An Accelerated Newton Process to Solve Riccati Equation via Matrix Sign Function

Abstract

This pacer deals with the numerical solution of the algebraic matrix Riccati equation PA + ATP - PSP + Q = O. Many approaches have been developed to solve this problem ranging from integration of differential equations, Hamiltonian algebraic eigenvalue problem, non linear equations solver, explicit transformation to an equivalent discrete formulation and, recently, a nice numerical method based upon the real ordered Schur form of the Hamiltonian matrix. The new fast algorithm, which is discussed here, enjoys the same good numerical properties as the last one and can be implemented in a much more elementary code using only a linear equation solver. Some of the main properties of our Riccati solver algorithm are : convergence speed independent of the dimension of P (say n) and unaffected by the stability or unstability of the open loop system - an average cost of 60 n3 multiplications. Finally, several numerical tests are reported with n varying from 2 to 25, which must clearly show how the algorithm runs.