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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 6 C. KIPARISSIDES et al. which minimizes the quadratic performance index is obtained from the solution of the following Riccati Eqs. (Sage, 1968): At each sampling interval, the matrix difference Eqs. (26)-(27) are solved backward in time (k = N, N - 1, . . . , 1) with the final condition given by Eq. (28). Note that the gain matrix, K,, can be calculated off-line and applied to the physical system as it runs forward in real-time according to the "sub-optimal" control law: uk = -K,Y(k) (29) In this work a final value for PN = 0 was used. Q was chosen to be a (3 x 3) diagonal matrix with elements q,, = 1, q,, = 5 x lo7 and q,, = 0. The zero value for q,, implies that no explicit control on the deviation of p, from its nominal value is applied. Finally, the value of the weighting scalar parameter R was chosen to vary in the range 1 x 10-4-5 x DYNAMIC MATRIX CONTROL Dynamic Matrix Control (DMC) is a multivariable control algorithm which was developed during the last decade by several industrial work groups. In DMC, a non-parametric convolution representation of the process is employed to design a digital controller based on a long-term predictor of future process outputs. Model Development If the dynamics of the process are assumed to be linear, the following discrete convolution model can be invoked to describe the one-step ahead output of a SISO process (Morshedi et al., 1985) where {a,) (j = 1, 2, . . . , N) are the "unit step response coefficients" of the system and N is the number of time intervals required for the output to reach steady state after a step change in the input. Note that for k Z-N, a, =aN. yo denotes the process initial condition and Sui is the jth change in the process input (buj = u, - uj-,). Finally, dk accounts for the contribution of unmeasured disturbances and modeling errors. Assuming a time horizon of M future sampling intervals and fixing the number of future input changes to L moves (L s M), we obtain the following prediction

Downloaded By: [HEAL-Link Consortium] At: 12:27 29 July 2008 POLYMERIZATION REACTOR CONTROL 7 equations for the yk+i and jk+i process outputs. yk+i and jk+i denote the future process output values based on the combined past-future moves and the past only moves, respectively. The prediction Eq. (31) can be readily expressed in the more convenient vector-matrix form 6y=(y-y)=A6u (33) where A is an (M X L) "dynamic matrix" composed of the unit step response coefficients, ai. The above model developments can easily be extended to multi-input, multi-output processes (MIMO). Thus, for a MIMO process with n-inputs and r-outputs, Eq. (33) becomes. A, represents a dynamic submatrix of the MIMO process with respect to the r-output and n-input, 6u, is the vector of future control moves associated with the n-input variable and the vector 6y, denotes the change in the r-output caused by current and future control moves. Multistep Predictive Conrrol By setting the future output values, yk+,, equal to the desired setpoint values, y;+,, in Eq. (33), we obtain an expression for the open-loop error prediction vector, E. Note that the error prediction vector, E, is simply expressed in terms of the known future output values, h+i. Assuming closed-loop behaviour of the process, that is, including the effects of both past and future input moves on the process output, we can obtain an estimate of the so-called closed-loop error prediction vector, E, directly from Eq. (31). E=(~'-~)=E-:\~U (36) where e = [(Y;+I- Y~+,) ...(Y;+~- Y~+M:II' The calculation of the future control moves, 6u, is obtained by minimizing a

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