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Residual bounds of approximate solutions of the discrete-time algebraic Riccati equation
Umeå University, Faculty of Science and Technology, Departement of Computing Science.
1998 (English)In: Numerische Mathematik, Vol. 78, no 3, 463-478 p.Article in journal (Refereed) Published
Abstract [en]

Let (X) over tilde be a Hermitian matrix which approximates the unique Hermitian positive semi-definite solution X to the discrete-time algebraic Riccati equation (DARE) X - (FXF)-X-H + F(H)XG(1)(G(2) + G(1)(H)XG(1))(-1)G(1)(H)XF + (CC)-C-H = 0; where F is an element of C-nxn, C-2 is an element of C-mxm is Hermitian positive definite, G(1) is an element of C-nxm,C is an element of C-rxn, the pair (F, G(1)) is stabilizable, and the pair (C, F) is detectable. Assume that I + G (X) over tilde is nonsingular, and (I + G (X) over tilde)F-1 is stable. Let G = G(1)G(2)(-1)G(1)(H), H = 2 (CC)-C-H, and let (R) over cap = (X) over tilde - F-H (X) over tilde(I + G (X) over tilde)F-1 - H be the residual of the DARE with respect to (X) over tilde. Define the linear operator L by LW = W - F-H(I + (X) over tilde G)W-1(I + G (X) over tilde)F-1, W = W-H is an element of C-nxn. The main result of this paper is: If epsilon = parallel to L-1(R) over cap parallel to less than or equal to l/gamma(2 phi(2) + 2 phi root phi(2) + l + l)' where parallel to parallel to denotes any unitarily invariant norm, and l = parallel to L(-1)parallel to(-1), phi = parallel to(I + G (X) over tilde)F-1 parallel to(2), gamma = parallel to(I + G (X) over tilde)(-1)G parallel to(2), then \\(X) over tilde - X\\ less than or equal to 2l epsilon/(1 + gamma epsilon)l + root(1 + gamma epsilon)(2)l(2) - 4(phi(2) + l)gamma l epsilon(-) less than or equal to 2\\L-1(R) over cap\\/1 + gamma\\L-1(R) over cap R\\(.)

Place, publisher, year, edition, pages
1998. Vol. 78, no 3, 463-478 p.
Identifiers
URN: urn:nbn:se:umu:diva-22009ISBN: 0029-599X OAI: oai:DiVA.org:umu-22009DiVA: diva2:212272
Available from: 2009-04-21 Created: 2009-04-21 Last updated: 2009-04-21

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