6 EXPANDING-MEMORY (GROWING-MEMORY) POLYNOMIAL FILTERS 6.1 INTRODUCTION The fixed-memory flter described in Chapter 5 has two important disadvan- tages. First, all the data obtained over the last L þ 1 observations have to be stored. This can result in excessive memory requirements in some instances. Second, at each new observation the last L þ 1 data samples have to be reprocessed to obtain the update estimate with no use being made of the previous estimate calculations. This can lead to a large computer load. When these disadvantages are not a problem, the fixed-memory filter would be used generally. Two filters that do not have these two disadvantages are the expanding-memory filter and the fading memory filter. The expanding memory filter is, as discussed in Section 1.2.10 and later in Section 7.6, suitable for track initiation and will be covered in detail in this chapter. The fading memory filter as discussed in Chapter 1 is used for steady state tracking, as is the fixed-memory filter, and will be covered in detail in Chapter 7. Before proceeding it is important to highlight the advantages of the fixed-memory filter. First, if bad data is acquired, the effect on the filter will only last for a finite time because the filter has a finite memory of duration L þ 1; that is, the fixed-memory filter has a finite transient response. Second, fixed-memory filters of short duration have the advantage of allowing simple processor models to be used when the actual process model is complex or even unknown because simple models can be used over short observation intervals. These two advantages are also obtained when using a short memory for the fading-memory filter discussed in Chapter 7. 233 Tracking and Kalman Filtering Made Easy. Eli Brookner Copyright # 1998 John Wiley & Sons, Inc. ISBNs: 0-471-18407-1 (Hardback); 0-471-22419-7 (Electronic) 6.2 EXTRAPOLATION FROM FIXED-MEMORY FILTER RESULTS All the results given in Chapter 5 for the fixed-memory filter apply directly to the expanding-memory filter except now L is increasing with time instead of being fixed. To allow for the variation of L, it is convenient to replace the variable L by n and to have the first observation y nÀL be designated as y 0 . The measurement vector Y ðnÞ of (5.2-1) becomes Y ðnÞ ¼½y n ; y nÀ1 ; .; y 0  T ð6:2-1Þ where n is now an increasing variable. The filter estimate is now based on all the n þ 1 past measurements. All the equations developed in Chapter 5 for the fixed-memory state vector estimate covariance matrix [such as (5.6-4), (5.6-7), and (5.8-1)] and systematic error [such as (5.10-2) and (5.10-3)] apply with L replaced by n. The least-squares polynomial fit equations given by (5.3-11), (5.3-13), and (5.5-3) also applies with L again replaced by n. In this form the smoothing filter has the disadvantage, as already mentioned, of generally not making use of any of the previous estimate calculations in order to come up with the newest estimate calculation based on the latest measurement. An important characteristic of the expanding-memory filter, for which n increases, is that it can be put in a recursive form that allows it to make use of the last estimate plus the newest observation y n to derive the latest estimate with the past measurements ð y 0 ; y 1 ; .; y nÀ1 Þ not being needed. This results in a considerable savings in computation and memory requirements because the last n measurements do not have to be stored, only the most recent state vector estimate, X à n;nÀ1 . This estimate contains all the information needed relative to the past measurements to provide the next least-squares estimate. The next section gives the recursive form of the least-squares estimate orthogonal Legendre filter. 6.3 RECURSIVE FORM It can be shown [5, pp. 348–362] after quite some manipulation that the filter form given by (5.3-13) can be put in the recursive forms of Table 6.3-1 for a one-state predictor when m ¼ 0; 1; 2; 3. The results are given in terms of the scaled state vector Z à nþ1;n [see (5.4-12)]. As indicated before only the last one- state update vector Z à n;nÀ1 has to be remembered to do the update. This is an amazing result. It says that the last one-step update state vector Z à n;nÀ1 of dimension m þ 1 contains all the information about the previous n observations in order to obtain the linear least-squares polynomial fit to the past data Y ðnÀ1Þ and the newest measurement y n . Stated another way, the state vector Z à n;nÀ1 is a sufficient statistic [8, 9, 100]. 234 EXPANDING-MEMORY (GROWING-MEMORY) POLYNOMIAL FILTERS TABLE 6.3-1. Expanding-Memory Polynomial Filter Define z à 0 z à 1 z à 2 z à 3 0 B B B B B B @ 1 C C C C C C A nþ1;n ¼ x à T _ x à T 2 2!  x à T 3 3! _x à 0 B B B B B B @ 1 C C C C C C A nþ1;n " n  y n À z à 0 ÀÁ n;nÀ1 Degree 0 a : z à 0 ÀÁ nþ1;n ¼ z à 0 ÀÁ n;nÀ1 þ 1 n þ 1 " n Degree 1 a : z à 1 ÀÁ nþ1;n ¼ z à 1 ÀÁ n;nÀ1 þ 6 ðn þ 2Þðn þ 1Þ " n z à 0 ÀÁ nþ1;n ¼ z à 0 ÀÁ n;nÀ1 þ z à 1 ÀÁ nþ1;n þ 2ð2n þ 1Þ ðn þ 2Þðn þ 1Þ " n Degree 2 a : z à 2 ÀÁ nþ1;n ¼ z à 2 ÀÁ n;nÀ1 þ 30 ðn þ 3Þðn þ 2Þðn þ 1Þ " n z à 1 ÀÁ nþ1;n ¼ z à 1 ÀÁ n;nÀ1 þ 2 z à 2 ÀÁ nþ1;n þ 18ð2n þ 1Þ ðn þ 3Þðn þ 2Þðn þ 1Þ " n z à 0 ÀÁ nþ1;n ¼ z à 0 ÀÁ n;nÀ1 þðz à 1 Þ nþ1; n À z à 2 ÀÁ nþ1;n þ 3ð3n 2 þ 3n þ 2Þ ðn þ 3Þðn þ 2Þðn þ 1Þ " n Degree 3 a : z à 3 ÀÁ nþ1;n ¼ z à 3 ÀÁ n;nÀ1 þ 140 ðn þ 4Þðn þ 3Þðn þ 2Þðn þ 1Þ " n z à 2 ÀÁ nþ1;n ¼ z à 2 ÀÁ n;nÀ1 þ 3 z à 3 ÀÁ nþ1;n þ 120ð2n þ 1Þ ðn þ 4Þðn þ 3Þðn þ 2Þðn þ 1Þ " n z à 1 ÀÁ nþ1;n ¼ z à 1 ÀÁ n;nÀ1 þ 2 z à 2 ÀÁ nþ1;n À 3 z à 3 ÀÁ nþ1;n þ 20ð6n 2 þ 6n þ 5Þ ðn þ 4Þðn þ 3Þðn þ 2Þðn þ 1Þ " n z à 0 ÀÁ nþ1;n ¼ z à 0 ÀÁ n;nÀ1 þ z à 1 ÀÁ nþ1;n À z à 2 ÀÁ nþ1;n þðz à 3 Þ nþ1;n þ 8ð2n 3 þ 3n 2 þ 7n þ 3Þ ðn þ 4Þðn þ 3Þðn þ 2Þðn þ 1Þ " n a In all cases, n starts at zero. Source: From Morrison [5]. RECURSIVE FORM 235 The filter equations given in Table 6.3-1 for m ¼ 1 are exactly the same as those of the g–h growing-memory filter originally given in Section 1.2.10 for track initiation; compare (1.2-38a) and (1.2-38b) with the expressions for g and h given in Table 6.3-1 for m ¼ 1; see problem 6.5-2. The filter of Section 1.2.10 and Table 6.3-1 for m ¼ 1 are for a target characterized as having a constant- velocity dynamics model. The equations in Table 6.3-1 for m ¼ 2 are for when the target dynamics has a constant acceleration and corresponds to the g–h–k growing-memory filter. The equations for m ¼ 3 are the corresponding equations for when the target dynamics have a constant jerk, that is, a constant rate of change of acceleration. Practically, filters of higher order than m ¼ 3 are not warranted. Beyond jerk are the yank and snatch, respectively, the fourth and fifth derivatives of position. The equation for m ¼ 0 is for a stationary target. In this case the filter estimate of the target position is simply an average of the n þ 1 measurements as it should be; see (4.2-23) and the discussion immediately before it. Thus we have developed the growing-memory g–h filter and its higher and lower order forms from the theory of least-squares estimation. In the next few sections we shall present results relative to the growing-memory filter with respect to its stability, track initiation, estimate variance, and systematic error. 6.4 STABILITY Recursive differential equations such as those of Table 6.3-1 are called stable if any transient responses induced into them die out eventually. (Stated more rigorously, a differential equation is stable if its natural modes, when excited, die out eventually.) It can be shown that all the recursive differential expanding- memory filter equations of Table 6.3-1 are stable. 6.5 TRACK INITIATION The track initiation of the expanding-memory filters of Table 6.3-1 needs an initial estimate of Z à n;nÀ1 for some starting n. If no a prori estimate is available, then the first m þ 1 data points could be used to obtain an estimate for Z à m;mÀ1 , where m is the order of the expanding-memory filter being used. This could be done by simply fitting an mth-order polynomial filter through the first m þ 1 data points, using, for example, the Lagrange interpolation method [5]. However, an easier and better method is available. It turns out that we can pick any arbitrary value for Z à 0;À1 and the growing memory filter will yield the right value for the scaled state vector Z à mþ1;m at time m. In fact the estimate Z à mþ1;m will be least-squares mth-order polynomial fit to the first m þ 1 data samples independent of the value chosen for Z à 0;À1 ; see problems 6.5-1 and 6.5-2. This is what we want. Filters having this property are said to be self- starting. 236 EXPANDING-MEMORY (GROWING-MEMORY) POLYNOMIAL FILTERS 6.6 VARIANCE REDUCTION FACTOR For large n the VRF for the expanding-memory filter can be obtained using (5.8-1) and Tables (5.8-1) and (5.8-2) with L replaced by n. Expressions for the VRF for arbitrary n are given in Table 6.6-1 for the one-step predictor when m ¼ 0, 1, 2, 3. Comparing the one-step predictor variance of Table 6.6-1 for m ¼ 1 with that given in Section 1.2.10 for the growing-memory filter indicates that they are identical, as they should be; see (1.2-42). Also note that the same variance is obtained from (5.6-5) for the least-squares fixed-memory filter. TABLE 6.6-1. VRF for Expanding-Memory One-Step Predictors a (Diagonal Elements of S à nþ1;n ) Degree (m) Output VRF 0 x à nþ1;n 1 ðn þ 1Þ ð1Þ 1 _ x à nþ1;n 12 T 2 ðn þ 2Þ ð3Þ x à nþ1;n 2ð2n þ 3Þ ðn þ 1Þ ð2Þ 2 x à nþ1;n 720 T 4 ðn þ 3Þ ð5Þ _ x à nþ1;n 192n 2 þ 744n þ 684 T 2 ðn þ 3Þ ð5Þ x à nþ1;n 9n 2 þ 27n þ 24 ðn þ 1Þ ð3Þ 3 _x à nþ1;n 100; 800 T 6 ðn þ 4Þ ð7Þ  x à nþ1;n 25; 920n 2 þ 102; 240n þ 95; 040 T 4 ðn þ 4Þ ð7Þ _ x à nþ1;n 1200n 4 þ 10; 200n 3 þ 31; 800n 2 þ 43; 800n þ 23; 200 T 2 ðn þ 4Þ ð7Þ x à nþ1;n 16n 3 þ 72n 2 þ 152n þ 120 ðn þ 1Þ ð4Þ a Recall that x ðmÞ ¼ xðx À 1Þðx À 2ÞÁÁÁðx À m þ 1Þ; see (5.3-4a). Source: (From Morrison [5].) VARIANCE REDUCTION FACTOR 237 6.7 SYSTEMATIC ERRORS Because the systematic error of the expanding-memory filter grows as n grows (see Section 5.10), this filter cannot be cycled indefinitely. The fixed-memory filter of Chapter 5 and g–h fading-memory filter of Section 1.2.6 do not have this problem. The g–h fading-memory filter and its higher order forms are developed in the next section from the least-squares estimate theory results developed in Section 4.1. 238 EXPANDING-MEMORY (GROWING-MEMORY) POLYNOMIAL FILTERS . memory for the fading-memory filter discussed in Chapter 7. 233 Tracking and Kalman Filtering Made Easy. Eli Brookner Copyright # 1998 John Wiley & Sons,. disadvantages are the expanding-memory filter and the fading memory filter. The expanding memory filter is, as discussed in Section 1.2.10 and later in Section