch]{} -- (addn.north);
|
|
\draw[->] (adddy.east) -- (G-|endpos) node[above left]{$y$};
|
|
\draw[<-] (addfb.west) -- ++(-\cdist, 0) node[above right](r){$r$};
|
|
\draw[->] (addn.west) -- (Hl.east);
|
|
\draw[->] (Hl.west) -| (addfb.south);
|
|
\draw[<-] (adddy.north) -- ++(0, \cdist) node[below right]{$d_y$};
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={(Hl.south east) (r.north west)}, inner sep=4pt, draw, fill=black!20!white, dashed, label={RT controller}] (Kfb) {};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_cf_arch_class
|
|
#+caption: Equivalent classical feedback control architecture
|
|
#+RESULTS:
|
|
[[file:figs/detail_control_cf_arch_class.png]]
|
|
|
|
\begin{subequations}\label{eq:detail_control_cf_sf_cl_tf_K_inf}
|
|
\begin{align}
|
|
y &= \frac{ H_H dy + G^{\prime} G^{-1} r - G^{\prime} G^{-1} H_L n }{H_H + G^\prime G^{-1} H_L} \label{eq:detail_control_cf_cl_system_y}\\
|
|
u &= \frac{ -G^{-1} H_L dy + G^{-1} r - G^{-1} H_L n }{H_H + G^\prime G^{-1} H_L} \label{eq:detail_control_cf_cl_system_u}
|
|
\end{align}
|
|
\end{subequations}
|
|
|
|
At frequencies where the model accurately represents the physical plant ($G^{-1} G^{\prime} \approx 1$), the denominator simplifies to $H_H + G^\prime G^{-1} H_L \approx H_H + H_L = 1$, and the closed-loop transfer functions are described by eqref:eq:detail_control_cf_sf_cl_tf_K_inf_perfect.
|
|
|
|
\begin{subequations}\label{eq:detail_control_cf_sf_cl_tf_K_inf_perfect}
|
|
\begin{alignat}{5}
|
|
y &= H_H dy &&+ r &&- H_L n \label{eq:detail_control_cf_cl_performance_y} \\
|
|
u &= -G^{-1} H_L dy &&+ G^{-1} r &&- G^{-1} H_L n \label{eq:detail_control_cf_cl_performance_u}
|
|
\end{alignat}
|
|
\end{subequations}
|
|
|
|
The sensitivity transfer function equals the high-pass filter $S = \frac{y}{dy} = H_H$, and the complementary sensitivity transfer function equals the low-pass filter $T = \frac{y}{n} = H_L$.
|
|
Hence, when the plant model closely approximates the actual system, the closed-loop behavior becomes fully determined by the designed complementary filters, enabling direct translation of performance requirements into filter design.
|
|
|
|
** Translating the performance requirements into the shapes of the complementary filters
|
|
<<ssec:detail_control_cf_trans_perf>>
|
|
**** Introduction :ignore:
|
|
Performance specifications in feedback systems can be expressed as upper bounds on the magnitudes of closed-loop transfer functions such that the sensitivity $|S(j\omega)|$ and complementary sensitivity $|T(j\omega)|$ transfer functions [[cite:&bibel92_guidel_h]].
|
|
The design of a controller $K(s)$ to achieve desired shapes of these closed-loop transfer functions is known as closed-loop shaping.
|
|
|
|
In the proposed control architecture, the closed-loop transfer functions eqref:eq:detail_control_cf_sf_cl_tf_K_inf are expressed in terms of the complementary filters $H_L(s)$ and $H_H(s)$ rather than directly through the controller $K(s)$.
|
|
Therefore, performance requirements must be translated into constraints on the shapes of these complementary filters.
|
|
|
|
**** Nominal Stability (NS)
|
|
A closed-loop system is stable when all its elements (here $K$, $G^\prime$, and $H_L$) are stable and the sensitivity function $S = \frac{1}{1 + G^\prime K H_L}$ is stable.
|
|
For the nominal system ($G^\prime = G$), the sensitivity transfer function equals the high-pass filter: $S(s) = H_H(s)$.
|
|
|
|
Nominal stability is therefore guaranteed when $H_L$, $H_H$, and $G$ are stable, and both $G$ and $H_H$ are minimum phase (ensuring $K$ is stable).
|
|
Consequently, stable and minimum phase complementary filters must be employed.
|
|
|
|
**** Nominal Performance (NP)
|
|
|
|
Performance specifications can be formalized using weighting functions $w_H$ and $w_L$, where performance is achieved when eqref:eq:detail_control_cf_weights is satisfied.
|
|
|
|
\begin{subequations}\label{eq:detail_control_cf_weights}
|
|
\begin{align}
|
|
|w_H(j\omega) S(j\omega)| &\le 1 \quad \forall\omega\\
|
|
|w_L(j\omega) T(j\omega)| &\le 1 \quad \forall\omega
|
|
\end{align}
|
|
\end{subequations}
|
|
|
|
For the nominal system, where $S = H_H$ and $T = H_L$, nominal performance is ensured by satisfying eqref:eq:detail_control_cf_nominal_performance.
|
|
|
|
\begin{equation}\label{eq:detail_control_cf_nominal_performance}
|
|
\Aboxed{\text{NP} \Longleftrightarrow {\begin{cases*}
|
|
|w_H(j\omega) H_H(j\omega)| \le 1 & \forall\omega \\
|
|
|w_L(j\omega) H_L(j\omega)| \le 1 & \forall\omega
|
|
\end{cases*}}}
|
|
\end{equation}
|
|
|
|
Typical performance requirements can therefore be translated into constraints on the complementary filters.
|
|
For disturbance rejection, the magnitude of the sensitivity function $|S(j\omega)| = |H_H(j\omega)|$ should be minimized, particularly at low frequencies where disturbances are usually most prominent.
|
|
Similarly, for noise attenuation, the magnitude of the complementary sensitivity function $|T(j\omega)| = |H_L(j\omega)|$ should be minimized, especially at high frequencies where measurement noise typically dominates.
|
|
The closed-loop bandwidth can be effectively limited by ensuring that $|T(j\omega)|$ remains below $\frac{1}{\sqrt{2}}$ at frequencies above the maximum desired bandwidth.
|
|
By carefully selecting the shapes of these complementary filters, nominal performance specifications can be directly addressed in an intuitive manner.
|
|
|
|
Classical stability margins (gain and phase margins) are also related to the maximum amplitude of the sensitivity transfer function.
|
|
Typically, maintaining $|S|_{\infty} \le 2$ ensures a gain margin of at least 2 and a phase margin of at least $\SI{29}{\degree}$.
|
|
|
|
**** Robust Stability (RS)
|
|
|
|
Robust stability refers to a control system's ability to maintain stability despite discrepancies between the actual system $G^\prime$ and the model $G$ used for controller design.
|
|
These discrepancies may arise from unmodeled dynamics or nonlinearities.
|
|
|
|
To represent these model-plant differences, input multiplicative uncertainty as illustrated in Figure ref:fig:detail_control_cf_input_uncertainty is employed.
|
|
The set of possible plants $\Pi_i$ is described by eqref:eq:detail_control_cf_multiplicative_uncertainty.
|
|
With the weighting function $w_I$ selected such that all possible plants $G^\prime$ are contained within the set $\Pi_i$.
|
|
|
|
\begin{equation}\label{eq:detail_control_cf_multiplicative_uncertainty}
|
|
\Pi_i: \quad G^\prime(s) = G(s)\big(1 + w_I(s)\Delta_I(s)\big); \quad |\Delta_I(j\omega)| \le 1 \ \forall\omega
|
|
\end{equation}
|
|
|
|
#+begin_src latex :file detail_control_cf_input_uncertainty.pdf
|
|
\tikzset{block/.default={0.8cm}{0.6cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\def\cdist{0.7}
|
|
|
|
\begin{tikzpicture}
|
|
% Blocs
|
|
\node[block] (G) {$G$};
|
|
|
|
\node[addb, left= of G] (addi) {};
|
|
\node[block, above left=0.3 and 0.3 of addi] (deltai) {$\Delta_I$};
|
|
\node[block, left= of deltai] (wi) {$w_I$};
|
|
\node[branch] (branch) at ($(wi.west|-addi)+(-0.4, 0)$) {};
|
|
|
|
% Connections and labels
|
|
\draw[->] (branch.center) |- (wi.west);
|
|
\draw[->] ($(branch)+(-0.6, 0)$) -- (addi.west);
|
|
\draw[->] (wi.east) -- (deltai.west);
|
|
\draw[->] (deltai.east) -| (addi.north);
|
|
\draw[->] (addi.east) -- (G.west);
|
|
\draw[->] (G.east) -- ++(0.6, 0);
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={(branch|-wi.north) (G.south east)}, inner sep=6pt, draw, dashed, fill=black!20!white] (Gp) {};
|
|
\node[below left] at (Gp.north east) {$G\prime$};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_cf_input_uncertainty_nyquist
|
|
#+caption: Input multiplicative uncertainty to model the differences between the model and the physical plant (\subref{fig:detail_control_cf_input_uncertainty}). Effect of this uncertainty is displayed on the Nyquist plot (\subref{fig:detail_control_cf_nyquist_uncertainty})
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_input_uncertainty}Input multiplicative uncertainty}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_cf_input_uncertainty.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_nyquist_uncertainty}Nyquist plot - Effect of multiplicative uncertainty}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_cf_nyquist_uncertainty.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
When considering input multiplicative uncertainty, robust stability can be derived graphically from the Nyquist plot (illustrated in Figure ref:fig:detail_control_cf_nyquist_uncertainty), yielding to eqref:eq:detail_control_cf_robust_stability_graphically, as demonstrated in [[cite:&skogestad07_multiv_feedb_contr, chapt. 7.5.1]].
|
|
|
|
\begin{equation}\label{eq:detail_control_cf_robust_stability_graphically}
|
|
\text{RS} \Longleftrightarrow \left|w_I(j\omega) L(j\omega) \right| \le \left| 1 + L(j\omega) \right| \quad \forall\omega
|
|
\end{equation}
|
|
|
|
After algebraic manipulation, robust stability is guaranteed when the low-pass complementary filter $H_L$ satisfies eqref:eq:detail_control_cf_condition_robust_stability.
|
|
|
|
\begin{equation}\label{eq:detail_control_cf_condition_robust_stability}
|
|
\boxed{\text{RS} \Longleftrightarrow |w_I(j\omega) H_L(j\omega)| \le 1 \quad \forall \omega}
|
|
\end{equation}
|
|
|
|
**** Robust Performance (RP)
|
|
|
|
Robust performance ensures that performance specifications eqref:eq:detail_control_cf_weights are met even as plant dynamics varies within specified bounds.
|
|
This requires the performance condition to be valid for all possible plants in the defined uncertainty set $\Pi_i$:
|
|
|
|
\begin{equation}\label{eq:detail_control_cf_robust_perf_S}
|
|
\text{RP} \Longleftrightarrow |w_H(j\omega) S(j\omega)| \le 1 \quad \forall G^\prime \in \Pi_I, \ \forall\omega
|
|
\end{equation}
|
|
|
|
Transforming this condition into constraints on the complementary filters yields:
|
|
|
|
\begin{equation}\label{eq:detail_control_cf_robust_performance}
|
|
\boxed{\text{RP} \Longleftrightarrow | w_H(j\omega) H_H(j\omega) | + | w_I(j\omega) H_L(j\omega) | \le 1, \ \forall\omega}
|
|
\end{equation}
|
|
|
|
The robust performance condition effectively combines both nominal performance eqref:eq:detail_control_cf_nominal_performance and robust stability conditions eqref:eq:detail_control_cf_condition_robust_stability.
|
|
If both NP and RS conditions are satisfied, robust performance will be achieved within a factor of 2 [[cite:&skogestad07_multiv_feedb_contr, chapt. 7.6]].
|
|
Therefore, for SISO systems, ensuring robust stability and nominal performance is typically sufficient.
|
|
|
|
** Complementary filter design
|
|
<<ssec:detail_control_cf_analytical_complementary_filters>>
|
|
|
|
As proposed in Section ref:sec:detail_control_sensor, complementary filters can be shaped using standard $\mathcal{H}_{\infty}$ synthesis techniques.
|
|
This approach is particularly well-suited since performance requirements were expressed as upper bounds on the magnitude of the complementary filters (Section ref:ssec:detail_control_cf_trans_perf).
|
|
|
|
Alternatively, analytical formulas for complementary filters may be employed.
|
|
For some applications, first-order complementary filters as shown in Equation eqref:eq:detail_control_cf_1st_order are sufficient.
|
|
|
|
\begin{subequations}\label{eq:detail_control_cf_1st_order}
|
|
\begin{align}
|
|
H_L(s) &= \frac{1}{1 + s/\omega_0} \\
|
|
H_H(s) &= \frac{s/\omega_0}{1 + s/\omega_0}
|
|
\end{align}
|
|
\end{subequations}
|
|
|
|
These filters can be transformed into the digital domain using the Bilinear transformation, resulting in the digital filter representations shown in Equation eqref:eq:detail_control_cf_1st_order_z.
|
|
|
|
\begin{subequations}\label{eq:detail_control_cf_1st_order_z}
|
|
\begin{align}
|
|
H_L(z^{-1}) &= \frac{T_s \omega_0 + T_s \omega_0 z^{-1}}{T_s \omega_0 + 2 + (T_s \omega_0 - 2) z^{-1}} \\
|
|
H_H(z^{-1}) &= \frac{2 - 2 z^{-1}}{T_s \omega_0 + 2 + (T_s \omega_0 - 2) z^{-1}}
|
|
\end{align}
|
|
\end{subequations}
|
|
|
|
A significant advantage of using analytical formulas for complementary filters is that key parameters such as $\omega_0$ can be modified in real-time, as illustrated in Figure ref:fig:detail_control_cf_arch_tunable_params.
|
|
This real-time tunability allows rapid testing of different control bandwidths to evaluate performance and robustness characteristics.
|
|
|
|
#+begin_src latex :file detail_control_cf_arch_tunable_params.pdf
|
|
\tikzset{block/.default={0.8cm}{0.6cm}}
|
|
\tikzset{addb/.append style={scale=0.7}}
|
|
\tikzset{node distance=0.6}
|
|
\def\cdist{0.7}
|
|
|
|
\begin{tikzpicture}
|
|
\node[addb={+}{}{}{}{-}] (addfb) at (0, 0){};
|
|
\node[block, right=of addfb] (Hh){$H_H^{-1}$};
|
|
\node[block, right=of Hh] (Ginv){$G^{-1}$};
|
|
\node[block, right=of Ginv] (G){$G^\prime$};
|
|
\node[addb={+}{}{}{}{}, right=of G] (adddy){};
|
|
\node[addb={+}{}{}{}{}, below right=1.2 and 0.3 of adddy] (addn) {};
|
|
\node[block] (Hl) at (Hh|-addn) {$H_L$};
|
|
|
|
\node[color=colorred] (wb) at ($0.5*(Hh.south) + 0.5*(Hl.north)$) {$\bullet$};
|
|
\draw[-, color=colorred] ($(wb) + (-0.6, 0)$)node[left]{$\omega_0$} -- (wb.center);
|
|
\draw[->, color=colorred] (wb.center) -- (Hh.south);
|
|
\draw[->, color=colorred] (wb.center) -- (Hl.north);
|
|
|
|
\draw[->] (addfb.east) -- (Hh.west);
|
|
\draw[->] (Hh.east) -- (Ginv.west);
|
|
\draw[->] (Ginv.east) -- (G.west) node[above left]{$u$};
|
|
\draw[->] (G.east) -- (adddy.west);
|
|
\draw[<-] (addn.east) -- ++(\cdist, 0) coordinate[](endpos) node[above left]{$n$};
|
|
\draw[->] (G-|addn)node[branch]{} -- (addn.north);
|
|
\draw[->] (adddy.east) -- (G-|endpos) node[above left]{$y$};
|
|
\draw[<-] (addfb.west) -- ++(-\cdist, 0) node[above right](r){$r$};
|
|
\draw[->] (addn.west) -- (Hl.east);
|
|
\draw[->] (Hl.west) -| (addfb.south);
|
|
\draw[<-] (adddy.north) -- ++(0, \cdist) node[below right]{$d_y$};
|
|
|
|
\begin{scope}[on background layer]
|
|
\node[fit={(Hl.south -| Ginv.east) (r.north west)}, inner sep=4pt, draw, fill=black!20!white, dashed, label={RT controller}] (Kfb) {};
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_cf_arch_tunable_params
|
|
#+caption: Implemented digital complementary filters with parameter $\omega_0$ that can be changed in real time
|
|
#+RESULTS:
|
|
[[file:figs/detail_control_cf_arch_tunable_params.png]]
|
|
|
|
For many practical applications, first order complementary filters are not sufficient.
|
|
Specifically, a slope of $+2$ at low frequencies for the sensitivity transfer function (enabling accurate tracking of ramp inputs) and a slope of $-2$ for the complementary sensitivity transfer function are often desired.
|
|
For these cases, the second-order complementary filters presented in Equation eqref:eq:detail_control_cf_2nd_order are proposed.
|
|
|
|
\begin{subequations}\label{eq:detail_control_cf_2nd_order}
|
|
\begin{align}
|
|
H_L(s) &= \frac{(1+\alpha) (\frac{s}{\omega_0})+1}{\left((\frac{s}{\omega_0})+1\right) \left((\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1\right)}\\
|
|
H_H(s) &= \frac{(\frac{s}{\omega_0})^2 \left((\frac{s}{\omega_0})+1+\alpha\right)}{\left((\frac{s}{\omega_0})+1\right) \left((\frac{s}{\omega_0})^2 + \alpha (\frac{s}{\omega_0}) + 1\right)}
|
|
\end{align}
|
|
\end{subequations}
|
|
|
|
The influence of parameters $\alpha$ and $\omega_0$ on the frequency response of these complementary filters is illustrated in Figure ref:fig:detail_control_cf_analytical_effect.
|
|
The parameter $\alpha$ primarily affects the damping characteristics near the crossover frequency, while $\omega_0$ determines the frequency at which the transition between high-pass and low-pass behavior occurs.
|
|
These filters can also be implemented in the digital domain with analytical formulas, preserving the ability to adjust $\alpha$ and $\omega_0$ in real-time.
|
|
|
|
The presented analytical formulations offer an attractive balance between design simplicity and performance.
|
|
This capability to tune parameters in real-time is particularly valuable during commissioning of the controller.
|
|
|
|
#+begin_src matlab :exports none :results none
|
|
%% Analytical Complementary Filters - Effect of alpha
|
|
freqs_study = logspace(-2, 2, 1000);
|
|
alphas = [0.1, 1, 10];
|
|
w0 = 2*pi*1;
|
|
s = tf('s')
|
|
|
|
figure;
|
|
hold on;
|
|
for i = 1:length(alphas)
|
|
alpha = alphas(i);
|
|
Hh2 = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
|
|
Hl2 = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
|
|
plot(freqs_study, abs(squeeze(freqresp(Hh2, freqs_study, 'Hz'))), 'color', colors(i,:), 'DisplayName', sprintf('$\\alpha = %g$', alphas(i)));
|
|
plot(freqs_study, abs(squeeze(freqresp(Hl2, freqs_study, 'Hz'))), 'color', colors(i,:), 'HandleVisibility', 'off');
|
|
end
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Relative Frequency $\frac{\omega}{\omega_0}$'); ylabel('Magnitude');
|
|
hold off;
|
|
ylim([1e-3, 20]);
|
|
leg = legend('location', 'northeast', 'FontSize', 8);
|
|
leg.ItemTokenSize(1) = 18;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_cf_analytical_effect_alpha.pdf', 'width', 'half', 'height', 'normal');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results none
|
|
%% Analytical Complementary Filters - Effect of w0
|
|
freqs_study = logspace(-1, 3, 1000);
|
|
alpha = [1];
|
|
w0s = [2*pi*1, 2*pi*10, 2*pi*100];
|
|
s = tf('s')
|
|
|
|
figure;
|
|
hold on;
|
|
for i = 1:length(w0s)
|
|
w0 =w0s(i);
|
|
Hh2 = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
|
|
Hl2 = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
|
|
plot(freqs_study, abs(squeeze(freqresp(Hh2, freqs_study, 'Hz'))), 'color', colors(i,:), 'DisplayName', sprintf('$\\omega_0 = %g$ Hz', w0/2/pi));
|
|
plot(freqs_study, abs(squeeze(freqresp(Hl2, freqs_study, 'Hz'))), 'color', colors(i,:), 'HandleVisibility', 'off');
|
|
end
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
|
hold off;
|
|
xlim([freqs_study(1), freqs_study(end)]); ylim([1e-3, 20]);
|
|
leg = legend('location', 'southeast', 'FontSize', 8);
|
|
leg.ItemTokenSize(1) = 18;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_cf_analytical_effect_w0.pdf', 'width', 'half', 'height', 'normal');
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_cf_analytical_effect
|
|
#+caption: Shape of proposed analytical complementary filters. Effect of $\alpha$ (\subref{fig:detail_control_cf_analytical_effect_alpha}) and $\omega_0$ (\subref{fig:detail_control_cf_analytical_effect_w0}) are shown.
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_analytical_effect_alpha}Effect of $\alpha$}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :width 0.95\linewidth
|
|
[[file:figs/detail_control_cf_analytical_effect_alpha.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_analytical_effect_w0}Effect of $\omega_0$}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :width 0.95\linewidth
|
|
[[file:figs/detail_control_cf_analytical_effect_w0.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
** Numerical Example
|
|
<<ssec:detail_control_cf_simulations>>
|
|
**** Procedure :ignore:
|
|
|
|
To systematically apply the proposed control technique, the following procedure is recommended:
|
|
|
|
1. Identify the plant to be controlled to obtain the plant model $G$.
|
|
2. Design the weighting function $w_I$ such that all possible plants $G^\prime$ are contained in the uncertainty set $\Pi_i$.
|
|
3. Translate performance requirements into upper bounds on the complementary filters as explained in Section ref:ssec:detail_control_cf_trans_perf.
|
|
4. Design the weighting functions $w_H$ and $w_L$ and generate the complementary filters using $\mathcal{H}_{\infty}\text{-synthesis}$ as described in Section ref:ssec:detail_control_sensor_hinf_method.
|
|
If the synthesis fails to produce filters satisfying the defined upper bounds, either revise the requirements or develop a more accurate model $G$ that will allow for a smaller $w_I$.
|
|
For simpler cases, the analytical formulas for complementary filters presented in Section ref:ssec:detail_control_cf_analytical_complementary_filters can be employed.
|
|
5. If $K(s) = H_H^{-1}(s) G^{-1}(s)$ is not proper, add low-pass filters with sufficiently high corner frequencies to ensure realizability.
|
|
|
|
**** Plant
|
|
|
|
To evaluate this control architecture, a simple test model representative of many synchrotron positioning stages is utilized (Figure ref:fig:detail_control_cf_test_model).
|
|
In this model, a payload with mass $m$ is positioned on top of a stage.
|
|
The objective is to accurately position the sample relative to the X-ray beam.
|
|
|
|
The relative position $y$ between the payload and the X-ray is measured, which typically involves measuring the relative position between the focusing optics and the sample.
|
|
Various disturbance forces affect positioning stability, including stage vibrations $d_w$ and direct forces applied to the sample $d_F$ (such as cable forces).
|
|
The positioning stage itself is characterized by stiffness $k$, internal damping $c$, and a controllable force $F$.
|
|
|
|
The model of the plant $G(s)$ from actuator force $F$ to displacement $y$ is described by Equation eqref:eq:detail_control_cf_test_plant_tf.
|
|
|
|
\begin{equation}\label{eq:detail_control_cf_test_plant_tf}
|
|
G(s) = \frac{1}{m s^2 + c s + k}
|
|
\end{equation}
|
|
|
|
The parameter values are set to $m = \SI{20}{\kg}$, $k = 1\si{\N/\mu\m}$, and $c = 10^2\si{\N\per(\m\per\s)}$.
|
|
|
|
The plant dynamics include uncertainties related to limited support compliance, unmodeled flexible dynamics, payload dynamics, and other factors.
|
|
These uncertainties are represented using a multiplicative input uncertainty weight eqref:eq:detail_control_cf_test_plant_uncertainty., which specifies the magnitude of uncertainty as a function of frequency:
|
|
|
|
\begin{equation}\label{eq:detail_control_cf_test_plant_uncertainty}
|
|
w_I(s) = 10 \cdot \frac{(s+100)^2}{(s+1000)^2}
|
|
\end{equation}
|
|
|
|
Figure ref:fig:detail_control_cf_bode_plot_mech_sys illustrates both the nominal plant dynamics and the complete set of possible plants $\Pi_i$ encompassed by the uncertainty model.
|
|
|
|
#+begin_src matlab
|
|
%% Test model
|
|
freqs = logspace(0, 3, 1000); % Frequency Vector [Hz]
|
|
|
|
m = 20; % mass [kg]
|
|
k = 1e6; % stiffness [N/m]
|
|
c = 1e2; % damping [N/(m/s)]
|
|
|
|
% Plant dynamics
|
|
G = 1/(m*s^2 + c*s + k);
|
|
|
|
% Uncertainty weight
|
|
wI = generateWF('n', 2, 'w0', 2*pi*50, 'G0', 0.1, 'Ginf', 10, 'Gc', 1);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results none
|
|
%% Bode plot of the plant with dynamical uncertainty
|
|
figure;
|
|
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
% Magnitude
|
|
ax1 = nexttile([2,1]);
|
|
hold on;
|
|
plot(freqs, abs(squeeze(freqresp(G, freqs, 'Hz'))), 'k-', 'DisplayName', 'G');
|
|
plotMagUncertainty(wI, freqs, 'G', G, 'DisplayName', '$\Pi_i$');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Magnitude [m/N]'); set(gca, 'XTickLabel',[]);
|
|
ylim([1e-8, 7e-5]);
|
|
hold off;
|
|
leg = legend('location', 'northeast', 'FontSize', 8);
|
|
leg.ItemTokenSize(1) = 18;
|
|
|
|
% Phase
|
|
ax2 = nexttile;
|
|
hold on;
|
|
plotPhaseUncertainty(wI, freqs, 'G', G);
|
|
plot(freqs, 180/pi*unwrap(angle(squeeze(freqresp(G, freqs, 'Hz')))), 'k-');
|
|
set(gca,'xscale','log');
|
|
yticks(-360:90:90);
|
|
ylim([-270 45]);
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
hold off;
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([freqs(1), freqs(end)]);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_cf_bode_plot_mech_sys.pdf', 'width', 'half', 'height', 450);
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_cf_test_model_plant
|
|
#+caption: Schematic of the test system (\subref{fig:detail_control_cf_test_model}). Bode plot of the transfer function $G(s)$ from $F$ to $y$ and the associated uncertainty set (\subref{fig:detail_control_cf_bode_plot_mech_sys}).
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_test_model}Test model}
|
|
#+attr_latex: :options {0.3\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_cf_test_model.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_bode_plot_mech_sys}Bode plot of $G(s)$ and associated uncertainty set}
|
|
#+attr_latex: :options {0.66\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 1
|
|
[[file:figs/detail_control_cf_bode_plot_mech_sys.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
**** Requirements and choice of complementary filters
|
|
|
|
As discussed in Section ref:ssec:detail_control_cf_trans_perf, nominal performance requirements can be expressed as upper bounds on the shapes of the complementary filters.
|
|
For this example, the requirements are:
|
|
- to track ramp inputs (constant velocity scans) with zero steady-state error, which necessitates a $+2$ slope at low frequencies for the magnitude of the sensitivity function $|S(j\omega)|$
|
|
- filtering of measurement noise above $\SI{300}{Hz}$, where sensor noise is significant (requiring a filtering factor of approximately 100 above this frequency)
|
|
- maximizing disturbance rejection
|
|
|
|
Additionally, robust stability must be ensured, requiring the closed-loop system to remain stable despite the dynamic uncertainties modeled by $w_I$.
|
|
This condition is satisfied when the magnitude of the low-pass complementary filter $|H_L(j\omega)|$ remains below the inverse of the uncertainty weight magnitude $|w_I(j\omega)|$, as expressed in Equation eqref:eq:detail_control_cf_condition_robust_stability.
|
|
|
|
Robust performance is achieved when both nominal performance and robust stability conditions are simultaneously satisfied.
|
|
|
|
All requirements imposed on $H_L$ and $H_H$ are visualized in Figure ref:fig:detail_control_cf_specs_S_T.
|
|
While $\mathcal{H}_\infty$ synthesis could be employed to design the complementary filters, analytical formulas were used for this relatively simple example.
|
|
Specifically, the second-order complementary filters from Equation eqref:eq:detail_control_cf_2nd_order were selected, providing the desired $+2$ and $-2$ slopes, with parameters $\alpha = 1$ and $\omega_0 = 2\pi \cdot 20$ Hz.
|
|
|
|
#+name: fig:detail_control_cf_specs_S_T_obtained_filters
|
|
#+caption: Performance requirement and complementary filters used (\subref{fig:detail_control_cf_specs_S_T}). Obtained controller from the complementary filters and the plant inverse is shown in (\subref{fig:detail_control_cf_bode_Kfb}).
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_specs_S_T}Specifications and complementary filters}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :width 0.95\linewidth
|
|
[[file:figs/detail_control_cf_specs_S_T.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_bode_Kfb}Bode plot of $K(s) \cdot H_L(s)$}
|
|
#+attr_latex: :options {0.48\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :width 0.95\linewidth
|
|
[[file:figs/detail_control_cf_bode_Kfb.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
#+begin_src matlab
|
|
%% Analytical Complementary Filters
|
|
w0 = 2*pi*20;
|
|
alpha = 1;
|
|
|
|
Hh = (s/w0)^2*((s/w0)+1+alpha)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
|
|
Hl = ((1+alpha)*(s/w0)+1)/(((s/w0)+1)*((s/w0)^2 + alpha*(s/w0) + 1));
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results none
|
|
%% Specifications
|
|
figure;
|
|
hold on;
|
|
plot([1, 100], [0.01, 100], ':', 'color', colors(2,:));
|
|
plot([300, 1000], [0.01, 0.01], ':', 'color', colors(1,:));
|
|
plot(freqs, 1./abs(squeeze(freqresp(wI, freqs, 'Hz'))), ':', 'color', colors(1,:));
|
|
plot(freqs, abs(squeeze(freqresp(Hl, freqs, 'Hz'))), '-', 'color', colors(1,:));
|
|
plot(freqs, abs(squeeze(freqresp(Hh, freqs, 'Hz'))), '-', 'color', colors(2,:));
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
|
hold off;
|
|
xlim([freqs(1), freqs(end)]);
|
|
ylim([1e-3, 10]);
|
|
xticks([0.1, 1, 10, 100, 1000]);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
% exportFig('figs/detail_control_cf_specs_S_T.pdf', 'width', 'half', 'height', 'normal');
|
|
#+end_src
|
|
|
|
**** Controller analysis
|
|
|
|
The controller to be implemented takes the form $K(s) = \tilde{G}^{-1}(s) H_H^{-1}(s)$, where $\tilde{G}^{-1}(s)$ represents the plant inverse, which must be both stable and proper.
|
|
To ensure properness, low-pass filters with high corner frequencies are added as shown in Equation eqref:eq:detail_control_cf_test_plant_inverse.
|
|
|
|
\begin{equation}\label{eq:detail_control_cf_test_plant_inverse}
|
|
\tilde{G}^{-1}(s) = \frac{m s^2 + c s + k}{1 + \frac{s}{2\pi \cdot 1000} + \left( \frac{s}{2\pi \cdot 1000} \right)^2}
|
|
\end{equation}
|
|
|
|
The Bode plot of the controller multiplied by the complementary low-pass filter, $K(s) \cdot H_L(s)$, is presented in Figure ref:fig:detail_control_cf_bode_Kfb.
|
|
The frequency response reveals several important characteristics:
|
|
- The presence of two integrators at low frequencies, enabling accurate tracking of ramp inputs
|
|
- A notch at the plant resonance frequency (arising from the plant inverse)
|
|
- A lead component near the control bandwidth of approximately 20 Hz, enhancing stability margins
|
|
|
|
#+begin_src matlab
|
|
%% Obtained controller
|
|
omega = 2*pi*1000;
|
|
|
|
K = 1/(Hh*G) * 1/((1+s/omega+(s/omega)^2));
|
|
K = zpk(minreal(K));
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results none
|
|
%% Bode plot of the controller K
|
|
figure;
|
|
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
|
|
|
|
% Magnitude
|
|
ax1 = nexttile([2, 1]);
|
|
plot(freqs, abs(squeeze(freqresp(K*Hl, freqs, 'Hz'))), 'k-');
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
|
|
ylim([8e3, 1e8])
|
|
|
|
% Phase
|
|
ax2 = nexttile;
|
|
plot(freqs, 180/pi*angle(squeeze(freqresp(K*Hl, freqs, 'Hz'))), 'k-');
|
|
set(gca,'xscale','log');
|
|
yticks(-180:45:180);
|
|
ylim([-180 45]);
|
|
xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([freqs(1), freqs(end)]);
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_cf_bode_Kfb.pdf', 'width', 'half', 'height', 500);
|
|
#+end_src
|
|
|
|
**** Robustness and Performance analysis
|
|
|
|
Robust stability is assessed using the Nyquist plot shown in Figure ref:fig:detail_control_cf_nyquist_robustness.
|
|
Even when considering all possible plants within the uncertainty set, the Nyquist plot remains sufficiently distant from the critical point $(-1,0)$, indicating robust stability with adequate margins.
|
|
|
|
Performance is evaluated by examining the closed-loop sensitivity and complementary sensitivity transfer functions, as illustrated in Figure ref:fig:detail_control_cf_robust_perf.
|
|
It is shown that the sensitivity transfer function achieves the desired $+2$ slope at low frequencies and that the complementary sensitivity transfer function nominally provides the wanted noise filtering.
|
|
|
|
#+begin_src matlab
|
|
num_delta_points = 50;
|
|
theta = linspace(0, 2*pi, num_delta_points);
|
|
delta_points = exp(1j * theta);
|
|
|
|
% Get frequency responses for all components
|
|
G_resp = squeeze(freqresp(G, freqs, 'Hz'));
|
|
K_resp = squeeze(freqresp(K, freqs, 'Hz'));
|
|
Hl_resp = squeeze(freqresp(Hl, freqs, 'Hz'));
|
|
wI_resp = squeeze(freqresp(wI, freqs, 'Hz'));
|
|
|
|
% Calculate nominal responses
|
|
nom_L = G_resp .* K_resp .* Hl_resp;
|
|
nom_S = 1 ./ (1 + nom_L);
|
|
nom_T = nom_L ./ (1 + nom_L);
|
|
|
|
% Store all the points in the complex plane that L can take
|
|
loop_region_points = zeros(length(freqs), num_delta_points);
|
|
|
|
% Initialize arrays to store magnitude bounds
|
|
S_mag_min = ones(length(freqs), 1) * inf;
|
|
S_mag_max = zeros(length(freqs), 1);
|
|
T_mag_min = ones(length(freqs), 1) * inf;
|
|
T_mag_max = zeros(length(freqs), 1);
|
|
|
|
% Calculate magnitude bounds for all delta values
|
|
for i = 1:num_delta_points
|
|
% Perturbed loop gain
|
|
loop_perturbed = nom_L .* (1 + wI_resp .* delta_points(i));
|
|
loop_region_points(:,i) = loop_perturbed;
|
|
|
|
% Perturbed sensitivity function
|
|
S_perturbed = 1 ./ (1 + loop_perturbed);
|
|
S_mag = abs(S_perturbed);
|
|
|
|
% Update S magnitude bounds
|
|
S_mag_min = min(S_mag_min, S_mag);
|
|
S_mag_max = max(S_mag_max, S_mag);
|
|
|
|
% Perturbed complementary sensitivity function
|
|
T_perturbed = loop_perturbed ./ (1 + loop_perturbed);
|
|
T_mag = abs(T_perturbed);
|
|
|
|
% Update T magnitude bounds
|
|
T_mag_min = min(T_mag_min, T_mag);
|
|
T_mag_max = max(T_mag_max, T_mag);
|
|
end
|
|
|
|
% At frequencies where |wI| > 1, T min is zero
|
|
T_mag_min(abs(wI_resp)>1) = 1e-10;
|
|
#+end_src
|
|
|
|
#+begin_src matlab
|
|
%% Nyquist plot to check Robust Stability
|
|
figure;
|
|
hold on;
|
|
plot(real(squeeze(freqresp(G*K*Hl, freqs, 'Hz'))), imag(squeeze(freqresp(G*K*Hl, freqs, 'Hz'))), 'k', 'DisplayName', '$L(j\omega)$ - Nominal');
|
|
plot(alphaShape(real(loop_region_points(:)), imag(loop_region_points(:)), 0.1), 'FaceColor', [0, 0, 0], 'EdgeColor', 'none', 'FaceAlpha', 0.3, 'DisplayName', '$L(j\omega)$ - $\forall G \in \Pi_i$');
|
|
plot(-1, 0, 'k+', 'MarkerSize', 5, 'HandleVisibility', 'off');
|
|
hold off;
|
|
grid on;
|
|
axis equal
|
|
xlim([-1.4, 0.2]); ylim([-1.2, 0.4]);
|
|
xticks(-1.4:0.2:0.2); yticks(-1.2:0.2:0.4);
|
|
xlabel('Real Part'); ylabel('Imaginary Part');
|
|
leg = legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
|
|
leg.ItemTokenSize(1) = 18;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_cf_nyquist_robustness', 'width', 'half', 'height', 'normal');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none :results none
|
|
%% Robust Performance
|
|
figure;
|
|
hold on;
|
|
plot(freqs, abs(nom_S), 'color', colors(2,:), 'DisplayName', '$|S|$ - Nom.');
|
|
plot(freqs, abs(nom_T), 'color', colors(1,:), 'DisplayName', '$|T|$ - Nom.');
|
|
|
|
patch([freqs, fliplr(freqs)], [S_mag_max', fliplr(S_mag_min')], colors(2,:), 'FaceAlpha', 0.2, 'EdgeColor', 'none', 'HandleVisibility', 'off');
|
|
patch([freqs, fliplr(freqs)], [T_mag_max', fliplr(T_mag_min')], colors(1,:), 'FaceAlpha', 0.2, 'EdgeColor', 'none', 'HandleVisibility', 'off');
|
|
|
|
plot([1, 100], [0.01, 100], ':', 'color', colors(2,:), 'DisplayName', 'Specs.');
|
|
plot([300, 1000], [0.01, 0.01], ':', 'color', colors(1,:), 'DisplayName', 'Specs.');
|
|
plot(freqs, 1./abs(squeeze(freqresp(wI, freqs, 'Hz'))), ':', 'color', colors(1,:), 'HandleVisibility', 'off');
|
|
|
|
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
|
|
hold off;
|
|
xlabel('Frequency [Hz]'); ylabel('Magnitude');
|
|
xlim([freqs(1), freqs(end)]);
|
|
ylim([1e-4, 5]);
|
|
xticks([0.1, 1, 10, 100, 1000]);
|
|
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 3);
|
|
leg.ItemTokenSize(1) = 18;
|
|
#+end_src
|
|
|
|
#+begin_src matlab :tangle no :exports results :results file replace
|
|
exportFig('figs/detail_control_cf_robust_perf.pdf', 'width', 'half', 'height', 'normal');
|
|
#+end_src
|
|
|
|
#+name: fig:detail_control_cf_simulation_results
|
|
#+caption: Validation of Robust stability with the Nyquist plot (\subref{fig:detail_control_cf_nyquist_robustness}) and validation of the nominal and robust performance with the magnitude of the closed-loop transfer functions (\subref{fig:detail_control_cf_robust_perf})
|
|
#+attr_latex: :options [htbp]
|
|
#+begin_figure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_nyquist_robustness}Robust Stability}
|
|
#+attr_latex: :options {0.49\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 0.8
|
|
[[file:figs/detail_control_cf_nyquist_robustness.png]]
|
|
#+end_subfigure
|
|
#+attr_latex: :caption \subcaption{\label{fig:detail_control_cf_robust_perf}Nominal and Robust performance}
|
|
#+attr_latex: :options {0.49\textwidth}
|
|
#+begin_subfigure
|
|
#+attr_latex: :scale 0.8
|
|
[[file:figs/detail_control_cf_robust_perf.png]]
|
|
#+end_subfigure
|
|
#+end_figure
|
|
|
|
** Conclusion
|
|
:PROPERTIES:
|
|
:UNNUMBERED: t
|
|
:END:
|
|
|
|
In this section, a control architecture in which complementary filters are used for closed-loop shaping has been presented.
|
|
This approach differs from traditional open-loop shaping in that no controller is manually designed; rather, appropriate complementary filters are selected to achieve the desired closed-loop behavior.
|
|
The method shares conceptual similarities with mixed-sensitivity $\mathcal{H}_{\infty}\text{-synthesis}$, as both approaches aim to shape closed-loop transfer functions, but with notable distinctions in implementation and complexity.
|
|
|
|
While $\mathcal{H}_{\infty}\text{-synthesis}$ synthesis offers greater flexibility and can be readily generalized to MIMO plants, the presented approach provides a simpler alternative that requires minimal design effort.
|
|
Implementation only necessitates extracting a model of the plant and selecting appropriate analytical complementary filters, making it particularly interesting for applications where simplicity and intuitive parameter tuning are valued.
|
|
|
|
Due to time constraints, an extensive literature review comparing this approach with similar existing architectures, such as Internal Model Control [[cite:&saxena12_advan_inter_model_contr_techn]], was not conducted.
|
|
Consequently, it remains unclear whether the proposed architecture offers significant advantages over existing methods in the literature.
|
|
|
|
The control architecture has been presented for SISO systems, but can be applied to MIMO systems when sufficient decoupling is achieved.
|
|
It will be experimentally validated with the NASS during the experimental phase.
|
|
|
|
* Conclusion
|
|
:PROPERTIES:
|
|
:UNNUMBERED: t
|
|
:END:
|
|
<<sec:detail_control_conclusion>>
|
|
|
|
|
|
* Bibliography :ignore:
|
|
#+latex: \printbibliography[heading=bibintoc,title={Bibliography}]
|
|
|
|
* Matlab Functions :noexport:
|
|
** =generateWF=: Generate Weighting Functions
|
|
|
|
#+begin_src matlab :tangle matlab/src/generateWF.m :comments none :mkdirp yes :eval no
|
|
function [W] = generateWF(args)
|
|
% generateWF -
|
|
%
|
|
% Syntax: [W] = generateWeight(args)
|
|
%
|
|
% Inputs:
|
|
% - n - Weight Order (integer)
|
|
% - G0 - Low frequency Gain
|
|
% - G1 - High frequency Gain
|
|
% - Gc - Gain of the weight at frequency w0
|
|
% - w0 - Frequency at which |W(j w0)| = Gc [rad/s]
|
|
%
|
|
% Outputs:
|
|
% - W - Generated Weighting Function
|
|
|
|
%% Argument validation
|
|
arguments
|
|
args.n (1,1) double {mustBeInteger, mustBePositive} = 1
|
|
args.G0 (1,1) double {mustBeNumeric, mustBePositive} = 0.1
|
|
args.Ginf (1,1) double {mustBeNumeric, mustBePositive} = 10
|
|
args.Gc (1,1) double {mustBeNumeric, mustBePositive} = 1
|
|
args.w0 (1,1) double {mustBeNumeric, mustBePositive} = 1
|
|
end
|
|
|
|
% Verification of correct relation between G0, Gc and Ginf
|
|
mustBeBetween(args.G0, args.Gc, args.Ginf);
|
|
|
|
%% Initialize the Laplace variable
|
|
s = zpk('s');
|
|
|
|
%% Create the weighting function according to formula
|
|
W = (((1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
|
|
(args.G0/args.Gc)^(1/args.n))/...
|
|
((1/args.Ginf)^(1/args.n)*(1/args.w0)*sqrt((1-(args.G0/args.Gc)^(2/args.n))/(1-(args.Gc/args.Ginf)^(2/args.n)))*s + ...
|
|
(1/args.Gc)^(1/args.n)))^args.n;
|
|
|
|
%% Custom validation function
|
|
function mustBeBetween(a,b,c)
|
|
if ~((a > b && b > c) || (c > b && b > a))
|
|
eid = 'createWeight:inputError';
|
|
msg = 'Gc should be between G0 and Ginf.';
|
|
throwAsCaller(MException(eid,msg))
|
|
end
|
|
#+end_src
|
|
|
|
** =generateCF=: Generate Complementary Filters
|
|
|
|
#+begin_src matlab :tangle matlab/src/generateCF.m :comments none :mkdirp yes :eval no
|
|
function [H1, H2] = generateCF(W1, W2, args)
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|
% generateCF -
|
|
%
|
|
% Syntax: [H1, H2] = generateCF(W1, W2, args)
|
|
%
|
|
% Inputs:
|
|
% - W1 - Weighting Function for H1
|
|
% - W2 - Weighting Function for H2
|
|
% - args:
|
|
% - method - H-Infinity solver ('lmi' or 'ric')
|
|
% - display - Display synthesis results ('on' or 'off')
|
|
%
|
|
% Outputs:
|
|
% - H1 - Generated H1 Filter
|
|
% - H2 - Generated H2 Filter
|
|
|
|
%% Argument validation
|
|
arguments
|
|
W1
|
|
W2
|
|
args.method char {mustBeMember(args.method,{'lmi', 'ric'})} = 'ric'
|
|
args.display char {mustBeMember(args.display,{'on', 'off'})} = 'on'
|
|
end
|
|
|
|
%% The generalized plant is defined
|
|
P = [W1 -W1;
|
|
0 W2;
|
|
1 0];
|
|
|
|
%% The standard H-infinity synthesis is performed
|
|
[H2, ~, gamma, ~] = hinfsyn(P, 1, 1,'TOLGAM', 0.001, 'METHOD', args.method, 'DISPLAY', args.display);
|
|
|
|
%% H1 is defined as the complementary of H2
|
|
H1 = 1 - H2;
|
|
#+end_src
|
|
|
|
** =plotMagUncertainty=
|
|
|
|
#+begin_src matlab :tangle matlab/src/plotMagUncertainty.m :comments none :mkdirp yes :eval no
|
|
function [p] = plotMagUncertainty(W, freqs, args)
|
|
% plotMagUncertainty -
|
|
%
|
|
% Syntax: [p] = plotMagUncertainty(W, freqs, args)
|
|
%
|
|
% Inputs:
|
|
% - W - Multiplicative Uncertainty Weight
|
|
% - freqs - Frequency Vector [Hz]
|
|
% - args - Optional Arguments:
|
|
% - G
|
|
% - color_i
|
|
% - opacity
|
|
%
|
|
% Outputs:
|
|
% - p - Plot Handle
|
|
|
|
arguments
|
|
W
|
|
freqs double {mustBeNumeric, mustBeNonnegative}
|
|
args.G = tf(1)
|
|
args.color_i (1,1) double {mustBeInteger, mustBeNonnegative} = 0
|
|
args.opacity (1,1) double {mustBeNumeric, mustBeNonnegative} = 0.3
|
|
args.DisplayName char = ''
|
|
end
|
|
|
|
% Get defaults colors
|
|
colors = get(groot, 'defaultAxesColorOrder');
|
|
|
|
p = patch([freqs flip(freqs)], ...
|
|
[abs(squeeze(freqresp(args.G, freqs, 'Hz'))).*(1 + abs(squeeze(freqresp(W, freqs, 'Hz')))); ...
|
|
flip(abs(squeeze(freqresp(args.G, freqs, 'Hz'))).*max(1 - abs(squeeze(freqresp(W, freqs, 'Hz'))), 1e-6))], 'w', ...
|
|
'DisplayName', args.DisplayName);
|
|
|
|
if args.color_i == 0
|
|
p.FaceColor = [0; 0; 0];
|
|
else
|
|
p.FaceColor = colors(args.color_i, :);
|
|
end
|
|
p.EdgeColor = 'none';
|
|
p.FaceAlpha = args.opacity;
|
|
|
|
end
|
|
#+end_src
|
|
|
|
** =plotPhaseUncertainty=
|
|
|
|
#+begin_src matlab :tangle matlab/src/plotPhaseUncertainty.m :comments none :mkdirp yes :eval no
|
|
function [p] = plotPhaseUncertainty(W, freqs, args)
|
|
% plotPhaseUncertainty -
|
|
%
|
|
% Syntax: [p] = plotPhaseUncertainty(W, freqs, args)
|
|
%
|
|
% Inputs:
|
|
% - W - Multiplicative Uncertainty Weight
|
|
% - freqs - Frequency Vector [Hz]
|
|
% - args - Optional Arguments:
|
|
% - G
|
|
% - color_i
|
|
% - opacity
|
|
%
|
|
% Outputs:
|
|
% - p - Plot Handle
|
|
|
|
arguments
|
|
W
|
|
freqs double {mustBeNumeric, mustBeNonnegative}
|
|
args.G = tf(1)
|
|
args.unwrap logical {mustBeNumericOrLogical} = false
|
|
args.color_i (1,1) double {mustBeInteger, mustBeNonnegative} = 0
|
|
args.opacity (1,1) double {mustBeNumeric, mustBePositive} = 0.3
|
|
args.DisplayName char = ''
|
|
end
|
|
|
|
% Get defaults colors
|
|
colors = get(groot, 'defaultAxesColorOrder');
|
|
|
|
% Compute Phase Uncertainty
|
|
Dphi = 180/pi*asin(abs(squeeze(freqresp(W, freqs, 'Hz'))));
|
|
Dphi(abs(squeeze(freqresp(W, freqs, 'Hz'))) > 1) = 360;
|
|
|
|
% Compute Plant Phase
|
|
if args.unwrap
|
|
G_ang = 180/pi*unwrap(angle(squeeze(freqresp(args.G, freqs, 'Hz'))));
|
|
else
|
|
G_ang = 180/pi*angle(squeeze(freqresp(args.G, freqs, 'Hz')));
|
|
end
|
|
|
|
p = patch([freqs flip(freqs)], [G_ang+Dphi; flip(G_ang-Dphi)], 'w', ...
|
|
'DisplayName', args.DisplayName);
|
|
|
|
if args.color_i == 0
|
|
p.FaceColor = [0; 0; 0];
|
|
else
|
|
p.FaceColor = colors(args.color_i, :);
|
|
end
|
|
p.EdgeColor = 'none';
|
|
p.FaceAlpha = args.opacity;
|
|
|
|
end
|
|
#+end_src
|
|
* Helping Functions :noexport:
|
|
** Initialize Path
|
|
#+NAME: m-init-path
|
|
#+BEGIN_SRC matlab
|
|
%% Path for functions, data and scripts
|
|
addpath('./matlab/src/'); % Path for functions
|
|
addpath('./matlab/'); % Path for scripts
|
|
#+END_SRC
|
|
|
|
#+NAME: m-init-path-tangle
|
|
#+BEGIN_SRC matlab
|
|
%% Path for functions, data and scripts
|
|
addpath('./src/'); % Path for functions
|
|
#+END_SRC
|
|
|
|
** Initialize other elements
|
|
#+NAME: m-init-other
|
|
#+BEGIN_SRC matlab
|
|
%% Colors for the figures
|
|
colors = colororder;
|
|
#+END_SRC
|
|
|
|
* Footnotes
|
|
[fn:detail_control_2]$n$ corresponds to the number of degrees of freedom, here $n = 3$
|
|
[fn:detail_control_1]A set of two complementary filters are two transfer functions that sum to one.
|