Numerical solution of the optimization problem

As it may be hard (or even impossible) to compute the derivatives of F with respect to the manipulated variables, approximations are normally provided for both H and VF in Equations 8.25 and 8.26. (The use of numerical procedures based on variational principles was very popular in the past [ 161 ]. In order to solve variational problems numerically, standard Newton-Raphson procedures are generally used to solve the resulting two-boimdary value problem that is associated with the variational formulation. For this reason, optimization of dynamic problems based on variational principles is also included in this set of SQP-related numerical techniques.) 

The performances of SQP and RSA were compared during the optimization of grade transitions in solution MMA polymerizations performed in CSTRs [174]. As it was shown that multiple optima were possible, the use of RSA is advantageous because it leads to the global optimum solution more frequently than the SQP and is less sensitive to initialization of the numerical procedure. 

A numerical technique that has become very popular in the control field for optimization of dynamic problems is the IDP (iterative dynamic programming) technique. For application of the IDP procedure, the dynamic trajectory is divided first into NS piecewise constant discrete trajectories. Then, the Bellman s theory of dynamic programming [175] is used to divide the optimization problem into NS smaller optimization problems, which are solved iteratively backwards from the desired target values to the initial conditions. Both SQP and RSA can be used for optimization of the NS smaller optimization problems. IDP has been used for computation of optimum solutions in different problems for different purposes. For example, it was used to minimize energy consumption and byproduct formation in poly(ethylene terephthalate) processes [ 176]. It was also used to develop optimum feed rate policies for the simultaneous control of copolymer composition and MWDs in emulsion reactions [36, 37]. 

Assume that an optimum feed flow rate trajectory is searched for the emulsion copolymerization problem described in Section 8.3.3. In this case, also assume that the copolymer composition should be constant throughout the batch. Also assume that the final monomer conversion should be as close as possible to 1, to allow for rninirnization of the residual monomer. In order to achieve the control objectives, one is allowed to manipulate the feed flow rate of monomer 1, which is assumed to be the most reactive monomer, and the initial monomer concentrations. In this particular case, one may write the following objective function  

Equation 8.27 assumes that the batch time tp can be discretized into NU time intervals, where the feed flow rate is kept constant Tp represents the desired copolymer composition ki are the weighting functions or the relative importance of controlling the instantaneous copolymer composition, when compared to controlling the final residual monomer. (As discussed previously, the proposed control strategy may lead to thermal runaway, if serious heat transfer limitations are present. If this is the case, the energy balance must be included in the model and constraints should be imposed on temperature values and temperature profiles.) The process model can be written as 

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Figure 3.6 shows the lines of constant Gibbs energy determined by Equation 3.81 as a function of the two reaction extents. We see immediately that the minimum is unique. Notice in Figure 3.6 that G is defined only in the region specified by the constraints. The numerical solution of the optimization problem is... 

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