Geometric output tracking of nonlinear distributed parameter systems via adaptive model reduction

We focus on the output tracking problem of distributed parameter systems (DPSs) which can be described by a set of nonlinear dissipative partial differential equations (PDEs). The infinite-dimensional modal representation of such systems in appropriate subspaces can be decomposed to finite-dimensional slow and probably unstable, and infinite-dimensional fast and stable subsystems. Taking advantage of this decomposition, adaptive model reduction techniques and specifically adaptive proper orthogonal decomposition (APOD) can be used for the recursive construction of locally accurate low dimensional reduced order models (ROMs). The proposed geometric APOD-based control structure is the combination of a nonlinear Luenberger-like geometric dynamic observer and a globally linearizing controller (GLC) designed for tracking the desired output. The proposed geometric control approach is successfully illustrated on the output tracking of target thermal dynamics for a catalytic reactor. Specifically, the geometric output tracking strategy is used to reduce the hot spot temperature and manage the thermal energy distribution through reactor length during process evolution with limited number of actuators and sensors.