Residual bounds of approximate solutions of the discrete-time algebraic Riccati equation

Let be a Hermitian matrix which approximates the unique Hermitian positive semi-definite solution to the discrete-time algebraic Riccati equation (DARE) where , is Hermitian positive definite, , the pair is stabilizable, and the pair is detectable. Assume that is nonsingular, and is stable. Let , and let be the residual of the DARE with respect to . Define the linear operator by The main result of this paper is: If where denotes any unitarily invariant norm, and then Received June 7, 1995 / Revised version received February 28, 1996