(747d) Output Feedback Control of Transport Reaction Processes Using Adaptive Proper Orthogonal Decomposition

Most of the processes in the chemical process industry necessitate the consideration of transport phenomena often coupled with chemical reactions. Examples include distillation in petroleum processing, plasma enhanced chemical vapor deposition, etching and metallorganic vapor phase epitaxy (MOVPE) in semiconductor manufacturing and tin float bath in glass production.

These processes’ control problem is nontrivial due to the spatially distributed nature of the associated system dynamics. Typically, this problem is addressed through model reduction, where the controller design is based on finite dimensional approximations to the original infinite dimensional PDE system. The computation of basis functions that are subsequently utilized to obtain finite dimensional ordinary differential equation (ODE) models is the most important part of this design. A common approach to this task is the Karhunen-Loeve expansion combined with the method of snapshots, then using the method of weighted residuals to construct the finite ODE. Current research focus is on the recursive computation of eigenfunctions as additional data from the process becomes available due to  a) the computational load reduction and  b) to circumvent the issue of a priori availability of a sufficiently large ensemble of PDE solution data. Initially, an ensemble of eigenfunctions is constructed based on a relatively small number of snapshots. The dominant eigenspace is then utilized to compute the empirical eigenfunctions required for model reduction. This dominant eigenspace is recomputed with the addition of each snapshot with possible increase or decrease in its dimensionality [1].

In this paper, our recent results on control of distributed parameter systems with limited state measurements (i.e., there are only limited regions where information is available continuously) employing new approach in adaptive proper-orthogonal-decomposition (APOD) methodology will be presented. In previous APOD approaches, the ensemble of the snapshots that is used to update the basis functions just includes the most recent snapshots of the system due to computational load reduction that has the important snapshots loss possibility. To avoid this possible loss, we use a new procedure to keep the most important snapshots in the ensemble. Also, the exact reduced order model (ROM) of the system is not accessible because we do not access to all states continuously. Initially, the limited measurements outputs are used to construct a dynamic observer that is used to construct ROM. The use of APOD methodology with some motivations along with the dynamic observer 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 adaptive model reduction and control methodology is applied to a typical diffusion reaction process that exhibits nonlinear dynamic behavior; we consider an elementary exothermic reaction taking place on a thin catalytic rod [2]. The temperature of the rod is adjusted by means of limited actuators located along the length of the rod. The spatial profile of the temperature of the rod is described by a parabolic PDE. The proposed controller structure successfully stabilizes the system at a spatially uniform steady-state profile even though only limited information of the state is known.

References:

1. S. Pitchaiah, and A. Armaou, “Feedback control of dissipative distributed parameter systems using adaptive model reduction”, Ind. & Eng. Chem. Res., 55, 906-918, 2010.
2. A. Varshney, S. Pitchaiah and A. Armaou, “Feedback control of dissipative PDE systems using adaptive model reduction”, AICHE Journal., 55, 906-918, 2009.