Perturbation Analysis of Perturbation Analysis of
Algebraic Riccati Equations

Ji-guang Sun
Department of Computing Science
Umeå University
S-901 87 Umeå , Sweden
jisun@cs.umu.se

Consider the continuous-time Riccati equation (CARE)

Q+ATX+XA-XBR-1BTX = 0,
and the discrete-time algebraic Riccati equation (DARE)
X-ATXA+ATXB(R+BTXB)-1BTXA-Q = 0.
The algebraic Riccati equations (AREs) arise naturally in control, systems, and signals, and play an important role. Some special hypotheses on the coefficient matrices can be made to ensure certain properties of the solution X; for instance, X is a unique symmetric positive semidefinite (p.s.d.) solution.

The central question of forward perturbation analysis for an ARE is: How does the symmetric p.s.d. solution change when the coefficient matrices are subject to perturbations? The central question of backward perturbation analysis for an ARE is: Is an computed symmetric p.s.d. solution of an ARE the exact solution of a slightly perturbed ARE? The interest in these topics is motivated by the fact that each ARE is usually subject to perturbations in the coefficient matrices reflecting various errors in the formulation of the problem and in its solution by a computer. In this talk, we present a unifying framework and effective techniques for creating perturbation theory for AREs, and present improved perturbation results which include perturbation bounds, condition numbers, backward errors, and residual bounds for AREs. The theoretical results are illustrated by numerical examples.