Model-plant mismatch

Once the optimal feed rates were obtained, they were applied to the actual process (i.e. simulation by the mechanistic model of the process). Table 2 shows the difference between the amounts of the final product and by-product on neural network model and the actual process. It can be seen from Table 2 that the actual amounts of product and by-product rmder these optimal control policies are quite different from the neural network model predictions. This indicates that the single neural network based optimal control policies are only optimal on the neural network model and are not optimal on the real process. Hence, they are not reliable. This is mainly due to the model plant mismatches, which is rmavoidable in data based modelling. 

A method to overcome the impact of model plant mismatch on optimisation performance was previously investigated by Zhang [8] where model prediction confidence bormds are incorporated as a penalty in the objective function. Therefore, the objective function can be modified as... 

Figure 4.15 Evolution of the process composition variables for a 10% increase in the production rate at t = 0, under plant-model parameter mismatch, (a) Product purity and (b) reactor impurity level.

Figure 5.20 Evolution of the process stream flow rates for a 25% unmeasured increase in the inlet impurity levels yi0 occurring at t = 0, under plant-model parameter mismatch. The reaction rate and the mass-transfer coefficient /CB in the controller model are assumed to be overestimated by 10% compared with their values in the plant, (a) Effluent and recycle flow rates, and (b) product flow rate.

Figures 7.23-7.27 show the closed-loop profiles for a 10% increase in the production rate at operating point I (attained by increasing Fo), and a decrease in the purity setpoint to Cb,Sp = 1.888 mol/1 - this reduction is necessary since the nominal purity is beyond the maximum attainable purity for the increased throughput. Although controller design was carried out to account for the inverse response exhibited by the system at operating points II and III, and in spite of the plant-model parameter mismatch, the proposed control structure clearly yields good performance at operating point I as well.

Figure 7.23 Evolution of the coolant flow rate for a 10% rise in the production rate at operating point I, under plant-model parameter mismatch. The heat transfer coefficient U in the controller model is overestimated by 10% compared with its value in the plant.

Simulation studies carried out to compare the amount of desired product C obtained from on-line dynamic optimization strategy with that from off-line strategy, are cases where the perfect model (all parameters correctly specified) is used (nominal case), and where plant/model mismatch is introduced by changing parameters in actual plant i.e. pre-exponential rate constant (kf) decreased by 50% and activation energy (Ea) increased by 20% from their nominal values, as shown in Table 5. 

For the mismatch in the value of the desired product C = 7.8851 can be achieved at the end of batch for the on-line optimization strategy which is higher than that obtained from off-line strategy where the mismatch is not noticed (C = 7.6751). Similar results can be observed under the case of plant model mismatch in k as shown in Table 5. These results indicate clearly that the performance of batch reactor operation is improved via the proposed strategy. Due to similarity in their control responses, only the result for change in k is shown in Fig. 7. 

Finally, with a change in both k (—50% ko) and k (+20% Ea) in plant model, the results using the on-line optimization strategy show that the GMC controller is able to accommodate this change very well as can be seen in Fig. 8(a). Fig. 8(b) presents the performance of the EKF for estimation of k and k. Since the EKF estimates these parameters close to the true values, the mismatch is eliminated. That leads to high product C obtained at the final batch time (C = 10.2137) compared to the value of C = 8.5827 obtained from the off-line optimization strategy. 

Figure 3.11 Closed-loop response of the product composition and reactor holdup for a 15% increase in production rate and 1.5% decrease of the product-purity setpoint, in the presence of plant-model mismatch, (a) Product stream composition and (b) reactor holdup and setpoint.

In Greaves et al. (2001) and Greaves (2003), instead of using a rigorous model (as in the methodology described above), an actual pilot plant batch distillation column is used. The differences in predictions between the actual plant and the simple model (Type III and also in Mujtaba, 1997) are defined as the dynamic process-model mismatches. The mismatches are modelled using neural network techniques as described in earlier sections and are incorporated in the simple model to develop the hybrid model that represents the predictions of the actual column. 

B show the model and pilot plant predictions respectively. Figure 12.6 clearly shows that there are large process-model mismatches in the composition profiles although for a given batch time of tdiS = 220 min the amount of distillate achieved by the experiment was the same as that obtained by the simulation. These process-model mismatches can be attributed to factors such as use of constant Vmodei instead of a dynamic one constant relative volatility parameter used in the model and uncertainties associated with it actual efficiency of the plates. 

Keywords Measurement-based optimization Real-time optimization Plant-model mismatch Model adaptation Model parameterization. 

It is weU known that the interaction between the model-update and reoptimization steps must be considered carefully for the two-step approach to achieve optimal performance. In the absence of plant-model mismatch and when the parameters are structurally and practically identifiable, convergence to the plant optimum may be achieved in one iteration. However, in the presence of plant-model mismatch, whether the scheme converges, or to which operating point the scheme converges, becomes anybody s guess. This is due to the fact that the update objective might be unrelated to the cost or constraints in the optimization problem, and minimizing the mean-square error in y may not help in our quest for feasibility and optimality. To alleviate this difficulty, Srinivasan and Bonvin [23] presented an approach where the criterion in the update problem is modified to account for the subsequent optimization objective. 

Convergence under plant-model mismatch has been addressed by several authors [3,8] it has been shown that an optimal operating point is reached if model adaptation leads to a matching of the KKT conditions for the model and the plant. 

In order to overcome the modeling deficiencies and to handle plant-model mismatch, several variants of the two-step approach have been presented in the literature. Generically, they consist in modifying for the cost and constraints of the optimization problem for the KKT conditions of the model and the plant to match. The optimization problem with modifiers can be written as follows ... 

This last class of methods provides a way of avoiding the repeated optimization of a process model by transforming it into a feedback control problem that directly manipulates the input variables. This is motivated by the fact that practitioners like to use feedback control of selected variables as a way to cormteract plant-model mismatch and plant disturbances, due to its simphcity and reliability compared to on-line optimization. The challenge is to find functions of the measured variables which, when held constant by adjusting the input variables, enforce optimal plant performance [19,21]. Said differently, the goal of the control structure is to achieve a similar steady-state performance as would be realized by an (fictitious) on-line optimizing controller. 

Parameter identification is complicated by several factors (i) the complexity of the models and the nonconvexity of the parameter estimation problems, and (ii) the need for the model parameters to be identifiable from the available measurements. Moreover, in the presence of structural plant-model mismatch, parameter identification does not necessarily lead to model improvement. In order to avoid the task of identifying a model on-line, fixed-model methods have been proposed. The idea therein is to utilize both the available measurements and a (possibly inaccurate) steady-state model to drive the process towards a desirable operating point. In constraint-adaptation schemes (Forbes and Marlin, 1994 Chachuat et al., 2007), for instance, the measurements are used to correct the constraint functions in the RTO problem, whereas a process model is used to... 

Model predictive control node. MPC has several components. It has a model (usually an identified linear) of the world. It has a KD where past values of the manipulated variables (MVs) and controlled variables (CVs) are stored. In this KD other information is stored as MVs and CVs limitations, weighting factors, etc. The model uses the inputs to predict the future. This state is used in Behavior Generation module. In this module an optimization is performed to select the best action plan. This plan (a set of movements for the MVs along with CVs values) is set and sent to the regulatory level. Some preprocessing is implemented as well. The MPC module implements also a feedback loop to correct model errors (due to model mismatch with the actual plant). 

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