A family of BDF algorithms for solving Differential Matrix Riccati Equations using adaptive techniques

Abstract

Abstract Differential Matrix Riccati Equations play a fundamental role in control theory, for example, in optimal control, filtering and estimation, decoupling and order reduction, etc. One of the most popular codes to solve stiff Differential Matrix Riccati Equations (DMREs) is based on Backward Differentiation Formula (BDF). In previous papers the authors of this paper showed two algorithms for solving DMREs based on an iterative Generalized Minimum RESidual (GMRES) approach and on a Fixed-Point approach. In this paper we present two contributions to improve the above algorithms. Firstly six variants of previous algorithms are carried out by using one of above algorithms in the first step and another algorithm to carry out the other steps until reaching convergence. Numerous tests on four case studies have been done comparing both precision and computational costs of MATLAB implementations of the above algorithms. Experimental results show that in some cases these algorithms improve on the speed and convergence of the original algorithms. Secondly, using the previous experimental results and since all algorithms have a similar structure and there is no best algorithm to solve all problems, two general-purpose adaptive algorithms have been designed for selecting the most appropriate algorithm, which can be chosen using a parameter that indicates the stiffness of the DMRE to be solved.