Most industrial batch processes are operated through open-loop application of an off-line optimized input profile, such as feed or temperature. This is because modeling accuracy is typically poor (Juba and Hamer, 19861, and direct concentration measurements that would allow to cope with the frequently encountered lack of reproducibility are rare.
However, when on-line measurement information gives access to the system state, on-line reoptimization promises considerable improvement. Since industrial on-line measurements typically do not immediately reveal perfect information on the entire system state, on-line state and parameter estimators need to be used. Their estimates often contain non-negligible uncertainty due to the system state being inferred from indirect, so-called model-based measurements, which can be subject to both stochastic measurement noise and structural measurement-model mismatch.
Given such estimates, it is common practice to perform on-line optimization ignoring their uncertainty. Still, this is optimal theoretically only for ideal linear systems rather than possibly strongly nonlinear batch processes. Another approach is to begin with the open-loop optimal profile and switch to on-line optimization either empirically after a given number of measurements or when estimate uncertainty is sufficiently small. This leaves the problem of finding a good compromise between waiting for good estimates and reacting sufficiently early, as sensitivity of the final operation objective to input changes usually decreases rapidly.
The present contribution suggests a cautious on-line correction mechanism that replaces the switching from open-loop to closed-loop operation with a smooth transition controlled by estimate uncertainty. Its single scalar tuning parameter represents the desired degree of cautiousness or boldness with which current estimates are used for on-line correction of open-loop optimized input profiles. In the limiting cases of no information (large uncertainty) and perfect information (certainty), the corrector naturally reduces to optimal openloop and optimal feedback operation, respectively.