By C.K. Chui
This publication offers an intensive dialogue of the mathematical conception of Kalman filtering. The filtering equations are derived in a chain of straightforward steps allowing the optimality of the method to be understood. It presents a entire therapy of varied significant subject matters in Kalman-filtering conception, together with uncorrelated and correlated noise, coloured noise, steady-state concept, nonlinear platforms, platforms id, numerical algorithms, and real-time functions. a sequence of difficulties for the scholar, including an entire set of strategies, also are integrated. the fashion of the publication is casual, and the math simple yet rigorous, making it available to all people with a minimum wisdom of linear algebra and platforms conception. during this moment version, as well as a few minor corrections and up-dating, the part on real-time method identity has been increased and a quick creation to wavelet research incorporated.
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All charts together A := (Φι , Vι ) : ι ∈ J constitute an atlas of the manifold. When Φj : Vj → Uj ∩ M with j = 1, 2 represent two charts of the atlas A such that W1,2 := M ∩ U1 ∩ U2 = ∅ is correct, then we consider the parameter transformation Φ2,1 := Φ−1 2 ◦ Φ1 . If the functional determinant satisfies JΦ2,1 > 0 on Φ−1 (W ) for such ar1,2 1 §4 The Stokes integral theorem for manifolds 33 bitrarily chosen charts from the atlas, the manifold is oriented by the atlas. ✛✘ U2 ✛✘ U1 V1 M ❈❖❈ ✚✙ ✚✙ ❈ ❇▼ ✕ W1,2 ❇ ✁ ✁ ❇ ✁ ❇ Φ2 Φ1 ✁ −1 ❇ ✁ Φ2 ◦ Φ1 ❇ ✁ ❇ V2 ( ) ( ) −1 −1 Φ2 (W1,2 ) Φ1 (W1,2 ) Definition 2.
We call (Φ, V ) a chart of the manifold. All charts together A := (Φι , Vι ) : ι ∈ J constitute an atlas of the manifold. When Φj : Vj → Uj ∩ M with j = 1, 2 represent two charts of the atlas A such that W1,2 := M ∩ U1 ∩ U2 = ∅ is correct, then we consider the parameter transformation Φ2,1 := Φ−1 2 ◦ Φ1 . If the functional determinant satisfies JΦ2,1 > 0 on Φ−1 (W ) for such ar1,2 1 §4 The Stokes integral theorem for manifolds 33 bitrarily chosen charts from the atlas, the manifold is oriented by the atlas.
N− 1 and the property 1. Evidently, the condition |ξ| = 1 is valid on Ω˙ ∩ U . Therefore, it remains to show the property 3. When 0 < | | < 0 holds true, we infer the inequality n Ψ (x + ξ) = Ψ (x + ξ) − Ψ (x) = Ψxi (x + κ ξ)ξi i=1 = 1 |∇Ψ (x)| < 0 if − n Ψxi (x + κ ξ)Ψxi (x) 0 > 0 if 0 < i=1 < <0 < 0 for all points x ∈ Ω˙ ∩ U ; with a quantity κ = κ( ) ∈ (0, 1). This implies Ω if − x+ ξ ∈ R \ Ω if 0 < n 0 < <0 < 0 . d. Remark: Let the surface patch F = Fl bounding Ω be given by the parametric representation X(t) = X(t1 , .