{"id":22537,"date":"2018-01-19T05:19:19","date_gmt":"2018-01-19T05:19:19","guid":{"rendered":"https:\/\/3-inst.toejvy8-liquidwebsites.com\/?p=22537"},"modified":"2020-04-23T22:00:31","modified_gmt":"2020-04-23T16:30:31","slug":"pid-controllers-graphical-analysis","status":"publish","type":"post","link":"https:\/\/instrumentationtools.com\/pid-controllers-graphical-analysis\/","title":{"rendered":"PID Controllers Graphical Analysis"},"content":{"rendered":"<p style=\"text-align: justify;\">A common exercise for students learning the <strong>function of PID controllers<\/strong> is to practice graphing a controller\u2019s output with given input (PV and SP) conditions, either qualitatively or quantitatively.<\/p>\n<p style=\"text-align: justify;\">This can be a frustrating experience for some students, as they struggle to accurately combine the effects of P, I, and\/or D responses into a single output trend. Here, I will present a way to ease the pain.<\/p>\n<h2>PID Controllers<\/h2>\n<p style=\"text-align: justify;\">Suppose for example you were tasked with graphing the response of a PD (proportional + derivative) <a href=\"https:\/\/instrumentationtools.com\/pid-controller-functions\/\" target=\"_blank\" rel=\"noopener noreferrer\">controller<\/a> to the following PV and SP inputs over time.<\/p>\n<p style=\"text-align: justify;\">You are told the controller has a gain of 1, a derivative time constant of 0.3 minutes, and is reverse-acting:<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-22538 size-full\" title=\"Proportional Derivative Controller Graph\" src=\"https:\/\/instrumentationtools.com\/wp-content\/uploads\/2018\/01\/PID-Controller-Graph-1.png\" alt=\"Proportional Derivative Controller Graph\" width=\"1022\" height=\"670\" \/><\/p>\n<p style=\"text-align: justify;\">My first recommendation is to qualitatively sketch the individual <strong>P and D responses<\/strong>. Simply draw two different trends, each one right above or below the given PV\/SP trends, showing the shapes of each response over time.<\/p>\n<p style=\"text-align: justify;\">You might even find it easier to do if you re-draw the original PV and SP trends on a piece of non-graph paper with the qualitative P and D trends also sketched on the same piece of non-graph paper.<\/p>\n<p style=\"text-align: justify;\">The purpose of the <strong>qualitative sketches<\/strong> is to separate the task of determining shapes from the task of determining numerical values, in order to simplify the process.<\/p>\n<p style=\"text-align: justify;\">After sketching the separate P and D trends, label each one of the \u201cfeatures\u201d (changes either up or down) in these qualitative trends. This will allow you to more easily combine the effects into one output trend later:<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-22539 size-full\" title=\"PID Controller Graph\" src=\"https:\/\/instrumentationtools.com\/wp-content\/uploads\/2018\/01\/PID-Controller-Graph-2.png\" alt=\"PID Controller Graph\" width=\"1052\" height=\"541\" \/><\/p>\n<p style=\"text-align: justify;\">Now, you may qualitatively sketch an output trend combining each of these \u201cfeatures\u201d into one graph.<\/p>\n<p style=\"text-align: justify;\">Be sure to label each ramp or step originating with the separate <strong>P or D trends<\/strong>, so you know where each \u201cfeature\u201d of the combined output graph originates from:<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-22540 size-full\" title=\"PD Controller Graph Analysis\" src=\"https:\/\/instrumentationtools.com\/wp-content\/uploads\/2018\/01\/PID-Controller-Graph-3.png\" alt=\"PD Controller Graph Analysis\" width=\"1034\" height=\"814\" \/><\/p>\n<p style=\"text-align: justify;\">Once the general shape of the output has been qualitatively determined, you may go back to the separate P and D trends to calculate numerical values for each of the labeled \u201cfeatures.\u201d<\/p>\n<p style=\"text-align: justify;\">Note that each of the PV ramps is 15% in height, over a time of 15 seconds (one-quarter of a minute). With a controller gain of 1, the proportional response to each of these ramps will also be a ramp that is 15% in height.<\/p>\n<p style=\"text-align: justify;\">Taking our given derivative time constant of 0.3 minutes and multiplying that by the PV\u2019s rate-of-change ( dPV\/dt ) during each of its ramping periods (15% per one-quarter minute, or 60% per minute) yields a derivative response of 18% during each of the ramping periods. Thus, each derivative response \u201cstep\u201d will be 18% in height.<\/p>\n<p style=\"text-align: justify;\">Going back to the qualitative <strong>sketches of P and D actions<\/strong>, and to the combined (qualitative) output sketch, we may apply the calculated values of 15% for each proportional ramp and 18% for each derivative step to the labeled \u201cfeatures.\u201d<\/p>\n<p style=\"text-align: justify;\">We may also label the starting value of the output trend as given in the original problem (35%), to calculate actual output values at different points in time.<\/p>\n<p style=\"text-align: justify;\">Calculating output values at specific points in the graph becomes as easy as cumulatively adding and subtracting the P and D \u201cfeature\u201d values to the starting output value:<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-22541 size-full\" title=\"P+D Controller Graph\" src=\"https:\/\/instrumentationtools.com\/wp-content\/uploads\/2018\/01\/PID-Controller-Graph-4.png\" alt=\"P+D Controller Graph\" width=\"1056\" height=\"902\" \/><\/p>\n<p style=\"text-align: justify;\">Now that we know the output values at all the critical points, we may quantitatively <strong>sketch the output trend<\/strong> on the original graph:<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-22542\" src=\"https:\/\/instrumentationtools.com\/wp-content\/uploads\/2018\/01\/PID-Controller-Graph-5.png\" alt=\"PID Controller Graph - 5\" width=\"1019\" height=\"674\" \/><\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #808080;\">Credits : Tony R. Kuphaldt \u2013 Creative Commons Attribution 4.0 License<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>A common exercise for students learning the function of PID controllers is to practice graphing a controller\u2019s output with given input (PV and SP) conditions, either qualitatively or quantitatively. This can be a frustrating experience for some students, as they struggle to accurately combine the effects of P, I, and\/or D responses into a single [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":22540,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_mo_disable_npp":"","footnotes":""},"categories":[12],"tags":[4290,47173,47172],"class_list":{"0":"post-22537","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-control-systems","8":"tag-pid-controller-graph","9":"tag-pid-controllers-analysis","10":"tag-pid-controllers-graphical-analysis"},"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.4 (Yoast SEO v27.4) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>PID Controllers Graphical Analysis - InstrumentationTools<\/title>\n<meta name=\"description\" content=\"Analyze the PID controllers controller\u2019s output with given input (PV and SP) conditions, either qualitatively or quantitatively using graphical analysis.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/instrumentationtools.com\/pid-controllers-graphical-analysis\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"PID Controllers Graphical Analysis - 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