On the design of equation-free controllers for dissipative PDEs via DEIM

We propose an equation free control method to control dissipative distributed parameter systems, in which the dynamics of the system are unknown while the effect of the control action is. A static observer is used to estimate the state using proper orthogonal decomposition (POD) so that a complete profile of the system can be estimated when a limited number of point sensors are available. Sensor locations are determined by interpolation indices in discrete empirical interpolation method (DEIM). By using both velocity and state sensors an explicit form of the complete equation become superfluous, needing to only have a description of the actuator effect. The proposed method is successfully employed in a diffusion-reaction process with Dirichlet and Neumann boundary conditions. Feedback linearization is combined with the proposed method to regulate the system. Computational results demonstrate that this method can regulate a dissipative distributed parameter system without explicitly requiring a model of it and is robust to disturbances.