Abstract
Linear and quadratic (Riccati) matrix equations are fundamental for systems and control theory and its numerous applications. Generalized or standard Sylvester and Lyapunov equations and generalized Riccati equations for continuous- and discrete-time systems are considered. Essential applications in control are mentioned. The main solvability conditions and properties of these equations, as well as state-of-the-art solution techniques, are summarized. The continuous progress in this area paves the way for further developments and extensions to more complex control problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Bibliography
Benner P, Mehrmann V, Sima V, Van Huffel S, Varga A (1999) SLICOT – A subroutine library in systems and control theory. In: Datta BN (ed) Applied and computational control, signals, and circuits, vol 1, chapter 10. Birkhäuser, Boston, pp 499–539
Benner P, Mehrmann V, Sorensen D (eds) (2005) Dimension reduction of large-scale systems. Lecture notes in computational science and engineering, vol 45. Springer, Berlin/Heidelberg
Benner P, Sima V, Voigt M (2016) Algorithm 961: Fortran 77 subroutines for the solution of skew-Hamiltonian/Hamiltonian eigenproblems. ACM Trans Math Softw (TOMS) 42(3):1–26
Bini DA, Iannazzo B, Meini B (2012) Numerical solution of algebraic Riccati equations. SIAM, Philadelphia
Francis BA (1987) A course in H∞ control theory. Lecture notes in control and information sciences, vol 88. Springer, New York
Golub GH, Van Loan CF (2013) Matrix computations, 4th edn. The Johns Hopkins University Press, Baltimore
Hammarling SJ (1982) Numerical solution of the stable, non-negative definite Lyapunov equation. IMA J Numer Anal 2(3):303–323
Lancaster P, Rodman L (1995) The algebraic Riccati equation. Oxford University Press, Oxford
Mehrmann V (1991) The autonomous linear quadratic control problem: theory and numerical solution. Lecture notes in control and information sciences, vol 163. Springer, Berlin
Penzl T (1998) Numerical solution of generalized Lyapunov equations. Adv Comput Math 8:33–48
Sima V (1996) Algorithms for linear-quadratic optimization. Pure and applied mathematics: a series of monographs and textbooks, vol 200. Marcel Dekker, Inc., New York
Simoncini V (2016) Computational methods for linear matrix equations. SIAM Rev 58(3):377–441
Van Dooren P (1981) A generalized eigenvalue approach for solving Riccati equations. SIAM J Sci Stat Comput 2(2):121–135
Van Huffel S, Sima V, Varga A, Hammarling S, Delebecque F (2004) High-performance numerical software for control. IEEE Control Syst Mag 24(1):60–76
Varga A (2017) Solving fault diagnosis problems: linear synthesis techniques. Springer, Berlin
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Rights and permissions
Copyright information
© 2019 Springer-Verlag London Ltd., part of Springer Nature
About this entry
Cite this entry
Sima, V. (2019). Matrix Equations in Control. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London. https://doi.org/10.1007/978-1-4471-5102-9_100053-1
Download citation
DOI: https://doi.org/10.1007/978-1-4471-5102-9_100053-1
Published:
Publisher Name: Springer, London
Print ISBN: 978-1-4471-5102-9
Online ISBN: 978-1-4471-5102-9
eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering