An iterative algorithm to solve Algebraic Riccati Equations with an indefinite quadratic term

Alexander Lanzon; Yantao Feng; Brian D.O. Anderson

doi:10.23919/ecc.2007.7068239

An iterative algorithm to solve Algebraic Riccati Equations with an indefinite quadratic term

Alexander Lanzon, Yantao Feng^*, Brian D.O. Anderson

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

15 Citations (Scopus)

Abstract

In this paper, an iterative algorithm to solve Algebraic Riccati Equations (ARE) arising from, for example, a standard H_∞ control problem is proposed. By constructing two sequences of positive semidefinite matrices, we reduce an ARE with an indefinite quadratic term to a series of AREs with a negative semidefinite quadratic term which can be solved more easily by existing iterative methods (e.g. Kleinman algorithm in [2]). We prove that the proposed algorithm is globally convergent and has local quadratic rate of convergence. Numerical examples are provided to show that our algorithm has better numerical reliability when compared with some traditional algorithms (e.g. Schur method in [5]). Some proofs are omitted for brevity and will be published elsewhere.

Original language	English
Title of host publication	2007 European Control Conference, ECC 2007
Place of Publication	Kos, Greece
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	3033-3039
Number of pages	7
Edition	Peer Reviewed
ISBN (Electronic)	9783952417386
ISBN (Print)	9783952417386
DOIs	https://doi.org/10.23919/ecc.2007.7068239
Publication status	Published - 2007
Event	9th European Control Conference (ECC 2007) - Kos Greece Duration: 1 Jan 2007 → … http://ieeexplore.ieee.org/document/7068220/

Publication series

Name	2007 European Control Conference, ECC 2007

Conference

Conference	9th European Control Conference (ECC 2007)
Period	1/01/07 → …
Other	July 2-5 2007
Internet address	http://ieeexplore.ieee.org/document/7068220/

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10.23919/ecc.2007.7068239

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Lanzon, A., Feng, Y., & Anderson, B. D. O. (2007). An iterative algorithm to solve Algebraic Riccati Equations with an indefinite quadratic term. In 2007 European Control Conference, ECC 2007 (Peer Reviewed ed., pp. 3033-3039). Article 7068239 (2007 European Control Conference, ECC 2007). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.23919/ecc.2007.7068239

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title = "An iterative algorithm to solve Algebraic Riccati Equations with an indefinite quadratic term",

abstract = "In this paper, an iterative algorithm to solve Algebraic Riccati Equations (ARE) arising from, for example, a standard H∞ control problem is proposed. By constructing two sequences of positive semidefinite matrices, we reduce an ARE with an indefinite quadratic term to a series of AREs with a negative semidefinite quadratic term which can be solved more easily by existing iterative methods (e.g. Kleinman algorithm in [2]). We prove that the proposed algorithm is globally convergent and has local quadratic rate of convergence. Numerical examples are provided to show that our algorithm has better numerical reliability when compared with some traditional algorithms (e.g. Schur method in [5]). Some proofs are omitted for brevity and will be published elsewhere.",

keywords = "ARE, Indefinite quadratic term, Iterative",

author = "Alexander Lanzon and Yantao Feng and Anderson, {Brian D.O.}",

note = "Publisher Copyright: {\textcopyright} 2007 EUCA.; 9th European Control Conference (ECC 2007) ; Conference date: 01-01-2007",

year = "2007",

doi = "10.23919/ecc.2007.7068239",

language = "English",

isbn = "9783952417386",

series = "2007 European Control Conference, ECC 2007",

publisher = "Institute of Electrical and Electronics Engineers Inc.",

pages = "3033--3039",

booktitle = "2007 European Control Conference, ECC 2007",

address = "United States",

edition = "Peer Reviewed",

url = "http://ieeexplore.ieee.org/document/7068220/",

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Lanzon, A, Feng, Y & Anderson, BDO 2007, An iterative algorithm to solve Algebraic Riccati Equations with an indefinite quadratic term. in 2007 European Control Conference, ECC 2007. Peer Reviewed edn, 7068239, 2007 European Control Conference, ECC 2007, Institute of Electrical and Electronics Engineers Inc., Kos, Greece, pp. 3033-3039, 9th European Control Conference (ECC 2007), 1/01/07. https://doi.org/10.23919/ecc.2007.7068239

An iterative algorithm to solve Algebraic Riccati Equations with an indefinite quadratic term. / Lanzon, Alexander; Feng, Yantao; Anderson, Brian D.O.
2007 European Control Conference, ECC 2007. Peer Reviewed. ed. Kos, Greece: Institute of Electrical and Electronics Engineers Inc., 2007. p. 3033-3039 7068239 (2007 European Control Conference, ECC 2007).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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T1 - An iterative algorithm to solve Algebraic Riccati Equations with an indefinite quadratic term

AU - Lanzon, Alexander

AU - Feng, Yantao

AU - Anderson, Brian D.O.

PY - 2007

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N2 - In this paper, an iterative algorithm to solve Algebraic Riccati Equations (ARE) arising from, for example, a standard H∞ control problem is proposed. By constructing two sequences of positive semidefinite matrices, we reduce an ARE with an indefinite quadratic term to a series of AREs with a negative semidefinite quadratic term which can be solved more easily by existing iterative methods (e.g. Kleinman algorithm in [2]). We prove that the proposed algorithm is globally convergent and has local quadratic rate of convergence. Numerical examples are provided to show that our algorithm has better numerical reliability when compared with some traditional algorithms (e.g. Schur method in [5]). Some proofs are omitted for brevity and will be published elsewhere.

AB - In this paper, an iterative algorithm to solve Algebraic Riccati Equations (ARE) arising from, for example, a standard H∞ control problem is proposed. By constructing two sequences of positive semidefinite matrices, we reduce an ARE with an indefinite quadratic term to a series of AREs with a negative semidefinite quadratic term which can be solved more easily by existing iterative methods (e.g. Kleinman algorithm in [2]). We prove that the proposed algorithm is globally convergent and has local quadratic rate of convergence. Numerical examples are provided to show that our algorithm has better numerical reliability when compared with some traditional algorithms (e.g. Schur method in [5]). Some proofs are omitted for brevity and will be published elsewhere.

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KW - Indefinite quadratic term

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DO - 10.23919/ecc.2007.7068239

M3 - Conference contribution

SN - 9783952417386

T3 - 2007 European Control Conference, ECC 2007

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BT - 2007 European Control Conference, ECC 2007

PB - Institute of Electrical and Electronics Engineers Inc.

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T2 - 9th European Control Conference (ECC 2007)

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ER -

An iterative algorithm to solve Algebraic Riccati Equations with an indefinite quadratic term

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