Abstract :
Roberts matrix sign function solution to the ARE is defined so as to speed convergence and reduce storage requirements; our work extends ideas proposed by R. Byers, ref(8). Features of the sign function presented here are: (a) our formulation of the Roberts-Byers algorithm recurses on the symmetric transformed Hamiltonian, which reduces storage requirements, (b) the symmetric indefinite matrix inversion required by the algorithm is carried out using LINPACK (and our excellent numerical results reflect the wisdom of this choice); corrections to the computed (approximate) solution are obtained by applying the same algorithms to the translated problem (which improves upon the linear Lyapunov equation correction that has been used), and (d) simple (but somewhat ad hoc) convergence criteria are proposed to reduce computation. The algorithm described in this work has been tested on a variety of continuous time ARE test problems, and the results have been very satisfactory. Tests on numerically ill-conditioned problems produced results of comparable accuracy with those obtained by the Shur vector RICPACK method. Our sign function iterative ARE solution demonstrates numerical robustness, accurate results, rapid (super linear) convergence, algorithmic simplicity, and modest storage requirements. Our work shows that iterative ARE solutions offer a viable alternative to the Shur eigenvector approach that is a generally accepted reference.
