Robust control of dissipative PDE systems in the presence of uncertainty using adaptive model reduction

The problem of robust feedback control of spatially distributed processes described by highly 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-Loeve 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 a representative example of dissipative PDEs, to demonstrate the effectiveness of the approach to design robust controllers.