1 00:00:04,320 --> 00:00:09,480 We will now have a look at different mathematical ways of representing different states, based systems 2 00:00:10,470 --> 00:00:15,720 in the previous video we have seen, there are two different ways of representing the time domain of 3 00:00:15,720 --> 00:00:16,340 the system. 4 00:00:16,740 --> 00:00:22,920 We can break the system down into a continuous time system or either a discrete time system. 5 00:00:24,000 --> 00:00:26,130 Now, these are the different ways of representing time. 6 00:00:26,250 --> 00:00:32,190 We can also break them down into different subcategories as well for the continuous time system. 7 00:00:32,490 --> 00:00:38,850 We can also break it down into a non-linear or general system, or we can also break it down into a 8 00:00:38,850 --> 00:00:39,770 linear system. 9 00:00:40,680 --> 00:00:43,050 We can do the same thing with the discrete system. 10 00:00:43,050 --> 00:00:49,350 We can break it down into a nonlinear or the general system, and we can also break it down into a linear 11 00:00:49,350 --> 00:00:49,950 system. 12 00:00:51,570 --> 00:00:55,860 So now let's have a close look at each of these different systems and what they actually mean. 13 00:00:57,000 --> 00:01:02,430 So let's start by looking at the continuous non-linear or general time system as shown here. 14 00:01:04,040 --> 00:01:09,980 The continuous non-linear time model can be represented by a general function here that can take any 15 00:01:09,980 --> 00:01:16,280 form, it can be anything, it can be linear, it can be non-linear is just any general function that 16 00:01:16,280 --> 00:01:20,210 can be a input of time because it can be time varying. 17 00:01:20,480 --> 00:01:23,150 It can be a function of the input of the current state. 18 00:01:23,150 --> 00:01:26,480 X can be input of the current control input. 19 00:01:26,480 --> 00:01:32,420 You and the general function F here outputs the continuous time state writes. 20 00:01:33,760 --> 00:01:39,970 So this function here is a very general way of representing that this function can be any function with 21 00:01:39,970 --> 00:01:43,630 these inputs that produces the right output. 22 00:01:45,300 --> 00:01:49,680 So now let's have a look at this continuous time, linear system category. 23 00:01:51,400 --> 00:01:58,120 The continuous time linear model is also, again, a function of the current state x the current input 24 00:01:58,120 --> 00:01:58,540 you. 25 00:01:58,930 --> 00:02:05,500 It is also time Verinata is a function of T and it produces an output of the state rights. 26 00:02:05,500 --> 00:02:07,430 So X as a function of T. 27 00:02:07,960 --> 00:02:10,420 So this is why is a continuous time system. 28 00:02:11,050 --> 00:02:13,210 Now this model here is a linear model. 29 00:02:13,210 --> 00:02:19,810 So we make the assumption that the output of the system, state rights is a linear combination of the 30 00:02:19,810 --> 00:02:24,200 states and the inputs based on this matrix multiplications here. 31 00:02:25,000 --> 00:02:32,080 So we have this state matrix A and we have the state vector X, we have this input matrix B and the 32 00:02:32,080 --> 00:02:33,070 input vector you. 33 00:02:34,120 --> 00:02:38,570 So the state rates here can be a function of this whole operation here. 34 00:02:39,010 --> 00:02:43,990 So anything that can be represented as a matrix multiplication of the states and a matrix multiplication 35 00:02:43,990 --> 00:02:47,740 of the inputs can be represented as a linear model. 36 00:02:49,760 --> 00:02:53,720 Now, each of these coefficients in The Matrix can also be time varying because you can see they're 37 00:02:53,770 --> 00:03:01,220 a function of T and the same thing for the control matrix B, but any system that can be represented 38 00:03:01,220 --> 00:03:06,020 in this form can be considered as a continuous time, linear, dynamic system. 39 00:03:08,850 --> 00:03:11,950 So those are the two continuous time representations. 40 00:03:12,300 --> 00:03:17,910 Now let's have a look at the discrete time representation starting off with the discrete non-linear 41 00:03:17,910 --> 00:03:18,870 model shown here. 42 00:03:20,690 --> 00:03:26,330 So we already know that this is great time model, instead of calculating a rate, it calculates the 43 00:03:26,330 --> 00:03:27,080 next state. 44 00:03:27,080 --> 00:03:29,360 So you can see here this is a general function. 45 00:03:29,360 --> 00:03:35,690 If that can be any function that can take the input of the current time, the current state and the 46 00:03:35,690 --> 00:03:36,410 current input. 47 00:03:36,770 --> 00:03:42,440 And this function here calculates the next state for the discrete time, step one in the future. 48 00:03:43,930 --> 00:03:49,990 Now, this is a very generous representation because it is the most generous representation, any function 49 00:03:49,990 --> 00:03:57,340 that takes in the current time state and input or one or more of these and outputs the next time in 50 00:03:57,340 --> 00:04:02,440 this great time is a general non-linear model for this great time. 51 00:04:04,450 --> 00:04:10,390 And again, we can break the discrete time system down into a more restrictive category of being the 52 00:04:10,390 --> 00:04:13,540 linear, discrete time system as shown here. 53 00:04:14,870 --> 00:04:21,080 And again, the linear model is just a way of representing the function in a more structured form, 54 00:04:21,380 --> 00:04:29,540 so a linear system again is any system that is a modification of this vector and the input vector as 55 00:04:29,540 --> 00:04:35,780 a matrix multiplication system here is a linear system, is a combination of this F matrix multiplied 56 00:04:35,780 --> 00:04:39,100 by the state plus this G matrix multiplied by the input. 57 00:04:39,860 --> 00:04:45,140 And again, this is a linear operation because of the matrix times, a vector and again a matrix times 58 00:04:45,140 --> 00:04:45,860 the vector here. 59 00:04:46,880 --> 00:04:54,560 Now each of these matrix F and G, they can vary with time because we have this subscript K here. 60 00:04:54,740 --> 00:04:56,630 You can see that for every discrete time. 61 00:04:56,630 --> 00:05:01,400 Update this whole function here and this whole function here can change with time. 62 00:05:02,270 --> 00:05:04,270 But as you can see, this is linear function. 63 00:05:04,580 --> 00:05:07,070 It is a function of the states and the inputs. 64 00:05:07,220 --> 00:05:13,340 That's a linear multiplication and outputs the next state for the next timestep. 65 00:05:15,020 --> 00:05:19,010 So these are all the different ways we can represent states based systems. 66 00:05:19,040 --> 00:05:25,120 So, again, it can be in a continuous time domain or convenience, great demand time domain. 67 00:05:25,130 --> 00:05:30,620 The continuous systems calculate the state rates as seen here, the extent of the derivatives with respect 68 00:05:30,620 --> 00:05:37,550 to time, the discrete time systems just calculate the next time for the next discrete point where you 69 00:05:37,550 --> 00:05:39,930 can break it down into linear and general systems. 70 00:05:39,930 --> 00:05:42,180 So you can say here we have a linear systems here. 71 00:05:42,680 --> 00:05:46,880 So the state vector and the input vector are multiplied by a matrix. 72 00:05:47,390 --> 00:05:51,740 Or we can have the general or nonlinear functions here where we just have a general function that can 73 00:05:51,740 --> 00:05:54,240 be anything, including a linear function. 74 00:05:55,130 --> 00:05:58,760 Now can also break it down into time varying and time invariant. 75 00:05:59,150 --> 00:06:04,310 So time varying systems means that the function can change if time is a function of time, or we can 76 00:06:04,310 --> 00:06:07,880 see the linear systems, the matrix and B here change over time. 77 00:06:08,030 --> 00:06:11,960 Same thing with the F and G in a time varying systems. 78 00:06:11,960 --> 00:06:15,080 It just means that the function is not a function of time. 79 00:06:15,590 --> 00:06:17,660 So this function F does not have the time. 80 00:06:17,660 --> 00:06:21,860 Input here is always a constant function and the same thing for the linear system. 81 00:06:21,860 --> 00:06:23,450 So you can see the linear system here. 82 00:06:23,750 --> 00:06:30,110 The A and B matrix are fixed for all time and the same thing for this great time system, F and G do 83 00:06:30,110 --> 00:06:31,410 not change the constants. 84 00:06:32,060 --> 00:06:35,480 So this just means that this is a time invariant system here. 85 00:06:35,480 --> 00:06:38,060 The system dynamics do not change with time. 86 00:06:38,390 --> 00:06:41,510 They just change with the state and inputs.