00:01
Hi, in this question, given that x dot equals 6x plus u and v equals integral over 0 to 1 x square plus 2u square into dt.
00:14
In part a, we have to determine the state function of boundary gain h.
00:20
So, which can be written as h equals lambda x cap plus x square plus 2u square and where x cap equals 6x plus u.
00:33
On substituting in this, then we get h equals 6 lambda x plus u lambda plus x square plus 2u square.
00:45
On differentiating with respect to u, then we get 0 plus lambda plus 0 plus 4u.
00:53
Hence, conclude that dou h by dou u equals lambda plus 4u.
01:05
Hence, we can write it as u dash equals minus lambda by 4.
01:16
Next, move on to part b.
01:18
Here, we need to determine the optimal input u naught.
01:22
We know that u dash equals minus lambda by 4.
01:26
So, h of x comma lambda comma t equals 6 lambda x plus u into lambda plus x square plus 2u square.
01:36
On substituting in this, then we get 6 lambda u plus u into lambda s lambda u s minus lambda by 4 plus x square plus 2 into minus lambda by 4 the whole square which is equal to 6 lambda x minus lambda square divided by 8 plus x square.
02:07
Let this as equation number 2.
02:11
In part c, we have to determine the equation governing the control system in terms of v and lambda.
02:19
So, x equals dou h into x comma h comma t divided by dou x.
02:27
So, here x can be written as 6x minus 2 lambda divided by 8 plus 0.
02:34
X dash equals 6x minus lambda by 4 which implies here capital x dash equals minus dou h dash x comma h lambda comma t divided by dou x.
02:52
From equation 2, we can write it as x dash equals minus 6 lambda minus 0 plus 2u which is equal to minus 6 lambda minus 2u...