By Bruce P. Gibbs
This booklet offers a whole rationalization of estimation idea and software, modeling methods, and version assessment. every one subject starts off with a transparent rationalization of the idea (often together with historic context), via software concerns that are supposed to be thought of within the layout. various implementations designed to handle particular difficulties are provided, and diverse examples of various complexity are used to illustrate the concepts.This ebook is meant basically as a guide for engineers who needs to layout functional systems. Its primary goal is to provide an explanation for all very important features of Kalman filtering and least-squares conception and application. dialogue of estimator layout and version improvement is emphasised in order that the reader may well improve an estimator that meets all software specifications and is strong to modeling assumptions. because it is usually tricky to a priori be sure the simplest version constitution, use of exploratory information research to outline version constitution is discussed. equipment for identifying the "best" version also are awarded. A moment objective is to give little recognized extensions of least squares estimation or Kalman filtering that offer suggestions on version constitution and parameters, or make the estimator extra powerful to adjustments in real-world behavior.A 3rd objective is dialogue of implementation matters that make the estimator extra exact or effective, or that make it versatile in order that version possible choices may be simply compared.The fourth aim is to supply the designer/analyst with advice in comparing estimator functionality and in determining/correcting problems.The ultimate objective is to supply a subroutine library that simplifies implementation, and versatile normal function high-level drivers that permit either effortless research of other versions 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
For GOES, each of the five misalignment parameters is modeled as the sum of a linear or quadratic term plus a Fourier series of 4 to 13 harmonics, depending on the parameter. For example, instrument roll φ (t) is modeled as φ(t ) = x1 + x2 t + x3 cos θ + x4 sin θ + x5 cos 2θ + x6 sin 2θ + … + xn sin(n / 2 − 1)θ where θ = 2π t/1440 and t is minutes of the solar day. The other four misalignment parameters are modeled similarly using different x coefficients. Then the instrument-observed scan angles to various stars and landmarks are modeled using a nonlinear function of the five misalignment angles.
2-3), without the qc term, is integrated to obtain the desired sensitivities. If the model is entirely linear, the model becomes: SYSTEM DYNAMICS AND MODELS x(t ) = F(t ) x(t ) + B(t ) u(t ) + G(t ) q c (t ). 2-3) with suitable redefinition of terms. 2-4) must be time-integrated over intervals ti to ti+1. For a deterministic linear system, the solution can be represented as the sum of the response to an initial condition on x(ti), called the homogeneous solution, and the response due to the driving (forcing) terms u(t) and q(t) for ti < t ≤ ti+1, called the particular or forced solution, that is, x(ti + 1 ) = x H (ti + 1 ) + x P (ti + 1 ).
There are several choices for defining attitude. Euler angles are commonly used for representing aircraft attitude, and are sometimes used to define spacecraft attitude. If the Euler angles are defined in 3-2-1 order (as used for aircraft), the first rotation is about axis-3 (yaw or ψ), the next rotation is about axis-2 (pitch or θ), and the final rotation is about axis-1 (roll or φ). 2-39) 30 ADVANCED KALMAN FILTERING, LEAST-SQUARES AND MODELING: A PRACTICAL HANDBOOK where Cθ = cos θ, Sθ = sin θ, etc.