### Abstract

In the last chapter, it was discussed that the algebraic Riccati equation (ARE) need be solved in order to obtain the state-space solutions of the normalized coprime factorizations. In Chap. 2, the Lyapunov equation was employed to determine the controllability and observability gramians of a system. Both the algebraic Riccati and Lyapunov equations play prominent roles in the synthesis of robust and optimal control as well as in the stability analysis of control systems. In fact, the Lyapunov equation is a special case of the ARE. The ARE indeed has wide applications in control system analysis and synthesis. For example, the state-space formulation for particular coprime factorizations with a J-lossless (or dual J-lossless) numerator requires solving an ARE; in turn, the J-lossless and dual J-lossless systems are essential in the synthesis of robust controllers using the CSD approach. In this chapter, the ARE will be formally introduced. Solution procedures to AREs and their various properties will be discussed. Towards the end of this chapter, the coprime factorization approach to solve several spectral factorization problems is to be considered.

Original language | English |
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Title of host publication | Advances in Industrial Control |

Publisher | Springer International Publishing |

Pages | 171-209 |

Number of pages | 39 |

Edition | 9781447162568 |

DOIs | |

Publication status | Published - 2014 Jan 1 |

### Publication series

Name | Advances in Industrial Control |
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Number | 9781447162568 |

ISSN (Print) | 1430-9491 |

ISSN (Electronic) | 2193-1577 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Automotive Engineering
- Aerospace Engineering
- Industrial and Manufacturing Engineering

### Cite this

*Advances in Industrial Control*(9781447162568 ed., pp. 171-209). (Advances in Industrial Control; No. 9781447162568). Springer International Publishing. https://doi.org/10.1007/978-1-4471-6257-5_7

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*Advances in Industrial Control.*9781447162568 edn, Advances in Industrial Control, no. 9781447162568, Springer International Publishing, pp. 171-209. https://doi.org/10.1007/978-1-4471-6257-5_7

**Algebraic riccati equations and spectral factorizations.** / Tsai, Mi-Ching; Gu, Da Wei.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - Algebraic riccati equations and spectral factorizations

AU - Tsai, Mi-Ching

AU - Gu, Da Wei

PY - 2014/1/1

Y1 - 2014/1/1

N2 - In the last chapter, it was discussed that the algebraic Riccati equation (ARE) need be solved in order to obtain the state-space solutions of the normalized coprime factorizations. In Chap. 2, the Lyapunov equation was employed to determine the controllability and observability gramians of a system. Both the algebraic Riccati and Lyapunov equations play prominent roles in the synthesis of robust and optimal control as well as in the stability analysis of control systems. In fact, the Lyapunov equation is a special case of the ARE. The ARE indeed has wide applications in control system analysis and synthesis. For example, the state-space formulation for particular coprime factorizations with a J-lossless (or dual J-lossless) numerator requires solving an ARE; in turn, the J-lossless and dual J-lossless systems are essential in the synthesis of robust controllers using the CSD approach. In this chapter, the ARE will be formally introduced. Solution procedures to AREs and their various properties will be discussed. Towards the end of this chapter, the coprime factorization approach to solve several spectral factorization problems is to be considered.

AB - In the last chapter, it was discussed that the algebraic Riccati equation (ARE) need be solved in order to obtain the state-space solutions of the normalized coprime factorizations. In Chap. 2, the Lyapunov equation was employed to determine the controllability and observability gramians of a system. Both the algebraic Riccati and Lyapunov equations play prominent roles in the synthesis of robust and optimal control as well as in the stability analysis of control systems. In fact, the Lyapunov equation is a special case of the ARE. The ARE indeed has wide applications in control system analysis and synthesis. For example, the state-space formulation for particular coprime factorizations with a J-lossless (or dual J-lossless) numerator requires solving an ARE; in turn, the J-lossless and dual J-lossless systems are essential in the synthesis of robust controllers using the CSD approach. In this chapter, the ARE will be formally introduced. Solution procedures to AREs and their various properties will be discussed. Towards the end of this chapter, the coprime factorization approach to solve several spectral factorization problems is to be considered.

UR - http://www.scopus.com/inward/record.url?scp=85021082984&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021082984&partnerID=8YFLogxK

U2 - 10.1007/978-1-4471-6257-5_7

DO - 10.1007/978-1-4471-6257-5_7

M3 - Chapter

AN - SCOPUS:85021082984

T3 - Advances in Industrial Control

SP - 171

EP - 209

BT - Advances in Industrial Control

PB - Springer International Publishing

ER -