The measurements are represented by Eq. (4.95), which can be modified if the inputs u directly affect the measurements. The Kalman filter, and the recursive relations to compute the filter gain matrix Kfc, and the covariance propagation matrices P^ and P j are given by Eqs. (4.101), (4.103), (4.102), and (4.99) or (4.83), respectively.

A challenge in this approach is the selection of covariance matrices Qfc, Rfc, and Po- Process knowledge and simulation studies must be used to find an acceptable set of these tuning parameters to prevent biased and poor estimates of state variables. Knowledge of the initial state xo affects the accuracy of estimates as well. If xq initially unknown, the EKF can be restarted from the beginning with each new measurement using the updated estimate Sto\k. Convergence to the unknown xo is usually achieved during the early states of the estimation, but there is a substantial increase in computational load. If feedback control is used during the batch run, rapid convergence to initially unknown disturbance and parameter states can be achieved using this reiterative Kalman filter. One approach to implement the reiterative estimation of Xo is to combine a recursive nonlinear parameter estimation procedure with the EKF [300].

Most batch processes are nonlinear to some degree and may be represented better by nonlinear models. An alternative is to develop a series of local, preferably linear models to describe parts of the batch operation, then link these models to describe the whole batch operation. These models are referred to as local models, operating-regime based models [156, 260], or linear parameter-varying models [312].

Operating-regime Based Models.

Consider the general state space representation in Eqs. (4.44-4.45)

where x, u, y are the state, input (control) and output (measurement) vectors, w and v are disturbance and measurement noise vectors, and f and g are nonlinear function systems. Assume that the batch run can be partitioned into i operating regimes that can be represented sufficiently well by local model structures parameterized with the vector Oi. Each local model will be valid in its particular operating regime. Denote by (pi the operating point (described by some x, u, y) representing a specific regime The whole batch run (the full range of operation) is composed of N regimes: {$i, • ■ ■ , $jv} = The selection of variables to characterize an operating regime will be process dependent, containing a subset of state variables, inputs, and disturbances. Assume the existence of a smooth model validity function p^ that has a value close to 1 for operating points where the model % of Eq.(4.113) is a good description of the process, and close to 0 otherwise. Define an interpolation x = f(x, u, w) y = h(x,u,v)

function cut with range [0,1] that is the normalization of pl:

such that <^>i{4>) = 1- To guarantee a global model, not all local model validity functions should vanish at any operating point <j>.

The modeling framework consists of three tasks [156]:

• Decompose the operating range of the process into a number of operating regimes that completely cover the whole range of operation (complete batch run). This can be achieved based on process knowledge or by using computerized decomposition tools on the basis of an informative data sequence [156].

• Develop a local model structure using process knowledge and data. Assign local model validity functions.

• Identify local model parameters. The unknown parameter sets

• ■ ■ , 6 n are identified. If the models are linear, many model identification methods and tools that are readily available can be used. Attention must be paid during data collection to generate data that contain significant information for all operating regimes.

The local models approach was extended such that a linear parameter varying (LPV) model was obtained by interpolation between multiple local models to emulate the behavior of a nonlinear batch process [312]. First consider the nonlinear process model Eq. 4.112 without disturbances.

Batch processes are operated to track an optimal profile obtained by offline optimization or empirical methods using historical information from previous batches. Local models are obtained by linearizing the model at different points in time on the optimal profile. Denote the optimal profile by (x0,i,y0,i), i = 1, • ■ • ,N where N represents the number of operating points. Linearize Eq. 4.115 using Taylor series expansion and neglecting terms higher than first order. The resulting model is similar to Eqs. 4.61-

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