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CHAPTER 5 OPTIMAL STATE ESTIMATION y = 1 0 0 1 p v a + b
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where var( ) = R = 10 m2 , P SD v = 25 10 3 m3 and P SD a = 10 s 2 10 6 m5 The measurement bias b is a scalar Gauss-Markov process with s 2 1 = 300 sec 1 and P SD b = Qb = 4 m s The designer decides not to estimate the sensor bias and to use the Schmidt-Kalman lter implementation approach Figure 55 plots the lter gains in the right column and the predicted error standard deviations in the left column Even though the position sensor s additive white noise has standard deviation equal to 32m, the accuracy predicted for the position estimate is approximately 25m due to the unestimated sensor bias The approach of this example should be compared with that of Example 61
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Decoupling
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Let the state vector x of dimension n be decomposable into two weaklycoupled (ie, 12 and 21 approximately 0) subvectors x1 and x2 of dimensions n1 and n2 such that n1 + n2 = n Then the di erence equation can be approximated as xk+1 = x1 x2 =
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where subscript k s have been dropped in the right-hand side expressions In this case, the error covariance can be time propagated according to P11 P21 P12 P22 =
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11 P11 11 22 P21 11
11 P12 22 22 P22 22
Qd1 0
0 Qd2
The decoupled equations require 3 n3 + 1 n2 + 3 n3 + 1 n2 + n2 n2 + n1 n2 1 2 2 1 2 1 2 2 2 2 FLOP s The full equation requires 3 n3 + 1 n2 FLOP s Therefore, the 2 2 decoupled equations require 7 n2 n2 + n1 n2 +n1 n2 fewer FLOP s per time 1 2 2 iteration than the full covariance propagation equations Similar special purpose covariance equations can be derived for the case where only one of the coupling terms is non-zero
O -line Gain Calculation
The covariance update and Kalman gain calculations do not depend on the measurement data; therefore, they could be computed o -line Since
510 SUBOPTIMAL FILTERING
these calculations represent the greatest portion of the lter computations, this approach can greatly decrease the on-line computational requirements, possibly at the expense of greater memory requirements The memory requirements depend on which of three approaches is used when storing the Kalman gains The Kalman gains can be stored for each measurement epoch, curve t, or stored as steady-state values Pre-calculation of the Kalman gains will always involve at least the risk of RMS estimation error performance loss In the case where only the steady-state gains are stored, the initial transient response of the implemented system will deteriorate as will the settling time Even in the case where the entire gain sequence is stored, the absence of expected measurements will result in discrepancies between the actual and expected error covariance matrices The e ect of such events should be thoroughly analyzed prior to committing to a given approach Example 510 Consider using the steady-state Kalman lter gains to estimate the state of the system de ned as: x1 x2 yk = = 00 10 10 02 10 00 x1 x2 x1 x2
00 10
160 00 00 10 ,
+ k , with P0 =
P SD = 10 and var( k ) = 10 The measurements are available at a 10 Hz rate The performance of the steady-state lter (dashed-dotted) relative to the performance of the Kalman lter with time-varying gains (solid) is displayed in Figure 56 The two estimators have the same performance until the instant of the rst measurement at t = 10 s The time-varying Kalman lter makes a much larger correction initially due to the large initial error covariance During the initial transient, the relative performance 2 2 di erence is large with 1 + 2 always smaller for the Kalman lter than for the steady state lter Since both lters are stable and both lters use the same gains in steady-state, the steady-state performance of the steady state lter is identical to that of the full Kalman lter implementation Note that the steady state lter approach will fail to apply directly if either the Kalman gains are zero in steady state or do not approach constant values as t
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