Order reduction approaches for the algebraic Riccati equation and the LQR problem
For control theorists and engineers, this provides a more efficient method for solving ARE/LQR problems with large-scale systems, though it is an incremental extension of existing Galerkin methods.
The paper proposes Petrov-Galerkin strategies that simultaneously reduce the dynamical system and approximately solve the algebraic Riccati equation (ARE) for the linear-quadratic regulator (LQR) problem, showing advantages over classical approaches for large matrices.
We explore order reduction techniques for solving the algebraic Riccati equation (ARE), and investigating the numerical solution of the linear-quadratic regulator problem (LQR). A classical approach is to build a surrogate low dimensional 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 that simultaneously reduce the dynamical system while approximately solving the ARE by projection. This methodology significantly generalizes a recently developed Galerkin method by using a pair of projection spaces, as it is often done in model order reduction of dynamical systems. Numerical experiments illustrate the advantages of the new class of methods over classical approaches when dealing with large matrices.