I I I ING THE KALMAN FILTER IN TERMS OF CORRELATION COEFFICXENTS ETC (U) fl:la5sifiejyjjjj ELLPGvTR ~~~~~ S8!A!E0o0 j

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1 - - - * NAVAL RESEARCH LAB WASHINGTON D C FIG 12/i ING THE KALMAN FILTER IN TERMS OF CORRELATION COEFFICXENTS ETC (U) fl:la5sifiejyjjjj ELLPGvTR ~~~~~ S8!A!E0o0 j NL I I I a I

2 ~~~~ 1 10 ~ 28 IHH ~ 5 ~~~~ L ~~ 1 1 ~ ~ HD ~ 1_. L ~ 11Iuu.25 UW ~ 1IL 1*1M ~ L NATIONAL BUREAU OF STANDARDS MICROCOPY RESOLUTIO I TEST CHSR T

3 NRL ~~~~ J u;9 ~~~ Updating the Kalman Filter in Terms of Correlation Coefficients and Stand ard Deviations B. H. CANTRELL and G. V. TRUNK Radar Analysis Staff Radar Division C-, May 17, 1978 DDC U JUL 1 197e 5 ~ 4 I C$ ~~ q NAVAL RESEARCH LABORATORY Washington, D.C. Approved for public release; distr ibutio n unlimited.

4 . ~~~~~~~ ~~~~~~~~~~ ~~~ -- - ~~~~~~~~~~ SECURITY CLASSIFICA TION OF THIS PAGE (IS?,.., 0.i. EnI.,.d) I R EPORT NUMBER REPOJtj DOCUMENTATION PAGE NRL Report ~~~~~~~~~~~~~~~~~~~~~~~~ READ INSTRUCTIONS BEFORE COMPLETING FORM GOVT ACCESSION NO. 3. RECI PIENT S CATALOG NUMBER 12. Interim report on a continuing NRL Problem ~~~~~~~~~ (.d ScbIIII S PE OF REPORT & PERIOD COVERED ~~~~~ ~ I ~ ~~~~~~ THE I ALMAN FILTER I ~ RMS OF ~ CORRELATION COEFFICIENTS AND ~~~~~~~~~~~~~ DEVIATIONS,. ~~~~~~~~~~~ RUNG ORG. REPORT N ~~~~~~~ R 7 A ~ ) T$ ~~ B(.) S CONTRACT OR GRANT NUMBER(.). -- B.H/CantreL1 ~~~ G.V./l runk - S PERFORMING ORGANIZATION NAME AND ADDRESS._._ ~ 10. PROGRAM ELEMENT. PROJECT. TASK AREA & WORK UNIT NUMBERS Naval Research Laboratory Washington, D.C /.. NRL Problem R02.97 Project RR Program Element 61153N-21 I I. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Department of the Navy May 17, 1978 Office of Naval Research s. w Oke ww ~ AG ~ S Arlington, VA MB i ,...S..UA&IE & A OORESS(l1 dllf., I ((0.,. ConIrollInS Office) l5. ~ ~~ 5f ~~ U 1! UNCLASSIFIED ~ - 1 / / r (,! ISI. R.pO,I) ~~~~~~~~~~~ IS. JiO I j ~ (of Thu. c pe(i) DECLASSIFICATION/DOWNGRADING SCHEDULE Approved for public release; distribution unlimited IT DISTR ~ $L ~ TION STATEM ENT (Of (he.b.i,.c(.,i.e.d In Block 20. i dli. Ho.. R.po.t) / / y Ia SuPPL EMENTA RY NOTES I K EY WORDS (Co,,Iinoe on ~~~ lters Kalman filters Tracking Optimum filters SRIF (square-root information filter) filtering Id. I n.c...a,y ~~ d ld.nhi fy by block n. ~~.b 1) 20 AWSTR ACT (ConiiflOe 0.,eo, ~ e cud. I n.c...ry a.d Id.nuIIy by block noo. b..) A factorization method of updating the Kalman filter is examined. The covariances are factored in terms of correlation coefficients and standard deviations. The covariances of the Kalman filter are then updated in terms of the factors. The technique is similar in principle to Carison s UDU method but differs in the covartance factorization. DD ~~~~~~~~ 1473 EDITION OF I PiOV SSISOSSOLEIE S/N GS02-0S4 ~~ 650S I SECURITY CLASSIFICA TION OF THIS PAGE (531., Dci. lni., sd)

5 CONTBNTS INTRODUCTION 1 ONE-BY- ONE UPDATING OF THE KALMAN FILTER 1 UPDATING OF THE KALMAN FILTER BY UPDATING THE CORRELATION COEFFICIENTS AND STANDARD DEVIATIONS 5 SUMMARY 7 REFERENCES 7 ~~~ ESSION ~~~~~~ NTIS Sect ofl ~ ~~ : ~~~ on 0 DNiNNOUMC ~~ 0 ~ JSTIFICATION / BY U ~ T1IO1IONIAYAILABILUY codes DISL ~~~~~~,. oiid/0r S ~~~~~ U ±TT BLkI*L.NoT 7IU~ffD

6 UPDATING THE KALMAN FILTER IN TERMS OF CORRELATION COEFFICiENTS AND STANDARD DEVIATIONS INTRODUCTION The problem of estimating a set of parameters or variables from a set of measurements has long been a problem of interest. Kalman in the early sixties provided a simple recursive estimation procedure by introducing the concept of state and state transition. This procedure in some instances provided simpler implementation than batching techniques. Since Kalman s work a number of numerical procedures have been developed. An excellent account of these procedures as well as historical notes can be found in Bierman s book [1]. Two basic techniques described in Ref. 1 are the SRIF filter (square-root information filter) and the UDU filter. (Actually UDU should be written UDU, where D is a diagonal matrix, U is an upper triangular matrix, and U is the transpose.) In both cases the data are prewhitened using Cholesky factorization. The SRIF filter is then based on the Householder transform. The UDU filter is based on a Cholesky factorization of the smoothed covariance update using one measurement at a time and th ~ modified Gram-Schmidt for updating the predicted covariances. These numerical technir ~ u.os are claimed to have better numerical stability than use of the Kalman filter equations directly and may be more amenable to hardware implementation. In this report another factorization is considered for updating the Kalman filter. The factors representing the correlation coefficients and standard deviations are updated in the Kalman filter rather than the factors themselves. Before these results are obtained a method of updating the Kalman filter using only matrix multiplications and additions is first described. ONE-BY-ONE UPDATING OF THE KALMAN FILTER The Kalman filter is obtainu ~ d from modeling the process as state equations, defining a measurement procedure, and best estimating the states of the systems. The state equation and measurement process are defined as and X(h) = 4 ~ (k) X(k - 1) + 1 (k)w(k) Xm (k) H(k)X(k) 4 V(k), where it is desired to best estimate the n-by-i state vector X(k). The remaining quantities are an n-by-n state transition matrix 4(k), an n-by-p matrix r(k), an n-by-rn measurement matrix H(k), an rn-by-i measurement vector Xm (k), and W(k) and V(k), which are Gaussian noises with the following properties: Manuscri pt submitted March 17, L ~~ ~~~

7 - ~~~~~ -- ~~~~~~~~ -- and CANTRELL AND TRUNK E[ W(k)] =0, EI W(k) W (J) ~ = S(k)ö 11,, EE V(k)1 = 0, E[ V(k)V (jf l = E[ W(k)V(j) ) =0, where is 1 when j = k and is 0 otherwise. The covanance matrices S(k) and Q(k) are of dimension p by p and m by m respectively. The best estimate of X(k), denoted by X (k) in the standard Kalman-filter format, is where K(k) is the filter gain, given by X(k) = X(k ) + K(k)[ X 01 (k) H(k)X( k) J, K (k) = in which P (k) is the smoothed covariance matrix, given by P(k) is the predicted covariance matrix, and The prediction is P ~~ (k) = P (k)+ H (k)q 1 (k)h(k). P (k + 1) = ~ I (k + i )P (k) ~ I ~ (k + i) + [ (k + 1 )S (k + i)[ (k + 1). X(k + 1) = b(k + 1)X(k). The filter operates in a predict and correct fashion. The Kalman fi lter can be written in a slightly different form by prewhitening the measurement noise. This is a standard practice which can be achieved using Cholesky factorization as described in Ref. 1. The covariance Q(k) of the noise V(k) is factored into the form of or Q(k) = LEL (la) Q ~~ (k) = (L ~~ 1 E L 1, (i b) 2 - ~~~~ -~~~~~~~ ~~~~. -. ~~-- ~~

8 NRL REPORT 8229 where L is a lower triangular matrix with l s on the diagonal and E is a diagonal matrix with diagonal elements e1. The algorithm for obtaining L and E is ~ JJ =1 = q 3 ~, k j + 1,..., n, and ~~~~~~~~~~~~~~~ k=j +1,..., n and i k,..., n. With use of equations (la) and (ib) the Kalman filter can be rewritten as and X(k) = X(k) + K(k) [Z, ~ (k) ~ (k) X(k) J, (2) K(k) = P (k) ii (k)e ~~ (k), (3) P ~ (k) = P (h) + ~ t (k)e ~ (k) ft(k), (4) P(k + i) = (l (k + i )P (k)1 ~ (k + 1) + l (k + 1)S(! ~ + 1)1 (k + 1), (5) where + 1) ~ I (k + 1)X(k), (6) Z 01 (k) = L ~~ X m(k) and = L H(k). Equations (2), (3), (5), and (6) are easily updated by matrix multiplications and additions. A means of updating equation (4) which does not involve a matrix inverse will now be described. The matrix identity used in converting forms of the Kalman filter [21 is + B C~~ Bi = A - AB (BAB + Cy BA, ( 7) where A and C represent covariances. For the special case in which BAR + C is a scaler, equation (7) becomes (A +B C Br =A - (C + BAB ) (8) 3

9 I This result will be used subsequently. CANTRELL AND TRUNK The smoothed covariance update, equation (4), is rewritten as P ~~ = ~~~ - ~~ + h E ~~ J {, (9) where the kth-sample notation has been dropped for notational convenience. The prewhitened measurement matrix is defined as H 1 ~~ = H 2 (10) H. where, represents the ith row. Equation (9) can then be rewritten as ~~ 1 =~~~~~ +[R ~ H ~~... H. ~~ ] ~ H 2 (11) em Equation (ii) can then be rewritten as P =P + ~ ( + H 2 R H ~ ~~~~ (12) Equation (12) can be placed in recursive form by P_I.o pf l 1 + H ; - ~~ K,, i=l,..., m, (13) where = P ~ and ~; 1 = ~~~~. The matrix identity mentioned previously in equation (8) is applied to equation (13), yielding ~ i-l H; R 1~ 1_ 1 P 1 =P ~~1, i i,..., m, (14) where e, + H i ~ -i H is a scaler, because the R ~ as defined in (10) are row vectors. Consequently equation (14) shows that no matrix inverses need be found to update the smoothed covariance. Equation (14) suggests the title of this section : One-by-One Updating of the 4 L.. - ~~~~~~~~~~ -

10 ~~~~~~ ~ ~~~~~~~~~~~~~~~~~~~ - ~~~~~ NRL REPORT 8229 Kalman Filter. Each measurement variance, taken one at a time, is prewhitened and used to modify the smoothed covariance. A means of updating the filter in terms of correlation coefficients and s dard deviations is considered next.. UPDATING OF THE KALMAN FILTER BY UPDATING THE CORRELATION COEFFIC IE NTS AND STANDARD DEVIATIONS A covariance matrix written in terms of correlation coefficients and standard deviations is = 1,..., n and j = 1,., n) and the ele- where the elements of U are u 1~ = ments of D are d 1 = ~~~~~~~~ ~~~ with the matrix 1) being a diagonal matrix with diagonal elements d ~ which represent the standard deviations. The diagonal elements of U are i s and the offdiagonal elements are correlation coefficients with values between - 1 and 1. The one-byone updating of the Kalman filter is modified to reflect the factorization (14). The Kalman-filter equations (2) through (6) will flow be reexamined, with the kthsample notation being eliminated for notational convenience. Equations (2) and (6) remain the same, since they do not involve covariances. These equations are (15) X X+K[Zm I ~ X ~ -~ and X (. The filter gain (equation (3)) i ~ written as where each element of the gain matrix K is K=P J ( E 1, k 11 ~~ (P 12 ~~12)/e~, i 1,..., n andj i,..., m. (16) Using the factorization in equations (15) and (16) yields the gains k., ~ n ~~ (u,q) ~fq d 1 d ~ e The prediction covariance updat: in equation (5) i: next examined for the case of zero process noise. The equation is P4 4 s. 5 L. --~~~-~~~

11 CANTRELL AND TRUNK The factorization described in equation (15) is used in yielding Since D = D, equation (17) can be rewritten as ñiri5 = 4 ~ 1Yfl 4 ~. (17) DUD ~~~~~~~~~~~~~~~~~~~~~~~~ (18) The bracketed term of equation (18) is factored in the form of (15), yielding The forms (18) and (19) then yield D=DR, which is the simple product of two diagonal matrices. = RUR. (19) The last equation to consider is the smoothed covariance, which relies on the one-byone update previously described - The equation is p 1-1 H; 1< - (20) e + ) 4 ~ P1_1 R ~ for I = 1,..., m and P 0 = P. By use of the factorization (15), equation (20) becomes ~~~ i ~~~ -i H; H 1 D ~~1~~~ 1\ D ~ U 1 D ~ =D ~ _1 ( U ~ _1 J D._ 1, (21) \ e, + R 1 D1_1 U ~ _1D ~ _1 k ~ J where U 0 f ~ i 1 ~ o = 1), ~ - ~ rn = U, and D rn = D. The vector V1_1 is defined as vi_1 - _ _ ~~~ _.. + H 1 D1_1 U1_1 D1_1 K; Equation (21) can then be written as D 1 LT ~ D 1 D 1_1 ( ~~ _1 - v;vjn _ 1. (22) Equation (22) is factored into the correlation coefficient form, where U ~~1 -V,~ V ~ =RO,R. Therefore D, =D, 1 R, which is the product of two diagonal matrices. This completes the development. 6

12 NRL REPORT 8229 The role the DUD factorization, in terms of standard deviations and correlation coefficients, plays in the Kalman filter is that of the scale factors, and this factorizatio ~ is numerically different than a straightforward implementation. This may be an advantage in some problems in which the range of standard deviations are known and fixed-point hardware could probably be used to an advantage. It should be noted that in the computations the values of the elements of U are usually near 1 and the values of the elements of D usually change only slightly from the values of the elements of the previous D. This m dicates that for some problems good numerical accuracy may be achieved. The updating of the Kalman filter in terms of correlation coefficients and standard deviations is similar in many respects to Carlson s UD U-filter update found in Ref. 1. Both rely on the same form of the Kalman filter in the one-by-one update form as previously presented. Both filters update factors of the covariances, and both require a refactorization of a portion of covariance equations to obtain the desired form. They are different in that the factorization is different. SUMMARY The Kalman filter was briefly reviewed, and a one-by-one covariance update technique was shown to operate such that the entire Kalman filter can be updated with only matrix multiplications and additions, given that the noise is prewhitened. The one-by-one update operates on one measurement at a time in a simple fashion, and after all measurements are used, the Kalman filter is updated. The covariances are factored in terms of correlation coefficients and standard deviations, and the Kalman filter is updated in terms of these factors. The technique is similar in principle to Carison s technique but with a different factorization. REFERENCES 1. G.J. Bierman, Factorization Methods for Discrete Sequential Estimation, Academic Press, New York, M. Aoki, Optimization of Stochastic Systems: Topics in Discrete Time Systems, Academic Press, New York,