Observer based LQ optimal boundary control of 2D heat flow by order reduction

Linear Quadratic (LQ) optimal boundary control of a 2D heat flow is studied. The design is carried out on a reduced order model of the Partial Differential Equation (PDE) process. For this purpose, Proper Orthogonal Decomposition (POD) technique is utilized and the Low Dimensional (LD) model is derived. The boundary controller is developed using the state information obtained via an observer. An infinite dimensional version of the observer is developed first and its finite dimensional counterpart is derived according to POD procedure. Having obtained the states of the system, a LQ optimal control approach is followed to demonstrate that the entire design works satisfactorily under the presence of noise, uncertainty and disturbances. The contribution of the paper is to draw a clear path between a spatially continuous process and an optimal boundary controller minimizing a quadratic cost, and the emphasis on the merits of POD based designs.