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42 3. Orthogonal Projection and Kalman Filter we have .... v Pk,k ==(1 - GkCk)Pk,k-l(I - GkCk) T.... + (I - .. GkCk)Pk,k-l(GkCk) ==(1 - OkCk)Pk,k-l . T (3.23) Therefore, combining (3.13), (3.16), (3.18), (3.21), (3.22) and (3.23), together with Po,o == II x o - xOloll~ == Var(xo) , (3.24) we obtain the Kalman filtering equations which agree with the ones we derived in Chapter 2. That is, we have xklk == xklk' xklk-l == xklk-l and Ok == Gk as follows: Po,o == Var(xo) Pk,k-l == Ak-lP
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   3.25 42 3. OrthogonalProjection andKalman Filter we have ....v   T Pk,k == 1 - GkCk)Pk,k-l I - GkCk) +  I - GkCk)Pk,k-l GkCk) == 1 - OkCk)Pk,k-l .  3.23 Therefore,combining 3.13 , 3.16 , 3.18 , 3.21 , 3.22 and  3.23 ,togetherwith Po,o   x o- xOloll~   Var(xo) , (3.24) we obtain the Kalmanfilteringequationswhichagreewith the ones we derivedin Chapter 2. That is, we have xklk   xklk xklk l   xklk l and Ok   Gk asfollows: Po,o   Var(xo) Pk,k-l   Ak-lPk-l,k-lAl-l + fk l k lfr l Gk   Pk,k-l C  : (CkPk,k-1C-: + Rk)-l Pk,k   I - GkCk)Pk,k-l xOlo   E(xo) xklk l   Ak-1Xk-llk-l xklk   xklk l + Gk(Vk - CkXk1k-l)k   1,2, .... Of course, the Kalmanfilteringequations 2.18 derivedinSection2.4for the generallineardeterministic/stochasticsystem { Xk l   AkXk + BkUk + rk~k Vk   CkXk + DkUk+ 1k canalsobeobtainedwithout the assumptionon the invertibilityof the matrices Ak, VarC~k,j etc.  cf. Exercise3.6 . 3.5 Real-Time Tracking Toillustrate the applicationof the Kalmanfilteringalgorithmdescribedby 3.25 ,letusconsider an exampleofreal-timetracking.Let x t , 0 :::; t < 00, denote the trajectory inthree-dimensionalspaceofaflyingobject,where t denotes the timevariable cf.Fig.3.1 .Thisvector-valuedfunction is discretizedbysampling and quantizingwithsamplingtime h > 0 to yield Xk   x kh , k   0,1,···.  3.5Real-Time  fracking 43  ig 3.1.   ,   -- x t - I I ã x O) Forpracticalpurposes, x t can be assumed to havecontinuousfirst and secondorderderivatives,denotedby x t and x t , respectively,so that forsmallvaluesof h, the position and velocityvectors Xk and Xk   x kh aregovernedby the equations { h·1h2 .. ~k+l = ~k + ~k + 2 Xk Xk+l = Xk + hXk , where Xk   x kh and k = 0,1,···. In addition,in many applicationsonly the position vector of the flyingobjectisobserved at each time instant,so that Vk = CXk with C =   0 0] ismeasured. In viewofExercise3.8, to facilitateourdiscussion we onlyconsider the trackingmodel  3.26  44 3. OrthogonalProjectionandKalmanFilter to be zero-meanGaussianwhitenoisesequencessatisfying: E ~k = 0, E( 1k) = 0, E ~k~; = Qk 6 kl, E( 1k 1l) = rk6 kl, E xo~; = 0, E(Xo 1k) = 0, where Qk is anon-negativedefinitesymmetric matrixand rk > 0 forall k. It is furtherassumed that initialconditions E xo) and Var xo) aregiven.Forthistrackingmodel, the Kalmanfilteringalgorithmcanbespecifiedasfollows:Let Pk := Pk,k and let P[i, j] denote the  i, j th entryof P. Then we have Pk,k-l[l,l] = Pk-l[l,  ] + 2hPk-l[1,  ] + h2Pk_l[1,  ] + h2 Pk-l[2, 2] h 4 + h 3 Pk-l[2, 3] + 4Pk-1[3, 3] + Qk-l[l, 1], Pk ,k-l[1,2] = Pk,k-l[2, 1] 3h 2 = Pk-l[l,  ] + hPk-l[l,  ] + hPk-l[2,  ] + TPk-1[2 3] h 3 + 2Pk-1[3, 3] + Qk-l[l, 2], Pk,k-l[2,2] = Pk-l[2,  ] + 2hPk-l[2,  ] + h2 Pk-l[3,  ] + Qk-l[2, 2], Pk,k-l[1,3] = Pk,k-l[3, 1] h 2 = Pk-1 [1 ] + hP k-1 [2 ] + 2Pk-1[3,  ] + Qk-l[l, 3], Pk,k-l [2 3] = Pk,k-l [3 2] = Pk-l[2,  ] + hPk-l[3,  ] + Qk-l[2, 3], Pk,k-l [3 3] = Pk-l [3 3] + Qk-l [3 3] , with Po,o = Var xo) ,  with :Kala = E(xo).  xer ises Exercises 45  3.27 3.1.Let A =f. 0 be anon-negativedefinite and symmetric constant matrix.Show that trA > o.  Hint:Decompose A as A = BB T with B  f 0.) 3.2.Let j-I ej = Cj Xj - Yj-I) = Cj (Xi - L Pi-l,iVi) ,  1.=0 where Pj I i aresomeconstantmatrices.UseAssumption2.1 to show that forall / ? j. 3.3.Forrandomvectors WO, , W r, define Y(Wo, , w r  r y= LPiWi, i=O Po, ...  Pr constant matrices}. Let j-I Zj = Vj - Cj L Pj-l iVi i=O be definedasin 3.4 and ej = Ilzjll lzj Show that 3.4.Let j-I Yj-I = L Pj-l iVi i=O and j-I Zj = Vj - Cj L Pj-l iVi . i=O Show that j = 0,1, . ,k - 1.

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