## Algebraic Riccati EquationsThis book provides a careful treatment of the theory of algebraic Riccati equations. It consists of four parts: the first part is a comprehensive account of necessary background material in matrix theory including careful accounts of recent developments involving indefinite scalar products and rational matrix functions. The second and third parts form the core of the book and concern the solutions of algebraic Riccati equations arising from continuous and discrete systems. The geometric theory and iterative analysis are both developed in detail. The last part of the book is an exciting collection of eight problem areas in which algebraic Riccati equations play a crucial role. These applications range from introductions to the classical linear quadratic regulator problems and the discrete Kalman filter to modern developments in HD*W*w control and total least squares methods. |

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### Contents

1 | |

INDEFINITE SCALAR PRODUCTS | 31 |

SKEWSYMMETRIC SCALAR PRODUCTS | 71 |

MATRIX THEORY AND CONTROL | 83 |

LINEAR MATRIX EQUATIONS | 97 |

RATIONAL MATRIX FUNCTIONS | 107 |

CONTINUOUS ALGEBRAIC RICCATI | 147 |

THE REAL CASE | 215 |

PERTURBATION THEORY FOR DISCRETE | 329 |

APPLICATIONS AND CONNECTIONS | 347 |

THE DISCRETE KALMAN FILTER | 371 |

THE TOTAL LEAST SQUARES TECHNIQUE | 387 |

CANONICAL FACTORIZATION | 397 |

H CONTROL PROBLEMS | 409 |

CONTRACTIVE RATIONAL | 421 |

THE MATRIX SIGN FUNCTION | 437 |

CONSTRUCTIVE EXISTENCE AND COMPARISON | 231 |

HERMITIAN SOLUTIONS AND FACTORIZATIONS | 247 |

PERTURBATION THEORY | 257 |

DISCRETE ALGEBRAIC | 269 |

CONSTRUCTIVE EXISTENCE AND COMPARISON | 307 |

STRUCTURED STABILITY RADIUS | 447 |

459 | |

LIST OF NOTATION AND CONVENTIONS | 473 |

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### Common terms and phrases

A-invariant A*XA algebraic Riccati equation analytic Assume B*XB c-stable canonical form chapter coefficients Conversely Corollary d-stabilizable DARE defined deflating subspace denote equality equivalent example follows from Theorem formula given Gohberg graph subspace H-self-adjoint Hence hermitian matrix hermitian solution imaginary axis imaginary or zero implies indefinite scalar product invariant subspaces invertible matrix Jordan block Jordan form Lemma linear transformation matrix pencil maximal hermitian solution maximal solution minimal factorization minimal realization n x n matrix n-dimensional nonnegative nonzero Observe obtain orthogonal partial multiplicities poles polynomial positive semidefinite projector proof of Theorem properties Proposition prove pure imaginary rational matrix function real eigenvalues real matrices real symmetric solutions resp respect result satisfies Section sequence shows sign characteristic sign controllable sign function skew-symmetric skew-symmetric matrix spectral subspace stability radius stabilizable stabilizing solution statement unit circle vector zero eigenvalues