{"id":1660,"date":"2025-12-21T08:01:27","date_gmt":"2025-12-21T07:01:27","guid":{"rendered":"https:\/\/control-advent.com\/?p=1660"},"modified":"2025-12-21T08:01:28","modified_gmt":"2025-12-21T07:01:28","slug":"21-how-to-fix-the-countdown-till-christmas","status":"publish","type":"post","link":"https:\/\/control-advent.com\/?p=1660","title":{"rendered":"21 &#8211; How to fix the countdown till Christmas?"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Only&nbsp;<strong>4 days left<\/strong>, and a digital Christmas countdown clock was designed very thoroughly by an elf. Starting from today, it digitally generates the sequence<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">x<sub>1<\/sub>x<sub>2<\/sub> = 11\u2005\u200a\u2192\u2005\u200a10\u2005\u200a\u2192\u2005\u200a01\u2005\u200a\u2192\u2005\u200a00<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">with a sampling time of&nbsp;<strong>24 hours<\/strong>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Knowing discrete-time state-space models from control theory, she implemented the clock as<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">x<sub>1<\/sub>(k+1) = f<sub>1<\/sub>(x<sub>1<\/sub>(k),x<sub>2<\/sub>(k)),<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">x<sub>2<\/sub>(k+1)= f<sub>2<\/sub>(x<sub>1<\/sub>(k),x<sub>2<\/sub>(k)).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">With some AI, the clock will finally sound&nbsp;<em>\u201cHurray hurray, it\u2019s Christmas day!\u201d<\/em>&nbsp;\ud83c\udf84<br>Unfortunately, the Easter Bunny from December 2 is very angry about all the attention Santa gets\u2014and also about a dent in his Ferrari. Thus (for no real reason), he has taken away the parameter configuration file for this clock.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since the elf knows how to restore it but has more important duties right now, could you please help and determine&nbsp;<strong>which functions to choose<\/strong>?<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">a. \ud83c\udf84\ud83d\udd22\ud83e\udde0 f<sub>1<\/sub> = 1\u2212x<sub>2<\/sub>, f<sub>2<\/sub> = x<sub>1<\/sub>\u22c5x<sub>2<\/sub><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">b. \ud83e\udd16\ud83c\udf85\ud83d\udcd0 f<sub>1<\/sub> = x<sub>1<\/sub>\u22c5x<sub>2<\/sub>, f<sub>2<\/sub> = 1\u2212x<sub>2<\/sub><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">c. \ud83d\udd01\ud83d\udcca\u2728 f<sub>1<\/sub> = x<sub>2<\/sub>, f<sub>2<\/sub> = x<sub>1<\/sub><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">d. \ud83c\udf81\ud83d\udd04\u2744\ufe0f f<sub>1<\/sub> = 1\u2212x<sub>1<\/sub>, f<sub>2<\/sub> = 1\u2212x<sub>2<\/sub><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">e. \ud83d\udcc9\ud83d\udeab\ud83d\udcd8 The options&nbsp;a.\u2013d.&nbsp;are no valid state-space models, because there is no system matrix&nbsp;A.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">f. \ud83e\udddd\u200d\u2640\ufe0f\ud83d\udcbb\ud83d\udcaa I\u2019ll do the work for the elf so that she can do the configuration \ud83d\ude09<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Hint:<\/strong><br>It is given that we start today from x<sub>1<\/sub>(0) = 1, x<sub>2<\/sub>(0) = 1.<\/p>\n\n\n<div class=\"yop-poll-container\" data-yop-poll-id=\"58\"><script type=\"application\/json\" 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Starting from today, it digitally generates the sequence x1x2 = 11\u2005\u200a\u2192\u2005\u200a10\u2005\u200a\u2192\u2005\u200a01\u2005\u200a\u2192\u2005\u200a00 with a sampling time of&nbsp;24 hours. Knowing discrete-time state-space models from control theory, &hellip; <span class=\"more-link\"><a href=\"https:\/\/control-advent.com\/?p=1660\" class=\"readmore\">Continue reading<span class=\"screen-reader-text\">21 &#8211; How to fix the countdown till Christmas?<\/span><\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":1661,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[22],"tags":[111],"class_list":["post-1660","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\/1660","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=1660"}],"version-history":[{"count":3,"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/posts\/1660\/revisions"}],"predecessor-version":[{"id":1668,"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/posts\/1660\/revisions\/1668"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=\/wp\/v2\/media\/1661"}],"wp:attachment":[{"href":"https:\/\/control-advent.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1660"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1660"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/control-advent.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1660"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}