1 00:00:03,790 --> 00:00:08,890 In this video, we are going to look at some implementation notes, so we have covered the theory of 2 00:00:08,920 --> 00:00:13,270 the common filter, but there's a few little things that we should know when we come to implementing 3 00:00:13,270 --> 00:00:14,070 the common filter. 4 00:00:14,320 --> 00:00:17,750 It is not always as straightforward as the theory makes us believe. 5 00:00:18,100 --> 00:00:22,060 So we're going to cover a few things that we should consider when we're implementing the common filter 6 00:00:22,060 --> 00:00:23,830 in a real world applications. 7 00:00:24,890 --> 00:00:31,430 The equations used the covariance matrix after the update process assumes that the computational implementation 8 00:00:31,430 --> 00:00:33,680 has a perfect numerical accuracy. 9 00:00:34,520 --> 00:00:37,390 So computers do not have infinite precision or accuracy. 10 00:00:37,520 --> 00:00:42,830 They represent floating point numbers only to a certain level of number of significant figures. 11 00:00:43,160 --> 00:00:48,410 Numerical errors can cause a covariance matrix to lose its covariance matrix properties such as being 12 00:00:48,410 --> 00:00:51,280 symmetrical and positive, positive, semi definite. 13 00:00:52,250 --> 00:00:57,560 When this happens, it can cause the filter to fail or the calculations to crash because we're trying 14 00:00:57,560 --> 00:01:00,200 to do the inverse of a non-convertible matrix. 15 00:01:00,860 --> 00:01:02,690 So these issues become very critical. 16 00:01:02,690 --> 00:01:06,890 When we start running the common Philidor for a long period of time, these numerical inaccuracies can 17 00:01:06,890 --> 00:01:09,680 build up, which then can cause the filter to fail. 18 00:01:10,530 --> 00:01:13,630 So numerical inadequacies can cause stability issues. 19 00:01:14,420 --> 00:01:19,100 So the equations shown on the slide here is a covariance matrix update for the update step. 20 00:01:19,550 --> 00:01:22,370 So this is only one way that we can represent this equation. 21 00:01:23,120 --> 00:01:28,360 Another way is to use a Joseph stabilized version of the common Thawra update equation. 22 00:01:28,670 --> 00:01:32,540 So this version shown on Slide now is more stable and more robust. 23 00:01:32,900 --> 00:01:36,790 Yes, it does take more calculations, but it's less prone to numerical errors. 24 00:01:37,100 --> 00:01:42,080 So when we're implementing the common filter and we come into stability issues, one potential solution 25 00:01:42,080 --> 00:01:44,930 is to use this form of the covariance matrix update. 26 00:01:45,950 --> 00:01:50,550 Theoretically, the common filter is an attractive choice for estimation and data fusion. 27 00:01:50,990 --> 00:01:56,330 However, when the common filter is implemented in real systems, it may not always work, even though 28 00:01:56,330 --> 00:01:58,160 the theory and equations are correct. 29 00:01:58,580 --> 00:02:01,460 Now these issues can be caused by the finite position. 30 00:02:01,460 --> 00:02:06,980 So the computational limits which we've covered in the previous slide and modelling errors which come 31 00:02:06,980 --> 00:02:08,920 up from incorrect system modeling. 32 00:02:08,930 --> 00:02:14,080 So we have incorrect matrixes or or we make the wrong assumptions inside the process model such as we 33 00:02:14,090 --> 00:02:17,540 assuming the noise is Gaussian or it has some properties. 34 00:02:18,950 --> 00:02:25,820 So potential solutions to these problems, if we encounter them first, we might increase the computational 35 00:02:25,820 --> 00:02:26,330 precision. 36 00:02:26,330 --> 00:02:31,490 So instead of using 32 bit numbers, we might bump it up to using 64 bit numbers so we can represent 37 00:02:31,730 --> 00:02:35,510 more significant figures or more decimal places inside the calculations. 38 00:02:36,720 --> 00:02:41,730 We also might use a form of square root, common filtering, so we didn't cover this in this course, 39 00:02:42,030 --> 00:02:47,450 but basically we break down the covariance matrix into a multiplication of two matrices. 40 00:02:48,270 --> 00:02:53,260 And when we're using this square root, common filtering form, it means the calculations are more stable. 41 00:02:54,900 --> 00:02:59,460 We can also force the covariance matrix to be symmetrical after each update step. 42 00:02:59,700 --> 00:03:03,690 There's a few different ways of doing this, but if we force it to be symmetrical after each step, 43 00:03:03,810 --> 00:03:07,170 there's a greater chance the filter will stay stable for longer. 44 00:03:08,250 --> 00:03:13,860 We also want to initialize the covariance matrix as best as possible to stop large jumps in the matrix 45 00:03:13,860 --> 00:03:14,880 during operations. 46 00:03:15,240 --> 00:03:20,910 So when we initialize a matrix, ideally we don't want to initialize if very large numbers sort of telling 47 00:03:20,910 --> 00:03:22,400 the filter that we're really uncertain. 48 00:03:22,650 --> 00:03:26,420 We want to set the uncertainty to be the correct level of uncertainty. 49 00:03:26,430 --> 00:03:28,590 So we should have a reasonable estimate. 50 00:03:28,620 --> 00:03:29,550 Well, that should be. 51 00:03:31,740 --> 00:03:38,550 We can also use a form of a fading memory filter, so as the common thread runs, the covariance matrix 52 00:03:38,550 --> 00:03:40,240 captures all these correlations. 53 00:03:40,240 --> 00:03:46,110 For all the past historical measurements and properties of the system, we can use a form of fading 54 00:03:46,110 --> 00:03:49,370 memory which causes the filter to forget some of the past information. 55 00:03:49,770 --> 00:03:51,060 So this can help for issues. 56 00:03:51,060 --> 00:03:57,930 When we have incorrect modelling, it can start to forget what it's done in the past to help it update 57 00:03:57,930 --> 00:03:58,670 in the future. 58 00:04:00,270 --> 00:04:04,950 And we've also already covered this one, we can also increase the price, this model, Nonoy, so increasing 59 00:04:04,950 --> 00:04:09,420 the size of the values inside the key metrics to account for any modelling errors. 60 00:04:11,410 --> 00:04:16,330 So these are some of the strategies you can use if you start to run in problems when you're implementing 61 00:04:16,330 --> 00:04:19,680 the common filter in real life to solve real world problems.