{"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 – How to fix the countdown till Christmas?"},"content":{"rendered":"\n

Only 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

x1<\/sub>x2<\/sub> = 11\u2005\u200a\u2192\u2005\u200a10\u2005\u200a\u2192\u2005\u200a01\u2005\u200a\u2192\u2005\u200a00<\/p>\n\n\n\n

with a sampling time of 24 hours<\/strong>.<\/p>\n\n\n\n

Knowing discrete-time state-space models from control theory, she implemented the clock as<\/p>\n\n\n\n

x1<\/sub>(k+1) = f1<\/sub>(x1<\/sub>(k),x2<\/sub>(k)),<\/p>\n\n\n\n

x2<\/sub>(k+1)= f2<\/sub>(x1<\/sub>(k),x2<\/sub>(k)).<\/p>\n\n\n\n

With some AI, the clock will finally sound \u201cHurray hurray, it\u2019s Christmas day!\u201d<\/em> \ud83c\udf84
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

Since the elf knows how to restore it but has more important duties right now, could you please help and determine which functions to choose<\/strong>?<\/p>\n\n\n\n

a. \ud83c\udf84\ud83d\udd22\ud83e\udde0 f1<\/sub> = 1\u2212x2<\/sub>, f2<\/sub> = x1<\/sub>\u22c5x2<\/sub><\/p>\n\n\n\n

b. \ud83e\udd16\ud83c\udf85\ud83d\udcd0 f1<\/sub> = x1<\/sub>\u22c5x2<\/sub>, f2<\/sub> = 1\u2212x2<\/sub><\/p>\n\n\n\n

c. \ud83d\udd01\ud83d\udcca\u2728 f1<\/sub> = x2<\/sub>, f2<\/sub> = x1<\/sub><\/p>\n\n\n\n

d. \ud83c\udf81\ud83d\udd04\u2744\ufe0f f1<\/sub> = 1\u2212x1<\/sub>, f2<\/sub> = 1\u2212x2<\/sub><\/p>\n\n\n\n

e. \ud83d\udcc9\ud83d\udeab\ud83d\udcd8 The options a.\u2013d. are no valid state-space models, because there is no system matrix A.<\/p>\n\n\n\n

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

Hint:<\/strong>
It is given that we start today from x1<\/sub>(0) = 1, x2<\/sub>(0) = 1.<\/p>\n\n\n

\n\t\t\t\t\t\t\t

<\/p><\/div>

<\/div>
<\/div><\/div>
<\/div><\/div>
<\/div><\/div>
<\/div><\/div>
<\/div><\/div><\/div><\/div><\/div>
Choose an option:<\/div>