Approximate Kalman Filtering by Guan Rong Chen PDF

By Guan Rong Chen

ISBN-10: 981021359X

ISBN-13: 9789810213596

Kalman filtering set of rules offers optimum (linear, independent and minimal error-variance) estimates of the unknown kingdom vectors of a linear dynamic-observation process, below the commonplace stipulations resembling excellent info details; whole noise facts; designated linear modelling; excellent will-conditioned matrices in computation and strictly centralized filtering. In perform, even though, a number of of the aforementioned stipulations will not be chuffed, in order that the normal Kalman filtering set of rules can't be without delay used, and therefore ''approximate Kalman filtering'' turns into priceless. within the final decade, loads of awareness has been eager about enhancing and/or extending the traditional Kalman filtering strategy to deal with such abnormal situations. This ebook is a suite of numerous survey articles summarizing contemporary contributions to the sphere, alongside the road of approximate Kalman filtering with emphasis on its useful features

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The matrix F mentioned in Definition 3 satisfies the condition 2FR = A C T , (18) Fisher Initialization 29 where A is an n x n matrix. Proof: From (6) and condition (6) of Definition 3 we have it/ = F C x + FT? , and by (6) of the same definition, x/ = (C T )"x + Fry. , or equivalent ly, X / - x = - ( C T ) ' x + Fr7. (20) From equation (20) and part (a) of Definition 3 it follows that From equation (20) and part (a) of Definition 3 it follows that F { ( x / - x ) ( X / - x ) T } = FRFT + ( C T ) ' F { x x T } ( C T ) ' (21) In a manner analogous to equation (10), equation (21) implies that In a manner analogous to equation (10), equation (21) implies that \\xf-xf=tr{FRFT}+\\(CTyx\\2 (22) In order to find an F that minimizes || x^ — x || subject to the constraint (16) we form the Lagrangian C(F, A) = tr{FRFT}+ || (C T )'x f +tr{((CT)" - FC)AT} , (23) where the Lagrance multiplier A in (23) is a square matrix.

Nevertheless, Bayesian techniques have proven to be very successful in the analysis of many, many statistical problems. Certainly the Kalman filter, an inherently Bayesian scheme, provides ample evidence of the validity of such a statement. The problem of choosing priors is, therefore, a real problem with which one must deal. One of the most compelling approaches to this problem is, in our opinion, provided by the work of Jaynes [7]. A thorough discussion of Jaynes' work would take us too far afield here, but suffice it to say that Jaynes proposes choosing priors that maximize entropy [7,8].

68) Because S is invertible ( 5 " 1 = Pf + P[), if follows that (C2SC2T)" = Cf. Thus if we multiply (68) on the right by (C2SC2)+ we obtain F2C2' = SCj(C2SCj)+ + XP[C2T{C2SC2T)+ (69) From (54) we note that F\C\P[ = 0, so multiplying (49) on the right by P{ we obtain F2C2P[ = (CT)"P[ . P[C2\C2SC2)+, we have that AP 1 'C 2 T (C 2 5C 2 T ) + = {CT)"P,l{P'1C2T{C2SC2T)+C2P[)+P[C2T{C2SCj)+ - SC2T(C2SC2T)+C2P{(P{C2T(C2SCj)+C2P{)+P{Cj(C2SCj)+ (71) Substituting (71) into (69) we obtain our final expression for F2 as stated in the theorem.

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Approximate Kalman Filtering by Guan Rong Chen

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