The LTV model can be extended for state estimation. An adaptive technique using the Bayesian framework has been developed [40]. Consider the discrete time version of the state space model Eq. (4.116) with zero-mean Gaussian process and measurement noises w and v, respectively

Xfe+i = F(pt)xfc + G(pi)uk + wfc yk = C(p,)xfc +T>(Pi)uk +vfc (4.122)

with noise covariance matrices Q^ and Rfc for w and v, respectively. A moving horizon estimator that updates the piS based on a past window of data of length ne can be developed. At the start of the batch, the number of data samples is fewer than ne and the measurement data history is given by data of different length

Let p(j¡Yfc) represent the probability that model j is describing the batch process based on measurements collected until time k. Bayes theorem can be used to compute the posterior probability p(j|Yfc) given the prior probability (computation of p(j) before measurements at time k are used)

/(yfclj, Yfc-i)p(j|Yfc-i) £?/(yfc|i.Yfc_i)p(i|Yfc_i)

where /(yfc|j, Yfc_i) is the probability distribution function (PDF) of the outputs at time k computed by using model j and the measurement history Yfc_i collected until time k — 1. A Kalman filter is designed for each model in order to evaluate the PDFs. The jth Kalman filter assumes that the jth model matches the plant and any mismatch is caused by process and measurement noises with covariance matrices Qj and Rr The update of state variables of the jth model is

where i = ^2f=iPj,k^-j,k\k- Following the derivations of Kalman filters in Section (4.3.3), measurements at time k provide a correction to the updates

Xj,fc|fc = Xj,fc|fc-i + Kj (yfc - Cj.fcX^fcifc-! - D^fcUfc) (4.126)

where Kj denotes the Kalman filter gain for the jth model. The outputs of the jth model are yj,k = + • (4.127)

If the assumptions about the accuracy of the jth model and noise characteristics hold, then the model residuals £Jtk = Yk — fj,k will have zero mean and covariance ilj — CjPjCj + Rj. Here Pj is the state covariance matrix from the jth Kalman filter. The PDF is computed using [312]

^pHf^Vflfc)

Estimates of model probabilities are then obtained by substituting recursively Eq. (4.128) into Eq. (4.124) [312], To reduce the time required for the computations to converge to the correct probability model, the probabilities are initialized as [312]:

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