Nonlinear control of dissipative PDE systems employing adaptive model reduction

The problem of feedback control of distributed processes is considered. Typically this problem is addressed through model reduction where finite dimensional approximations to the original infinite dimensional system are derived. The key step in this approach is the computation of basis functions that are subsequently utilized to obtain finite dimensional ODE models using the method of weighted residuals. The most common approach for this task is the Karhunen-Loeve expansion combined with the method of snapshots. However, this approach requires a priori availability of a sufficiently large ensemble of PDE solution data, a requirement which is difficult to satisfy. In this work we focus on the recursive computation of eigenfunctions using a relatively small number of snapshots. The empirical eigenfunctions are continuously modified as additional data from the process becomes available. We use ideas from the recursive projection method to keep track of the dominant invariant eigenspace of the covariance matrix which is subsequently utilized to compute the empirical eigenfunctions required for model reduction. This dominant eigenspace is continuously modified with the addition of each snapshot with possible increase or decrease in its dimensionality, while simultaneously the computational burden is kept relatively small. The proposed approach is applied to control temperature in a jacketed tubular reactor where first order chemical reaction is taking place and the closed-loop system is successfully stabilized at an unstable steady-state.