Balancing-Related Model Reduction for Parabolic Control Systems

Balanced truncation (BT) is a well established model reduction technique for linear ordinary differential equations. If the original dynamics are described by an instationary PDE, then BT is usually applied to the large-scale linear system resulting from a spatial semi-discretization using finite elements/volumes/differences. We will discuss this approach as well as a variant of BT based on balancing the solution of the linear-quadratic Gaussian (LQG) algebraic Riccati equations, called LQG BT, allowing the reduction of unstable systems and yielding a stabilizing feedback controller as a by-product. The error between reduced-order model and original system is split into discretization and model reduction components. We discuss the resulting error bounds and ways how to exploit this in order to adaptively choose the reduced system order. Numerical examples support our findings.