Adams–Bashforth and Adams–Moulton methods for solving differential Riccati equations

Abstract

Differential Riccati equations play a fundamental role in control theory, for example, optimal control, filtering and estimation, decoupling and order reduction, etc. In this paper several algorithms for solving differential Riccati equations based on Adams–Bashforth and Adams–Moulton methods are described. The Adams–Bashforth methods allow us explicitly to compute the approximate solution at an instant time from the solutions in previous instants. In each step of Adams–Moulton methods an algebraic matrix Riccati equation (AMRE) is obtained, which is solved by means of Newton’s method. Nine algorithms are considered for solving the AMRE: a Sylvester algorithm, an iterative generalized minimum residual (GMRES) algorithm, a fixed-point algorithm and six combined algorithms. Since the above algorithms have a similar structure, it is possible to design a general and efficient algorithm that uses one algorithm or another depending on the considered differential matrix Riccati equation. MATLAB versions of the above algorithms are developed, comparing precision and computational costs, after numerous tests on five case studies.