Chapter 13 - Numerical Solutions and Conditioning of Algebraic Riccati Equations

Abstract

This chapter is devoted to the study of numerical solutions of the continuous-time algebraic Riccati equation (CARE) and its discrete counterpart (DARE). Historically, algebraic Riccati equations started as an important tool in the solution of Linear Quadratic Optimization problems. In recent years, they became a subject of intensive study, both from theoretical and computational viewpoints, because of their important roles in state-space solutions of H∞ and robust control problems. The chapter discusses the following computational methods for the CARE and DARE that are widely known in the literature: the eigenvector methods; the Schur methods and tile structure-preserving Schur methods; the generalized eigenvector, the generalized Schur, and inverse-free generalized methods; the matrix sign function methods; and Newton's methods. The eigenvector methods are well known to have numerical difficulties in case the Hamiltonian matrix associated with the CARE or the Symplectic matrix associated with the DARE has some multiple or near-multiple eigenvalues. Newton's methods are iterative in nature and are usually used as iterative refinement techniques for solutions obtained by the Schur methods or the matrix sign function methods. The chapter also presents a comparison of the different methods and the recommendation based on this comparison.