FADING-MEMORY (DISCOUNTED LEAST-SQUARES) FILTER DISCOUNTED LEAST-SQUARES ESTIMATE The fading-memory filter introduced in Chapter 1, is similar to the fixedmemory filter in that it has essentially a finite memory and is used for tracking a target in steady state. As indicated in Section , for the fading-memory filter the data vector is semi-infinite and given by Y ðnÞ ¼ ½ y n ; y nÀ1 ; . . . T ð7:1-1Þ The filter realizes essentially finite memory for this semi-infinite data set by having, as indicated in section , a fading-memory. As for the case of the fixed-memory filter in Chapter 5,. | 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 7 FADING-MEMORY DISCOUNTED LEAST-SQUARES FILTER DISCOUNTED LEAST-SQUARES ESTIMATE The fading-memory filter introduced in Chapter 1 is similar to the fixed-memory filter in that it has essentially a finite memory and is used for tracking a target in steady state. As indicated in Section for the fading-memory filter the data vector is semi-infinite and given by Y n yn yn-1 T 7-1-1 The filter realizes essentially finite memory for this semi-infinite data set by having as indicated in section a fading-memory. As for the case of the fixed-memory filter in Chapter 5 we now want to fit a polynomial p P r n see . to the semi-infinite data set given by . Here however it is essential that the old stale data not play as great a role in determining the polynomial fit to the data because we now has a semi-infinite set of measurements. For example if the latest measurement is at time n and the target made a turn at data sample n-10 then we do not want the samples prior to the n 10 affecting the polynomial fit as much. The least-squares polynomial fit for the fixed-memory filter minimized the sum of the squares of the errors given by . If we applied this criteria to our filter then the same importance or weight would be given an error resulting from the most recent measurement as well as one resulting for an old measurement. To circumvent this undesirable feature we now weight the error due to the old data less than that due to recent data. This is achieved using a discounted least-squares weighting as done in that is we 239 240 FADING-MEMORY DISCOUNTED LEAST-SQUARES FILTER minimize en j ijn-r p r n 3 7-1-2 r 0 where here positive r is now running backward in time and 0 e 1 The parameter d here determines the discounting of the old data errors as done in Section . For the most recent
