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id="toc-Linear_and_nonlinear_control_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analysis_techniques_-_frequency_domain_and_time_domain" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Analysis_techniques_-_frequency_domain_and_time_domain"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Analysis techniques - frequency domain and time domain</span> </div> </a> <ul id="toc-Analysis_techniques_-_frequency_domain_and_time_domain-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-System_interfacing_-_SISO_&_MIMO" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#System_interfacing_-_SISO_&_MIMO"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>System interfacing - SISO & MIMO</span> </div> </a> <button aria-controls="toc-System_interfacing_-_SISO_&_MIMO-sublist" class="cdx-button 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</li> </ul> </li> <li id="toc-Topics_in_control_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Topics_in_control_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Topics in control theory</span> </div> </a> <button aria-controls="toc-Topics_in_control_theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Topics in control theory subsection</span> </button> <ul id="toc-Topics_in_control_theory-sublist" class="vector-toc-list"> <li id="toc-Stability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stability"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Stability</span> </div> </a> <ul id="toc-Stability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Controllability_and_observability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Controllability_and_observability"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Controllability and observability</span> </div> </a> <ul id="toc-Controllability_and_observability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Control_specification" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Control_specification"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Control specification</span> </div> </a> <ul id="toc-Control_specification-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Model_identification_and_robustness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Model_identification_and_robustness"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>Model identification and robustness</span> </div> </a> <ul id="toc-Model_identification_and_robustness-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-System_classifications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#System_classifications"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>System classifications</span> </div> </a> <button aria-controls="toc-System_classifications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle System classifications subsection</span> </button> <ul id="toc-System_classifications-sublist" class="vector-toc-list"> <li id="toc-Linear_systems_control" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_systems_control"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Linear systems control</span> </div> </a> <ul id="toc-Linear_systems_control-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nonlinear_systems_control" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nonlinear_systems_control"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Nonlinear systems control</span> </div> </a> <ul id="toc-Nonlinear_systems_control-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Decentralized_systems_control" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Decentralized_systems_control"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Decentralized systems control</span> </div> </a> <ul id="toc-Decentralized_systems_control-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Deterministic_and_stochastic_systems_control" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Deterministic_and_stochastic_systems_control"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.4</span> <span>Deterministic and stochastic systems control</span> </div> </a> <ul id="toc-Deterministic_and_stochastic_systems_control-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Main_control_strategies" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Main_control_strategies"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Main control strategies</span> </div> </a> <ul id="toc-Main_control_strategies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-People_in_systems_and_control" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#People_in_systems_and_control"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>People in systems and control</span> </div> </a> <ul id="toc-People_in_systems_and_control-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Control theory</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 40 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-40" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">40 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D8%AA%D8%AD%D9%83%D9%85" title="نظرية التحكم – Arabic" lang="ar" hreflang="ar" data-title="نظرية التحكم" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_del_control" title="Teoría del control – Asturian" lang="ast" hreflang="ast" data-title="Teoría del control" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0dar%C9%99etm%C9%99_n%C9%99z%C9%99riyy%C9%99si" title="İdarəetmə nəzəriyyəsi – Azerbaijani" lang="az" hreflang="az" data-title="İdarəetmə nəzəriyyəsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A8%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A6%A8%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%A3_%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="নিয়ন্ত্রণ তত্ত্ব – Bangla" lang="bn" hreflang="bn" data-title="নিয়ন্ত্রণ তত্ত্ব" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_de_control" title="Teoria de control – Catalan" lang="ca" hreflang="ca" data-title="Teoria de control" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Teorie_%C5%99%C3%ADzen%C3%AD" title="Teorie řízení – Czech" lang="cs" hreflang="cs" data-title="Teorie řízení" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kontrolltheorie" title="Kontrolltheorie – German" lang="de" hreflang="de" data-title="Kontrolltheorie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Automaatika" title="Automaatika – Estonian" lang="et" hreflang="et" data-title="Automaatika" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CE%B5%CE%BB%CE%AD%CE%B3%CF%87%CE%BF%CF%85" title="Θεωρία ελέγχου – Greek" lang="el" hreflang="el" data-title="Θεωρία ελέγχου" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_del_control" title="Teoría del control – Spanish" lang="es" hreflang="es" data-title="Teoría del control" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Rega_teorio" title="Rega teorio – Esperanto" lang="eo" hreflang="eo" data-title="Rega teorio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Kontrolaren_teoria" title="Kontrolaren teoria – Basque" lang="eu" hreflang="eu" data-title="Kontrolaren teoria" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%DA%A9%D9%86%D8%AA%D8%B1%D9%84" title="نظریه کنترل – Persian" lang="fa" hreflang="fa" data-title="نظریه کنترل" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_du_contr%C3%B4le" title="Théorie du contrôle – French" lang="fr" hreflang="fr" data-title="Théorie du contrôle" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teor%C3%ADa_de_control" title="Teoría de control – Galician" lang="gl" hreflang="gl" data-title="Teoría de control" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%9C%EC%96%B4%EC%9D%B4%EB%A1%A0" title="제어이론 – Korean" lang="ko" hreflang="ko" data-title="제어이론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%82%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%A3_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="नियंत्रण सिद्धान्त – Hindi" lang="hi" hreflang="hi" data-title="नियंत्रण सिद्धान्त" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Controllo_automatico" title="Controllo automatico – Italian" lang="it" hreflang="it" data-title="Controllo automatico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%91%D7%A7%D7%A8%D7%94" title="תורת הבקרה – Hebrew" lang="he" hreflang="he" data-title="תורת הבקרה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Ir%C3%A1ny%C3%ADt%C3%A1stechnika" title="Irányítástechnika – Hungarian" lang="hu" hreflang="hu" data-title="Irányítástechnika" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teori_kawalan" title="Teori kawalan – Malay" lang="ms" hreflang="ms" data-title="Teori kawalan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Meet-_en_regeltechniek" title="Meet- en regeltechniek – Dutch" lang="nl" hreflang="nl" data-title="Meet- en regeltechniek" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%88%B6%E5%BE%A1%E7%90%86%E8%AB%96" title="制御理論 – Japanese" lang="ja" hreflang="ja" data-title="制御理論" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Teoria_sterowania" title="Teoria sterowania – Polish" lang="pl" hreflang="pl" data-title="Teoria sterowania" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teoria_de_controle" title="Teoria de controle – Portuguese" lang="pt" hreflang="pt" data-title="Teoria de controle" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D1%83%D0%BF%D1%80%D0%B0%D0%B2%D0%BB%D0%B5%D0%BD%D0%B8%D1%8F" title="Теория управления – Russian" lang="ru" hreflang="ru" data-title="Теория управления" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Teoria_e_kontrollit" title="Teoria e kontrollit – Albanian" lang="sq" hreflang="sq" data-title="Teoria e kontrollit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Tiur%C3%ACa_d%C3%BB_cuntrollu" title="Tiurìa dû cuntrollu – Sicilian" lang="scn" hreflang="scn" data-title="Tiurìa dû cuntrollu" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Control_theory" title="Control theory – Simple English" lang="en-simple" hreflang="en-simple" data-title="Control theory" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Te%C3%B3ria_riadenia" title="Teória riadenia – Slovak" lang="sk" hreflang="sk" data-title="Teória riadenia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Teorija_kontrole" title="Teorija kontrole – Serbian" lang="sr" hreflang="sr" data-title="Teorija kontrole" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Teorya_ng_kontrol" title="Teorya ng kontrol – Tagalog" lang="tl" hreflang="tl" data-title="Teorya ng kontrol" data-language-autonym="Tagalog" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Branch of engineering and mathematics</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about control theory in engineering. For control theory in linguistics, see <a href="/wiki/Control_(linguistics)" title="Control (linguistics)">control (linguistics)</a>. For control theory in psychology and sociology, see <a href="/wiki/Control_theory_(sociology)" title="Control theory (sociology)">control theory (sociology)</a> and <a href="/wiki/Perceptual_control_theory" title="Perceptual control theory">Perceptual control theory</a>.</div> <p class="mw-empty-elt"> </p><p><b>Control theory</b> is a field of <a href="/wiki/Control_engineering" title="Control engineering">control engineering</a> and <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a> that deals with the <a href="/wiki/Control_system" title="Control system">control</a> of <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a> in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any <i>delay</i>, <i>overshoot</i>, or <i>steady-state error</i> and ensuring a level of control <a href="/wiki/Stability_theory" title="Stability theory">stability</a>; often with the aim to achieve a degree of <a href="/wiki/Optimal_control" title="Optimal control">optimality</a>. </p><p>To do this, a <b>controller</b> with the requisite corrective behavior is required. This controller monitors the controlled <a href="/wiki/Process_variable" title="Process variable">process variable</a> (PV), and compares it with the reference or <a href="/wiki/Setpoint_(control_system)" title="Setpoint (control system)">set point</a> (SP). The difference between actual and desired value of the process variable, called the <i>error</i> signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are <a href="/wiki/Controllability" title="Controllability">controllability</a> and <a href="/wiki/Observability" title="Observability">observability</a>. Control theory is used in <a href="/wiki/Control_system_engineering" class="mw-redirect" title="Control system engineering">control system engineering</a> to design automation that have revolutionized manufacturing, aircraft, communications and other industries, and created new fields such as <a href="/wiki/Robotics" title="Robotics">robotics</a>. </p><p>Extensive use is usually made of a diagrammatic style known as the <a href="/wiki/Block_diagram" title="Block diagram">block diagram</a>. In it the <a href="/wiki/Transfer_function" title="Transfer function">transfer function</a>, also known as the system function or network function, is a mathematical model of the relation between the input and output based on the <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> describing the system. </p><p>Control theory dates from the 19th century, when the theoretical basis for the operation of governors was first described by <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Control theory was further advanced by <a href="/wiki/Edward_Routh" title="Edward Routh">Edward Routh</a> in 1874, <a href="/wiki/Jacques_Charles_Fran%C3%A7ois_Sturm" title="Jacques Charles François Sturm">Charles Sturm</a> and in 1895, <a href="/wiki/Adolf_Hurwitz" title="Adolf Hurwitz">Adolf Hurwitz</a>, who all contributed to the establishment of control stability criteria; and from 1922 onwards, the development of <a href="/wiki/PID_control" class="mw-redirect" title="PID control">PID control</a> theory by <a href="/wiki/Nicolas_Minorsky" title="Nicolas Minorsky">Nicolas Minorsky</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Although a major application of <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> control theory is in <a href="/wiki/Control_Systems_Engineering" class="mw-redirect" title="Control Systems Engineering">control systems engineering</a>, which deals with the design of <a href="/wiki/Process_control" class="mw-redirect" title="Process control">process control</a> systems for industry, other applications range far beyond this. As the general theory of feedback systems, control theory is useful wherever feedback occurs - thus control theory also has applications in life sciences, computer engineering, sociology and <a href="/wiki/Operations_research" title="Operations research">operations research</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Control_engineering#History" title="Control engineering">Control engineering § History</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Boulton_and_Watt_centrifugal_governor-MJ.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Boulton_and_Watt_centrifugal_governor-MJ.jpg/220px-Boulton_and_Watt_centrifugal_governor-MJ.jpg" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Boulton_and_Watt_centrifugal_governor-MJ.jpg/330px-Boulton_and_Watt_centrifugal_governor-MJ.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Boulton_and_Watt_centrifugal_governor-MJ.jpg/440px-Boulton_and_Watt_centrifugal_governor-MJ.jpg 2x" data-file-width="1152" data-file-height="1536" /></a><figcaption><a href="/wiki/Centrifugal_governor" title="Centrifugal governor">Centrifugal governor</a> in a <a href="/wiki/Boulton_%26_Watt_engine" class="mw-redirect" title="Boulton & Watt engine">Boulton & Watt engine</a> of 1788</figcaption></figure> <p>Although control systems of various types date back to antiquity, a more formal analysis of the field began with a dynamics analysis of the <a href="/wiki/Centrifugal_governor" title="Centrifugal governor">centrifugal governor</a>, conducted by the physicist <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> in 1868, entitled <i>On Governors</i>.<sup id="cite_ref-Maxwell1867_4-0" class="reference"><a href="#cite_note-Maxwell1867-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> A centrifugal governor was already used to regulate the velocity of windmills.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Maxwell described and analyzed the phenomenon of <a href="/wiki/Self-oscillation" title="Self-oscillation">self-oscillation</a>, in which lags in the system may lead to overcompensation and unstable behavior. This generated a flurry of interest in the topic, during which Maxwell's classmate, <a href="/wiki/Edward_John_Routh" class="mw-redirect" title="Edward John Routh">Edward John Routh</a>, abstracted Maxwell's results for the general class of linear systems.<sup id="cite_ref-Routh1975_6-0" class="reference"><a href="#cite_note-Routh1975-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Independently, <a href="/wiki/Adolf_Hurwitz" title="Adolf Hurwitz">Adolf Hurwitz</a> analyzed system stability using differential equations in 1877, resulting in what is now known as the <a href="/wiki/Routh%E2%80%93Hurwitz_theorem" title="Routh–Hurwitz theorem">Routh–Hurwitz theorem</a>.<sup id="cite_ref-Routh1877_7-0" class="reference"><a href="#cite_note-Routh1877-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Hurwitz1964_8-0" class="reference"><a href="#cite_note-Hurwitz1964-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>A notable application of dynamic control was in the area of crewed flight. The <a href="/wiki/Wright_brothers" title="Wright brothers">Wright brothers</a> made their first successful test flights on December 17, 1903, and were distinguished by their ability to control their flights for substantial periods (more so than the ability to produce lift from an airfoil, which was known). Continuous, reliable control of the airplane was necessary for flights lasting longer than a few seconds. </p><p>By <a href="/wiki/World_War_II" title="World War II">World War II</a>, control theory was becoming an important area of research. <a href="/wiki/Irmgard_Fl%C3%BCgge-Lotz" title="Irmgard Flügge-Lotz">Irmgard Flügge-Lotz</a> developed the theory of discontinuous automatic control systems, and applied the <a href="/wiki/Bang%E2%80%93bang_control" title="Bang–bang control">bang-bang principle</a> to the development of <a href="/wiki/Autopilot" title="Autopilot">automatic flight control equipment</a> for aircraft.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Other areas of application for discontinuous controls included <a href="/wiki/Fire-control_system" title="Fire-control system">fire-control systems</a>, <a href="/wiki/Guidance_system" title="Guidance system">guidance systems</a> and <a href="/wiki/Electronics" title="Electronics">electronics</a>. </p><p>Sometimes, mechanical methods are used to improve the stability of systems. For example, <a href="/wiki/Stabilizer_(ship)" title="Stabilizer (ship)">ship stabilizers</a> are fins mounted beneath the waterline and emerging laterally. In contemporary vessels, they may be gyroscopically controlled active fins, which have the capacity to change their angle of attack to counteract roll caused by wind or waves acting on the ship. </p><p>The <a href="/wiki/Space_Race" title="Space Race">Space Race</a> also depended on accurate spacecraft control, and control theory has also seen an increasing use in fields such as economics and artificial intelligence. Here, one might say that the goal is to find an <a href="/wiki/Internal_model_(motor_control)" title="Internal model (motor control)">internal model</a> that obeys the <a href="/wiki/Good_regulator" title="Good regulator">good regulator theorem</a>. So, for example, in economics, the more accurately a (stock or commodities) trading model represents the actions of the market, the more easily it can control that market (and extract "useful work" (profits) from it). In AI, an example might be a chatbot modelling the discourse state of humans: the more accurately it can model the human state (e.g. on a telephone voice-support hotline), the better it can manipulate the human (e.g. into performing the corrective actions to resolve the problem that caused the phone call to the help-line). These last two examples take the narrow historical interpretation of control theory as a set of differential equations modeling and regulating kinetic motion, and broaden it into a vast generalization of a <a href="/wiki/Controller_(control_theory)" class="mw-redirect" title="Controller (control theory)">regulator</a> interacting with a <a href="/wiki/Plant_(control_theory)" title="Plant (control theory)">plant</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Open-loop_and_closed-loop_(feedback)_control"><span id="Open-loop_and_closed-loop_.28feedback.29_control"></span>Open-loop and closed-loop (feedback) control</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=2" title="Edit section: Open-loop and closed-loop (feedback) control"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Control_loop#Open-loop_and_closed-loop" title="Control loop">Control loop § Open-loop and closed-loop</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Control_loop&action=edit#Open-loop_and_closed-loop">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <p>Fundamentally, there are two types of control loop: <i><a href="/wiki/Open-loop_control" class="mw-redirect" title="Open-loop control">open-loop control</a></i> (feedforward), and <i><a href="/wiki/Closed-loop_control" class="mw-redirect" title="Closed-loop control">closed-loop control</a></i> (feedback). </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Electromechanicaltimer.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Electromechanicaltimer.JPG/170px-Electromechanicaltimer.JPG" decoding="async" width="170" height="191" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Electromechanicaltimer.JPG/255px-Electromechanicaltimer.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Electromechanicaltimer.JPG/340px-Electromechanicaltimer.JPG 2x" data-file-width="401" data-file-height="450" /></a><figcaption>An electromechanical timer, normally used for open-loop control based purely on a timing sequence, with no feedback from the process</figcaption></figure> <p>In open-loop control, the control action from the controller is independent of the "process output" (or "controlled process variable"). A good example of this is a central heating boiler controlled only by a timer, so that heat is applied for a constant time, regardless of the temperature of the building. The control action is the switching on/off of the boiler, but the controlled variable should be the building temperature, but is not because this is open-loop control of the boiler, which does not give closed-loop control of the temperature. </p><p>In closed loop control, the control action from the controller is dependent on the process output. In the case of the boiler analogy this would include a thermostat to monitor the building temperature, and thereby feed back a signal to ensure the controller maintains the building at the temperature set on the thermostat. A closed loop controller therefore has a feedback loop which ensures the controller exerts a control action to give a process output the same as the "reference input" or "set point". For this reason, closed loop controllers are also called feedback controllers.<sup id="cite_ref-Control_loop_auto_11-0" class="reference"><a href="#cite_note-Control_loop_auto-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>The definition of a closed loop control system according to the <a href="/wiki/British_Standards_Institution" class="mw-redirect" title="British Standards Institution">British Standards Institution</a> is "a control system possessing monitoring feedback, the deviation signal formed as a result of this feedback being used to control the action of a final control element in such a way as to tend to reduce the deviation to zero."<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> Likewise; "A <i>Feedback Control System</i> is a system which tends to maintain a prescribed relationship of one system variable to another by comparing functions of these variables and using the difference as a means of control."<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></div></div> <div class="mw-heading mw-heading2"><h2 id="Classical_control_theory">Classical control theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=3" title="Edit section: Classical control theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1066933788"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Closed-loop_controller" title="Closed-loop controller">Closed-loop controller</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Closed-loop_controller&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Industrial_control_loop.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Industrial_control_loop.jpg/300px-Industrial_control_loop.jpg" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Industrial_control_loop.jpg/450px-Industrial_control_loop.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Industrial_control_loop.jpg/600px-Industrial_control_loop.jpg 2x" data-file-width="1500" data-file-height="1125" /></a><figcaption>Example of a single industrial control loop; showing continuously modulated control of process flow.</figcaption></figure> <p>A <a href="/wiki/Closed-loop_controller" title="Closed-loop controller">closed-loop controller</a> or feedback controller is a <a href="/wiki/Control_loop" title="Control loop">control loop</a> which incorporates <a href="/wiki/Feedback" title="Feedback">feedback</a>, in contrast to an <i><a href="/wiki/Open-loop_controller" title="Open-loop controller">open-loop controller</a></i> or <i>non-feedback controller</i>. A closed-loop controller uses feedback to control <a href="/wiki/State_(controls)" class="mw-redirect" title="State (controls)">states</a> or <a href="/wiki/Negative_feedback#Overview" title="Negative feedback">outputs</a> of a <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical system</a>. Its name comes from the information path in the system: process inputs (e.g., <a href="/wiki/Voltage" title="Voltage">voltage</a> applied to an <a href="/wiki/Electric_motor" title="Electric motor">electric motor</a>) have an effect on the process outputs (e.g., speed or torque of the motor), which is measured with <a href="/wiki/Sensor" title="Sensor">sensors</a> and processed by the controller; the result (the control signal) is "fed back" as input to the process, closing the loop.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>In the case of linear <a href="/wiki/Feedback" title="Feedback">feedback</a> systems, a <a href="/wiki/Control_loop" title="Control loop">control loop</a> including <a href="/wiki/Sensor" title="Sensor">sensors</a>, control algorithms, and actuators is arranged in an attempt to regulate a variable at a <a href="/wiki/Setpoint_(control_system)" title="Setpoint (control system)">setpoint</a> (SP). An everyday example is the <a href="/wiki/Cruise_control" title="Cruise control">cruise control</a> on a road vehicle; where external influences such as hills would cause speed changes, and the driver has the ability to alter the desired set speed. The <a href="/wiki/PID_algorithm" class="mw-redirect" title="PID algorithm">PID algorithm</a> in the controller restores the actual speed to the desired speed in an optimum way, with minimal delay or <a href="/wiki/Overshoot_(signal)" title="Overshoot (signal)">overshoot</a>, by controlling the power output of the vehicle's engine. Control systems that include some sensing of the results they are trying to achieve are making use of feedback and can adapt to varying circumstances to some extent. <a href="/wiki/Open-loop_controller" title="Open-loop controller">Open-loop control systems</a> do not make use of feedback, and run only in pre-arranged ways. </p><p>Closed-loop controllers have the following advantages over open-loop controllers: </p> <ul><li>disturbance rejection (such as hills in the cruise control example above)</li> <li>guaranteed performance even with <a href="/wiki/Mathematical_model" title="Mathematical model">model</a> uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact</li> <li><a href="/wiki/Instability" title="Instability">unstable</a> processes can be stabilized</li> <li>reduced sensitivity to parameter variations</li> <li>improved reference tracking performance</li> <li>improved rectification of random fluctuations<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></li></ul> <p>In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termed <i><a href="/wiki/Feed_forward_(control)" title="Feed forward (control)">feedforward</a></i> and serves to further improve reference tracking performance. </p><p>A common closed-loop controller architecture is the <a href="/wiki/PID_controller" class="mw-redirect" title="PID controller">PID controller</a>. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Ideal_feedback_model.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Ideal_feedback_model.svg/220px-Ideal_feedback_model.svg.png" decoding="async" width="220" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Ideal_feedback_model.svg/330px-Ideal_feedback_model.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Ideal_feedback_model.svg/440px-Ideal_feedback_model.svg.png 2x" data-file-width="733" data-file-height="269" /></a><figcaption> A basic feedback loop</figcaption></figure></div></div> <div class="mw-heading mw-heading2"><h2 id="Linear_and_nonlinear_control_theory">Linear and nonlinear control theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=4" title="Edit section: Linear and nonlinear control theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The field of control theory can be divided into two branches: </p> <ul><li><i><a href="/wiki/Linear_control_theory" class="mw-redirect" title="Linear control theory">Linear control theory</a></i> – This applies to systems made of devices which obey the <a href="/wiki/Superposition_principle" title="Superposition principle">superposition principle</a>, which means roughly that the output is proportional to the input. They are governed by <a href="/wiki/Linear_differential_equation" title="Linear differential equation">linear differential equations</a>. A major subclass is systems which in addition have parameters which do not change with time, called <i><a href="/wiki/Linear_time_invariant" class="mw-redirect" title="Linear time invariant">linear time invariant</a></i> (LTI) systems. These systems are amenable to powerful <a href="/wiki/Frequency_domain" title="Frequency domain">frequency domain</a> mathematical techniques of great generality, such as the <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a>, <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>, <a href="/wiki/Z_transform" class="mw-redirect" title="Z transform">Z transform</a>, <a href="/wiki/Bode_plot" title="Bode plot">Bode plot</a>, <a href="/wiki/Root_locus" class="mw-redirect" title="Root locus">root locus</a>, and <a href="/wiki/Nyquist_stability_criterion" title="Nyquist stability criterion">Nyquist stability criterion</a>. These lead to a description of the system using terms like <a href="/wiki/Bandwidth_(signal_processing)" title="Bandwidth (signal processing)">bandwidth</a>, <a href="/wiki/Frequency_response" title="Frequency response">frequency response</a>, <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a>, <a href="/wiki/Gain_(electronics)" title="Gain (electronics)">gain</a>, <a href="/wiki/Resonant_frequency" class="mw-redirect" title="Resonant frequency">resonant frequencies</a>, <a href="/wiki/Zeros_and_poles" title="Zeros and poles">zeros and poles</a>, which give solutions for system response and design techniques for most systems of interest.</li> <li><i><a href="/wiki/Nonlinear_control_theory" class="mw-redirect" title="Nonlinear control theory">Nonlinear control theory</a></i> – This covers a wider class of systems that do not obey the superposition principle, and applies to more real-world systems because all real control systems are nonlinear. These systems are often governed by <a href="/wiki/Nonlinear_differential_equation" class="mw-redirect" title="Nonlinear differential equation">nonlinear differential equations</a>. The few mathematical techniques which have been developed to handle them are more difficult and much less general, often applying only to narrow categories of systems. These include <a href="/wiki/Limit_cycle" title="Limit cycle">limit cycle</a> theory, <a href="/wiki/Poincar%C3%A9_map" title="Poincaré map">Poincaré maps</a>, <a href="/wiki/Lyapunov_function" title="Lyapunov function">Lyapunov stability theorem</a>, and <a href="/wiki/Describing_function" title="Describing function">describing functions</a>. Nonlinear systems are often analyzed using <a href="/wiki/Numerical_method" title="Numerical method">numerical methods</a> on computers, for example by <a href="/wiki/Simulation" title="Simulation">simulating</a> their operation using a <a href="/wiki/Simulation_language" title="Simulation language">simulation language</a>. If only solutions near a stable point are of interest, nonlinear systems can often be <a href="/wiki/Linearization" title="Linearization">linearized</a> by approximating them by a linear system using <a href="/wiki/Perturbation_theory" title="Perturbation theory">perturbation theory</a>, and linear techniques can be used.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Analysis_techniques_-_frequency_domain_and_time_domain">Analysis techniques - frequency domain and time domain</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=5" title="Edit section: Analysis techniques - frequency domain and time domain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematical techniques for analyzing and designing control systems fall into two different categories: </p> <ul><li><i><a href="/wiki/Frequency_domain" title="Frequency domain">Frequency domain</a></i> – In this type the values of the <a href="/wiki/State_variable" title="State variable">state variables</a>, the mathematical <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> representing the system's input, output and feedback are represented as functions of <a href="/wiki/Frequency" title="Frequency">frequency</a>. The input signal and the system's <a href="/wiki/Transfer_function" title="Transfer function">transfer function</a> are converted from time functions to functions of frequency by a <a href="/wiki/Transform_(mathematics)" class="mw-redirect" title="Transform (mathematics)">transform</a> such as the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>, <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a>, or <a href="/wiki/Z_transform" class="mw-redirect" title="Z transform">Z transform</a>. The advantage of this technique is that it results in a simplification of the mathematics; the <i><a href="/wiki/Differential_equation" title="Differential equation">differential equations</a></i> that represent the system are replaced by <i><a href="/wiki/Algebraic_equation" title="Algebraic equation">algebraic equations</a></i> in the frequency domain which is much simpler to solve. However, frequency domain techniques can only be used with linear systems, as mentioned above.</li> <li><i><a href="/wiki/Time-domain_state_space_representation" class="mw-redirect" title="Time-domain state space representation">Time-domain state space representation</a></i> – In this type the values of the <a href="/wiki/State_variable" title="State variable">state variables</a> are represented as functions of time. With this model, the system being analyzed is represented by one or more <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>. Since frequency domain techniques are limited to <a href="/wiki/Linear_function" title="Linear function">linear</a> systems, time domain is widely used to analyze real-world nonlinear systems. Although these are more difficult to solve, modern computer simulation techniques such as <a href="/wiki/Simulation_language" title="Simulation language">simulation languages</a> have made their analysis routine.</li></ul> <p>In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain <a href="/wiki/State_space_(controls)" class="mw-redirect" title="State space (controls)">state space</a> representation,<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (December 2022)">citation needed</span></a></i>]</sup> a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. To abstract from the number of inputs, outputs, and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the latter only being possible when the dynamical system is linear). The state space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information about a system. Unlike the frequency domain approach, the use of the state-space representation is not limited to systems with linear components and zero initial conditions. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a point within that space.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="System_interfacing_-_SISO_&_MIMO"><span id="System_interfacing_-_SISO_.26_MIMO"></span>System interfacing - SISO & MIMO</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=6" title="Edit section: System interfacing - SISO & MIMO"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Control systems can be divided into different categories depending on the number of inputs and outputs. </p> <ul><li><a href="/wiki/Single-input_single-output_system" title="Single-input single-output system">Single-input single-output</a> (SISO) – This is the simplest and most common type, in which one output is controlled by one control signal. Examples are the cruise control example above, or an <a href="/wiki/Audio_system" class="mw-redirect" title="Audio system">audio system</a>, in which the control input is the input audio signal and the output is the sound waves from the speaker.</li> <li><a href="/wiki/Multiple-input_multiple-output_system" class="mw-redirect" title="Multiple-input multiple-output system">Multiple-input multiple-output</a> (MIMO) – These are found in more complicated systems. For example, modern large <a href="/wiki/Telescope" title="Telescope">telescopes</a> such as the <a href="/wiki/Keck_telescopes" class="mw-redirect" title="Keck telescopes">Keck</a> and <a href="/wiki/MMT_Observatory" title="MMT Observatory">MMT</a> have mirrors composed of many separate segments each controlled by an <a href="/wiki/Actuator" title="Actuator">actuator</a>. The shape of the entire mirror is constantly adjusted by a MIMO <a href="/wiki/Active_optics" title="Active optics">active optics</a> control system using input from multiple sensors at the focal plane, to compensate for changes in the mirror shape due to thermal expansion, contraction, stresses as it is rotated and distortion of the <a href="/wiki/Wavefront" title="Wavefront">wavefront</a> due to turbulence in the atmosphere. Complicated systems such as <a href="/wiki/Nuclear_reactor" title="Nuclear reactor">nuclear reactors</a> and human <a href="/wiki/Cell_(biology)" title="Cell (biology)">cells</a> are simulated by a computer as large MIMO control systems.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Classical_SISO_system_design">Classical SISO system design</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=7" title="Edit section: Classical SISO system design"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The scope of classical control theory is limited to <a href="/wiki/Single-input_single-output_system" title="Single-input single-output system">single-input and single-output</a> (SISO) system design, except when analyzing for disturbance rejection using a second input. The system analysis is carried out in the time domain using <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a>, in the complex-s domain with the <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a>, or in the frequency domain by transforming from the complex-s domain. Many systems may be assumed to have a second order and single variable system response in the time domain. A controller designed using classical theory often requires on-site tuning due to incorrect design approximations. Yet, due to the easier physical implementation of classical controller designs as compared to systems designed using modern control theory, these controllers are preferred in most industrial applications. The most common controllers designed using classical control theory are <a href="/wiki/PID_controller" class="mw-redirect" title="PID controller">PID controllers</a>. A less common implementation may include either or both a Lead or Lag filter. The ultimate end goal is to meet requirements typically provided in the time-domain called the step response, or at times in the frequency domain called the open-loop response. The step response characteristics applied in a specification are typically percent overshoot, settling time, etc. The open-loop response characteristics applied in a specification are typically Gain and Phase margin and bandwidth. These characteristics may be evaluated through simulation including a dynamic model of the system under control coupled with the compensation model. </p> <div class="mw-heading mw-heading3"><h3 id="Modern_MIMO_system_design">Modern MIMO system design</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=8" title="Edit section: Modern MIMO system design"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Modern control theory is carried out in the <a href="/wiki/State_space_(controls)" class="mw-redirect" title="State space (controls)">state space</a>, and can deal with <a href="/wiki/Multiple-input_multiple-output_system" class="mw-redirect" title="Multiple-input multiple-output system">multiple-input and multiple-output</a> (MIMO) systems. This overcomes the limitations of classical control theory in more sophisticated design problems, such as fighter aircraft control, with the limitation that no frequency domain analysis is possible. In modern design, a system is represented to the greatest advantage as a set of decoupled first order <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> defined using <a href="/wiki/State_variables" class="mw-redirect" title="State variables">state variables</a>. <a href="/wiki/Nonlinear_control" title="Nonlinear control">Nonlinear</a>, <a href="/w/index.php?title=Multivariable_control&action=edit&redlink=1" class="new" title="Multivariable control (page does not exist)">multivariable</a>, <a href="/wiki/Adaptive_control" title="Adaptive control">adaptive</a> and <a href="/wiki/Robust_control" title="Robust control">robust control</a> theories come under this division. Matrix methods are significantly limited for MIMO systems where linear independence cannot be assured in the relationship between inputs and outputs.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="No citation or explanation on this generic claim (November 2019)">citation needed</span></a></i>]</sup> Being fairly new, modern control theory has many areas yet to be explored. Scholars like <a href="/wiki/Rudolf_E._K%C3%A1lm%C3%A1n" title="Rudolf E. Kálmán">Rudolf E. Kálmán</a> and <a href="/wiki/Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Aleksandr Lyapunov</a> are well known among the people who have shaped modern control theory. </p> <div class="mw-heading mw-heading2"><h2 id="Topics_in_control_theory">Topics in control theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=9" title="Edit section: Topics in control theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Stability">Stability</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=10" title="Edit section: Stability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>stability</i> of a general <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical system</a> with no input can be described with <a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov stability</a> criteria. </p> <ul><li>A <a href="/wiki/Linear_system" title="Linear system">linear system</a> is called <a href="/wiki/BIBO_stability" title="BIBO stability">bounded-input bounded-output (BIBO) stable</a> if its output will stay <a href="/wiki/Bounded_function" title="Bounded function">bounded</a> for any bounded input.</li> <li>Stability for <a href="/wiki/Nonlinear_system" title="Nonlinear system">nonlinear systems</a> that take an input is <a href="/wiki/Input-to-state_stability" title="Input-to-state stability">input-to-state stability</a> (ISS), which combines Lyapunov stability and a notion similar to BIBO stability.</li></ul> <p>For simplicity, the following descriptions focus on continuous-time and discrete-time <b>linear systems</b>. </p><p>Mathematically, this means that for a causal linear system to be stable all of the <a href="/wiki/Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">poles</a> of its <a href="/wiki/Transfer_function" title="Transfer function">transfer function</a> must have negative-real values, i.e. the real part of each pole must be less than zero. Practically speaking, stability requires that the transfer function complex poles reside </p> <ul><li>in the open left half of the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> for continuous time, when the <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a> is used to obtain the transfer function.</li> <li>inside the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> for discrete time, when the <a href="/wiki/Z-transform" title="Z-transform">Z-transform</a> is used.</li></ul> <p>The difference between the two cases is simply due to the traditional method of plotting continuous time versus discrete time transfer functions. The continuous Laplace transform is in <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> where the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> axis is the real axis and the discrete Z-transform is in <a href="/wiki/Circular_coordinates" class="mw-redirect" title="Circular coordinates">circular coordinates</a> where the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> axis is the real axis. </p><p>When the appropriate conditions above are satisfied a system is said to be <a href="/wiki/Asymptotic_stability" class="mw-redirect" title="Asymptotic stability">asymptotically stable</a>; the variables of an asymptotically stable control system always decrease from their initial value and do not show permanent oscillations. Permanent oscillations occur when a pole has a real part exactly equal to zero (in the continuous time case) or a <a href="/wiki/Absolute_value#Complex_numbers" title="Absolute value">modulus</a> equal to one (in the discrete time case). If a simply stable system response neither decays nor grows over time, and has no oscillations, it is <a href="/wiki/Marginal_stability" title="Marginal stability">marginally stable</a>; in this case the system transfer function has non-repeated poles at the complex plane origin (i.e. their real and complex component is zero in the continuous time case). Oscillations are present when poles with real part equal to zero have an imaginary part not equal to zero. </p><p>If a system in question has an <a href="/wiki/Impulse_response" title="Impulse response">impulse response</a> of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x[n]=0.5^{n}u[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mn>0.5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>u</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x[n]=0.5^{n}u[n]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3fe9bf89c5cffaf461081935fd41745dc768063" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.905ex; height:2.843ex;" alt="{\displaystyle \ x[n]=0.5^{n}u[n]}"></span></dd></dl> <p>then the Z-transform (see <a href="/wiki/Z-transform#Example_2_(causal_ROC)" title="Z-transform">this example</a>), is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ X(z)={\frac {1}{1-0.5z^{-1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>X</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mn>0.5</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ X(z)={\frac {1}{1-0.5z^{-1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f65c87b92a3611ffc58d3df40792ff4e5e69b65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.79ex; height:5.676ex;" alt="{\displaystyle \ X(z)={\frac {1}{1-0.5z^{-1}}}}"></span></dd></dl> <p>which has a pole in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>0.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=0.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9267ea9749ae11b3cde9c735c21f2507de379f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.158ex; height:2.176ex;" alt="{\displaystyle z=0.5}"></span> (zero <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary part</a>). This system is BIBO (asymptotically) stable since the pole is <i>inside</i> the unit circle. </p><p>However, if the impulse response was </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x[n]=1.5^{n}u[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mn>1.5</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>u</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x[n]=1.5^{n}u[n]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b769b726a2a55b9fc5e5c8d800187d7715cf84cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.905ex; height:2.843ex;" alt="{\displaystyle \ x[n]=1.5^{n}u[n]}"></span></dd></dl> <p>then the Z-transform is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ X(z)={\frac {1}{1-1.5z^{-1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>X</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mn>1.5</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ X(z)={\frac {1}{1-1.5z^{-1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81dd2f0c11aecabe8b5bf08505c7d52974f4ec35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.79ex; height:5.676ex;" alt="{\displaystyle \ X(z)={\frac {1}{1-1.5z^{-1}}}}"></span></dd></dl> <p>which has a pole at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>1.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=1.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f374dfe3fe7ce1355695d53ea69482094aa8935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.158ex; height:2.176ex;" alt="{\displaystyle z=1.5}"></span> and is not BIBO stable since the pole has a modulus strictly greater than one. </p><p>Numerous tools exist for the analysis of the poles of a system. These include graphical systems like the <a href="/wiki/Root_locus" class="mw-redirect" title="Root locus">root locus</a>, <a href="/wiki/Bode_plot" title="Bode plot">Bode plots</a> or the <a href="/wiki/Nyquist_plot" class="mw-redirect" title="Nyquist plot">Nyquist plots</a>. </p><p>Mechanical changes can make equipment (and control systems) more stable. Sailors add ballast to improve the stability of ships. Cruise ships use <a href="/wiki/Ship_stability#Stabilizer_fins" title="Ship stability">antiroll fins</a> that extend transversely from the side of the ship for perhaps 30 feet (10 m) and are continuously rotated about their axes to develop forces that oppose the roll. </p> <div class="mw-heading mw-heading3"><h3 id="Controllability_and_observability">Controllability and observability</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=11" title="Edit section: Controllability and observability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Controllability" title="Controllability">Controllability</a> and <a href="/wiki/Observability" title="Observability">Observability</a></div> <p><a href="/wiki/Controllability" title="Controllability">Controllability</a> and <a href="/wiki/Observability" title="Observability">observability</a> are main issues in the analysis of a system before deciding the best control strategy to be applied, or whether it is even possible to control or stabilize the system. Controllability is related to the possibility of forcing the system into a particular state by using an appropriate control signal. If a state is not controllable, then no signal will ever be able to control the state. If a state is not controllable, but its dynamics are stable, then the state is termed <i>stabilizable</i>. Observability instead is related to the possibility of <i>observing</i>, through output measurements, the state of a system. If a state is not observable, the controller will never be able to determine the behavior of an unobservable state and hence cannot use it to stabilize the system. However, similar to the stabilizability condition above, if a state cannot be observed it might still be detectable. </p><p>From a geometrical point of view, looking at the states of each variable of the system to be controlled, every "bad" state of these variables must be controllable and observable to ensure a good behavior in the closed-loop system. That is, if one of the <a href="/wiki/Eigenvalues" class="mw-redirect" title="Eigenvalues">eigenvalues</a> of the system is not both controllable and observable, this part of the dynamics will remain untouched in the closed-loop system. If such an eigenvalue is not stable, the dynamics of this eigenvalue will be present in the closed-loop system which therefore will be unstable. Unobservable poles are not present in the transfer function realization of a state-space representation, which is why sometimes the latter is preferred in dynamical systems analysis. </p><p>Solutions to problems of an uncontrollable or unobservable system include adding actuators and sensors. </p> <div class="mw-heading mw-heading3"><h3 id="Control_specification">Control specification</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=12" title="Edit section: Control specification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Several different control strategies have been devised in the past years. These vary from extremely general ones (PID controller), to others devoted to very particular classes of systems (especially <a href="/wiki/Robotics" title="Robotics">robotics</a> or aircraft cruise control). </p><p>A control problem can have several specifications. Stability, of course, is always present. The controller must ensure that the closed-loop system is stable, regardless of the open-loop stability. A poor choice of controller can even worsen the stability of the open-loop system, which must normally be avoided. Sometimes it would be desired to obtain particular dynamics in the closed loop: i.e. that the poles have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Re[\lambda ]<-{\overline {\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>e</mi> <mo stretchy="false">[</mo> <mi>λ</mi> <mo stretchy="false">]</mo> <mo><</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Re[\lambda ]<-{\overline {\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acd3c480f7bd6fa14fd42e56521994a3b4ad8e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.873ex; height:3.509ex;" alt="{\displaystyle Re[\lambda ]<-{\overline {\lambda }}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de9d2ee37eb0cd550de8d8f34b9d13700f79384e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.47ex; height:3.009ex;" alt="{\displaystyle {\overline {\lambda }}}"></span> is a fixed value strictly greater than zero, instead of simply asking that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Re[\lambda ]<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>e</mi> <mo stretchy="false">[</mo> <mi>λ</mi> <mo stretchy="false">]</mo> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Re[\lambda ]<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57bd3912e4d0e7aafac442e28a10f4748da7b90d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.757ex; height:2.843ex;" alt="{\displaystyle Re[\lambda ]<0}"></span>. </p><p>Another typical specification is the rejection of a step disturbance; including an <a href="/wiki/Integrator" title="Integrator">integrator</a> in the open-loop chain (i.e. directly before the system under control) easily achieves this. Other classes of disturbances need different types of sub-systems to be included. </p><p>Other "classical" control theory specifications regard the time-response of the closed-loop system. These include the <a href="/wiki/Rise_time" title="Rise time">rise time</a> (the time needed by the control system to reach the desired value after a perturbation), peak <a href="/wiki/Overshoot_(signal)" title="Overshoot (signal)">overshoot</a> (the highest value reached by the response before reaching the desired value) and others (<a href="/wiki/Settling_time" title="Settling time">settling time</a>, quarter-decay). Frequency domain specifications are usually related to <a href="/wiki/Robust_control" title="Robust control">robustness</a> (see after). </p><p>Modern performance assessments use some variation of integrated tracking error (IAE, ISA, CQI). </p> <div class="mw-heading mw-heading3"><h3 id="Model_identification_and_robustness">Model identification and robustness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=13" title="Edit section: Model identification and robustness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A control system must always have some robustness property. A <a href="/wiki/Robust_control" title="Robust control">robust controller</a> is such that its properties do not change much if applied to a system slightly different from the mathematical one used for its synthesis. This requirement is important, as no real physical system truly behaves like the series of differential equations used to represent it mathematically. Typically a simpler mathematical model is chosen in order to simplify calculations, otherwise, the true <a href="/wiki/System_dynamics" title="System dynamics">system dynamics</a> can be so complicated that a complete model is impossible. </p> <dl><dt>System identification</dt></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/System_identification" title="System identification">System identification</a></div> <p>The process of determining the equations that govern the model's dynamics is called <a href="/wiki/System_identification" title="System identification">system identification</a>. This can be done off-line: for example, executing a series of measures from which to calculate an approximated mathematical model, typically its <a href="/wiki/Transfer_function" title="Transfer function">transfer function</a> or matrix. Such identification from the output, however, cannot take account of unobservable dynamics. Sometimes the model is built directly starting from known physical equations, for example, in the case of a <a href="/wiki/Mass-spring-damper_model" title="Mass-spring-damper model">mass-spring-damper</a> system we know that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\ddot {x}}(t)=-Kx(t)-\mathrm {B} {\dot {x}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>K</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\ddot {x}}(t)=-Kx(t)-\mathrm {B} {\dot {x}}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eb22dba60937019ecc7cd8dd87f67d6977c2209" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.435ex; height:2.843ex;" alt="{\displaystyle m{\ddot {x}}(t)=-Kx(t)-\mathrm {B} {\dot {x}}(t)}"></span>. Even assuming that a "complete" model is used in designing the controller, all the parameters included in these equations (called "nominal parameters") are never known with absolute precision; the control system will have to behave correctly even when connected to a physical system with true parameter values away from nominal. </p><p>Some advanced control techniques include an "on-line" identification process (see later). The parameters of the model are calculated ("identified") while the controller itself is running. In this way, if a drastic variation of the parameters ensues, for example, if the robot's arm releases a weight, the controller will adjust itself consequently in order to ensure the correct performance. </p> <dl><dt>Analysis</dt></dl> <p>Analysis of the robustness of a SISO (single input single output) control system can be performed in the frequency domain, considering the system's transfer function and using <a href="/wiki/Nyquist_diagram" class="mw-redirect" title="Nyquist diagram">Nyquist</a> and <a href="/wiki/Bode_diagram" class="mw-redirect" title="Bode diagram">Bode diagrams</a>. Topics include <a href="/wiki/Bode_plot#Gain_margin_and_phase_margin" title="Bode plot">gain and phase margin</a> and amplitude margin. For MIMO (multi-input multi output) and, in general, more complicated control systems, one must consider the theoretical results devised for each control technique (see next section). I.e., if particular robustness qualities are needed, the engineer must shift their attention to a control technique by including these qualities in its properties. </p> <dl><dt>Constraints</dt></dl> <p>A particular robustness issue is the requirement for a control system to perform properly in the presence of input and state constraints. In the physical world every signal is limited. It could happen that a controller will send control signals that cannot be followed by the physical system, for example, trying to rotate a valve at excessive speed. This can produce undesired behavior of the closed-loop system, or even damage or break actuators or other subsystems. Specific control techniques are available to solve the problem: <a href="/wiki/Model_predictive_control" title="Model predictive control">model predictive control</a> (see later), and <a href="/wiki/Anti-wind_up_system_(control)" class="mw-redirect" title="Anti-wind up system (control)">anti-wind up systems</a>. The latter consists of an additional control block that ensures that the control signal never exceeds a given threshold. </p> <div class="mw-heading mw-heading2"><h2 id="System_classifications">System classifications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=14" title="Edit section: System classifications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Linear_systems_control">Linear systems control</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=15" title="Edit section: Linear systems control"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/State_space_(controls)" class="mw-redirect" title="State space (controls)">State space (controls)</a></div> <p>For MIMO systems, pole placement can be performed mathematically using a <a href="/wiki/State_space_(controls)" class="mw-redirect" title="State space (controls)">state space representation</a> of the open-loop system and calculating a feedback matrix assigning poles in the desired positions. In complicated systems this can require computer-assisted calculation capabilities, and cannot always ensure robustness. Furthermore, all system states are not in general measured and so observers must be included and incorporated in pole placement design. </p> <div class="mw-heading mw-heading3"><h3 id="Nonlinear_systems_control">Nonlinear systems control</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=16" title="Edit section: Nonlinear systems control"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Nonlinear_control" title="Nonlinear control">Nonlinear control</a></div> <p>Processes in industries like <a href="/wiki/Robotics" title="Robotics">robotics</a> and the <a href="/wiki/Aerospace_industry" class="mw-redirect" title="Aerospace industry">aerospace industry</a> typically have strong nonlinear dynamics. In control theory it is sometimes possible to linearize such classes of systems and apply linear techniques, but in many cases it can be necessary to devise from scratch theories permitting control of nonlinear systems. These, e.g., <a href="/wiki/Feedback_linearization" title="Feedback linearization">feedback linearization</a>, <a href="/wiki/Backstepping" title="Backstepping">backstepping</a>, <a href="/wiki/Sliding_mode_control" title="Sliding mode control">sliding mode control</a>, trajectory linearization control normally take advantage of results based on <a href="/wiki/Lyapunov%27s_theory" class="mw-redirect" title="Lyapunov's theory">Lyapunov's theory</a>. <a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a> has been widely used as a tool for generalizing well-known linear control concepts to the nonlinear case, as well as showing the subtleties that make it a more challenging problem. Control theory has also been used to decipher the neural mechanism that directs cognitive states.<sup id="cite_ref-Shi_Gu_et_al_19-0" class="reference"><a href="#cite_note-Shi_Gu_et_al-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Decentralized_systems_control">Decentralized systems control</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=17" title="Edit section: Decentralized systems control"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Distributed_control_system" title="Distributed control system">Distributed control system</a></div> <p>When the system is controlled by multiple controllers, the problem is one of decentralized control. Decentralization is helpful in many ways, for instance, it helps control systems to operate over a larger geographical area. The agents in decentralized control systems can interact using communication channels and coordinate their actions. </p> <div class="mw-heading mw-heading3"><h3 id="Deterministic_and_stochastic_systems_control">Deterministic and stochastic systems control</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=18" title="Edit section: Deterministic and stochastic systems control"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Stochastic_control" title="Stochastic control">Stochastic control</a></div> <p>A stochastic control problem is one in which the evolution of the state variables is subjected to random shocks from outside the system. A deterministic control problem is not subject to external random shocks. </p> <div class="mw-heading mw-heading2"><h2 id="Main_control_strategies">Main control strategies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=19" title="Edit section: Main control strategies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every control system must guarantee first the stability of the closed-loop behavior. For <a href="/wiki/Linear_system" title="Linear system">linear systems</a>, this can be obtained by directly placing the poles. Nonlinear control systems use specific theories (normally based on <a href="/wiki/Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Aleksandr Lyapunov</a>'s Theory) to ensure stability without regard to the inner dynamics of the system. The possibility to fulfill different specifications varies from the model considered and the control strategy chosen. </p> <dl><dt>List of the main control techniques</dt></dl> <ul><li><a href="/wiki/Optimal_control" title="Optimal control">Optimal control</a> is a particular control technique in which the control signal optimizes a certain "cost index": for example, in the case of a satellite, the jet thrusts needed to bring it to desired trajectory that consume the least amount of fuel. Two optimal control design methods have been widely used in industrial applications, as it has been shown they can guarantee closed-loop stability. These are <a href="/wiki/Model_Predictive_Control" class="mw-redirect" title="Model Predictive Control">Model Predictive Control</a> (MPC) and <a href="/wiki/Linear-quadratic-Gaussian_control" class="mw-redirect" title="Linear-quadratic-Gaussian control">linear-quadratic-Gaussian control</a> (LQG). The first can more explicitly take into account constraints on the signals in the system, which is an important feature in many industrial processes. However, the "optimal control" structure in MPC is only a means to achieve such a result, as it does not optimize a true performance index of the closed-loop control system. Together with PID controllers, MPC systems are the most widely used control technique in <a href="/wiki/Process_control" class="mw-redirect" title="Process control">process control</a>.</li> <li><a href="/wiki/Robust_control" title="Robust control">Robust control</a> deals explicitly with uncertainty in its approach to controller design. Controllers designed using <i>robust control</i> methods tend to be able to cope with small differences between the true system and the nominal model used for design.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> The early methods of <a href="/wiki/Hendrik_Wade_Bode" title="Hendrik Wade Bode">Bode</a> and others were fairly robust; the state-space methods invented in the 1960s and 1970s were sometimes found to lack robustness. Examples of modern robust control techniques include <a href="/wiki/H-infinity_loop-shaping" title="H-infinity loop-shaping">H-infinity loop-shaping</a> developed by Duncan McFarlane and <a href="/wiki/Keith_Glover" title="Keith Glover">Keith Glover</a>, <a href="/wiki/Sliding_mode_control" title="Sliding mode control">Sliding mode control</a> (SMC) developed by <a href="/wiki/Vadim_Utkin" title="Vadim Utkin">Vadim Utkin</a>, and safe protocols designed for control of large heterogeneous populations of electric loads in Smart Power Grid applications.<sup id="cite_ref-TCL1_21-0" class="reference"><a href="#cite_note-TCL1-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Robust methods aim to achieve robust performance and/or <a href="/wiki/Stability_theory" title="Stability theory">stability</a> in the presence of small modeling errors.</li> <li><a href="/wiki/Stochastic_control" title="Stochastic control">Stochastic control</a> deals with control design with uncertainty in the model. In typical stochastic control problems, it is assumed that there exist random noise and disturbances in the model and the controller, and the control design must take into account these random deviations.</li> <li><a href="/wiki/Adaptive_control" title="Adaptive control">Adaptive control</a> uses on-line identification of the process parameters, or modification of controller gains, thereby obtaining strong robustness properties. Adaptive controls were applied for the first time in the <a href="/wiki/Aerospace_industry" class="mw-redirect" title="Aerospace industry">aerospace industry</a> in the 1950s, and have found particular success in that field.</li> <li>A <a href="/wiki/Hierarchical_control_system" title="Hierarchical control system">hierarchical control system</a> is a type of <a href="/wiki/Control_system" title="Control system">control system</a> in which a set of devices and governing software is arranged in a <a href="/wiki/Hierarchical" class="mw-redirect" title="Hierarchical">hierarchical</a> <a href="/wiki/Tree_(data_structure)" class="mw-redirect" title="Tree (data structure)">tree</a>. When the links in the tree are implemented by a <a href="/wiki/Computer_network" title="Computer network">computer network</a>, then that hierarchical control system is also a form of <a href="/wiki/Networked_control_system" title="Networked control system">networked control system</a>.</li> <li><a href="/wiki/Intelligent_control" title="Intelligent control">Intelligent control</a> uses various AI computing approaches like <a href="/wiki/Artificial_neural_networks" class="mw-redirect" title="Artificial neural networks">artificial neural networks</a>, <a href="/wiki/Bayesian_probability" title="Bayesian probability">Bayesian probability</a>, <a href="/wiki/Fuzzy_logic" title="Fuzzy logic">fuzzy logic</a>,<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>, <a href="/wiki/Evolutionary_computation" title="Evolutionary computation">evolutionary computation</a> and <a href="/wiki/Genetic_algorithms" class="mw-redirect" title="Genetic algorithms">genetic algorithms</a> or a combination of these methods, such as <a href="/wiki/Neuro-fuzzy" title="Neuro-fuzzy">neuro-fuzzy</a> algorithms, to control a <a href="/wiki/Dynamic_system" class="mw-redirect" title="Dynamic system">dynamic system</a>.</li> <li><a href="/wiki/Self-organized_criticality_control" title="Self-organized criticality control">Self-organized criticality control</a> may be defined as attempts to interfere in the processes by which the <a href="/wiki/Self-organized" class="mw-redirect" title="Self-organized">self-organized</a> system dissipates energy.</li></ul> <div class="mw-heading mw-heading2"><h2 id="People_in_systems_and_control">People in systems and control</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=20" title="Edit section: People in systems and control"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/People_in_systems_and_control" class="mw-redirect" title="People in systems and control">People in systems and control</a></div> <p>Many active and historical figures made significant contribution to control theory including </p> <ul><li><a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> invented the <a href="/wiki/Z-transform" title="Z-transform">Z-transform</a> in his work on <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, now used to solve discrete-time control theory problems. The Z-transform is a discrete-time equivalent of the <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a> which is named after him.</li> <li><a href="/wiki/Irmgard_Flugge-Lotz" class="mw-redirect" title="Irmgard Flugge-Lotz">Irmgard Flugge-Lotz</a> developed the theory of <a href="/wiki/Bang-bang_control" class="mw-redirect" title="Bang-bang control">discontinuous automatic control</a> and applied it to <a href="/wiki/Autopilot" title="Autopilot">automatic aircraft control systems</a>.</li> <li><a href="/wiki/Alexander_Lyapunov" class="mw-redirect" title="Alexander Lyapunov">Alexander Lyapunov</a> in the 1890s marks the beginning of <a href="/wiki/Stability_theory" title="Stability theory">stability theory</a>.</li> <li><a href="/wiki/Harold_Stephen_Black" title="Harold Stephen Black">Harold S. Black</a> invented the concept of <a href="/wiki/Negative_feedback_amplifier" class="mw-redirect" title="Negative feedback amplifier">negative feedback amplifiers</a> in 1927. He managed to develop stable negative feedback amplifiers in the 1930s.</li> <li><a href="/wiki/Harry_Nyquist" title="Harry Nyquist">Harry Nyquist</a> developed the <a href="/wiki/Nyquist_stability_criterion" title="Nyquist stability criterion">Nyquist stability criterion</a> for feedback systems in the 1930s.</li> <li><a href="/wiki/Richard_Bellman" class="mw-redirect" title="Richard Bellman">Richard Bellman</a> developed <a href="/wiki/Dynamic_programming" title="Dynamic programming">dynamic programming</a> in the 1940s.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Warren_E._Dixon" title="Warren E. Dixon">Warren E. Dixon</a>, control theorist and a professor</li> <li><a href="/w/index.php?title=Kyriakos_G._Vamvoudakis&action=edit&redlink=1" class="new" title="Kyriakos G. Vamvoudakis (page does not exist)">Kyriakos G. Vamvoudakis</a>, developed synchronous reinforcement learning algorithms to solve optimal control and game theoretic problems</li> <li><a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Andrey Kolmogorov</a> co-developed the <a href="/wiki/Wiener_filter" title="Wiener filter">Wiener–Kolmogorov filter</a> in 1941.</li> <li><a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a> co-developed the Wiener–Kolmogorov filter and coined the term <a href="/wiki/Cybernetics" title="Cybernetics">cybernetics</a> in the 1940s.</li> <li><a href="/wiki/John_R._Ragazzini" title="John R. Ragazzini">John R. Ragazzini</a> introduced <a href="/wiki/Digital_control" title="Digital control">digital control</a> and the use of <a href="/wiki/Z-transform" title="Z-transform">Z-transform</a> in control theory (invented by Laplace) in the 1950s.</li> <li><a href="/wiki/Lev_Pontryagin" title="Lev Pontryagin">Lev Pontryagin</a> introduced the <a href="/wiki/Pontryagin%27s_minimum_principle" class="mw-redirect" title="Pontryagin's minimum principle">maximum principle</a> and the <a href="/wiki/Bang-bang_control" class="mw-redirect" title="Bang-bang control">bang-bang principle</a>.</li> <li><a href="/wiki/Pierre-Louis_Lions" title="Pierre-Louis Lions">Pierre-Louis Lions</a> developed <a href="/wiki/Viscosity_solutions" class="mw-redirect" title="Viscosity solutions">viscosity solutions</a> into stochastic control and <a href="/wiki/Optimal_control" title="Optimal control">optimal control</a> methods.</li> <li><a href="/wiki/Rudolf_E._K%C3%A1lm%C3%A1n" title="Rudolf E. Kálmán">Rudolf E. Kálmán</a> pioneered the <a href="/wiki/State-space" class="mw-redirect" title="State-space">state-space</a> approach to systems and control. Introduced the notions of <a href="/wiki/Controllability" title="Controllability">controllability</a> and <a href="/wiki/Observability" title="Observability">observability</a>. Developed the <a href="/wiki/Kalman_filter" title="Kalman filter">Kalman filter</a> for linear estimation.</li> <li><a href="/wiki/Ali_H._Nayfeh" title="Ali H. Nayfeh">Ali H. Nayfeh</a> who was one of the main contributors to nonlinear control theory and published many books on perturbation methods</li> <li><a href="/wiki/Jan_Camiel_Willems" title="Jan Camiel Willems">Jan C. Willems</a> Introduced the concept of dissipativity, as a generalization of <a href="/wiki/Lyapunov_function" title="Lyapunov function">Lyapunov function</a> to input/state/output systems. The construction of the storage function, as the analogue of a Lyapunov function is called, led to the study of the <a href="/wiki/Linear_matrix_inequality" title="Linear matrix inequality">linear matrix inequality</a> (LMI) in control theory. He pioneered the behavioral approach to mathematical systems theory.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=21" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><span><img alt="diagram" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Complex-adaptive-system.jpg/32px-Complex-adaptive-system.jpg" decoding="async" width="32" height="23" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Complex-adaptive-system.jpg/48px-Complex-adaptive-system.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Complex-adaptive-system.jpg/64px-Complex-adaptive-system.jpg 2x" data-file-width="2100" data-file-height="1500" /></span></span></span><span class="portalbox-link"><a href="/wiki/Portal:Systems_science" title="Portal:Systems science">Systems science portal</a></span></li></ul> <dl><dt>Examples of control systems</dt></dl> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Automation" title="Automation">Automation</a></li> <li><a href="/wiki/Deadbeat_controller" class="mw-redirect" title="Deadbeat controller">Deadbeat controller</a></li> <li><a href="/wiki/Distributed_parameter_systems" class="mw-redirect" title="Distributed parameter systems">Distributed parameter systems</a></li> <li><a href="/wiki/Fractional-order_control" title="Fractional-order control">Fractional-order control</a></li> <li><a href="/wiki/H-infinity_loop-shaping" title="H-infinity loop-shaping">H-infinity loop-shaping</a></li> <li><a href="/wiki/Hierarchical_control_system" title="Hierarchical control system">Hierarchical control system</a></li> <li><a href="/wiki/Model_predictive_control" title="Model predictive control">Model predictive control</a></li> <li><a href="/wiki/Optimal_control" title="Optimal control">Optimal control</a></li> <li><a href="/wiki/Process_control" class="mw-redirect" title="Process control">Process control</a></li> <li><a href="/wiki/Robust_control" title="Robust control">Robust control</a></li> <li><a href="/wiki/Servomechanism" title="Servomechanism">Servomechanism</a></li> <li><a href="/wiki/State_space_(controls)" class="mw-redirect" title="State space (controls)">State space (controls)</a></li> <li><a href="/wiki/Vector_control_(motor)" title="Vector control (motor)">Vector control</a></li></ul> </div> <dl><dt>Topics in control theory</dt></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col"> <ul><li><a href="/wiki/Coefficient_diagram_method" title="Coefficient diagram method">Coefficient diagram method</a></li> <li><a href="/wiki/Control_reconfiguration" title="Control reconfiguration">Control reconfiguration</a></li> <li><a href="/wiki/Feedback" title="Feedback">Feedback</a></li> <li><a href="/wiki/H_infinity" class="mw-redirect" title="H infinity">H infinity</a></li> <li><a href="/wiki/Hankel_singular_value" title="Hankel singular value">Hankel singular value</a></li> <li><a href="/wiki/Krener%27s_theorem" title="Krener's theorem">Krener's theorem</a></li> <li><a href="/wiki/Lead-lag_compensator" class="mw-redirect" title="Lead-lag compensator">Lead-lag compensator</a></li> <li><a href="/wiki/Minor_loop_feedback" title="Minor loop feedback">Minor loop feedback</a></li> <li><a href="/wiki/Minor_loop_feedback" title="Minor loop feedback">Multi-loop feedback</a></li> <li><a href="/wiki/Positive_systems" title="Positive systems">Positive systems</a></li> <li><a href="/wiki/Radial_basis_function" title="Radial basis function">Radial basis function</a></li> <li><a href="/wiki/Root_locus" class="mw-redirect" title="Root locus">Root locus</a></li> <li><a href="/wiki/Signal-flow_graph" title="Signal-flow graph">Signal-flow graphs</a></li> <li><a href="/wiki/Stable_polynomial" title="Stable polynomial">Stable polynomial</a></li> <li><a href="/wiki/State_space_representation" class="mw-redirect" title="State space representation">State space representation</a></li> <li><a href="/wiki/Steady_state" title="Steady state">Steady state</a></li> <li><a href="/wiki/Transient_response" title="Transient response">Transient response</a></li> <li><a href="/wiki/Transient_state" title="Transient state">Transient state</a></li> <li><a href="/wiki/Underactuation" title="Underactuation">Underactuation</a></li> <li><a href="/wiki/Youla%E2%80%93Kucera_parametrization" title="Youla–Kucera parametrization">Youla–Kucera parametrization</a></li> <li><a href="/wiki/Markov_chain_approximation_method" title="Markov chain approximation method">Markov chain approximation method</a></li></ul> </div> <dl><dt>Other related topics</dt></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col"> <ul><li><a href="/wiki/Adaptive_system" title="Adaptive system">Adaptive system</a></li> <li><a href="/wiki/Automation_and_remote_control" class="mw-redirect" title="Automation and remote control">Automation and remote control</a></li> <li><a href="/wiki/Bond_graph" title="Bond graph">Bond graph</a></li> <li><a href="/wiki/Control_engineering" title="Control engineering">Control engineering</a></li> <li><a href="/wiki/Control%E2%80%93feedback%E2%80%93abort_loop" title="Control–feedback–abort loop">Control–feedback–abort loop</a></li> <li><a href="/wiki/Controller_(control_theory)" class="mw-redirect" title="Controller (control theory)">Controller (control theory)</a></li> <li><a href="/wiki/Cybernetics" title="Cybernetics">Cybernetics</a></li> <li><a href="/wiki/Intelligent_control" title="Intelligent control">Intelligent control</a></li> <li><a href="/wiki/Mathematical_system_theory" class="mw-redirect" title="Mathematical system theory">Mathematical system theory</a></li> <li><a href="/wiki/Negative_feedback_amplifier" class="mw-redirect" title="Negative feedback amplifier">Negative feedback amplifier</a></li> <li><a href="/wiki/Outline_of_management" title="Outline of management">Outline of management</a></li> <li><a href="/wiki/People_in_systems_and_control" class="mw-redirect" title="People in systems and control">People in systems and control</a></li> <li><a href="/wiki/Perceptual_control_theory" title="Perceptual control theory">Perceptual control theory</a></li> <li><a href="/wiki/Systems_theory" title="Systems theory">Systems theory</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=22" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free 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"Robustness of Adaptation in Controlled Self-Adjusting Chaotic Systems". <i>Fluctuation and Noise Letters</i>. <b>02</b> (4): L285–L292. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0219477502000919">10.1142/S0219477502000919</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fluctuation+and+Noise+Letters&rft.atitle=Robustness+of+Adaptation+in+Controlled+Self-Adjusting+Chaotic+Systems&rft.volume=02&rft.issue=4&rft.pages=L285-L292&rft.date=2002&rft_id=info%3Adoi%2F10.1142%2FS0219477502000919&rft.aulast=Melby&rft.aufirst=Paul&rft.au=et.%2C+al.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></span> </li> <li id="cite_note-TCL1-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-TCL1_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFN._A._Sinitsyn._S._Kundu,_S._Backhaus2013" class="citation journal cs1">N. A. Sinitsyn. S. Kundu, S. Backhaus (2013). "Safe Protocols for Generating Power Pulses with Heterogeneous Populations of Thermostatically Controlled Loads". <i><a href="/wiki/Energy_Conversion_and_Management" title="Energy Conversion and Management">Energy Conversion and Management</a></i>. <b>67</b>: 297–308. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1211.0248">1211.0248</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013ECM....67..297S">2013ECM....67..297S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.enconman.2012.11.021">10.1016/j.enconman.2012.11.021</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:32067734">32067734</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Energy+Conversion+and+Management&rft.atitle=Safe+Protocols+for+Generating+Power+Pulses+with+Heterogeneous+Populations+of+Thermostatically+Controlled+Loads&rft.volume=67&rft.pages=297-308&rft.date=2013&rft_id=info%3Aarxiv%2F1211.0248&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A32067734%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.enconman.2012.11.021&rft_id=info%3Abibcode%2F2013ECM....67..297S&rft.au=N.+A.+Sinitsyn.+S.+Kundu%2C+S.+Backhaus&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiuWilson_WangFarid_GolnaraghiEric_Kubica2010" class="citation journal cs1">Liu, Jie; Wilson Wang; Farid Golnaraghi; Eric Kubica (2010). "A novel fuzzy framework for nonlinear system control". <i>Fuzzy Sets and Systems</i>. <b>161</b> (21): 2746–2759. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.fss.2010.04.009">10.1016/j.fss.2010.04.009</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fuzzy+Sets+and+Systems&rft.atitle=A+novel+fuzzy+framework+for+nonlinear+system+control&rft.volume=161&rft.issue=21&rft.pages=2746-2759&rft.date=2010&rft_id=info%3Adoi%2F10.1016%2Fj.fss.2010.04.009&rft.aulast=Liu&rft.aufirst=Jie&rft.au=Wilson+Wang&rft.au=Farid+Golnaraghi&rft.au=Eric+Kubica&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichard_Bellman1964" class="citation magazine cs1"><a href="/wiki/Richard_Bellman" class="mw-redirect" title="Richard Bellman">Richard Bellman</a> (1964). "Control Theory". <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i>. Vol. 211, no. 3. pp. 186–200. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican0964-186">10.1038/scientificamerican0964-186</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=Control+Theory&rft.volume=211&rft.issue=3&rft.pages=186-200&rft.date=1964&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican0964-186&rft.au=Richard+Bellman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=23" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevine1996" class="citation book cs1">Levine, William S., ed. (1996). <i>The Control Handbook</i>. New York: CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8493-8570-4" title="Special:BookSources/978-0-8493-8570-4"><bdi>978-0-8493-8570-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Control+Handbook&rft.place=New+York&rft.pub=CRC+Press&rft.date=1996&rft.isbn=978-0-8493-8570-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarl_J._ÅströmRichard_M._Murray2008" class="citation book cs1">Karl J. Åström; Richard M. Murray (2008). <a rel="nofollow" class="external text" href="http://www.cds.caltech.edu/~murray/books/AM08/pdf/am08-complete_28Sep12.pdf"><i>Feedback Systems: An Introduction for Scientists and Engineers</i></a> <span class="cs1-format">(PDF)</span>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-13576-2" title="Special:BookSources/978-0-691-13576-2"><bdi>978-0-691-13576-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Feedback+Systems%3A+An+Introduction+for+Scientists+and+Engineers.&rft.pub=Princeton+University+Press&rft.date=2008&rft.isbn=978-0-691-13576-2&rft.au=Karl+J.+%C3%85str%C3%B6m&rft.au=Richard+M.+Murray&rft_id=http%3A%2F%2Fwww.cds.caltech.edu%2F~murray%2Fbooks%2FAM08%2Fpdf%2Fam08-complete_28Sep12.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChristopher_Kilian2005" class="citation book cs1">Christopher Kilian (2005). <i>Modern Control Technology</i>. Thompson Delmar Learning. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4018-5806-3" title="Special:BookSources/978-1-4018-5806-3"><bdi>978-1-4018-5806-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Control+Technology&rft.pub=Thompson+Delmar+Learning&rft.date=2005&rft.isbn=978-1-4018-5806-3&rft.au=Christopher+Kilian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVannevar_Bush1929" class="citation book cs1">Vannevar Bush (1929). <i>Operational Circuit Analysis</i>. John Wiley and Sons, Inc.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Operational+Circuit+Analysis&rft.pub=John+Wiley+and+Sons%2C+Inc.&rft.date=1929&rft.au=Vannevar+Bush&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_F._Stengel1994" class="citation book cs1">Robert F. Stengel (1994). <i>Optimal Control and Estimation</i>. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-68200-6" title="Special:BookSources/978-0-486-68200-6"><bdi>978-0-486-68200-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optimal+Control+and+Estimation&rft.pub=Dover+Publications&rft.date=1994&rft.isbn=978-0-486-68200-6&rft.au=Robert+F.+Stengel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFranklin2002" class="citation book cs1">Franklin; et al. (2002). <i>Feedback Control of Dynamic Systems</i> (4 ed.). New Jersey: Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-032393-4" title="Special:BookSources/978-0-13-032393-4"><bdi>978-0-13-032393-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Feedback+Control+of+Dynamic+Systems&rft.place=New+Jersey&rft.edition=4&rft.pub=Prentice+Hall&rft.date=2002&rft.isbn=978-0-13-032393-4&rft.au=Franklin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoseph_L._HellersteinDawn_M._TilburySujay_Parekh2004" class="citation book cs1">Joseph L. Hellerstein; <a href="/wiki/Dawn_Tilbury" title="Dawn Tilbury">Dawn M. Tilbury</a>; Sujay Parekh (2004). <i>Feedback Control of Computing Systems</i>. John Wiley and Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-26637-2" title="Special:BookSources/978-0-471-26637-2"><bdi>978-0-471-26637-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Feedback+Control+of+Computing+Systems&rft.pub=John+Wiley+and+Sons&rft.date=2004&rft.isbn=978-0-471-26637-2&rft.au=Joseph+L.+Hellerstein&rft.au=Dawn+M.+Tilbury&rft.au=Sujay+Parekh&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDiederich_Hinrichsen_and_Anthony_J._Pritchard2005" class="citation book cs1"><a href="/wiki/Diederich_Hinrichsen" title="Diederich Hinrichsen">Diederich Hinrichsen</a> and Anthony J. Pritchard (2005). <i>Mathematical Systems Theory I – Modelling, State Space Analysis, Stability and Robustness</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-44125-0" title="Special:BookSources/978-3-540-44125-0"><bdi>978-3-540-44125-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Systems+Theory+I+%E2%80%93+Modelling%2C+State+Space+Analysis%2C+Stability+and+Robustness&rft.pub=Springer&rft.date=2005&rft.isbn=978-3-540-44125-0&rft.au=Diederich+Hinrichsen+and+Anthony+J.+Pritchard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSontag1998" class="citation book cs1"><a href="/wiki/Eduardo_D._Sontag" title="Eduardo D. Sontag">Sontag, Eduardo</a> (1998). <a rel="nofollow" class="external text" href="http://www.sontaglab.org/FTPDIR/sontag_mathematical_control_theory_springer98.pdf"><i>Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition</i></a> <span class="cs1-format">(PDF)</span>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98489-6" title="Special:BookSources/978-0-387-98489-6"><bdi>978-0-387-98489-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Control+Theory%3A+Deterministic+Finite+Dimensional+Systems.+Second+Edition&rft.pub=Springer&rft.date=1998&rft.isbn=978-0-387-98489-6&rft.aulast=Sontag&rft.aufirst=Eduardo&rft_id=http%3A%2F%2Fwww.sontaglab.org%2FFTPDIR%2Fsontag_mathematical_control_theory_springer98.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoodwin2001" class="citation book cs1">Goodwin, Graham (2001). <i>Control System Design</i>. Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-958653-8" title="Special:BookSources/978-0-13-958653-8"><bdi>978-0-13-958653-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Control+System+Design&rft.pub=Prentice+Hall&rft.date=2001&rft.isbn=978-0-13-958653-8&rft.aulast=Goodwin&rft.aufirst=Graham&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChristophe_Basso2012" class="citation book cs1">Christophe Basso (2012). <a rel="nofollow" class="external text" href="http://cbasso.pagesperso-orange.fr/Spice.htm"><i>Designing Control Loops for Linear and Switching Power Supplies: A Tutorial Guide</i></a>. Artech House. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1608075577" title="Special:BookSources/978-1608075577"><bdi>978-1608075577</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Designing+Control+Loops+for+Linear+and+Switching+Power+Supplies%3A+A+Tutorial+Guide.&rft.pub=Artech+House&rft.date=2012&rft.isbn=978-1608075577&rft.au=Christophe+Basso&rft_id=http%3A%2F%2Fcbasso.pagesperso-orange.fr%2FSpice.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoris_J._LuriePaul_J._Enright2019" class="citation book cs1">Boris J. Lurie; Paul J. Enright (2019). <i>Classical Feedback Control with Nonlinear Multi-loop Systems</i> (3 ed.). CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-1385-4114-6" title="Special:BookSources/978-1-1385-4114-6"><bdi>978-1-1385-4114-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Feedback+Control+with+Nonlinear+Multi-loop+Systems&rft.edition=3&rft.pub=CRC+Press&rft.date=2019&rft.isbn=978-1-1385-4114-6&rft.au=Boris+J.+Lurie&rft.au=Paul+J.+Enright&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></li></ul> <dl><dt>For Chemical Engineering</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLuyben1989" class="citation book cs1">Luyben, William (1989). <i>Process Modeling, Simulation, and Control for Chemical Engineers</i>. McGraw Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-039159-8" title="Special:BookSources/978-0-07-039159-8"><bdi>978-0-07-039159-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Process+Modeling%2C+Simulation%2C+and+Control+for+Chemical+Engineers&rft.pub=McGraw+Hill&rft.date=1989&rft.isbn=978-0-07-039159-8&rft.aulast=Luyben&rft.aufirst=William&rfr_id=info%3Asid%2Fen.wikipedia.org%3AControl+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Control_theory&action=edit&section=24" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media 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srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Control_theory" class="extiw" title="commons:Category:Control theory">Control theory</a></span>.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></span></span></div> <div class="side-box-text plainlist">Wikibooks has a book on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Control_Systems" class="extiw" title="wikibooks:Control Systems">Control Systems</a></b></i></div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://ctms.engin.umich.edu/CTMS/index.php?aux=Home">Control Tutorials for Matlab</a>, a set of worked-through control examples solved by several different methods.</li> <li><a rel="nofollow" class="external text" href="https://controlguru.com/">Control Tuning and Best Practices</a></li> <li><a rel="nofollow" class="external text" href="https://www.pidlab.com/">Advanced control structures, free on-line simulators explaining the control theory</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul 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title="Edit this template">e</abbr></a></li></ul></div><div id="Control_theory" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Control theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Branches</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_control" title="Adaptive control">Adaptive control</a></li> <li><a class="mw-selflink selflink">Control theory</a></li> <li><a href="/wiki/Digital_control" title="Digital control">Digital control</a></li> <li><a href="/wiki/Energy-shaping_control" title="Energy-shaping control">Energy-shaping control</a></li> <li><a href="/wiki/Fuzzy_control" class="mw-redirect" title="Fuzzy control">Fuzzy control</a></li> <li><a href="/wiki/Hybrid_computer" title="Hybrid computer">Hybrid</a> control</li> <li><a href="/wiki/Intelligent_control" title="Intelligent control">Intelligent control</a></li> <li><a href="/wiki/Model_predictive_control" title="Model predictive control">Model predictive control</a></li> <li><a href="/w/index.php?title=Multivariable&action=edit&redlink=1" class="new" title="Multivariable (page does not exist)">Multivariable</a> control</li> <li><a href="/wiki/Neural" class="mw-redirect" title="Neural">Neural</a> control</li> <li><a href="/wiki/Nonlinear_control" title="Nonlinear control">Nonlinear control</a></li> <li><a href="/wiki/Optimal_control" title="Optimal control">Optimal control</a></li> <li><a href="/wiki/Real-time_control" class="mw-redirect" title="Real-time control">Real-time control</a></li> <li><a href="/wiki/Robust_control" title="Robust control">Robust control</a></li> <li><a href="/wiki/Stochastic_control" title="Stochastic control">Stochastic control</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">System properties</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bode_plot" title="Bode plot">Bode plot</a></li> <li><a href="/wiki/Block_diagram" title="Block diagram">Block diagram</a></li> <li><a href="/wiki/Closed-loop_transfer_function" title="Closed-loop transfer function">Closed-loop transfer function</a></li> <li><a href="/wiki/Controllability" title="Controllability">Controllability</a></li> <li><a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a></li> <li><a href="/wiki/Frequency_response" title="Frequency response">Frequency response</a></li> <li><a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a></li> <li><a href="/wiki/Negative_feedback" title="Negative feedback">Negative feedback</a></li> <li><a href="/wiki/Observability" title="Observability">Observability</a></li> <li><a href="/wiki/Performance" title="Performance">Performance</a></li> <li><a href="/wiki/Positive_feedback" title="Positive feedback">Positive feedback</a></li> <li><a href="/wiki/Root_locus_analysis" title="Root locus analysis">Root locus</a> ethod</li> <li><a href="/wiki/Servomechanism" title="Servomechanism">Servomechanism</a></li> <li><a href="/wiki/Signal-flow_graph" title="Signal-flow graph">Signal-flow graph</a></li> <li><a href="/wiki/State_space_representation" class="mw-redirect" title="State space representation">State space representation</a></li> <li><a href="/wiki/Stability_theory" title="Stability theory">Stability</a> theory</li> <li><a href="/wiki/Steady_state" title="Steady state">Steady state</a> analysis & design</li> <li><a href="/wiki/System_dynamics" title="System dynamics">System dynamics</a></li> <li><a href="/wiki/Transfer_function" title="Transfer function">Transfer function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digital control</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Discrete-time_signal" class="mw-redirect" title="Discrete-time signal">Discrete-time signal</a></li> <li><a href="/wiki/Digital_signal_processing" title="Digital signal processing">Digital signal processing</a></li> <li><a href="/wiki/Quantization_(signal_processing)" title="Quantization (signal processing)">Quantization</a></li> <li><a href="/wiki/Real-time_computing" title="Real-time computing">Real time</a> software</li> <li><a href="/wiki/Sample_(statistics)" class="mw-redirect" title="Sample (statistics)">Sampled</a> data</li> <li><a href="/wiki/System_identification" title="System identification">System identification</a></li> <li><a href="/wiki/Z-transform" title="Z-transform">Z-transform</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Advanced techniques</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural network</a></li> <li><a href="/wiki/Coefficient_diagram_method" title="Coefficient diagram method">Coefficient diagram method</a></li> <li><a href="/wiki/Control_reconfiguration" title="Control reconfiguration">Control reconfiguration</a></li> <li><a href="/wiki/Distributed_parameter_systems" class="mw-redirect" title="Distributed parameter systems">Distributed parameter systems</a></li> <li><a href="/wiki/Fractional-order_control" title="Fractional-order control">Fractional-order control</a></li> <li><a href="/wiki/Fuzzy_logic" title="Fuzzy logic">Fuzzy logic</a></li> <li><a href="/wiki/H-infinity_loop-shaping" title="H-infinity loop-shaping">H-infinity loop-shaping</a></li> <li><a href="/wiki/Hankel_singular_value" title="Hankel singular value">Hankel singular value</a></li> <li><a href="/wiki/Kalman_filter" title="Kalman filter">Kalman filter</a></li> <li><a href="/wiki/Krener%27s_theorem" title="Krener's theorem">Krener's theorem</a></li> <li><a href="/wiki/Least_squares" title="Least squares">Least squares</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov stability</a></li> <li><a href="/wiki/Minor_loop_feedback" title="Minor loop feedback">Minor loop feedback</a></li> <li><a href="/wiki/Perceptual_control_theory" title="Perceptual control theory">Perceptual control theory</a></li> <li><a href="/wiki/State_observer" title="State observer">State observer</a></li> <li><a href="/wiki/Vector_control_(motor)" title="Vector control (motor)">Vector control</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Controllers</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Embedded_system" title="Embedded system">Embedded controller</a></li> <li><a href="/wiki/Closed-loop_controller" title="Closed-loop controller">Closed-loop controller</a></li> <li><a href="/wiki/Lead-lag_compensator" class="mw-redirect" title="Lead-lag compensator">Lead-lag compensator</a></li> <li><a href="/wiki/Numerical_control" title="Numerical control">Numerical control</a></li> <li><a href="/wiki/PID_controller" class="mw-redirect" title="PID controller">PID controller</a></li> <li><a href="/wiki/Programmable_logic_controller" title="Programmable logic controller">Programmable logic controller</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Control applications</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><i><a href="/wiki/Automation_and_Remote_Control" title="Automation and Remote Control">Automation and Remote Control</a></i></li> <li><a href="/wiki/Distributed_control_system" title="Distributed control system">Distributed control system</a></li> <li><a href="/wiki/Electric_motor" title="Electric motor">Electric motors</a></li> <li><a href="/wiki/Industrial_control_systems" class="mw-redirect" title="Industrial control systems">Industrial control systems</a></li> <li><a href="/wiki/Mechatronics" title="Mechatronics">Mechatronics</a></li> <li><a href="/wiki/Motion_control" title="Motion control">Motion control</a></li> <li><a href="/wiki/Process_control" class="mw-redirect" title="Process control">Process control</a></li> <li><a href="/wiki/Robotics" title="Robotics">Robotics</a></li> <li><a href="/wiki/SCADA" title="SCADA">Supervisory control (SCADA)</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Subfields_of_and_cyberneticians_involved_in_cybernetics" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Cybernetics" title="Template:Cybernetics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Cybernetics" title="Template talk:Cybernetics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Cybernetics" title="Special:EditPage/Template:Cybernetics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Subfields_of_and_cyberneticians_involved_in_cybernetics" style="font-size:114%;margin:0 4em">Subfields of and cyberneticians involved in <a href="/wiki/Cybernetics" title="Cybernetics">cybernetics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Subfields</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Artificial_intelligence" title="Artificial intelligence">Artificial intelligence</a></li> <li><a href="/wiki/Biocybernetics" title="Biocybernetics">Biological cybernetics</a></li> <li><a href="/wiki/Biomedical_cybernetics" title="Biomedical cybernetics">Biomedical cybernetics</a></li> <li><a href="/wiki/Biorobotics" title="Biorobotics">Biorobotics</a></li> <li><a href="/wiki/Biosemiotics" title="Biosemiotics">Biosemiotics</a></li> <li><a href="/wiki/Brain%E2%80%93computer_interface" title="Brain–computer interface">Neurocybernetics</a></li> <li><a href="/wiki/Catastrophe_theory" title="Catastrophe theory">Catastrophe theory</a></li> <li><a href="/wiki/Computational_neuroscience" title="Computational neuroscience">Computational neuroscience</a></li> <li><a href="/wiki/Connectionism" title="Connectionism">Connectionism</a></li> <li><a class="mw-selflink selflink">Control theory</a></li> <li><a href="/wiki/Conversation_theory" title="Conversation theory">Conversation theory</a></li> <li><a href="/wiki/Cybernetics_in_the_Soviet_Union" title="Cybernetics in the Soviet Union">Cybernetics in the Soviet Union</a></li> <li><a href="/wiki/Decision_theory" title="Decision theory">Decision theory</a></li> <li><a href="/wiki/Emergence" title="Emergence">Emergence</a></li> <li><a href="/wiki/Engineering_cybernetics" title="Engineering cybernetics">Engineering cybernetics</a></li> <li><a href="/wiki/Homeostasis" title="Homeostasis">Homeostasis</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Management_cybernetics" title="Management cybernetics">Management cybernetics</a></li> <li><a href="/wiki/Medical_cybernetics" title="Medical cybernetics">Medical cybernetics</a></li> <li><a href="/wiki/Second-order_cybernetics" title="Second-order cybernetics">Second-order cybernetics</a></li> <li><a href="/wiki/Cybersemiotics" class="mw-redirect" title="Cybersemiotics">Cybersemiotics</a></li> <li><a href="/wiki/Sociocybernetics" title="Sociocybernetics">Sociocybernetics</a></li> <li><a href="/wiki/Synergetics_(Haken)" title="Synergetics (Haken)">Synergetics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Cyberneticist" title="Cyberneticist">Cyberneticians</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alexander_Lerner" title="Alexander Lerner">Alexander Lerner</a></li> <li><a href="/wiki/Alexey_Lyapunov" title="Alexey Lyapunov">Alexey Lyapunov</a></li> <li><a href="/wiki/Alfred_Radcliffe-Brown" title="Alfred Radcliffe-Brown">Alfred Radcliffe-Brown</a></li> <li><a href="/wiki/Allenna_Leonard" title="Allenna Leonard">Allenna Leonard</a></li> <li><a href="/wiki/Anthony_Wilden" title="Anthony Wilden">Anthony Wilden</a></li> <li><a href="/wiki/Buckminster_Fuller" title="Buckminster Fuller">Buckminster Fuller</a></li> <li><a href="/wiki/Charles_Fran%C3%A7ois_(systems_scientist)" title="Charles François (systems scientist)">Charles François</a></li> <li><a href="/wiki/Genevieve_Bell" title="Genevieve Bell">Genevieve Bell</a></li> <li><a href="/wiki/Margaret_Boden" title="Margaret Boden">Margaret Boden</a></li> <li><a href="/wiki/Claude_Bernard" title="Claude Bernard">Claude Bernard</a></li> <li><a href="/wiki/Cliff_Joslyn" title="Cliff Joslyn">Cliff Joslyn</a></li> <li><a href="/wiki/Erich_von_Holst" title="Erich von Holst">Erich von Holst</a></li> <li><a href="/wiki/Ernst_von_Glasersfeld" title="Ernst von Glasersfeld">Ernst von Glasersfeld</a></li> <li><a href="/wiki/Francis_Heylighen" title="Francis Heylighen">Francis Heylighen</a></li> <li><a href="/wiki/Francisco_Varela" title="Francisco Varela">Francisco Varela</a></li> <li><a href="/wiki/Frederic_Vester" title="Frederic Vester">Frederic Vester</a></li> <li><a href="/wiki/Geoffrey_Vickers" title="Geoffrey Vickers">Charles Geoffrey Vickers</a></li> <li><a href="/wiki/Gordon_Pask" title="Gordon Pask">Gordon Pask</a></li> <li><a href="/wiki/Gordon_S._Brown" title="Gordon S. Brown">Gordon S. Brown</a></li> <li><a href="/wiki/Gregory_Bateson" title="Gregory Bateson">Gregory Bateson</a></li> <li><a href="/wiki/Heinz_von_Foerster" title="Heinz von Foerster">Heinz von Foerster</a></li> <li><a href="/wiki/Humberto_Maturana" title="Humberto Maturana">Humberto Maturana</a></li> <li><a href="/wiki/I._A._Richards" title="I. A. Richards">I. A. Richards</a></li> <li><a href="/wiki/Igor_Aleksander" title="Igor Aleksander">Igor Aleksander</a></li> <li><a href="/wiki/Jacque_Fresco" title="Jacque Fresco">Jacque Fresco</a></li> <li><a href="/wiki/Jakob_Johann_von_Uexk%C3%BCll" title="Jakob Johann von Uexküll">Jakob von Uexküll</a></li> <li><a href="/wiki/Jason_Jixuan_Hu" title="Jason Jixuan Hu">Jason Jixuan Hu</a></li> <li><a href="/wiki/Jay_Wright_Forrester" title="Jay Wright Forrester">Jay Wright Forrester</a></li> <li><a href="/wiki/Jennifer_Wilby" title="Jennifer Wilby">Jennifer Wilby</a></li> <li><a href="/wiki/John_N._Warfield" title="John N. Warfield">John N. Warfield</a></li> <li><a href="/wiki/Kevin_Warwick" title="Kevin Warwick">Kevin Warwick</a></li> <li><a href="/wiki/Ludwig_von_Bertalanffy" title="Ludwig von Bertalanffy">Ludwig von Bertalanffy</a></li> <li><a href="/wiki/Maleyka_Abbaszadeh" title="Maleyka Abbaszadeh">Maleyka Abbaszadeh</a></li> <li><a href="/wiki/Manfred_Clynes" title="Manfred Clynes">Manfred Clynes</a></li> <li><a href="/wiki/Margaret_Mead" title="Margaret Mead">Margaret Mead</a></li> <li><a href="/wiki/Marian_Mazur" title="Marian Mazur">Marian Mazur</a></li> <li><a href="/wiki/N._Katherine_Hayles" title="N. Katherine Hayles">N. Katherine Hayles</a></li> <li><a href="/wiki/Natalia_Bekhtereva" title="Natalia Bekhtereva">Natalia Bekhtereva</a></li> <li><a href="/wiki/Niklas_Luhmann" title="Niklas Luhmann">Niklas Luhmann</a></li> <li><a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a></li> <li><a href="/wiki/Petro_Grigorenko" title="Petro Grigorenko">Pyotr Grigorenko</a></li> <li><a href="/wiki/Qian_Xuesen" title="Qian Xuesen">Qian Xuesen</a></li> <li><a href="/wiki/Ranulph_Glanville" title="Ranulph Glanville">Ranulph Glanville</a></li> <li><a href="/wiki/Robert_Trappl" title="Robert Trappl">Robert Trappl</a></li> <li><a href="/wiki/Sergei_P._Kurdyumov" class="mw-redirect" title="Sergei P. Kurdyumov">Sergei P. Kurdyumov</a></li> <li><a href="/wiki/Stafford_Beer" title="Stafford Beer">Anthony Stafford Beer</a></li> <li><a href="/wiki/Stuart_Kauffman" title="Stuart Kauffman">Stuart Kauffman</a></li> <li><a href="/wiki/Stuart_Umpleby" title="Stuart Umpleby">Stuart Umpleby</a></li> <li><a href="/wiki/Talcott_Parsons" title="Talcott Parsons">Talcott Parsons</a></li> <li><a href="/wiki/Ulla_Mitzdorf" title="Ulla Mitzdorf">Ulla Mitzdorf</a></li> <li><a href="/wiki/Valentin_Turchin" title="Valentin Turchin">Valentin Turchin</a></li> <li><a href="/wiki/Valentino_Braitenberg" title="Valentino Braitenberg">Valentin Braitenberg</a></li> <li><a href="/wiki/W._Ross_Ashby" title="W. Ross Ashby">William Ross Ashby</a></li> <li><a href="/wiki/Walter_Bradford_Cannon" title="Walter Bradford Cannon">Walter Bradford Cannon</a></li> <li><a href="/wiki/Walter_Pitts" title="Walter Pitts">Walter Pitts</a></li> <li><a href="/wiki/Warren_Sturgis_McCulloch" title="Warren Sturgis McCulloch">Warren McCulloch</a></li> <li><a href="/wiki/William_Grey_Walter" title="William Grey Walter">William Grey Walter</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Systems_science" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Systems_science" title="Template:Systems science"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Systems_science" title="Template talk:Systems science"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Systems_science" title="Special:EditPage/Template:Systems science"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Systems_science" style="font-size:114%;margin:0 4em"><a href="/wiki/Systems_science" title="Systems science">Systems science</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/System" title="System">System</a><br />types</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Systems_art" title="Systems art">Art</a></li> <li><a href="/wiki/Biological_system" title="Biological system">Biological</a></li> <li><a href="/wiki/Complex_system" title="Complex system">Complex</a></li> <li><a href="/wiki/Coupled_human%E2%80%93environment_system" title="Coupled human–environment system">Coupled human–environment</a></li> <li><a href="/wiki/Ecosystem" title="Ecosystem">Ecological</a></li> <li><a href="/wiki/Economic_system" title="Economic system">Economic</a></li> <li><a href="/wiki/Information_system" title="Information system">Information</a></li> <li><a href="/wiki/Multi-agent_system" title="Multi-agent system">Multi-agent</a></li> <li><a href="/wiki/Nervous_system" title="Nervous system">Nervous</a></li> <li><a href="/wiki/Recommender_system" title="Recommender system">Recommender</a></li> <li><a href="/wiki/Social_system" title="Social system">Social</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Doubling_time" title="Doubling time">Doubling time</a></li> <li><a href="/wiki/Twelve_leverage_points" title="Twelve leverage points">Leverage points</a></li> <li><a href="/wiki/Limiting_factor" title="Limiting factor">Limiting factor</a></li> <li><a href="/wiki/Negative_feedback" title="Negative feedback">Negative feedback</a></li> <li><a href="/wiki/Positive_feedback" title="Positive feedback">Positive feedback</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theoretical<br />fields</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Control theory</a></li> <li><a href="/wiki/Cybernetics" title="Cybernetics">Cybernetics</a></li> <li><a href="/wiki/Earth_system_science" title="Earth system science">Earth system science</a></li> <li><a href="/wiki/Living_systems" title="Living systems">Living systems</a></li> <li><a href="/wiki/Sociotechnical_system" title="Sociotechnical system">Sociotechnical system</a></li> <li><a href="/wiki/Systemics" title="Systemics">Systemics</a></li> <li><a href="/wiki/Urban_metabolism" title="Urban metabolism">Urban metabolism</a></li> <li><a href="/wiki/World-systems_theory" title="World-systems theory">World-systems theory</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Systems_analysis" title="Systems analysis">Analysis</a></li> <li><a href="/wiki/Systems_biology" title="Systems biology">Biology</a></li> <li><a href="/wiki/System_dynamics" title="System dynamics">Dynamics</a></li> <li><a href="/wiki/Systems_ecology" title="Systems ecology">Ecology</a></li> <li><a href="/wiki/Systems_engineering" title="Systems engineering">Engineering</a></li> <li><a href="/wiki/Systems_neuroscience" title="Systems neuroscience">Neuroscience</a></li> <li><a href="/wiki/Systems_pharmacology" title="Systems pharmacology">Pharmacology</a></li> <li><a href="/wiki/Systems_philosophy" title="Systems philosophy">Philosophy</a></li> <li><a href="/wiki/Systems_psychology" title="Systems psychology">Psychology</a></li> <li><a href="/wiki/Systems_theory" title="Systems theory">Theory</a> (<a href="/wiki/Systems_thinking" title="Systems thinking">Systems thinking</a>)</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_systems_scientists" title="List of systems scientists">Scientists</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Russell_L._Ackoff" title="Russell L. Ackoff">Russell L. Ackoff</a></li> <li><a href="/wiki/Victor_Aladjev" title="Victor Aladjev">Victor Aladjev</a></li> <li><a href="/wiki/William_Ross_Ashby" class="mw-redirect" title="William Ross Ashby">William Ross Ashby</a></li> <li><a href="/wiki/Ruzena_Bajcsy" title="Ruzena Bajcsy">Ruzena Bajcsy</a></li> <li><a href="/wiki/B%C3%A9la_H._B%C3%A1n%C3%A1thy" title="Béla H. Bánáthy">Béla H. Bánáthy</a></li> <li><a href="/wiki/Gregory_Bateson" title="Gregory Bateson">Gregory Bateson</a></li> <li><a href="/wiki/Anthony_Stafford_Beer" class="mw-redirect" title="Anthony Stafford Beer">Anthony Stafford Beer</a></li> <li><a href="/wiki/Richard_E._Bellman" title="Richard E. Bellman">Richard E. Bellman</a></li> <li><a href="/wiki/Ludwig_von_Bertalanffy" title="Ludwig von Bertalanffy">Ludwig von Bertalanffy</a></li> <li><a href="/wiki/Margaret_Boden" title="Margaret Boden">Margaret Boden</a></li> <li><a href="/wiki/Alexander_Bogdanov" title="Alexander Bogdanov">Alexander Bogdanov</a></li> <li><a href="/wiki/Kenneth_E._Boulding" title="Kenneth E. Boulding">Kenneth E. Boulding</a></li> <li><a href="/wiki/Murray_Bowen" title="Murray Bowen">Murray Bowen</a></li> <li><a href="/wiki/Kathleen_Carley" title="Kathleen Carley">Kathleen Carley</a></li> <li><a href="/wiki/Mary_Cartwright" title="Mary Cartwright">Mary Cartwright</a></li> <li><a href="/wiki/C._West_Churchman" title="C. West Churchman">C. West Churchman</a></li> <li><a href="/wiki/Manfred_Clynes" title="Manfred Clynes">Manfred Clynes</a></li> <li><a href="/wiki/George_Dantzig" title="George Dantzig">George Dantzig</a></li> <li><a href="/wiki/Edsger_W._Dijkstra" title="Edsger W. Dijkstra">Edsger W. Dijkstra</a></li> <li><a href="/wiki/Fred_Emery" title="Fred Emery">Fred Emery</a></li> <li><a href="/wiki/Heinz_von_Foerster" title="Heinz von Foerster">Heinz von Foerster</a></li> <li><a href="/wiki/Stephanie_Forrest" title="Stephanie Forrest">Stephanie Forrest</a></li> <li><a href="/wiki/Jay_Wright_Forrester" title="Jay Wright Forrester">Jay Wright Forrester</a></li> <li><a href="/wiki/Barbara_J._Grosz" title="Barbara J. Grosz">Barbara Grosz</a></li> <li><a href="/wiki/Charles_A._S._Hall" title="Charles A. S. Hall">Charles A. S. Hall</a></li> <li><a href="/wiki/Mike_Jackson_(systems_scientist)" title="Mike Jackson (systems scientist)">Mike Jackson</a></li> <li><a href="/wiki/Lydia_Kavraki" title="Lydia Kavraki">Lydia Kavraki</a></li> <li><a href="/wiki/James_J._Kay" title="James J. Kay">James J. Kay</a></li> <li><a href="/wiki/Faina_Mihajlovna_Kirillova" class="mw-redirect" title="Faina Mihajlovna Kirillova">Faina M. Kirillova</a></li> <li><a href="/wiki/George_Klir" title="George Klir">George Klir</a></li> <li><a href="/wiki/Allenna_Leonard" title="Allenna Leonard">Allenna Leonard</a></li> <li><a href="/wiki/Edward_Norton_Lorenz" title="Edward Norton Lorenz">Edward Norton Lorenz</a></li> <li><a href="/wiki/Niklas_Luhmann" title="Niklas Luhmann">Niklas Luhmann</a></li> <li><a href="/wiki/Humberto_Maturana" title="Humberto Maturana">Humberto Maturana</a></li> <li><a href="/wiki/Margaret_Mead" title="Margaret Mead">Margaret Mead</a></li> <li><a href="/wiki/Donella_Meadows" title="Donella Meadows">Donella Meadows</a></li> <li><a href="/wiki/Mihajlo_D._Mesarovic" title="Mihajlo D. Mesarovic">Mihajlo D. Mesarovic</a></li> <li><a href="/wiki/James_Grier_Miller" title="James Grier Miller">James Grier Miller</a></li> <li><a href="/wiki/Radhika_Nagpal" title="Radhika Nagpal">Radhika Nagpal</a></li> <li><a href="/wiki/Howard_T._Odum" title="Howard T. Odum">Howard T. Odum</a></li> <li><a href="/wiki/Talcott_Parsons" title="Talcott Parsons">Talcott Parsons</a></li> <li><a href="/wiki/Ilya_Prigogine" title="Ilya Prigogine">Ilya Prigogine</a></li> <li><a href="/wiki/Qian_Xuesen" title="Qian Xuesen">Qian Xuesen</a></li> <li><a href="/wiki/Anatol_Rapoport" title="Anatol Rapoport">Anatol Rapoport</a></li> <li><a href="/wiki/John_Seddon" title="John Seddon">John Seddon</a></li> <li><a href="/wiki/Peter_Senge" title="Peter Senge">Peter Senge</a></li> <li><a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a></li> <li><a href="/wiki/Katia_Sycara" title="Katia Sycara">Katia Sycara</a></li> <li><a href="/wiki/Eric_Trist" title="Eric Trist">Eric Trist</a></li> <li><a href="/wiki/Francisco_Varela" title="Francisco Varela">Francisco Varela</a></li> <li><a href="/wiki/Manuela_M._Veloso" title="Manuela M. Veloso">Manuela M. Veloso</a></li> <li><a href="/wiki/Kevin_Warwick" title="Kevin Warwick">Kevin Warwick</a></li> <li><a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a></li> <li><a href="/wiki/Jennifer_Wilby" title="Jennifer Wilby">Jennifer Wilby</a></li> <li><a href="/wiki/Anthony_Wilden" title="Anthony Wilden">Anthony Wilden</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Systems_theory_in_anthropology" title="Systems theory in anthropology">Systems theory in anthropology</a></li> <li><a href="/wiki/Systems_theory_in_archaeology" title="Systems theory in archaeology">Systems theory in archaeology</a></li> <li><a href="/wiki/Systems_theory_in_political_science" title="Systems theory in political science">Systems theory in political science</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Organizations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_systems_sciences_organizations" title="List of systems sciences organizations">List</a></li> <li><a href="/wiki/Principia_Cybernetica" title="Principia Cybernetica">Principia Cybernetica</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <b><a href="/wiki/Category:Systems_science" title="Category:Systems science">Category</a></b></li> <li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Symbol_portal_class.svg" class="mw-file-description" title="Portal"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span> <b><a href="/wiki/Portal:Systems_science" title="Portal:Systems science">Portal</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" 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style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Areas_of_mathematics" title="Template:Areas of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Areas_of_mathematics" title="Template talk:Areas of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Areas_of_mathematics" title="Special:EditPage/Template:Areas of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_mathematics_areas" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li> <li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebra" title="Algebra">Algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary</a></li> <li><a href="/wiki/Group_theory" title="Group theory">Group theory</a></li> 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title="Differential equation">Differential equations</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Geometry" 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