Synthesis of Equation-Free Control Structures for Dissipative Distributed Parameter Systems Using Proper Orthogonal Decomposition and Discrete Empirical Interpolation Methods

We describe an equation-free control framework for the regulation of dissipative distributed parameter systems, with emphasis on improving the accuracy of the estimation by using a correction term. This control method is capable of regulating systems that have unknown dynamics but known effect of the control action. The system state and the dynamics are estimated by using the offline observations (snapshots ensemble) and the online continuous measurement of a restricted number of point sensors. First, we construct a reduced order model (ROM) with unknown terms using Galerkin/proper orthogonal decomposition (POD). Then the state of the ROM is estimated by a static observer with the information from the state sensors, and the mapping between the dynamics of the system and velocity sensors are generated using a similar approach. A discrete empirical interpolation method (DEIM) is employed to determine the sensor locations. To improve the accuracy of the estimation, a correction term is updated consistently. The proposed equation-free control framework is illustrated through a diffusion-reaction process and the performance of the proposed method is evaluated by simulation.