linear discrete time system

𝐱𝐱 𝑘𝑘 + 1 =
0.9974 0.0539
−0.1078 1.1591 𝐱𝐱 𝑘𝑘 + 0.0013
0.0539 𝑢𝑢 𝑘𝑘
Consider the following linear discrete time system with initial state [2, 1].
which is to be controlled to minimize the follow performance measure,
𝐽𝐽 = 1
2 �
𝑘𝑘=0
𝑁𝑁−1
0.25𝑥𝑥1
2(𝑘𝑘) + 0.05𝑥𝑥2
2(𝑘𝑘) + 0.05𝑢𝑢2(𝑘𝑘) .
Determine the optimal control law.
Use the Q, R given in the above, and let H=identity matrix.
Plot 3 sets of results for N=50, 100, 200, respectively.
Each set of the results should include the feedback gain, the optimal control, and the states. Also
determine the optimal cost for each of the three N values.
Also attach a copy of your simulation code.