Order Reduction Approaches for the Algebraic Riccati Equation and the LQR Problem

We explore order reduction techniques to solve the algebraic Riccati equation (ARE), and investigate the numerical solution of the linear-quadratic regulator problem (LQR). A classical approach is to build a low dimensional surrogate model of the dynamical system, for instance by means of balanced truncation, and then solve the corresponding ARE. Alternatively, iterative methods can be used to directly solve the ARE and use its approximate solution to estimate quantities associated with the LQR. We propose a class of Petrov-Galerkin strategies based on Krylov subspaces that simultaneously reduce the dynamical system while approximately solving the ARE by projection. This methodology significantly generalizes a recently developed Galerkin method, based on Krylov subspaces, by using a pair of projection spaces, as it is often done in model order reduction (MOR) of dynamical systems. Numerical experiments illustrate the advantages of the new class of methods over classical approaches when dealing with large matrices.