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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 10 C. KIPARISSIDES et al. uk, uk+,, . . . , u,+,-~ so that the extended horizon **control** criterion, E,+, satisfied. That is, = 0, is There are several possible **control** policies that may satisfy the above equation. In the present work, the **control** moves are calculated by m**in**imiz**in**g the **control** effort, Ch1 u:+~-? Ydstie et al. (1985) showed that the solution to this problem is given by In practice, only the first **control** move is implemented at each sampl**in**g **in**terval, which means that only the uk move needs to be computed. It can be shown that the use **of** an **in**cremental model, Eq. (47), leads naturally to **in**tegral action **in** the derived **control**ler. eT = (ah, a;, . . . , a;-I; p;, . . . , p;+L-l) As mentioned previously, **in** the present application a s**in**gle manipulated variable is used to **control** two process outputs. Hence, two **in**cremental models are derived to relate the two state variables, monomer conversion and zero moment **of** the MWD, to a s**in**gle manipulated variable, the **polymerization** temperature. Thus, y1 denotes the monomer conversion **in** the **batch** **reactor**, y2 is the normalized zero moment, po, **of** the MWD and u is the **polymerization** temperature. At each sampl**in**g **in**terval both models, Eqs. (48)-(49), are solved to calculate the **control** policies, u: and u:. Note that the f**in**al **control** move applied to the **polymerization** **reactor** is, actually, a **weight**ed average **of** the two **in**dependently calculated temperatures, u1 and u2 where g, (OSg, 5 1) and g2 (=1 -gl) are **weight****in**g factors which allow different **weight**s to be placed on the two calculated **control** actions, u: and u:.

Downloaded By: [HEAL-L**in**k Consortium] At: 12:27 29 July 2008 POLYMERIZATION REACTOR CONTROL 11 Adaptive Controller Implementation The parameters **of** the prediction models, Eqs. (48)-(49), can be estimated on-l**in**e from **in**put-output data by us**in**g a robust recursive least-squares estimator. To improve parameter adaptivity **in** a nonl**in**ear environment, a variable forgett**in**g factor is **in**troduced. (Fortesque et al., 1981). Implementation **of** the algorithm requires the follow**in**g steps. Given; {Po = 106*1, e0 = 0, L 2 1 and Zo) 1. Obta**in** a measurement **of** the process output, y,. 2. Calculate the prediction error, ek. 3. Determ**in**e the variable forgett**in**g factor, A,. 4. Update the covariance matrix, Pk, and the ga**in** Kk. 5. Update the parameter estimates, 8, 8, = i)k-l + Kkek 6. Calculate the predicted output value, Yk+,. 7. Calculate the **in**tegral **control** action uk. The parameter Xo is an "effective asymptotic memory length" that **control**s the speed **of** parameter adaptation. A large value **of** Z, will result **in** a slow adaptation, whereas a short memory length will give fast adaptation but noisy estimates. DISCUSSION AND RESULTS For the economic production **of** polymers, the reaction **in**gredients (i.e. monomers, **in**itiators, solvents, transfer agents) are commonly fed to the **reactor** without any prior purification. As a result **in**ert and reactive impurities are

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