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

Downloaded By: [HEAL-L**in**k Consortium] At: 12:27 29 July 2008 8 C. KIPARISSIDES et al. modified performance **in**dex **of** the follow**in**g form: m**in** J = E ~QE + 6uTR 6u 6u where Q and R are positive def**in**ite matrices which allow different **weight**s to be assigned on the deviation vector, E, and the future **control** moves, 6u. The unconstra**in**ed solution to this m**in**imization problem can be shown to be (Marchetti et al., 1983): In practice, only the current **control** move, auk, is calculated by multiply**in**g the first row **of** the pseudo-**in**verse 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 updat**in**g procedure **of** the error prediction vector, E (Balh**of**f and Lau, 1985). The pr**in**cipal tun**in**g parameters **in**volved **in** a DMC **control**ler are: the dimension **of** the convolution model times the sampl**in**g **in**terval, Nz, the prediction horizon, M, the **control** horizon, L, the **weight****in**g matrix, Q and the move suppresion matrix, R, **in** Eq. (37). General guidel**in**es for the selection **of** these tun**in**g parameters are given by Garcia and Morari, (1982) and Maurath et al. (1985). EXTENDED SELF-TUNING CONTROLLER The extended self-tun**in**g regulator (ESTR) is a more robust implementation **of** the classical STR algorithm, which was developed by Astrom et al. (1977) to **control** time-**in**variant l**in**ear systems with unknown or uncerta**in** models operat**in**g under the **in**fluence **of** stochastic disturbances. Ydstie et al. (1985) modified the basic STR algorithm by **in**troduc**in**g the concepts **of** the variable forgett**in**g factor and **of** the extended **control** horizon to improve the adaptivity and robustness **of** the basic STR. The variable forgett**in**g factor is **in**troduced **in** the least squares algorithm to keep a measure **of** the **in**formation content **of** the estimation constant by discount**in**g the limited **in**formation **of** past data. The use **of** an extended **control** horizon enhances the **control**ler's robustness **in** the presence **of** time-vary**in**g delays and nonm**in**imum phase system characteristics. As it will become apparent **in** the follow**in**g section, our closed-loop **control** objective is to ma**in**ta**in** the monomer conversion and number-average **molecular** **weight** along some desired trajectories despite the presence **of** process disturbances **in** the **in**itiator concentration by manipulat**in**g the **polymerization** temperature. As a result, a s**in**gle manipulated variable is used to **control** two output variables.

Downloaded By: [HEAL-L**in**k Consortium] At: 12:27 29 July 2008 Model Development POLYMERIZATION REACTOR CONTROL 9 Assume that an autoregressive mov**in**g 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 **in**terval, u, is the **control** **in**put delayed by D samples (zl), b is a bias parameter, and vk is an **in**dependent noise sequence with zero mean and variance sk. In order to derive the **control**ler algorithm, Eq. (40) is expressed **in** an equivalent prediction model form. Follow**in**g 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 **in**formation available at time k and y",, is the desired setpo**in**t 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 m**in**imum value ( D = 1). Accord**in**gly, the prediction model can be cast **in**to the follow**in**g 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 **in**terest**in**g 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 **in**terpreted 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 obta**in** an estimate **of** the closed-loop prediction error, q+,, by sett**in**g the future output value, yk+,, equal to the desired setpo**in**t value, y;+,. The calculation **of** the future **control** moves is obta**in**ed by select**in**g the values **of**

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