Book Corrections for Optimal Estimation of Dynamic Systems, 2 nd Edition

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1 Boo Correcions for Opimal Esimaion of Dynamic Sysems, nd Ediion John L. Crassidis and John L. Junins November 17, 017 Chaper 1 This documen provides correcions for he boo: Crassidis, J.L., and Junins, J.L., Opimal Esimaion of Dynamics Sysems, nd Ediion, CRC Press, Boca Raon, FL, 011. Any oher correcions are welcome via o he auhors. The upper limi for he index i in Equaions (1.105) and (1.1) should be n + 1 insead of n, so hey should read, respecively, as and m ˆx i = h i ( j ) ˆx i = T 1 m h i ( j )ỹ j, i = 1,,..., n y()h i() d T 0 [h, i = 1,,..., n + 1 i()] d Also, he paragraph under Equaion (1.106) should read Orhogonaliy of he basis funcions of Equaion (1.106) means ha he coefficiens ˆx i are compued independenly as raios of inner producs in Equaion (1.105), so adding addiional erms o he series in Equaion (1.10) does no require re-compuaion of he previously compued erms. This allows adapaion wherein addiional erms can be added, so long as n (m 1)/, unil some convergence crierion is me. The gradien of Equaion (1.15) is acually evaluaed a he curren esimae. So, Equaion (1.153) is given by and Equaion (1.154) is given by ˆx J xc = H T W [ỹ f(x c )] H T W y c H f CUBRC Professor in Space Siuaional Awareness, Deparmen of Mechanical & Aerospace Engineering, Universiy a Buffalo, Sae Universiy of New Yor, Amhers, NY johnc@buffalo.edu. Royce E. Wisenbaer 39 Chair Professor of Aerospace Engineering, Deparmen of Aerospace Engineering, Texas A&M Universiy, College Saion, TX junins@amu.edu. xc 1

2 Chaper On page 70 jus before Eq. (.6), we are maing use of Eq. (.58b) insead of Eq. (.58a). Afer Eq. (.83) i saes ha he assumpion s n > s n+1 mus be valid. This condiion is in fac required for he validiy of he TLS soluion iself in Eq. (.80). This is shown as Theorem 4.1 in he paper Golub, G.H. and Van Loan, C.F., An Analysis of he Toal Leas Squares Problem, SIAM Journal on Numerical Analysis, Vol. 17, No. 6, Dec. 1980, pp Exercise.11 should read: Prove ha he Cramér-Rao inequaliy given by Equaion (.100) achieves he equaliy if and only if ln[p(ỹ x)] = F (x)(x ˆx) where F (x) is he Fisher informaion marix explicily shown as a funcion of x. Alhough exercise.1 is correc, anoher way o sae he problem is as follows: Suppose ha an esimaor of a non-random scalar x is biased, wih bias denoed by b(x), so ha E{ˆx} = x + b(x). Show ha a lower bound on he variance of he esimae ˆx is given by ( var(ˆx) 1 + b(x) ) J 1 where J = E { [ ] } ln[p(ỹ x)] Chaper 3 On page 173, Eq. (3.188) should read d < 1 E { (x ˆx) } = 0 d > 1 R d = 1 On page 148, he innovaions covariance in Eqs. (3.55b) and (3.56) should be H P HT + R.

3 Chaper 4 Equaion (4.110) on page 54 is incorrec. I should read: = c (j) 1 p (ỹ ˆx (j) ) w(j) M The code has been updaed for Example 4.6. Noe ha resuls do no change much from he original code resuls. On pages 83 and 84 Sep for Sysemaic Resampling and Sraified Resampling are incorrec. The correced versions are:. Se i = 1. Perform he nex seps for j = 1,,..., N. Execue a while loop: while z (i) < u (j) i i + 1 end while where denoes replacemen; choose he resuling i afer he while loop as he new index and replace x (j) wih x (i). The MATLAB codes for Examples 4.11 and 4.1 are incorrec, so he Figures 4.14 and are no correc. Originally he poserior densiies were ploed using surf(xii,,f) bu his is only correc for he las se of poins, given by xii. This has been correced by using waerfall(xi,repma(,1,100),f) for boh examples. Noe ha he auhors do no now how o plo he resuls correcly using he surf command. Please le he auhors now if he reader nows how o plo he poserior densiies using he surf command. Chaper 6 The MATLAB code for Example 6.1 is incorrec. The boresigh of he sar camera sensor is along he body z-axis. Thus an ideniy quaernion would align he body z-axis wih he inerial z-axis, which causes an incorrec moion compared o wha is described in he example. I is assumed ha he Earh-poining spacecraf is in an equaorial 350 m circular orbi, which is equivalen o a 91.5 minue orbial period. The spacecraf s z-axis is poined in he nadir direcion, he y-axis is poined in he negaive orbi momenum s vecor, and he x-axis is poined in he orbi velociy direcion. The rue angular velociy is given by ω() = [ ] T rad/sec. Firs roae +90 degrees abou x-body axis. Then roae 180 degrees abou he new x-body axis, which correcly places he boresigh in he ani-nadir direcion [ ]T. (i.e. he radial direcion). The iniial quaernion is hen given by q 0 = Also, a magniude of 6 is chosen for he sars. There are imes when he number of available sars is less han. A hese imes a soluion is no possible. 3

4 Chaper 7 The MATLAB code for Example 7.1 is incorrec. The boresigh of he sar camera sensor is along he body z-axis. Thus an ideniy quaernion would align he body z-axis wih he inerial z-axis, which causes an incorrec moion compared o wha is described in he example. I is assumed ha he Earh-poining spacecraf is in an equaorial 350 m circular orbi, which is equivalen o a 91.5 minue orbial period. The spacecraf s z-axis is poined in he nadir direcion, he y-axis is poined in he negaive orbi momenum s vecor, and he x-axis is poined in he orbi velociy direcion. The rue angular velociy is given by ω() = [ ] T rad/sec. Firs roae +90 degrees abou x-body axis. Then roae 180 degrees abou he new x-body axis, which correcly places he boresigh in he ani-nadir direcion [ ]T. (i.e. he radial direcion). The iniial quaernion is hen given by q 0 = Also, a magniude of 6 is chosen for he sars. There are imes when he number of available sars is less han. The exended Kalman filer sill provides an updae even when only 1 sar is available. Equaion (7.60) should read p θθ = p + θθ σ c = 1/4 σn 1/ (σ v + σ u σ n 1/) 1/4 (1) The σ v erm in he original σ u σ v 1/ should be σ n. Equaion (7.83a) should read Z 11 = v D R ϕ + h, Z 1 = v E an ϕ R λ + h ω e sin ϕ, Z 13 = v N R ϕ + h The ω e sin ϕ erm in Z 1 should be subraced no added, and Z 13 was originally labeled as Z 1. The MATLAB code for Example 7. has been correced. Appendix A The derivaion of Equaion (A.111) is no correc. The goal is o deermine he iniial condiion x( 0 ), so replace 0 wih, and replace wih f in Equaion (A.109) W o (, f ) f Φ T (τ, ) H T (τ) H(τ) Φ(τ, ) dτ Noe ha he noaion for W o () has changed here, and ha he inegraions in Equaions (A.107) o (A.109) should be done from 0 and f. The ime derivaive of Φ(τ, ) = Φ 1 (, τ) will be needed. Tae he ime derivaive of V V 1 = I for some marix V : V V 1 + V V 1 = 0 V 1 = V 1 V V 1 Leing V Φ(, τ) and noing V 1 = Φ(τ, ) leads o Φ(τ, ) = Φ(τ, ) Φ(, τ) Φ(τ, ) = Φ(τ, ) F () Φ(, τ) Φ(τ, ) = Φ(τ, ) F () 4

5 where he following ideniies were used Φ(, τ) = F () Φ(, τ) Φ(, τ) Φ(τ, ) = Φ(, ) = I Then he derivaive of he observabiliy Gramian is given by Ẇ o (, f ) = Φ T (, ) H T () H() Φ(, ) f F T () Φ T (τ, ) H T (τ) H(τ) Φ(τ, ) dτ } {{ } f W o(, f ) Φ T (τ, ) H T (τ) H(τ) Φ(τ, ) dτ F () } {{ } W o(, f ) Thus, Ẇ o (, f ) = F T () W o (, f ) W o (, f ) F () H T () H() which is inegraed bacwards wih W o ( f, f ) = 0. The conrollabiliy Gramian can be derived in a similar fashion bu is inegraed forward in ime. Appendix C Above Eq. (C.3) i should read Two processes, {x( )} and {y( )}, are uncorrelaed if E { x( i ) y T ( j ) } = E {x( i )} E { y T ( j ) } for all i and j. Appendix D Below Eq. (D.16) i should read Equaions (D.16) provide hree equaions... The necessary condiions in Example D. are given by ϕ y = 1 18λ(y 4) = 0 ϕ z = 1 8λ(z 5) = 0 3 ψ(x) = 9(y 4) + 4(z 5) 36 = 0 The equaion for ϕ/ z is incorrec in he boo. 5