1 00:00:00,880 --> 00:00:05,680 We will now look at different representations of time, so when we normally think about time, we normally 2 00:00:05,680 --> 00:00:07,180 think about continuous time. 3 00:00:07,540 --> 00:00:11,860 But there's an important concept when we're dealing with dynamic systems called discrete time. 4 00:00:11,860 --> 00:00:13,360 And we're going to have a look at that now. 5 00:00:14,520 --> 00:00:19,590 The differential equations that have been shown so far, all have been in continuous time. 6 00:00:20,160 --> 00:00:26,190 This just means that they're defined with respect to an independent variable time t, which varies continuously 7 00:00:26,190 --> 00:00:26,940 and smoothly. 8 00:00:27,400 --> 00:00:32,520 But for many practical purposes, especially in engineering, we only really need to know the state 9 00:00:32,520 --> 00:00:38,310 of the system at this great point in time, or we only need to calculate or operate on that system in 10 00:00:38,310 --> 00:00:39,300 this great point of time. 11 00:00:39,930 --> 00:00:45,990 So basically, we only need to do different operations at TI, not T1, T2 all the way up. 12 00:00:46,290 --> 00:00:52,530 And we represent this is Teekay minus one for the time set before Teekay for the current time and Teekay 13 00:00:52,530 --> 00:00:54,600 plus one for the timestep in feature. 14 00:00:55,590 --> 00:01:01,620 They can be a discrete instances in time, so there could be zero point one point two all the way up 15 00:01:01,620 --> 00:01:04,440 in continuous increasing segments. 16 00:01:04,680 --> 00:01:09,810 Or it could be something such as zero two four six eight alé up. 17 00:01:10,350 --> 00:01:12,320 So this is what we call discrete time. 18 00:01:12,480 --> 00:01:13,740 This is where we take time. 19 00:01:13,740 --> 00:01:19,380 Values fix a discrete value to them and incrementally increase them with time. 20 00:01:19,920 --> 00:01:24,420 So when we talk about this great time, we use this K notation here. 21 00:01:25,050 --> 00:01:28,560 So K is an integer value representing different steps and times. 22 00:01:28,560 --> 00:01:34,650 Every time we take a step in time, the time value actually increases by a constant amount to Kansi. 23 00:01:34,860 --> 00:01:37,320 Between these values here is increasing by point. 24 00:01:37,320 --> 00:01:41,070 One of the second between these values here is increasing by two seconds. 25 00:01:41,490 --> 00:01:43,730 So again, this is discrete time. 26 00:01:43,980 --> 00:01:47,920 This is breaking up continuous time into discrete time points. 27 00:01:48,810 --> 00:01:51,940 So this is easier saying up on this graph here. 28 00:01:51,960 --> 00:01:54,450 So here we have an example of continuous times. 29 00:01:54,450 --> 00:01:55,920 You can see at times varying. 30 00:01:55,920 --> 00:02:00,090 So he starts out at zero seconds and it continuously varies with time. 31 00:02:00,690 --> 00:02:05,730 So this is the first Orlistat equations written with continuous time. 32 00:02:07,840 --> 00:02:14,380 But we can break this equation up and represent in discrete time so he can see we have the dotted line, 33 00:02:14,380 --> 00:02:19,290 here is the continuous time, but we have breaking up the response into discrete time. 34 00:02:19,300 --> 00:02:22,030 So basically have discrete time steps. 35 00:02:22,030 --> 00:02:25,660 K, so K not K one, K two all the way up. 36 00:02:26,470 --> 00:02:31,590 And we evaluate the output of the state space and the d different time points. 37 00:02:31,640 --> 00:02:33,170 We have this value here. 38 00:02:33,190 --> 00:02:35,700 The value here, the value here and so forth. 39 00:02:36,310 --> 00:02:43,270 So basically the time steps forward by this Delta T amount and so we can calculate the continuous time 40 00:02:43,390 --> 00:02:45,110 just by being our energy. 41 00:02:45,130 --> 00:02:46,730 K times is timestep. 42 00:02:47,350 --> 00:02:55,680 So here we can have be one times DTI, here will be two times DETI all the way up and K being the timestep. 43 00:02:55,990 --> 00:03:01,180 So when we represent things in this form here, we can represent the dynamic system in this discrete 44 00:03:01,180 --> 00:03:01,870 time here. 45 00:03:03,460 --> 00:03:10,000 So basically where we have t we basically replace it with T, OK, so you can see here we have T of 46 00:03:10,000 --> 00:03:10,930 K being time. 47 00:03:10,930 --> 00:03:12,720 So it's a time varying dynamic system. 48 00:03:13,090 --> 00:03:16,120 We have the vector at this point in time. 49 00:03:16,120 --> 00:03:19,060 We have the input vector at this distinct point in time. 50 00:03:19,640 --> 00:03:22,120 But this function here does not calculate a state. 51 00:03:22,120 --> 00:03:22,420 Right. 52 00:03:22,420 --> 00:03:23,900 So it does not calculate the derivative. 53 00:03:24,400 --> 00:03:29,130 Instead, it calculates the next point, the state at the next point in time. 54 00:03:29,980 --> 00:03:36,280 So it's not a not a first order differential equation is now turned it into a recursive equation. 55 00:03:36,290 --> 00:03:37,630 So it moves and steps. 56 00:03:37,750 --> 00:03:39,220 It does not move in rates. 57 00:03:39,550 --> 00:03:43,210 So this is the discrete time representation of a dynamic system. 58 00:03:43,570 --> 00:03:47,410 This is the continuous time representation of a dynamic system.