1 00:00:03,900 --> 00:00:09,630 In this video, we are going to look at the simulation of this great time and continuous time models. 2 00:00:11,180 --> 00:00:17,150 In order to solve the differential equations with respect to time, the integral of the equations need 3 00:00:17,150 --> 00:00:18,050 to be computed. 4 00:00:18,890 --> 00:00:24,530 This process is called integration and in simulation, it is usually carried out numerically because 5 00:00:24,530 --> 00:00:27,620 an exact analytical solution cannot usually be found. 6 00:00:28,460 --> 00:00:35,330 In the previous video, we presented the result for the analytical solution for the continuous time 7 00:00:35,570 --> 00:00:37,960 time invariant linear system. 8 00:00:38,570 --> 00:00:40,730 But this is only a very specific case. 9 00:00:40,910 --> 00:00:47,480 A analytical solution can be found in general, and analytical solution is usually pretty hard to find 10 00:00:47,480 --> 00:00:48,560 and cannot be found. 11 00:00:50,210 --> 00:00:56,120 The simulation on numerical integration of continuous systems and discrete systems will be explained 12 00:00:56,120 --> 00:00:56,950 in this video. 13 00:00:58,390 --> 00:01:05,710 So let's first consider the continuous time simulation, continuous time simulation solves this differential 14 00:01:05,710 --> 00:01:06,430 equation here. 15 00:01:06,460 --> 00:01:12,700 So this is the state's best representation, a continuous time, non-linear states based system. 16 00:01:14,790 --> 00:01:21,480 So if we break this down and have a look at the state X with respect to time as shown in this graph 17 00:01:21,480 --> 00:01:25,190 here, it has some trajectory or profile. 18 00:01:25,830 --> 00:01:27,670 So let's consider this time point here. 19 00:01:27,690 --> 00:01:34,800 So this time for time, T is going to have a value of X of T, and now we would like to work out what 20 00:01:34,800 --> 00:01:36,940 the state is as sometime in the future. 21 00:01:37,590 --> 00:01:41,910 So the first step in this process is to look at the derivative. 22 00:01:41,910 --> 00:01:47,500 So the derivative or the rate of change of the state X respectively is a slope of this graph. 23 00:01:48,240 --> 00:01:52,830 Now, the slope of this graph at this point here is exactly the solution here. 24 00:01:52,840 --> 00:01:56,250 So this is the derivative of the state respective time. 25 00:01:57,580 --> 00:02:02,110 So what we would like to do is to look at the state at some time in the future. 26 00:02:02,590 --> 00:02:09,370 So let's say we want to propagate the time to a very small time in the future of Delta T. So we want 27 00:02:09,370 --> 00:02:14,500 to step forward by dignity to work out the state at time T plus Delta T.. 28 00:02:15,400 --> 00:02:22,120 So if we assume that this directive is concerned over this time period date here, all we have to do 29 00:02:22,120 --> 00:02:26,420 is just propagate this derivative up to this point over here. 30 00:02:26,890 --> 00:02:27,880 So we look at this. 31 00:02:27,880 --> 00:02:34,090 We can use our gradient rule and just say that decks of ADT times date. 32 00:02:34,780 --> 00:02:38,580 The dates here are going to cancel out, leaving UTX or Delta X. 33 00:02:38,590 --> 00:02:43,220 So this is going to tell us how much X has changed over this time period here. 34 00:02:44,170 --> 00:02:52,120 So if we do that, we can propagate our X value for up to here to work out our X at time T plus Delta 35 00:02:52,120 --> 00:02:55,110 T or we write this in equation form. 36 00:02:55,450 --> 00:03:00,760 We can say that the value of the state at time t possibility is equal. 37 00:03:00,760 --> 00:03:07,930 The value of the state, AT&T T plus the derivative of the state at 20 times the ability. 38 00:03:08,650 --> 00:03:12,700 So this is a first order approximation of the system dynamics. 39 00:03:13,390 --> 00:03:19,150 This basically steps forward a small amount of time, assuming that the state right is content to work 40 00:03:19,150 --> 00:03:22,300 out the new type to work out new state at the new timestep. 41 00:03:24,100 --> 00:03:30,730 So this is known as our first order integration, so we can say here that we use the dynamics of the 42 00:03:30,730 --> 00:03:37,450 system to work out our state rights or extort, then we can use our other integration, which basically 43 00:03:37,450 --> 00:03:39,540 propagates forward in small chunks of time. 44 00:03:40,000 --> 00:03:43,690 So we have the current state, AT&T, plus our state. 45 00:03:43,690 --> 00:03:46,760 Right at times our timestep here. 46 00:03:47,080 --> 00:03:51,270 This is going to work out in our new state at some small time step in the future. 47 00:03:51,790 --> 00:03:55,870 And what we want to do is that we want to select a timestep that is very small. 48 00:03:56,020 --> 00:04:01,240 The smaller that this state is here, the better solution is, the more that it approximates the real 49 00:04:01,450 --> 00:04:02,800 time, the main solution. 50 00:04:04,010 --> 00:04:08,230 So what we do is we first assume an initial condition, so X of naught. 51 00:04:08,450 --> 00:04:13,820 So this is our starting condition, our starting state, then we use this to calculate the state right 52 00:04:13,820 --> 00:04:14,320 equation. 53 00:04:14,330 --> 00:04:19,850 So we feel in this equation here and evaluate it using the current state, the current input to work 54 00:04:19,850 --> 00:04:22,760 out how quickly the state is changing for the current time. 55 00:04:23,480 --> 00:04:29,900 Then we can integrate the timestep using our first order integration, using this equation here to work 56 00:04:29,900 --> 00:04:34,970 out the new state at the time step in the future and then we can probably get the time forward by small 57 00:04:34,970 --> 00:04:38,230 DETI and all we do is just be continuing. 58 00:04:38,750 --> 00:04:45,590 We continuously repeat steps two and three as needed until we get to the end time of the simulation 59 00:04:45,590 --> 00:04:46,070 that we want. 60 00:04:46,990 --> 00:04:49,870 Now, all our integration is a seamless integration method. 61 00:04:49,900 --> 00:04:55,810 There is, however, it is the least accurate, the time step must be small to capture the dynamics 62 00:04:55,810 --> 00:04:57,190 of the nonlinear system. 63 00:04:58,090 --> 00:05:00,150 It can also be numerically unstable. 64 00:05:00,160 --> 00:05:00,730 The solution? 65 00:05:00,730 --> 00:05:04,180 Accuracy gets exponentially, exponentially worse with time. 66 00:05:04,570 --> 00:05:05,680 And this can be a problem. 67 00:05:06,960 --> 00:05:13,210 How about this is the most simplest form of numerical integration and it's easy to understand. 68 00:05:13,230 --> 00:05:14,400 So it's worth considering. 69 00:05:15,450 --> 00:05:21,480 So now let's consider a second order mechanical system and specifically, let's consider the enforced, 70 00:05:21,480 --> 00:05:27,300 which means zero input, second order mass, bring down the system as shown on the slide here. 71 00:05:27,930 --> 00:05:32,120 So this system has a mass that can freely move in and out, back and forth. 72 00:05:32,550 --> 00:05:35,430 It is attached to a spring and a daffner. 73 00:05:35,820 --> 00:05:40,700 And what we want to do, the states of the system are going to be x the position and the velocity. 74 00:05:40,710 --> 00:05:42,630 So how quickly is moving back and forth? 75 00:05:43,500 --> 00:05:47,310 Now, this spring here, it's going to resist any displacement. 76 00:05:47,310 --> 00:05:51,960 So the amount of force that's applied to this mass here is going to be a function of displacement. 77 00:05:51,960 --> 00:05:57,930 So the displacement X Tomsky is going to give you the amount of restoring full force. 78 00:05:58,740 --> 00:06:01,950 This Depner over here is going to resist the velocity. 79 00:06:02,280 --> 00:06:07,980 So the velocity times is Dettman coefficient B is going to give you how much force is being applied 80 00:06:07,980 --> 00:06:08,700 by the dapena. 81 00:06:09,240 --> 00:06:14,890 And the state space system can be represented as a linear first order system as shown here. 82 00:06:15,510 --> 00:06:22,680 So we have the state's velocity and position and this is the system dynamics with the spring coefficient. 83 00:06:22,680 --> 00:06:25,740 We have the Dutney coefficient and we have the mass of the object. 84 00:06:26,370 --> 00:06:28,710 And over here we have the state rates of the system. 85 00:06:28,720 --> 00:06:34,950 So we have velocity dot also known as acceleration and Eckstut also known as velocity. 86 00:06:36,470 --> 00:06:42,650 So these equations here, we can work out the enforced response to the system, the enforced response 87 00:06:42,650 --> 00:06:45,520 just looks at the system dynamics without any input. 88 00:06:45,860 --> 00:06:47,690 So we consider this system over here. 89 00:06:47,720 --> 00:06:55,400 This is a mess that the system we move the mess and amount of X, and from here we let go and it should 90 00:06:55,400 --> 00:06:57,410 spring back to its initial position. 91 00:06:57,770 --> 00:07:03,530 And what we want to do is we want to look at how the position and velocity evolve with time as it's 92 00:07:03,530 --> 00:07:04,070 released. 93 00:07:05,980 --> 00:07:12,040 So using the system dynamics, we can generate the simulated response for a given configuration, so 94 00:07:12,040 --> 00:07:18,550 we'll assume a massive one kilogram adapting coefficient of one, a spring coefficient of 10, we will 95 00:07:18,550 --> 00:07:24,530 assume that the initial velocity zero and we'll give an initial displacement or point two meters. 96 00:07:24,820 --> 00:07:31,090 So that's basically saying we move this mess out here by point ten meters, we hold a stationary and 97 00:07:31,090 --> 00:07:31,840 then we let go. 98 00:07:31,840 --> 00:07:34,030 And then we look at the response of the system. 99 00:07:34,030 --> 00:07:39,910 So we simulate what is going to look like and we can see what the response is over on the results of 100 00:07:39,910 --> 00:07:41,870 the simulation over here, dysgraphia. 101 00:07:42,340 --> 00:07:44,050 So this top graph is a position. 102 00:07:44,590 --> 00:07:46,090 This graph here is a velocity. 103 00:07:46,090 --> 00:07:51,760 So you can see at time zero, we have a displacement or point two and we have a zero velocity. 104 00:07:52,480 --> 00:07:57,810 And as we let it go, the system springs back towards its zero position is neutral position. 105 00:07:58,390 --> 00:08:00,850 So initially we have zero velocity. 106 00:08:00,850 --> 00:08:01,450 We let go. 107 00:08:01,450 --> 00:08:05,950 It starts to accelerate backwards and then it oscillates back and forth. 108 00:08:06,400 --> 00:08:11,080 So we can see here also back and forth, but eventually it converges and dies out. 109 00:08:13,350 --> 00:08:18,480 So this system, the response here was done using all our integration with a really small timestep, 110 00:08:18,480 --> 00:08:26,100 you can see it here is a very small fraction of a second, but this system here can be very susceptible 111 00:08:26,100 --> 00:08:27,620 to the timestep size. 112 00:08:27,630 --> 00:08:33,360 So you can see here, if we use the timestep of point one seconds, we get a totally different spring 113 00:08:33,360 --> 00:08:34,020 response. 114 00:08:34,540 --> 00:08:40,170 So this is because the systems will inherently spring systems, spring damper systems are very stiff, 115 00:08:40,890 --> 00:08:43,010 so they're very susceptible to the timestep. 116 00:08:43,500 --> 00:08:48,140 So if we use a timestep, appoint one second, you can see that they get a very different response. 117 00:08:48,150 --> 00:08:51,850 We actually get a A system that just keeps oscillating back and forth. 118 00:08:52,320 --> 00:08:53,840 So this is incorrect. 119 00:08:54,090 --> 00:08:56,450 This is not a good simulation of the system. 120 00:08:56,460 --> 00:08:58,620 This is because the timestep is too large. 121 00:08:59,280 --> 00:09:05,040 So it is important to select a timestep that small enough to capture all the dynamics when we're using 122 00:09:05,040 --> 00:09:06,870 others first order integration. 123 00:09:07,350 --> 00:09:12,440 And if we were to use a time set even larger, we might actually get an unstable response, which we're 124 00:09:12,720 --> 00:09:17,460 talking about before, which just means that the position and velocity might distort oscillating and 125 00:09:17,460 --> 00:09:19,950 increasing larger and larger and larger in time. 126 00:09:21,380 --> 00:09:26,540 So it is important to select the correct integration method, and if we're using other integration, 127 00:09:26,540 --> 00:09:32,600 it is important to select a very small time period such that we can capture all the system dynamics. 128 00:09:33,940 --> 00:09:40,450 This allows us to be certain that the simulation results are a good approximation of the actual continuous 129 00:09:40,450 --> 00:09:42,210 time dynamics of the system. 130 00:09:44,030 --> 00:09:47,660 Now, in the previous example, we looked at the continuous time simulation. 131 00:09:47,720 --> 00:09:54,050 Now we're going to look at the discrete time simulation since the system dynamics is already in discrete 132 00:09:54,050 --> 00:09:57,080 time, the simulation of the model is very straightforward. 133 00:09:57,350 --> 00:09:59,750 We don't need to do any numerical integration. 134 00:10:00,110 --> 00:10:05,720 We only need to use the recursive solution to step forward in time from the start of the simulation 135 00:10:05,900 --> 00:10:08,390 to the end time or the desired time. 136 00:10:09,330 --> 00:10:15,360 To do this, we first assume an initial condition for the state are ignored and then we step to the 137 00:10:15,360 --> 00:10:18,560 solution using the discrete time equation of the system. 138 00:10:19,230 --> 00:10:26,550 So using our ethanol, we substitute in for X, OK, we substitute in for input at this time, set to 139 00:10:26,550 --> 00:10:30,430 get at our new time step solution for the time set in the future. 140 00:10:30,870 --> 00:10:32,190 We also propagate forward. 141 00:10:32,240 --> 00:10:35,540 Okay, so initially K is going to be equal to zero. 142 00:10:35,550 --> 00:10:36,870 So this is be ignored. 143 00:10:37,410 --> 00:10:40,750 This will be, you know, and we calculate what our X1 is. 144 00:10:41,040 --> 00:10:47,060 We also say X, we also say that K goes from zero to one and then repeat the solution again. 145 00:10:47,170 --> 00:10:48,420 Now we have X one. 146 00:10:48,630 --> 00:10:56,720 You want to work out what our x2 is and we probably get Foord K, so K goes from one plus one to two 147 00:10:56,730 --> 00:10:59,280 and we just keep repeating this step forward in time. 148 00:11:00,930 --> 00:11:05,550 So, again, if we look at the same second order mechanical system as we did for the continuous system, 149 00:11:06,330 --> 00:11:12,570 using our conversion from continuous time to discrete time, using a timestamp or point, one gives 150 00:11:12,570 --> 00:11:14,820 us this value for the F matrix here. 151 00:11:15,630 --> 00:11:22,470 So this basically is a system that it works out how we work out the new velocity and position, given 152 00:11:22,470 --> 00:11:24,330 the current velocity and position. 153 00:11:24,990 --> 00:11:30,990 And if we step forward in time and simulate it, we get a response that looks like this, which is very 154 00:11:31,170 --> 00:11:36,690 which is exactly what we got, or looks very similar as what we got in the continuous time solution. 155 00:11:36,930 --> 00:11:40,950 Except at this time, we're not we're not using a very small time set. 156 00:11:40,950 --> 00:11:42,780 We're just using TimeStep 0.01. 157 00:11:42,780 --> 00:11:50,220 So we get a solution at zero point one point two point three all the way up in this case to ten seconds. 158 00:11:51,870 --> 00:11:56,710 So in this example, we've used a time, seven point five, and we end up with a different F matrix. 159 00:11:57,510 --> 00:12:01,960 This is because we evaluated the Matrix exponential using a different Delta team. 160 00:12:02,670 --> 00:12:04,750 And again, we get a response that looks like this. 161 00:12:05,490 --> 00:12:11,790 Now, this looks now this does not look as similar as a continuous time system as we've had in the past, 162 00:12:12,180 --> 00:12:18,000 but it's still very valid, is still the perfect representation of the continuous time system, except 163 00:12:18,000 --> 00:12:22,750 that we're only calculating the solution at point five increments in time. 164 00:12:23,160 --> 00:12:25,050 So we have a solution for zero seconds. 165 00:12:25,050 --> 00:12:28,300 Point five one one point five two. 166 00:12:28,560 --> 00:12:29,190 So forth. 167 00:12:29,190 --> 00:12:30,220 All the way up to 10. 168 00:12:31,710 --> 00:12:37,200 So this is a perfectly valid representation of the continuous time system, but just in discrete form, 169 00:12:37,410 --> 00:12:41,940 we only care about the solution at point five second increments. 170 00:12:43,840 --> 00:12:50,050 And if we could take it to even more extreme, so if we just criticize the system using a date of two 171 00:12:50,050 --> 00:12:52,320 seconds, we end up with this matrix here. 172 00:12:52,600 --> 00:12:58,230 And this is the response we get when we use the recursive recursive solution and step forward in time. 173 00:12:58,750 --> 00:13:03,370 So we have a solution at zero at two, four, six, eight, 10. 174 00:13:04,120 --> 00:13:09,400 And again, you might look at this and say this looks like nothing like the continuous time solution, 175 00:13:10,000 --> 00:13:15,690 but in fact, it is a perfect representation of the continuous time solution we're only looking at at 176 00:13:15,700 --> 00:13:16,780 different time points. 177 00:13:17,350 --> 00:13:23,440 So if we go back and overlay the continuous time solution on the solution, we end up with a graph that 178 00:13:23,440 --> 00:13:24,120 looks like this. 179 00:13:24,580 --> 00:13:26,970 So the blue line is the continuous time solution. 180 00:13:26,980 --> 00:13:28,240 You can say it's nice and smooth. 181 00:13:29,140 --> 00:13:33,600 The orange line here is the discrete solution with a time set of two. 182 00:13:33,880 --> 00:13:41,890 And what you can see here is that at the zero two four six eight 10, the value of the continuous time 183 00:13:41,890 --> 00:13:44,270 and discrete time system perfectly match. 184 00:13:44,950 --> 00:13:50,410 This is because when we work out this F matrix, we're using the Matrix exponential or the integration. 185 00:13:51,160 --> 00:13:54,430 It perfectly captures all the dynamics in between here. 186 00:13:54,700 --> 00:13:59,380 But we don't need to use all our integration and simulate all the individual points. 187 00:13:59,710 --> 00:14:04,180 We can just jump from one time into the next time and into the next time. 188 00:14:04,630 --> 00:14:08,210 So this is the power of representing things as a discrete system. 189 00:14:08,680 --> 00:14:11,650 It means that we don't have to consider all the intermediate points. 190 00:14:11,920 --> 00:14:16,690 We can just consider the output or the state of the system at a time steps that we actually want to 191 00:14:16,690 --> 00:14:17,500 do something with. 192 00:14:18,250 --> 00:14:23,440 So whether this is looking at calculating a control solution or doing some data fusion, we don't need 193 00:14:23,440 --> 00:14:25,260 to know the continuous system dynamics. 194 00:14:25,270 --> 00:14:31,600 We only need to know the dynamics or the state of the system at certain points in time, which makes 195 00:14:31,600 --> 00:14:35,800 perfect sense to represent the system in discrete form because this is how it's actually going to get 196 00:14:35,800 --> 00:14:37,210 implemented in real life. 197 00:14:39,310 --> 00:14:44,950 So we've covered a very basic introduction into how to simulate continues and simulate discrete systems, 198 00:14:45,230 --> 00:14:51,220 you can see that using the this great system, it perfectly represents the continuous dynamics, except 199 00:14:51,220 --> 00:14:56,350 instead of having to integrate the system with a very small time set, we can capture all that dynamics 200 00:14:56,350 --> 00:14:59,920 inside the state transition matrix, the F and G matrixes. 201 00:15:00,370 --> 00:15:06,130 And we can calculate the system for a given times in the future and we can just step forward in discrete 202 00:15:06,130 --> 00:15:09,610 chunks of time without worrying about all the intermediate time points. 203 00:15:10,300 --> 00:15:16,840 This is the power of discrete time representation and it is the most natural way of representing a system 204 00:15:17,200 --> 00:15:21,460 when we want to do something with it, whether it's control or filtering or anything to do with the 205 00:15:21,460 --> 00:15:23,790 actual implementation in a real world system.