Only 4 days left, and a digital Christmas countdown clock was designed very thoroughly by an elf. Starting from today, it digitally generates the sequence
x1x2 = 11 → 10 → 01 → 00
with a sampling time of 24 hours.
Knowing discrete-time state-space models from control theory, she implemented the clock as
x1(k+1) = f1(x1(k),x2(k)),
x2(k+1)= f2(x1(k),x2(k)).
With some AI, the clock will finally sound “Hurray hurray, it’s Christmas day!” 🎄
Unfortunately, the Easter Bunny from December 2 is very angry about all the attention Santa gets—and also about a dent in his Ferrari. Thus (for no real reason), he has taken away the parameter configuration file for this clock.
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?
a. 🎄🔢🧠 f1 = 1−x2, f2 = x1⋅x2
b. 🤖🎅📐 f1 = x1⋅x2, f2 = 1−x2
c. 🔁📊✨ f1 = x2, f2 = x1
d. 🎁🔄❄️ f1 = 1−x1, f2 = 1−x2
e. 📉🚫📘 The options a.–d. are no valid state-space models, because there is no system matrix A.
f. 🧝♀️💻💪 I’ll do the work for the elf so that she can do the configuration 😉
Hint:
It is given that we start today from x1(0) = 1, x2(0) = 1.
Related control theory topic: multilinear systems, state space models