1 00:00:00,360 --> 00:00:01,230 ‫Welcome back. 2 00:00:01,830 --> 00:00:08,580 ‫Now, if you remember from the previous course, then we really wanted to keep the changes of control 3 00:00:08,580 --> 00:00:10,880 ‫inputs as small as possible. 4 00:00:11,130 --> 00:00:16,600 ‫And I'm talking about the changes of control inputs, the deltas, the increments. 5 00:00:17,370 --> 00:00:24,480 ‫So if you remember, we modified our cost function in such a way that in the vehicle case, we started 6 00:00:24,480 --> 00:00:34,170 ‫treating the increment of the steering wheel angle, Delta Delta at K plus I as our control input, 7 00:00:35,040 --> 00:00:39,240 ‫but we had to keep track of the real steering wheel angle as well. 8 00:00:39,780 --> 00:00:50,970 ‫For that we define our Delta Delta at Key like this, Delta at K, minus Delta at K, minus one. 9 00:00:51,690 --> 00:01:00,510 ‫And so we decided that we would keep Delta at K minus one, the control input from one time sample before 10 00:01:00,840 --> 00:01:09,570 ‫we were treated as an additional state and then Delta at K would be an additional state one time sample 11 00:01:09,570 --> 00:01:10,110 ‫later. 12 00:01:10,980 --> 00:01:20,280 ‫So this thing here would be actually part of our present state vector and this would be part of our 13 00:01:21,210 --> 00:01:26,430 ‫future state vector at K plus one. 14 00:01:27,440 --> 00:01:36,050 ‫So the state vector, it looked like this in the previous course, so why not at K K side on that K 15 00:01:36,090 --> 00:01:46,340 ‫and then the big inertial Y at K, but then we also added our Delta at K minus one here, and this vector 16 00:01:46,340 --> 00:01:53,000 ‫became an augmented state vector and augmented state vector. 17 00:01:53,690 --> 00:02:00,860 ‫And the notation that we use for an augment a state vector was that we added a tilde here and then the 18 00:02:00,860 --> 00:02:07,970 ‫state vector at K plus one would equal like this the same states but at K plus one. 19 00:02:08,420 --> 00:02:18,890 ‫But then we added the delta at K here and it became an augmented future state vector at K plus one. 20 00:02:19,910 --> 00:02:29,120 ‫And so to do that, to augment our state vectors, to expand them, we have to augment or expand our 21 00:02:29,120 --> 00:02:36,830 ‫system itself to incorporate this and this into our state's base equations. 22 00:02:38,070 --> 00:02:43,920 ‫And so the same thing will happen in the ugly case as well, instead of. 23 00:02:44,960 --> 00:02:59,180 ‫The control input use of key plus I, our control input will be dealt the vector at K plus I, or you 24 00:02:59,180 --> 00:03:00,940 ‫can write it out like this as well. 25 00:03:01,700 --> 00:03:12,340 ‫Delta you to add K plus I Delta U three at K plus I and Delta U for at K plus I. 26 00:03:13,070 --> 00:03:16,040 ‫And so we will define our delta. 27 00:03:16,040 --> 00:03:25,940 ‫You at K like this, the U vector at K minus the U vector at cave minus one. 28 00:03:26,840 --> 00:03:37,880 ‫We can then rewrite it like this you vector at K which is this one here equals U vector at K minus one 29 00:03:38,600 --> 00:03:39,830 ‫which is this one. 30 00:03:40,800 --> 00:03:51,360 ‫Plus, Delta, you, Victor and Kay, which is this one here, so our discrete states, base equations 31 00:03:51,360 --> 00:04:01,470 ‫can be rewritten like this, your state vector at Cape plus one equals a matrix times, your state vector 32 00:04:01,470 --> 00:04:08,500 ‫at K plus B times your regional control input vector atk. 33 00:04:09,510 --> 00:04:20,340 ‫But then you can rewrite the same thing like this where instead of use up k, you write it out in this 34 00:04:20,340 --> 00:04:30,760 ‫form like it is over here you add K minus one plus Delta U at K like this. 35 00:04:31,620 --> 00:04:40,080 ‫So this thing in the brackets that would be your use up K and of course you can open the brackets so 36 00:04:40,080 --> 00:04:55,650 ‫X and K plus one equals eight times X and K plus B times you add K minus one and plus B times Delta 37 00:04:55,650 --> 00:04:57,480 ‫you add K. 38 00:04:58,690 --> 00:05:03,560 ‫Then I'm also going to take this equation and I'm going to write it here as well. 39 00:05:04,060 --> 00:05:20,320 ‫You at K equals you at K minus one, plus Delta, you add K and this is a system of equations and now 40 00:05:20,320 --> 00:05:22,600 ‫you can augment your system like this. 41 00:05:23,880 --> 00:05:34,710 ‫This is your ex and Kate plus one, this is your you at K, you pack it in in one vector. 42 00:05:35,860 --> 00:05:37,900 ‫Then you will have a matrix here. 43 00:05:39,030 --> 00:05:50,400 ‫And you will have a vector here, so here you will have X and K, U at K minus one and then you will 44 00:05:50,400 --> 00:05:57,120 ‫have another Matrix here times Delta, you at K. 45 00:05:58,090 --> 00:06:04,630 ‫So this belongs to this, then this belongs to this. 46 00:06:05,620 --> 00:06:15,070 ‫Then this here belongs to this and this belongs to this, and also you have your you at K minus one 47 00:06:15,070 --> 00:06:19,330 ‫here, so that also belongs to this place here. 48 00:06:20,150 --> 00:06:27,240 ‫And then Delta you at K in both equations, they belonged to this one here. 49 00:06:28,090 --> 00:06:38,920 ‫And don't forget, you also had an output vector Y at K and that equals some kind of matrix and then 50 00:06:39,160 --> 00:06:41,290 ‫the augmented state vector. 51 00:06:42,070 --> 00:06:53,620 ‫So X at K would be here and then you add K minus one would be here and then you had another Matrix here 52 00:06:54,430 --> 00:07:04,960 ‫times Delta, you and K and now this is your exercice fill in this matrix here, this matrix here, 53 00:07:05,260 --> 00:07:08,020 ‫this matrix here and this matrix here. 54 00:07:08,680 --> 00:07:09,970 ‫What elements. 55 00:07:09,970 --> 00:07:15,170 ‫Those matrices would have tried yourself and then will continue in the next to you. 56 00:07:15,280 --> 00:07:16,210 ‫Thank you very much.