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A Priori Bounds

  • R. S. Bucy
Part of the Signal Processing and Digital Filtering book series (SIGNAL PROCESS)

Abstract

We want to study the equation
$${P_{n + 1}} = {\phi _{n + 1}}\left( {{P_n}{{H'}_n}{{({H_n}{P_n}{{H'}_n} + {R_n})}^{ - 1}}{H_n}{P_n}} \right){\phi '_{n + 1}} + {G_{n + 1}}{Q_n}{G'_{n + 1}}$$
where Pno = Г. Recall that Pn+1 represents the prediction error covariance at time n + l given data to time n, while \({P_n} - {P_n}{H'_n}{\left( {{H_n}{P_n}{{H'}_n} + {R_n}} \right)^{ - 1}}{H_n}{P_n}\) represents the filter error covariance, the error covariance matrix of the signal process x at time n given data to time n. We have and will assume the matrix Φ n is invertible for all n. When the processes arise from sampling of continuous time diffusion processes this assumption is fulfilled. Define the mapping τn+1 as
$${\tau _{n + 1}}({P_n})\mathop = \limits^\vartriangle {\phi _{n + 1}}{\left( {{P_n} - {P_n}{{H'}_n}({H_n}{P_n}{{H'}_n}) + {R_n}} \right)^{ - 1}}{H_n}{P_n}){\phi '_{n + 1}} + {G_{n + 1}}{Q_n}{G'_{n + 1}}$$

Keywords

Riccati Equation Error Covariance Matrix Nonlinear Difference Equation Gaussian Random Vector Joint Information 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • R. S. Bucy
    • 1
  1. 1.Department of Aerospace EngineeringUniversity of Southern CaliforniaLos AngelesUSA

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