1 00:00:03,770 --> 00:00:08,290 In this video, we are going to cover the prediction step equations for the extended common filter. 2 00:00:08,690 --> 00:00:14,180 We will cover all the equations in a summary form, and then in later, videos will go back and look 3 00:00:14,180 --> 00:00:16,280 at the derivation of these equations. 4 00:00:17,240 --> 00:00:22,910 So to start off, we begin the estimation step of the extended coming field up with the initial condition 5 00:00:23,210 --> 00:00:27,860 X not which is the best estimate of the current initial state of the system. 6 00:00:29,730 --> 00:00:33,880 We want to probably get Tom Ford to find our estimate at TimeStep one. 7 00:00:33,900 --> 00:00:39,870 So we want to find out estimated state X1, which is just the estimated value of the current system 8 00:00:39,870 --> 00:00:44,970 at timeworn, so we can use the nonlinear motion model of the system to do this. 9 00:00:45,770 --> 00:00:51,750 So using the state transition matrix, if we can substitute in our current state for export our current 10 00:00:51,750 --> 00:00:58,320 control vector, if you want and we can set prices will always be zero because we estimated we want 11 00:00:58,320 --> 00:01:02,630 to estimate the mean to get the predicted machine, which is going to be our X1 here. 12 00:01:03,360 --> 00:01:07,330 We can write this in general as this equation here. 13 00:01:07,890 --> 00:01:13,950 So we basically take our state transition matrix f subject in our current state for time K minus one 14 00:01:14,880 --> 00:01:21,750 substitute in our current control, our UK we set the price is one and zero and we can use this to calculate 15 00:01:21,750 --> 00:01:25,260 our current state estimate ICSC for the current timestep. 16 00:01:28,240 --> 00:01:33,280 So this is the prediction step or the time update step of the extent to come, if we use the process 17 00:01:33,280 --> 00:01:39,310 model of the system, we want to estimate to predict the estimate a state which is just going to be 18 00:01:39,310 --> 00:01:43,920 the main of the Gaussian of the probability density function of the state estimation error. 19 00:01:44,140 --> 00:01:46,480 And we use this to predict it forward in time. 20 00:01:48,920 --> 00:01:54,470 So this equation is the state prediction step for the external karma filter, and it can be simply written 21 00:01:54,470 --> 00:01:54,940 like this. 22 00:01:56,210 --> 00:01:58,520 So exactly the same as linear common. 23 00:01:58,520 --> 00:02:04,630 Philidor, what this equation is doing is taking the current best estimate state for the previous timestep. 24 00:02:04,680 --> 00:02:12,650 So we have the set, we have the A posterior, I estimate our X plus for K minus one, which is also 25 00:02:12,650 --> 00:02:19,010 just a shorthand way of writing our estimate state times of K minus one, given all the information 26 00:02:19,010 --> 00:02:20,210 up to K minus one. 27 00:02:20,510 --> 00:02:26,000 So this is given all the information up to the current timestep, which can be probabilistically written 28 00:02:26,000 --> 00:02:28,200 as the expectation of Raida. 29 00:02:28,220 --> 00:02:36,920 So this parameter here is just the expected value of the state at K minus one, given all the measurements 30 00:02:36,920 --> 00:02:38,030 up to minus one. 31 00:02:38,690 --> 00:02:47,060 So we take this parameter here and we use our process model S to transform this into our apriority state 32 00:02:47,060 --> 00:02:55,010 for the constant timestep K so we can say here we transform our previous timestep into our current timestep, 33 00:02:55,580 --> 00:02:58,220 which is a shorthand way of writing it this way. 34 00:02:58,220 --> 00:02:59,750 Again, sorry to go over here. 35 00:02:59,750 --> 00:03:04,490 We have the estimated state at times of K given all the information up to this one. 36 00:03:05,390 --> 00:03:09,010 So you can say that we've probably gone forward in time said he's came on this one. 37 00:03:09,230 --> 00:03:14,080 This is K, but we haven't included we haven't included any extra information. 38 00:03:14,360 --> 00:03:17,660 So all the measurements are still for K minus one came on this one. 39 00:03:18,170 --> 00:03:20,320 So this is why this is the A. 40 00:03:20,720 --> 00:03:26,790 This is this is why this is the a priori state, because we haven't included any any extra information. 41 00:03:27,440 --> 00:03:30,190 And again, this can be written in the expectation form. 42 00:03:30,500 --> 00:03:36,500 We're looking at the expected value of the state of the system, like given all the information up to 43 00:03:36,500 --> 00:03:37,320 came on this one. 44 00:03:37,700 --> 00:03:42,530 So we're still using the same set of information in the we've only predicted the state forward. 45 00:03:45,080 --> 00:03:49,730 So we've looked at the prediction set for the state of the system or the main of the estimate. 46 00:03:50,210 --> 00:03:53,250 We also want to propagate the covariance forward in time as well. 47 00:03:53,630 --> 00:03:59,090 So to begin with, just like in the linear coming up, we start with the initial covariance matrix peno, 48 00:03:59,420 --> 00:04:03,380 which is the initial Aruca variance for the state estimate ex not 49 00:04:06,170 --> 00:04:06,380 so. 50 00:04:06,380 --> 00:04:11,430 The definition of Pinart again is just going to be the expected value of the estimation error. 51 00:04:11,450 --> 00:04:17,300 So we can say this time, here is the true state minus the estimated state. 52 00:04:17,330 --> 00:04:20,370 So this is the estimation error and then we get the square of that. 53 00:04:20,390 --> 00:04:26,540 So this term here, multiply the transpose of the same time is going to give us the estimation error 54 00:04:26,540 --> 00:04:27,090 squared. 55 00:04:27,410 --> 00:04:29,150 So this is the covariance matrix. 56 00:04:32,280 --> 00:04:38,370 So given Peno, we want to propagate forward in time to find out Paswan so we can do this using the 57 00:04:38,370 --> 00:04:43,930 linnear transform just like in the Linnear midfielder, but said the transformations are going to be 58 00:04:43,930 --> 00:04:46,510 the linear approximation of the non-linear model. 59 00:04:46,950 --> 00:04:49,320 So that gives us this equation shown here. 60 00:04:50,160 --> 00:04:55,860 So this looks very similar to the linear shown below, except now we're going to have these new matrixes 61 00:04:55,860 --> 00:04:59,070 here and replace of the F and L Matrixes. 62 00:04:59,640 --> 00:05:03,090 And these matrices are actually called Jacobean matrixes. 63 00:05:03,300 --> 00:05:06,600 So these are a linear approximation of the nonlinear model. 64 00:05:07,410 --> 00:05:14,070 So this Eucumbene here is the Jacobean of the process model F with respect to the state X, and you 65 00:05:14,070 --> 00:05:20,280 can see again we have this this Jacobean matrix here and it's transposed just like we have no f f transpose 66 00:05:20,940 --> 00:05:24,720 and we have the Jacobean of the process model with respect to the noise. 67 00:05:25,500 --> 00:05:31,170 And again, we have that matrix here and it's transpose over here just like we have the L Matrix and 68 00:05:31,170 --> 00:05:32,980 the Matrix transpose over here. 69 00:05:33,630 --> 00:05:41,310 So basically we've changed the linear equations and we replace the linear matrix with these linear approximations 70 00:05:41,310 --> 00:05:42,480 of the nonlinear model. 71 00:05:46,410 --> 00:05:52,860 So in general, we can write the predictions for the covariance in this form here, so we have the covariance 72 00:05:52,860 --> 00:05:58,830 matrix for the error for the previous timestep, and we use it to transform it forward using our Deucalion 73 00:05:58,830 --> 00:06:04,920 matrix of the process model with respect to the state and to get our predicted state covariance for 74 00:06:04,920 --> 00:06:09,100 TimeStep came and we add onto the contribution for the noise. 75 00:06:09,120 --> 00:06:15,900 So we have our noise covariance matrix here and we use the and we transform it into the state space 76 00:06:16,380 --> 00:06:20,510 by using the Jakov in our process model with respect to the process model noise. 77 00:06:21,030 --> 00:06:22,810 So we use the transformation operator. 78 00:06:23,280 --> 00:06:29,210 So this basically gives us the prediction step for the covariance for the extended common filter. 79 00:06:29,850 --> 00:06:33,860 And these Yukari matrices here can be written in this form here. 80 00:06:33,870 --> 00:06:40,940 So the Jacobean of the process model F respect to X is just the partial derivative of the process model 81 00:06:40,950 --> 00:06:49,110 with respect to X, but we evaluate it at the current best estimate, so we evaluate it at our X, K 82 00:06:49,110 --> 00:06:52,410 minus one best estimate and very similar. 83 00:06:52,410 --> 00:06:58,110 We have our Jacobean for the process model with respect to noise, which is going to be the derivative 84 00:06:58,110 --> 00:07:02,250 of a process model F with respect to our noise vector. 85 00:07:03,060 --> 00:07:05,930 And again, it is going to be evaluated at our best estimate. 86 00:07:07,380 --> 00:07:13,680 So as time steps forward and times changes, we're going to have to keep revalidate reevaluating these 87 00:07:13,680 --> 00:07:18,840 Jacobins because these Jacobins are the linear approximation of the nonlinear system. 88 00:07:19,050 --> 00:07:22,770 As the system changes, we have to keep updating our linear ization point. 89 00:07:23,010 --> 00:07:30,120 So we have to keep updating these matrices here so that you can be Matrix's ethics. 90 00:07:30,120 --> 00:07:36,410 And if you a very similar to the Matrix in the linear coming filter and the matrix in the linear coming 91 00:07:36,420 --> 00:07:42,480 field, they have the same job, except now we're changing the linear model with as time goes on. 92 00:07:42,630 --> 00:07:47,580 So we have to keep updating them using this linear approximation of the nonlinear system. 93 00:07:50,010 --> 00:07:55,770 So we can think of the covariance prediction step in this form here, we start off with the a posterior 94 00:07:55,770 --> 00:07:57,800 eye Erica variance for K minus one. 95 00:07:58,110 --> 00:08:04,530 So we have our covariance matrix for TOMPSETT came on this one, given all the information up to came 96 00:08:04,530 --> 00:08:10,170 on this one, which is another way of writing it out in the expectation of radiophone. 97 00:08:10,410 --> 00:08:15,510 So we have the estimated state error, have the estimated state error over here again. 98 00:08:15,510 --> 00:08:20,790 So it's estimating state R-squared, given all the information or the measurements up to came on this 99 00:08:20,790 --> 00:08:21,010 one. 100 00:08:21,450 --> 00:08:25,890 So this is the kind of going back to the expectation operator. 101 00:08:25,890 --> 00:08:28,400 This is the covariance of the state estimation error. 102 00:08:30,270 --> 00:08:36,930 We can use our Jacobean of the system to transform this process, to transform this covariance matrix 103 00:08:37,350 --> 00:08:44,880 and propagate it forward in time to find the current covariance matrix so we can find out P minus K, 104 00:08:44,880 --> 00:08:50,160 which is going to be various metrics for the current TimeStep, given all the information up to the 105 00:08:50,160 --> 00:08:51,990 previous TimeStep K minus one. 106 00:08:52,680 --> 00:08:58,490 So again, just like in our transformation with me, we've increased the timestep for the prediction. 107 00:08:58,500 --> 00:09:02,640 So we've gone from minus one to carry, but we've kept the same amount of information. 108 00:09:02,640 --> 00:09:07,520 So we're only used in measurements up to Cayman's one K minus one. 109 00:09:08,220 --> 00:09:10,470 So you can say it again in this estimation. 110 00:09:11,130 --> 00:09:15,960 So I can tell you again in the expectation operator again, another way of expressing what the P matrix 111 00:09:15,960 --> 00:09:16,260 is. 112 00:09:16,800 --> 00:09:23,970 We've increased our prediction to K, but we've kept out all the information in the same set that we're 113 00:09:23,970 --> 00:09:24,510 still using. 114 00:09:24,510 --> 00:09:27,060 al-Said came on this one, said K minus one. 115 00:09:31,950 --> 00:09:34,930 So for comparison, when you look at the process model predictions. 116 00:09:35,190 --> 00:09:41,130 So that's the prediction for the state and we can compare it between the Lynnae Common Filter and extend 117 00:09:41,130 --> 00:09:41,880 the common filter. 118 00:09:42,540 --> 00:09:44,180 So Interlinear come and fill it up. 119 00:09:44,520 --> 00:09:48,990 We use the linear system model F and G to propagate and step forward. 120 00:09:48,990 --> 00:09:52,590 So we had the input and we had demesne for the previous timestep. 121 00:09:52,980 --> 00:09:58,650 We put them into this equation here and we use this process model to predict forward to get the prediction 122 00:09:58,650 --> 00:09:59,670 at the current TOMPSETT. 123 00:10:00,870 --> 00:10:03,710 Now when we use the extended common here is very similar. 124 00:10:03,900 --> 00:10:05,850 We have the input for the current timestep. 125 00:10:06,090 --> 00:10:11,460 We have the state estimate for the previous times that we put them into the nonlinear process model 126 00:10:11,850 --> 00:10:15,290 and we use that to predict forward to get a current prediction at times. 127 00:10:15,390 --> 00:10:20,670 Okay, so the only difference between the two steps here is that we're using the full nonlinear system 128 00:10:20,670 --> 00:10:20,900 here. 129 00:10:21,300 --> 00:10:23,820 While up here we were using the linear system. 130 00:10:26,440 --> 00:10:32,200 Now, if we look at the process model predictions that for the government, so looking at the linnear 131 00:10:32,200 --> 00:10:35,250 coming, shouldn't we use this linear transformation here? 132 00:10:35,470 --> 00:10:42,730 We used the previous covariance p we substituted it in using the state transition matrixes and state 133 00:10:43,150 --> 00:10:43,810 noise matrix. 134 00:10:43,990 --> 00:10:51,130 So if an elementary ccis and low as well as our noise covariance matrix, we put it into this equation 135 00:10:51,130 --> 00:10:56,870 here and we transformed from the previous timestep to the current time step to get our covariance matrix 136 00:10:56,880 --> 00:10:59,920 for K the external incoming filter. 137 00:11:00,070 --> 00:11:07,000 Again, very similar, except now we've changed our F and O matrixes to be these giacobbe matrixes. 138 00:11:07,000 --> 00:11:12,910 So Jacobean of the state transition matrix with respect to X and the Jacobean of the state transition 139 00:11:12,910 --> 00:11:18,850 matrix, respect to the noise we take in our previous covariance predictor forward to get the current 140 00:11:18,850 --> 00:11:19,540 covariance. 141 00:11:20,080 --> 00:11:25,000 And every time we do this, every time we step forward in time, we have to re evaluate the psychobilly 142 00:11:25,000 --> 00:11:28,300 matrixes as we've changed our linear ization point. 143 00:11:28,300 --> 00:11:34,300 And Jacobean is a linear approximation, which is only valid at a given state of the system, the linear 144 00:11:34,300 --> 00:11:34,950 regression point. 145 00:11:35,410 --> 00:11:39,430 So we have to keep updating that as we estimate as the time goes on. 146 00:11:40,900 --> 00:11:44,830 So in summary, we have the prediction step equation for the extended common field. 147 00:11:45,400 --> 00:11:49,490 We have the prediction state estimate, which is this equation here. 148 00:11:49,510 --> 00:11:50,460 So we've covered this. 149 00:11:51,040 --> 00:11:57,580 This takes this uses the first model F takes in the previous state and the current control to predict 150 00:11:57,580 --> 00:11:59,050 forward to get the current state. 151 00:12:00,190 --> 00:12:02,630 We also have the predicted covariance equation estimate. 152 00:12:02,980 --> 00:12:09,580 So this takes in the previous covariance uses the Jacobean of the system and the Jacobin of the noise 153 00:12:09,880 --> 00:12:13,670 and predicts forward from the previous timestep K minus one to the current times. 154 00:12:13,700 --> 00:12:14,220 Okay. 155 00:12:16,750 --> 00:12:21,280 And finally, we have our definition of the Jacobins, the actual process. 156 00:12:21,290 --> 00:12:22,330 And what does this mean? 157 00:12:22,330 --> 00:12:23,680 We've got to cover in a little bit here. 158 00:12:23,920 --> 00:12:29,230 But basically the Jakobsson is just the partial derivative of the function or the state transition matrix 159 00:12:29,350 --> 00:12:35,530 respect to the current state evaluated at something linear as Asian Point or the current estimate. 160 00:12:36,400 --> 00:12:40,920 And we have the same thing for the noise giacobbe into the Caribbean of the state transition metrics. 161 00:12:40,930 --> 00:12:42,100 But we respect the noise. 162 00:12:42,110 --> 00:12:44,870 So the difference here is ones with respect to noise. 163 00:12:44,890 --> 00:12:52,050 This one is with respect to the state so that you gravimetric these can be thought of a linear transformation. 164 00:12:52,060 --> 00:12:58,300 So instead of using the only system we can use, this linear approximation of the system, however, 165 00:12:58,300 --> 00:13:05,650 is only valid at the state that it is linear rise about and is only valid at small probation's around 166 00:13:05,650 --> 00:13:06,240 that state.