Output feedback control of the FitzHugh-Nagumo equation using adaptive model reduction

This work addresses the problem of tracking and stabilization of FitzHugh-Nagumo equation (FHN) subject to Neumann boundary conditions via static output feedback control using adaptive model reduction methodology and specifically the adaptive proper orthogonal decomposition (APOD) approach. Initially, an ensemble of eigenfunctions is constructed based on a relatively small data ensemble using method of snapshots and Karhunen-Lòeve expansions (KLE). We then recursively update the eigenfunctions as additional data from the process become available periodically, thus relaxing the need for a representative ensemble in KLE. An accurate reduced-order model (ROM) is constructed and periodically refined via nonlinear Galerkin's method based on the eigenfunctions. Using the ROM and continuous measurements available from restricted number of sensors a static output feedback controller is subsequently designed. This controller is used achieve the desired control objective of stabilizing the FHN equation at a desired reference trajectory. The success of the adaptive model reduction and output-feedback controller design methodology are illustrated using computer simulations.