A complex line homotopy algorithm for coupled algebraic Riccati equations

Abstract

This paper presents an algorithm based on a complex line homotopy mapping for the numerical solution of systems of coupled algebraic matrix Riccati equations which often met in continuous time control system design problems. Well-defined complex homotopy paths lead to solutions globally from independent initial conditions, and they are computed via continuous dynamic flow algorithms (ordinary differential equation solvers). Although the homotopy mapping is complex it is shown how it is possible in practice to obtain desired real solutions of certain definiteness even from zero initial conditions. Moreover, a scaling method is deployed within the algorithms for numerical enhancement and its effect to homotopy paths, convergence and stability is experimentally studied. In addition, a first-order perturbation analysis framework for the assessment of computed solutions is provided. Comparisons with other existing methods from the literature are made through several numerical examples considering mixed H|$_{2}$|/H|$_{\infty }$|⁠, Nash games and stochastic control system design problems, where the efficiency of the proposed algorithm is illustrated. It is worth noticing that algorithmic implementation and numerical experimentation with complex homotopies for the coupled algebraic Riccati equations under study appears to be absent the literature. Hence, the contribution of the present paper to related problems in systems and control.