### Abstract

Continuous-time algebraic Riccati equations (CAREs) can be transformed, à la Cayley, to discrete-time algebraic Riccati equations (DAREs). The efficient structure-preserving doubling algorithm (SDA) for DAREs, from [E.K.-W. Chu, H.-Y. Fan, W.-W. Lin, A structure-preserving doubling algorithm for periodic discrete-time algebraic Riccati equations, preprint 2002-28, NCTS, National Tsing Hua University, Hsinchu 300, Taiwan, 2003; E.K.-W. Chu, H.-Y. Fan, W.-W. Lin, C.-S. Wang, A structure-preserving doubling algorithm for periodic discrete-time algebraic Riccati equations, preprint 2002-18, NCTS, National Tsing Hua University, Hsinchu 300, Taiwan, 2003], can then be applied. In this paper, we develop the structure-preserving doubling algorithm from a new point of view and show its quadratic convergence under assumptions which are weaker than stabilizability and detectability, as well as practical issues involved in the application of the SDA to CAREs. A modified version of the SDA, developed for DAREs with a "doubly symmetric" structure, is also presented. Extensive numerical results show that our approach is efficient and competitive.

Original language | English |
---|---|

Pages (from-to) | 55-80 |

Number of pages | 26 |

Journal | Linear Algebra and Its Applications |

Volume | 396 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - 2005 Feb 1 |

### Fingerprint

### Keywords

- Cayley transform
- Continuous-time algebraic Riccati equation
- Doubling algorithm
- Matrix sign function
- Structure-preserving

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

*Linear Algebra and Its Applications*,

*396*(1-3), 55-80. https://doi.org/10.1016/j.laa.2004.10.010

**A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations.** / Chu, E. K.W.; Fan, Hung-Yuan; Lin, W. W.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 396, no. 1-3, pp. 55-80. https://doi.org/10.1016/j.laa.2004.10.010

}

TY - JOUR

T1 - A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations

AU - Chu, E. K.W.

AU - Fan, Hung-Yuan

AU - Lin, W. W.

PY - 2005/2/1

Y1 - 2005/2/1

N2 - Continuous-time algebraic Riccati equations (CAREs) can be transformed, à la Cayley, to discrete-time algebraic Riccati equations (DAREs). The efficient structure-preserving doubling algorithm (SDA) for DAREs, from [E.K.-W. Chu, H.-Y. Fan, W.-W. Lin, A structure-preserving doubling algorithm for periodic discrete-time algebraic Riccati equations, preprint 2002-28, NCTS, National Tsing Hua University, Hsinchu 300, Taiwan, 2003; E.K.-W. Chu, H.-Y. Fan, W.-W. Lin, C.-S. Wang, A structure-preserving doubling algorithm for periodic discrete-time algebraic Riccati equations, preprint 2002-18, NCTS, National Tsing Hua University, Hsinchu 300, Taiwan, 2003], can then be applied. In this paper, we develop the structure-preserving doubling algorithm from a new point of view and show its quadratic convergence under assumptions which are weaker than stabilizability and detectability, as well as practical issues involved in the application of the SDA to CAREs. A modified version of the SDA, developed for DAREs with a "doubly symmetric" structure, is also presented. Extensive numerical results show that our approach is efficient and competitive.

AB - Continuous-time algebraic Riccati equations (CAREs) can be transformed, à la Cayley, to discrete-time algebraic Riccati equations (DAREs). The efficient structure-preserving doubling algorithm (SDA) for DAREs, from [E.K.-W. Chu, H.-Y. Fan, W.-W. Lin, A structure-preserving doubling algorithm for periodic discrete-time algebraic Riccati equations, preprint 2002-28, NCTS, National Tsing Hua University, Hsinchu 300, Taiwan, 2003; E.K.-W. Chu, H.-Y. Fan, W.-W. Lin, C.-S. Wang, A structure-preserving doubling algorithm for periodic discrete-time algebraic Riccati equations, preprint 2002-18, NCTS, National Tsing Hua University, Hsinchu 300, Taiwan, 2003], can then be applied. In this paper, we develop the structure-preserving doubling algorithm from a new point of view and show its quadratic convergence under assumptions which are weaker than stabilizability and detectability, as well as practical issues involved in the application of the SDA to CAREs. A modified version of the SDA, developed for DAREs with a "doubly symmetric" structure, is also presented. Extensive numerical results show that our approach is efficient and competitive.

KW - Cayley transform

KW - Continuous-time algebraic Riccati equation

KW - Doubling algorithm

KW - Matrix sign function

KW - Structure-preserving

UR - http://www.scopus.com/inward/record.url?scp=11044235476&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=11044235476&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2004.10.010

DO - 10.1016/j.laa.2004.10.010

M3 - Article

VL - 396

SP - 55

EP - 80

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 1-3

ER -