271658 Model Adequacy for Real-Time Optimization
Optimization is important in science and engineering as a way of finding ”optimal” situations, designs or operating conditions. Optimization is typically performed on the basis of a mathematical model of the process under investigation. In practice, optimization is complicated by the presence of uncertainty in the form of plant-model mismatch and unknown disturbances. Without uncertainty, one could use the model at hand, optimize it numerically off-line and implement the optimal inputs in an open-loop fashion. However, because of uncertainty, the inputs need to be re-computed or adjusted in real-time based on measurements. This is the field of real-time optimization, which is labeled RTO for static optimization problems . A standard way of implementing RTO is via the so-called two-step approach of repeated parameter estimation and optimization. In this scheme, measurements are used to adapt the model parameters, and the updated model is used for optimization. The fixed point of this iterative scheme is an important issue.
The quality of the model is often overlooked in real-time optimization. The model is typically inaccurate due to lack of time for detailed modeling or shear complexity. More surprising is the fact that a model can predict the plant outputs well but be inadequate to push the plant to its optimum, which means that the model does not adequately represent the optimality conditions of the plant. The problem of model selection in the two-step RTO approach has been discussed in . If the model is structurally correct and the parameters are identifiable, convergence to the plant optimum can be achieved in a single iteration. However, in the presence of plant-model mismatch, whether the scheme converges, or to which point it does converge, becomes anyone’s guess. This is due to the fact that the objective for parameter adaptation might be unrelated to the cost and constraint values and gradients that drive optimality in the optimization problem. Hence, minimizing the mean-square error of the plant outputs may not help in our quest for feasibility and optimality. Convergence under plant-model mismatch has been addressed in  and , where it has been shown that optimal operation is reached if model adaptation leads to matched KKT conditions for the model and the plant.
This contribution addresses the convergence of two-step RTO schemes in the presence of structural plant-model mismatch. We propose to investigate the parameter estimation and optimization problems in the light of the second-order sufficient conditions of optimality to show that, in general, there are too few degrees of freedom to be able to reach plant optimality. A possible solution to this problem consists in reconciling the objectives of the parameter estimation and optimization problems along the lines of “modeling for optimization” [5,6]. One could for example use as “measured outputs” in the parameter estimation problem estimates of the necessary conditions of optimality for the optimization problem. These issues will be investigated both theoretically and via the simulation of chemical processes at steady state.
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