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 Karhunen-Loéve expansion (KLE) 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. Recently, the adaptive model reduction methodology [1] was proposed as a solution to the above problem. In this methodology the eigenfunctions required for the model reduction were computed recursively as new snapshots from the process becomes available. Initially eigenfunctions 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 eigenfunctions required for model reduction.

In this work, we extend the applicability of the above methodology to processes where snapshots of the state evolution become available only periodically. We assume that point measurements from a limited number of sensors are continuously available. The reduced-order model of the PDE system is obtained using the empirical eigenfunctions. We then design a static output observer, based on the continuous point measurements, to estimate the modes of the reduced order model. The associated empirical eigenfunctions are recursively updated, when a new snapshot from the process becomes available, using the adaptive model reduction methodology. A nonlinear controller is designed using the estimated reduced-order model and the parameters of the controller are continuously revised as the process evolves based on changes in the estimated reduced-order model.

The proposed approach is applied to control temperature in a jacketed tubular reactor where first order chemical reaction is taking place. We design an output feedback controller, based on estimated modes from noisy point measurements, to successfully stabilize the process at an open-loop unstable steady state.

[1] A Varshney, S Pitchaiah and A Armaou, “Feedback control of dissipative PDEs using adaptive model reduction”, submitted to AICHE journal, 2008.