Adaptive model reduction and feedback control of transport reaction processes

The problem of robust feedback control of spatially distributed processes described by dissipative partial differential equations (PDEs) is considered. Typically, this problem is addressed through model reduction where finite dimensional approximations to the original PDE system are derived. A common approach to this task is the Karhunen-Loève expansion combined with the method of snapshots. To circumvent the issue of a priori availability of a sufficiently large ensemble of PDE solution data, we focus on the recursive computation of eigenfunctions as additional data from the process become available. Initially, an ensemble of eigenfunctions is constructed with a relatively small number of snapshots. The dominant eigenspace of this ensemble is then identified to compute the empirical eigenfunctions required for model reduction. This dominant eigenspace is reevaluated with the addition of new snapshots the dominant eigenspace is reevaluated and its dimensionality may increase or decrease. Because this dimensionality is typically small the computational burden is also small. This approach is applied to representative example of a thin catalytic rod, where zero-th order exothermic reaction is taking place. The available model for heat transfer in this catalytic rod was assumed to contain uncertain parameters (heat of reaction) and as a result standard control design techniques cannot be used. We demonstrate the effectiveness of the proposed methodology in this case by designing robust controllers to stabilize an unsteady process operating condition in the thin-catalytic rod.