Discrete Input/Output Maps and their Relation to Proper Orthogonal Decomposition

Current control design techniques require system models of moderate size to be applicable. The generation of such models is challenging for complex systems which are typically described by partial differential equations (PDEs), and model-order reduction or low-order-modeling techniques have been developed for this purpose. Many of them heavily rely on the state space models and their discretizations. However, in control applications, a sufficient accuracy of the models with respect to their input/output (I/O) behavior is typically more relevant than the accurate representation of the system states. Therefore, a discretization framework has been developed and is discussed here, which heavily focuses on the I/O map of the original PDE system and its direct discretization in the form of an I/O matrix and with error bounds measuring the relevant I/O error. We also discuss an SVD-based dimension reduction for the matrix representation of an I/O map and how it can be interpreted in terms of the Proper Orthogonal Decomposition (POD) method which gives rise to a more general POD approach in time capturing. We present numerical examples for both, reduced I/O map s and generalized POD.