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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 6 C. KIPARISSIDES et al. which m**in**imizes the quadratic performance **in**dex is obta**in**ed from the solution **of** the follow**in**g Riccati Eqs. (Sage, 1968): At each sampl**in**g **in**terval, the matrix difference Eqs. (26)-(27) are solved backward **in** time (k = N, N - 1, . . . , 1) with the f**in**al condition given by Eq. (28). Note that the ga**in** matrix, K,, can be calculated **of**f-l**in**e and applied to the physical system as it runs forward **in** real-time accord**in**g to the "sub-optimal" **control** law: uk = -K,Y(k) (29) In this work a f**in**al 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 nom**in**al value is applied. F**in**ally, the value **of** the **weight****in**g 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 dur**in**g the last decade by several **in**dustrial work groups. In DMC, a non-parametric convolution representation **of** the process is employed to design a digital **control**ler based on a long-term predictor **of** future process outputs. Model Development If the dynamics **of** the process are assumed to be l**in**ear, the follow**in**g discrete convolution model can be **in**voked 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 **in**tervals required for the output to reach steady state after a step change **in** the **in**put. Note that for k Z-N, a, =aN. yo denotes the process **in**itial condition and Sui is the jth change **in** the process **in**put (buj = u, - uj-,). F**in**ally, dk accounts for the contribution **of** unmeasured disturbances and model**in**g errors. Assum**in**g a time horizon **of** M future sampl**in**g **in**tervals and fix**in**g the number **of** future **in**put changes to L moves (L s M), we obta**in** the follow**in**g prediction

Downloaded By: [HEAL-L**in**k 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 comb**in**ed 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-**in**put, multi-output processes (MIMO). Thus, for a MIMO process with n-**in**puts and r-outputs, Eq. (33) becomes. A, represents a dynamic submatrix **of** the MIMO process with respect to the r-output and n-**in**put, 6u, is the vector **of** future **control** moves associated with the n-**in**put variable and the vector 6y, denotes the change **in** the r-output caused by current and future **control** moves. Multistep Predictive Conrrol By sett**in**g the future output values, yk+,, equal to the desired setpo**in**t values, y;+,, **in** Eq. (33), we obta**in** 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. Assum**in**g closed-loop behaviour **of** the process, that is, **in**clud**in**g the effects **of** both past and future **in**put moves on the process output, we can obta**in** 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 obta**in**ed by m**in**imiz**in**g a

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