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Table 55: Discrete-time Kalman lter equations Computation of the matrices k and Qdk is discussed in Section 471 any stabilizing state feedback gain vector L The approaches described in eqns (557), (562), and (568) apply only for the Kalman lter gain Kk
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To show the equivalence of eqns (557-558) to eqns (561-562), we will apply the Matrix Inversion Lemma (see Appendix B) to eqn (557) with 1 , and D = HP The A = P , B = P H , C = R + HP H result is P+
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= = = =
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P P H P P H P
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R + HP H R + HP H R + HP H R + HP H
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1 1 1 1
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(563) (564) (565)
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I P H
H P
Substituting eqn (564) into eqn (558) yields K = P+ H R 1 = P H I R + HP H = = P H P H R + HP H R + HP H
1 1 1
HP H
R 1 R 1 (566)
R + HP H HP H
54 KALMAN FILTER DERIVATION SUMMARY Substituting eqn (566) into eqn (565), simpli es the latter to P+ = [I KH] P where eqns (566 567) correspond to eqns (561-562)
(567)
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P+ = P K R + HP H
An alternative use of eqns (566) and (563) is to show that K (568)
Lastly, eqn (567) can be manipulated as shown below: P+ = [I KH] P + P H K P H K = [I KH] P + K R + HP H K P H K = P KHP + KRK + KHP H K P H K = [I KH] P [I KH] + KRK (569) where eqn (566) was used in the transition from the rst to second line Eqn (569) is referred to as the Joseph form of the covariance propagation equations [34]
Kalman Filter Examples
This section presents a few Kalman lter examples The following example is useful in various applications including the estimation of the GPS ionospheric error using data from a two frequency receiver as is discussed in Section 86 and carrier smoothing of code measurements as is discussed in Section 87 Example 54 Consider the scenario where at each measurement epoch two measurements are available yk zk = = xk + nk xk + vk + B (570) (571)
1, B is a constant, and xk where nk N (0, 2 ), vk N (0, ( )2 ) with can change arbitrarily from one epoch to another, and all the variables in the righthand sides of the above equations are mutually uncorrelated The objective is to estimate xk One solution using a complementary lter is illustrated in Figure 53 The residual measurement rk = zk yk is equivalent to rk = B + wk
CHAPTER 5 OPTIMAL STATE ESTIMATION
zk + yk S B+wk + S ^+ xk
^+ Bk Kalman Filter
Figure 53: Measurement bias complementary lter for Example 54 where wk = vk nk and wk N (0, (1 + 2 ) 2 ) Assuming that there is no prior information about B, this is the problem of estimating a constant from independent noise corrupted measurements of the constant that was discussed in Example 53 The solution is 1 r Bk = Bk 1 + ( k Bk 1 ) k starting at k = 1 with B0 = 0 Note that the value of B0 is immaterial 2 The estimation error variance for Bk is PB (k) = (1 + 2 ) Given this k estimate of B, using eqn (571), the value of xk can be computed as xk = zk Bk ,
1 where the error variance for xk is Px (k) = (1 2 ) k + 2 2 Initially, 2 the variance of x is as the number of measurements k increases, the variance of x decreases toward 2 2 This example has relevance to GPS applications for estimation of ionospheric delay and for carrier smoothing of the pseudorange observable
This example is further investigated in Exercise 511 That exercise considers the case where the measurement error process nk is not white Example 55 Section 494 presented a one dimension aided INS example using a state estimation gain vector designed via the pole placement approach That example is continued here using a Kalman lter The state space description and noise covariance matrices are speci ed in Section 494 In the example, the initial error covariance matrix P (0) is assumed to be a zero matrix The performance of the estimator is indicated by error standard devia2 tion plots are shown in the left column of Figure 54 where var( ) = p , p 2 2 b) var( ) = v , and var( = b The portion of this curve that appears as v a wide band in the left column of gures is due to the covariance growth between position measurements followed by the covariance decrease at the measurement instant If the time axis was magni ed, this growth and decrease would be clearly evident as it was in Figure 47 The resulting timevarying Kalman lter gains are plotted in the right column The Kalman
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