{"id":1605,"date":"2025-12-16T08:10:02","date_gmt":"2025-12-16T07:10:02","guid":{"rendered":"https:\/\/control-advent.com\/?p=1605"},"modified":"2025-12-16T08:10:19","modified_gmt":"2025-12-16T07:10:19","slug":"16-how-to-fix-the-magical-solution","status":"publish","type":"post","link":"https:\/\/control-advent.com\/?p=1605","title":{"rendered":"16 &#8211; How to fix the magical solution?"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">In the gift factory of the North Pole, many processes use vapor pressure to work.<br>The vapor pressure is controlled by opening and closing a valve.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In the past, this valve was controlled using a simple analog <strong>on\/off controller<\/strong>. Unfortunately, this caused <strong>sustained oscillations<\/strong>: the vapor pressure kept going up and down instead of settling at a steady value.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To fix this problem, the elves decided to try a more sophisticated and magical solution: the <strong>Pinecone\u2013Icicle\u2013Dancing (PID) controller<\/strong>. It contains three parts:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Pinecone (P):<\/strong> The magical pinecone reacts to the difference between the desired pressure and the current pressure. It amplifies or attenuates this difference by a factor <strong>K<\/strong>.<br><\/li>\n\n\n\n<li><strong>Icicle (I):<\/strong> The icicle grows or melts depending on the same pressure difference.<br>The time constant <strong>Ti<\/strong> determines how fast this happens: a larger <strong>Ti<\/strong> means the icicle grows more slowly.<br><\/li>\n\n\n\n<li><strong>Dancing elves (D):<\/strong> The dancing elves react to how fast the pressure is changing.<br>The time constant <strong>Td<\/strong> determines how strongly they react to quick changes: a larger <strong>Td<\/strong> means the elves try harder to anticipate where the pressure is heading.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The controller output is the sum of the effects of the pinecone, the icicle, and the dancing elves.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The elves enthusiastically replace the old on\/off controller with this new PID controller and switch the system back to automatic mode. To their great disappointment, the vapor pressure is <strong>still oscillating<\/strong>!<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Assume that:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>the pressure sensor is clean (not noisy), and<\/li>\n\n\n\n<li>the oscillations are caused by the controller reacting too aggressively.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The elves now wonder whether adjusting the pinecone gain <strong>K<\/strong>, the icicle growing time <strong>Ti<\/strong>, and the dancing time <strong>Td<\/strong> could calm things down.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>How should these parameters be modified to decrease the oscillations?<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">a. \ud83d\udcc8\ud83c\udf32\u23f1\ufe0f Increase K, increase Ti, and increase Td.<br>b. \ud83d\udcc8\ud83c\udf32\u2b07\ufe0f Increase K, increase Ti, and decrease Td.<br>c. \ud83d\udcc8\u2744\ufe0f\u23f1\ufe0f Increase K, decrease Ti, and increase Td.<br>d. \ud83d\udcc8\u2744\ufe0f\u2b07\ufe0f Increase K, decrease Ti, and decrease Td.<br>e. \ud83d\udcc9\ud83c\udf32\u23f1\ufe0f Decrease K, increase Ti, and increase Td.<br>f. \ud83d\udcc9\ud83c\udf32\u2b07\ufe0f Decrease K, increase Ti, and decrease Td.<br>g. \ud83d\udcc9\u2744\ufe0f\u23f1\ufe0f Decrease K, decrease Ti, and increase Td.<br>h. \ud83d\udcc9\u2744\ufe0f\u2b07\ufe0f Decrease K, decrease Ti, and decrease Td.<br>i. \ud83e\udd37\u200d\u2642\ufe0f\ud83d\udd04\u274c None of the above: if a system oscillates, there is no way to stop it.<\/p>\n\n\n<div class=\"yop-poll yop-poll-container\" data-template=\"classic\" data-yop-poll-id=\"53\" data-yop-state=\"ballot\" data-yop-tracking=\"https:\/\/control-advent.com\" data-yop-page-id=\"0\" data-yop-msg-auth-failed=\"Sign-in failed. Please try again.\" data-yop-msg-no-provider=\"That sign-in method is unavailable.\" data-yop-msg-not-started=\"This poll is not accepting votes yet\" data-yop-msg-ended=\"This poll is no longer accepting votes\" data-yop-perm=\"wordpress\" style=\"background:#b70004;border-width:0px;border-style:none;border-color:#b70004;border-radius:0px;padding-top:10px;padding-bottom:10px;padding-left:10px;padding-right:10px;--yop-radio-color:black;--yop-ans-ts:14px;--yop-ans-tw:normal;--yop-ans-tc:#ffffff;--yop-ans-pt:0px;--yop-ans-pb:0px;--yop-ans-pl:0px;--yop-ans-pr:0px;--yop-poll-bg:#b70004;--yop-msg-bcs:#008000;--yop-msg-bce:#ff0000;--yop-msg-blw:10px;--yop-msg-pt:0px;--yop-msg-pb:0px;--yop-msg-pl:10px;--yop-msg-pr:10px;--yop-msg-tc:#000000;--yop-msg-ts:14px;--yop-msg-tw:normal;--yop-msg-br:0px;--yop-btn-bg:#ffffff;--yop-btn-tc:#000000;--yop-btn-bw:1px;--yop-btn-bs:solid;--yop-btn-bc:#000000;--yop-btn-br:0px;--yop-btn-pt:5px;--yop-btn-pb:5px;--yop-btn-pl:10px;--yop-btn-pr:10px;--yop-btn-ts:14px;--yop-btn-tw:normal;--yop-el-pt:10px;--yop-el-pb:10px;--yop-el-pl:0px;--yop-el-pr:0px;--yop-el-ta:center;--yop-q-tc:#ffffff;--yop-q-ts:14px;--yop-q-tw:normal\"><div class=\"yop-poll-decoy\" aria-hidden=\"true\"><input type=\"text\" name=\"website\" tabindex=\"-1\" autocomplete=\"off\" value=\"\" \/><\/div><div class=\"yop-poll-message\" data-yop-message role=\"status\" aria-live=\"polite\"><\/div><div class=\"yop-poll-ballot\" data-yop-page=\"0\"><div class=\"yop-poll-element\" data-element-type=\"question-text\" data-yop-el=\"53\" data-yop-required=\"1\" data-yop-msg-required=\"No answer selected\"><fieldset class=\"yop-poll-fieldset\"><legend id=\"yop-poll-53-53-label\" class=\"yop-poll-question\">Choose an option:<\/legend><div class=\"yop-poll-answers yop-poll-answers--horizontal\"><div class=\"yop-poll-answer-row\"><label class=\"yop-poll-answer\"><input type=\"radio\" name=\"yop-poll-53-53\" value=\"520\" data-yop-sub=\"520\" \/><span class=\"yop-poll-answer-text\"><span>a<\/span><\/span><\/label><\/div><div class=\"yop-poll-answer-row\"><label class=\"yop-poll-answer\"><input type=\"radio\" name=\"yop-poll-53-53\" value=\"521\" data-yop-sub=\"521\" \/><span class=\"yop-poll-answer-text\"><span>b<\/span><\/span><\/label><\/div><div class=\"yop-poll-answer-row\"><label class=\"yop-poll-answer\"><input type=\"radio\" name=\"yop-poll-53-53\" value=\"522\" data-yop-sub=\"522\" \/><span class=\"yop-poll-answer-text\"><span>c<\/span><\/span><\/label><\/div><div class=\"yop-poll-answer-row\"><label class=\"yop-poll-answer\"><input type=\"radio\" name=\"yop-poll-53-53\" value=\"526\" data-yop-sub=\"526\" \/><span class=\"yop-poll-answer-text\"><span>d<\/span><\/span><\/label><\/div><div class=\"yop-poll-answer-row\"><label class=\"yop-poll-answer\"><input type=\"radio\" name=\"yop-poll-53-53\" value=\"540\" data-yop-sub=\"540\" \/><span class=\"yop-poll-answer-text\"><span>e<\/span><\/span><\/label><\/div><div class=\"yop-poll-answer-row\"><label class=\"yop-poll-answer\"><input type=\"radio\" name=\"yop-poll-53-53\" value=\"541\" data-yop-sub=\"541\" \/><span class=\"yop-poll-answer-text\"><span>f<\/span><\/span><\/label><\/div><div class=\"yop-poll-answer-row\"><label class=\"yop-poll-answer\"><input type=\"radio\" name=\"yop-poll-53-53\" value=\"542\" data-yop-sub=\"542\" \/><span class=\"yop-poll-answer-text\"><span>g<\/span><\/span><\/label><\/div><div class=\"yop-poll-answer-row\"><label class=\"yop-poll-answer\"><input type=\"radio\" name=\"yop-poll-53-53\" value=\"543\" data-yop-sub=\"543\" \/><span class=\"yop-poll-answer-text\"><span>h<\/span><\/span><\/label><\/div><div class=\"yop-poll-answer-row\"><label class=\"yop-poll-answer\"><input type=\"radio\" name=\"yop-poll-53-53\" value=\"544\" data-yop-sub=\"544\" \/><span class=\"yop-poll-answer-text\"><span>i<\/span><\/span><\/label><\/div><\/div><\/fieldset><\/div><div class=\"yop-poll-footer yop-poll-footer--center\"><div class=\"yop-poll-actions\"><button type=\"button\" class=\"yop-poll-vote-button has-background has-text-color\" data-yop-vote data-yop-perm=\"wordpress\" data-yop-signed-in-label=\"\u2192\">Sign in with WordPress<\/button><\/div><\/div><\/div><noscript><p class=\"yop-poll-noscript\">Voting in this poll requires JavaScript.<\/p><\/noscript><\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Related control theory topic:<\/strong> <strong>PID controller<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In the gift factory of the North Pole, many processes use vapor pressure to work.The vapor pressure is controlled by opening and closing a valve. In the past, this valve was controlled using a simple analog on\/off controller. Unfortunately, this &hellip; <span class=\"more-link\"><a href=\"https:\/\/control-advent.com\/?p=1605\" class=\"readmore\">Continue reading<span class=\"screen-reader-text\">16 &#8211; How to fix the magical solution?<\/span><\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":1616,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[22],"tags":[111],"class_list":["post-1605","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-english","tag-2025-en"],"_links":{"self":[{"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/posts\/1605","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1605"}],"version-history":[{"count":6,"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/posts\/1605\/revisions"}],"predecessor-version":[{"id":1618,"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/posts\/1605\/revisions\/1618"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/media\/1616"}],"wp:attachment":[{"href":"https:\/\/control-advent.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1605"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1605"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1605"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}