By Charles K. Chui, Guanrong Chen (auth.)
Kalman Filtering with Real-Time functions provides a radical dialogue of the mathematical idea and computational schemes of Kalman filtering. The filtering algorithms are derived through diverse methods, together with an immediate process inclusive of a sequence of easy steps, and an oblique approach in response to innovation projection. different themes contain Kalman filtering for platforms with correlated noise or coloured noise, restricting Kalman filtering for time-invariant structures, prolonged Kalman filtering for nonlinear platforms, period Kalman filtering for doubtful structures, and wavelet Kalman filtering for multiresolution research of random signs. such a lot filtering algorithms are illustrated by utilizing simplified radar monitoring examples. the fashion of the e-book is casual, and the maths is user-friendly yet rigorous. The textual content is self-contained, appropriate for self-study, and obtainable to all readers with a minimal wisdom of linear algebra, chance thought, and approach engineering.
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Additional resources for Kalman Filtering: with Real-Time Applications
Example text
Derive the Kalman filtering algorithm for this model. 7. Consider a simplified radar tracking model where a largeamplitude and narrow-width impulse signal is transmitted by an antenna. The impulse signal propagates at the speed of light c, and is reflected by a flying object being tracked. t is obtained. The range (or distance) d from the radar to the object is then given by d == c~t/2. The impulse signal is transmitted periodically with period h. Assume that the object is traveling at a constant velocity w with random disturbance ~ ~ N(O, q), so that the range d satisfies the difference equation d k+1 == dk + h(Wk + ~k) .
At each step, we only use the incoming bit of the data information so that very little storage of the data is necessary. This is what is usually called the Kalman filtering algorithm. 3 Prediction-Correction Formulation To compute Xk in real-time, we will derive the recursive formula ~klk = ~klk-l ~ Gk(Vk - CkXklk-l) { Xklk-l - Ak-1Xk-llk-l , where Gk will be called the Kalman gain matrices. The starting point is the initial estimate Xo = xOlo. Since Xo is an unbiased estimate of the initial state Xo, we could use Xo = E(xo), which is a constant vector.
3. 3. Orthogonal Projection and Kalman Filter The elementary approach to the derivation of the optimal Kalman filtering process discussed in Chapter 2 has the advantage that the optimal estimate Xk == xklk of the state vector Xk is easily understood to be a least-squares estimate of Xk with the properties that (i) the transformation that yields Xk from the data Vk == [vci··· vl]T is linear, (ii) Xk is unbiased in the sense that E(Xk) == E(Xk), and (iii) it yields a minimum variance estimate with (VarCfk,k))-l as the optimal weight.