By Bruce P. Gibbs
This ebook offers an entire clarification of estimation concept and program, modeling ways, and version evaluate. each one subject begins with a transparent clarification of the speculation (often together with ancient context), through software matters that are supposed to be thought of within the layout. assorted implementations designed to deal with particular difficulties are awarded, and various examples of various complexity are used to illustrate the concepts.This e-book is meant basically as a instruction manual for engineers who needs to layout sensible systems. Its primary goal is to give an explanation for all vital elements of Kalman filtering and least-squares conception and application. dialogue of estimator layout and version improvement is emphasised in order that the reader could boost an estimator that meets all program requisites and is powerful to modeling assumptions. because it is usually tricky to a priori make sure the simplest version constitution, use of exploratory facts research to outline version constitution is discussed. equipment for deciding upon the "best" version also are offered. A moment target is to give little recognized extensions of least squares estimation or Kalman filtering that offer counsel on version constitution and parameters, or make the estimator extra powerful to adjustments in real-world behavior.A 3rd target is dialogue of implementation concerns that make the estimator extra exact or effective, or that make it versatile in order that version choices might be simply compared.The fourth aim is to supply the designer/analyst with suggestions in comparing estimator functionality and in determining/correcting problems.The ultimate target is to supply a subroutine library that simplifies implementation, and versatile common function high-level drivers that let either effortless research of different types and entry to extensions of the elemental filtering.
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Extra resources for Advanced Kalman Filtering, Least-Squares and Modeling: A Practical Handbook
203), that is, tb = hb + w b × hb . = I b w b + w b × (I b w b ) Rearranging yields: ⎡ hb1 ⎤ ⎡τ b1 ⎤ ⎡ω b 2 hb 3 − ω b 3 hb 2 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ hb 2 ⎥ = ⎢τ b 2 ⎥ − ⎢ ω b 3 hb1 − ω b1 hb 3 ⎥ . 2-38) Notice that this model is nonlinear in ωb. If the body of interest contains other rotating bodies, such as momentum or reaction wheels, the separate angular momentum of those rotating bodies must be included as part of the total and additional states modeling the separate angular rates (or equivalently angular momentum) must be included.
If τ and h are referenced to the coordinate system of the rotating body (the b-system), then hb = Ibωb where Ib is the inertia tensor of the body and ωb is the rotational rate. Since the reference frame is rotating, the change in angular momentum of the rotating coordinate system must be accounted for as part of the angular acceleration (Goldstein 1950, section 4-8; Housner and Hudson 1959, p. 203), that is, tb = hb + w b × hb . = I b w b + w b × (I b w b ) Rearranging yields: ⎡ hb1 ⎤ ⎡τ b1 ⎤ ⎡ω b 2 hb 3 − ω b 3 hb 2 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ hb 2 ⎥ = ⎢τ b 2 ⎥ − ⎢ ω b 3 hb1 − ω b1 hb 3 ⎥ .
2002; Bronzino 2006) applications. 2 CONTINUOUS-TIME DYNAMIC MODELS Continuous time dynamic models are usually defined by a set of linear or nonlinear first-order differential state equations. Differential equations are linear if coefficients multiplying derivatives of the dependent variable x are not functions of x. The assumption of first-order differential equations is not restrictive since higher order differential equations can be written as a set of first-order equations. For example, the stochastic linear second-order system x1 (t ) = −k2 x1 (t ) − k1 x1 (t ) + q(t ) can be written as x1 (t ) = x2 (t ) .
Advanced Kalman Filtering, Least-Squares and Modeling: A Practical Handbook by Bruce P. Gibbs