Lancaster Peter, Rodman Leiba. Algebraic Riccati Equations

Monograph. — Oxford: Clarendon Press, 1995. — 480 p.The first concern of this book is with the solution of quadratic equations in matrices, wich are calld the "algebraic Riccati equations". Important role played by these equations in optimal filter design and control theory and has resulted in a prodigious number of research publications which have steadily increased our understanding of the equations and their solution sets. Although the algebraic Riccati equations in our purview are truly nonlinear, they are amenable to solution by methods which rely heavily on linear algebra and the theory of matrices.Matrix theory.
Preliminaries from the theory of matrices.
Indefinite scalar products.
Skew-symmetric scalar products.
Matrix theory and control.
Linear matrix equations.
Rational matrix functions.
Continuous algebraic Riccati equations.
Geometric theory: the complex case.
Geometric theory: the real case.
Constructive existence and comparison theorems.
Hermitian solutions and factorizations.
Perturbation theory.
Discrete algebraic Riccati equations.
Geometric theory for the discrete algebraic Riccati Equation.
Constructive existence and comparison theorems.
Perturbation theory for discrete algebraic Riccati equations.
Discrete algebraic riccati equations and matrix pencils.
Applications and connections.
Linear-quadratic regulator problems.
The discrete Kalman filter.
The total least squares technique.
Canonical factorization.
H^{\infty} Control problems.
Contractive rational matrix functions.
The matrix sign function.
Structured stability radius.