On solving periodic Riccati equations

Numerically reliable algorithms to compute the periodic non-negative definite stabilizing solutions of the periodic differential Riccati equation (PRDE) and discrete-time periodic Riccati equation (DPRE) are proposed. For the numerical solution of PRDEs, a new multiple shooting-type algorithm is developed to compute the periodic solutions in an arbitrary number of time moments within one period by employing suitable discretizations of the continuous-time problems. In contrast to single shooting periodic generator methods, the multiple shooting-type methods have the main advantage of being able to address problems with larger periods. Three methods are discussed to solve DPREs. Two of the methods represent extensions of the periodic QZ algorithm to non-square periodic pairs, whereas the third method represents an extension of a quotient-product swapping and collapsing ‘fast’ algorithm. All proposed approaches are completely general, being applicable to periodic Riccati equations with time-varying dimensions as well as with singular weighting matrices. Copyright q 2008 John Wiley & Sons, Ltd.