where pi > (1 — pi)/(N — 1) and N is the number of local models. The relative magnitude of pi with respect to p, depends on the expected magnitude of disturbances, for large disturbances pi is closer to pi [312].

Local model dynamics are affected by disturbances entering the batch process, and the PDFs of various local models may become identical. The proximity of model outputs to the optimal profiles may be used to select the best local model with a moving horizon Bayesian estimator (MHBE) with time-varying tuning parameters [312]. The aim of the MHBE is to assign greater credibility to a model when plant outputs are closer to the outputs around which the model is identified. This reduces the covariance of model residuals flj^ for model i at time k. This approach is implemented by reducing the noise covariance matrices Q- t and Rt,fc which may be used as the tuning parameters for the respective Kalman filters. Relating these covariances to deviations from optimal output trajectories

where yo>; is the optimal output profile for local model i and a is a tuning parameter, Qj^ and R, ^ of the most likely local model is reduced. The Euclidian norm in Eqs. 4.130 is defined as || x ||2= xTx. Consequently, the residual covariance matrix flitk is reduced as well and the probability of model i is increased. The parameter a reflects the trust in the model and at higher values promotes rapid transition between models [312]. Case studies reported in [312] indicate that model predictive control of a batch reactor using LVS models provided better control than model predictive control with extended Kalman filters.

A different way of approximating the nonlinear process behavior is based on the use of linear ARX models for each sampling instant. Define the input (u), output (y) and disturbance (d) sequences over the entire batch ne

ne run as u = [uT(0) uT(l) y = [yr(l) yT(2) d = [dT(l) dT(l)

where N is the batch run data length. Given the initial condition y^, = y(0), a nonlinear model relating the outputs to inputs and disturbances is expressed as y = A4(u,yini,d) = 7V(u,d) + Aimi(yi,

The nonlinear model JV(u, d) can be approximated with one linear model for each sampling instant [70]. Denote a specific or optimal output trajectory by yo, the batch index by k and define the output error trajectory ey = yo ~ yfc- The state equations for the output error are

where Aufe+i = uk+i-\ik, Ayini,fe+i = ymi.k+i-yini.fc, and v^ are zero-mean, independently and identically distributed random noise sequences with respect to k, and eyk is the noise-free (cannot be measured) part of the error trajectory. Matrices G'7 and Gfni are linear system approximations. The same modeling approach can be applied to secondary outputs s (outputs that are not used in control systems) and quality variables q and the resulting models can be combined. Define the error trajectory vector for the controlled outputs, secondary outputs and quality variables as e* =

Ayini.fc

The resulting combined model then is [70]

ëfc+1 = efc - GAufc+i - Ginieini^+i + wk efc = ëfc + vfe .

To enable the use of a Kalman filter for estimation of initial conditions ejni, the initial condition can be modeled with a batchwise random walk model [70]:

®ini,ic+l — 6ini,fc "I" "ini.fc zini ,k = eini,fc +

where is a batchwise white disturbance, z^i^. is the measured value of einiand £ini k is a batchwise white measurement noise.

The general form of the model has a very large dimension. Structural information is used in [70, 200] to reduce the dimensions of the model.

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