## Kalman Filtering for Linear Discrete-time Dynamic Systems |

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algorithm applied approach assumed asymptotically stable best linear estimate Bk+1 BkQkBkT calculated Cholesky decomposition Ck+1 computed conditional expectation considered covariance matrix defined derivation diagonal dimension discrete discussed disturbances dynamic system EE|Y eigenvalues elements filtering equations follows g-inverse Gaussian given gives gonal Householder transformation implementation information filter k+l|k Kalman covariance Kalman estimation Kalman filter Kk+1 linear function linear manifold spanned loss function matrix inverses measurement equation measurement update minimizes nonlinear nonsingular notation observations obtained optimal estimate optimal state estimate orthogonal projections Pk+l|k+l Pk|k probability density function process noise proof of Theorem pseudoinverse random vectors recursively Rk+1 scalar Section sequential processing shows Sk|k solution solved square square-root filter Substitution Theorem 17 Theorem 20 tion uncorrelated uncorrelatedness update equations variables xk+l xk|k y-Cx zero mean