1 00:00:04,080 --> 00:00:10,050 We have discussed in the past that most of the real world processes or dynamics can be represented as 2 00:00:10,050 --> 00:00:11,370 differential equations. 3 00:00:11,670 --> 00:00:18,210 Now, differential equations are inherently continuous time systems because they use a derivative of 4 00:00:18,210 --> 00:00:19,690 a state with respect to time. 5 00:00:20,670 --> 00:00:24,840 This makes them very easy to represent as continuous time processes. 6 00:00:25,620 --> 00:00:32,160 So if we have a look at a continuous time invariant linear system as shown on this slide here, somehow 7 00:00:32,160 --> 00:00:35,610 we want to represent this as a discrete time system. 8 00:00:36,420 --> 00:00:41,640 This is because for engineering purposes, we don't really care about the system state and every single 9 00:00:41,640 --> 00:00:46,100 time we only really care about the state at specific times in the future. 10 00:00:46,560 --> 00:00:49,610 So we can represent this using a discrete system. 11 00:00:50,610 --> 00:00:58,500 So if we look at a discrete time, time invariant linear system as shown here, we want to somehow convert 12 00:00:58,500 --> 00:01:01,710 a continuous time system into a discrete time system. 13 00:01:01,980 --> 00:01:06,090 And we're going to again look at our time invariant and linear systems. 14 00:01:06,700 --> 00:01:12,240 So somehow we need a mapping process which can convert the A and B matrixes for the continuous time 15 00:01:12,240 --> 00:01:16,530 system into the energy matrix for the discrete time system. 16 00:01:17,340 --> 00:01:22,200 So this is what we're going to look at in this video, our continuous to screen time conversions. 17 00:01:23,390 --> 00:01:30,020 So if we take our continuous time invariant linear system as shown here, we know that the solution 18 00:01:30,020 --> 00:01:32,870 to this system is given by this equation here. 19 00:01:34,510 --> 00:01:39,910 Now, all of this equation is saying that we can work out the state of the system at some time in the 20 00:01:39,910 --> 00:01:45,160 future based on this matrix exponential equation and this integration here. 21 00:01:45,790 --> 00:01:52,720 So this part of the equation here, we can see that it only depends on our system dynamics A and the 22 00:01:52,720 --> 00:01:56,640 state at TI not so this is the free response to the system. 23 00:01:56,920 --> 00:02:00,040 So this is what happens if we have no other inputs into the system. 24 00:02:00,220 --> 00:02:02,230 This will be how the time this is. 25 00:02:02,350 --> 00:02:04,240 This is how this is me rolls of time. 26 00:02:05,440 --> 00:02:11,380 But if we have some input here, so if you have some input here, this gets added onto the system response 27 00:02:11,380 --> 00:02:17,980 so we can separate the system and dynamics, the free response with the force response or the input 28 00:02:17,980 --> 00:02:19,280 response to the system. 29 00:02:20,080 --> 00:02:25,810 So this is integral here between Time Teno and the current time that we want T or what this equation 30 00:02:25,810 --> 00:02:31,360 is doing is basically summing up the different impulse responses for all the time in between these two 31 00:02:31,360 --> 00:02:32,150 time periods. 32 00:02:33,040 --> 00:02:39,250 So if we have, for example, our input at time t not is going to have more of an effect than the input 33 00:02:39,250 --> 00:02:42,880 at time t just because it acts over a longer time period. 34 00:02:43,240 --> 00:02:47,680 So this is what this integral is doing, is just summing up all the different inputs for all the different 35 00:02:47,680 --> 00:02:51,580 impulses over the time period in between T not and T. 36 00:02:53,200 --> 00:03:00,250 So if we consider a discreet time step, so we have the current time, so let's say it is equal to take 37 00:03:00,700 --> 00:03:06,790 the previous time, step is going to be out not which is going to be a T minus one, and we'll have 38 00:03:06,790 --> 00:03:08,410 our discrete time set aside. 39 00:03:08,420 --> 00:03:14,050 So Delta T is just a difference between the time at Teekay and the time at K minus one. 40 00:03:14,380 --> 00:03:16,600 So this is just the difference between our two time points. 41 00:03:17,730 --> 00:03:24,540 So if we would take these discrete time representations for T and T not and substitute them into our 42 00:03:24,540 --> 00:03:30,330 above solution for the continuous time system here, we end up with this equation down here. 43 00:03:31,140 --> 00:03:36,300 So this is basically saying we want to evaluate the system dynamics over a very small time interval. 44 00:03:36,300 --> 00:03:41,780 Delta T, so we have Teekay minus one and we're working at the state for Altec. 45 00:03:42,270 --> 00:03:47,790 And again, it looks very similar as a continuous time solution up here, except now we're going to 46 00:03:47,790 --> 00:03:52,800 instead of using our T minus tonight, we're just going to use our ability and we're going to use that 47 00:03:52,800 --> 00:03:54,370 K minus one and Otik. 48 00:03:54,870 --> 00:03:58,020 So again, we have a free response to the system. 49 00:03:58,050 --> 00:04:03,360 So what happens if we have no inputs and we just evolve the state of time and then we have the force 50 00:04:03,360 --> 00:04:03,800 response? 51 00:04:03,840 --> 00:04:06,810 And what happens if we adding a forcing input to the system? 52 00:04:08,470 --> 00:04:14,950 So if we consider our equation from the last slide here, which is just the propagation of our current 53 00:04:14,950 --> 00:04:22,180 state to our future state or from state, Teekay minus one to take, we can look at this equation here 54 00:04:22,180 --> 00:04:27,790 and evaluate it with respect to this discrete system matrix matrix equation here. 55 00:04:29,430 --> 00:04:34,740 So it is fairly easy to see this is a factor here, multiply by this matrix so we can easily create 56 00:04:34,740 --> 00:04:39,030 this matrix, this matrix expansion into our F system matrix. 57 00:04:39,870 --> 00:04:41,490 Looking at this side of the equation here. 58 00:04:41,520 --> 00:04:42,510 We can do the same thing. 59 00:04:42,510 --> 00:04:49,620 So if we group all of this together, multiply it by our control input, you we can see that this is 60 00:04:49,620 --> 00:04:51,690 basically this matrix here. 61 00:04:54,390 --> 00:05:02,430 So we can break it out into this here, Alef Matrix is our Matrix exponential of a Delta T g matrix. 62 00:05:02,430 --> 00:05:08,400 A forcing response is just going to be our free response multiplied by the integral from zero to Delta 63 00:05:08,400 --> 00:05:11,010 T of this part of the equation here. 64 00:05:12,530 --> 00:05:18,520 Now, this part of the equation here can be simplified, so if our continuous time, Matrix A. is convertible, 65 00:05:18,530 --> 00:05:25,640 so if Inverse A exists, then we can simplify this part down into this equation here so we don't have 66 00:05:25,640 --> 00:05:26,680 to do the integral. 67 00:05:26,870 --> 00:05:31,580 We can just assume that if a convertible, then this is the response. 68 00:05:33,350 --> 00:05:38,900 So these equations are how we convert a continuous time system into a discrete time system, we can 69 00:05:38,900 --> 00:05:45,410 use the and B matrix and then using a specified timestep, we can convert the system into a discrete 70 00:05:45,410 --> 00:05:48,280 time system, working out alef energy matrix. 71 00:05:48,890 --> 00:05:55,100 And if A is invariable, then we can use this identity here to help simplify the equation so we don't 72 00:05:55,100 --> 00:05:56,210 have to do the integration. 73 00:05:57,810 --> 00:06:02,730 So now you might be wondering about this matrix exponential and how do we actually calculate this matrix 74 00:06:02,730 --> 00:06:03,420 exponential? 75 00:06:04,810 --> 00:06:11,590 So this matrix exponential can be written as an infinite sum, and if we expand this out, you can see 76 00:06:11,590 --> 00:06:16,390 that each of the eight is just going to be at the power of zero eight to power. 77 00:06:16,390 --> 00:06:22,970 One plus eight is the power or two over two factorial and so on and so forth all the way up to infinity. 78 00:06:24,820 --> 00:06:26,350 So this is a useful energy. 79 00:06:26,440 --> 00:06:30,700 It allows us to make an approximation of what this infinite sum is going to be. 80 00:06:31,120 --> 00:06:36,550 So depending on the matrix dynamics of A, we might be able to make a few approximations. 81 00:06:36,820 --> 00:06:42,220 So if the matrix dynamics is appropriate, we can make a first order approximation. 82 00:06:42,940 --> 00:06:49,870 So basically we can use a first order approximation in some cases so the exponential amte can be approximated 83 00:06:50,100 --> 00:06:53,380 by the identity matrix plus eight times T.. 84 00:06:54,010 --> 00:06:55,830 So this is a first order approximation. 85 00:06:55,840 --> 00:06:58,340 It just looks at these first two terms here. 86 00:06:58,360 --> 00:07:01,270 So a T to the power of zero is just identity. 87 00:07:01,540 --> 00:07:05,460 And then we just have a taste of how one is just able to it. 88 00:07:05,920 --> 00:07:07,750 So this is where it comes out here. 89 00:07:08,470 --> 00:07:13,390 Now, if the dynamics of A are such that this is a very bad approximation, well, then we'll have to 90 00:07:13,390 --> 00:07:16,360 use a different approximation method to calculate this matrix. 91 00:07:16,360 --> 00:07:17,050 Exponential.