Aixtt

To complete the filtering algorithm, update equations that account for new measurement information must be developed. Assume that the estimate of x(t) and its associated covariance matrix are propagated using Eqs. (4.98) and (4.99) and denote the solutions at time tk by xk and PjJT. When a new measurement y*. is received the updated estimates are

with

The same approach as the linear case is used to determine the optimal filter gain matrix:

where

resulting from Taylor series expansion of hjt(xfe) = hfc(xjjT) + H.k(x^)(xk —

EKF equations for a continuous process with discrete-time measurements are summarized in Table 4.3. EKFs for continuous measurements and more advanced filters for nonlinear systems are discussed in [17, 182]. Applications of EKFs to batch processes are illustrated in [72, 120].

Kalman Filters for Processes with Nonstationary Stochastic Disturbances

The literature on Kalman filters focuses mostly on deterministic systems subjected to arbitrary noise to account for modeling error (Eq. 4.70) and measurement error (Eq. 4.71). This framework implies that the true process states can never drift away for prolonged periods from their values predicted by the deterministic model equations [356]. Consequently, the Kalman filter will not contain any integration terms and will not track a

Table 4.3. Summary of extended Kalman filter equations for continuous process with discrete-time measurements

Description

Equation

Other

Process model Measurement model

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