Many industrially relevant diffusion-convection-reaction processes are characterized by the presence of strong spatial variations due to coupling between diffusive and convective mechanisms with chemical reactions. Typical examples of such processes include fluidized-bed and packed-bed reactors, rapid thermal processing, plasma reactors, and chemical vapor deposition. The feedback control problem for these processes is nontrivial owing to the spatially distributed nature of their dynamics. This leads to infinite dimensional dynamic descriptions in appropriate spaces which necessitate the use of model reduction in order to design practically implementable controllers. The key step in this approach is the computation of basis functions that are subsequently utilized to obtain finite dimensional approximations to the original infinite dimensional system. A common approach for this task, when the spatial dynamics are described by nonlinear operators, is the proper orthogonal decomposition (POD) combined with method of snapshots. However, this approach requires the a priori availability of a sufficiently large ensemble of PDE solution data (snapshots) which excites all of the possible spatial modes in the solution of the PDE system, a requirement which may be difficult to satisfy. It is also important to note that the complete spatial profile of the state is needed. Recently, the adaptive POD methodology [1] was proposed as a solution to the above problem. In this methodology the basis functions required for the model reduction were computed recursively as new snapshots from the process becomes available. Initially basis functions were computed using a relatively small number of snapshots; we then kept track of the dominant eigenspace of the covariance matrix which was subsequently utilized to compute the empirical basis functions required for model reduction.

In this talk, we will be presenting our recent results on stabilization of distributed-parameter-systems with partial state measurements (i.e., there are spatial regions where information is unavailable) employing our adaptive proper-orthogonal-decomposition (APOD) methodology. Initially, the incomplete measurements are reconstructed using a gappy-reconstruction procedure and are then utilized to derive and periodically update a reduced-order-model (ROM). The use of APOD methodology along with the gappy procedure allows the development of accurate low-dimensional ROMs for controller synthesis thus resulting in a computationally-efficient alternative to using large-dimensional models with global validity. The proposed gappy-APOD methodology is applied to address the problem of stabilization of an open-loop unstable system modeled by Kuramoto-Sivashinksy equation. The proposed controller successfully stabilizes the system at a spatially uniform steady-state even though only partial information of the state profiles is known.

Reference:

1. Pitchaiah, and A. Armaou, "Feedback control of dissipative distributed parameter systems using adaptive model reduction," Ind. & Eng. Chem. Res., **55**, 906-918, 2010.

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