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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 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 minimizing 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 sampling interval, which means that only the uk move needs to be computed. It can be shown that the use of an incremental model, Eq. (47), leads naturally to integral action in the derived controller. eT = (ah, a;, . . . , a;-I; p;, . . . , p;+L-l) As mentioned previously, in the present application a single manipulated variable is used to control two process outputs. Hence, two incremental models are derived to relate the two state variables, monomer conversion and zero moment of the MWD, to a single 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 sampling interval both models, Eqs. (48)-(49), are solved to calculate the control policies, u: and u:. Note that the final control move applied to the polymerization reactor is, actually, a weighted average of the two independently calculated temperatures, u1 and u2 where g, (OSg, 5 1) and g2 (=1 -gl) are weighting factors which allow different weights to be placed on the two calculated control actions, u: and u:.

Downloaded By: [HEAL-Link 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-line from input-output data by using a robust recursive least-squares estimator. To improve parameter adaptivity in a nonlinear environment, a variable forgetting factor is introduced. (Fortesque et al., 1981). Implementation of the algorithm requires the following steps. Given; {Po = 106*1, e0 = 0, L 2 1 and Zo) 1. Obtain a measurement of the process output, y,. 2. Calculate the prediction error, ek. 3. Determine the variable forgetting factor, A,. 4. Update the covariance matrix, Pk, and the gain Kk. 5. Update the parameter estimates, 8, 8, = i)k-l + Kkek 6. Calculate the predicted output value, Yk+,. 7. Calculate the integral control action uk. The parameter Xo is an "effective asymptotic memory length" that controls 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 ingredients (i.e. monomers, initiators, solvents, transfer agents) are commonly fed to the reactor without any prior purification. As a result inert and reactive impurities are

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