Model reduction for control of a multiphysics system: Coupled Burgers' equation

Abstract

We consider control and design for coupled, multiphysics systems governed by partial differential equations (PDEs). The numerical solution of the control problem involves large systems of ordinary differential equations arising from a spatial discretization scheme, which can be prohibitively expensive. Utilizing reduced order surrogate models evolved as a way to circumvent this computational problem. While many reduced order models work well for simulation, the task of control adds additional complexity. We investigate the effects of different reduced order models on the optimal feedback control. We propose to use a structure-preserving surrogate model, constructed by computing dominant subspaces for each physical quantity separately. This method addresses the different scaling of variables commonly found in multiphysics problems. As a test example, a coupled Burgers' equation multiphysics PDE model is considered. In the numerical study, we find that the feedback gains obtained from the standard proper orthogonal decomposition for the combined variables fail to converge, while the physics-based method produces convergent control feedback matrices.