A parametrization of the nonnegative definite solutions of the algebraic Riccati equation
type: Item
instance of: scholarly article
title: A parametrization of the nonnegative definite solutions of the algebraic Riccati equation
zbMATH Open document ID: 0916.93060
DOI: 10.1016/S0005-1098(97)00193-3
author: Domenico D'Alessandro
published in: Automatica
publication date: 1999-07-12T00:00:00Z
review text: The author studies the set \(\mathcal P\) of Hermitian nonnegative definite solutions of the continuous time algebraic Riccati equation \(A^{*}P+PA-PBB^{*}P+C^{*}C=0\). Using the parametrization result of \textit{H. K. Wimmer} [SIAM J. Control Optim. 32, No. 4, 995-1007 (1994; Zbl 0799.93034)], he proves a theorem according to which any solution in the set \(\mathcal P\) can be represented as the product of the largest solution and a projection. This extends the results of \textit{F. M. Callier} and \textit{J. L. Willems} [IEEE Trans. Automat. Control AC-26, 1232-1242 (1981; Zbl 0472.93032)] and \textit{T. Ando} [Matrix quadratic equations (1988; Zbl 0645.34001)] proved under the hypothesis of stabilizability of the pair \((A,B)\). The analysis of the present paper only assumes the existence of a largest solution in \(\mathcal P\). The results are illustrated by a numerical example.
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 49N10
Mathematics Subject Classification ID: 93C99
zbMATH DE Number: 1192896
zbMATH Keywords: algebraic Riccati equation
zbMATH Keywords: nonnegative definite solutions
zbMATH Keywords: classification of solutions
zbMATH Keywords: representation formulas
zbMATH Keywords: parametrization
zbMATH Keywords: projection
zbMATH Keywords: stabilizability
zbMATH Keywords: largest solution
reviewed by: É. Gyurkovics
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1192896