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.<\/p>\n\n\n\n
In the past, this valve was controlled using a simple analog on\/off controller<\/strong>. Unfortunately, this caused sustained oscillations<\/strong>: the vapor pressure kept going up and down instead of settling at a steady value.<\/p>\n\n\n\n To fix this problem, the elves decided to try a more sophisticated and magical solution: the Pinecone\u2013Icicle\u2013Dancing (PID) controller<\/strong>. It contains three parts:<\/p>\n\n\n\n The controller output is the sum of the effects of the pinecone, the icicle, and the dancing elves.<\/p>\n\n\n\n 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 still oscillating<\/strong>!<\/p>\n\n\n\n Assume that:<\/p>\n\n\n\n The elves now wonder whether adjusting the pinecone gain K<\/strong>, the icicle growing time Ti<\/strong>, and the dancing time Td<\/strong> could calm things down.<\/p>\n\n\n\n How should these parameters be modified to decrease the oscillations?<\/strong><\/p>\n\n\n\n a. \ud83d\udcc8\ud83c\udf32\u23f1\ufe0f Increase K, increase Ti, and increase Td.\n
<\/li>\n\n\n\n
The time constant Ti<\/strong> determines how fast this happens: a larger Ti<\/strong> means the icicle grows more slowly.
<\/li>\n\n\n\n
The time constant Td<\/strong> determines how strongly they react to quick changes: a larger Td<\/strong> means the elves try harder to anticipate where the pressure is heading.<\/li>\n<\/ul>\n\n\n\n\n
b. \ud83d\udcc8\ud83c\udf32\u2b07\ufe0f Increase K, increase Ti, and decrease Td.
c. \ud83d\udcc8\u2744\ufe0f\u23f1\ufe0f Increase K, decrease Ti, and increase Td.
d. \ud83d\udcc8\u2744\ufe0f\u2b07\ufe0f Increase K, decrease Ti, and decrease Td.
e. \ud83d\udcc9\ud83c\udf32\u23f1\ufe0f Decrease K, increase Ti, and increase Td.
f. \ud83d\udcc9\ud83c\udf32\u2b07\ufe0f Decrease K, increase Ti, and decrease Td.
g. \ud83d\udcc9\u2744\ufe0f\u23f1\ufe0f Decrease K, decrease Ti, and increase Td.
h. \ud83d\udcc9\u2744\ufe0f\u2b07\ufe0f Decrease K, decrease Ti, and decrease Td.
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