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CHAPTER 5 OPTIMAL STATE ESTIMATION
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Recalling that P 1 is the information matrix for xm , eqn (537) shows m the amount that each new measurement increases the information matrix Note that if all measurements are noisy (ie, Rj = 0 for any j), then 1 Hm+1 Rm+1 Hm+1 is always nite and positive; the information matrix is monotonically increasing; and, the variance of the estimate (ie, Pm+1 ) monotonically decreases with each new measurement Multiplying eqn (537) on the right by xm , we obtain P 1 xm m+1 =
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1 P 1 xm + Hm+1 Rm+1 ym+1 m
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(538)
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Multiplying on the left by Pm+1 and rearranging yields Pm+1 P 1 xm m
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1 = xm Pm+1 Hm+1 Rm+1 ym+1
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(539)
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which will be needed in the following derivation The new estimate is calculated from (527) as xm+1 = Pm+1 I Hm+1 Pm 0 0 Rm+1
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xm ym+1
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1 = Pm+1 P 1 xm + Hm+1 Rm+1 ym+1 m 1 y = xm + Pm+1 Hm+1 Rm+1 ( m+1 ym+1 ) = xm + Km+1 ( m+1 ym+1 ) y
xm+1
(540)
1 where Km+1 = Pm+1 Hm+1 Rm+1 and eqn (539) was used to obtain the second to the last line of the derivation Although a linear estimator with the form of eqn (533) was initially stated as an objective, such a linear relationship was never imposed as a constraint The linear update relationship of eqn (540) is a natural consequence of the problem formulation Eqns (537) and (540) provide recursive formulas for the estimation of the vector x With proper initialization, the estimate is exactly the same as that attained by use of a batch approach for m+1 measurements using eqns (527) and (528) Table 51 shows that, for large m, the memory and computational requirements of the batch algorithm are O(n2 m) and O(nm), respectively These requirements are necessary even if the estimate is known for (m 1) measurements prior to the m-th measurement The computational and memory requirements for incorporating a single additional scalar measurement using the recursive algorithm of eqns (537) and (540) are evaluated in Table 52 For the recursive algorithm, the memory and computational requirements for incorporating each measurement are determined only by the dimension of the estimated vector Eqn (537) computes P 1 where Pm+1 is required for the calculation m+1 of K; therefore, the algorithm as written requires matrix inversion which is not desirable Table 52 shows that the matrix inversion plays a dominant
53 FROM WLS TO THE KALMAN FILTER Workspace Memory 1 n
1 2 (n 1 2 (n
181 Permanent Memory n
Computation Hm+1 r = ym+1 Hm+1 xm d1 = Fm+1 = Fm + d1 Hm+1 Pm+1 = F 1 m+1 K = Pm+1 d1 xm+1 = xm + Kr Total
Hm+1 Rm+1
Flops n n
1 2 (n + 1)n 3 n + 1 n2 + 1 n 2 2 2
+ 1)n
+ 1)n n n
5 2n
n + 2n + 4n
1 2 2n
1 2 2n
+ 5n 2
Table 52: Computational and memory requirements for the recursive least squares algorithm of eqns (537) and (540) with x Rn , Hm+1 R1 n , y R, and Rm+1 R The matrix Fm+1 Rn n represents the information matrix
role in determining the amount of required computation Section 54 will show that eqns (537) and (540) are equivalent to Km+1 xm+1 Pm+1 = = = Pm Hm+1 Rm+1 + Hm+1 Pm Hm+1 xm + Km+1 ( m+1 Hm+1 xm ) y (I Km+1 Hm+1 ) Pm
(541) (542) (543)
The computational requirements for this set of equations are evaluated in Table 53 The workspace and computational requirements are signi cantly reduced relative to the previous algorithms In addition to requiring smaller amounts of memory and computation, the recursive least squares algorithm provides iterative estimates immediately following the measurement time This has the potential to provide estimates with reduced delay over an approach which waits to accumulate a xed sized batch of m samples before calculating an estimate Example 53 This example reconsiders the problem stated in Example 51 using the algorithm of eqns (541 543) For this problem, n = 1, Hm+1 = 1 and Rm+1 = 2 Using the rst measurement and eqns (529 530) to initialize the state and covariance, we have x1 = y1 and P1 = 2 The recursion then starts at m = 2: Km+1 Pm+1 Pm 2 + Pm = (1 Km+1 ) Pm =
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