Volume 41, pp. 396-419, 2014.
A structure-preserving algorithm for semi-stabilizing solutions of generalized algebraic Riccati equations
Tiexiang Li and Delin Chu
Abstract
In this paper, a structure-preserving algorithm is developed for
the computation of a semi-stabilizing solution of a Generalized Algebraic
Riccati Equation (GARE). The semi-stabilizing solution of
GAREs has been used to characterize the solvability of the -spectral factorization problem in control theory for general
rational matrices which may have poles and zeros on the extended
imaginary axis. The main difficulty in solving such a GARE lies in the fact that
its associated Hamiltonian/skew-Hamiltonian pencil has eigenvalues
on the extended imaginary axis. Consequently, it is not clear which
eigenspace of the associated Hamiltonian/skew-Hamiltonian pencil can
characterize the desired semi-stabilizing solution. That is, it is
not clear which eigenvectors and principal vectors corresponding to
the eigenvalues on the extended imaginary axis should be contained
in the eigenspace that we wish to compute. Hence, the well-known
generalized eigenspace approach for the classical algebraic Riccati
equations cannot be employed directly. The proposed algorithm
consists of a structure-preserving doubling algorithm (SDA) and a
postprocessing procedure to determine the desired
eigenvectors and principal vectors corresponding to the purely
imaginary and infinite eigenvalues. Under mild assumptions,
linear convergence of rate for the SDA is proved. Numerical
experiments illustrate that the proposed algorithm performs efficiently
and reliably.
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Key words
Generalized Algebraic Riccati Equation, structure-preserving doubling algorithm, semi-stabilizing solution
AMS subject classifications
15A15, 15A09, 15A23
