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8-3. 9 EMPIRICAL MODEL IDENTIFICATION Subsequent discussions assume that any random process under study has been modeled as a linear system driven by gaussian white noise. In this section, methods for determining the best linear-gaussian model to fit an observed sequence of data are discussed. As illustrated in Fig. 9-1, a model for the process is produced by combining empirical data and prior knowledge of underlying physical mechanisms. The derived model and the data are then used in an estimation process as described in succeeding chapters.

Between its allowable limits), this estimate is equal to the maximum likelihood estimate. If the object is to find a generalized minimum variance Bayes' estimate, that is, to minimize the cost functional i i where S is an arbitrary, positive semidefiiite matrix, we simply set a J / a i = Q to find, independent of S, that - - - =l_ 1- . g & l z J d x ~ d x , . . 0-9) which is the conditional mean estimate. J: .. L @ p @ l ~ d x , d x 2 . . 0- 10) where L a ) isa scalar "loss function" of the estimation error 106 where Po is the a priori covariance matrix of&.

In the sequel, we define the error in the estimate of a state vector to be the difference between the estimated 6) and actual (9 values: The covariance of a, designated P, is then given by It provides a statistical measure of the uncertainty in J. The notation permits us to discuss the properties of the covariance matrix independently of the mean value of the state. Some features of the covariance matrix can be seen by treating the error in knowledge of two random system state variables, The covariance matrix of & is r k ~ k l Notice that the covariance matrix of an n-state vector is an n X n symmemc matrix; this fact will be used repeatedly in subsequent chapters.