(389d) Robust Controller Design Based On Adaptive Model Reduction for Transport-Reaction Processes

Most of the processes relevant to the chemical process industry necessitate the consideration of transport phenomena (fluid flow, heat and mass transfer) often coupled with chemical reactions. Examples range from reactive distillation in petroleum processing to plasma enhanced chemical vapor deposition, etching and metallorganic vapor phase epitaxy in semiconductor manufacturing. Mathematical descriptions of these transport-reaction processes can be derived from dynamic conservation equations and usually involve highly dissipative (typically parabolic) partial differential equation (PDEs) systems. The problem of feedback control of such processes is nontrivial owing to these spatially distributed mathematical descriptions.

Typically, this problem is addressed through model reduction where finite dimensional approximations to the original PDE system are derived. A common approach used for this task is the method of snapshots. This method has been extensively utilized to ``empirically'' compute the eigenfunctions using an ensemble of solution data obtained either through experimental observations or from detailed numerical simulations. However, the a priori availability of a large ensemble of snapshots is necessary to accurately characterize the behavior of any new trends that become available during the course of process evolution. Generating such an ensemble is not straightforward as it necessitates using a suitably designed input to excite all the modes. This is especially relevant when control of the process is considered as there is a need to extend the basis function set in order to describe the new trends in the data. To address these concerns we formulated the adaptive model reduction methodology, wherein initially the basis functions required for the construction of the reduced order model were derived from a small ensemble of snapshots that does not represent the entire process. These basis functions were updated, in a computationally efficient way, as new data from the process becomes available.

In the present work, the applicability adaptive model reduction methodology is extended to design robust controllers that address the issue of model uncertainty. In general, robust control methodologies for dissipative PDEs like Lyapunov redesign require the off-line availability of the dominant eigenvalues and eigenfunctions of the process. This restrictive requirement is relaxed in our method by identifying these dominant eigenvalues and eigenfunctions on demand as new information from the process becomes available. Due to the computational efficiency of the proposed method this can be achieved on-line. We evaluate the effectiveness of this approach numerically through a representative diffusion-reaction process involving multiple steady states.