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Additional resources for Applied Optimal Estimation
G & l z J d x ~ d x , . . 0-9) which is the conditional mean estimate. J: .. L @ p @ l ~ d x , d x 2 . . 0- 10) where L a ) isa scalar "loss function" of the estimation error 106 where Po is the a priori covariance matrix of&. In comparing the various estimation methods just discussed, we note that if there is little or no a priori information, PC1 is very small and Eq. 12) becomes Eq. 0-5). , R = a21), Eq. 0-5) reducer to Eq. 0-3). In his important work, Kalman (Ref. 2) formulated and solved the Wiener problem for gauss-markov sequences through use of state-space representation and the viewpoint of conditional distributions and expectations.
In broad terms, a very stable system* will cause the first term in Eq. 7-10) to be smaller than the covariance Pk. No restrictions regarding the stability of the system were made in the development above and the error covariance of an unstable system will grow unbounded in the absence of As At -+ 0, the equation becomes &t)= ~ ( t ) ~ (+t p(t)~T(t) ) + ~ ( t ) a t )cT(t) *That is, one whose F-matrix only has elgenvalues with large, megatwe real parts 'That is, one whore Pmatrix has aome purely ~magiruryeigenvalue pairs.
0-3). In his important work, Kalman (Ref. 2) formulated and solved the Wiener problem for gauss-markov sequences through use of state-space representation and the viewpoint of conditional distributions and expectations. His results also reduce to those given above. Therefore, the important conclusion is reached that for gaussian random variables, identical results are obtained by all these methods as long os the assumptions are the same in each case (Ref. 3). This property motivates us to consider primarily Bayesian minimum variance estimators.