Contributions to the solution of discrete-time invariant and periodic algebraic Riccati equations on multiprocessors

This work remains in the research field of parallel and high performance computing applied to the solution of control problems. This research field has had a great growth in the last years, and provides control engineers with the possibility to apply methods for solving control problems that until now are impossible to apply in practice. Our work is focussed to provide basic tools for solving the discrete-time algebraic Riccati equation, in the case of invariant formulation and in the case of periodic formulation. This two equations arises when the solution of the lineal-quadractic optimal control is desired. When we try to solve this problem for large systems we must use parallel computing, because with the sequential computation there is not enough power to perform all the computations and, normally, we can have storage problems.; The proposed methods for the solution of the discrete-time algebraic Riccati equation, in the invariant and periodic case, obtains a base of an invariant subspace of a simplectic pencil of matrices associated to the equation. Our solvers, for the invariant case, obtain an initial solution of the Riccati equation via the disc function method, using an inverse-free iteration, and then refine this solution using the Newton's method, in order to obtain the greatest possible precision for the condition of the problem and with the precision of the computer. At each iteration of the Newton's method a Stein equation is solved using the Smith iteration.; In the periodic case, two approaches are presented. The first one uses the periodic Schur form and the second one a reordering of a product of periodic matrices associated to the problem that provides a direct solution to the periodic equation.; All the implementations of the proposed methods has been done over a cluster of personal computers with a Myrinet interconnection network.