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control of molecular weight in a batch polymerization reactor using ...

control of molecular weight in a batch polymerization reactor using ...

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Downloaded By: [HEAL-Link Consortium] At: 12:27 29 July 2008 8 C. KIPARISSIDES et al. modified performance index of the following form: min J = E ~QE + 6uTR 6u 6u where Q and R are positive definite matrices which allow different weights to be assigned on the deviation vector, E, and the future control moves, 6u. The unconstrained solution to this minimization problem can be shown to be (Marchetti et al., 1983): In practice, only the current control move, auk, is calculated by multiplying the first row of the pseudo-inverse matrix, (ATQA + R)-'A~Q, by the error prediction vector, E. That is, The Control Algorithm The most important feature of the DMC algorithm is the recursive updating procedure of the error prediction vector, E (Balhoff and Lau, 1985). The principal tuning parameters involved in a DMC controller are: the dimension of the convolution model times the sampling interval, Nz, the prediction horizon, M, the control horizon, L, the weighting matrix, Q and the move suppresion matrix, R, in Eq. (37). General guidelines for the selection of these tuning parameters are given by Garcia and Morari, (1982) and Maurath et al. (1985). EXTENDED SELF-TUNING CONTROLLER The extended self-tuning regulator (ESTR) is a more robust implementation of the classical STR algorithm, which was developed by Astrom et al. (1977) to control time-invariant linear systems with unknown or uncertain models operating under the influence of stochastic disturbances. Ydstie et al. (1985) modified the basic STR algorithm by introducing the concepts of the variable forgetting factor and of the extended control horizon to improve the adaptivity and robustness of the basic STR. The variable forgetting factor is introduced in the least squares algorithm to keep a measure of the information content of the estimation constant by discounting the limited information of past data. The use of an extended control horizon enhances the controller's robustness in the presence of time-varying delays and nonminimum phase system characteristics. As it will become apparent in the following section, our closed-loop control objective is to maintain the monomer conversion and number-average molecular weight along some desired trajectories despite the presence of process disturbances in the initiator concentration by manipulating the polymerization temperature. As a result, a single manipulated variable is used to control two output variables.

Downloaded By: [HEAL-Link Consortium] At: 12:27 29 July 2008 Model Development POLYMERIZATION REACTOR CONTROL 9 Assume that an autoregressive moving average (ARMA) model representation can be utilized to describe the process dynamics: where ~(z-') and B(z-l) are polynomials in the back shift operator z-I, yk is the process output at the kth interval, u, is the control input delayed by D samples (zl), b is a bias parameter, and vk is an independent noise sequence with zero mean and variance sk. In order to derive the controller algorithm, Eq. (40) is expressed in an equivalent prediction model form. Following the theoretical developments of Ydstie et al. (1985), an extended control horizon, L, is chosen so that E{yk+,/Yk) = y",,,, where Yk denotes all the information available at time k and y",, is the desired setpoint value at time k + L. Note that the prediction horizon L is greater than the true process time delay, D. As a result, D can be chosen to be equal to its minimum value ( D = 1). Accordingly, the prediction model can be cast into the following vector-matrix form: a; and /?, are the unknown coefficients of the polynomials A and B, which might be of the same order N. 6 is a modified bias parameter in the prediction model and q+, is a closed-loop prediction error with zero mean and variance rk. It is interesting to notice that, similarly to the DMC Eqs. (31)-(32), the prediction Eq. (41) can be written as, We see that yk+, and jk+, can be interpreted as the prediction output values based on the past-future control moves and the past only moves, respectively. Extended Horizon Control From Eq. (42), we can obtain an estimate of the closed-loop prediction error, q+,, by setting the future output value, yk+,, equal to the desired setpoint value, y;+,. The calculation of the future control moves is obtained by selecting the values of

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